Diffraction grating

DIFFRACTION GRATING
HANDBOOK
sixth edition

Christopher Palmer
Newport Corporation

Erwin Loewen (first edition)

NEWPORT CORPORATION
705 St. Paul Street, Rochester, New York 14605 USA
tel: 585/262-1331, fax: 585/454-1568
e-mail: gratings.sales@newport.com
web site: http://www.newport.com/

Copyright 2005, Newport Corporation, All Rights Reserved

In 2004, Spectra -Physics, including Richardson Gratings, Corio n Filters, Hilger Crystals and
Oriel Instruments, was acquired by Newport Corporation.

CONTENTS

PREFACE TO THE SIXTH EDITION 12

1. SPECTROSCOPY AND GRATINGS 14
1.0. INTRODUCTION 14
1.1. THE DIFFRACTION GRATING 15
1.2. A BRIEF HISTORY OF GRATING DEVELOPMENT 16
1.3. HISTORY OF THE NEWPORT GRATINGS OPERATION 17
1.4. DIFFRACTION GRATINGS FROM NEWPORT 18

2. THE PHYSICS OF DIFFRACTION GRATINGS 20
2.1. THE GRATING EQUATION 20
2.2. DIFFRACTION ORDERS 25
2.2.1. Existence of diffraction orders 25
2.2.2. Overlapping of diffracted spectra 26
2.3. DISPERSION 28
2.3.1. Angular dispersion 28
2.3.2. Linear dispersion 29
2.4. RESOLVING POWER, SPECTRAL RESOLUTION, AND
BANDPASS 31
2.4.1. Resolving power 31
2.4.2. Spectral resolution 33
2.4.3. Bandpass 34
2.4.4. Resolving power vs. resolution 34
2.5. FOCAL LENGTH AND ƒ/NUMBER 35
2.6. A NAMORPHIC M AGNIFICATION 37
2.7. FREE SPECTRAL RANGE 38
2.8. ENERGY DISTRIBUTION (GRATING EFFICIENCY) 38
2.9. SCATTERED AND STRAY LIGHT 41
2.10. SIGNAL-TO -NOISE RATIO (SNR) 41

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3. RULED GRATINGS 44
3.1. RULING ENGINES 44
3.1.1. The Michelson engine 45
3.1.2. The Mann engine 45
3.1.3. The MIT 'B' engine 46
3.2. THE RULING PROCESS 47
3.3. VARIED LINE-SPACE (VLS) GRATINGS 48

4. HOLOGRAPHIC GRATINGS 50
4.1. PRINCIPLE OF M ANUFACTURE 51
4.1.1. Formation of an interference pattern 51
4.1.2. Formation of the grooves 52
4.2. CLASSIFICATION OF HOLOGRAPHIC GRATINGS 53
4.2.1. Single-beam interference 53
4.2.2. Double-beam interference 54
4.3. THE RECORDING PROCESS 56
4.4. DIFFERENCES BETWEEN RULED AND HOLOGRAPHIC
GRATINGS 57
4.4.1. Differences in grating efficiency 57
4.4.2. Differences in scattered light 58
4.4.3. Differences and limitations in the groove profile 58
4.4.4. Limitations in obtainable groove frequencies 60
4.4.5. Differences in the groove patterns 60
4.4.6. Differences in the substrate shapes 61
4.4.7. Differences in the size of the master substrate 62
4.4.8. Differences in generation time for master gratings 62

5. REPLICATED GRATINGS 64
5.1. THE REPLICATION PROCESS 64
5.2. REPLICA GRATINGS VS. M ASTER GRATINGS 69
5.3. STABILITY OF REPLICATED GRATINGS 71

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6. PLANE GRATINGS AND THEIR MOUNTS 76
6.1. GRATING M OUNT TERMINOLOGY 76
6.2. PLANE GRATING M ONOCHROMATOR M OUNTS 76
6.2.1. The Czerny-Turner monochromator 77
6.2.2. The Ebert-Fastie monochromator 78
6.2.3. The Monk-Gillieson monochromator 79
6.2.4. The Littrow monochromator 80
6.2.5. Double & triple monochromators 81
6.2.6. The constant-scan monochromator 83
6.3. PLANE GRATING SPECTROGRAPH M OUNTS 84

7. CONCAVE GRATINGS AND THEIR MOUNTS 86
7.1. CLASSIFICATION OF GRATING TYPES 86
7.1.1. Groove patterns 87
7.1.2. Substrate (blank) shapes 88
7.2. CLASSICAL CONCAVE GRATING IMAGING 89
7.3. NONCLASSICAL CONCAVE GRATING IMAGING 96
7.4. REDUCTION OF A BERRATIONS 99
7.5. CONCAVE GRATING M OUNTS 102
7.5.1. The Rowland circle spectrograph 102
7.5.2. The Wadsworth spectrograph 104
7.5.3. Flat-field spectrographs 105
7.5.4. Imaging spectrographs and monochromators 106
7.5.5. Constant-deviation monochromators 107

8. IMAGING PROPERTIES OF GRATING SYSTEMS 110
8.1. CHARACTERIZATION OF IMAGING QUALITY 110
8.1.1. Geometric raytracing and spot diagrams 110
8.1.2. Linespread calculations 112
8.2. INSTRUMENTAL IMAGING 113
8.2.1. Magnification of the entrance aperture 113
8.2.2. Effects of the entrance aperture dimensions 116
8.2.3. Effects of the exit aperture dimensions 118
8.3. INSTRUMENTAL BANDPASS 122

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9. EFFICIENCY CHARACTERISTICS OF DIFFRACTION GRATINGS 124
9.1. GRATING EFFICIENCY AND GROOVE SHAPE 128
9.2. EFFICIENCY CHARACTERISTICS FOR TRIANGULAR-GROOVE
GRATINGS 129
9.3. EFFICIENCY CHARACTERISTICS FOR SINUSOIDAL-GROOVE
GRATINGS 135
9.4. THE EFFECTS OF FINITE CONDUCTIVITY 139
9.5. DISTRIBUTION OF ENERGY BY DIFFRACTION ORDER 140
9.6. USEFUL W AVELENGTH RANGE 143
9.7. BLAZING OF RULED TRANSMISSION GRATINGS 144
9.8. BLAZING OF HOLOGRAPHIC REFLECTION GRATINGS 144
9.9. OVERCOATING OF REFLECTION GRATINGS 145
9.11. THE RECIPROCITY THEOREM 147
9.12. CONSERVATION OF ENERGY 148
9.13. GRATING A NOMALIES 149
9.13.1. Rayleigh anomalies 149
9.13.2. Resonance anomalies 150
9.14. GRATING EFFICIENCY CALCULATIONS 152

1 0 . STRAY LIGHT CHARACTERISTICS OF GRATINGS AND
GRATING SYSTEMS 155
10.1. GRATING SCATTER 156
10.1.1. Surface irregularities in the grating coating 157
10.1.2. Dust, scratches & pinholes on the surface of the grating 157
10.1.3. Irregularities in the position of the grooves 157
10.1.4. Irregularities in the depth of the grooves 158
10.1.5. Spurious fringe patterns due to the recording system 159
10.1.6. The perfect grating 159
10.2. INSTRUMENTAL STRAY LIGHT 160
10.2.1. Grating scatter 160
10.2.2. Other diffraction orders from the grating 161
10.2.3. Overfilling optical surfaces 161
10.2.4. Direct reflections from other surfaces 162

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10.2.5. Optical effects due to the sample or sample cell 163
10.2.6. Thermal emission 163
10.3. A NALYSIS OF OPTICAL RAY PATHS IN A GRATING-BASED
INSTRUMENT 164
10.4. DESIGN CONSIDERATIONS FOR REDUCING STRAY LIGHT 167

11. TESTING AND CHARACTERIZING DIFFRACTION GRATINGS 171
11.1. THE M EASUREMENT OF SPECTRAL DEFECTS 171
11.1.1. Rowland ghosts 172
11.1.2. Lyman ghosts 174
11.1.3. Satellites 174
11.2. THE M EASUREMENT OF GRATING EFFICIENCY 176
11.3. THE M EASUREMENT OF DIFFRACTED W AVEFRONT
QUALITY 177
11.3.1. The Foucault knife-edge test 177
11.3.2. Direct wavefront testing 179
11.4. THE M EASUREMENT OF RESOLVING POWER 181
11.5. THE M EASUREMENT OF SCATTERED LIGHT 183
11.6. THE M EASUREMENT OF INSTRUMENTAL STRAY LIGHT 185
11.6.1. The use of cut-off filters 185
11.6.2. The use of monochromatic light 187
11.6.3. Signal-to-noise and errors in absorbance readings 188

12. SELECTION OF DISPERSING SYSTEMS 190
12.1. REFLECTION GRATING SYSTEMS 190
12.1.1. Plane reflection grating systems 190
12.1.2. Concave reflection grating systems 191
12.2. TRANSMISSION GRATING SYSTEMS 192
12.3. GRATING PRISMS (GRISMS) 194
12.4. GRAZING INCIDENCE SYSTEMS 196
12.5. ECHELLES 196

13. APPLICATIONS OF DIFFRACTION GRATINGS 202
13.1. GRATINGS FOR INSTRUMENTAL A NALYSIS 202
13.1.1. Atomic and molecular spectroscopy 202
13.1.2. Fluorescence spectroscopy 204

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13.1.3. Colorimetry 204
13.1.4. Raman spectroscopy 205
13.2. GRATINGS IN LASER SYSTEMS 205
13.2.1. Laser tuning 206
13.2.2. Pulse stretching and compression 208
13.3. GRATINGS IN A STRONOMICAL A PPLICATIONS 209
13.3.1. Ground-based astronomy 210
13.3.2. Space-borne astronomy 213
13.4. GRATINGS IN SYNCHROTRON RADIATION BEAMLINES 214
13.5. SPECIAL USES FOR GRATINGS 214
13.5.1. Gratings as filters 214
13.5.2. Gratings in fiber-optic telecommunications 215
13.5.3 Gratings as beam splitters 217
13.5.4 Gratings as optical couplers 218
13.5.5 Gratings in metrological applications 218

14. ADVICE TO GRATING USERS 219
14.1. CHOOSING A SPECIFIC GRATING 219
14.2. A PPEARANCE 220
14.2.1. Ruled gratings 220
14.2.2 Holographic gratings 221
14.3. GRATING M OUNTING 221
14.4. GRATING SIZE 221
14.5. SUBSTRATE M ATERIAL 222
14.6. GRATING COATINGS 222

15. HANDLING GRATINGS 223
15.1. THE GRATING SURFACE 223
15.2. PROTECTIVE COATINGS 223
15.3. GRATING COSMETICS AND PERFORMANCE 224
15.4. UNDOING DAMAGE TO THE GRATING SURFACE 225
15.5. GUIDELINES FOR HANDLING GRATINGS 226

16. GUIDELINES FOR SPECIFYING GRATINGS 227
16.1. REQUIRED SPECIFICATIONS 227
16.2. SUPPLEMENTAL SPECIFICATIONS 231

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16.3. A DDITIONAL REQUIRED SPECIFICATIONS FOR CONCAVE
A BERRATION-REDUCED GRATINGS 232

APPENDIX A. SOURCES OF ERROR IN MONOCHROMATOR-MODE
EFFICIENCY MEASUREMENTS OF PLANE DIFFRACTION
GRATINGS 237
A.0. INTRODUCTION 237
A.1. OPTICAL SOURCES OF ERROR 239
A.1.1. Wavelength error 239
A.1.2. Fluctuation of the light source intensity 241
A.1.3. Bandpass 241
A.1.4. Superposition of diffracted orders 242
A.1.5. Degradation of the reference mirror 243
A.1.6. Collimation 244
A.1.7. Stray light or “optical noise” 244
A.1.8. Polarization 245
A.1.9. Unequal path length 246
A.2. M ECHANICAL SOURCES OF ERROR 246
A.2.1. Alignment of incident beam to grating rotation axis 246
A.2.2. Alignment of grating surface to grating rotation axis 247
A.2.3. Orientation of the grating grooves (tilt adjustment) 247
A.2.4. Orientation of the grating surface (tip adjustment) 247
A.2.5. Grating movement 248
A.3. ELECTRICAL SOURCES OF ERROR 248
A.3.1. Detector linearity 248
A.3.2. Changes in detector sensitivity 249
A.3.3. Sensitivity variation across detector surface 250
A.3.4. Electronic noise 250
A.4. ENVIRONMENTAL FACTORS 250
A.4.1. Temperature 250
A.4.2. Humidity 251
A.4.3. Vibration 251
A.5. SUMMARY 252

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APPENDIX B. LIE ABERRATION THEORY FOR GRATING SYSTEMS 253

FURTHER READING 257

GRATING PUBLICATIONS BY NEWPORT CORPORATION
PERSONNEL 260

INDEX 266

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PREFACE TO THE SIXTH EDITION

“No single tool has contributed more to the progress of
modern physics than the diffraction grating …”1

Newport Corporation is proud to build upon the heritage of technical
excellence that began when Bausch & Lomb produced its first high-quality
master grating in the late 1940s. A high-fidelity replication process was
subsequently developed to make duplicates of the tediously generated master
gratings. This process became the key to converting diffraction gratings from
academic curiosities to commercially-available optical components, which in turn
enabled gratings to essentially replace prisms as the optical dispersing element
of choice in modern laboratory instrumentation.
For several years, since its introduction in 1970, the Diffraction Grating
Handbook was the primary source of information of a general nature regarding
diffraction gratings. In 1982, Dr. Michael Hutley of the National Physical
Laboratory published Diffraction Gratings, a monograph that addresses in more
detail the nature and uses of gratings, as well as their manufacture. In 1997, Dr.
Erwin Loewen, emeritus director of the Bausch & Lomb grating laboratory who
wrote the original Handbook thirty-five years ago, wrote with Dr. Evgeny Popov
(now with the Laboratoire d’Optique Électromagnétique) a very thorough and
complete monograph entitled Diffraction Gratings and Applications. Readers of
this Handbook who seek additional insight into the many aspects of diffraction
grating behavior, manufacture and use are encouraged to turn to these two
books.

Christopher Palmer
Newport Corporation
Rochester, New York

January 2005

1 G. R. Harrison, “The production of diffraction gratings. I. Development of the rulin g art,”
J. Opt. Soc. Am. 39, 413-426 (1949).

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1. SPECTROSCOPY AND GRATINGS

“It is difficult to point to another single device that has brought more
important experimental information to every field of science than the
diffraction grating. The physicist, the astronomer, the chemist, the
biologist, the metallurgist, all use it as a routine tool of unsurpassed
accuracy and precision, as a detector of atomic species to determine the
characteristics of heavenly bodies and the presence of atmospheres in
the planets, to study the structures of molecules and atoms, and to
obtain a thousand and one items of information without which modern
science would be greatly handicapped.”

 J. Strong, “The Johns Hopkins University and diffraction gratings,”
J. Opt. Soc. Am. 50, 1148-1152 (1960), quoting G. R. Harrison.

1.0. INTRODUCTION

Spectroscopy is the study of electromagnetic spectra – the wavelength
composition of light – due to atomic and molecular interactions. For many years,
spectroscopy has been important in the study of physics, and it is now equally
important in astronomical, biological, chemical, metallurgical and other analytical
investigations. The first experimental tests of quantum mechanics involved
verifying predictions regarding the spectrum of hydrogen with grating
spectrometers. In astrophysics, diffraction gratings provide clues to the
composition of and processes in stars and planetary atmospheres, as well as
offer clues to the large-scale motions of objects in the universe. In chemistry,
toxicology and forensic science, grating-based instruments are used to determine
the presence and concentration of chemical species in samples. In telecom-
munications, gratings are being used to increase the capacity of fiber-optic
networks using wavelength division multiplexing (WDM). Gratings have also

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found many uses in tuning and spectrally shaping laser light, as well as in
chirped pulse amplification applications.
The diffraction grating is of considerable importance in spectroscopy, due to
its ability to separate (disperse) polychromatic light into its constituent
monochromatic components. In recent years, the spectroscopic quality of
diffraction gratings has greatly improved, and Newport has been a leader in this
development.
The extremely high accuracy required of a modern diffraction grating
dictates that the mechanical dimensions of diamond tools, ruling engines, and
optical recording hardware, as well as their environmental conditions, be con-
trolled to the very limit of that which is physically possible. A lower degree of
accuracy results in gratings that are ornamental but have little technical or scien-
tific value. The challenge to produce precision diffraction gratings has attracted
the attention of some of the world's most capable scientists and technicians.
Only a few have met with any appreciable degree of success, each limited by the
technology available.

1.1. THE DIFFRACTION GRATING

A diffraction grating is a collection of reflecting (or transmitting) elements
separated by a distance comparable to the wavelength of light under study. It
may be thought of as a collection of diffracting elements, such as a pattern of
transparent slits (or apertures) in an opaque screen, or a collection of reflecting
grooves on a substrate (also called a blank ). In either case, the fundamental
physical characteristic of a diffraction grating is the spatial modulation of the
refractive index. Upon diffraction, an electromagnetic wave incident on a grating
will have its electric field amplitude, or phase, or both, modified in a predictable
manner, due to the periodic variation in refractive index in the region near the
surface of the grating.
A reflection grating consists of a grating superimposed on a reflective
surface, whereas a transmission grating consists of a grating superimposed on a
transparent surface.
A master grating (also called an original) is a grating whose surface-relief
pattern is created “from scratch”, either by mechanical ruling (see Chapter 3) or
holographic recording (see Chapter 4). A replica grating is one whose surface-
relief pattern is generated by casting or molding the relief pattern of another
grating (see Chapter 5).

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1.2. A BRIEF HISTORY OF GRATING DEVELOPMENT

The first diffraction grating was made by an American astronomer, David
Rittenhouse, in 1785, who reported constructing a half-inch wide grating with
fifty-three apertures.2 Apparently he developed this prototype no further, and
there is no evidence that he tried to use it for serious scientific experiments.
In 1821, most likely unaware of the earlier American report, Joseph von
Fraunhofer began his work on diffraction gratings.3 His research was given
impetus by his insight into the value that grating dispersion could have for the
new science of spectroscopy. Fraunhofer's persistence resulted in gratings of
sufficient quality to enable him to measure the absorption lines of the solar
spectrum, now generally referred to as the Fraunhofer lines. He also derived the
equations that govern the dispersive behavior of gratings. Fraunhofer was in-
terested only in making gratings for his own experiments, and upon his death, his
equipment disappeared.
By 1850, F.A. Nobert, a Prussian instrument maker, began to supply
scientists with gratings superior to Fraunhofer's. About 1870, the scene of
grating development returned to America, where L.M. Rutherfurd, a New York
lawyer with an avid interest in astronomy, became interested in gratings. In just
a few years, Rutherfurd learned to rule reflection gratings in speculum metal that
were far superior to any that Nobert had made. Rutherfurd developed gratings
that surpassed even the most powerful prisms. He made very few gratings,
though, and their uses were limited.
Rutherfurd's part-time dedication, impressive as it was, could not match the
tremendous strides made by H.A. Rowland, professor of physics at the Johns
Hopkins University. Rowland's work established the grating as the primary
optical element of spectroscopic technology.4 Rowland constructed sophis -
ticated ruling engines and invented the concave grating, a device of spectacular

2 D. Rittenhouse, “Explanation of an optical deception,” Trans. Amer. Phil. Soc. 2, 37-42
(1786).
3 J. Fruanhofer, “Kurtzer Bericht von den Resultaten neuerer Versuche über die Sesetze des
lichtes, und die Theorie derselbem,” Ann. D. Phys. 74, 337-378 (1823).
4 H. Rowland, “Preliminary notice of results accomplished on the manufacture and theory of
gratings for optical purposes,” Phil. Mag. Suppl. 13, 469-474 (1882); G. R. Harrison and E.
G. Loewen, “Ruled gratings and wavelength tables,” Appl. Opt. 15, 1744-1747 (1976).

16

value to modern spectroscopists. He continued to rule gratings until his death in
1901.
After Rowland's great success, many people set out to rule diffraction
gratings. The few who were successful sharpened the scientific demand for
gratings. As the advantages of gratings over prisms and interferometers for
spectroscopic work became more apparent, the demand for diffraction gratings
far exceeded the supply.

1.3. HISTORY OF THE NEWPORT GRATINGS OPERATION

In 1947, Bausch & Lomb decided to make precision gratings available
commercially. In 1950, through the encouragement of Prof. George R. Harrison of
MIT, David Richardson and Robert Wiley of Bausch & Lomb succeeded in pro-
ducing their first high quality grating. This was ruled on a rebuilt engine that had
its origins in the University of Chicago laboratory of Prof. Albert A. Michelson.
A high fidelity replication process was subsequently developed, which was
crucial to making replicas, duplicates of the painstakingly-ruled master gratings.
A most useful feature of modern gratings is the availability of an enormous range
of sizes and groove spacings (up to 10,800 grooves per millimeter), and their en-
hanced quality is now almost taken for granted. In particular, the control of
groove shape (or blazing) has increased spectral efficiency dramatically. In
addition, interferometric and servo control systems have made it possible to
break through the accuracy barrier previously set by the mechanical constraints
inherent in the ruling engines.5
During the subsequent decades, we have produced thousands of master
gratings and many times that number of high quality replicas. In 1985, Milton
Roy Company acquired Bausch & Lomb's gratings and spectrometer operations;
in 1995 it sold these operations to Life Sciences International plc as part of
Spectronic Instruments, Inc. – at this time, the gratings operations took the name
Richardson Grating Laboratory. In 1997, Spectronic Instruments was acquired
by Thermo Electron Corporation, and the gratings operation was called Thermo
RGL for a time before being transferred to Thermo Electron’s subsidiary,
Spectra-Physics.

5 G. R. Harrison and G. W. Stroke, “Interferometric control of grating ruling with continuous
carriage advance,” J. Opt. Soc. Am. 45, 112-121 (1955).

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In 2004, Spectra-Physics was acquired by Newport Corporation, a leading
global supplier of advanced-technology products and systems to the
semiconductor, communications, electronics, research and life and health
sciences markets. Newport provides components and integrated subsystems to
manufacturers of semiconductor processing equipment, biomedical instru-
mentation and medical devices, advanced automated assembly and test systems
to manufacturers of communications and electronics devices, and a broad array
of high-precision systems, components and instruments to commercial, academic
and government customers worldwide. Newport’s innovative solutions leverage
its expertise in photonics instrumentation, lasers and light sources, precision
robotics and automation, sub-micron positioning systems, vibration isolation,
optical components and optical subsystems to enhance the capabilities and pro-
ductivity of its customers’ manufacturing, engineering and research applications.
During these changes in corporate ownership, we have continued to uphold
the traditions of precision and quality established by Bausch & Lomb over fifty
years ago.

1.4. DIFFRACTION GRATINGS FROM NEWPORT

The gratings operation of Newport Corporation, which is known throughout
the world as “the Grating Lab”, is a unique facility in Rochester, New York,
containing not only the Newport ruling engines and holographic recording
chambers (both of which are used for making master gratings) but the replication
and associated testing and inspection facilities for supplying replicated gratings
in commercial quantities.
To achieve the high practical resolution characteristic of high-quality
gratings, a precision of better than 1 nm (= 0.001 µm) in the spacing of the
grooves must be maintained. Such high precision requires extraordinary control
over temperature fluctuation and vibration in the ruling engine environment.
This control has been established by the construction of specially-designed
ruling cells that provide environments in which temperature stability is
maintained at ±0.01 °C for weeks at a time, as well as vibration isolation that
suppresses ruling engine displacement to less than 0.025 µm. The installation
can maintain reliable control over the important environmental factors for periods
in excess of six weeks, the time required to rule large, finely-spaced gratings.
Newport has facilities for coating and testing master and replica substrates,
as well as special areas for the controlled replication process itself. In order to

18

produce the finest gratings with maximum control and efficiency, even storage,
packing and shipping of finished gratings are part of the same facility.
In addition to burnishing gratings with a diamond tool, an optical
interference pattern can be used to produce holographic gratings. Master
holographic gratings require strict maintenance of the recording optical system
to obtain the best contrast and fringe structure. Newport produces holographic
gratings in its dedicated recording facility, in whose controlled environment
thermal gradients and air currents are minimized and fine particulates are filtered
from the air. These master gratings are replicated in a process identical to that
for ruled master gratings.

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2. THE PHYSICS OF DIFFRACTION
GRATINGS

2.1. THE GRATING EQUATION

When monochromatic light is incident on a grating surface, it is diffracted
into discrete directions. We can picture each grating groove as being a very
small, slit-shaped source of diffracted light. The light diffracted by each groove
combines to form set of diffracted wavefronts. The usefulness of a grating
depends on the fact that there exists a unique set of discrete angles along which,
for a given spacing d between grooves, the diffracted light from each facet is in
phase with the light diffracted from any other facet, leading to constructive
interference.
Diffraction by a grating can be visualized from the geometry in Figure 2-1,
which shows a light ray of wavelength λ incident at an angle α and diffracted by
a grating (of groove spacing d, also called the pitch) along at set of angles {βm}.
These angles are measured from the grating normal, which is shown as the
dashed line perpendicular to the grating surface at its center. The sign con-
vention for these angles depends on whether the light is diffracted on the same
side or the opposite side of the grating as the incident light. In diagram (a),
which shows a reflection grating, the angles α > 0 and β1 > 0 (since they are
measured counter-clockwise from the grating normal) while the angles β0 < 0 and
β–1 < 0 (since they are measured clockwise from the grating normal). Diagram (b)
shows the case for a transmission grating.
By convention, angles of incidence and diffraction are measured from the
grating normal to the beam. This is shown by arrows in the diagrams. In both
diagrams, the sign convention for angles is shown by the plus and minus
symbols located on either side of the grating normal. For either reflection or
transmission gratings, the algebraic signs of two angles differ if they are mea-
sured from opposite sides of the grating normal. Other sign conventions exist,
so care must be taken in calculations to ensure that results are self-consistent.

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grating normal
(a)
+ –
incident light reflected light

β
0
α
β
1 diffracted light
diffracted light
β
–1

d

(b) grating normal

+ –
incident light

α

d

β–1

β1
β0
diffracted light

+ –
Figure 2-1. Diffraction by a plane grating. A beam of monochromatic light of wavelength
λ is incident on a grating and diffracted along several discrete paths. The triangular
grooves come out of the page; the rays lie in the plane of the page. The sign convention
for the angles α and β is shown by the + and – signs on either side of the grating normal.
(a) A reflection grating: the incident and diffracted rays lie on the same side of the grating.
(b) A transmission grating: the diffracted rays lie on the opposite side of the grating from
the incident ray.

Another illustration of grating diffraction, using wavefronts (surfaces of
constant phase), is shown in Figure 2 -2. The geometrical path difference

21

between light from adjacent grooves is seen to be d sinα + d sinβ. [Since β < 0,
the term d sinβ is negative.] The principle of constructive interference dictates
that only when this difference equals the wavelength λ of the light, or some
integral multiple thereof, will the light from adjacent grooves be in phase (leading
to constructive interference). At all other angles the wavelets originating from
the groove facets will interfere destructively.

grating
normal
ray 2
ray 1
+ –

α

A β

B

d sinβ d sinα

d

Figure 2-2. Geometry of diffraction, for planar wavefronts. Two parallel rays, labeled 1
and 2, are incident on the grating one groove spacing d apart and are in phase with each
other at wavefront A. Upon diffraction, the principle of constructive interference implies
that these rays are in phase at diffracted wavefront B if the difference in their path
lengths, dsinα + dsinβ, is an integral number of wavelengths; this in turn leads to the
grating equation.

These relationships are expressed by the grating equation

mλ = d (sinα + sinβ), (2-1)

which governs the angular locations of the principal intensity maxima when light
of wavelength λ is diffracted from a grating of groove spacing d. Here m is the
diffraction order (or spectral order), which is an integer. For a particular

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wavelength λ, all values of m for which |mλ/d| < 2 correspond to propagating
(rather than evanescent) diffraction orders. The special case m = 0 leads to the
law of reflection β = –α.
It is sometimes convenient to write the grating equation as

Gmλ = sinα + sinβ, (2-2)

where G = 1/d is the groove frequency or groove density, more commonly called
"grooves per millimeter".
Eq. (2-1) and its equivalent Eq. (2-2) are the common forms of the grating
equation, but their validity is restricted to cases in which the i cident and
n
diffracted rays lie in a plane which is perpendicular to the grooves (at the center
of the grating). The majority of grating systems fall within this category, which is
called classical (or in-plane) diffraction. If the incident light beam is not
perpendicular to the grooves, though, the grating equation must be modified:

Gmλ = cosε (sinα + sinβ). (2-3)

Here ε is the angle between the incident light path and the plane perpendicular to
the grooves at the grating center (the plane of the page in Figure 2-2). If the
incident light lies in this plane, ε = 0 and Eq. (2-3) reduces to the more familiar Eq.
(2-2). In geometries for which ε ? 0, the diffracted spectra lie on a cone rather
than in a plane, so such cases are termed conical diffraction.
For a grating of groove spacing d, there is a purely mathematical relationship
between the wavelength and the angles of incidence and diffraction. In a given
spectral order m, the different wavelengths of polychromatic wavefronts incident
at angle α are separated in angle:

 mλ 
β(λ) = sin −1 − sin α  . (2-4)
 d 

When m = 0, the grating acts as a mirror, and the wavelengths are not separated
(β = –α for all λ); this is called specular reflection or simply the zero order.
A special but common case is that in which the light is diffracted back
toward the direction from which it came (i.e., α = β); this is called the Littrow
configuration, for which the grating equation becomes

23

mλ = 2d sinα, in Littrow. (2-5)

In many applications a constant-deviation monochromator mount is used, in
which the wavelength λ is changed by rotating the grating about the axis
coincident with its central ruling, with the directions of incident and diffracted
light remaining unchanged. The deviation angle 2K between the incidence and
diffraction directions (also called the angular deviation) is

2K = α – β = constant, (2-6)

while the scan angle φ, which varies with λ and is measured from the grating
normal to the bisector of the beams, is

2φ = α + β. (2-7)

Note that φ changes with λ (as do α and β). In this case, the grating equation can
be expressed in terms of φ and the half deviation angle K as

mλ = 2d cosK sinφ. (2-8)

This version of the grating equation is useful for monochromator mounts (see
Chapter 7). Eq. (2-8) shows that the wavelength diffracted by a grating in a
monochromator mount is directly proportional to the sine of the scan angle φ
through which the grating rotates, which is the basis for monochromator drives
in which a sine bar rotates the grating to scan wavelengths (see Figure 2-3).
For the constant-deviation monochromator mount, the incidence and
diffraction angles can be expressed simply in terms of the scan angle φ and the
half-deviation angle K via

α(λ) = φ(λ) + K (2-9)

and

β(λ) = φ(λ) – K , (2-10)

where we show explicitly that α, β and φ depend on the wavelength λ.

24

axis of grating rotation
(out of page)

grating x
φ

screw

Figure 2-3. A sine bar mechanism for wavelength scanning. As the screw is extended
linearly by the distance x shown, the grating rotates through an angle φ in such a way that
sinφ is proportional to x.

2.2. DIFFRACTION ORDERS

Generally several integers m will satisfy the grating equation – we call each
of these values a diffraction order.

2.2.1. Existence of diffraction orders

For a particular groove spacing d, wavelength λ and incidence angle α, the
grating equation (2-1) is generally satisfied by more than one diffraction angle β.
In fact, subject to restrictions discussed below, there will be several dis crete
angles at which the condition for constructive interference is satisfied. The
physical significance of this is that the constructive reinforcement of wavelets
diffracted by successive grooves merely requires that each ray be retarded (or
advanced) in phase with every other; this phase difference must therefore
correspond to a real distance (path difference) which equals an integral multiple
of the wavelength. This happens, for example, when the path difference is one
wavelength, in which case we speak of the positive first diffraction order (m = 1)
or the negative first diffraction order (m = –1), depending on whether the rays are
advanced or retarded as we move from groove to groove. Similarly, the second
order (m = 2) and negative second order (m = –2) are those for which the path

25

difference between rays diffracted from adjacent grooves equals two
wavelengths.
The grating equation reveals that only those spectral orders for which |mλ/d|
< 2 can exist; otherwise, |sinα + sinβ | > 2, which is physically meaningless. This
restriction prevents light of wavelength λ from being diffracted in more than a
finite number of orders. Specular reflection (m = 0) is always possible; that is, the
zero order always exists (it simply requires β = –α). In most cases, the grating
equation allows light of wavelength λ to be diffracted into both negative and
positive orders as well. Explicitly, spectra of all orders m exist for which

–2d < mλ < 2d, m an integer. (2-11)

For λ/d << 1, a large number of diffracted orders will exist.
As seen from Eq. (2-1), the distinction between negative and positive
spectral orders is that

β > –α for positive orders (m > 0),

β < –α for negative orders (m < 0), (2-12)

β = –α for specular reflection (m = 0).

This sign convention for m requires that m > 0 if the diffracted ray lies to the left
(the counter-clockwise side) of the zero order (m = 0), and m < 0 if the diffracted
ray lies to the right (the clockwise side) of the zero order. This convention is
shown graphically in Figure 2-4.

2.2.2. Overlapping of diffracted spectra

The most troublesome aspect of multiple order behavior is that successive
spectra overlap, as shown in Figure 2-5. It is evident from the grating equation

26

grating normal
+ − m =0

α

positive negative
orders orders

m>0 m<0

Figure 2-4. Sign convention for the spectral order m. In this example α is positive.

grating
normal
m =0 incident
light
α
300 200 100
m = +2
nm nm nm

600 400 200
m = +1
nm nm nm

Figure 2-5. Overlapping of spectral orders. The light for wavelengths 100, 200 and 300
nm in the second order is diffracted in the same direction as the light for wavelengths 200,
400 and 600 nm in the first order. In this diagram, the light is incident from the right, so α
< 0.

27

that light of wavelength λ diffracted by a grating along direction β will be
accompanied by integral fractions λ/2, λ/3, etc.; that is, for any grating instrument
configuration, the light of wavelength λ diffracted in the m = 1 order will coincide
with the light of wavelength λ/2 diffracted in the m = 2 order, etc. In this example,
the red light (600 nm) in the first spectral order will overlap the ultraviolet light
(300 nm) in the second order. A detector sensitive at both wavelengths would
see both simultaneously. This superposition of wavelengths, which would lead
to ambiguous spectro scopic data, is inherent in the grating equation itself and
must be prevented by suitable filtering (called order sorting), since the detector
cannot generally distinguish between light of different wavelengths incident on
it (within its range of sensitivity). [See also Section 2.7 below.]

2.3. DISPERSION

The primary purpose of a diffraction grating is to dis perse light spatially by
wavelength. A beam of white light incident on a grating will be separated into its
component wavelengths upon diffraction from the grating, with each wavelength
diffracted along a different direction. Dispersion is a measure of the separation
(either angular or spatial) between diffracted light of different wavelengths.
Angular dispersion expresses the spectral range per unit angle, and linear reso-
lution expresses the spectral range per unit length.

2.3.1. Angular dispersion

The angular spread ∆β of a spectrum of order m between the wavelength λ
and λ + ∆λ can be obtained by differentiating the grating equation, assuming the
incidence angle α to be constant. The change D in diffraction angle per unit
wavelength is therefore

dβ m m
D= = = secβ = Gm secβ, (2-13)
dλ d cosβ d

where β is given by Eq. (2-4). The quantity D is called the angular dispersion.
As the groove frequency G = 1/d increases, the angular dispersion increases
(meaning that the angular separation between wavelengths increases for a given
order m).

28

In Eq. (2-13), it is important to realize that the quantity m/d is not a ratio
which may be chosen independently of other parameters; substitution of the
grating equation into Eq. (2-13) yields the following general equation for the
angular dispersion:

dβ sin α + sin β
D= = . (2-14)
dλ λ cos β

For a given wavelength, this shows that the angular dispersion may be
considered to be solely a function of the angles of incidence and diffraction.
This becomes even more clear when we consider the Littrow configuration
(α = β), in which case Eq. (2-14) reduces to

dβ 2
D= = tan β , in Littrow. (2-15)
dλ λ

When |β| increases from 10° to 63° in Littrow use, the angular dispersion can be
seen from Eq. (2-15) to increase by a factor of ten, regardless of the spectral order
or wavelength under consideration. Once the diffraction angle β has been
determined, the choice must be made whether a fine-pitch grating (small d)
should be used in a low diffraction order, or a coarse-pitch grating (large d) such
as an echelle grating (see Section 12.5) should be used in a high order. [The fine-
pitched grating, though, will provide a larger free spectral range; see Section 2.7
below.]

2.3.2. Linear dispersion

For a given diffracted wavelength λ in order m (which corresponds to an
angle of diffraction β), the linear dispersion of a grating system is the product of
the angular dispersion D and the effective focal length r'(β) of the system:

dβ mr ′ mr ′
r' D = r' = = sec β = Gmr' secβ. (2-16)
dλ d cos β d

The quantity r' ∆β = ∆l is the change in position along the spectrum (a real
distance, rather than a wavelength). We have written r'(β) for the focal length to

29

show explicitly that it may depend on the diffraction angle β (which, in turn,
depends on λ).
The reciprocal linear dispersion, sometimes called the plate factor P, is
more often considered; it is simply the reciprocal of r' D, usually measured in
nm/mm:

d cos β
P= . (2-14’)
mr ′

P is a measure of the change in wavelength (in nm) corresponding to a change in
location along the spectrum (in mm). It should be noted that the terminology
plate factor is used by some authors to represent the quantity 1/sinΦ, where Φ
is the angle the spectrum makes with the line perpendicular to the diffracted rays
(see Figure 2 -6); in order to avoid confusion, we call the quantity 1/sinΦ the
obliquity factor. When the image plane for a particular wavelength is not
perpendicular to the diffracted rays (i.e., when Φ ? 90°), P must be multiplied by
the obliquity factor to obtain the correct reciprocal linear dispersion in the image
plane.

Φ

diffracted ray

plane of spectral image

Figure 2-6. The obliquity angle Φ . The spectral image recorded need not lie in the plane
perpendicular to the diffracted ray (i.e., Φ ? 90°).

30

2.4. RESOLVING POWER, SPECTRAL RESOLUTION, AND
BANDPASS

2.4.1. Resolving power

The resolving power R of a grating is a measure of its ability to separate
adjacent spectral lines of average wavelength λ. It is usually expressed as the di-
mensionless quantity

λ
R= . (2-17)
∆λ

Here ∆λ is the limit of resolution, the difference in wavelength between two lines
of equal intensity that can be distinguished (that is, the peaks of two
wavelengths λ1 and λ2 for which the separation |λ1 – λ2| < ∆λ will be ambiguous).
Often the Rayleigh criterion is used to determine ∆λ – that is, the intensity
maxima of two neighboring wavelengths are resolvable (.e., identifiable as
i
distinct spectral lines) if the intensity maximum of one wavelength coincides with
the intensity minimum of the other wavelength.6
The theoretical resolving power of a planar diffraction grating is given in
elementary optics textbooks as

R = mN, (2-18)

where m is the diffraction order and N is the total number of grooves illuminated
on the surface of the grating. For negative orders (m < 0), the absolute value of R
is considered.
A more meaningful expression for R is derived below. The grating equation
can be used to replace m in Eq. (2-18):

Nd (sin α + sin β )
R= . (2-19)
λ

6 D. W. Ball, The Basics of Spectroscopy, SPIE Press (2001), ch. 8.

31

If the groove spacing d is uniform over the surface of the grating, and if the
grating substrate is planar, the quantity Nd is simply the ruled width W of the
grating, so

W (sin α + sin β )
R= . (2-20)
λ

As expressed by Eq. (2-20), R is not dependent explicitly on the spectral order or
the number of grooves; these parameters are contained within the ruled width
and the angles of incidence and diffraction. Since

| sinα + sinβ | < 2 , (2-21)

the maximum attainable resolving power is

2W
RMAX = , (2-22)
λ

regardless of the order m or number of grooves N under illumination. This maxi-
mum condition corresponds to the grazing Littrow configuration, i.e., |α| ˜ 90°
(grazing incidence) and α ˜ β (Littrow).
It is useful to consider the resolving power as being determined by the
maximum phase retardation of the extreme rays diffracted from the grating.7
Measuring the difference in optical path lengths between the rays diffracted from
opposite sides of the grating provides the maximum phase retardation; dividing
this quantity by the wavelength λ of the diffracted light gives the resolving
power R.
The degree to which the theoretical resolving power is attained depends not
only on the angles α and β, but also on the optical quality of the grating surface,
the uniformity of the groove spacing, the quality of the associated optics in the
system, and the width of the slits (or detector elements). Any departure of the
diffracted wavefront greater than λ/10 from a plane (for a plane grating) or from a
sphere (for a spherical grating) will result in a loss of resolving power due to
aberrations at the image plane. The grating groove spacing m be kept ust

7 N. Abramson, “Principle of least wave change,” J. Opt. Soc. Am. A6, 627-629 (1989).

32

constant to within about one percent of the wavelength at which theoretical
performance is desired. Experimental details, such as slit width, air currents, and
vibrations can seriously interfere with obtaining optimal results.
The practical resolving power is limited by the spectral width of the spectral
lines emitted by the source. For this reason, systems with revolving powers
greater than R = 500,000 are not usually required except for the study of spectral
line shapes, Zeeman effects, and line shifts, and are not needed for separating
individual spectral lines.
A convenient test of resolving power is to examine the isotopic structure of
the mercury emission line at λ = 546.1 nm (see Section 11.4). Another test for
resolving power i to examine the line profile generated in a spectrograph or
s
scanning spectrometer when a single mode laser is used as the light source. The
full width at half maximum intensity (FWHM) can be used as the criterion for ∆λ.
Unfortunately, resolving power measurements are the convoluted result of all
optical elements in the system, including the locations and dimensions of the en-
trance and exit slits and the auxiliary lenses and mirrors, as well as the quality of
these elements. Their effects on resolving power measurements are necessarily
superimposed on those of the grating.

2.4.2. Spectral resolution

While resolving power can be considered a characteristic of the grating and
the angles at which it is used, the ability to resolve two wavelengths λ1 and
λ2 = λ1 + ∆λ generally depends not only on the grating but on the dimensions
and locations of the entrance and exit slits (or detector elements), the aberrations
in the images, and the magnification of the images. The minimum wavelength dif-
ference ∆λ (also called the limit of resolution, or simply resolution) between two
wavelengths that can be resolved unambiguously can be determined by
convoluting the image of the entrance aperture (at the image plane) with the exit
aperture (or detector element). This measure of the ability of a grating system to
resolve nearby wavelengths is arguably more relevant than is resolving power,
since it takes into account the image effects of the system. While resolving
power is a dimensionless quantity, resolution has spectral units (usually
nanometers).

33

2.4.3. Bandpass

The (spectral) bandpass B of a spectroscopic system is the wavelength
interval of the light that passes through the exit slit (or falls onto a detector
element). It is often defined as the difference in wavelengths between the points
of half-maximum intensity on either side of an intensity maximum.
For an optical system in which the width of the image of the entrance slit is
roughly equal to the width of the exit slit, an estimate for bandpass is the product
of the exit slit width w' and the reciprocal linear dispersion P:

B ˜ w' P. (2-23)

An instrument with smaller bandpass can resolve wavelengths that are closer
together than an instrument with a larger bandpass. Bandpass can be reduced
by decreasing the width of the exit slit (to a certain limit; see Chapter 8), but
usually at the expense of decreasing light intensity as well.
See Section 8.3 for additional comments on instrumental bandpass.

2.4.4. Resolving power vs. resolution

In the literature, the terms resolving power and resolution are sometimes in-
terchanged. While the word power has a very specific meaning (energy per unit
time), the phrase resolving power does not involve power in this way; as
suggested by Hutley, though, we may think of resolving power as “ability to re-
solve”.8
The comments above regarding resolving power and resolution pertain to
planar classical gratings used in collimated light (plane waves). The situation is
complicated for gratings on concave substrates or with groove patterns
consisting of unequally spaced lines, which restrict the usefulness of the
previously defined simple formulas, though they may still yield useful approxima-
tions. Even in these cases, though, the concept of maximum retardation is still a
useful measure of the resolving power, and the convolution of the image and the
exit slit is still a useful measure of resolution.

8 M. C. Hutley, Diffraction Gratings, Academic Press (New York, New York: 1982), p. 29.

34

2.5. FOCAL LENGTH AND ƒ/NUMBER

For gratings (or grating systems) that image as well as diffract light, or
disperse light that is not collimated, a focal length may be defined. If the beam
diffracted from a grating of a given wavelength λ and order m converges to a
focus, then the distance between this focus and the grating center is the focal
length r'(λ). [If the diffracted light is collimated, and then focused by a mirror or
lens, the focal length is that of the refocusing mirror or lens and not the distance
to the grating.] If the diffracted light is diverging, the focal length may still be
defined, although by convention we take it to be negative (indicating that there
is a virtual image behind the grating). Similarly, the incident light may diverge
toward the grating (so we define the incidence or entrance slit distance r(λ) > 0)
or it may converge toward a focus behind the grating (for which r(λ) < 0).
Usually gratings are used in configurations for which r does not depend on
wavelength (though in such cases r' usually depends on λ).
In Figure 2-7, a typical concave grating configuration is shown; the
monochromatic incident light (of wavelength λ) diverges from a point source at A
and is diffracted toward B. Points A and B are distances r and r', respectively,
from the grating center O. In this figure, both r and r' are positive.

source point
incident light
A

r
α
W O GN
β
r'
B
image point
diffracted light

Figure 2-7. Geometry for focal distances and focal ratios (ƒ/numbers). GN is the grating
normal (perpendicular to the grating at its center, O), W is the width of the grating (its
dimension perpendicular to the groove direction, which is out of the page), and A and B
are the source and image points, respectively.

35

Calling the width (or diameter) of the grating (in the dis persion plane) W
allows the input and output ƒ/numbers (also called focal ratios) to be defined:

r r ′(λ )
ƒ/noINPUT = , ƒ/noOUTPUT = . (2-24)
W W

Usually the input ƒ/number is matched to the ƒ/number of the light cone leaving
the entrance optics (e.g., an entrance slit or fiber) in order to use as much of the
grating surface for diffraction as possible. This increases the amount of
diffracted energy while not overfilling the grating (which would generally con-
tribute to instrumental stray light; see Chapter 10).
For oblique (non-normal) incidence or diffraction, Eqs. (2-25) are often
modified by replacing W with the projected width of the grating:

r r ′ (λ )
ƒ/noINPUT = , ƒ/noOUTPUT = . (2-25)
W cos α W cos β

These equations account for the reduced width of the grating as seen by the
entrance and exit slits; moving toward oblique angles (i.e., increasing |α| or |β|)
decreases the projected width and therefore increases the ƒ/number.
The focal length is an important parameter in the design and specification of
grating spectrometers, since it governs the overall size of the optical system
(unless folding mirrors are used). The ratio between the input and output focal
lengths determines the projected width of the entrance slit that must be matched
to the exit slit width or detector element size. The ƒ/number is also important, as
it is generally true that spectral aberrations decrease as ƒ/number increases.
Unfortunately, increasing the input ƒ/number results in the grating subtending a
smaller solid angle as seen from the entrance slit; this will reduce the amount of
light energy the grating collects and consequently reduce the intensity of the
diffracted beams. This trade-off prohibits the formulation of a simple rule for
choosing the input and output ƒ/numbers, so sophisticated design procedures
have been developed to minimize aberrations while maximizing collected energy.
See Chapter 7 for a discussion of the imaging properties and Chapter 8 for a
description of the efficiency characteristics of grating systems.

36

2.6. ANAMORPHIC M AGNIFICATION

For a given wavelength λ, we may consider the ratio of the width of a
collimated diffracted beam to that of a collimated incident beam to be a measure
of the effective magnification of the grating (see Figure 2-8). From this figure we
see that this ratio is

b cos β
= . (2-26)
a cosα

Since α and β depend on λ through the grating equation (2-1), this magnification
will vary with wavelength. The ratio b/a is called the anamorphic magnification;
for a given wavelength λ, it depends only on the angular configuration in which
the grating is used.

grating
normal

a
α
β
b

grating
surface

Figure 2-8. Anamorphic magnification. The ratio b/a of the beam widths equals the
anamorphic magnification; the grating equation (2-1) guarantees that this ratio will not
equal unity unless m = 0 (specular reflection) or α = β (the Littrow configuration).

The magnification of an object not located at infinity (so that the incident
rays are not collimated) is discussed in Chapter 8.

37

2.7. FREE SPECTRAL RANGE

For a given set of incidence and diffraction angles, the grating equation is
satisfied for a different wavelength for each integral diffraction order m. Thus
light of several wavelengths (each in a different order) will be diffracted along the
same direction: light of wavelength λ in order m is diffracted along the same
direction as light of wavelength λ/2 in order 2m, etc.
The range of wavelengths in a given spectral order for which superposition
of light from adjacent orders does not occur is called the free spectral range Fλ .
It can be calculated directly from its definition: in order m, the wavelength of light
that diffracts along the direction of λ in order m+1 is λ + ∆λ, where

m +1
λ + ∆λ = λ, (2-27)
m

from which

λ
Fλ = ∆λ = . (2-28)
m

The concept of free spectral range applies to all gratings capable of operation in
more than one diffraction order, but it is particularly important in the case of
echelles, because they operate in high orders with correspondingly short free
spectral ranges.
Free spectral range and order sorting are intimately related, since grating
systems with greater free spectral ranges may have less need for filters (or cross-
dispersers) that absorb or diffract light from overlapping spectral orders. This is
one reason why first-order applications are widely popular.

2.8. ENERGY DISTRIBUTION (GRATING EFFICIENCY)

The distribution of power of a given wavelength diffracted by a grating into
the various spectral order depends on many parameters, including the power and
polarization of the incident light, the angles of incidence and diffraction, the
(complex) index of refraction of the materials at the surface of the grating, and the
groove spacing. A complete treatment of grating efficiency requires the vector

38

formulation of electromagnetic theory (.e., Maxwell's equations) applied to
i
corrugated surfaces, which has been studied in detail over the past few decades.
While the theory does not yield conclusions easily, certain rules of thumb can be
useful in making approximate predictions.
The simplest and most widely used rule of thumb regarding grating
efficiency (for reflection gratings) is the blaze condition

mλ = 2dsinθΒ, (2-29)

where θΒ (often called the blaze angle of the grating) is the angle between the
face of the groove and the plane of the grating (see Figure 2-9). When the blaze
condition is satisfied, the incident and diffracted rays follow the law of reflection
when viewed from the facet; that is, we have

α − θΒ = β − θΒ. (2-30)

Because of this relationship, it is often said that when a grating is used at the
blaze condition, the facets act as tiny mirrors – this is not strictly true (since the
facet “mirror” is roughly of the same dimensions as the wavelength itself, ray
optics does not provide an adequate physical model), but it is a useful way to
remember the conditions under which a grating can be used to enhance
efficiency.
Eq. (2-29) generally leads to the highest efficiency when the following
condition is also satisfied:

2K = α – β = 0, (2-31)

where 2K was defined above as the angle between the incident and diffracted
beams (see Eq. (2-6)). Eqs. (2-29) and (2-31) collectively define the Littrow blaze
condition. When Eq. (2-31) is not satisfied (i.e., α ≠ β and therefore the grating
is not used in the Littrow configuration), efficiency is generally seen to decrease
as one moves further off Littrow (i.e., as 2K increases).

39

GN
FN

θΒ

α
β

θΒ

Figure 2-9. Blaze condition. The angles of incidence α and diffraction β are shown in
relation to the facet angle θΒ for the blaze condition. GN is the grating normal and FN is
the facet normal. When the facet normal bisects the angle between the incident and
diffracted rays, the blaze condition (Eq. (2-29)) is satisfied.

For a given blaze angle θΒ, the Littrow blaze condition provides the blaze
wavelength λΒ, the wavelength for which the efficiency is maximal when the
grating is used in the Littrow configuration:

2d
λΒ = sinθΒ, in Littrow. (2-32)
m

Many grating catalogs specify the first-order Littrow blaze wavelength for each
grating:

λΒ = 2d sinθΒ, in Littrow (m = 1). (2-33)

Unless a diffraction order is specified, quoted values of λΒ are generally assumed
to be for the first diffraction order, in Littrow.

40

Recently, computer codes have become commercially available that
accurately predict grating efficiency for a wide variety of groove profiles over
wide spectral ranges.
The topic of grating efficiency is addressed more fully in Chapter 9.

2.9. SCATTERED AND STRAY LIGHT

All light that reaches the detector of a grating-based instrument from
anywhere other than the grating, by any means other than d iffraction as
governed by Eq. (2-1), for any order other than the primary diffraction order of
use, is called instrumental stray light (or more commonly, simply stray light).
All components in an optical system contribute stray light, as will any baffles,
apertures, and partially reflecting surfaces. Unwanted light originating from an
illuminated grating itself is often called scattered light or grating scatter.
Instrumental stray light can introduce inaccuracies in the output of an
absorption spectrometer used for chemical analysis. These instruments usually
employ a “white light” (broad spectrum) light source and a monochromator to
isolate a narrow spectral range from the white light spectrum; however, some of
the light at other wavelengths will generally reach the detector, which will tend to
make an absorbance reading too low (i.e., the sample will seem to be slightly
more transmissive than it would in the absence of stray light). In most
commercial benchtop spectrometers, such errors are on the order of 0.1 to 1
percent (and can be much lower with proper instrument design) but in certain
circumstances (e.g., in Raman spectroscopy), instrumental stray light can lead to
significant errors. Grating scatter and instrumental stray light are addressed in
more detail in Chapter 10.

2.10. SIGNAL-TO-NOISE RATIO (SNR)

The signal-to-noise ratio (SNR) is the ratio of diffracted energy to unwanted
light energy. While we might be tempted to think that increasing diffraction
efficiency will increase SNR, stray light usually plays the limiting role in the
achievable SNR for a grating system.
Replicated gratings from ruled master gratings generally have quite high
SNRs, though holographic gratings sometimes have even higher SNRs, since

41

they have no ghosts due to periodic errors in groove location and lower
interorder stray light.
As SNR is a property of the optical instrument, not of the grating only, there
exist no clear rules of thumb regarding what type of grating will provide higher
SNR.

42

3. RULED GRATINGS

3.0. INTRODUCTION

The first diffraction gratings made for commercial use were mechanically
ruled, manufactured by burnishing grooves individually with a diamond tool
against a thin coating of evaporated metal applied to a plane or concave surface.
Such ruled gratings comprise the majority of diffraction gratings used in
spectroscopic instrumentation.

3.1. RULING ENGINES

The most vital component in the production of ruled diffraction gratings is
the apparatus, called a ruling engine, on which master gratings are ruled. At
present, Newport has three ruling engines in full-time operation, each producing
a substantial number of high-quality master gratings every year. Each of these
engines produces gratings with very low Rowland ghosts, high resolving power,
and high efficiency uniformity.
Selected diamonds, whose crystal axis is oriented for optimum behavior, are
used to shape the grating grooves. The ruling diamonds are carefully shaped by
skilled diamond toolmakers to produce the exact groove profile required for each
grating. The carriage that carries the diamond back and forth during ruling must
maintain its position to better than a few nanometers for ruling periods that may
last for one day or as long as six weeks.
The mechanisms for advancing the grating carriages on all Newport engines
are designed to make it possible to rule gratings with a wide choice of groove
spacings. Newport’s Diffraction Grating Catalog shows the range of groove
spacings available.

44

3.1.1. The Michelson engine

In 1947 Bausch & Lomb acquired its first ruling engine from the University
of Chicago; this engine was originally designed by Michelson in the 1910s and
rebuilt by Gale. It underwent further refinement, which greatly improved its
performance, and has produced a continuous supply of high-quality gratings of
up to 200 x 250 mm ruled area.
The Michelson engine originally used an interferometer system to plot the
error curve of the lead screw, from which an appropriate mechanical correction
cam was derived. In 1990, this system was superseded by the addition of a
digital computer servo control system based on a laser interferometer. The
Michelson engine is unusual in that it covers the widest range of groove
spacings of any ruling engine: it can rule gratings as coarse as 20 grooves per
millimeter (g/mm) and as fine as 10,800 g/mm.

3.1.2. The Mann engine

The second ruling engine installed at Newport has been producing gratings
since 1953, was originally built by the David W. Mann Co. of Lincoln, Mas-
sachusetts. Bausch & Lomb equipped it with an interferometric control system
following the technique of Harrison of MIT.9 The Mann engine can rule areas up
to 110 x 110 mm, with virtually no detectable ghosts and nearly theoretical
resolving power.
While the lead screws of the ruling engines are lapped to the highest
precision attainable, there are always residual errors in both threads and bearings
that must be compensated to produce the highest quality gratings. The Mann
engine is equipped with an automatic interferometer servo system that
continually adjusts the grating carriage to the correct position as each groove is
ruled. In effect, the servo system simulates a perfect screw.

9 G. R. Harrison and J. E. Archer, “Interferometric calibration of precision screws and
control of ruling engines,” J. Opt. Soc. Am. 41, 495 (1951); G. R. Harrison and G. W.
Stroke, “Interferometric control of grating ruling with continuous carriage advance,” J. Soc.
Opt. Am. 45, 112 (1955); G. R. Harrison, N. Sturgis, S. C. Baker and G. W. Stroke, “Ruling
of large diffraction grating with interferometric control”, J. Opt. Soc. Am. 47, 15 (1957) .

45

3.1.3. The MIT 'B' engine

The third ruling engine at Newport was built by Harrison and moved to
Rochester in 1968. It has the capacity to rule plane gratings to the greatest
precision ever achieved; these gratings may be up to 420 mm wide, with grooves
(between 20 and 1500 per millimeter) up to 320 mm long. It uses a double interfer-
ometer control system, based on a frequency-stabilized laser, to monitor not only
table position but to correct residual yaw errors as well. This engine produces
gratings with nearly theoretical resolving powers, virtually eliminating Rowland
ghosts and minimizing stray light. It has also ruled almost perfect echelle grat-
ings, the most demanding application of a ruling engine.

Figure 3-1. Newport MIT ‘B’ Engine. This ruling engine, now in operation at Newport,
is shown with its cover removed.

46

3.2. THE RULING PROCESS

Master gratings are ruled on carefully selected well-annealed substrates of
several different materials. The choice is generally between BK-7 optical glass,
special grades of fused silica, or a special grade of ZeroDur®. The optical sur-
faces of these substrates are polished to closer than λ/10 for green light (about
50 nm), then coated with a reflective film (usually aluminum or gold).
Compensating for changes in temperature and atmospheric pressure is
especially important in the environment around a ruling engine. Room tem-
perature must be held constant to within 0.01 °C for small ruling engines (and to
within 0.005 °C for larger engines). Since the interferometric control of the ruling
process uses monochromatic light, whose wavelength is sensitive to the
changes of the refractive index of air with pressure fluctuations, atmo spheric
pressure must be compensated for by the system. A change in pressure of
2.5 mm of mercury results in a corresponding change in wavelength of one part
per million.10 This change is negligible if the optical path of the interferometer is
near zero, but becomes significant as the optical path increases during the ruling.
If this effect is not compensated, the carriage control system of the ruling engine
will react to this change in wavelength, causing a variation in groove spacing.
The ruling engine must also be isolated from those vibrations that are easily
transmitted to the diamond. This may be done by suspending the engine mount
from springs that isolate vibrations between frequencies from 2 or 3 Hz (which
are of no concern) to about 60 Hz, above which vibration amplitudes are usually
too small to have a noticeable effect.11
The actual ruling of a master grating is a long, slow and painstaking process.
The set-up of the engine, prior to the start of the ruling, requires great skill and
patience. This critical alignment is impossible without the use of a high-power
interference microscope, or an electron microscope for more finely spaced
grooves.
After each microscopic examination, the diamond is readjusted until the
operator is satisfied that the groove shape is appropriate for the particular

10H. W. Babcock, “Control of a ruling engine by a modulated interferometer,” Appl. Opt. 1,
415-420 (1962).
11G. R. Harrison, “Production of diffraction gratings. I. Development of the ruling art,” J.
Opt. Soc. Am. 39, 413-426 (1949).

47

grating being ruled. This painstaking adjustment, although time consuming,
results in very "bright" gratings with nearly all the diffracted light energy
concentrated in a specific angular range of the spectrum. This ability to con-
centrate the light selectively at a certain part of the spectrum is what distin-
guishes blazed diffraction gratings from all others.
Finished master gratings are carefully tested to be certain that they have met
specifications completely. The wide variety of tests run to evaluate all the impor-
tant properties include spectral resolution, efficiency, Rowland ghost intensity,
and surface accuracy. Wavefront interferometry is used when appropriate. If a
grating meets all specifications, it is then used as a master for the production of
our replica gratings.

3.3. VARIED LINE-SPACE (VLS) GRATINGS

For over a century, great effort has been expended in keeping the spacing
between successive grooves uniform as a master grating is ruled. In an 1893
paper, Cornu realized that variations in the groove spacing modified the cur-
vature of the diffracted wavefronts.12 While periodic and random variations were
understood to produce stray light, a uniform variation in groove spacing across
the grating surface was recognized by Cornu to change the location of the focus
of the spectrum, which need not be considered a defect if properly taken into ac-
count. He determined that a planar classical grating, which by itself would have
no focusing properties if used in collimated incident light, would focus the
diffracted light if ruled with a systematic 'error' in its groove spacing. He was
able to verify this by ruling three gratings whose groove positions were specified
to vary as each groove was ruled. Such gratings, in which the pattern of straight
parallel grooves has a variable yet well-defined (though not periodic) spacing
between successive grooves, are now called varied line-space (VLS) gratings.
The Michelson engine, which has digital computer control, can readily rule
VLS gratings. Any groove spacing d(y) that varies reasonably as a function of
position y along the grating surface (and no more than about ±100 nm from the
nominal groove spacing) can be programmed into the computer. The relationship
between groove spacing (and curvature) and imaging is discussed in Chapter 7.

12 M. A. Cornu, “ Vérifications numériques relatives aux propriétés focales des réseaux
diffringents plans,” Comptes Rendus Acad. Sci. 117, 1032-1039 (1893).

48

4. HOLOGRAPHIC GRATINGS

4.0. INTRODUCTION

Since the late 1960s, a method distinct from mechanical ruling has also been
used to manufacture diffraction gratings. This method involves the
photographic recording of a stationary interference fringe field. Such
interference gratings, more commonly (though inaccurately) known as holo-
graphic gratings, have several characteristics that distinguish them from ruled
gratings.
In Aimé Cotton produced experimental holographic gratings,13 fifty years
before the concepts of holography were developed by Gabor. A few decades
later, Michelson considered the interferometric generation of diffraction gratings
obvious, but recognized that an intense monochromatic light source and a
photosensitive material of sufficiently fine granularity did not then exist.14 In the
mid 1960s, ion lasers and photoresists (grainless photosensitive materials)
became available; the former provided a strong monochromatic line, and the latter
was photoactive at the molecular level, rather than at the crystalline level (unlike,
for example, photographic film).
In the late 1960s, researchers independently produced the first holographic
diffraction gratings of spectroscopic quality. 15

13 A. Cotton, “Resaux obtenus par la photographie des ordes stationaires,” Seances Soc.
Fran. Phys. 70-73 (1901).
14A. A. Michelson, Studies in Optics (U. Chicago, 1927; reprinted by Dover Publications,
1995).
15 D. Rudolph and G. Schmahl, “Verfahren zur Herstellung von Röntgenlinsen und
Beugungsgittern,” Umschau Wiss. Tech. 78, 225 (1967); G. Schmahl, “Holographically made
diffraction gratings for the visible, UV and soft x-ray region,” J. Spectrosc. Soc. Japan 23, 3-
11 (1974); A. Labeyrie and J. Flamand, “Spectroscopic performance of holographically
made diffraction gratings,” Opt. Commun. 1, 5 (1969).

50

4.1. PRINCIPLE OF M ANUFACTURE

4.1.1. Formation of an interference pattern

When two sets of coherent equally polarized monochromatic optical plane
waves of equal intensity intersect each other, a standing wave pattern will be
formed in the region of intersection if both sets of waves are of the same
wavelength λ (see Figure 4-1).16 The combined intensity dis tribution forms a set
of straight equally-spaced fringes (bright and dark lines). Thus a photographic
plate would record a fringe pattern, since the regions of zero field intensity would
leave the film unexposed while the regions of maximum intensity would leave the
film maximally exposed. Regions between these extremes, for which the com-
bined intensity is neither maximal nor zero, would leave the film partially exposed.
The combined intensity varies sinusoidally with position as the interference
pattern is scanned along a line. If the beams are not of equal intensity, the
minimum intensity will no longer be zero, thereby decreasing the contrast be-
tween the fringes. As a consequence, all portions of the photographic plate will
be exposed to some degree.
The centers of adjacent fringes (that is, adjacent lines of maximum intensity)
are separated by a distance d, where

λ
d= (4-1)
2 sin θ

and θ is the half the angle between the beams. A small angle between the beams
will produce a widely spaced fringe pattern (large d), whereas a larger angle will
produce a fine fringe pattern. The lower limit for d is λ/2, so for visible recording
light, thousands of fringes per millimeter may be formed.

16 Most descriptions of holographic grating recording stipulate coherent beams, but such
gratings may also be made using incoherent light; see M. C. Hutley, “Improvements in or
relating to the formation of photographic records,” UK Patent no. 1384281 (1975).

51

4.1.2. Formation of the grooves

Master holographic diffraction gratings are recorded in photoresist, a mate-
rial whose intermolecular bonds are either strengthened or weakened by ex-
posure to light. Commercially available photoresists are more sensitive to some
wavelengths than others; the recording laser line must be matched to the type of
photoresist used. The proper combination of an intense laser line and a pho-
toresist that is highly sensitive to this wavelength will reduce exposure time.
Photoresist gratings are chemically developed after exposure to reveal the
fringe pattern. A photoresist may be positive or negative, though the latter is
rarely used. During chemical development, the portions of a substrate covered in
positive photoresist that have been exposed to light are dis solved, while for
negative photoresist the unexposed portions are dissolved. Upon immersion in
the chemical developer, a surface relief pattern is formed: for positive photoresist,
valleys are formed where the bright fringes were, and ridges where the dark
fringes were. At this stage a master holographic grating has been produced;

2θ

substrate

Figure 4-1. Formation of interference fringes. Two collimated beams of wavelength λ
form an interference pattern composed of straight equally spaced planes of intensity
maxima (shown as the horizontal lines). A sinusoidally varying interference pattern is
found at the surface of a substrate placed perpendicular to these planes.

52

its grooves are sinusoidal ridges. This grating may be coated and replicated like
master ruled gratings.
Lindau has developed simple theoretical models for the groove profile
generated by making master gratings holographically, and shown that even the
application of a thin metallic coating to the holographically-produced groove
profile can alter that profile.17

4.2. CLASSIFICATION OF HOLOGRAPHIC GRATINGS

4.2.1. Single-beam interference

An interference pattern can be generated from a single collimated
monochromatic coherent light beam if it is made to reflect back upon itself. A
standing wave pattern will be formed, with intensity maxima forming planes
parallel to the wavefronts. The intersection of this interference pattern with a
photoresist-covered substrate will yield on its surface a pattern of grooves,
whose spacing d depends on the angle θ between the substrate surface and the
planes of maximum intensity (see Figure 4-2)18 ; the relation between d and θ is
identical to Eq. (4-1), though it must be emphasized that the recording geometry
behind the single-beam holographic grating (or Sheridon grating) is different
from that of the double-beam geometry for which Eq. (4-1) was derived.
The groove depth h for a Sheridon grating is dictated by the separation
between successive planes of maximum intensity (nodal planes); explicitly,

λ0
h= , (4-2)
2n

where λ0 is the wavelength of the recording light and n the refractive index of the
photoresist. This severely limits the range of available blaze wavelengths,
typically to those between 200 and 250 nm.

17S. Lindau, “The groove profile formation of holographic gratings,” Opt. Acta 29, 1371-
1381 (1982).
18 N. K. Sheridon, “Production of blazed holograms,” Appl. Phys. Lett. 12, 316-318 (1968).

53

θ

mirror substrate

Figure 4-2. Sheridon recording method. A collimated beam of light, incident from the
right, is retroreflected by a plane mirror, which forms a standing wave pattern whose
intensity maxima are shown. A transparent substrate, inclined at an angle θ to the fringes,
will have its surfaces exposed to a sinusoidally varying intensity pattern.

4.2.2. Double-beam interference

The double-beam interference pattern shown in Figure 4 is a series of
-1
straight parallel fringe planes, whose intensity maxima (or minima) are equally
spaced throughout the region of interference. Placing a substrate covered in
photoresist in this region will form a groove pattern defined by the intersection
of the surface of the substrate with the fringe planes. If the substrate is planar,
the grooves will be straight, parallel and equally spaced, though their spacing
will depend on the angle between the substrate surface and the fringe planes. If
the substrate is concave, the grooves will be curved and unequally spaced,
forming a series of circles of different radii and spacings. Regardless of the
shape of the substrate, the intensity maxima are equally spaced planes, so the
grating recorded will be a classical equivalent holographic grating (more often
called simply a classical grating). This name recognizes that the groove pattern
(on a planar surface) is identical to that of a planar classical ruled grating. Thus
all holographic gratings formed by the intersection of two sets of plane waves
are called classical equivalents, even if their substrates are not planar (and
therefore groove patterns are not straight equally spaced parallel lines).
If two sets of spherical wavefronts are used instead, as in Figure 4-3, a first
generation holographic grating is recorded. The surfaces of maximum intensity

54

are now confocal hyperboloids (if both sets of wavefronts are converging, or if
both are diverging) or ellipsoids (if one set is converging and the other
diverging). This interference pattern can be obtained by focusing the recording
laser light through pinholes (to simulate point sources). Even on a planar sub-
strate, the fringe pattern will be a collection of unequally spaced curves. Such a
groove pattern will alter the curvature of the diffracted wavefronts, regardless of
the substrate shape, thereby providing focusing. Modification of the curvature
and spacing of the grooves can be used to reduce aberrations in the spectral
images; as there are three degrees of freedom in such a recording geometry, three
aberrations can be reduced (see Chapter 6).

Β

substrate

Α

Figure 4-3. First-generation recording method. Laser light focused through pinholes at
A and B forms two sets of spherical wavefronts, which diverge toward the grating
substrate. The standing wave region is shaded; the intensity maxima are confocal
hyperboloids.

The addition of auxiliary concave mirrors or lenses into the recording beams
can render the recording wavefronts toroidal (that is, their curvature in two
perpendicular directions will generally differ). The grating thus recorded is a

55

second generation holographic grating.19 The additional degrees of freedom in
the recording geometry (e.g., the location, orientation and radii of the auxiliary
mirrors) provide for the reduction of additional aberrations above the three
provided by first generation holographic gratings.20
The use of aspheric recording wavefronts can be further accomplished by
using aberration-reduced gratings in the recording system; the first set of
gratings is designed and recorded to produce the appropriate recording
wavefronts to make the second grating.21 Another technique is to illuminate the
substrate with light from one real source, and reflect the light that passes
through the substrate by a mirror behind it, so that it interferes with itself to
create a stationary fringe pattern.22 Depending on the angles involved, the
curvature of the mirror and the curvature of the front and back faces of the
substrate, a number of additional degrees of freedom may be used to reduce
high-order aberrations. [Even more degrees of freedom are available if a lens is
placed in the recording system thus described.23 ]

4.3. THE RECORDING PROCESS

Holographic gratings are recorded by placing a light-sensitive surface in an
interferometer. The generation of a holographic grating of spectroscopic quality
requires a stable optical bench and laser as well as high-quality optical compo-
nents (mirrors, collimating optics, etc.). Ambient light must be eliminated so that
fringe contrast is maximal. Thermal gradients and air currents, which change the

19 C. Palmer, “Theory of second-generation holographic gratings,” J. Opt. Soc. Am. A6,
1175-1188 (1989); T. Namioka and M. Koike, “Aspheric wavefront recording optics for
holographic gratings,” Appl. Opt. 34, 2180-2186 (1995).
20M. Duban, “Holographic aspheric gratings printed with aberrant waves,” Appl. Opt. 26,
4263-4273 (1987).
21 E. A. Sokolova, “Concave diffraction gratings recorded in counterpropagating beams,” J.
Opt. Technol. 66, 1084-1088 (1999); E. A. Sokolova, “New-generation diffraction
gratings,” J. Opt. Technol. 68, 584-589 (2001).
22 E. A. Sokolova, “Geometric theory of two steps recorded holographic diffraction
gratings,” Proc. SPIE 3540, 113-324 (1998); E. Sokolova, B. Kruizinga, T. Valkenburg and
J. Schaarsberg, “Recording of concave diffraction gratings in counterpropagating beams using
meniscus blanks,” J. Mod. Opt. 49, 1907-1917 (2002).
23 E. Sokolova, B. Kruizinga and I. Golubenko, “Recording of concave diffraction gratings in
a two-step process using spatially incoherent light,” Opt. Eng. 43, 2613-2622 (2004).

56

local index of refraction in the beams of the interferometer, must be avoided.
Newport records master holographic gratings in a clean room specially-designed
to meet these requirements.
During the recording process, the components of the optical system must be
of nearly diffraction-limited quality, and mirrors, pinholes and spatial filters must
be adjusted as carefully as possible. Any object in the optical system receiving
laser illumination may scatter this light toward the grating, which will contribute
to stray light. Proper masking and baffling during recording are essential to the
successful generation of a holographic grating, as is single-mode operation of
the laser throughout the duration of the exposure.
The substrate on which the master holographic grating is to be produced
must be coated with a highly uniform, virtually defect-free coating of photoresist.
Compared with photographic film, photoresists are somewhat insensitive to light
during exposure, due to the molecular nature of their interaction with light. As a
result, typical exposures may take from minutes to hours, during which time an
extremely stable fringe pattern (and, therefore, optical system) is required. After
exposure, the substrate is immersed in a developing agent, which forms a surface
relief fringe pattern; coating the substrate with metal then produces a master
holographic diffraction grating.

4.4. DIFFERENCES BETWEEN RULED AND HOLOGRAPHIC
GRATINGS

Due to the distinctions between the fabrication processes for ruled and
holographic gratings, each type of grating has advantages and disadvantages
relative to the other, some of which are described below.

4.4.1. Differences in grating efficiency

The efficiency curves of ruled and holographic gratings generally differ
considerably, though this is a direct result of the differences in groove profiles
and not strictly due to method of making the master grating. For example,
holographic gratings made using the Sheridon method described in Section 4.2.1
above have nearly triangular groove profiles, and therefore have efficiency

57

curves that look more like those of ruled gratings than those of sinusoidal-
groove holographic gratings.
There exist no clear rules of thumb for describing the differences in
efficiency curves between ruled and holographic gratings; the best way to gain
insight into these differences is to look at representative curves of each grating
type. Chapter 9 in this Handbook contains a number of curves; the paper24 on
which this chapter is based contains even more curves, and the book Diffraction
Gratings and Applications25 by Loewen and Popov has an extensive collection
of efficiency curves and commentary regarding the efficiency behavior of plane
reflection gratings, transmission gratings, echelle gratings and concave gratings.

4.4.2. Differences in scattered light

Since holographic gratings do not involve burnishing grooves into a thin
layer of metal, the surface irregularities on its grooves differ from those of
mechanically ruled gratings. Moreover, errors of ruling, which a a mani-re
festation of the fact that ruled gratings have one groove formed after another, are
nonexistent in interferometric gratings, for which all grooves are formed
simultaneously. Holographic gratings, if properly made, can be entirely free of
both small periodic and random groove placement errors found on even the best
mechanically ruled gratings. Holographic gratings may offer advantages to
spectroscopic systems in which light scattered from the grating surface is per-
formance-limiting, such as in the study of the Raman spectra of solid samples,
though proper instrumental design is essential to ensure that the performance of
the optical system is not limited by other sources of stray light.

4.4.3. Differences and limitations in the groove profile

The groove profile has a significant effect on the light intensity diffracted
from the grating (see Chapter 9). While ruled gratings may have triangular or
trapezoidal groove profiles, holographic gratings usually have sinusoidal (or

24 E. G. Loewen, M. Nevière and D. Maystre, "Grating efficiency theory as it applies to
blazed and holographic gratings," Appl. Opt. 16, 2711-2721 (1977).
25 E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Marcel Dekker, Inc.
(1997).

58

nearly sinusoidal) groove profiles (see Figure 4 -4). A ruled grating and a
holographic grating, identical in every way except in groove profile, will have
demonstrably different efficiencies (diffraction intensities) for a given
wavelength and spectral order. Moreover, ruled gratings are more easily blazed
(by choosing the proper shape of the burnishing diamond) than are holographic
gratings, which are usually blazed by ion bombardment (ion etching).
Differences in the intensity diffracted into the order in which the grating is to be
used implies differences in the intensities in all other orders as well; excessive
energy in other orders usually makes the suppression of stray light more difficult.

(a)

(b)

.
Figure 4 -4. Ideal groove profiles for ruled and holographic gratings. (a) Triangular
grooves, representing the profile of a mechanically ruled grating. (b) Sinusoidal grooves,
representing the profile of a holographic grating.

The distribution of groove profile characteristics across the surface of a
grating may also differ between ruled and holographic gratings. For a ruled
concave grating, the facet angles are not aligned identically and the effective
blaze wavelength varies from one side of the grating to the other. A holographic
grating, on the other hand, usually demonstrates much less variation in effi-
ciency characteristics across its surface. Gratings have been ruled by changing
the facet angle at different places on the substrate during ruling. These so-called
"multipartite" gratings, in which the ruling is interrupted and the diamond
reoriented at different places across the width of the grating, demonstrate
enhanced efficiency but do not provide the resolving power expected from an

59

uninterrupted ruling (since each section of grooves may be out of phase with the
others).26

4.4.4. Limitations in obtainable groove frequencies

Limits on the number of grooves per millimeter differ between ruled and
holographic gratings: ruled gratings offer a much wider range of groove
spacings. Below a few hundred grooves per millimeter, the recording optical
system necessary to generate holographic gratings becomes cumbersome, while
ruled gratings can have as few as thirty grooves per millimeter. As an upper limit,
holographic gratings can be recorded with several thousand grooves per
millimeter, producing a groove density almost as high as those ruled gratings
with over 10,000 grooves per millimeter.

4.4.5. Differences in the groove patterns

Classical ruled plane gratings, which constitute the vast majority of ruled
gratings, have straight equally-spaced grooves. Classical ruled concave gratings
have unequally spaced grooves that form circular arcs on the grating surface, but
this groove pattern, when projected onto the plane tangent to the grating at its
center, is still a set of straight equally spaced lines. [It is the projected groove
pattern that governs imaging.27 ] Even ruled varied line-space (VLS) gratings (see
Chapter 3) do not contain curved grooves, except on curved substrates. The
aberration reduction possible with ruled gratings is therefore limited to that
possible with straight grooves, though this limitation is due to the mechanical
motions possible with present-day ruling engines rather than with the burnishing
process itself.
Holographic gratings, on the other hand, need not have straight grooves.
Groove curvature can be modified to reduce aberrations in the spectrum, thereby
improving the throughput and spectral resolution of imaging spectrometers. A
common spectrometer mount is the flat-field spectrograph, in which the

26M. C. Hutley and W. R. Hunter, "Variation of blaze of concave diffraction gratings," Appl.
Opt. 20, 245-250 (1981).
27 C. Palmer and W. R. McKinney, "Imaging theory of plane-symmetric varied line-space
grating systems," Opt. Eng. 33, 820-829 (1994).

60

spectrum is imaged onto a flat detector array and several wavelengths are
monitored simultaneously. Holographic gratings can significantly improve the
imaging of such a grating system, whereas classical ruled gratings are not
suitable for forming well-focused planar spectra without auxiliary optics.

4.4.6. Differences in the substrate shapes

The interference pattern used to record holographic gratings is not
dependent on the substrate shape or dimension, so gratings can be recorded
interferometrically on substrates of low ƒ/number more easily than they can be
mechanically ruled on these substrates. Consequently, holographic concave
gratings lend themselves more naturally to systems with short focal lengths.
Holographic gratings of unusual curvature can be recorded easily; of course,
there may still remain technical problems associated with the replication of such
gratings.
The substrate shape affects both the grating efficiency characteristics its
imaging performance.

• Grating efficiency depends on the groove profile as well as the angle at
which the light is incident and diffracted; for a concave grating, both
the groove profile and the local angles vary with position on the grating
surface. This leads to the efficiency curve being the sum of the various
efficiency curves for small regions of the grating, each with its own
groove profile and incidence and diffraction angles.
• Grating imaging depends on the directions of the diffracted rays over
the surface of the grating, which in turn are governed by the local
groove spacing and curvature (i.e., the groove pattern) as well as the
local incidence angle. For a conventional plane grating used in
collimated light, the groove pattern is the same everywhere on the
grating surface, as is the incidence angle, so all diffracted ray are
parallel. For a grating on a concave substrate, though, the groove
pattern is generally position-dependent, as is the local incidence angle,
so the diffracted rays are not parallel – thus the grating has focal
(imaging) properties as well as dispersive properties.

61

4.4.7. Differences in the size of the master substrate

While ruled master gratings can generally be as large as 320 x 420 mm,
holographic master gratings are rarely this large, due to the requirement that the
recording apparatus contain very large, high-quality lenses or mirrors, and well
as due to the decrease in optical power farther from the center of the master
grating substrate.

4.4.8. Differences in generation time for master gratings

A ruled master grating is formed by burnishing each groove individually; to
do so, the ruling diamond may travel a very large distance to rule one grating.
For example, a square grating of dimensions 100 x 100 mm with 1000 grooves per
millimeter will require the diamond to move 10 km (over six miles), which may take
several weeks to rule.
In the fabrication of a master holographic grating, on the other hand, the
grooves are created simultaneously. Exposure times vary from a few minutes to
tens of minutes, depending on the intensity of the laser light used and the
spectral response (sensitivity) of the photoresist at this wavelength. Even
counting preparation and development time, holographic master gratings are
produced much more quickly than ruled master gratings. Of course, an extremely
stable and clean optical recording environment is necessary to record precision
holographic gratings. For plane gratings, high-grade collimating optics are
required, which can be a limitation for larger gratings.

62

5. REPLICATED GRATINGS

5.0. INTRODUCTION

Decades of research and development at Newport have contributed to the
process for manufacturing replicated diffraction gratings (replicas) of
spectroscopic quality. This process is capable of producing thousands of
duplicates of master gratings which equal the quality and performance of the
master gratings themselves. The replication process has reduced the price of a
typical diffraction grating by a factor of one hundred or more, compared with the
cost of acquiring a master grating, as well as greatly increasing their commercial
availability.

5.1. THE REPLICATION PROCESS

The process for making replica gratings results in a grating whose grooves
are formed in a very thin layer of resin that adheres strongly to the surface of the
substrate material. The optical surface of a reflection replica is usually coated
with aluminum (Al), but gold (Au) or silver (Ag) is recommended for greater dif-
fracted energy in certain spectral regions. Transmission gratings have no
reflective coating.
Most commercially-available surface-relief gratings are made using a casting
process, which faithfully reproduces the three-dimension nature of the grating
surface. [It is for this reason that photographic replication techniques are not
generally sufficient.28 ]
The casting process for the production of a replicated diffraction grating is a
series of sequential steps:

(1997), p. 577.

64

• Submaster selection. The replication process starts with the selection
of a suitable submaster grating that has the desired specifications
(groove frequency, blaze angle, size, etc.). [A submaster grating is a
grating replicated from a master, or from another submaster, but is itself
used not as a final optical product but as a mold for the replication of
product gratings; for this reason, it is not strictly required that a
submaster grating meet all of the performance specifications of the
product grating (e.g., it need not have a suitably reflective coating).]

• Application of parting agent. A parting agent is applied to the surface
of the submaster grating. The parting agent serves no optical purpose
and should have no deleterious optical effects but aids in the separation
of the delicate submaster and product grating surfaces. Since the
replicated optical surface is intended to match that of the submaster as
closely as possible, the parting agent must be very thin and conformal
to the surface of the submaster.29

• Application of transfer coating. After the parting agent is applied, a
reflective coating (usually aluminum) is applied to the surface of the
submaster. This coating will form the optical surface of the product
grating upon separation. To obtain an optical quality coating, this step
is performed in a vacuum deposition chamber. [Since this coating is
applied to the submaster, but transfers to the product grating upon
separation, it is called a transfer coating.] Typical transfer coating
thicknesses are about one micron.

• Cementing. A substrate is then cemented with a layer of resin to the
grooved surface of the master grating; this layer can vary in thickness,
but it is usually tens of microns thick. It is the resin that holds the
groove profile and replicates it from the submaster to the product; the
transfer coating is much too thin for this purpose. The “sandwich”
formed by the substrate and submaster cemented together is shown in
Figure 5-1.

29 E. G. Loewen, Replication of Mirrors and Diffraction Gratings, SPIE Tutorial T10 (1983).

65

Since the resin is in the liquid state when it is applied to the
submaster, it must harden sufficiently to ensure that it can maintain the
groove profile faithfully when the product grating is separated from the
submaster. This hardening, or curing, is usually accomplished by a
room-temperature cure period (lasting from hours to days) or by heating
the resin to accelerate the curing, though gratings can also be replicated
using a UV-curable resin.30

product substrate product resin layer

transfer coating

parting agent

submaster metallic
coating

submaster substrate submaster resin layer

Figure 5-1. The replication “sandwich”, showing the substrates, the resin layers, the
metallic coatings, and the parting agent.

• Separation. After the resin is fully cured, the groove profile is faithfully
replicated in the resin when the submaster and product are separated.
The parting agent serves as the weak interface and allows the
separation to take place between the submaster coating and the transfer
metallic coating. The groove profile on the product is the inverse of the
groove profile on the submaster; if this profile is not symmetric with

30 S. D. Fantone, “Replciating optical surfaces using UV curing cements: a method,” Appl.
Opt. 22, 764 (1983); R. J. M. Zwiers and G. C. M. Dortant, “Aspheric lenses produced by a
fast high-precision replication process using UV-curable coatings,” Appl. Opt. 24, 4483-4488
(1985).

66

respect to this inversion, the efficiency characteristics of the product
grating will generally differ from those of the submaster grating. In such
cases, an additional replication must be done to invert the inverted
profile, resulting in a profile identical to that of the original submaster.
However, for certain types of gratings, inversion of the groove
increases efficiency significantly.
At this stage, if a transmission grating is desired, the transfer
coating is removed from the product, leaving the groove structure intact
in the transparent resin.

• Inspection. After separation, both the submaster and the product
gratings are inspected for surface or substrate damage. The product
grating may also be tested for key performance characteristics (e.g.,
efficiency, wavefront flatness (or curvature), scattered light, alignment
of the grooves to a substrate edge) depending on requirements.

The product grating formed by this replication process may be used as an
optical component, or it may serve as a mold (replication tool) by being
considered a submaster. In this way, a single master grating can make several
submasters, each of which can make several more submasters, etc., to form a
replication tree (see Figure 5-2).
The replication tree shown in Figure 5-2 illustrates two important features of
replication: extension horizontally (within a generation) and vertically (to
subsequent generations). Replication within a generation is accomplished by the
successive replication of a single grating (much as a parent can have many
children). Replication to additional generations is accomplished by forming a
replica (child), which itself forms a replica (grandchild), etc. Thus replication can
extend both within generations (X-1, X -2, X-3, X …) and to subsequent
-4,
generations (X-1, X -1-3, X -1-3-1, X-1-3-1-4, …) to create a large number of
replicas from a single master grating.
As an example, consider a master grating X from which five first-generation
replicas are made (X-1 through X-5). Each of these is used as a submaster to
form five replicas: X-1 forms X-1-1 through X-1-5, X-2 forms X-2-1 through X-2-5,
and so on. This forms twenty-five second generation replicas. If each of these
replicas is itself replicated five times, we arrive at 125 third-generation products

67

(X-1-1-1, X-1-1-2, …, through X -5-5-5). This example illustrates that a large
number of replicas can be made from a single master grating, assuming a
conservative number of replicas and a reasonable number of generations.

X master grating

X-1 X-2 X-3 first
generation
replicas

second generation replicas
X-1-1 X-1-3

X-1-3-1 third generation replicas

Figure 5-2. A replication tree. Master X is replicated to create several first-generation
replicas (X-1, X-2, …), which themselves are replicated to form second-generation
replicas (X-1-1, …), etc.

The number N of replicas of a particular generation that can be made from a
single master can be estimated using the following formula,

N = Rg , (5-1)

where R is the number of replications per generation and g is the number of
generations. Reasonable values of R are 5 to 10 (though values well above 20 are
not unheard of), and g generally ranges from 3 to 9. Conservatively, then, for R =
5 and g = 3, we have N = 125 third-generation replicas; at the other end of the
ranges we have R = 10 and g = 9 so that N = 1,000,000,000 ninth-generation
replicas. Of course, one billion replicas of a single grating has never been
required, but even if it were, Eq. (5-1) assumes that each replica in every
generation (except the last) is replicated R times, whereas in practice most
gratings cannot be replicated too many times before being damaged or otherwise

68

rendered unusable. That is, some branches of the replication tree are truncated
prematurely. Consequently, Eq. (5-1) must be taken as an upper limit, which
becomes unrealistically high as either R or g increase. In practice, N can be in the
thousands, and can be even higher if care is taken to ensure that the submasters
in the replication tree are not damaged.

5.2. REPLICA GRATINGS VS. M ASTER GRATINGS

There are two fundamental differences between master gratings and replica
gratings: how they are made and what they are made of.

Manufacturing process. Replica gratings are made by the replication
process outlined in Section 5.1 above – they are resin castings of master
gratings. The master gratings themselves, though, are not castings: their
grooves are created either by burnishing (in the case of ruled gratings) or by
optical exposure and chemical development (in the case of holographic
gratings).

Composition. Replica gratings are composed of a metallic coating on a resin
layer, which itself rests on a substrate (usually glass). Master gratings also
usually have glass substrates, but have no resin (the grooves of a ruled
master are contained entirely within a metallic layer on the substrate, and
those of a holographic master are contained entirely within a layer of
photoresist or similar photosensitive material).

The differences in manufacturing processes for master gratings and replica
gratings naturally provide an advantage in both production time and unit cost to
replica gratings, thereby explaining their popularity, but the replication process
itself must be designed and carried out to ensure that the performance
characteristics of the replicated grating match those of the master grating.
Exhaustive experimentation has shown how to eliminate loss of resolution
between master and replica – this is done by ensuring that the surface figure of
the replica matches that of the master, and that the grooves are not displaced as
a result of replication. The efficiency of a replica matches that of its master when
the groove profile is reproduced faithfully. Other characteristics, such as

69

scattered light, are generally matched as well, provided care is taken during the
transfer coating step to ensure a dense metallic layer. [Even if the layer were not
dense enough, so that its surface roughness caused increased scattered light
from the replica when compared with the master, this would be diffuse scatter;
scatter in the dispersion plane, due to irregularities in the groove spacing, would
be faithfully replicated by the resin and does not depend significantly on the
quality of the coating.] Circumstances in which a master grating is shown to be
superior to a replicated grating are quite rare, and can often be attributed to flaws
or errors in the particular replication process used, not to the fact that the grating
was replicated.
In one respect, replicated gratings can provide an advantage over master
gratings: those cases where the ideal groove profile is not obtainable in a master
grating, but the inverse profile is obtainable. Echelle gratings, for example, are
ruled so that their grooves exhibit a sharp trough but a relatively less sharp peak.
By replicating, the groove profile is inverted, leaving a first-generation replica
with a sharp peak. The efficiency of the replica will be considerably higher than
the efficiency of the master grating. In such cases, only odd-generation replicas
are used as products, since the even-generation replicas have the same groove
profile (and therefore the same efficiency characteristics) as the master itself.†
The most prominent hazard to a grating during the replication process, either
master or replica, is scratching, since the grating surface consists of a thin metal
coating on a resin layer. Scratches involve damage to the groove profile, which
generally leads to increased stray light, though in some applications this may be
tolerable. Scratches faithfully replicate from master to submaster to product, and
cannot be repaired, since the grating surface is not a polished surface, and an
overcoating will not repair the damaged grooves.
Another hazard during replication is surface contamination from fin-
gerprints; should this happen, a grating can sometimes (but not always) be
cleaned or recoated to restore it to its original condition. [In use, accidentally
evaporated contaminants, typical of vacuum spectrometry pumping systems, can
be especially harmful when baked on the surface of the grating with ultraviolet
radiation.]

† By convention, a master grating is designated as the zeroth (0) generation, so the second-,
fourth- and subsequent even-generation submasters from the master will have the same
groove profile, and the first -, third- and subsequent odd-generation submasters will have
inverted groove profiles.

70

5.3. STABILITY OF REPLICATED GRATINGS

Temperature. There is no evidence of deterioration or change in standard
replica gratings with age or when exposed to thermal variations from the boiling
point of nitrogen (77 K = –196 °C) to 50 °C. Gratings that must withstand higher
temperatures can be made with a special resin whose glass transition temperature
is high enough to prevent the resin from flowing at high temperatures (thereby
distorting the grooves). In addition to choosing the appropriate resin, the cure
cycle can be modified to result in a grating whose grooves will not distort under
high temperature.
Gratings replicated onto substrates made of low thermal expansion materials
behave as the substrate dictates: the resin and aluminum, which have m uch
higher thermal expansion coefficients, are present in very thin layers compared
with the substrate thickness and therefore do not expand and contract
appreciably with temperature changes since they are fixed rigidly to the
substrate.
Relative Humidity. Standard replicas generally do not show signs of
degradation in normal use in high relative humidity environments, but some
applications (e.g., fiber-optic telecommunications) require extended exposure to
very high humidity environments. Coatings and epoxies that resist the effects of
water vapor are necessary for these applications.
Instead of a special resin, the metallic coating on a reflection grating made
with standard resin is often sufficient to protect the underlying resin from the
effects of water vapor. A transmission grating that requires protection from
environmental water vapor can be so protected by applying a dielectric coating
(e.g., SiO) to its grooved surface.31
Temperature and Relative Humidity. Recent developments in fiber optic
telecommunications require diffraction gratings that meet harsh environmental
standards, particularly those in the Telcordia (formerly Bellcore) document GR-
1221, “Generic Reliability Assurance Requirements for Passive Optical
Components”. Special resin materials, along with specially-designed proprietary
replication techniques, have been developed to produce replicated gratings that
can meet this demanding requirement with no degradation in performance.

(1997), p. 582.

71

High Vacuum. Even the highest vacuum, such as that of outer space, has
no effect on replica gratings. Concerns regarding outgassing from the resin are
addressed by recognizing that the resin is fully cured. However, some
outgassing may occur in high vacuum, which may be a problem for gratings used
in synchrotron beamlines; in certain cases ruled master gratings are used instead.
Energy Density of the Beam. For applications in which the energy density at
the surface of the grating is very high (as in some pulsed laser applications),
enough of the energy incident on the grating surface may be absorbed to cause
damage to the surface. In these cases, it may be necessary to make the transfer
coat thicker than normal, or to apply a second metallic layer (an overcoat) to
increase the opacity of the metal film(s) sufficiently to protect the underlying
resin from exposure to the light and to permit the thermal energy absorbed from
the pulse to be dissipated without damaging the groove profile. Using a metal
rather than glass substrate is also helpful in that it permits the thermal energy to
be dissipated; in some cases, a water-cooled metal substrate is used for
additional benefit.32
Pulsed lasers often require optical components with high damage
thresholds, due to the short pulse duration and high energy of the pulsed beam.
For gratings used in the infrared, gold is generally used as the reflective coating
(since it is more reflective than aluminum in the near IR).
A continuous-wave laser operating at λ = 10.6 µm was reported by Huguley
and Loomis 33 to generate damage to the surface of replicated grating at about
150 kW/cm2 or above.
Gill and Newnam34 undertook a detailed experimental study of laser-induced
damage of a set of master gratings and a set of replicated gratings using 30-ps
pulses at λ = 1.06 µm. They reported that the damage threshold for the
holographic gratings they tested was a factor of 1.5 to 5 times higher than for the
ruled gratings they tested. Differences in the damage threshold for S- vs. P-

32 F. M. Anthony, “High heat load optics: an historical overview,” Opt. Eng. 34, 313-320
(1995).
33C. A. Huguley and J. S. Loomis, “Optical material damage from 10.6 µm CW radiation,” in
Damage in Laser Materials, A. J. Glass and A. H. Guenther, eds., Nat. Bur. Stand. (U.S.)
Spec. Publ. 435 (1975).
34 D. H. Gill and B. E. Newnam, “Picosecond-pulse damage studies of diffraction gratings,” in
Damage in Laser Materials, H. E. Bennett, A. H. Guenther, D. Milam and B. E. Newnam,
eds., Nat. Bur. Stand. (U.S.) Spec. Publ. 727 (1986), pp. 154-161.

72

polarized light were also observed: the threshold for S-polarized light was 1.5 to 6
times higher than for P -polarized light, though how this correlates to grating
efficiency in these polarization states is not clear. The (holographic) master
gratings tested exhibited lower damage thresholds than did the replicated
gratings. Some of the experimental results reported by Gill and Newnam are
reproduced in Table 5-1.

Damage Threshold (J/cm2)
P polarization S polarization
Au coating Al coating Au coating Al coating
1800 g/mm
2.6 0.3 1.2 0.1
holographic #1

#2 1.0 0.3 0.8 0.1

600 g/mm ruled 1.1 0.2 0.4 0.1

300 g/mm ruled 0.5 0.3 0.1 0.1

Table 5-1. Damage thresholds reported by Gill and Newnam. For these gratings, the
difference in dame threshold measurements between Au and Al coatings, between P- and
S-polarization, and between the 1800 g/mm holographic gratings and 300 and 600 g/mm
ruled gratings are evident.

Increasing the thickness of the reflective layer can, in certain circumstances,
greatly increase the damage threshold of a replicated grating used in pulsed
beams, presumably by reducing the maximum temperature which the metallic
coating reaches during illumination.35
Experimental damage thresholds for continuous wave (cw) beams, reported
by Loewen and Popov36 , are given in Table 5-2.

35 R. W. C. Hansen, “Replica grating radiation da mage in a normal incidence
monochromator,” Rev. Sci. Instrum. 67 (9), 1-5 (September 1996)
(1997), p. 485.

73

Coating defects can play a critical role in the incidence of laser damage, as
reported by Steiger and Brausse,37 who studied optical components illuminated
by a pulsed Nd:YAG laser operating at λ = 1.06 µm.

Grating type Damage Threshold
(energy density)

Standard replica grating
40 to 80 W/cm2
on glass substrate
c. 100 W/cm2
on copper substrate
150 to 250 W/cm2
on water-cooled copper substrate

Table 5-2. Damage thresholds for continuous wave (cw) beams.

37B. Steiger and H. Brausse, “Interaction of laser radiation with coating defects,” Proc. SPIE
2428, 559-567 (1995).

74

6. PLANE GRATINGS AND THEIR MOUNTS

6.1. GRATING M OUNT TERMINOLOGY

The auxiliary collimating and focusing optics that modify the wavefronts
incident on and diffracted by a grating, as well as the angular configuration in
which it is used, is often called its mount. Grating mounts are a class of
spectrometer, a term that usually refers to any spectroscopic instrument, re-
gardless of whether it scans wavelengths individually or entire spectra
simultaneously, or whether it employs a prism or grating. For this discussion we
consider grating spectrometers only.
A monochromator is a spectrometer that images a single wavelength or
wavelength band at a time onto an exit slit; the spectrum is scanned by the
relative motion of the entrance and/or exit optics (usually slits) with respect to
the grating. A spectrograph is a spectrometer that images a range of wave-
lengths simultaneously, either onto photographic film or a series of detector el-
ements, or through several exit slits (sometimes called a polychromator). The
defining characteristic of a spectrograph is that an entire section of the spectrum
is recorded at once.

6.2. PLANE GRATING M ONOCHROMATOR M OUNTS

A plane grating is one whose surface is flat. Plane gratings are normally
used in collimated incident light, which is dispersed by wavelength but is not
focused. Plane grating mounts generally require auxiliary optics, such as lenses
or mirrors, to collect and focus the energy. Some simplified plane grating mounts
illuminate the grating with converging light, though the focal properties of the
system will then depend on wavelength. For simplicity, only plane reflection
grating mounts are dis cussed below, though each mount may have a
transmission grating analogue.

76

6.2.1. The Czerny-Turner monochromator 38

This design involves a classical plane grating illuminated by collimated light.
The incident light is usually diverging from a source or slit, and collimated by a
concave mirror (the collimator), and the diffracted light is focused by a second
concave mirror (the camera); see Figure 6-1. Ideally, since the grating is planar
and classical, and used in collimated incident light, no aberrations should be
introduced into the diffracted wavefronts. In practice, since spherical mirrors are
often used, aberrations are contributed by their use off-axis.39

collimator
entrance slit

grating

exit slit
camera

Figure 6-1. The Czerny-Turner mount. The plane grating provides dispersion and the
concave mirrors provide focusing.

38 A. Shafer, L. Megil and L. Droppelman, “Optimization of Czerny-Turner spectrometers,”
J. Opt. Soc. Am. 54, 879-888 (1964); J. M. Simon, M. A. Gil and A. N. Fantino, “Czerny-
Turner monochromator: astigmatism in the classical and in the crossed beam dispositions,”
Appl. Opt. 25, 3715-3720 (1986); K. M. Rosfjord, R. A. Villalaz and T. K. Gaylord,
“Constant-bandwidth scanning of the Czerny-Turner monochromator,” Appl. Opt. 39, 568-
572 (2000).
39R. F. James and R. S. Sternberg, The Design of Optical Spectrometers, Chapman and Hall
(London: 1969).

77

Like all monochromator mounts, the wavelengths are imaged individually.
The spectrum is scanned by rotating the grating; this moves the grating normal
relative to the incident and diffracted beams, which (by Eq. (2-1)) changes the
wavelength diffracted toward the second mirror. Since the light incident on and
diffracted by the grating is collimated, the spectrum remains at focus at the exit
slit for each wavelength, since only the grating can introduce wavelength-de-
pendent focusing properties.
Aberrations† caused by the auxiliary mirrors include astigmatism and
spherical aberration (each of which is contributed additively by the mirrors); as
with all concave mirror geometries, astigmatism increases as the angle of
reflection increases. Coma, though generally present, can be eliminated at one
wavelength through proper choice of the angles of reflection at the mirrors; due
to the anamorphic (wavelength-dependent) tangential magnification of the
grating, the images of the other wavelengths experience higher-order coma
(which becomes troublesome only in special systems).

6.2.2. The Ebert-Fastie monochromator 40

This design is a special case of a Czerny-Turner mount in which a single
relatively large concave mirror serves as both the collimator and the camera
(Figure 6-2). Its use is limited, since stray light and aberrations are difficult to
control – the latter effect being a consequence of the relatively few degrees of
freedom in design (compared with a Czerny-Turner monochromator). This can be
seen by recognizing that the Ebert-Fastie monochromator is a special case of the
Czerny-Turner monochromator in which both concave mirror radii are the same,
and for which their centers of curvature coincide. However, an advantage that
the Ebert-Fastie mount provides is the avoidance of relative misalignment of the
two mirrors.

† See Chapter 7 for a discussion of aberrations.
40 W. G. Fastie, “A small plane grating monochromator,” J. Opt. Soc. Am. 42, 641-647
(1952); G. Fastie, “Image forming properties of the Ebert monochromator,” J. Opt. Soc.
Am . 42, 647-652 (1952).

78

entrance slit

grating

exit slit mirror

Figure 6-2. The Ebert-Fastie mount. A single concave mirror replaces the two concave
mirrors found in Czerny-Turner mounts.

6.2.3. The Monk-Gillieson monochromator 41

In this mount (see Figure 6-3), a plane grating is illuminated by converging
light. Usually light diverging from an entrance slit (or fiber) is rendered con-
verging by off-axis reflection from a concave mirror (which introduces
aberrations, so the light incident on the grating is not composed of perfectly
spherical converging wavefronts). The grating diffracts the light, which con-
verges toward the exit slit; the spectrum is scanned by rotating the grating to
bring different wavelengths into focus at or near the exit slit. Often the angles of
reflection (from the primary mirror), incidence and diffraction are small (measured
from the appropriate surface normals), which keeps aberrations (especially off-
axis astigmatism) to a minimum.
Since the incident light is not collimated, the grating introduces wavelength-
dependent aberrations into the diffracted wavefronts (see Chapter 7).

41 G. S. Monk, “A mounting for the plane grating,” J. Opt. Soc. Am . 17, 358 (1928); A.
Gillieson, “A new spectrographic diffraction grating monochromator,” J. Sci. Instr. 26, 334-
339 (1949); T. Kaneko, T. Namioka and M. Seya, “Monk-Gillieson monochromator,” Appl.
Opt. 10, 367-381 (1971); M. Koike and T. Namioka, “Grazing-incidence Monk-Gillieson
monochromator based on surface normal rotation of a varied line-spacing grating,” Appl.
Opt. 41, 245-257 (2002).

79

Consequently the spectrum cannot remain in focus at a fixed exit slit when the
grating is rotated (unless this rotation is about an axis displaced from the central
groove of the grating42 ). For low-resolution applications, the Monk-Gillieson
mount enjoys a certain amount of popularity, since it represents the simplest and
least expensive spectrometric system imaginable.

incident light

grating
mirror

exit slit

Figure 6-3. The Monk-Gillieson mount. A plane grating is used in converging light.

6.2.4. The Littrow monochromator 43

A grating used in the Littrow or autocollimating configuration diffracts light
of wavelength λ back along the incident light direction (Figure 6-4). In a Littrow
monochromator, the spectrum is scanned by rotating the grating; this reorients
the grating normal, so the angles of incidence α and diffraction β change (even
though α = β for all λ). The same auxiliary optics can be used as both collimator

42 D. J. Schroeder, "Optimization of converging-beam grating monochromators," J. Opt.
Soc. Am. 60, 1022 (1970).
43 J. F. James and R. S. Sternberg, The Design of Optical Spectrometers, Chapman and Hall
(London: 1969); R. Masters, C. Hslech and H. L. Pardue, “Advantages of an off-Littrow
mounting of an echelle grating,” Appl. Opt. 27, 3895-3897 (1988).

80

and camera, since the diffracted rays retrace the incident rays. Usually the
entrance slit and exit slit (or image plane) will be offset slightly along the di-
rection parallel to the grooves so that they do not coincide; this will generally
introduce out-of-plane aberrations. True Littrow monochromators are quite
popular in laser tuning applications (see Chapter 13).

grating

entrance slit

exit slit

mirror

Figure 6-4. The Littrow monochromator mount. The entrance and exit slits are slightly
above and below the dispersion plane, respectively; they are shown separated for clarity.

6.2.5. Double & triple monochromators 44

Two monochromator mounts used in series form a double monochromator.
The exit slit of the first monochromator usually serves as the entrance slit for the
second monochromator (see Figure 6-5), though some systems have been
designed without an intermediate slit. Stray light in a double monochromator

44R. L. Christensen and R. J. Potter, “Double monochromator systems,” Appl. Opt. 2, 1049-
1054 (1963); F. R. Lipsett, G. Oblinsky and S. Johnson, “Varioilluminator (subtractive
double monochromator with variable bandpass),” Appl. Opt. 12, 818 (1973).

81

with an intermediate slit is much lower than in a single monochromator: it is
approximately the product of ratios of stray light intensity to parent line intensity
for each single monochromator.
A double monochromator may be designed to have either additive
dispersion or subtractive dispersion.
• In the case of additive dispersion, the reciprocal linear dispersion of the
entire system is the sum of the reciprocal linear dispersions of each
monochromator: that is, the spectrum that is dispersed by the first
monochromator is further dispersion in passing through the second
monochromator.

mirror
exit slit

grating
1 grating
2
intermediate
slit

entrance slit

mirror

Figure 6-5. A double monochromator mount.

82

• In the case of subtractive dispersion, the entire system is designed so that
the spectral dispersion at the exit slit of the second monochromator is
essentially zero; in this case, the dispersion of the double monochromator
system is that of the first monochromator. A subtractive-dispersion
monochromator has the property that the light leaving its exit slit is
spectrally uniform: the homogeneous combination of all wavelengths is
transmitted through the intermediate slit, instead of a spectrum of
continuous varying wavelength as seen in single monochromators and
additive-dispersion double monochromators. Such instruments have found
use in fluorescence and luminescence excitation.45

A triple monochromator mount consists of three monochromators in series.
These mounts are used only when the demands to reduce instrumental stray
light are extraordinarily severe (e.g., Raman spectroscopy).46

6.2.6. The constant-scan monochromator

The vast majority of monochromator mounts are of the constant deviation
variety: the grating is rotated to bring different wavelengths into focus at the
(stationary) exit slit. This mount has the practical advantage of requiring a single
rotation stage and no other moving parts, but it has the disadvantage of being
“on blaze” at only one wavelength – at other wavelengths, the incidence and
diffraction angles do not satisfy the blaze condition

mλ = d (sinα + sinβ) = 2dsinθB, (2-29)

where θB is the facet angle.
An alternative design that may be considered is the constant-scan
monochromator), so called because in the grating equation

45 F. R. Lipsett, G. Bechtold, F. D. Blair, F. V. Cairns and D. H. O’Hara, “Apparatus for
measurement of luminescence spectra wi th a digital recording systems,” Appl. Opt. 9, 1312
(1970).
46 A. Walsh and J. B. Willis, “Multiple monochromators. IV. A triple monochromator and
its application to near infrared, visible and ultraviolet spectroscopy,” J. Opt. Soc. Am. 43,
989-993 (1953).

83

mλ = 2d cosK sinφ (2-8)

it is the scan angle φ rather than the half-deviation angle K that remains fixed. In
this mount, the bisector of the entrance and exit arms must remain at a constant
angle to the grating normal as the wavelengths are scanned; the angle 2K =
α − β between the two arms must expand and contract to change wavelength
(see Figure 2-9).
Constant-scan plane grating monochromators have been designed47 but
have not been widely adopted, probably due to the complexity of the required
mechanisms for the precise movement of the slits. Hunter described a constant-
scan monochromator for the vacuum ultraviolet in which the entrance and exit
slits moved along the Rowland circle (see Section 7.2 below).48 The imaging
properties of the constant-scan monochromator with fixed entrance and exit arms
have not been fully explored, but since each wavelength remains on blaze, there
may be applications where this design proves advantageous. [As noted in
Section 2.8, though, the efficiency will drop as 2K increases, i.e. as the
monochromator is used off-Littrow.]

6.3. PLANE GRATING SPECTROGRAPH M OUNTS49

The plane grating monochromator mounts described in Section 6.2 have an
exit slit through which a narrow spectral region passes; the center wavelength of
this spectra region is changed by rotating the grating. Alternatively, a wide

47 C. Kunz, R. Haensel and B. Sonntag, “Grazing-incidence vacuum-ultraviolet
monochromator with fixed exit slit for use with distant sources,” J. Opt. Soc. Am. 58, 1415
(1968); H. Deitrich and C. Kunz, “A grazing incidence vacuum ultraviolet monochromator
with fixed exit slit,” Rev. Sci. Inst. 43, 434-442 (1972).
48 W. R. Hunter, “On-blaze scanning monochromator for the vacuum ultraviolet,” Appl.
Opt. 21 1634-1642 (1982); W. R. Hunter and J. C. Rife, “Higher-order suppression in an on-
blaze plane-grating monochromator,” Appl. Opt. 23, 293-299 (1984).
49 R. F. Jarrell, “Stigmatic plane grating spectrograph with order sorter,” J. Opt. Soc. Am. 45,
259-269 (1955); J. Reader, “Optimizing Czerny-Turner spectrographs: a comparison
between analytic theory and ray tracing,” J. Opt. Am. Soc. 59, 1189-1196 (1969); M. A.
Gil, J. M. Simon and A. N. Fantino, “Czerny-Turner spectrograph with a wide spectral
range,” Appl. Opt. 27, 4069-4072 (1988); N. C. Das, “Aberration properties of a Czerny-
Turner spectrograph using plane-holographic diffraction grating,” Appl. Opt. 30, 3589-3597
(1991).

84

spectral region can be imaged at once by leaving the grating fixed and using a
series of exits slits (or an array of detector elements) in a focal plane. Such
optical systems are call spectrographs.
Often the imaging properties of a plane grating spectrograph (with no
auxiliary optics) are acceptable over only a portion of the spectrum of interest,
which requires the use of additional lenses or mirrors to provide additional
focusing power to render the focal curve as close to the line (or curve)
represented by the slits or detector array.

85

7. CONCAVE GRATINGS AND THEIR
MOUNTS

7.0. INTRODUCTION

A concave reflection grating can be modeled as a concave mirror that
disperses; it can be thought to reflect and focus light by virtue of its concavity,
and to disperse light by virtue of its groove pattern.
Since their invention by Henry Rowland over one hundred years ago,50 con-
cave diffraction gratings have played an important role in spectrometry.
Compared with plane gratings, they offer one important advantage: they provide
the focusing (imaging) properties to the grating that otherwise must be supplied
by separate optical elements. For spectroscopy below 110 nm, for which the
reflectivity of available mirror coatings is low, concave gratings allow for systems
free from focusing mirrors that would reduce throughput two or more orders of
magnitude.
Many configurations for concave spectrometers have been designed. Some
are variations of the Rowland circle, while some place the spectrum on a flat field,
which is more suitable for charge-coupled device (CCD) array instruments. The
Seya-Namioka concave grating monochromator is especially suited for scanning
the spectrum by rotating the grating around its own axis.

7.1. CLASSIFICATION OF GRATING TYPES

The imaging characteristics of a concave grating system are governed by
the size, location and orientation of the entrance and exit optics (the mount), the
aberrations due to the grating, and the aberrations due to any auxiliary optics in
the system. [In this chapter we address only simple systems, in which the

50 H. A. Rowland, “Preliminary notice of the results accomplished in the manufacture and
theory of gratings for optical purposes,” Philos. Mag. 13, 469 (1882).

86

concave grating is the single optical element; auxiliary mirrors and lenses are not
considered.] The imaging properties of the grating itself are determined com-
pletely by the shape of its substrate (its curvature or figure) and the spacing and
curvature of the grooves (its groove pattern).
Gratings are classified both by their groove patterns and by their substrate
curvatures. In Chapter 6, we restricted our attention to plane classical gratings
and their mounts. In this chapter, more general gratings and grating systems are
considered.

7.1.1. Groove patterns

A classical grating is one whose grooves, when projected onto the tangent
plane, form a set of straight equally-spaced lines. Until the last few decades, the
vast majority of gratings were classical, in that any departure from uniform
spacing, groove parallelism or groove straightness was considered a flaw.
Classical gratings are made routinely both by mechanical ruling and inter-
ferometric (holographic) recording.
A first generation holographic grating has its grooves formed by the
intersection of a family of confocal hyperboloids (or ellipsoids) with the grating
substrate. When pro jected onto the tangent plane, these grooves have both un-
equal spacing and curvature. First generation holographic gratings are formed
by recording the master grating in a field generated by two sets of spherical
wavefronts, each of which may emanate from a point source or be focused
toward a virtual point.
A second generation holographic grating has the light from its point
sources reflected by concave mirrors (or transmitted through lenses) so that the
recording wavefronts are toroidal.51
A varied line-space (VLS) grating is one whose grooves, when projected
onto the tangent plane, form a set of straight parallel lines whose spacing varies
from groove to groove. Varying the groove spacing across the surface of the

51 C. Palmer, “Theory of second-generation holographic diffraction gratings,” J. Opt. Soc.
Am. 6, 1175-1188 (1989); T. Namioka and M. Koike, “Aspheric wavefront recording optics
for holographic gratings,” Appl. Opt. 34, 2180-2186 (1995).

87

grating moves the tangential focal curve, while keeping the groove straight and
parallel keeps the sagittal focal curve fixed.†

7.1.2. Substrate (blank) shapes

A concave grating is one whose surface is concave, regardless of its
groove pattern or profile, or the mount in which it is used. Examples are spherical
substrates (whose surfaces are portions of a sphere, which are definable with
one radius) and toroidal substrates (definable by two radii). Spherical substrates
are by far the most common type of concave substrates, since they are easily
manufactured and toleranced, and can be replicated in a straightforward manner.
Toroidal substrates are much more difficult to align, tolerance and replicate, but
astigmatism (see below) can generally be corrected better than by using a
spherical substrate.52 More general substrate shapes are also possible, such as
ellipsoidal or paraboloidal substrates 53 , but tolerancing and replication
complications relegate these grating surfaces out of the mainstream. Moreover,
the use of aspheric substrates whose surfaces are more general than those of the
toroid do not provide any additional design freedom for the two lowest-order
aberrations (defocus and astigmatism; see below)54 ; as a consequence, there
have been very few cases (for commercial instrumentation) in which the
improved imaging due to aspheric substrates has been worth the cost.
The shape of a concave grating (considering only spheres & toriods) can be
characterized either by its radii or its curvatures. The radii of the slice of the
substrate in the principal (dispersion) plane is called the tangential radius R,
while that in the plane parallel to the grooves at the grating center is called the
sagittal radius ρ. Equivalently, we can define the tangential curvature 1/R and
the sagittal curvature 1/ρ. For a spherical substrate, R = ρ.

† The tangential and sagittal focal curves are defined in Section 7.1.2 below.
52 H. Haber, “The torus grating,” J. Opt. Soc. Am. 40, 153-165 (1950).
53 T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. 51, 4-12
(1961).
54C. Palmer, “Limitations of aberration correction in spectrometer imaging,” Proc. SPIE
1055, 359-369 (1989).

88

A plane grating is one whose surface is planar. While plane gratings can
be thought of as a special case of concave gratings (for which the radii of
curvature of the substrate become infinite), we treat them separately here (see the
previous chapter). In the equations that follow, the case of a plane grating is
found simply by letting R (and ρ) → ∞.

7.2. CLASSICAL CONCAVE GRATING IMAGING

In Figure 7-1, a classical grating is shown; the Cartesian axes are defined as
follows: the x-axis is the outward grating normal to the grating surface at its
center (point O), the y-axis is tangent to the grating surface at O and perpen-
dicular to the grooves there, and the z-axis completes the right-handed triad of
axes (and is therefore parallel to the grooves at O). Light from point source
A(ξ, η, 0) is incident on a grating at point O; light of wavelength λ in order m is
diffracted toward point B(ξ', η', 0). Since point A was assumed, for simplicity, to
lie in the xy plane, to which the grooves are perpendicular at point O, the image
point B will lie in this plane as well; this plane is called the principal plane (also
called the tangential plane or the dispersion plane (see Figure 7-2). Ideally, any
point P(x, y, z) located on the grating surface will also diffract light from A to B.

z

y

P B

O A
x
grating surface

Figure 7-1. Use geometry. The grating surface centered at O diffracts light from point A
to point B. P is a general point on the grating surface. The x-axis points out of the grating

89

from its center, the z-axis points along the central groove, and the y-axis completes the
right-handed triad.

The plane through points O and B perpendicular to the principal plane is
called the sagittal plane, which is unique for this wavelength. The grating tan-
gent plane is the plane tangent to the grating surface at its center point O (i.e.,
the yz plane). The imaging effects of the groove spacing and curvature can be
completely separated from those due to the curvature of the substrate if the
groove pattern is projected onto this plane.
The imaging of this optical system can be investigated by considering the
optical path difference OPD between the pole ray AOB (where O is the center of
the grating) and the general ray APB (where P is an arbitrary point on the
grating surface). Application of Fermat's principle to this path difference, and the
subsequent expansion of the results in power series of the coordinates of the
tangent plane (y and z), yields expressions for the aberrations of the system.

y
B

P A

O
x

Figure 7-2. Use geometry – the principal plane. Points A, B and O lie in the xy
(principal) plane; the general point P on the grating surface may lie outside this plane.
The z-axis comes out of the page at O.

The optical path difference is

OPD = <APB> – <AOB> + Nmλ, (7-1)

90

where <APB> and <AOB> are the geometric lengths of the general and pole rays,
respectively (both multiplied by the index of refraction), m is the diffraction order,
and N is the number of grooves on the grating surface between points O and P.
The last term in Eq. (7-1) accounts for the fact that the distances <APB> and
<AOB> need not be exactly equal for the light along both rays to be in phase at
B: due to the wave nature of light, the light is in phase at B even if there are an
integral number of wavelengths between these two dis tances. If points O and P
are one groove apart (N = 1), the number of wavelengths in the difference <APB>
– <AOB> determines the order of diffraction m.
From geometric considerations, we find

<APB> = <AP> + <PB>

= (ξ − x )2 + (η − y )2 + z 2 + (ξ ′ − x) 2 + (η ′ − y )2 + z 2 , (7-2)

and similarly for <AOB>, if the medium of propagation is air (n ˜ 1). The optical
path difference can be expressed more simply if the coordinates of points A and
B are plane polar rather than Cartesian: letting

<AO> = r, <OB> = r', (7-3)

we may write

ξ = r cosα, η = r sinα;
(7-4)
ξ' = r' cosβ, η' = r' sinβ,

where the angles of incidence and diffraction (α and β) follow the sign
convention described in Chapter 2.
The power series for OPD can be written in terms of the grating surface
point coordinates y and z:

∞ ∞
OPD = ∑ ∑ Fij y i z j , (7-5)
i = 0 j =0

91

where Fij, the expansion coefficient of the (i,j) term, describes how the rays (or
wavefronts) diffracted from point P toward the ideal image point B differ (in
direction, or curvature, etc.) in proportion to y i z j from those from point O. The
x-dependence of OPD has been suppressed by writing

∞ ∞
x = x(y,z) = ∑ ∑ aij y i z j . (7-6)
i =0 j =0

This equation makes use of the fact that the grating surface is usually a regular
function of position, so x is not independent of y and z (e.g., if it is a spherical
surface of radius R, then ( x − R ) 2 + y 2 + z 2 = R 2 ).
By analogy with the terminology of lens and mirror optics, we call each term
in series (7-5) an aberration, and Fij its aberration coefficient. An aberration is
absent from the image of a given wavelength (in a given diffraction order) if its
associated coefficient Fij is zero.
Since we have imposed a plane of symmetry on the system (the principal (xy)
plane), all terms Fij for which j is odd vanish. Moreover, F00 = 0, since the expan-
sion (7-5) is about the origin O. The lowest- (first-) order terms F10 and F01 in the
expansion must equal zero in accordance with Fermat’s principle. Setting F10 = 0
yields the grating equation:

mλ = d (sinα + sinβ). (2-1)

By Fermat's principle, we may take this equation to be satis fied for all images.
Setting F01 = 0 yields the law of reflection in the plane perpendicular to the
dispersion plane. Thus, the second-order aberration terms F20 and F02 are those
of lowest order that need not necessarily vanish.
The generally accepted terminology is that a stigmatic image has vanishing
second-order coefficients even if higher-order aberrations are still present. The
second order terms describe the tangential and sagittal focusing:

 cos α   cos β 
F20 = cos α  − a20  + cos β  − a20  ≡ T(r, α) + T(r', β), (7-7)
 2r   2r ′ 

 1   1 
F02 =  − a02 cos α  +  − a02 cos β  ≡ S(r, α) + S(r', β). (7-8)
 2r   2r ′ 

92

The coefficient F20 governs the tangential (or spectral) focusing of the grating
system, while F02 governs the sagittal focusing. The associated aberrations are
called defocus and astigmatism, respectively. These equations may be seen to
be generalizations of the Coddington equations that describe the second-order
focal properties of an aspheric mirror.55
The two second-order aberrations describe the extent of a monochromatic
image: defocus pertains to the blurring of the image – its extent of the image
along the dis persion direction (i.e., in the tangential plane). Astigmatism pertains
to the extent of the image in the direction perpendicular to the dis persion
direction. In more common (but sometimes mis leading) terminology, defocus
applies to the "width" of the image in the spectral (dispersion) direction, and
astigmatism applies to the "height" of the spectral image; these terms imply that
the xy (tangential) plane be considered as horizontal and the yz (sagittal) plane
as vertical.
Actually astigmatism more correctly defines the condition in which the
tangential and sagittal foci are not coincident, which implies a line image at the
tangential focus. It is a general result of the off-axis use of a concave mirror (and,
by extension, a concave reflection grating as well). A complete three-
dimensional treatment of the optical path difference [see Eq. (7.1)] shows that the
image is actually a conical arc; image points away from the center of the ideal
image are diffracted toward the longer wavelengths. This effect, which
technically is not an aberration, is called (spectral) line curvature, and is most
noticeable in the spectra of Paschen-Runge mounts (see later in this chapter).56
Figure 7 shows astigmatism in the image of a wavelength diffracted off-axis
-3
from a concave grating, ignoring line curvature.
Since grating images are generally astigmatic, the focal distances r' in Eqs.
(7-7) and (7-8) should be distinguished. Calling r'T and r'S the tangential and
sagittal focal distances, respectively, we may set these equations equal to zero
and solve for the focal curves r'T(λ) and r'S(λ):

cos2 β
r'T(λ) = , (7-9)
A + B cos β

55 W. J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 2000), p. 317.
56 M. C. Hutley, Diffraction Gratings, Academic Press (New York, 1970), pp. 224 ff.

93

1
r'S(λ) = . (7-10)
D + E cos β

TF
SF

A
grating

Figure 7-3. Astigmatic focusing of a concave grating. Light from point A is focused into
a line parallel to the grooves at TF (the tangential focus) and perpendicular to the grooves
at SF (the sagittal focus). Spectral resolution is maximized at TF.

Here we have defined

cos2 α
A = B cosα − , B = 2 a 20,
r
(7-11)
1
D = E cosα − , E = 2 a 02,
r

where a 20 and a 02 are the coefficients in Eq. (7-6) (e.g., a 20 = a 02 = 1/(2R) for a
spherical grating of radius R). Eqs. (7-9) and (7-10) are completely general for
classical grating systems; that is, they apply to any type of grating mount or
configuration.
Of the two primary (second-order) focal curves, that corresponding to
defocus ( 20) is of greater importance in spectroscopy, since it is spectral
F
resolution that is most crucial to grating systems. For this reason we do not con-
cern ourselves with locating the image plane at the "circle of least confusion";

94

rather, we try to place the image plane at or near the tangential focus (where F20 =
0). For concave gratings (a 20 ? 0), there are two well-known solutions to the
defocus equation F20 = 0: those of Rowland and Wadsworth.
The Rowland circle is a circle whose diameter is equal to the tangential
radius of the grating substrate, and which passes through the grating center
(point O in Figure 7-5). If the point source A is placed on this circle, the tangen-
tial focal curve also lies on this circle. This solution is the basis for the Rowland
circle and Paschen-Runge mounts. For the Rowland circle mount,

cosα
r= = R cosα,
2a20
(7-12)
cos β
r'S = = R cosβ.
2a 20

The sagittal focal curve is

−1
 cosα + cos β 1 
r'S = 
 −  (7-13)
 ρ R cosα 


(where ρ is the sagittal radius of the grating), which is always greater than r'T
(even for a spherical substrate, for which ρ = R) unless α = β = 0. Consequently
this mount suffers from astigmatism, which in some cases is considerable.
The Wadsworth mount is one in which the incident light is collimated
(r → 8), so that the tangential focal curve is given by

cos2 β R cos2 β
r'T = = , (7-14)
2a20 (cosα + cos β ) cosα + cos β

and the sagittal focal curve is

1 ρ
r'S = = . (7-15)
2a02 (cosα + cos β ) cosα + cos β

95

In this mount, the imaging from a classical spherical grating (ρ = R) is such that
the astigmatism of the image is zero only for β = 0, though this is true for any
incidence angle α.
While higher-order aberrations are usually of less importance than defocus
and astigmatism, they can be significant. The third-order aberrations, primary or
tangential coma F30 and secondary or sagittal coma F12, are given by

sin α sin β
F30 = T(r, α) + T(r', β) – a 30 (cosα + cosβ) , (7-16)
r r′

sin α sin β
F12 = S(r, α) + S(r', β) – a 12 (cosα + cosβ) , (7-17)
r r′

where T and S are defined in Eqs. (7-7) and (7-8). Often one or both of these
third-order aberrations is significant in a spectral image, and must be minimized
with the second-order aberrations.

7.3. NONCLASSICAL CONCAVE GRATING IMAGING

For nonclassical groove patterns, the aberration coefficients Fij must be
generalized to account for the image-modifying effects of the variations in
curvature and spacing of the grooves, as well as for the focusing effects of the
concave substra te:

mλ
Fij = Mij + Hij ≡ Mij + H' ij . (7-18)
λ0

The terms Mij are simply those Fij coefficients for classical concave grating
mounts, dis cussed in Section 7.2 above. The H' ij coefficients describe how the
groove pattern differs from that of a classical grating (for classical gratings, H' ij =
0 for all terms of order two or higher (i + j = 2)). The tangential and sagittal focal
distances (Eqs. (7-9) and (7-10)) must now be generalized:

cos 2 β
r'T(λ) = , (7-19)
A + B cos β + C sin β

96

1
r'S(λ) = , (7-20)
D + E cos β + F sin β

where in addition to Eqs. (7-11) we have

C = – 2 H' 20, F = – 2 H' 02. (7-21)

Here H' 20 and H' 02 are the terms that govern the effect of the groove pattern on
the tangential and sagittal focusing. For a first generation holographic grating,
for example, the Hij coefficients may be written in terms of the parameters of the
recording geometry (see Figure 7-4):

H' 20 = – T(rC, γ) + T(rD, δ), (7-22)

H'02 = – S(rC, γ) + S(rD, δ), (7-23)

where C(rC, γ) and D(rD, δ) are the plane polar coordinates of the recording points.
These equations are quite similar to Eqs. (7-7) and (7-8), due to the similarity
between Figures 7-4 and 7-2.

y

rD D
O δ
γ x

rC C

Figure 7-4. Recording parameters. Spherical waves emanate from point sources C and
D; the interference pattern forms fringes on the concave substrate centered at O.

97

Nonclassical concave gratings are generally produced holographically, but
for certain applications, they can be made by mechanical ruling as well, by
changing the groove spacing from one groove to the next during ruling57 , by
curving the grooves 58 , or both.59 For such varied line-space (VLS) gratings (see
Chapter 4), the terms Hij are written in terms of the groove spacing coefficients
rather than in terms of recording coordinates.60
Several important conclusions may be drawn from the formalism developed
above for grating system imaging.61

• The imaging effects of the shape of the grating substrate (manifest in
the coefficients a ij) and the groove pattern (manifest in the coefficients
Hij) are completely separable.
• The imaging effects of the shape of the grating substrate are contained
completely in terms that are formally identical to those for the identical
mirrors substrate, except that the diffraction angle is given by the
grating equation (Eq. (2-1)) rather than the law of reflection.
• The imaging effects of the groove pattern are dictated completely by the
spacing and curvature of the grooves when projected onto the plane
tangent to the grating surface at its center.
• The y-dependence of the groove pattern governs the local groove
spacing, which in turn governs the tangential aberrations of the system.
• The z-dependence of the groove pattern governs the local groove
curvature, which in turn governs the sagittal aberrations of the system.
More details on the imaging properties of gratings systems can be found in
Namioka 62 and Noda et al.63

57 Y. Sakayanagi, “A stigmatic concave gating with varying spacing,” Sci. Light 16, 129-137
(1967).
58 Y. Sakayanagi, Sci. Light 3, 1 (1954).
59 T. Harada, S. Moriyama and T. Kita, “Mechanically ruled stigmatic concave gratings,”
Japan. J. Appl. Phys. 14, 175-179 (1974).
60 C. Palmer and W. R. McKinney, "Equivalence of focusing conditions for holographic and
varied line-space gratings," Appl. Opt. 29, 47-51 (1990).
61 C. Palmer and W. R. McKinney, "Imaging theory of plane-symmetric varied line-space
grating systems," Opt. Eng. 33, 820-829 (1994).
62 T. Namioka, "Theory of the concave grating," J. Opt. Soc. Am. 49, 446 (1959).

98

7.4. REDUCTION OF ABERRATIONS

In the design of grating systems, there exist several degrees of freedom
whose values may be chosen to optimize image quality. For monochromators,
the locations of the entrance slit A and exit slit B relative to the grating center O
provide three degrees of freedom (or four, if no plane of symmetry is imposed);
the missing degree of freedom is restricted by the grating equation, which sets
the angular relationship between the lines AO and BO. For spectrographs, the
location of the entrance slit A as well as the location, orientation and curvature
of the image field provide degrees of freedom (though the grating equation must
be satisfied). In addition, the curvature of the grating substrate provides
freedom, and the aberration coefficients H' ij for a holographic grating (or the
equivalent terms for a VLS grating) can be chosen to improve imaging. Even in
systems for which the grating use geometry (the mount) has been specified,
there exist several degrees of freedom due to the aberration reduction possibili-
ties of the grating itself.
Algebraic techniques can find sets of design parameter values that minimize
image size at one or two wavelengths, but to optimize the imaging of an entire
spectral range is usually so complicated that computer implementation of a
design procedure is essential. Newport has developed a set of proprietary
computer programs that are used to design and analyze grating systems. These
programs allow selected sets of parameter values governing the use and record-
ing geometries to vary within prescribed limits. Optimal imaging is found by
comparing the imaging properties for systems with different sets of parameters
values.
Design techniques for grating systems that minimize aberrations may be
classified into two groups: those that consider wavefront aberrations and those
that consider ray deviations. The wavefront aberration theory of grating
systems was developed by Beutler64 and Namioka 65 , and was presented in
Section 7.2. The latter group contains both the familiar raytrace techniques used
in commercial optical design software and the Lie aberration theory developed by

63 H. Noda, T. Namioka and M. Seya, "Geometric theory of the grating," J. Opt. Soc. Am.
64, 1031-1036 (1974).
64 H. G. Beutler, "The theory of the concave grating,” J. Opt. Soc. Am. 35, 311-350 (1945).
65 T. Namioka, "Theory of the concave grating,” J. Opt. Soc. Am. 49, 446-460 (1959).

99

Dragt.66 The principles of optical raytrace techniques are widely known and
taught in college courses, and are the basis of a number of commercially-
available optical design software packages, so they will not be addressed here,
but the concepts of Lie aberration theory are not widely known – for the
interested reader they are summarized in Appendix B.
Design algorithms generally identify a merit function, an expression that
returns a single value for any set of design parameter arguments; this allows two
different sets of design parameter values to be compared quantitatively.
Generally, merit functions are designed so that lower values correspond to better
designs – that is, the ideal figure of merit is zero.
For grating system design, a number of merit functions may be defined. The
Newport proprietary design software uses the function

M = w’ + ch’, (7-24)

where w’ and h’ are the width (in the dispersion plane) and height (perpendicular
to the dispersion plane) of the image, and c is a constant weighting factor.67
Minimizing M therefore reduces both the width and the height of the diffracted
image. Since image width (which affects spectral resolution) is almost always
more important to reduce than image height, c is generally chosen to be much
less than unity. If w’ is expressed not as a geometric width (say, in millimeters)
but a spectral width (in nanometers), then M will have these units as well; since
h’ is in millimeters (there being no dispersion in the direction in which h’ is
measured), c will have the units of reciprocal linear dispersion (e.g., nm/mm) but it
is not a measure of reciprocal linear dispersion – c is merely a weighting factor
introduced in Eq. (7-24) to ensure that image width and image height are properly
weighted in the optimization routine.

66 A. J. Dragt, "Lie algebraic theory of geometrical optics and optical aberrations,” J. Opt.
Soc. Am. 72, 372-379 (1982); K. Goto and T. Kurosaki, "Canonical formulation for the
geometrical optics of concave gratings,” J. Opt. Soc. Am. A10, 452-465 (1993); C. Palmer,
W. R. McKinney and B. Wheeler, "Imaging equations for spectroscopic systems using Lie
Transformations. Part I – Theoretical foundations,” Proc. SPIE 3450, 55-66 (1998); C.
Palmer, B. Wheeler and W. R. McKinney, "Imaging equations for spectroscopic systems
using Lie transformations. Part II - Multi-element systems," Proc. SPIE 3450, 67-77 (1998).
67 W. R. McKinney and C. Palmer, "Numerical design method for aberration-reduced
concave grating spectrometers,” Appl. Opt. 26, 3018-3118 (1987).

100

For optimization over a spectral range λ1 ≤ λ ≤ λ2, Eq. (7-24) can be
generalized to define the merit function as the maximum value of w’ + ch’ over all
wavelengths:

M = sup{w(λ ) + ch ′(λ )} , (7-25)
λ

where the supremum function sup{} returns the maximum value of all of its
arguments. Defining a merit function in the form of Eq. (7-25) minimizes the
maximum value of w’ + ch’ over all wavelengths considered. [A more general
form would allow the weighting factor to be wavelength-specific, i.e., c → c(λ).]
Eqs. (7-24) and (7-25) consider the ray deviations in the image plane,
determined either by direct ray tracing or by converting wavefront aberrations
into ray deviations. An alternative merit function may be defined using Eqs.
(7-19) and (7-20), the expressions for the tangential and sagittal focal distances.
Following Schroeder68 , we define the quantity ∆(λ) as

1 1
∆(λ) = − , (7-26)
rT (λ ) rS (λ )
′ ′

leading to the following merit function:

M = sup{? (λ )} . (7-27)
λ

This version of M will consider second-order aberrations only (i.e., F20 (defocus)
and F02 (astigmatism)) to minimize the distances between the tangential and
sagittal focal curves for each wavelength in the spectrum. 69
Noda et al.70 have suggested using as the merit function the integral of the
square of an aberration coefficient,

68 D. J. Schroeder, Astronomical Optics (Academic Press, New York, 1987), pp. 64 & 263.
69 C. Palmer, "Deviation of second-order focal curves in common plane-symmetric
spectrometer mounts,” J. Opt. Soc Am. A7, 1770-1778 (1990).
70 H. Noda, T. Namioka and M. Seya, "Geometric theory of the grating,” J. Opt. Soc Am.
64, 1031-1036 (1974).

101

∫ dλ (F (λ )) ,
2
M= ij (7-28)
λ

where the integration is over the spectrum of interest (λ1 ≤ λ ≤ λ2). Choosing
defocus (F20) as the aberration term would, however, not require the design
routine to minimize astigmatism as well. A number N of aberrations may be
considered, but this requires the simultaneous minimization of N merit functions
of the form given by Eq. (7-28).71
Two other merit functions have been used in the design of spectrometer
systems are the Strehl ratio 72 and the quality factor.73

7.5. CONCAVE GRATING M OUNTS

As with plane grating mounts, concave grating mounts can be either
monochromators or spectrographs.

7.5.1. The Rowland circle spectrograph

The first concave gratings of spectroscopic quality were ruled by Rowland,
who also designing their first mounting. Placing the ideal source point on the
Rowland circle (see Eqs. (7-12) and Figure 7-5) forms spectra on that circle free
from defocus and primary coma at all wavelengths (i.e., F20 = F30 = 0 for all λ);
while spherical aberration is residual and small, astigmatism is usually severe.
Originally a Rowland circle spectrograph employed a photographic plate bent
along a circular arc on the Rowland circle to record the spectrum in its entirety.
Today it is more common for a series of exit slits to be cut into a circular
mask to allow the recording of several discrete wavelengths photoelectrically;

71 E. Sokolova, B. Kruizinga and I. Gulobenko, “Recording of concave diffraction gratings in
a two-step process using spatially incoherent light,” Opt. Eng. 43, 2613-2622 (2004).
72 W. T. Welford, “Aberration tolerances for spectrum line images,” Opt. Acta 10, 121-127
(1963).
73M. Pouey, “Comparison between far ultraviolet spect rometers,” Opt. Commun. 2, 339-
342 (1970).

102

this system is called the Paschen-Runge mount. Other configurations based on
the imaging properties of the Rowland circle are the Eagle mount and the Abney
mount, both of which are described by Hutley74 and by Meltzer.75
Unless the exit slits (or photographic plates) are considerably taller than the
entrance slit, the astigmatism of Rowland circle mounts usually prevents more
than a small fraction of the diffracted light from being recorded, which greatly
decreases the efficiency of the instrument. Increasing the exit slit heights helps
collect more light, but since the images are curved, the exit slits would have to be
curved as well to maintain optimal resolution. To complicate matters further, this
curvature depends on the diffracted wavelength, so each exit slit would require a
unique curvature. Few instruments have gone to such trouble, so most Rowland
circle grating mounts collect only a small portion of the light incident on the grat-
ing. For this reason these mounts are adequate for strong sources (such as the
observation of the solar spectrum) but not for less intense sources (such as
stellar spectra).

Rowland circle
entrance slit

λ2

λ1
grating spectrum

Figure 7-5. The Rowland Circle spectrograph. Both the entrance slit and the diffracted
spectrum lie on the Rowland circle, whose diameter equals the tangential radius of

74 M. C. Hutley, Diffraction Gratings, Academic Press (New York, 1970).
75 R. J. Meltzer, "Spectrographs and Monochromators," in Applied Optics and Optical
Engineering, vol. V (chapter 3), R. Shannon, ed., Academic Press (New York: 1969).

103

curvature R of the grating and that passes through the grating center. Light of two
wavelengths is shown focused at different points on the Rowland circle.

The imaging properties of instruments based on the Rowland circle
spectrograph, such as direct readers and atomic absorption instruments, can be
improved by the use of nonclassical gratings. By replacing the usual concave
classical gratings with concave aberration-reduced gratings, astigmatism can be
improved substantially. Rowland circle mounts modified in this manner direct
more diffracted light through the exit slits, though often at the expense of
degrading resolution to some degree.

7.5.2. The Wadsworth spectrograph

When a classical concave grating is illuminated with collimated light (rather
than from a point source on the Rowland circle), spectral astigmatism on and near
the grating normal is greatly reduced. Such a grating system is called the
Wadsworth mount (see Figure 7-6).76 The wavelength-dependent aberrations of
the grating are compounded by the aberration of the collimating optics, though
use of a paraboloidal mirror illuminated on-axis will reduce off-axis aberrations
and spherical aberrations. The Wadsworth mount suggests itself in situations in
which the light incident on the grating is naturally collimated (from, for example,
astronomical sources). In other cases, an off-axis parabolic mirror would serve
well as the collimating element.

grating

GN

spectrum

76 F. Wadsworth, “The modern spectroscope,” Astrophys. J. 3, 47-62 (1896).

104

Figure 7 -6. The Wadsworth spectrograph. Collimated light is incident on a concave
grating; light of two wavelengths is shown focused at different points. GN is the grating
normal.

7.5.3. Flat-field spectrographs

One of the advantages of changing the groove pattern (as on a first- or
second- generation holographic grating or a VLS grating) is that the focal curves
can be modified, yielding grating mounts that differ from the classical ones. A
logical improvement of this kind on the Rowland circle spectrograph is the flat-
field spectrograph, in which the tangential focal curve is removed from the
Rowland circle and rendered nearly linear over the spectrum of interest (see
Figure 7-7). While a grating cannot be made that images a spectrum perfectly on
a line, one that forms a spectrum on a sufficiently flat surface is ideal for use in
linear detector array instruments of moderate resolution. This development has
had a significant effect on spectrograph design.

entrance slit

λ2

λ1
flat
grating spectrum

Figure 7-7. A flat-field spectrograph. The spectrum from λ1 to λ2 (>λ1) is shown imaged
onto a line.

The relative displacement between the tangential and sagittal focal curves
can also be reduced via VLS or interferometric modification of the groove pattern.
In this way, the resolution of a flat-field spectrometer can be maintained (or
improved) while its astigmatism is decreased; the latter effect allows more light to

105

be transmitted through the exit slit (or onto the detector elements). An example
of the process of aberration reduction is shown in Figure 7-8.

(a) (b) (c)

Figure 7-8. Modification of focal curves. The primary tangential focal curve (F20 = 0) is
thick; the primary sagittal focal curve (F02 = 0) is thin. (a) Focal curves for a classical
(H20 = H02 = 0) concave grating, illuminated off the normal (α ? 0) – the dark curve is an
arc of the Rowland circle. (b) Choosing a suitable nonzero H20 value moves the tangential
focal arc so that part of it is nearly linear, suitable for a flat-field spectrograph detector.
(c) Choosing a suitable nonzero value of H02 moves the sagittal focal curve so that it
crosses the tangential focal curve, providing a stigmatic image.

7.5.4. Imaging spectrographs and monochromators 77

Concave gratings may also be used in imaging spectrographs, which are
instruments for which a spectrum is obtained for different spatial regions in the
object plane. For example, an imaging spectrometer may generate a two-
dimensional spatial image on a detector array, and for each such image,
a spectrum is scanned (over time); alternatively, a spectrum can be recorded for a
linear slice of the image, and the slice itself can be moved across the image to
provide the second spatial dimension (sometimes called the “push broom”
technique).

77 M. Descour and E. Dereliak, “Computed-tomography imaging spectrometer: experimental
calibration and reconstruction results,” Appl. Opt. 34, 4817-4826 (1995); P. Mouroulis, D.
W. Wilson, P. D. Maker and R. E. Muller, “Convex grating types for concentric imaging
spectrometers,” Appl. Opt. 37, 7200-7028 (1998); M. Beasley, C. Boone, N. Cunningham, J.
Green and E. Wilkinson, “Imaging spectrograph for interstellar shocks: a narrowband imaging
payload for the far ultraviolet,” Appl. Opt. 43, 4633-4642 (2004).

106

7.5.5. Constant-deviation monochromators

In a constant-deviation monochromator, the angle 2K between the entrance
and exit arms is held constant as the grating is rotated (thus scanning the
spectrum; see Figure 7-9). This angle is called the deviation angle or angular
deviation (AD). While plane or concave gratings can be used in constant-de-
viation mounts, only in the latter case can imaging be made acceptable over an
entire spectrum without auxiliary focusing optics.

Α

Β
2K

Figure 7-9. Constant-deviation monochromator geometry. To scan wavelengths, the
entrance slit A and exit slit B remain fixed as the grating rotates. The deviation angle 2K is
measured from the exit arm to the entrance arm. The Seya-Namioka monochromator is a
special case for which Eqs. (7-29) are satisfied.

The Seya-Namioka monochromator78 is a very special case of constant-
deviation mount using a classical spherical grating, in which the deviation angle
2K between the beams and the entrance and exit slit distances (r and r') are given
by

78 M. Seya, "A new mounting of concave grating suitable for a spectrometer," Science of
Light 2, 8-17 (1952); T. Namioka, “Theory of the concave grating. III. Seya-Namioka
monochromator,” J. Opt. Soc. Am. 49, 951-961 (1959).

107

2K = 70°30', r = r' = R cos(70°30'/2), (7-29)

where R is the radius of the spherical grating substrate. The only moving part in
this system is the grating, through whose rotation the spectrum is scanned.
Resolution may be quite good in part of the spectrum, though it degrades farther
from the optimal wavelength; astigmatism is high, but at an optimum. Replacing
the grating with a classical toroidal grating can reduce the astigmatism, if the
minor radius of the toroid is chosen judiciously. The reduction of astigmatism by
suitably designed holographic gratings is also helpful, though the best way to
optimize the imaging of a constant-deviation monochromator is to relax the
restrictions given by Eqs. (7-29) on the use geometry.

108

8. IMAGING PROPERTIES OF GRATING
SYSTEMS

8.1. CHARACTERIZATION OF IMAGING QUALITY

In Chapter 7, we formulated the optical imaging properties of a grating sys-
tem in terms of wavefront aberrations. After arriving at a design, though, this
approach is not ideal for observing the imaging properties of the system. Two
tools of image analysis – spot diagrams and linespread functions – are discussed
below.

8.1.1. Geometric raytracing and spot diagrams

Raytracing (using the laws of geometrical optics) is superior to wavefront
aberration analysis in the determination of image quality. Aberration analysis is
an approximation to image analysis, since it involves expanding quantities in
infinite power series and considering only a few terms. Raytracing, on the other
hand, does not involve approximations, but shows (in the absence of the
diffractive effects of physical optics) where each ray of light incident on the
grating will diffract. It would be more exact to design grating systems with
a raytracing procedure as well, though to do so would be computationally
cumbersome.
The set of intersections of the diffracted rays and the image plane forms
a set of points, called a spot diagram. In Figure 8-1, several simple spot diagrams
are shown; their horizontal axes are in the plane of dispersion (the tangential
plane), and their vertical axes are in the sagittal plane. In (a) an uncorrected (out-
of-focus) image is shown; (b) shows good tangential focusing, and (c) shows
virtually point-like imaging. All three of these images are simplistic in that they
ignore the effects of line curvature as well as higher-order aberrations (such as
coma and spherical aberration), which render typical spot diagrams asymmetric,
as in (d).

110

(a) (b)

(c) (d)

Figure 8-1. Spot diagrams. In (a) the image is out of focus. In (b), the image is well
focused in the tangential plane only; the line curvature inherent to grating-diffracted
images is shown. In (c) the image is well focused in both directions – the individual spots
are not discernible. In (d) a more realistic image is shown.

A straightforward method of evaluating the imaging properties of
a spectrometer at a given wavelength is to measure the tangential and sagittal
extent of an image (often called the width w' and height h' of the image,
respectively), as in Figure 8-2.
Geometric raytracing provides spot diagrams in good agreement with
observed spectrometer images, except for well-focused images, in which the
wave nature of light dictates a minimum size for the image. Even if the image of a
point object is completely without aberrations, it is not a point image, due to the
diffraction effects of the pupil (which is usually the perimeter of the grating). The
minimal image size, called the diffraction limit, can be estimated for agiven
wavelength as the diameter a of the Airy disk for a mirror in the same geometry:

r ' (λ )
a = 2.44λ ƒ/noOUTPUT = 2.44λ . (8-1)
W cos β

111

Here ƒ/noOUTPUT is the output focal ratio, r'(λ) is the focal dis tance for this
wavelength, and W is the width of the grating (see Eq. (2-25), Chapter 2). Results
from raytrace analyses that use the laws of geometrical optics only should not be
considered valid if the dimensions of the image are found to be near or below the
diffraction limit calculated from Eq. (8-1).

w'

h'

Figure 8-2. Image dimensions. The width w' and height h' of the image in the image
plane are the dimensions of the smallest rectangle that contains the spots. The sides of
the rectangle are taken to be parallel (w') and perpendicular (h') to the principal plane.

8.1.2. Linespread calculations

A fundamental problem with geometric raytracing procedures (other than
that they ignore the variations in energy density throughout a cross-section of
the diffracted beam and the diffraction efficiency of the grating) is its ignorance
of the effect that the size and shape of the exit aperture has on the measured
resolution of the instrument.
An alternative to merely measuring the extent of a spectral image is to
compute its linespread function, which is the convolution of the
(monochromatic) image of the entrance slit with the exit aperture (the exit slit in a
monochromator, or a detector element in a spectrograph). A close physical
equivalent is obtained by scanning the monochromatic image by moving the exit

112

aperture past it in the image plane, and recording the light intensity passing
through the slit as a function of position in this plane.
The linespread calculation thus described accounts for the effect that the
entrance and exit slit dimensions have on the resolution of the grating system.

8.2. INSTRUMENTAL IMAGING

With regard to the imaging of actual optical instruments, it is not sufficient
to state that ideal performance (in which geometrical aberrations are completely
eliminated and the diffraction limit is ignored) is to focus a point object to a point
image. All real sources are extended sources – that is, they have finite widths
and heights.

8.2.1. Magnification of the entrance aperture

The image of the entrance slit, ignoring aberrations and the diffraction limit,
will not have the same dimensions as the entrance slit itself. Calling w and h the
width and height of the entrance slit, and w' and h' the width and height of its im-
age, the tangential and sagittal magnifications χT and χS are

w′ r ′ cosα h′ r ′
χT ≡ = , χS ≡ = . (8-2)
w r cos β h r

These relations, which indicate that the size of the image of the entrance slit
will usually differ from that of the entrance slit itself, are derived below.
Figure 8-3 shows the plane of dispersion. The grating center is at O; the
x-axis is the grating normal and the y-axis is the line through the grating center
perpendicular to the grooves at O. Monochromatic light of wavelength λ leaves
the entrance slit (of width w) located at the polar coordinates (r, α) from the
grating center O and is diffracted along angle β. When seen from O, the entrance
slit subtends an angle ∆α = w/r in the dispersion (xy) plane. Rays from one edge
of the entrance slit have incidence angle α, and are diffracted along β; rays from

113

the other edge have incidence angle α + ∆α, and are diffracted along β – ∆β.†
The image (located a distance r' from O), therefore subtends an angle ∆β when
seen from O, has width w' = r'∆β. The ratio χT = w'/w is the tangential
magnification.

y
w'
r'
∆β
β
O x
α
∆α
r
w

Figure 8-3. Geometry showing tangential magnification. Monochromatic light from the
entrance slit, of width w, is projected by the grating to form an image of width w'.

We may apply the grating equation to the rays on either side of the entrance
slit:

Gmλ = sinα + sinβ, (8-3)

Gmλ = sin(α+∆α)+ sin(β–∆β). (8-4)

Here G (= 1/d) is the groove frequency along the y-axis at O, and m is the
diffraction order. Expanding sin(α+∆α) in Eq. (8-4) in a Taylor series about
∆α = 0, we obtain

sin(α+∆α) = sinα + (cosα )∆α + ..., (8-5)

† In this section, both ∆α and ∆β are taken to be positive incremental angles, so by Eq. (2-
1), a positive change in α will lead to a negative change in β.

114

where terms of order two or higher in ∆α have been truncated. Using Eq. (8-5)
(and its analogue for sin(β–∆β)) in Eq. (8-4), and subtracting it from Eq. (8-3), we
obtain

cosα ∆α = cosβ ∆β, (8-6)

and therefore

∆β cos α
= , (8-7)
∆α cos β

from which the first of Eqs. (8-2) follows.
Figure 8-4 shows the same situation in the sagittal plane, which is
perpendicular to the principal plane and contains the pole diffracted ray. The
entrance slit is located below the principal plane; consequently, its image is
above this plane. A ray from the top of the center of the entrance slit is shown.
Since the grooves are parallel to the sagittal plane at O, the grating acts as a
mirror in this plane, so the angles φ and φ' are equal in magnitude.

z

r' h'

ψ' φ' z'
PRINCIPAL
O
ψ φ
PLANE
z
r
h

Figure 8-4. Geometry showing sagittal magnification. Monochromatic light from the
entrance slit, of height h, is projected by the grating to form an image of height h'.

115

Ignoring signs, the tangents of these angles are equal as well:

z z′
tanφ = tanφ' → = , (8-8)
r r′

where z and z' are the distances from the entrance and exit slit points to the
principal plane. A ray from an entrance slit point a distance |z + h| from this plane
will image toward a point |z' + h'| from this plane, where h' now defines the height
of the image. As this ray is governed by reflection as well,

z + h z′ + h ′
tanψ = tanψ' → = . (8-9)
r r′

Simplifying this using Eq. (8-8) yields the latter of Eqs. (8-2).

8.2.2. Effects of the entrance aperture dimensions

Consider a spectrometer with a point source located in the principal plane:
the aberrated image of this point source has width δw' (in the dispersion
direction) and height δh' (see Figure 8-5). If the point source is located out of the
principal plane, it will generally be distorted, tilted and enlarged: its dimensions
are now δW' and δH'. Because a point source is considered, these i age m
dimensions are not due to any magnification effects of the system.

Point δw′
δh′

system
m
δH′

δW′

Figure 8-5. Point source imaging. A point source is imaged by the system; the upper
image is for a point source located at the center of the entrance slit (in the dispersion

116

plane), and the lower image shows how this image is tilted and distorted (and generally
gets larger) for a point source off the dispersion plane.

Now consider a rectangular entrance slit of width W0 (in the dispersion
plane) and height H0. If we ignore aberrations and line curvature (see Section
7.2) for the moment, we see that the image of the entrance slit is also a rectangle,
whose width W0' and height H0' are magnified:

W0' = χT W0,
(8-10)
H0' = χS H0

(see Figure 8-6).

Entrance Slit – without aberrations

H0 system H0'
m

W0 W0'

Figure 8-6. Entrance slit imaging (without aberrations). Ignoring aberrations and line
curvature, the image of a rectangular entrance slit is also a rectangle, one that has been
magnified in both directions.

Combining these two cases provides the following illustration (Figure 8-7).
From this figure, we can estimate the width W' and height H' of the image of the
entrance slit, considering both magnification effects and aberrations, as follows:

r ′ cosα
W' = χT W0 + δW' = W0 + δW',
r cos β
(8-11)
r′
H' = χS H0 + δH' = H 0 + δH'.
r

117

Eqs. (8-11) allow the imaging properties of a grating system with an entrance slit
of finite area to be estimated quite well from the imaging properties of the system
in which an infinitesimally small object point is considered. In effect, rays need
only be traced from one point in the entrance slit (which determines δW' and δH'),
from which the image dimensions for an extended entrance slit can be calculated
using Eqs. (8-10).†

Entrance Slit – with aberrations

H0 system H′
m

W0
W′

Figure 8-7. Entrance slit imaging (including aberrations). Superimposing the point-
source images for the four corners of the entrance slit onto the (unaberrated) image of the
entrance slit leads to the diagram above, showing that the rectangle in which the entire
image lies has width W′ and height H′.

8.2.3. Effects of the exit aperture dimensions

The linespread function for a spectral image, as defined above, depends on
the width of the exit aperture as well as on the width of the diffracted image itself.
In determining the optimal width of the exit slit (or single detector element), a rule
of thumb is that the width w" of the exit aperture should roughly match the width
w' of the image of the entrance aperture, as explained below.
Typical linespread curves for the same diffracted image scanned by three
different exit slit widths are shown in Figure 8-8. For simplicity, we have assumed
χT = 1 for these examples. The horizontal axis is position along the image plane,

† These equations fail to consider the effects of line curvature, so they must be regarded as
approximate, though their accuracy should be acceptable for plane-symmetric grating
systems (i.e., those whose entrance slit is centered in the dispersion plane) provided the
entrance slit is not too tall (H0 << r).

118

in the plane of dispersion. This axis can also be thought of as a wavelength axis
(that is, in spectral units); the two axes are related via the dispersion. The
vertical a is relative light intensity (or throughput) at the image plane; its
xis
bottom and top represent no intensity and total intensity (or no rays entering the
slit and all rays entering the slit), respectively. Changing the horizontal
coordinate represents scanning the monochromatic image by moving the exit slit
across it, in the plane of dis persion. This is approximately equivalent to
changing the wavelength while keeping the exit slit fixed in space.

1
FWHM
FWZH
(a)

0

1

(b)

0

1

(c)

0

Figure 8-8. Linespread curves for different exit slit widths. The vertical axis is relative
intensity at the exit aperture, and the horizontal axis is position along the image plane (in
the plane of dispersion). For a given curve, the dark horizontal line shows the FWHM
(the width of that portion of the curve in which its amplitude exceeds its half maximum);
the FWZH is the width of the entire curve. (a) w" < w'; (b) w" = w'; (c) w" > w'. In (a)
the peak is below unity. In (a) and (b), the FWHM are approximately equal. Severely
aberrated images will yield linespread curves that differ from those above (in that they
will be asymmetric), although their overall shape will be similar.

119

An exit slit that is narrower than the image ( " < w') will result in a
w
linespread graph such as that seen in Figure 8-8(a). In no position of the exit slit
(or, for no diffracted wavelength) do all diffracted rays fall within the slit, as it is
not wide enough; the relative intensity does not reach its maximum value of
unity. In (b), the exit slit width matches the width of the image: w" = w'.
At exactly one point during the scan, all of the diffracted light is contained within
the exit slit; this point is the peak (at a relative intensity of unity) of the curve. In
(c) the exit slit is wider than the image (w" > w'). The exit slit contains the entire
image for many positions of the exit slit.
In these figures the quantities FWZH and FWHM are shown. These are
abbreviations for full width at zero height and full width at half maximum. The
FWZH is simply the total extent of the linespread function, usually expressed in
spectral units. The FWHM is the spectral extent between the two extreme points
on the linespread graph that are at half the maximum value. The FWHM is often
used as a quantitative measure of image quality in grating systems; it is often
called the effective spectral bandwidth. The FWZH is sometimes called the full
spectral bandwidth. It should be noted that the terminology is not universal
among authors and sometimes quite confusing.
As the exit slit width w' is decreased, the effective bandwidth will generally
decrease. If w' is roughly equal to the image width w, though, further reduction
of the exit slit width will not reduce the bandwidth appreciably. This can be seen
in Figure 8-8, in which reducing w' from case (c) to case (b) results in a decrease
in the FWHM, but further reduction of w' to case (a) does not reduce the
FWHM.
The situation in w" < w' is undesirable in that diffracted energy is lost (the
peak relative intensity is low) since the exit slit is too narrow to collect all of the
diffracted light at once. The situation w" > w' is also undesirable, since the
FWHM is excessively large (or, similarly, an excessively wide band of
wavelengths is accepted by the wide slit). The situation w" = w' seems optimal:
when the exit slit width matches the width of the spectral image, the relative
intensity is maximized while the FWHM is minimized. An interesting curve is
shown in Figure 8 in which the ratio FWHM/FWZH is shown vs. the ratio
-9,
w"/w' for a typical grating system. This ratio reaches its single minimum near
w" = w'.
The height of the exit aperture has a more subtle effect on the imaging
properties of the spectrometer, since by 'height' we mean extent in the direction

120

perpendicular to the plane of dispersion. If the exit slit height is less than the
height (sagittal extent) of the image, some diffracted light will be lost, as it will not
pass through the aperture. Since diffracted images generally display

1.0
FWHM/FWZH

w"/w'
0.5
0 1 2

Figure 8-9. FWHM/FWZH vs. w"/w' for a typical system.

exit slit

w' w '*
Figure 8-10. Effect of exit slit height on image width. Both the width and the height of the
image are reduced by the exit slit chosen. Even if the width of the exit slit is greater than
the width of the image, truncating the height of the image yields w'* < w'. [Only the top
half of each image is shown.]
121

curvature, truncating the sagittal extent of the image by choosing a short exit slit
also reduces the width of the image (see Figure 8 -10). This latter effect is
especially noticeable in Paschen-Runge mounts.
In this discussion we have ignored the diffraction effects of the grating
aperture: the comments above consider only the effects of geometrical optics on
instrumental imaging. For cases in which the entrance and exit slits are equal in
width, and this width is two or three times the diffraction limit, the linespread
function is approximately Gaussian in shape rather than the triangle shown in
Figure 8-8(b).

8.3. INSTRUMENTAL BANDPASS

The instrumental bandpass of an optical spectrometer depends on both the
dimensions of the image of the entrance slit and the exit slit dimensions.
Ignoring the effects of the image height, the instrumental bandpass B is given by

B = P sup(w',w") (8-12)

where P is the reciprocal linear dispersion (see Eq. (2-14’)), w' is the image width,
w" is the width of the exit slit, and sup(w',w") is the greater of its arguments (i.e.,
the two-argument version of Eq. (7-25)):

 w′ if w′ > w′′
sup(w′, w′′) =  . (8-13)
 w′′ otherwise

As P is usually expressed in nm/mm, the widths w' and w" must be expressed in
millimeters to obtain the bandpass B in nanometers.
In cases where the image of the entrance slit is wider than the exit slit (that
is, w' > w"), the instrumental bandpass is said to be imaging limited, whereas in
those cases where the exit slit is wider than the image of the entrance slit
(w' < w"), the instrumental bandpass is said to be slit limited. [When an
imaging-limited optical system is imaging limited due primarily to the grating,
either because of the resolving power of the grating or due to its wavefront
errors, the system is said to be grating limited.]

122

In the design of optical spectrometers, the widths of the entrance and exit
slits are chosen by balancing spectral resolution (which improves as the slits
become narrower, to a limit) and optical throughput (which improves as the slits
widen, up to a limit). Ideally, the exit slit width is matched to the width of the
image of the entrance slit (case (b) in Figure 8-8: w' = w") – this optimizes both
resolution and throughput. This optimum may only be achievable for one
wavelength, the resolution of the other wavelengths generally being either slit-
limited or imaging-limited (with suboptimal throughput likely as well).

123

9. EFFICIENCY CHARACTERISTICS OF
DIFFRACTION GRATINGS

9.0. INTRODUCTION

Efficiency and its variation with wavelength and spectral order are important
characteristics of a diffraction grating. For a reflection grating, efficiency is
defined as the energy flow (power) of monochromatic light diffracted into the
order being measured, relative either to the energy flow of the incident light
(absolute efficiency) or to the energy flow of specular reflection from a polished
mirror substrate coated with the same material (relative efficiency). [Intensity
may substitute for energy flow in these definitions.] Efficiency is defined
similarly for transmission gratings, except that an uncoated substrate is used in
the measurement of relative efficiency.
High-efficiency gratings are desirable for several reasons. A grating with
high efficiency is more useful than one with lower efficiency in measuring weak
transition lines in optical spectra. A grating with high efficiency may allow the
reflectivity and transmissivity specifications for the other components in the
spectrometer to be relaxed. Moreover, higher diffracted energy may imply lower
instrumental stray light due to other diffracted orders, as the total energy flow for
a given wavelength leaving the grating is conserved (being equal to the energy
flow incident on it minus any scattering and absorption).
Control over the magnitude and variation of diffracted energy with
wavelength is called blazing, and it involves the manipulation of the micro-
geometry of the grating grooves. As early as 1874, Lord Rayleigh recognized
that the energy flow distribution (by wavelength) of a diffraction grating could
be altered by modifying the shape of the grating grooves.79 It was not until four
decades later that R.W. Wood showed this to be true when he ruled a grating on

79 J. W. Strutt, Lord Rayleigh, “On the manufacture and theory of diffraction gratings,”
Philos. Mag. 47, 193-205 (1874).

124

which he had controlled the groove shape, thereby producing the first deliber-
ately blazed diffraction grating.80
The choice of an optimal efficiency curve for a grating depends on the
specific application. For some cases the desired instrumental response is linear
with wavelength; that is, the ratio of intensity of light and the electronic signal
into which it is transformed is to be nearly constant across the spectrum. To
approach this as closely as possible, the spectral emissivity of the light source
and the spectral response of the detector should be considered, from which the
desired grating efficiency curve can be derived. Usually this requires peak
grating efficiency in the region of the spectrum where the detectors are least
sensitive; for example, a visible-light spectrometer using a silicon detector would
be much less sensitive in the blue than in the red, suggesting that the grating
itself be blazed to yield a peak efficiency in the blue.
A typical efficiency curve (a plot of absolute or relative diffracted efficiency
vs. diffracted wavelength λ) is shown in Figure 9-1. Usually such a curve shows

E

EP

λB

λ

Figure 9-1. A typical (simplified) efficiency curve. This curve shows the efficiency E of a
grating in a given spectral order m, measured vs. the diffracted wavelength λ. The peak
efficiency EP occurs at the blaze wavelength λB.

80 R. Wood, “The echellette grating for the infra-red,” Philos. Mag. 20 (series 6), 770-778
(1910).

125

a single maximum, at the peak wavelength (or blaze wavelength) λB. This curve
corresponds to a given diffraction order m; the peak of the curve decreases in
magnitude and shifts toward shorter wavelengths as |m| increases. The
efficiency curve also depends on the angles of use (i.e., the angles of incidence
and diffraction). Moreover, the curve depends on the groove spacing d (more
appropriately, on the dimensionless parameter λ/d) and the material with which
the grating is coated (for reflection gratings) or made (for transmission gratings).
In many instances the diffracted power depends on the polarization of the
incident light. P-plane or TE polarized light is polarized parallel to the grating
grooves, while S-plane or TM polarized light is polarized perpendicular to the
grating grooves (see Figure 9-2). For completely unpolarized incident light, the
efficiency curve will be exactly halfway between the P and S efficiency curves.

P

S

P
grating

S

Figure 9-2. S and P polarizations. The P polarization components of the incident and
diffracted beams are polarized parallel to the grating grooves; the S components are
polarized perpendicular to the P components. Both the S and P components are perpen-
dicular to the propagation directions.

Usually light from a single spectral order m is used in a spectroscopic
instrument, so a grating with ideal efficiency characteristics would diffract all of

126

the power incident on it into this order (for the wavelength range considered).
In practice, this is never true: the distribution of the power by the grating
depends in a complicated way on the groove spacing and profile, the spectral or-
der, the wavelength, and the grating material.
Anomalies are locations on an efficiency curve (efficiency plotted vs.
wavelength) at which the efficiency changes abruptly. First observed by R. W.
Wood, these sharp peaks and troughs in an efficiency curve are sometimes
referred to as Wood's anomalies. Anomalies are rarely observed in P polarization
efficiency curves, but they are often seen in S polarization curves (see Figure 9-
3). Anomalies are discussed in more detail in Section 9.13.

3 −2 2 −1

E

S

P
λ

Figure 9-3. Anomalies in the first order for a typical grating with triangular grooves.
The P efficiency curve (solid line) is smooth, but anomalies are evident in the S curve
(dashed line). The passing-off locations are identified by their spectral order at the top of
the figure.

127

9.1. GRATING EFFICIENCY AND GROOVE SHAPE

The maximum efficiency of a grating is typically obtained with a simple
smooth triangular groove profile, as shown in Figure 9-4, when the groove (or
blaze) angle θB is such that the specular reflection angle for the angle of inci-
dence is equal (in magnitude and opposite in sign) to the angle of diffraction (see
Section 2.8). Ideally, the groove facet should be flat with smooth straight edges,
and be generally free from irregularities on a scale comparable to a small fraction
(< 1/10) of the wavelength of light being diffracted.

Fraunhofer was well aware that the dis tribution of power among the various
diffraction orders depended on the shape of the individual grating grooves.
Wood, many decades later, was the first to achieve a degree of control over the
groove shape, thereby concentrating spectral energy into one angular region.
Wood's gratings were seen to light up, or blaze, when viewed at the correct
angle.

GN
FN

θB

α
β

θB
blaze arrow

Figure 9-4. Triangular groove geometry. The angles of incidence α and diffraction β are
shown in relation to the facet angle θΒ. GN is the grating normal and FN is the facet
normal. When the facet normal bisects the angle between the incident and diffracted rays,

128

the grating is used in the blaze condition. The blaze arrow (shown) points from GN to
FN.

9.2. EFFICIENCY CHARACTERISTICS FOR TRIANGULAR-GROOVE
GRATINGS

Gratings with triangular grooves can be generated by mechanical ruling, or
by blazing sinusoidal groove profiles by ion etching. The efficiency behavior of
gratings with triangular groove profiles may be divided into six families, de-
pending on the blaze angle:81

family blaze angle
very low blaze angle θB < 5°
low blaze angle 5° < θB < 10°
medium blaze angle 10° < θB < 18°
special low anomaly 18° < θB < 22°
high blaze angle 22° < θB < 38°
very high blaze angle θB > 38°

Very low blaze angle gratings (θB < 5°) exhibit efficiency behavior that is
almost perfectly scalar; that is, polarization effects are virtually nonexistent.
In this region, a simple picture of blazing is applicable, in which each groove
facet can be considered a simple flat mirror. The diffracted efficiency is greatest
for that wavelength that is diffracted by the grating in the same direction as it
would be reflected by the facets. This efficiency peak occurs in the m = 1 order
at λ/d = 2 sinθ (provided the angle between the incident and diffracted beams is
not excessive). At λB/2, where λB is the blaze wavelength, the diffracted
efficiency will be virtually zero (Figure 9-5) since for this wavelength the second-
order efficiency will be at its peak. Fifty-percent absolute efficiency is obtained
from roughly 0.7λB to 1.8λB.

81 E. G. Loewen, M. Nevière and D. Maystre, “Grating efficiency theory as it applies to
blazed and holographic gratings,” Appl. Opt. 16, 2711-2721 (1977).

129

Figure 9-5. First-order theoretical efficiency curve: 2° blaze angle and Littrow mounting
(2K = 0). Solid curve, S-plane; dashed curve, P-plane.

For low blaze angle gratings (5° < θB < 10°), polarization effects will occur
within their usable range (see Figure 9-6). In particular, a strong anomaly is seen
near λ/d = 2/3. Also observed is the theoretical S-plane theoretical efficiency
peak of 100% exactly at the nominal blaze, combined with a P-plane peak that is
lower and at a shorter wavelength. It is characteristic of all P-plane curves to
decrease monotonically from their peak toward zero as λ/d → 2, beyond which
diffraction is not possible (see Eq. (2-1)). Even though the wavelength band over
which 50% efficiency is attained in unpolarized light is from 0.67λB to 1.8λB,
gratings of this type (with 1200 groove per millimeter, for example) are widely
used, because they most effectively cover the wavelength range between 200
and 800 nm (in which most ultraviolet-visible (UV-Vis) spectrophotometers
operate).

130

Figure 9-6. Same as Figure 9-5, except 9° blaze angle.

A typical efficiency curve for a medium blaze angle grating (10° < θB < 18°)
is shown in Figure 9-7. As a reminder that for unpolarized light the efficiency is
simply the arithmetic average of the S- and P-plane efficiencies, such a curve is
shown in this figure only, to keep the other presentations simple.
The low-anomaly blaze angle region (18° < θB < 22°) is a special one.
Due to the fact that the strong anomaly that corresponds to the –1 and +2 orders
passing off (λ/d = 2/3) occurs just where these gratings have their peak
efficiency, this anomaly ends up being severely suppressed (Figure 9-8).
This property is quite well maintained over a large range of angular deviations
(the angle between the incident and diffracted beams), namely up to 2K = 25°, but
it depends on the grooves having an apex angle near 90°. The relatively low P-
plane efficiency of this family of blazed gratings holds the 50% efficiency band
from 0.7λB to 1.9λB.

131

Figure 9-7. Same as Figure 9-5, except 14° blaze angle. The curve for unpolarized light
(marked U) is also shown; it lies exactly halfway between the S and P curves.

Figure 9-8. Same as Figure 9-5, except 19° blaze angle.

132

High blaze angle gratings (22° < θB < 38°) are widely used, despite the
presence of a very strong anomaly in their efficiency curves (Figure 9-9). For
unpolarized light, the effect of this anomaly is greatly attenuated by its coinci-
dence with the P-plane peak. Another method for reducing anomalies for such
gratings is to use them at angular deviations 2K above 45°, although this in-
volves some sacrifice in efficiency and wavelength range. The 50% efficiency is
theoretically attainable in the Littrow configuration from 0.6λB to 2λB, but in
practice the long-wavelength end corresponds to such an extreme angle of
diffraction that instrumental difficulties arise.

Figure 9-9. Same as Figure 9-5, except 26° 45' blaze angle.

Theoretically, all gratings have a second high-efficiency peak in the S-plane
at angles corresponding to the complement of the blaze angle (90°–θB); in
practice, this peak is fully developed only on steeper groove-angle gratings, and
then only when the steep face of the groove is not too badly deformed by the
lateral plastic flow inherent in the diamond tool burnishing process. The strong
polarization observed at all high angles of diffraction limits the useable efficiency

133

in unpolarized light, but it makes such gratings very useful for tuning lasers,
especially molecular lasers. The groove spacing may be chosen so that the
lasing band corresponds to either the first or second of the S -plane high-
efficiency plateaus. The latter will give at least twice the dis persion (in fact the
maximum possible), as it is proportional to the tangent of the angle of diffraction
under the Littrow conditions typical of laser tuning.
Very-high blaze angle gratings (θB > 38°) are rarely used in the first order;
their efficiency curves are interesting only because of the high P-plane values
(Figure 9-10). In high orders they are often used in tuning dye lasers, where high

Figure 9-10. Same as Figure 9-5, except 46° blaze angle and 8° and 45° between the
incident and diffracted beams (shown as light and heavy lines, respectively).

dispersion is important and where tuning through several orders can cover a
wide spectral region with good efficiency. Efficiency curves for this family of
gratings are shown for two configurations. With an angular deviation of 2K = 8°,
the efficiency does not differ too much from Littrow; when 2K = 45°, the deep
groove results in sharp reductions in efficiency. Some of the missing energy
shows up in the zeroth order, but some of it can be absorbed by the grating.

134

9.3. EFFICIENCY CHARACTERISTICS FOR SINUSOIDAL-GROOVE
GRATINGS

A sinusoidal-groove grating can be obtained by the interferometric
(holographic) recording techniques described in Chapter 4. Sinusoidal gratings
have a somewhat different diffracted efficiency behavior than do triangular-
groove gratings, and are treated separately.
It is convenient to consider five domains of sinusoidal-groove gratings, 82
with progressively increasing modulation µ, where

h
µ= , (9-1)
d

h is the groove height and d is the groove spacing:

domain modulation
very low µ < 0.05
low 0.05 < µ < 0.15
medium 0.15 < µ < 0.25
high 0.25 < µ < 0.4
very high µ > 0.4

Very low modulation gratings (µ < 0.05) operate in the scalar domain,83
where the theoretical efficiency peak for sinusoidal grooves is only 34% (Figure
9-11). This figure may be readily scaled, and specification is a simple matter as
soon as it becomes clear that the peak wavelength always occurs at λB = 3.4h =
3.4µd. A blazed grating with an equivalent peak wavelength will require a groove
depth 1.7 times greater.

82 E. G. Loewen, M. Nevière and D. Maystre, "Grating efficiency theory as it applies to
83 E. G. Loewen, M. Nevière and D. Maystre, "On an asymptotic theory of diffraction
gratings used in the scalar domain," J. Opt. Soc. Am. 68, 496-502 (1978).

135

Low modulation gratings (0.05 < µ < 0.15) are quite useful in that they have
a low but rather flat efficiency over the range 0.35 < λ/d < 1.4 (Figure 9-12). This
figure includes not only the infinite conductivity values shown on all previous
ones, but includes the effects of finite conductivity by adding the curves for an
1800 g/mm aluminum surface. The most significant effect is in the behavior of the
anomaly, which is the typical result of the finite conductivity of real metals.

Figure 9-11. First-order theoretical efficiency curve: sinusoidal grating, µ = 0.05 and
Littrow mounting (2K = 0). Solid curve, S-plane; dashed curve, P-plane.

Figure 9-13 is a good example of a medium modulation grating (0.15 < µ <
0.25). It demonstrates an important aspect of such sinusoidal gratings, namely
that reasonable efficiency requirements confine first-order applications to values
of λ/d > 0.45, which makes them generally unsuitable for systems covering wide
spectral ranges.

136

Figure 9-12. First-order theoretical efficiency curve: sinusoidal grating, aluminum
coating, 1800 grooves per millimeter, µ = 0.14 and Littrow mounting. Solid curves, S-
plane; dashed curves, P -plane. For reference, the curves for a perfectly conducting
surface are shown as well (light curves).

Figure 9-13. Same as Figure 9-12, except µ = 0.22 and 8° between incident and diffracted
beams (2K = 8°).

137

Over this restricted region, however, efficiencies are comparable to those of tri-
angular-groove gratings, including the high degree of polarization. This figure
also demonstrates how a departure from Littrow to an angular deviation of 2K =
8° splits the anomaly into two branches, corresponding to the new locations of
the –1 and +2 order passing-off conditions.
High modulation gratings (0.25 < µ < 0.40), such as shown in Figure 9-14,
have the maximum useful first-order efficiencies of sinusoidal-groove gratings.
Provided they are restricted to the domain in which higher orders diffract (i.e., λ/d
> 0.65), their efficiencies are very similar to those of triangular-groove gratings
having similar groove depths (i.e., 26° < θB < 35°).

Figure 9-14. Same as Figure 9-12, except µ = 0.36.

Very-high modulation gratings (µ > 0.40), in common with equivalent
triangular-groove gratings, have little application in the first order due to their
relatively low efficiencies except perhaps over narrow wavelengths ranges and
for grazing incidence applications.

138

9.4. THE EFFECTS OF FINITE CONDUCTIVITY

For metal-coated reflection gratings, the finite conductivity of the metal is of
little importance for wavelengths of diffraction above 4 µm, but the complex
nature of the dielectric constant and the index of refraction begin to effect effi-
ciency behavior noticeably for wavelengths below 1 µm, and progressively more
so as the wavelength decreases. In the P-plane, the effect is a simple reduction
in efficiency, in direct proportion to the reflectance. In the S-plane, the effect is
more complicated, especially for deeper grooves and shorter wavelengths.
Figure 9-15 shows the first-order efficiency curve for a widely-used grating:
θ
1200 g/mm, triangular grooves, medium blaze angle ( B = 10°), coated with

Figure 9-15. First-order theoretical efficiency curve: triangular-groove grating,
aluminum coating, 1200 grooves per millimeter, 10° blaze angle and 2K = 8°. Solid
curves, S-plane; dashed curves, P-plane. For reference, the c urves for a perfectly
conducting surface are shown as well (light curves).

139

aluminum and used with an angular deviation of 8°. The finite conductivity of
the metal causes a reduction in efficiency; also, severe modification of the
anomaly is apparent. It is typical that the anomaly is broadened and shifted
toward a longer wavelength compared with the infinite conductivity curve. Even
for an angular deviation as small as 8°, the single anomaly in the figure is sepa-
rated into a double anomaly.
For sinusoidal gratings, the situation is shown in Figures 9-12 and 9-14.
Figure 9-13 is interesting in that it clearly shows a series of new anomalies that
are traceable to the role of aluminum.
With scalar domain gratings (either θB < 5° or µ < 0.10), the role of finite
conductivity is generally (but not always) to reduce the efficiency by the ratio of
surface reflectance.84

9.5. DISTRIBUTION OF ENERGY BY DIFFRACTION ORDER

Gratings are most often used in higher diffraction orders to extend the
spectral range of a single grating to shorter wavelengths than can be covered in
lower orders. For blazed gratings, the second-order peak will be at one-half the
wavelength of the nominal first-order peak, the third-order peak at one-third, etc.
Since the ratio λ/d will be progressively smaller as m| increases, polarization
|
effects will become less significant; anomalies are usually negligible in diffraction
orders for which |m| > 2. Figures 9-16 and 9-17 show the second- and third-order
theoretical Littrow efficiencies, respectively, for a blazed grating with θB = 26°45';
they are plotted as a function of mλ/d in order to demonstrate the proper angular
ranges of use. These curves should be compared with Figure 9-9 for correspond-
ing first-order behavior.
For gratings with sinusoidally shaped grooves, higher orders can also be
used, but if efficiency is important, the choice is likely to be a finer pitch first-
order grating instead. When groove modulations are very low (so that the
grating is used in the scalar domain), the second-order efficiency curve looks
similar to Figure 9-18, except that the theoretical peak value is about 23% (instead
of 34%) and occurs at a wavelength 0.6 times that of the first-order peak, which

84 E. G. Loewen, M. Nevière and D. Maystre, "On an asymptotic theory of diffraction
gratings used in the scalar domain," J. Opt. Soc. Am. 68, 496-502 (1978).

140

corresponds to 2.05h (instead of 3.41h), where h is the groove depth. Successive
higher-order curves for gratings with sinusoidal grooves are not only closer
together, but drop off more sharply with order than for gratings

Figure 9-16. Second-order theoretical efficiency curve: 26° 45' blaze angle and Littrow
mounting. Solid curve, S-plane; dashed curve, P-plane.

141

Figure 9-17. Same as Figure 9-16, except third order.

Figure 9-18. Second-order theoretical efficiency curve: sinusoidal grating, µ = 0.36 and
Littrow mounting. Solid curve, S-plane; dashed curve, P-plane.

142

Figure 9-19. Same as Figure 9-18, except third order.

with triangular grooves. For sufficiently deeply modulated sinusoidal grooves,
the second order can often be used effectively, though (as Figure 9-18 shows)
polarization effects are relatively strong. The corresponding third-order theo-
retical curve is shown in Figure 9-19.

9.6. USEFUL WAVELENGTH RANGE

The laws governing diffracted efficiency are quite complicated, but a very
rough rule of thumb can be used to estimate the useful range of wavelengths
available on either side of the blaze (peak) wavelength λB for triangular-groove
gratings.
For coarse gratings (for which d = 2λ), the efficiency in the first diffraction
order is roughly half its maximum (which is at λB) at 2λB/3 and 3λB/2. Curves of
similar shape are obtained in the second and third orders, but the efficiencies are
typically 20% less everywhere, as compared with the first order.
Grating of fine pitch (d ˜ λ) have a somewhat lower peak efficiency than do
coarse gratings, though the useful wavelength range is greater.

143

9.7. BLAZING OF RULED TRANSMISSION GRATINGS

Because they have no metallic overcoating, triangular-groove transmission
gratings display far simpler efficiency characteristics than do their ruled
counterparts. In particular, transmission gratings have efficiency curves almost
completely free of polarization effects.
The peak wavelength generally occurs when the direction of refraction of
the incident beam through a groove (thought of as a small prism) equals the
direction dictated by the grating equation. [This is in direct analogy with the
model of reflection grating blazing in that the grooves are thought of as tiny
mirrors; see Section 2.8.] Due to the index of refraction of the grating, though,
the groove angle exceeds the blaze angle for a transmis sion grating.
See Section 12.2 for more information on transmission gratings.

9.8. BLAZING OF HOLOGRAPHIC REFLECTION GRATINGS

Although holographic gratings generally do not have the triangular groove
profile found in ruled gratings, holographic gratings may still exhibit blazing
characteristics (see, for example, Figure 9-18). For this reason it is not correct to
say that all blazed gratings have triangular profiles, or that all blazed gratings are
ruled gratings – blazing refers to high diffraction efficiency, regardless of the
profile of the grooves or the method used to generate them.
This being said, there are some cases in which it would be preferable for a
holographic grating to have a triangular groove profile rather than a sinusoidal
profile. The method of using standing waves to record the grooves (see Section
4.2.1) was developed by Sheridon85 and improved by Hutley.86
Another useful technique for rendering sinusoidal groove profiles more
nearly triangular is ion etching. By bombarding a surface with energetic ions,
the material can be removed (etched) by an amount per unit time dependent on
the angle between the beam and the local surface normal. The etching of a

85 N. K. Sheridon, “Production of blazed holograms,” Appl. Phys. Lett. 12, 316-318 (1968).
86 M. C. Hutley, “Blazed interference diffraction gratings for the ultraviolet,” Opt. Acta 22,
1-13 (1975); M. C. Hutley and W. R. Hunter, “Variation of blaze of concave diffraction
gratings,” Appl. Opt. 20, 245-250 (1981).

144

sinusoidal profile by an ion beam provides a continuously varying angle
between the ion beam and the surface normal, which preferentially removes
material at some parts of the profile while leaving other parts hardly etched. The
surface evolves toward a triangular groove profile as the ions bombard it.87
Other method for generating blazed groove profiles have been developed,88
but the Sheridon method and the method of ion etching are those most
commonly used for commercially-available gratings.

9.9. OVERCOATING OF REFLECTION GRATINGS

The metallic coating on a reflection grating is evaporated onto the substrate.
This produces a surface whose reflectivity is higher than that of the same metal
electroplated onto the grating surface. The thickness of the metallic layer is
chosen to enhance the diffraction efficiency throughout the spectral region of
interest.
Most standard reflection gratings are furnished with an aluminum (Al)
reflecting surface. While no other metal has more general application, there are a
number of special situations where alternative surfaces or coatings are recom-
mended. Gratings coated with gold (Au) and silver (Ag) have been used for
some time for higher reflectivity in certain spectral regions, as have more exotic
materials such as iridium (Ir), osmium (Os) and platinum (Pt).89
The reflectivity of aluminum drops rather sharply for wavelengths below 170
nm. While freshly deposited, fast-fired pure aluminum in high vacuum maintains
its reflectivity to wavelengths shorter than 100 nm, the thin layer of oxide that

87 Y. Aoyagi and S. Namba, Japan. J. Appl. Phys. 15, 721 (1976); L. F. Johnson,
“Evolution of gratin g profiles under ion-beam erosion,” Appl. Opt. 18, 2559-2574 (1979);
C. Palmer, J. Olson and M. M. Dunn, “Blazed diffraction gratings obtained by ion-milling
sinusoidal photoresist gratings,” Proc. SPIE 2622, 112-121 (1995).
88 M. B. Fleming and M. C. Hutley, “Blazed diffractive optics,” Appl. Opt. 36, 4635-4643
(1997).
89 E.g., J. M. Bennett and E. J. Ashley, “Infrared reflectance and emittance of silver and
goldevaporated in ultrahigh vacuum,” Appl. Opt. 4, 221-224 (1965); R. F. Malina and W.
Cash, “Extreme ultraviolet reflection efficiencies of diamond-turned aluminum, polished
nickel, and evaporated gold surfaces,” Appl. Opt. 17, 3309-3313 (1978); M. R. Torr,
“Osmium coated diffraction grating in the Space Shuttle environment: performance,” Appl.
Opt. 24, 2959-2961 (1985).

145

grows on the aluminum (upon introduction of the coating to atmosphere) will
cause a reduction in efficiency below about 250 nm.90 Fortunately, a method
borrowed from mirror technology makes it possible to preserve the reflectivity of
aluminum to shorter wavelengths.91 The process involves overcoating the
grating with a thin layer of fast-fired aluminum, which is followed immediately by
a coating of magnesium fluoride (MgF2) approximately 25 nm thick. The main
purpose of the MgF2 coating is to protect the aluminum from oxidation. The
advantage of this coating is especially marked in the region between 120 and 200
nm. While reflectivity drops off sharply below this region, it remains higher than
that of gold and comparable to that of platinum, the most commonly used
alternative materials, down to 70 nm.
Overcoating gratings so that their surfaces are coated with two layers of
different metals sometimes leads to a change in diffraction efficiency over time.
Hunter et al.92 have found the cause of this change to be intermetallic diffusion.
For example, they measured a drastic decrease (over time) in efficiency at 122 nm
for gratings coated in Au and then overcoated in Al + MgF2; this decrease was
attributed to the formation of intermetallic compounds, primarily AuAl2 and
Au 2Al. Placing a suitable dielectric layer such as SiO between the two metallic
layers prevents this diffusion.
As mentioned elsewhere, fingerprints are a danger to aluminized optics. It is
possible to overcoat such optics, both gratings and mirrors, with dielectrics like
MgF2, to prevent finger acids from attacking the aluminum. These MgF2
coatings cannot be baked, as is customary for glass optics, and therefore must
not be cleaned with water. Spectrographic-grade organic solvents are the only
recommended cleaning agents, and they should be used sparingly and with care.
Single-layer and m ultilayer dielectric overcoatings, which are so useful in
enhancing plane mirror surfaces, are less generally applicable to diffraction
gratings, since in certain circumstances these coatings lead to complex guided

90 R. P. Madden, L. R. Canfield and G. Hass, “On the vacuum-ultraviolet reflectance of
evaporated aluminum before and during oxidation,” J. Opt. Soc. Am. 53, 620-625 (1963).
91G. Hass and R. Tousey, “Reflecting coatings for the extreme ultraviolet,” J. Opt. Soc. Am.
49, 593-602 (1959).
92 W. R. Hunter, T. L. Mikes and G. Hass, "Deterioration of Reflecting Coatings by
Intermetallic Diffusion," Appl. Opt. 11, 1594-1597 (1972).

146

wave effects.93 For wavelengths below 30 nm, in which grazing angles of
incidence and diffraction are common, multilayer coatings can enhance efficiency
considerably.94

9.11. THE RECIPROCITY THEOREM

A useful property of grating efficiency is that embodied in the reciprocity
theorem,95 which states that (under certain conditions) reversing the direction of
the beam diffracted by a grating will leave its diffraction efficiency unchanged.
Stated another way, the reciprocity theorem says that the grating efficiency for
wavelength λ in order m is unchanged under the transformation α ↔ β. This
equivalence follows from the periodic nature of the grating, and is strictly true for
perfectly-conducting gratings and lossless dielectric gratings illuminated by an
incident plane wave.96
Three consequences of the reciprocity theorem should be noted:97
• The zeroth-order efficiency E(λ,0) is a symmetric function (of angle α)
about α = 0.
• Rotation of the grating groove profile through 180° (while keeping α
constant) does not effect E(λ,0); moreover, if only two diffraction orders
are propagating (say, m = 0 and m = 1), the efficiency E(λ,1) will be
unchanged as well.
• The efficiency E(λ,m) for a given diffraction order m is a symmetric
function of vs. sinα about the Littrow condition (α = β).98

93 M. C. Hutley, J. F. Verrill and R. C. McPhedran, “The effect of a dielectric layer on the
diffraction anomalies of an optical grating,” Opt. Commun 11, 207-209 (1974).
94 J. C. Rife, W. R. Hunter, T. W. Barbee, Jr., and R. G. Cruddace, "Multilayer-coated blazed
grating performance in the soft x-ray region," Appl. Opt. 28, 1984 (1989).
95 R. Petit, “A tutorial introduction," in Electromagnetic Theory of Gratings, R. Petit, ed.
(Springer-Verlag, New York, 1980), p. 12.
96E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New
York, 1997), p. 38.
97 R. C. McPhedran and M. D. Waterworth, “A theoretical demonstration of properties of
grating anomalies (S-polarization),” Opt. Acta 19, 877-892 (1972).
York, 1997), p. 38..

147

9.12. CONSERVATION OF ENERGY

The principle of conservation of energy requires that all of the energy
incident on a diffraction grating be accounted for; this can be represented
mathematically (in terms of intensities), considering a single wavelength λ, as

Iin = Iout = ∑I diff (m) + Iabsorbed + Iscattered , (for a single λ) (9-2)
m

where the summation is over all diffraction orders m that propagate (i.e., for a
reflection grating, all orders for which the diffraction angle β(m) satisfies the
inequalities –90º ≤ β(m) ≤ +90º). Here the terms Iabsorbed and Iscattered are the
“losses” due to absorption of energy by the grating and by scattering,
respectively.
It is important to recognize that Eq. (9-2) holds only for the case in which the
grating and the incident beam are fixed in space, and each diffraction order is
diffracted through a unique diffraction angle given by Eq. (2-4) with constant α.
Eq. (9-2) does not hold true when the intensity of each diffraction order m is
measured in the Littrow configuration, since in this case the incidence angle α is
changed for each order. Care must be taken when adding intensities (or
efficiencies) in several orders for a single wavelength: the sum of these
intensities is not conserved according to Eq. (9-2) unless the grating and incident
beam remain fixed while a detector is moved (in angle) from one order to the next
to take the intensity measurements.
Eq. (9-2) can be used to advantage by designing an optical system for which
only two diffraction orders propagate: order m = 1 (or m = –1) and order m = 0
(which always exists: |α| < 90º implies |β(0)| < 90º, since β(0) = –? ). This case
α
?
requires a small groove spacing d and an incidence angle α such that the
diffraction angle for m = ±2 and for the other first order pass beyond 90º: from Eq.
(2-1) this requires

mλ
|sinβ| = − sin α > 1 (9-3)
d

for m = 2, m = –2 and m = –1 (assuming that m = 1 is the order chosen to
propagate). For such a system, Eq. (9-2) simplifies to become

148

Iin = Iout = Idiff(0) + Idiff(1) + Iabsorbed + Iscattered , (for a single λ) (9-4)

where the term Idiff(0) corresponds to the reflected intensity and Idiff(1)
corresponds to the intensity diffracted into the m = 1 order. Choosing a groove
profile that reduces the reflected intensity Idiff(0) will thereby increase the
diffracted intensity Idiff(1).
Eq. (9-2) is generally useful in measuring grating efficiency, but in the
presence of anomalies (see below) they can lead to considerable inaccuracies.

9.13. GRATING ANOMALIES

In 1902. R. W. Wood observed that the intensity of light diffracted by a
grating generally changed slowly as the wavelength was varied, but occasionally
a sharp change in intensity was observed at certain wavelengths.99 Called
anomalies, these abrupt changes in the grating efficiency curve were later
categorized into two groups: Rayleigh anomalies and resonance anomalies. 100

9.13.1. Rayleigh anomalies

Lord Rayleigh predicted the spectral locations where a certain set of
anomalies would be found: he suggested that these anomalies occur when light
of a given wavelength λ' and spectral order m' is diffracted at |β| = 90° from the
grating normal (i.e., it becomes an evanescent wave, passing over the grating
horizon). For wavelengths λ < λ', |β| < 90°, so propagation is possible in order m'
(and all lower orders), but for λ > λ' no propagation is possible in order m' (but it
is still possible in lower orders). Thus there is a discontinuity in the diffracted
power vs. wavelength in order m' at wavelength λ, and the power that would
diffract into this order for λ > λ' is redistributed among the other propagating

99 R. W. Wood, “On the remarkable case of uneven distribution of light in a diffraction
grating spectrum,” Philos. Mag. 4, 396-402 (1902); R. W. Wood, “Anomalous diffraction
gratings,” Phys. Rev. 48, 928-936 (1935); J. E. Stewart and W. S. Gallaway, “Diffraction
anomalies in grating spectrophotometers,” Appl. Opt. 1, 421-430 (1962).
100A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,”
Appl. Opt. 4, 1275-1297 (1965).

149

orders. This causes abrupt changes in the power diffracted into these other
orders.
These Rayleigh anomalies, which arise from the abrupt redistribution of
energy when a diffracted order changes from propagating (|β| < 90°) to
evanescent (|β| > 90°), or vice versa, are also called threshold anomalies.101

9.13.2. Resonance anomalies

The second class of anomalies, which are usually much more noticeable than
Rayleigh anomalies, are caused by resonance phenomena,102 the m well- ost
known of which are surface excitation effects.103 At the interface between
a dielectric and a metal, there are specific conditions under which a charge
density oscillation (“electron wave”) can be supported, which carries light
intensity away from the incident beam and therefore decreases the diffraction
efficiency of the grating. The efficiency curve would show a sharp drop in
intensity at the corresponding conditions (see Figure 9-20).

E

λ

York, 1997), ch. 8.
102U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on
metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. 31, 213-222 (1941).
103 R. H. Ritchie, E. T. Arakawa , J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance
effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).

150

Figure 9 -20. A typical (simplified) efficiency curve showing a sharp drop where the
conditions are met for surface plasmon resonance. A narrow spectral region is shown;
the efficiency curve would appear to increase monotonically if the resonance condition
were not met.

For a resonance anomaly to exist, a resonance condition must be met – this
places restrictions on the wavelengths (and incidence angles) that will exhibit
resonance effects for a given groove profile and refractive indices. This results
from the fact that in this phenomenon – the surface plasmon resonance (SPR)
effect104 – the electromagnetic field that propagates along the metal-dielectric
interface extends into each medium, so the characteristics of this propagating
wave depend on the material conditions near the interface. This useful feature of
SPR has led to its use in a number of sensing applications,105 such as
biosensing106 and gas sensing.107 SPR can also be used to characterize the
surface profile of the grating itself, especially by probing the diffraction effects
due to higher harmonics in the periodic structure on the surface of the grating.108
While diffraction gratings generally do not convert incident P-polarized light
to S-polarized light (or vice versa) upon diffraction, it has recently been observed
that such polarization conversion can occur if the grating is not illuminated in
the principal plane (.e., ε ≠ 0 in Eq. (2-3)).109 In this case, called conical
i
diffraction (see Section 2.1), resonance effects can lead to a strong polarization

104 R. H. Ritchie, E. T. Arakawa, J. J. Cowan and R. N. Hamm, “Surface-plasmon resonance
effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
105 J. Homola, S. S. Yee and G. Gauglitz, “Surface plasmon resonance sensors: review,”
Sensors and Actuators B 54, 3-15 (1999);
106 F. Caruso, M. J. Jory, G. W. Bradberry, J. R. Sambles and D. N. Furlong, “Acousto-optic
surface-plasmon-resonance measurements of thin films on gold,” J. Appl. Phys. 83, 5 (1983);
D. C. Cullen, R. G. W. Brown and R. C. Lowe, “Detection of immunocomplex formation via
surface plasmon resonance on gold-coated diffraction gratings,” Biosensors 3, 211 (1987); J.
M. Brockman and S. M. Fernández, “Grating-coupled surface plasmon resonance for rapid,
label-free, array-based sensing,” American Laboratory, 37-40 (June 2001).
107M. J. Jory, P. S. Cann and J. R. Sambles, “Surface-plasmon-polariton studies of 18-crown-
6 metal-free phthalocyanide,” J. Phys. D: Appl. Phys., 27, 169-174 (1994)
108 E. L. Wood, J. R. Sambles, N. P. Cotter and S. C. Kitson, “Diffraction grating
characterization using multiple-wavelength excitation of surface plasmon polaritons,” J.
Mod. Opt. 42, 1343-1349 (1995).
109 G. P. Bryan-Brown, J. R. Sambles and M. C. Hutley, “ Polarisation conversion through the
excitation of surface plasmons on a metallic grating”, J. Mod. Opt. 37, 1227-1232 (1990);
S. J. Elston, G. P. Bryan-Brown and J. R. Sambles, “Polarization conversion from diffraction
gratings,” Phys. Rev. B 44, 6393-6400 (1991).

151

conversion peak (e.g., a sharp trough in the S-polarized efficiency curve
coincident with a sharp peak in the P-polarized efficiency curve).

9.14. GRATING EFFICIENCY CALCULATIONS

Several techniques have been developed to calculate grating efficiencies,
most of which have two characteristics in common: they employ Maxwell’s
equations whose boundary conditions are applied at the corrugated grating
surface, and their difficulty in implementation varies in rough proportion to their
accuracy. In this section only a brief mention of these techniques is provided –
more details may be found in Petit 110 , Maystre 111 , and Loewen and Popov112 .
Grating efficiency calculations start with a description of the physical
situation: an electromagnetic wave is incident upon a corrugated surface, the
periodicity of which allows for a multiplicity of diffracted waves (each in
a different direction, corresponding to a unique diffraction order as described in
Chapter 2). Efficiency calculations seek to determine the distribution of the
incident energy into each of the diffraction orders.
Scalar theories of grating efficiency lead to accurate results in certain cases,
such as when the wavelength is much smaller than the groove spacing (d << λ);
the vectorial nature of optical radiation (manifest in the property of polarization)
is not taken into account in this formalism.
Vector or electromagnetic theories can be grouped into two categories.
Differential methods start from the differential form of Maxwell’s equations for
TE (P) and TM (S) polarization states, whereas integral methods start from the
integral form of these equations. Each of these categories contains a number of
methods, none of which is claimed to cover all circumstances.
Both differential and integral methods have been developed and studied
extensively, and both have been implemented numerically and thoroughly tested

110 Petit, R., ed., Electromagnetic Theory of Gratings, vol. 22 in “Topics in Current
Physics” series (Springer-Verlag, 1980).
111D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics, vol.
XXI, E. Wolf, ed. (Elsevier, 1984), pp. 2-67.
112E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Marcel Dekker, Inc.
(New York, 1987), ch. 10.

152

against a wide variety of experimental data. Some of these numerical
implementations are commercially available.

153

10. STRAY LIGHT CHARACTERISTICS OF
GRATINGS AND GRATING SYSTEMS

10.0. INTRODUCTION

An annoying characteristic of all optical surfaces is their ability to scatter
light. This undesirable light is often referred to as stray radiant energy (SRE).
When this light reaches the detector of an instrument designed to measure
an optical signal, the SRE contributes to the noise of the system and thereby
reduces the signal-to-noise ratio (SNR).
The terminology of SRE in grating systems is not standard, so for clarity we
refer to unwanted light arising from imperfections in the grating itself as
scattered light or grating scatter, and unwanted light reaching the detector of a
grating-based instrument as instrumental stray light or simply stray light. [We
choose this definition of scattered light so that it will vanish for a perfect grating;
we will see below that this does not generally cause the instrumental stray light
to vanish as well.] With these definitions, some scattered light will also be stray
light (if it reaches the detector); moreover, some stray light will not be scattered
light (since it will not have arisen from imperfections in the grating).113

113 This definition of stray light is not universal; while it is in agreement with K. D. Mielenz,
V. R. Weidner and R. W. Burke, “Heterochromic stray light in UV absorption spectrometry:
a new test method,” Appl. Opt. 21, 3354-3356 (1982), it is not in agreement with the
ASTM, which defines the quantity stray radiant power as being composed of wavelengths
outside the spectral bandwidth of the monochromator (ASTM standard E387, “Standard Test
Method for Estimating Stray Radiant Power Ratio of Dispersive Spectrophotometers by the
Opaque Filter Method,” 2004). [See also W. Kaye, “Stray light ratio measurements,” Anal.
Chem. 53, 2201-2206 (1981).] The ASTM definition does not account for light of the
correct wavelength that reaches the detector, but which does not follow the desired optical
path.

155

10.1. GRATING SCATTER

Of the radiation incident on the surface of a reflection grating, some will be
diffracted according to Eq. (2-1) and some will be absorbed by the grating itself.
The remainder is scattered light, which may arise from several factors, including
imperfections in the shape and spacing of the grooves and roughness on the
surface of the grating. An excellent analysis of grating scatter can be found in
Sharpe & Irish,114 and measured grating scatter was compared to predictions of
Beckmann’s scalar theory and Rayleigh’s vector theory by Marx et al.115
Two types of scattered light are often distinguished. Diffuse scattered light
is scattered into the hemisphere in front of the grating surface. It is due mainly to
grating surface microroughness. It is the primary cause of scattered light in
holographic gratings. For monochromatic light of wavelength λ incident on a
grating, the intensity of diffuse scattered light is higher near the diffraction
orders of λ than between these orders.† In-plane scatter is unwanted energy in
the dispersion plane. Due primarily to random variations in groove spacing or
groove depth, its intensity is generally higher than the background diffuse
scattered light.
Consider a diffraction grating consisting of a pattern of grooves whose
nominal spacing is d. We have defined scattered light as all light leaving the
grating due to its imperfections; this is equivalent to the operational definition
that scattered light is all light energy leaving the surface of a diffraction grating
that does not follow the grating equation for the nominal groove spacing d,

mλ = d (sin α + sin β ) . (2-1)

This is analogous to the concept of scattered light for a mirror, which may be
defined the light leaving its surface that does not follow the law of reflection for
the nominal mirror surface.

114 M. R. Sharpe and D. Irish, “Stray light in diffraction grating monochromators,” Opt. Acta
25, 861-893 (1978).
115 E. Marx, T. A. Germer, T. V. Vorburger and B. C. Park, “Angular distribution of light
scattered from a sinusoidal grating,” Appl. Opt. 39, 4473-4485 (2000).
† This observation has lead some to observe that grating scatter is “blazed”.

156

This definition of grating scatter – as being caused by imperfections in the
grating – does not consider light diffracted into different orders {m} as scattered
light. That is, diffraction into multiple orders is not an artifact of grating
imperfections, but a direct consequence of the phenomenon of constructive
interference on which the grating operates (see Section 2.1). However, light
diffracted into other orders can contribute to instrumental stray light (see Section
10.2 below).

10.1.1. Surface irregularities in the grating coating

A grating surface that is rough on the scale of the incident wavelength (or
somewhat smaller) will cause a small portion of the incident light to be scattered
diffusely (i.e., into all directions) with intensity that varies approximately with the
inverse fourth power of the wavelength.116 Surface roughness is due in part to
the surface quality of the master grating, either ruled or holographic, since the
metal coating of a ruled master, and the photoresist coating of a holographic
master, are not perfectly smooth. Moreover, the addition of a reflective coating
may contribute to the surface roughness due to the coating’s granular structure.

10.1.2. Dust, scratches & pinholes on the surface of the grating

Each speck of dust, tiny scratch, and pinhole void in the surface o f
a reflection grating will serve as a “scatter center” and cause diffuse scatter.
This is evident upon inspecting a grating under a bright light: dust, scratches,
pinholes etc. are easily visible and bright when looked at from many different
angles (hence the diffuse nature of their scattered light).

10.1.3. Irregularities in the position of the grooves

The presence of spatial frequencies in the groove pattern other than that of
the groove spacing d will give rise to constructive interference of the diffracted

116 M. R. Sharpe and D. Irish, "Stray light in diffraction grating monochromators," Opt. Acta
25, 861-893 (1978).

157

light at angles that do not follow the grating equation for the nominal groove
spacing d, but for different spacings d’ ≠ d.
Until the recent advent of interferometric control of ruling engines,
mechanically ruled gratings exhibited secondary spectra, called ghosts, due to
slight deviations in the placement of its grooves compared with their ideal
locations. Ghosts that are close to and symmetric about the parent diffracted line
are called Rowland ghosts, and are due to longer-term periodicities (on the order
of millimeters), whereas Lyman ghosts are farther from the parent line and are
caused by short -term periodicities (on the order of the groove spacing).
Both Rowland and Lyman ghosts appear at angular positions given by the
grating equation, but for spatial frequencies other than 1/d (see Section 11.1).
The presence of random (rather than periodic) irregularities in groove
placement leads to a faint background between orders, rather than sharp ghosts,
whose intensity varies roughly with the inverse square of the wavelength.117
This background is called grass because it resembles blades of grass when
observed using green Hg light.
Ghosts and grass are in-plane effects (that is, they are seen in and near the
dispersion plane) and lead to interorder scatter. Holographic gratings, whose
grooves are formed simultaneously, do not exhibit measurable groove placement
irregularities if made properly and therefore generally exhibit lower levels of
interorder scatter. With the use of sophisticated interferometric control systems
on modern ruling engines, though, this advantage has been reduced when
holographic gratings are compared with recently-ruled gratings.

10.1.4. Irregularities in the depth of the grooves

A distribution of groove depths about the nominal groove depth is a natural
consequence of the burnishing process and the elasticity of metal coatings (in
the case of ruled master gratings) or to local variations in exposure intensities
and developing conditions (in the case of holographic master gratings). These
variations have been shown to generate a continuous distribution of scattered
light that varies with the inverse cube of the wavelength.118

117 Ibid.
118 Ibid.

158

10.1.5. Spurious fringe patterns due to the recording system

For holographic gratings, care must be taken to suppress all unwanted
reflections and scattered light when producing the master grating. Light from
optical mounts, for example, may reach the master grating substrate during
exposure and leave a weak fringe pattern that causes scattered light when the
grating is coated with a metal and illuminated.119 A scratch on a lens in a
recording beam can create a “bulls -eye” pattern on the master grating that serves
as a scatter center for every replica made from that master. Recording the
holographic master in incoherent light can reduce the stray light attributable to
recording artifacts.120

10.1.6. The perfect grating

From the perspective of scattered light, a perfect grating would have
a pattern of perfectly placed grooves (no variation in spacing from any groove to
the next, and no additional pattern to the grooves leading to spacings d’ ≠ d),
each of the proper depth (no variation), and the surface irregularities on the
grooves would be so much smaller than the wavelength of incident light that
these irregularities would have no effect on the diffracted light. Moreover, this
perfect grating would have no scratches, digs, blemishes or other visible surface
features, and (if holographic) would contain no holographic artifacts of the
recording optical system. In this ideal case, we might be forgiven in thinking that
all light incident on the grating would leave according to the grating equation (2-
1) for the nominal groove spacing d.
A general expression for the light intensity from a perfect grating is given by
Sharpe and Irish121 as

119 M. C. Hutley, Diffraction Gratings, Academic Press (New York, 1970), p. 107.
120 M. C. Hutley, “Improvements in or relating to the formation of photographic records,”
UK Patent no. 1384281 (1975); E. Sokolova, B. Kruizinga and I. Golubenko, “Recording of
concave diffraction gratings in a two -step process using spatially incoherent light,” Opt. Eng.
43, 2613-2622 (2004).
121 M. R. Sharpe and D. Irish, "Stray light in diffraction grating monochromators," Opt. Acta
25, 861-893 (1978).

159

 λ - λB 
sinc2 
 λ   −1
( )
S λ, λ =
B
Nλ

 λB  
 1 − cos 2πλ



 (10-1)
sinc 1−   λ 

2
 λ 

where λ is the illumination wavelength, λ is the monochromator setting (which
determines the orientation of the grating: it is not a wavelength), λB is the blaze
wavelength, B is the spectral bandpass of the instrument, and N is the number of
grooves under illumination. We see that this equation is generally non-zero, so
we must abandon any hope that a perfect grating will have provide no radiant
flux anywhere except in its diffraction orders.

10.2. INSTRUMENTAL STRAY LIGHT

Consider a spectrometer aligned so that the detector records the analytical
wavelength λ in spectral order m. Our definition of instrumental stray light leads
to its operational definition as light of either the wrong wavelength λ′ ≠ λ or the
wrong spectral order m′ ≠ m that reaches the detector; this is generally a problem
because most detectors are not wavelength-selective and cannot distinguish
between light of wavelength λ and light of wavelength λ′ ≠ λ. Also included in
our definition of stray light is any light that reaches the detector that does not
follow the optical path for which the system was designed, even if this light is of
wavelength λ and diffraction order m.
Instrumental stray light can be attributed to a number of factors.

10.2.1. Grating scatter

Light scattered by the grating, as discussed in Section 10.1 above, may
reach the detector and contribute to instrumental stray light. This type of stray
light is absent for a “perfect” grating.

160

10.2.2. Other diffraction orders from the grating

Light of the analytical wavelength λ is not only diffracted into order m, but
into any other orders that propagate. The zero order, which always propagates
but is almost always of no value in the instrument, is particularly troublesome.
The other diffracted beams are not oriented toward the detector by the grating,
but if these beams are reflected by a wall, a mount or another optical component,
or if these beams scatter off any interior surfaces in the instrument, some fraction
of their intensity may reach the detector and contribute to instrumental stray
light. This type of stray light is not absent even for a perfect grating, and
requires proper instrumental design (e.g., baffles, light traps, order-sorting filters
etc.) to reduce.

10.2.3. Overfilling optical surfaces

Fraunhofer diffraction from the illuminated edges of optical surfaces can be a
significant cause of instrumental stray light. All optics in the path should be
underfilled (that is, the illuminated area on the surface of each optic should fall
within the edges of the optic), with masks or other apertures if necessary. Verrill
has suggested that the intensity in the incident beam fall off (from the center)
according to a Gaussian function, to avoid an abrupt cut-off of intensity at the
edge of the beam. 122
Another important contributor to the instrumental stray light in some optical
systems is the illumination of optical components downstream from the grating
by light of wavelengths in the same diffraction order near the analytical
wavelength λ (i.e., the wavelength for which the monochromator is tuned). For
example, in a Czerny-Turner monochromator (see Figure 6-1), the instrument may
be designed so that light of wavelength λ underfills the concave mirror after the
grating, but light of wavelength λ±∆λ will diffract at slightly different angles and
may impinge upon the edges of the grating: these rays will scatter from these
edges and may reach the detector.123

122 J. F. Verrill, “The specification and measurement of scattered light from diffraction
gratings,” Opt. Acta 25, 531-547 (1978).
123S. Brown and A. W. S. Tarrant, “Scattered light in monochromators,” Opt. Acta 25,
1175-1186 (1978).

161

10.2.4. Direct reflections from other surfaces

The dispersive quality of diffraction gratings causes each wavelength
incident on it to be diffracted into a different set of directions (according to the
grating equation), which in turn will illuminate the interior of the optical
system. 124 Light in another order m′ ≠ m or at another wavelength λ′ ≠ λ for
which m′λ′ ≠ mλ will not be diffracted toward the exit slit, but as in Section 10.2.2,
this light may be reflected or scattered by other optical components, mounts or
interior walls and directed toward the exit slit.
For certain wavelengths, light may reflect from another surface toward the
grating and be rediffracted to the detector (called multiply diffracted light).125
For example, in a Czerny-Turner monochromator (see Section 6.2.1), light can be
diffracted by the grating back toward the f irst concave mirror and reflected
toward the grating; this light will be diffracted again, and may reach the second
mirror and then the exit slit. [Of course, the analogous situation may arise
involving the second mirror instead of the first.] These possibilities can be
eliminated by proper system design,126 filtering,127 or the use of masks.128
Proper instrument design and the use of baffles and light traps can reduce
the effects of these unwanted reflections on instrumental stray light. Care in the
analysis of the causes of stray light is especially important for monochromators,
since all wavelengths in all diffraction orders (including the zero order) move as
the analytical wavelength is scanned, so a wall or mount that does not cause
stray light when the grating is in one orientation may be a major cause of stray
light when the grating is rotated to another orientation.

124 ASTM standard E387, “Standard Test Method for Estimating Stray Radiant Power Ratio
of Dispersive Spectrophotometers by the Opaque Filter Method” (2004).
125 J. J. Mitteldorf and D. O. Landon, “Multiply diffracted light in the Czerny-Turner
spectrometer,” Appl. Opt. 7, 1431-1435 (1968); R. C. Hawes, ”Multiply diffracted light in
the Czerny-Turner spectrometer,” Appl. Opt. 8, 1063 (1969); A. B. Shafer and D. O.
Landon, “Comments on Multiple Diffracted Light in a Czerny-Turner Spectrometer,” Appl.
Opt. 8, 1063-1064 (1969).
127A. Watanabe and G. C. Tabisz, “Multiply diffracted light in Ebert Monochromators,”
Appl. Opt. 6, 1132-1134 (1967).
128 C. M. Penchina, “Reduction of stray light in in -plane grating spectrometers,” Appl. Opt.
6 1029-1031 (1967).

162

Reflection (and diffuse scatter) from interior instrument walls can be reduced
by using highly-absorbing paint or coatings on these surfaces, and moving
these surfaces as far from the optical train as possible (for this reason, it is
generally more difficult to reduce stray light in smaller instruments).
Light can also scatter (or be reflected) by the exit slit.129
Tilting the detector element or array slightly, so that any reflections from its
surface propagate out of the dispersion plane, can reduce the effects of this
cause of stray light.

10.2.5. Optical effects due to the sample or sample cell

In analytical instruments, care must be taken to choose sample cells that are
properly designed (given the characteristics of the optical path) and made of
materials that do not fluoresce; otherwise the cell will be a source of stray light.
Moreover, some samples will themselves fluoresce.

10.2.6. Thermal emission

For work in the far infrared, the blackbody radiation of all components in the
instrument (as well as the instrument walls) will generate a background in the
same spectral range as that of the analytical wavelength (e.g., at room
temperature (293 K = 20 °C = 68 °F), objects radiate with a spectrum that peaks at
c. λ = 10 µm).130

It is clear that a spectrometer containing a perfect grating (one that exhibits
no detectable scattered light) will still have nonzero instrumental stray light. The
often-made statement “the grating is the greatest cause of stray light in the
system” may well be true, but even a perfect grating must obey the grating
equation.

129 Ibid.
130 J. E. Stewart, “The effect of extraneous radiation on photometric accuracy of infrared
spectrophotometers,” Appl. Opt. 2, 1141-1146 (1963).

163

10.3. ANALYSIS OF OPTICAL RAY PATHS IN A GRATING-BASED
INSTRUMENT

Although a thorough raytrace analysis of an optical system is generally
required to model the effects of scattered light, we may approach the case of
a simple grating-based instrument conceptually. We consider the case in which
the grating is illuminated with monochromatic light; the more general case in
which many wavelengths are present can be considered by extension.
A simple case is shown in Figure 10-1. Light of wavelength λ enters the
instrument through the entrance slit and diverges toward the grating, which
diffracts the incident light into a number of orders {m} given by the grating
equation (for all orders m for which β given by Eq. (2-1) is real). One of these
orders (say m = 1) is the analytical order, that which is designed to pass through
the exit slit. All other propagating orders, including the ever-present m = 0 order,
are diffracted away from the exit slit and generally strike an interior wall of the
instrument, which absorbs some of the energy, reflects some, and scatters some.

second order

ES

G
incident

first order

XS

zero order

Figure 10-1. A simple grating system. Monochromatic light enters the system through
the entrance slit ES and is diffracted by grating G into several orders by the grating; one of
these orders (the analytical order) passes through the exit slit XS. Also shown are various
rays other than that of the analytical order that may reflect or scatter off interior walls or
other optics and reach the exit slit. [For simplicity, focusing elements are not shown.]

164

Some of the light reaching the interior walls may reflect or scatter directly
toward the exit slit, but most of it does not; that which is reflected or scattered in
any other direction will eventually reach another interior wall or it will return to
the grating (and thereby be diffracted again).
This simple illustration allows us to draw a number of conclusions regarding
the relative intensities of the various rays reaching the exit slit. We call E(λ,m)
the diffraction efficiency of the grating (in this use geometry) at wavelength λ in
order m; therefore we choose a grating for which E(λ,1) is maximal in this use
geometry (which will minimize the efficiencies of the other propagating orders:
E(λ,0), E(λ,−1), etc.; see Section 9.12). We further call ε the fraction of light
incident on an interior wall that is reflected and φ the fraction that is scattered in
any given direction, and stipulate that both ε and φ are much less of unity (i.e.,
we have chosen the interior walls to be highly absorbing). [Generally ε and φ
depend on wavelength and incidence angle, and φ on the direction of scatter as
well, but for this analysis we ignore these dependencies.]
With these definitions, we can approximate total intensity I(λ,1) of the light
incident on the grating that reaches the exit slit when the system is tuned to
transmits wavelength λ in order m = 1 as

I(λ,1) = I0(λ) E(λ,1)

+ I0(λ) ∑ ε E (λ ,m) + I (λ) ∑ φ E (λ, m)
m
0
m
m≠1 m≠1

+ I0(λ) ∑ε
m
2
E ( λ , m) + I0(λ) ∑ εφ E(λ, m)
m
(10-2)

m≠1 m≠1

+ I0(λ) ∑φ
m
2
E( λ , m) + O(3)
m≠1

where I0(λ) is the intensity incident on the grating and O(3) represents terms of
order three or higher in ε and φ.
The first term in Eq. (10-2) is the intensity in the analytical wavelength and
diffraction order; in an ideal situation, this would be the only light passing

165

through the exit slit, so we may call this quantity the “desired signal”.
Subtracting this quantity from both sides of Eq. (10-2), dividing by it and
collecting terms yields the fractional stray light S(λ,1):

I ( λ ,1) − I 0 (λ )E (λ ,1)
S(λ,1) =
I 0 (λ )E (λ ,1)

= (ε +φ + ε 2
+ εφ+φ 2 + O(3) ) ∑ EE((λλ,,m)) .
1
(10-3)
m
m≠1

The first term in Eq. (10-3) is the sum, over all other propagating orders, of the
fraction of light in those diffracted orders that is reflected by an interior wall to
the exit slit, divided by the desired signal; each element in this sum is generally
zero unless that order strikes the wall at the correct angle. The second term is the
sum, over all other orders, of the fraction of light in those orders that is scattered
directly into the exit slit; the elements in this sum are generally nonzero, again
divided by the desired signal. Both of these sums are linear in ε or φ (both << 1)
and in E(λ,m≠1) (each of which is considerably smaller than E(λ,1) since we have
chosen the grating to be blazed in the analytical order). The third through fifth
sums represent light that is reflected off two walls into the exit slit, or scattered
off two walls into the exit slit, or reflected off one wall and scattered off another
wall to reach the exit slit – in all three cases, the terms are quadratic in either ε or φ
and can therefore be neglected (under our assumptions).
If we generalize this analysis for a broad-spectrum source, so that
wavelengths other than λ are diffracted by the grating, then we obtain

(ε +φ +ε )∑ ∫
λ2 ( m)
E ( λ ′, m)
S(λ,1) =
2
+ εφ+φ 2 + O(3) dλ ′ . (10-4)
E (λ ′,1)
m λ1
m≠1

Note that, in each term, the integral over wavelength is inside the sum, since the
upper limit of integration is limited by the grating equation (2-1) for each
diffraction order m. Of course, the integration limits may be further restricted if
the detector employed is insensitive in certain parts of the spectrum.

166

10.4. DESIGN CONSIDERATIONS FOR REDUCING STRAY LIGHT

From Section 10.3, we can identify some suggestions for designing
a grating-based system for which instrumental stray light is reduced.
First, start with a grating as close to the definition of “perfect” in Section
10.1.6 as possible (easier said than done), and blaze it so that E(λ,m=1) is as high
as possible and E(λ,m≠1) are as low as possible for all other m. Provided other
design considerations ( .g., dispersion) are met, it may be advantageous to
e
choose a groove spacing d such that only the first and zero orders propagate; by
the analysis in Section 10.3, this will reduce each sum in Eq. (10-2) and Eq. (10-4)
to one element each (for m = 0).
Use an entrance slit that is as small as possible, and an exit slit that is as
narrow as possible (without being narrower than the image of the entrance slit)
and as short as possible (without reducing the signal to an unacceptably low
level).
Underfill the grating and all other optical components, preferably by using
a beam with a Gaussian intensity distribution. This will ensure that essentially all
of the light incident on the grating will be diffracted according to the grating
equation (2-1).
Next, design the system to contain as few optical components between the
entrance slit and the exit slit (or detector element(s)), for two reasons: each optic
is a source of scatter, and each optic will pass less than 100% of the light
incident on it – both of these effects will reduce the signal-to-noise (SNR) ratio.
Specify optical components with very smooth surfaces (a specification which is
more important when a short wavelength is used, since scatter generally varies
inversely with wavelength to some power greater than unity131 ).
Design the optical system so that the resolution is slit-limited, rather than
imaging-limited (see Section 8.3); this will reduce the spectral bandwidth passing
through the exit slit (whose width, multiplied by the reciprocal linear dispersion,
will equal the entire spectral range passing through the slit; otherwise, the
imaging imperfections will allow some neighboring wavelengths outside this
range to pass through as well).

131Stover, J. C., Optical Scattering: Measurement and Analysis (McGraw-Hill, New York:
1990).

167

The choice of mounting (see chapters 6 and 7) can also affect instrumental
stray light. For example, a Czerny-Turner monochromator (with two concave
mirrors; see Section 6.2.1) will generally have lower stray light than a comparable
Littrow monochromator (with a single concave mirror; see Section 6.2.4) since the
former will allow the entrance and exit slits to be located father apart.132
Make the distances between the surfaces as large as possible to take
advantage of the inverse square law that governs intensity per unit area as light
propagates; an underused idea is to design the optical system in three dimension
rather than in a plane – this reduces the volume taken by the optical system and
also removes some optics from the dispersion plane (which will reduce stray light
due to reflections and multiply diffracted light).
Use order-sorting filters where necessary (or, for echelle systems, cross-
dispersers 133 ). Also, the use of high-pass or low-pass filters to eliminate
wavelengths emitted by the source but outside the wavelength range of the
instrument, and to which the detector is sensitive, will help reduce stray light by
preventing the detector from seeing these wavelengths.
It may be advantageous to make the interior walls not only highly absorbing
but reflecting rather than scattering (i.e., use a glossy black paint rather than
a flat black paint). The rationale for this counterintuitive suggestion is that if all
unwanted light cannot be absorbed, it is better to control the direction of the
remainder by reflection rather than to allow it to scatter diffusely; controlled
reflections from highly-absorbing surfaces (with only a few percent of the light
reflected at each surface) will quickly extinguish the unwanted light without
adding to diffuse scatter. Of course, care must be taken during design to ensure
that there are no direct paths (for one or two reflections) directly to the exit slit;
baffles can be helpful when such direct paths are not otherwise avoidable.
Avoid grazing reflections from interior walls, since at grazing angle even
materials that absorb at near-normal incidence are generally highly reflecting.
Ensure that the system between the entrance slit and the detector is
completely light-tight, meaning that room light cannot reach the detector, and
that only light passing through the entrance slit can reach the exit slit.

133 R. W. Wood, J. Opt. Soc. Am. 37, 733 (1947); G. R. Harrison, “The production of
diffraction gratings: II. The design of echelle gratings and spectrographs,” J. Opt. Soc. Am.
39, 522-528 (1949).

168

Finally, hide all mounting brackets, screws, motors, etc. – anything that
might scatter or reflect light. Any edges (including the slits) should be painted
with a highly absorbing material; this includes the edges of baffles.
While it is always best to reduce instrumental stray light as much as
possible, a lock-in detection scheme can be employed to significantly reduce the
effects of instrumental stray light. The technique involves chopping (alternately
blocking and unblocking) the principal diffraction order and using phase-
sensitive detection to retrieve the desired signal.134
A useful technique at the breadboard stage (or, if necessary, the product
stage) is to operate the instrument in a dark room, replace the exit slit or detector
with the eye, and look back into the instrument (taking adequate precautions if
intense light is used). What other than the last optical component can be seen?
Are there any obvious sources of scatter, or obvious undesirable reflections?
What changes as the wavelength is scanned? Before the availability of
commercial stray light analysis software, this technique was often used to
determine what surfaces needed to be moved, or painted black, or hidden from
“the view of the exit slit” by baffles and apertures; even today, optical systems
designed with such software should be checked in this manner.

134H. Field, “UV-VIS-IR spectral responsivity measurement system for solar cells,” National
Renewable Energy Laboratory pub. CP-520-25654 (November 1998).

169

11. TESTING AND CHARACTERIZING
DIFFRACTION GRATINGS

11.1. THE M EASUREMENT OF SPECTRAL DEFECTS135

It is fundamental to the nature of diffraction gratings that errors are relatively
easy to measure, although not all attributes are equally detectable or sometimes
even definable.
For examp le, a grating with low background (in the form of scatter or
satellites) can be simply tested for Rowland ghosts on an optical bench. With a
mercury lamp or a laser source, and a scanning slit connected to a detector and
recorder, a ghost having intensity 0.002% of the intensity of the main line can be
easily observed. The periodic error in the groove spacing giving rise to such a
ghost may be less than one nanometer.
Grating ghosts are measured at Newport by making the grating part of
a scanning spectrometer and illuminating it with monochromatic light, such as
that from a mercury isotope lamp (isotope 198 or 202) or a helium-neon laser. On
scanning both sides of the parent line, using a chart recorder and calibrated
attenuators, it is easy to identify all ghost lines and to measure their intensities
relative to the parent line.
The importance of ghosts in grating applications varies considerably. In
most spectrophotometers, and in work with low-intensity sources, ghosts play a
negligible role. In Raman spectroscopy, however, even the weakest ghost may
appear to be a Raman line, especially when investigating solid samples, and
hence these ghosts must be suppressed to truly negligible values.
Ghosts are usually classified as Rowland ghosts, Lyman ghosts and
satellites.

135 For additional reading, see E. G. Loewen and E. Popov, Diffraction Gratings and
Applications, Marcel Dekker, Inc. (1997), pp. 402-413.

171

11.1.1. Rowland ghosts

Rowland ghosts are spurious lines seen in some grating spectra that result
from large-scale (millimeter) periodic errors in the spacing of the grooves (see
Figure 11-1). These lines are usually located symmetrically with respect to each
strong spectral line at a (spectral) dis tance from it that depends on the period of
the error, and with an intensity that depends on the amp litude of this error.

Figure 11-1. ‘Ghost’ trace showing Rowland ghosts caused by the periodic error of 2.54
mm in the lead screw of the Newport MIT ‘B’ engine. MR215 is an echelle grating, with
52.67 g/mm, in this case tested in the 54th order.

If the curve of groove spacing error vs. position is not simply sinusoidal,
there will be a number of ghosts on each side of the parent line representing the
various orders from each of the harmonics of the error curve. On engines with

172

mechanical drives, Rowland ghosts are associated primarily with errors in the
lead or pitch of the precision screw, or with the bearings of the ruling engine.
As a consequence, their location depends upon the number of grooves ruled for
each complete turn of the screw. For example, if the ruling engine has a pitch of
2 mm, and a ruling is made at 1200 grooves/mm, 2400 grooves will be ruled per
turn of the screw, and the ghosts in the first order can be expected to lie at ∆λ = ±
λ/2400 from the parent line λ, with additional ghosts located at integral multiples
of ∆λ. In gratings ruled on engines with interferometric feedback correction
mechanisms, Rowland ghosts are usually much less intense, but they can arise
from the mechanisms used in the correction system if care is not taken to prevent
their occurrence.
If the character of the periodic errors in a ruling engine were simply
harmonic, which is rarely true in practice, the ratio of the diffracted intensities of
the first order Rowland ghost (IRG(m=1)) to that of the parent line (IPL) is

2
I RG( m=1)  πA sin α 
= 4  , (11-1)
I PL  λ 

where A is the peak simple harmonic error amplitude, α is the angle of incidence,
and λ is the diffracted wavelength.
The second-order Rowland ghost IRG(m=2) will be much less intense (note the
exponent):

4
I RG(m= 2)  πA sin α 
= 4  . (11-2)
I PL  λ 

Higher-order Rowland ghosts would be virtually invisible. The ghost intensity is
independent of the diffraction order m of the parent line, and of the groove
spacing d. In the Littrow configuration, Eq. (11-1) becomes

2
I RG(m=1)  πmA 
=  , in Littrow, (11-3)
I PL  d 

an expression derived in 1893 by Rowland.

173

These simple mathematical formulas do not always apply in practice when
describing higher-order ghost intensities, since the harmonic content of actual
error curves gives rise to complex amplitudes that must be added vectorially and
then squared to obtain intensity functions. Fortunately, the result of this
complication is that ghost intensities are generally smaller than those predicted
from the peak error amplitude.
The order of magnitude of the fundamental harmonic error amplitude can be
derived from Eq. (11-1) [or Eq. (11-3)]. For example, a 1200 g/mm grating used in
the m = 1 order in Littrow will show a 0.14% first-order ghost intensity, compared
with the parent line, for a fundamental harmonic error amplitude of A = 10 nm. For
some applications, this ghost intensity is unacceptably high, which illustrates
the importance of minimizing periodic errors of ruling. For Raman gratings and
echelles, the amplitude A of the periodic error must not exceed one nanometer;
the fact that this has been accomplished is a remarkable achievement.

11.1.2. Lyman ghosts

Ghost lines observed at large spectral dis tances from their parent lines are
called Lyman ghosts. They result from compounded periodic errors in the
spacing of the grating grooves; the period of Lyman ghosts is on the order of a
few times the groove spacing.
Lyman ghosts can be said to be in fractional-order positions (see Figure
11-2). Thus, if every other groove is misplaced so that the period contains just
two grooves, ghosts are seen in the half-order positions. The number of grooves
per period determines the fractional-order position of Lyman ghosts.
Usually it is possible to find the origin of the error in the ruling engine once
its periodicity is determined. It is important that Lyman ghosts be kept to a mini-
mum, because they are not nearly as easy to identify as Rowland ghosts.

11.1.3. Satellites

Satellites are false spectral lines usually occurring very close to the parent
line. Individual gratings vary greatly in the number and intensity of satellites
which they produce. In a poor grating, they give rise to much scattered light, re-
ferred to as grass (so called since this low intensity scattered light appears like

174

a strip of lawn when viewed with green mercury light). In contrast to Rowland
ghosts, which usually arise from errors extending over large areas of the grating,
each satellite usually originates from a small number of randomly mis placed
grooves in a localized part of the grating. With laser illumination, a relative
background intensity of 10–7 is easily observable with the eye.

Figure 11-2. ‘Ghost’ trace showing Lyman ghosts (the small spikes between orders 2 and
3), which can be associated with fractional order positions, e.g., an error every five
grooves corresponds to a fraction order of 1/5.

175

11.2. THE M EASUREMENT OF GRATING EFFICIENCY136

Grating efficiency measurements are generally performed with a double
monochromator system. The first monochromator supplies monochromatic light
derived from a tungsten lamp, mercury arc, or deuterium lamp, depending on the
spectral region involved. The grating being tested serves as the dispersing
element in the second monochromator. In the normal mode of operation, the
output is compared with that from a high-grade mirror coated with the same
material as the grating. The efficiency of the grating relative to that of the mirror
is reported (relative efficiency), although absolute efficiency values can also be
obtained (either by direct measurement or through knowledge of the variation of
mirror reflectance with wavelength). For plane reflection gratings, the
wavelength region covered is usually 190 nm to 2.5 µm; gratings blazed farther
into the infrared are often measured in higher orders. Concave reflection gratings
focus as well as disperse the light, so the entrance and exit slits of the second
monochromator are placed at the positions for which the grating was designed
(that is, concave grating efficiencies are measured in the geometry in which the
gratings are to be used). Transmission gratings are tested on the same equip-
ment, with values given as the ratio of diffracted intensity to the intensity falling
directly on the detector from the light source (i.e., absolute efficiency).
Curves of efficiency vs. wavelength for plane gratings are made routinely on
all new master gratings produced by Newport, both plane and concave, with light
polarized in the S and P planes to assess the presence and amplitudes (if any) of
anomalies. Such curves are furnished by Newport upon request (for an example,
see Figure 11-3).

136 For additional reading, see E. G. Loewen and E. Popov, Diffraction Gratings and
Applications, Marcel Dekker, Inc. (1997), pp. 413-423, and also D. J. Michels, T. L. Mikes
and W. R. Hunter, ”Optical grating evaluator: a device for detailed measurement of
diffraction grating efficiencies in the vacuum ultraviolet,“ Appl. Opt. 13, 1223-1229 (1974).

176

Figure 11-3. Example of an efficiency curve. This efficiency curve is specific to the
particular grating under test, as well as the conditions of illumination (the incidence and
diffraction angles).

11.3. THE M EASUREMENT OF DIFFRACTED WAVEFRONT QUALITY

11.3.1. The Foucault knife-edge test

One of the most critical tests an optical system can undergo is the Foucault
knife-edge test. This test not only reveals a great deal about wavefront
deficiencies but also locates specific areas (or zones) on the optical component
where they originate. The test is suited equally well to plane and concave
gratings (for the former, the use of very high grade collimating optics is required).
The sharper (i.e., more abrupt) its knife-edge cut-off, the more likely that a grating
will yield high resolution.

177

The sensitivity of the test depends on the radius of the concave grating (or
the focal length of the collimating system), and may exceed that of interferometric
testing, although the latter is more quantitative.
The Foucault test is a sensitive and powerful tool, but experience is required
to interpret each effect that it makes evident. All Newport master plane gratings,
large plane replicas and large-radius concave gratings are checked by this
method (see Figure 11-4).

Figure 11-4. A grating under test on the Newport 5 -meter test bench. Light from
a mercury source (not shown, about 5 meters to the right) is collimated by the lens
(shown) which illuminates the grating (shown on a rotation stage); the same lens refocuses
the diffracted light to a plane very near the light source, where the diffracted wavefront
can be inspected.

178

11.3.2. Direct wavefront testing

Any departure from perfect flatness of the surface of a plane grating, or from
a perfect sphere of the surface of a concave grating, as well as variations in the
groove spacing, depth or parallelism, will result in a diffracted wavefront that is
less than perfect. In order to maintain resolution, this departure from perfection
is generally held to λ/4 or less, where λ is the wavelength of the light used in the
test. To obtain an understanding of the magnitudes involved, it is necessary to
consider the angle at which the grating is used. For simplicity, consider this to
be the blaze angle, under Littrow conditions. Any surface figure error of height h
will cause a wavefront deformation of 2h cosθ, which decreases with increasing
|θ|. On the other hand, a groove position error p introduces a wavefront error of
2p sinθ, which explains why ruling parameters are more critical for gratings used
in high-angle configurations.
A plane grating may produce a slightly cylindrical wavefront if the groove
spacing changes linearly, or if the surface figure is similarly deformed. In this
special case, resolution is maintained, but focal distance will vary with
wavelength.
Wavefront testing can be done conveniently by mounting a grating at its
autocollimating angle (Littrow) in a Twyman-Green interferometer or a phase
measuring interferometer (PMI; see Figure 11-5). Newport interferometers have
apertures up to 150 mm (6 inches). With coherent laser light sources, however, it
is possible to make the same measurements with a much simpler Fizeau
interferometer, equipped with computer fringe analysis .
It should be noted that testing the reflected wavefront – that is, illuminating
the grating in zero order – is generally inadequate since this arrangement will
examine the flatness of the grating surface but tells nothing about the uniformity
of the groove pattern.
Periodic errors give rise to zig-zag fringe displacements. A sudden change
in groove position gives rise to a step in the fringe pattern; in the spectrum, this
is likely to appear as a satellite. Curved fringes due to progressive ruling error
can be distinguished from figure problems by observing fringes obtained in zero,
first and higher orders. Fanning error (non-parallel grooves) will cause spreading
fringes. Figure 11-6 shows a typical interferogram, for an echelle grating
measured in Littrow in the diffraction order of use (m = 33).

179

Figure 11-5. A plane grating under test on a phase measuring interferometer. The
grating is tested in the Littrow configuration so that the flatness of the diffracted
wavefront is evaluated.

Experience has shown that the sensitivity of standard interferograms for
grating deficiencies equals or exceeds that of other plane grating testing methods
only for gratings used at high angles. This is why the interferometric test is
especially appropriate for the testing of echelles and other gratings used in high
diffraction orders.

180

Figure 11-6a. Example of an interferogram and histogram generated by a Phase
Measuring Interferometer (PMI). In this example, a grating is illuminated in a circular
region 50 mm in diameter, and its diffracted wavefront at λ = 632.8 nm in the m = 33
order is recorded.

11.4. THE M EASUREMENT OF RESOLVING POWER137

Resolving power (defined in Section 2.4) is an crucial characteristic of
diffraction gratings since it is a measure of the fundamental property for which
gratings are used: it quantifies the ability of the grating (when used in an optical
system) to separate two nearby wavelengths. Often resolving power is specified
to be great enough that the resolution of the optical system will be slit limited
rather than grating limited (see Section 8.3).

137 For additional reading, see J. Strong, “New Johns Hopkins ruling engine,” J. Opt. Soc.
Am. 41, 3-15 (1951), J. F. Verrill, “The limitations of currently used methods for evaluating
the resolution of diffraction gratings,” Opt. Acta 28, 177-185 (1981), and E. G. Loewen and
E. Popov, Diffraction Gratings and Applications, Marcel Dekker, Inc. (1997), pp. 423-432.

181

Resolving power is generally measured in a spectrometer with a large focal
length and very narrow slits in which the light source has fine spectral structure;
an example is the hyperfine spectrum of natural Hg near 546.1 nm (see Figure 11-
6). The spectral lines are identified, and the wavelengths of those that are
distinguishable (‘resolvable’) are subtracted, and this difference ∆λ is divided
into λ = 546.1 nm according to Eq. (2-17); the smaller the wavelength difference,
the greater the resolving power.

Hyperfine Stucture of Natural Mercury

30
Relative Intensity (arb. units)

25

20

15

10

5

0
55

60

65

85

90
50

70

75

80

95

00
6.0

6.0

6.0

6.0

6.0
6.0

6.0

6.0

6.0

6.0

6.1
54

54

54

54

54
54

54

54

54

54

54

Wavelength (nm)

Figure 11-6b. Hyperfine structure of natural Hg near 546.1 nm. Several spectral lines are
identified. Visual identification of two distinct emission lines centered on λ and separated
by ∆λ implies a resolving power at least as great as λ/∆λ.

Resolving power is measured on Newport diffraction gratings using
a specially-designed Czerny-Turner spectrograph, whose concave mirrors have
very long focal lengths (10 m) so that very large astronomical gratings may be
tested (see Section 13.3).

182

11.5. THE M EASUREMENT OF SCATTERED LIGHT138

As discussed in Chapters 2 and 10, light that leaves a grating surface that
does not follow the grating equation (2-1) is called scattered light. Scattered light
is generally measured in one of two ways: either with cut-off filters (which absorb
one part of the spectrum while transmitting the other part) or by using
monochromatic light (from an atomic emission source or a laser, or by the use of
interference filters that transmit a narrow spectral range).
Newport has two specially-designed instruments to measure light scattered
from small regions on the surface of a mirror or grating: one uses red HeNe light
(λ = 632.8 nm) to illuminate the grating, and the uses a Hg source to illuminate the
grating (the light reaching the detector is filtered to transmit a narrow spectral
band around 254 nm). These “scatter checkers” provide several degrees of
freedom so that light scattered between diffraction orders (called inter-order
scatter) can be attributed to areas on the grating surface.
Figure 11-7 shows a simplified schematic diagram of the scatter checker. The
beam from a polarized HeNe laser is spatially filtered to remove speckle and is
then directed onto a concave focusing mirror that brings the beam to focus at the
detector plane. The detector is a photomultiplier that, in combination with a
programmable-gain current amplifier, provides eight decades of dynamic range.
A PC equipped with a data acquisition card is used to process and store the
detector signal.
Scatter measurements are made by first obtaining a reference beam profile
(see Figure 11-8), or “instrument signature,” by translating the test optic out of
the way and rotating the detector through the beam in incremental steps over
a predetermined angular range. The test optic is then translated into the beam
path and the detector passed through the reflected (or diffracted) beam from the
test optic over the same angular range used to make the reference measurement.
The sample and reference beam profiles are “mirror images” of one another, so it
is necessary to invert one before a comparison is made. Any difference between
the sample and reference beam profiles can be attributed to light scattered from
the optic under test.

138 For additional reading, see J. F. Verrill, “The specification and measurement of scattered
light from diffraction gratings,” Opt. Acta 25, 531-547 (1978).

183

Programmable Current Amplifier
PC with
A/D card
Fold Mirror
Detector Spatial Filter
HeNe Laser
Detector swing arm

Rotation stage
Optical Aperture Focusing Mirror
component
under test

Figure 11-7. Schematic of the Newport red HeNe scatter measuring apparatus. In order
to minimize the effects that other diffraction orders may have on the scattered light
readings, this instrument is not enclosed so that any light that leaves the grating in a
direction other than toward the detector will travel a long distance before encountering a
reflecting or scattering surface.

Scatter Checker Instrument Signature
Signal (arb. units)

1.E+05
1.E+03
1.E+01
1.E-01
1.E-03
1.E-05
-50 -40 -30 -20 -10 0 10 20 30 40 50
Detector Angle (degrees)

Figure 11-8. Typical plot of data obtained from the Newport red HeNe scatter measuring
instrument. This plot of the measured signal vs. angle of rotation of the detector (from a
diffracted order) shows the reference beam profile (the “instrument signature”).

184

It is important to apply the lessons of Chapter 10 to the interpretation of
grating scatter measurements. That is, even a “perfect” grating (as defined in
Section 10.1.6) illuminated with monochromatic light will cause other diffraction
orders to propagate, and some of this light energy may reach the detector of the
scatter measuring apparatus. This is important when comparing the scatter
characteristics of a grating with those of a high-quality mirror (using the latter as
a reference’); the mirror produces only the m = 0 order (specular reflection) and
will therefore exhibit lower scatter than even a “perfect” grating. This subtle
point must be considered in defining the instrument signature of a grating-based
optical system by using a mirror.
In analyzing grating scatter measurements, care must be taken to account for
any stray light that is due to the measurement apparatus rather than the grating,
as discussed in Sections 10.2 and 10.3.

11.6. THE M EASUREMENT OF INSTRUMENTAL STRAY LIGHT

The consequence of undesired energy reaching the detector in a
spectrometer is a reduction in photometric accuracy, since some light reaches the
detector that cannot be attributed to the transmission (or absorption) of the
sample at the analytical wavelength.
Instrumental stray light, like scattered light, is generally measured either with
cut-off filters or monochromatic light.

11.6.1. The use of cut-off filters 139

Instrumental stray light is commonly measured in by using a set of high-
pass cut-off optical filters (whose transmission curves look like that in Figure 11-
9). The spectrometer is then scanned toward shorter wavelengths and the

139 R. E. Poulson, “Test methods in sp ectrophotometry: stray-light determination,” Appl.
Opt. 3, 99-104 (1964); A. W. S. Tarrant, “Optical techniques for studying stray light in
spectrometers,” Opt. Acta 25, 1167-1174 (1978); ASTM standard E387, “Standard Test
Method for Estimating Stray Radiant Power Ratio of Dispersive Spectrophotometers by the
Opaque Filter Method” (2004).

185

transmittance measured; once the transmittance level has reached a fairly steady
minimum (a plateau), this reading is taken to be the stray light.140

transmission coefficient

100%

50%

0
λC wavelength

Figure 11-9. Transmission curve of a typical high-pass cut-off filter. A filter of this type
is generally specified by the cut-off wavelength λC, the wavelength at which its
transmission coefficient is 50%. The slope of the transmission curve near λC should be as
steep as possible.

The instrument is tuned to the analytical wavelength λ and a series of filters,
each with a successively higher cut-off wavelength λC (>λ), is placed in the beam
and intensity readings taken at the detector. [Generally λC should exceed λ by at
least 20 nm, in the visible spectrum, to ensure than virtually no light of the
analytical wavelength λ passes through the filter and complicates the readings.]
Nonzero readings indicate the presence of stray light. A proper study requires
measurements at more than one analytical wavelength since stray light properties
cannot be easily extrapolated (due to the different wavelength dependencies of
the causes of grating scatter and instrumental stray light noted above, and – for

140 K. D. Mielenz, V. R. Weidner and R. W. Burke, “Heterochromic stray light in UV
absorption spectrometry: a new test method,” Appl. Opt. 21, 3354-3356 (1982).

186

monochromators – the fact that all rays diffracted from or scattered by the
grating change direction as the grating is rotated).

11.6.2. The use of monochromatic light

Another method for measuring instrumental stray light is to replace the
polychromatic light source (used with cut-off filters) with a narrow-band
monochromatic light source. Atomic emission sources provide narrow spectral
emission lines that can be used for this purpose; lasers can be used; and broad-
spectrum sources can be used in conjunction with bandpass filters.
Kaye141 describes a technique in which monochromatic light is used to
determine the amount of power detected at all wavelength settings for a given
input wavelength; this quantity is called the slit function. The spectrometer
(with slit widths w) is illuminated by light whose central wavelength is λ, and
whose spectral width ∆λ is very narrow (∆λ << λ). Scanning through the full
wavelength range of the instrument (the wavelength setting being denoted by
λ ; see Section 10.1.6) and recording the power at each setting yields the slit
( )
function Sλ λ , w , which we may write as

( ) ( )
Sλ λ , w = cE λ M λ λ , w Rλ , (11-4)

( )
where Eλ is the power emitted by the source, M λ λ , w is the transmittance of
the optical system (between the source and the detector), Rλ is the sensitivity of
the detector, and c is a constant of proportionality. If we had knowledge of the
slit function for all input wavelengths λ and for all wavelength settings λ , we
would be able to write for any wavelength setting the following integral:

∞
( ) ∫ Sλ (λ , w ),
S λ ,w = (11-5)
0

141 W. Kaye, “Resolution and stray light in near infrared spectroscopy,” Appl. Opt. 14,
1977-1986 (1975).

187

which represents the total power (for all wavelengths) recorded at wavelength
setting λ . In practice, the bounds of integration are not 0 and ∞, but are instead
determined by the spectral sensitivity limits of the detector.
Stray light can then be expressed as the ratio of the intensities (powers) of
the scattered light and principal beam. 142

11.6.3. Signal-to-noise and errors in absorbance readings

Often the unwanted light in a spectrometer is quantified not by instrumental
stray light but by the signal-to-noise ratio (SNR), a dimensionless quantity of
more relevance to instrumental specification. The SNR is defined as the ratio of
the signal (the desired power incident on the detector) to the noise (the
undesired power, equivalent in our definition to the instrumental stray light).
Another specification of instrumental stray light is given in absorbance, a
dimensionless quantity defined by

 100 
A = log 10  , (11-6)
 T 

where T is the percent transmittance (0 ≤ T ≤ 100). Higher values of A correspond
to lower transmittances, and instrumental stray light plays an important role in
the highest value of A for which the readings are accurate; an instrument for
which the stray light is about 1% as intense as the signal at a given wavelength
cannot provide absorbance readings of any accuracy greater than A ≈ 2.
When the stray light power s is known (as a percentage of the signal), Eq.
(11-6) may be modified to be made more accurate:143

 100 − s 
A = log10  . (11-7)
 T −s 

142 A. W. S. Tarrant, “Optical techniques for studying stray light in spectrophotometers,”
Opt. Acta 25, 1167-1174 (1978); S. Brown and A. W. S. Tarrant, “Scattered light in
monochromators,” Opt. Acta 25, 1175-1186 (1978).
143 A. Opler, “Spectrophotometry in the presence of stray radiation: a table of log[(100-
k)/(T -k)],” J. Opt. Soc. Am. 40, 401-403 (1950).

188

12. SELECTION OF DISPERSING SYSTEMS

12.1. REFLECTION GRATING SYSTEMS

Reflection grating systems are much more common than transmission
grating systems. Optical systems can be 'folded' with reflection gratings, which
reflect as well as disperse, whereas transmission grating systems are 'in-line' and
therefore usually of greater length. Moreover, reflection gratings are not limited
by the transmission properties of the grating substrate (or resin), and can operate
at much higher angles of diffraction.

12.1.1. Plane reflection grating systems

The choice of existing plane reflection gratings is extensive and continually
increasing. Master gratings as large as 320 mm x 420 mm have been ruled. For
infrared spectra, plane reflection gratings are most suitable because of the avail-
ability of large gratings. While plane gratings have been used for visible and
ultraviolet spectra for some time, they are also used increasingly for wavelengths
as short as 110 nm, an extension made possible by special overcoatings that give
satisfactory reflectivity even at such short wavelengths (see Chapter 9).
The most popular arrangement for plane reflection gratings is the Czerny-
Turner mount, which uses two spherical concave mirrors between the grating
and the entrance and exit slits. A single mirror arrangement (the Ebert-Fastie
mount) can also be used. Both achieve spectral scanning through rotation of the
grating. Collimating lenses are rarely used, since mirrors are inherently achro-
matic. [See Chapter 6 for a discussion of plane grating mounts.]
For special purposes, plane reflection gratings can be made on unusual
materials, such as ceramics or metals, given special shapes, or supplied with
holes for Cassegrain and Coudé-type telescopic systems.

190

12.1.2. Concave reflection grating systems

The great advantage in using concave gratings lies in the fact that separate
collimating optics are unnecessary. This is particularly important in the far
vacuum ultraviolet region of the spectrum, for which there are no good reflectors.
Two mirrors, each reflecting 20% of the light incident on them, will reduce
throughput by a factor of twenty-five. Hence, concave systems dominate the
entire ultraviolet region, and at wavelengths less than 110 nm are used exclu-
sively. Their chief deficiency lies in astigmatism, which limits the exit slit size
(and, consequently, the energy throughput). The situation can be improved
somewhat by using toroidal grating substrates; however, their use is restricted
because of high costs.
Though most ruled gratings are flat, curved substrates can be ruled as well if
their curvatures are not extreme (c. ƒ/9 or greater). Concave gratings are not only
more difficult to rule than plane gratings, since the tool must swing through an
arc as it crosses the substrate, but they require extremely tight control over the
sphericity to the substrate as well. Since each radius of curvature is a new
parameter, there cannot be the large selection of rulings (in size and blaze angle)
for any one given radius that there is with plane gratings.
Another limitation of ruled concave gratings appears when they are ruled at
shallow groove angles. The ruled width is unfortunately limited by the radius of
the substrate, since the diamond cannot rule useful grooves when the slope
angle of the substrate exceeds the blaze angle. The automatic energy limitation
that is thereby imposed can be overcome by ruling multipartite gratings, a
Newport development. Here the ruling is interrupted once or twice, so the tool
can be reset at a different angle. The resulting bipartite or tripartite gratings are
very useful, as available energy is otherwise low in the short wavelength regions.
One must not expect such gratings to have a resolving power in excess of that of
any single section, for such an achievement would require phase matching
between the grating segments to a degree that is beyond the present state of the
art.
The advent of the holographic method of generating gratings has made the
manufacture of concave gratings commonplace. Since the fringe pattern formed
during the recording process is three-dimensional, a curved substrate placed in
this pattern will record fringes. Unlike ruled gratings, concave holographic
gratings can be generated on substrates whose radii are smaller than 100 mm.

191

12.2. TRANSMISSION GRATING SYSTEMS

In certain types of instrumentation, transmission gratings (see Figure 12-1)
are much more convenient to use than reflection gratings. The most common
configuration involves converting cameras into simple spectrographs by
inserting a grating in front of the lens. This configuration is often used for
studying the composition of falling meteors or the re-entry of space vehicles,
where the distant luminous streak becomes the entrance slit. Another
application where high-speed lenses and transmission gratings can be combined
advantageously is in the determination of spectral sensitivity of photographic
emulsions.

Figure 12-1. Diffraction by a plane transmission grating. A beam of monochromatic light
of wavelength λ is incident on a grating at angle α to the grating normal, and diffracted
along several discrete paths {β m}, for diffraction orders {m}. The incident and diffracted
rays lies on opposite sides of the grating. The configuration shown, in which the
transmission grating is illuminated from the back, is most common.

192

Transmission gratings can be made by stripping the aluminum film from the
surface of a reflection grating. However, since the substrate is now part of the
imaging optics, special substrates are used, made to tighter specifications for
parallelism, and those used in the visible region are given a magnesium fluoride
(MgF2) antireflection coating on the back to reduce light loss and internal
reflections. The material used to form the substrate must also be chosen for its
transmission properties and for the absence of bubbles, inclusions, striae and
other imperfections, none of which is a concern for reflection gratings.
In most cases, relatively coarse groove frequencies are preferred for
transmission gratings, although gratings up to 600 g/mm are furnished routinely.
Experimentally, transmission gratings of 1200 g/mm have been used. Energy
distribution on either side of the blaze peak is very similar to that of reflection
gratings in the scalar domain. For wavelengths between 220 and 300 nm, trans-
mission gratings are made on fused silica substrates with a special resin capable
of high transmission for these wavelengths.
Since transmission gratings do not have a delicate metal film they are much
more readily cleaned. However, they are limited to spectral regions where
substrates and resins transmit. Their main drawback is that they do not fold the
optical path conveniently as a reflection grating does. Moreover, to avoid total
internal reflection, their diffraction angles cannot be extreme. Even though the
surface of the substrate is antireflection coated, internal reflections from the
grating-air interface leads to some backward-propagating orders (that is, the
transmission grating will also behave as a weak reflection grating); this limits the
maximum efficiency to about 80%.144 The efficiency behavior of transmission
gratings can be modeled adequately over a wide spectral range and for a wide
range of groove spacing by using scalar efficiency theory.145
For a reflection grating of a given groove angle θB with first-order blaze
wavelength λB, the transmission grating with the same groove angle will be
blazed between λB/4 and λB/3, depending on the index of refraction of the resin.
This estimate is often very good, though it becomes less accurate for θB > 25°.

144M. Nevière, “Electromagnetic study of transmission gratings,” Appl. Opt. 30, 4540-4547
(1991).
145 E. K. Popov, L. Tsonev and E. G. Loewen, “Scalar theory of transmission relief
gratings,” Opt. Commun. 80, 307-311 (1991).

193

Although there are cases in which transmission gratings are applicable or
even desirable, they are not often used: reflection gratings are much more
prevalent in spectroscopic and laser systems, due primarily to the following
advantages:
• Reflection gratings may be used in spectral regions where glass
substrates and resins absorb light (e.g., the ultraviolet).
• Reflection gratings provide much higher resolving power than
equivalent transmission gratings, since the path difference between
neighboring beams (i.e., separated by a single groove) is higher in the
case of the reflection grating – therefore transmission gratings much
generally be wider (so that more grooves are illuminated) to obtain
comparable resolving power.
• Reflection grating systems are generally smaller than transmission
grating systems since the reflection grating acts as a folding mirror.

12.3. GRATING PRISMS (GRISMS )

For certain applications, such as a direct vision spectroscope, it is very
useful to have a dispersing element that will provide in-line viewing for one
wavelength. This can be done by replicating a transmission grating onto the hy-
potenuse face of a right-angle prism. The light diffracted by the grating is bent
back in-line by the refracting effect of the prism. The device is known as a
Carpenter prism, but is more commonly called a grism.
The derivation of the formula for computing the required prism angle follows
(refer to Figure 12-2). On introducing Snell's law, the grating equation becomes

mλ = d (n sinα + n' sinβ), (12-1)

where n and n' are the refractive indices of glass and air, respectively, and β < 0
since the diffracted ray lies on the opposite side of the normal from the incident
rays (α > 0).
Taking n' = 1 for air, and setting α = –β = φ, the prism angle, Eq. (12-1)
becomes

mλ = d (n–1) sinφ. (12-2)

194

In this derivation it is assumed that the refractive index n of the glass is the same
(or very nearly the same) as the index n E of the resin at the straight-through
wavelength λ. While this is not likely to be true, the resulting error is often quite
small.

θ
φ
GN

β = –α
α

n nE n'

Figure 12-2. Grating prism (grism). The ray path for straight-through operation at one
wavelength is shown. The refractive indices of the prism, resin and air are indicated as n,
nE and n′, respectively; also, φ is the prism angle and θ is the groove angle. The incidence
angle α and diffraction angle β are measured from GN, the grating normal.

The dispersion of a grating prism cannot be linear, owing to the fact that the
dispersive effects of the prism are superimposed on those of the grating. The
following steps are useful in designing a grism:

1. Select the prism material desired (e.g., BK-7 glass for visible
light or fused silica for ultraviolet light).
2. Obtain the index of refraction of the prism material for the
straight-through wavelength.

195

3. Select the grating constant d for the appropriate dis persion
desired.
4. Determine the prism angle φ from Eq. (12-2).
5. For maximum efficiency in the straight-through direction,
select the grating from the Newport Diffraction Grating
Catalog with groove angle θ closest to φ.

Design equations for grism spectrometers may be found in Traub.146

12.4. GRAZING INCIDENCE SYSTEMS 147

For work in the x -ray region (roughly the wavelength range 1 nm < λ <
25 nm), the need for high dispersion and the normally low reflectivity of materials
both demand that concave gratings be used at grazing incidence (i.e., |α| > 80°,
measured from the grating normal). Groove spacings of 600 to 1200 per millimeter
are very effective, but exceptional groove smoothness is required on these
gratings to achieve good results.

12.5. ECHELLES

A need has long existed for spectroscopic devices that give higher
resolution and dis persion than ordinary gratings, but with a greater free spectral
range than a Fabry-Perot étalon. This need is admirably filled by the echelle
grating, first suggested by Harrison.148 Echelles have been used in a number of

146 W. A. Traub, “Constant-dispersion grism spectrometer for channeled spectra,” J. Opt.
Soc. Am. A7, 1779-1791 (1990).
147 W. Cash, “Echelle spectrographs at grazing incidence,” Appl. Opt. 21, 710-717 (1982);
L. B. Mashev, E. K. Popov and E. G. Loewen, “Optimization of the grating efficiency in
grazing incidence,” Appl. Opt. 26, 4738-4741 (1987); L. Poletto, G. Tondello and P.
Villoresi, “Optical design of a spectrometer-monochromator for the extreme-ultraviolet and
soft x-ray emission of high -order harmonics,” Appl. Opt. 42, 6367-6373 (2003).
148 G. R. Harrison, “The production of diffraction gratings II: The design of echelle gratings
and spectrographs,” J. Opt. Soc. Am. 39, 522-528 (1949).

196

applications that require compact instruments with high angular dispersion and
high throughput.
Echelles are a special class of gratings, with high groove spacings (i.e., a
coarse groove pattern), used in high angles in high diffraction orders (rarely
below |m| = 5, and sometimes used in orders beyond 100). Because of order
overlap, some type of filtering is normally required with higher-order grating
systems. This can take several forms, such as cut-off filters, detectors
insensitive to longer wavelengths, or cross-dispersion in the form of prisms or
low-dispersion gratings. The latter approach leads to a square display format
suitable for corresponding types of array detectors; with such a system a large
quantity of spectroscopic data may be recorded simultaneously.149 First-order
design principles for echelle spectrometers using a cross-disperser have been
developed by Dantzler.150
As seen in Figure 12-3, an echelle looks like a coarse grating used at such a
high angle (typically 63° from the normal) that the steep side of the groove
becomes the optically active facet. Typical echelle groove spacings are 31.6, 79
and 316 g/mm, all blazed at 63°26' (although 76° is available for greater
dispersion). With these grating, resolving powers greater than 1,000,000 for
near-UV wavelengths can be obtained, using an echelle 10 inches wide. Corre-
spondingly high values can be obtained throughout the visible spectrum and to
20 µm in the infrared.
Since echelles generally operate close to the Littrow mode and at the blaze
condition, the incidence, diffraction and groove angles are equal (α = β = θ) and
the grating equation becomes

mλ = 2d sinβ = 2d sinθ = 2t, (12-3)

where t = d sinθ is the width of one echelle step (see Figure 12-3).
The free spectral range is

149 D. Dravins, “High -dispersion astronomical spectrographs with holographic and ruled
diffraction gratings,” Appl. Opt. 17, 404-414 (1978).
150 A. A. Dantzler, “Echelle spectrograph software design aid,” Appl. Opt. 24, 4504-4508
(1985).

197

λ
Fλ = , (2-28)
m

Figure 12-3. Echelle geometry for use in the Littrow blaze condition. The groove spacing
d, step width t and step height s are shown. The double-headed arrow indicates that the
grating is used in the Littrow configuration (α = β), and β was chosen to equal the groove
angle θ to satisfy the blaze condition. GN is the grating normal and FN is the facet
normal. The blaze arrow (shown) points from GN toward FN.

which can be very narrow for high diffraction orders. From Equation (12-3),
m = 2t/λ, so

λ2
Fλ = ; (12-4)
2t

for an echelle used in Littrow. In terms of wavenumbers † , the free spectral range
is

∆λ 1
Fσ = = . (12-5)
λ 2
2t

198

The linear dispersion of the spectrum is, from Eq. (2-16),

∂β mr ′ mr ′ r ′  2t 
r' = = =  , (12-6)
∂ λ d cos β s sλ 

where s = d cosβ = d cosθ is the step height of the echelle groove (see Figure
12-3). The dispersion of an echelle used in high orders can be as high as that of
fine-pitch gratings used in the first order.
The useful length l of spectrum between two consecutive diffraction orders
is equal to the product of the linear dispersion and the free spectral range:

r ′λ
l= . (12-7)
s

For example, consider a 300 g/mm echelle with a step height s = 6.5 µm, combined
with an r' = 1.0 meter focal length mirror, working at a wavelength of λ = 500 nm.
The useful length of one free spectral range of the spectrum is l = 77 mm.
Typically, the spectral efficiency reaches a peak in the center of each free
spectral range, and drops to about half of this value at the ends of the range.
Because the ratio λ/d is generally very small (<< 1) for an echelle used in high
orders ( >> 1), polarization effects are not usually pronounced and scalar
m
methods may be employed in many cases to compute echelle efficiency.151
Echelle efficiency has been addressed in detail by Loewen et al.152
The steep angles and the correspondingly high orders at which echelles are
used make their ruling much more difficult than ordinary gratings. Periodic errors
of ruling must especially be limited to a few nanometers or even less, which is

† A wavenumber is a unit proportional to inverse wavelength, and is often used in the
infrared. The definition of a wavenumber σ in inverse centimeters (cm–1 ) is σ = 10000/λ,
where λ is expressed in µm.
151 D. J. Schroeder and R. L. Hilliard, “Echelle efficiencies: theory and experiment,” Appl.
Opt. 19, 2833-2841 (1980); B. H. Kleeman and J. Erxmeyer, “Independent electromagnetic
optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle
grating in Littrow mount,” J. Mod. Opt. 51, 2093-2110 (2004).
152 E. G. Loewen, D. Maystre, E. Popov and L. Tsonev, “Echelles: scalar, electromagnetic
and real groove properties,” Appl. Opt. 34, 1707-1727 (1995); E. G. Loewen, D. Maystre,
E. Popov and L. Tsonev, “Diffraction efficiency of echelles working in extremely high
orders,” Appl. Opt. 35, 1700-1704 (1996);

199

attainable only by using interferometric control of the ruling engine. The task is
made even more difficult by the fact that the coarse, deep grooves require heavy
loads on the diamond tool. Only ruling engines of exceptional rigidity can hope
to rule echelles. This also explains why the problems escalate as the groove
spacing increases.
Echelles are often referred to by their "R numbers". This number is the
tangent of the blaze angle θ:

t
R number = tanθ = (12-8)
s

(where s and t are shown in Figure 12-2). An R2 echelle, for example, has a blaze
angle of tan–1(2) = 63.4°; an R5 echelle has a blaze angle of tan–1(5) = 78.7°.

R number Groove angle

R1 45.0°
R2 63.4°
R3 71.6°
R3.5 74.1°
R4 76.0°
R5 78.7°

Table of common R numbers. The R number is simply the tangent of the groove angle.

Instruments using echelles can be reduced in size if the echelles are
“immersed” in a liquid of high refractive index n (see Figure 12-4). This has the
effect of reducing the effective wavelength by n, which is equivalent to
increasing the diffraction order, resolving power and dispersion of the echelle
(compared with the same echelle that is not immersed).153 A prism is usually

153 D. Enard and B. Delabre, “Two design approaches for high-efficiency low-resolution
spectroscopy,” Proc. SPIE 445, 522-529 (1984); G. Wiedemann and D. E. Jennings,
“Immersion grating for infrared astronomy,” Appl. Opt. 32, 1176-1178 (1993).

200

employed to couple the light to the grating surface, since at high angles most of
the light incident from air to the high-index liquid would be reflected. Often an
antireflection (AR) coating is applied to the normal face of the prism to minimize
the amount of energy reflected from the prism. 154

AR coating prism high index liquid

incident light

diffracted light

echelle grating

Figure 12-4. An immersed echelle grating used near Littrow. In this example, the incident
beam enters the prism normally at its face, so the prism contributes no angular dispersion
to the grism (except for what little results from Snell’s Law when the diffracted ray leaves
the prism).

154C. G. Wynne, “Immersed gratings and associated phenomena. I,” Opt. Commun. 73, 419-
421 (1989); C. G. Wynne, “Immersed gratings and associated phenomena. II,” Opt.
Commun. 75, 1-3 (1990).

201

13. APPLICATIONS OF DIFFRACTION
GRATINGS

13.1. GRATINGS FOR INSTRUMENTAL ANALYSIS

The most common use for the diffraction grating is to serve as the
wavelength separation device in an analytical laboratory instrument in which
matter is analyzed by studying its interaction with light (this analysis is called
spectroscopy). The spectral separation of the wavelengths is not strictly
required for this interaction; instead its purpose is to provide data that can be
interpreted unambiguously.
The techniques of analytical chemistry (i.e., that branch of chemistry that
determines the chemical composition of a substance by measuring its physical
properties) may be considered qualitative or quantitative. A qualitative
technique seeks to identify what is present; a quantitative technique seeks to
determine how much is present. Grating-based optical systems can be used to
identify or to quantify, by using the properties of light that is absorbed or
emitted by a substance (called absorption spectroscopy and emission
spectroscopy, respectively).
Most instruments designed for absorption spectroscopy are composed of
four primary elements (see Figure 13-1): a light source, a monochromator, a
sample illumination system, and a detector. Sometimes the monochromator and
sample illumination system are interchanged (that is, some instruments disperse
the light before it interacts with the sample, and some do this afterwards). Some
instruments use a spectrograph instead of a monochromator so that the entire
spectrum may be recorded at once.

13.1.1. Atomic and molecular spectroscopy

The field of atomic spectroscopy started with the observation by Balmer that
the discrete spectral lines emitted by a hydrogen source in the ultraviolet had

202

wavelengths that could be predicted by a simple formula; with the development
of quantum physics, the existence of a unique and predictable set of discrete
emission (and absorption) wavelengths for each chemical element was
hypothesized and subsequently observed. We now defined atomic absorption
spectroscopy as the measurement of the light absorbed by ionized atoms, and
atomic emission spectroscopy as the measurement of light emitted by energized
atoms or ions. Both the wavelength and the intensity of the light can be
measured using monochromators and spectrographs to provide information
about the atomic species.

Light (a)
Monochromator Sample Detector
Source

Light Monochromator
Sample Detector (b)
Source or Spectrograph

Figure 13-1. Absorption Spectrometer. Light from a broad-spectrum source, such as
deuterium (D2) or tungsten (W), is transmitted through the absorbing sample to be
analyzed and focused through the entrance slit of a monochromator (or spectrograph), and
the intensity of the light at each wavelength is recorded, producing an absorption
spectrum. In (a) the sample is illuminated by light after it has been spectrally tuned by
the monochromator; in (b) the sample is illuminated by the broad spectrum.

Many atomic emission instrument use an inductively coupled plasma (ICP)
composed of the atoms to be studied as the light source.
Molecular spectroscopy instrumentation is generally designed to transmit
light through a molecular species (often in a liquid suspension) and measure the
absorption at each wavelength.
Atomic and molecular spectroscopy is usually undertaken in the UV, visible
and IR portions of the spectrum, since atomic and molecular transition energies
lie in this range. Generally one instrument will cover only a portion of this wide
spectral range, leading to the classifications of UV spectrometers, UV-visible
spectrometers, visible spectrometers and IR spectrometers.

203

Both the wavelength and the intensity of the light can be measured using
monochromators and spectrographs to provide information about the atomic
species.

13.1.2. Fluorescence spectroscopy

Many atomic and molecular species fluoresce; that is, they absorb energy in
the UV-visible spectral region and rapidly emit most of that energy (the remainder
being converted to heat or vibrational energy in the medium). Generally, this
emission takes place on the order of nanoseconds after absorption, and (because
of the energy loss) the emission spectrum will appear at higher wavelengths than
the excitation spectrum (or, usually, a single excitation spectral line).
Fluorescent compounds may be identified by their unique fluorescence spectra;
in some applications, a non-fluorescent material may be tagged with a fluorescent
dye or fluorophore so that the non-fluorescent material may be detected using
fluorescence instrumentation.
Fluorescence instrumentation generally contains an excitation
monochromator, serving as a tunable filter for the excitation light, and an
emission spectrometer to disperse the emission spectrum (see Figure 13-2).

13.1.3. Colorimetry155

Colorimetry is the measurement and specification of color, used in analytical
chemistry, color matching, color reproduction and appearance studies. Because
color as perceived cannot be associated with a single wavelength – it is a more
complicated function of how the three different light receptors in the human eye
respond to the entire visible spectrum when looking at an object – it is common
to use a multiwavelength instrument such as a grating spectrometer.

155 E.g., C. J. Kok and M. C. Boshoff, “New spectrophotometer and tristimulus mask
colorimeter,” Appl. Opt. 10, 2617-2620 (1971); T. H. Chao, S. L. Zhuang, S. Z. Mao and F.
T. S. Yu, “Broad spectral band color image deblurring,” Appl. Opt. 22, 1439-1444 (1983).

204

Light Excitation
Sample
Source Monochromator

Emission
Monochromator

Detector

Figure 13-2. Fluorescence Spectrometer. Light from a broad-spectrum source is
spectrally tuned by an excitation monochromator; a spectrally narrow beam emerging
from this monochromator is absorbed by the sample. The fluorescence spectrum of the
sample is viewed (generally at an angle perpendicular to the excitation beam) and resolved
by an emission monochromator.

13.1.4. Raman spectroscopy

The interaction of light with matter falls into two broad categories:
absorption (on which absorption spectroscopy and fluorescence spectroscopy
are based), and scattering. Light can scatter elastically (i.e., energy is conserved)
or inelastically – the latter is called Raman scattering, and the study of the
spectrum of inelastically scattered light from matter is called Raman
spectroscopy. Since the ratio of intensities of inelastically scattered light to
elastically scattered light is generally under 10–6, the reduction of instrumental
stray light in Raman spectrometers is of paramount importance.

13.2. GRATINGS IN LASER SYSTEMS

Diffraction gratings are also used in laser systems to perform a number of
functions: to tune the lasing wavelength, to narrow the distribution of
wavelengths in the laser, and to control the pulse shape (vs. time).

205

13.2.1. Laser tuning

Lasing media have characteristic gain curves that describe the lasing
intensity vs. wavelength. In order to “tune” the laser to a wavelength with higher
gain with the gain curve, a grating can be used at one end of the resonant cavity
(in place of a mirror); using a grating instead of a mirror will disperse the
wavelengths in the laser, and the grating can be oriented so that the desired
wavelength propagates back into the lasing medium. 156
External-cavity semiconductor diode lasers are often used for their single-
mode operation and spectral tunability. Plane reflection gratings can be used in
the Littrow configuration to tune the lasing wavelength, as shown in Figure 13-3,
or in the grazing-incidence mount.

G OC

dye laser cell

Figure 13-3. Tuning a dye laser – the grating as a total reflector in the Littrow
configuration. Light from the dye laser cell is diffracted by the grating G, which is
oriented so that light of the desired wavelength is redirected back toward the cell; the
output beam is transmitted by an output coupler OC (which reflects most of the light
back into the laser). The wavelength is tuned by rotating the grating.

In some systems a beam expander is used to illuminate a larger area on the
grating surface, in order to achieve high resolution. Since the grating will allow
the zero-order to propagate as well as the (Littrow) diffraction order, the output
beam may be taken from the grating as in Figure 13-4.

156 T. M. Hard, “Laser wavelength selection and output coupling by a grating,” Appl. Opt. 9,
1825-1830 (1970); A. Hardy and D. Treves, “Modes of a diffraction grating optical
resonator,” Appl. Opt. 14, 589-592 (1975); T . W. Hänsch, “Repetitively pulsed tunable dye
laser for high resolution spectroscopy,” Appl. Opt. 11, 895-898 (1971); S. O. Kanstad & G.
Wang, “Laser resonators folded by diffraction gratings,” Appl. Opt. 17, 87-90 (1978).

206

Grazing-incidence tuning with one grating a ssociated with a mirror (or a
second grating) can also be used to tune dye lasers without the need for a beam
expander, leading to a more compact laser cavity; this is called the Littman-
Metcalf design157 and is shown in Figure 13-5.

G M

dye laser cell

Figure 13-4. Tuning a dye laser – the grating as output reflector. In this case, the zero-
order from the grating G is the output beam, and the output coupler in Figure 13-3 is
replaced by a mirror. The wavelength is tuned by rotating the grating.

G OC

dye laser cell

M

Figure 13-5. The Littman-Metcalf arrangement. The light diffracted by grating G is
retroreflected by mirror M, which diffracts the light again back into the dye laser cell.

157 M. G. Littman and H. J. Metcalf, “Spectrally narrow pulsed dye laser without beam
expander,” Appl. Opt. 17, 2224-2227 (1978).

207

Molecular lasers, operating in either a pulsed or continuous-wave (cw)
mode, have their output wavelength tuned by Littrow-mounted gratings. High
efficiency is obtained by using the first diffraction order at diffraction angles
|β | > 20°. The output is polarized in the S-plane, since the efficiency in the P
plane is quite low.
Some molecular lasers operate at powers high enough to destroy gratings.
For pulsed laser tuning, extra-thick replica films may help, but at maximum power
only master gratings survive. Due to their far greater thermal conductivity,
replica gratings on metal substrates are superior to glass for cw laser
applications; in some cases, the grating substrates must be water-cooled to
prevent failure.
Excimer lasers – used in surgery, micromachining and photolithography –
generally select a narrow spectral range from the emission profile by using an
echelle grating in the Littrow configuration.158 A coarse echelle (d > 10 µm) is
used in very high diffraction orders (m >> 10) at very high incidence angles (α =
65° to 79°) in order to obtain high dispersion (see Eq. (2-15)). At such an oblique
angle, a beam with circular cross section will illuminate an ellipse on the grating
that is three to five times wider in the dispersion direction than it is in the cross-
dispersion direction.

13.2.2. Pulse stretching and compression159

For optical systems employing lasers with very high peak powers, such as
those that use temporally short (< 1 ps) yet energetic (≈ 1 J) pulses, the required
damage thresholds of the optical components in the system can exceed the
performance of state-of-the-art components. Strickland and Mourou160
demonstrated that such pulses can be stretched (in time) so that their pulse
energy is spread out over a large time period (thereby reducing the peak power)

158 R Buffa, P Burlamacchi, R Salimbeni and M Matera, “Efficient spectral narrowing of a
XeCl TEA laser,” J. Phys. D: Appl. Phys. 16 L125-L128 (1983); J. P. Partenen, “Multipass
grating interferometer applied to line narrowing in excimer lasers,” Appl. Opt. 25, 3810-
3815 (1986).
159 E. Treacy, “ Optical pulse compression with diffraction gratings,” IEEE J. Quantum Elec.
5, 454-458 (1969).
160D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt.
Comm. 56, 219-221 (1985).

208

and then compressed using a grating compressor to return the pulse to its
original temporal profile. Between the two operation (stretching and
compression), optical comp onents are exposed to much lower peak powers than
that of the original (or final) pulse. By amplifying the pulse between the stretcher
and compressor, higher peak power pulses may be obtained.
A dual-grating pulse stretcher is shown in Figure 13-6.

G2 G1
` `

M

Figure 13-6. A grating-based pulse stretcher. Intermediate lenses are not shown.

13.3. GRATINGS IN ASTRONOMICAL APPLICATIONS

Much of what we know of the universe is due to our analysis of light
reaching the earth from planets, stars and galaxies. Grating-based spectrometers
play a key role in astronomical measurements. For example, the spectroscopic
analysis of starlight allows us to determine the composition of stars as well as
their relative velocities. The analysis of absorption lines in starlight that passes
through nebulae allows us to determine the composition of the nebulae. From
the analyses of these emission and absorption spectra, we can infer ages of
stars, distances to galaxies, etc.

209

13.3.1. Ground-based astronomy161

Ground-based astronomical telescopes generally have quite large apertures,
to maximize the light energy collected from distant astronomical objects; this
leads to the need for very large gratings to spectrally disperse the light received.
Often these gratings are so large that their resolving power exceeds the value for
which the spectrometer’s resolution would be grating-limited; that is, in most
cases the grating is ‘better’ than the instrument’s resolution requires.
Newport’s 'B' engine can rule large echelles and echellettes up to 320 mm x
420 mm in size (which provides a ruled area of 308 mm x 408 mm), suitable for all
but the largest ground-based astronomical instruments.162 The requirement for
even larger gratings for ground-based astronomical telescopes has led to three
alternative solutions: a static fixture to hold smaller gratings in a larger
configuration, an adjustable fixture with optical feedback to move the gratings
with respect to each other (to maintain focus)163 , and a mosaic grating produced
by high-accuracy multiple replication onto a single substrate.164 Such monolithic
mosaic gratings have the advantage of long-term alignment stability over the
other two alternatives.
In the 1990s, Newport developed the capability to replicate two large
submaster gratings onto one monolithic substrate. Except for a “dead space”
between the two replicated areas, the entire face of the larger product substrate
contains the groove pattern. This mosaic grating must have its two grating areas
aligned to very high accuracy if the mosaic is to perform as one high-quality
grating. Typical specifications for two 308 mm x 408 mm ruled areas on a 320 mm

161 E.g., W. Liller, “High dispersion stellar spectroscopy with an echelle grating,” Appl. Opt.
9, 2332-2336 (1970); D. J. Schroeder, Astronomical Optics, Academic Press (San Diego,
California: 1987).
162 S. S. Vogt and G. D. Penrod, “HiRES: A high resolution echelle spectrometer for the
Keck 10-meter t elescope,” in Instrumentation for Ground-Based Astronomy, L. B. Robinson,
ed. (Springer-Verlag, New York: 1988), pp. 68-103.
163G. A. Brealey, J. M. Fletcher, W. A. Grundmann and E. H. Richardson, “Adjustable
mosaic grating mounts,” Proc. SPIE 240, 225-228 (1980).
164 H. Dekker and J. Hoose, “Very high blaze angle R4 echelle mosaic,” Proc. ESO Workshop
on High Resolution Spectroscopy, M.-H. Ulrich, ed., p. 261 (1992); J. Hoose et al., "Grand
Gratings: Bigger is Better, Thanks to Mosaic Technology," Photonics Spectra 29, 118-120
(December 1995); T. Blasiak and S. Zheleznyak, "History and construction of large mosaic
diffraction gratings," Proc. SPIE 4485, 370-377 (2002).

210

x 840 mm substrate are one arc second alignment of the groove directions, one
arc second tilt between the two faces, and one micron displacement between the
two grating planes.
A large mosaic echelle grating produced by Newport for the European
Southern Observatory is shown in Figure 13-7. Two submasters from Newport
master MR160 (a 31.6 g/mm echelle blazed at 75.1°) were independently replicated
onto a large monolithic substrate to form this mosaic grating; the two halves of
its surface are clearly seen in the photograph.
Figure 13-8 shows a six-inch aperture Fizeau interferogram of an echelle
mosaic (31.6 g/mm) in the m = 98th order, tested at λ = 632.8 nm. The grooves are
vertical in the photos and the blaze arrow is facing left. One fringe over this
aperture is 0.43 arc seconds. These measurements indicate that the two sides of
the mosaic are aligned to 0.3 arc seconds.
Figure 13-9 shows a focal plane scan on a ten-meter optical test bench using
a mode-stabilized HeNe laser (λ = 632.8 nm) as the light source. The entrance slit
width is 25 microns, and the exit slit is opened just enough to get signal through.
The grating is operating in the m = 97th order with full aperture illumination. The
image seems to be dominated by the wavefront characteristics of the individual
segments, but still indicates a system resolving power better than R = 900,000.

211

Figure 13-7. A large mosaic grating. A monolithic 214 x 840 mm replica mosaic grating
was produced from two 214 x 415 mm submasters.

Alignment perpendicular Alignment parallel
to grooves to grooves

Figure 13-8. Six-inch-aperture Fizeau interferograms of a 31.6 g/mm echelle mosaic
produced from two 214 x 415 mm submasters. The photograph on the left shows
alignment perpendicular to the grooves; that on the right shows alignment in the direction
of the grooves. These interferograms were taken in the 98th diffraction order.

212

MR160 HARPS Echelle mosaic resolution at 632.8 nm
1

0.9

0.8
Normalized picoameter current

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0
-20 -15 -10 -5 0 5 10 15 20
Scan Distance in microradians

Figure 13-9. Signal trace of a 31.6 g/mm echelle mosaic at 632.8 nm in the 97th order.

13.3.2. Space-borne astronomy165

Neither master nor replica gratings suffer in any measurable way over ex-
tended periods of time in a space environment. The advantage of replica
gratings lies not only in their greater availability and lower cost, but in making
possible the provision of exact duplicates whenever needed.
Since most space work involves the study of ultraviolet (UV) and extreme
ultraviolet (EUV) wavelengths, special problems exist in setting and aligning the
optics. For this purpose Newport can rule gratings matching the EUV grating but
with a groove spacing modified so that the mercury 546.1-nm line lies in the

165 E.g., J. F. Seely, M. P. Kowalski, W. R. Hunter, T. W. Barbee Jr., R. G. Cruddace and J. C.
Rife, “Normal-incidence efficiencies in the 115-340-Å wavelength region of replicas of the
Skylab 3600-line/mm grating with multilayer and gold coatings,” Appl. Opt. 34, 6453-6458
(1995).

213

spectrum just where the main wavelength under study will lie. Another
possibility is to rule a small section on the main grating with similar coarse
spacings and then mask off this area when the alignment is complete. Sometimes
special tolerances on substrate radii are required for complete interchangeability.

13.4. GRATINGS IN SYNCHROTRON RADIATION BEAMLINES

Synchrotron radiation is generated by electrons traveling in circular orbits at
relativistic speeds; this radiation covers the x-ray through infrared portions of
the electromagnetic spectrum and may be used to investigate the electronic
properties of matter. Synchrotron beamlines are optical systems oriented
tangentially to synchrotron rings, and often gratings are used to disperse the
portion of the radiation in the extreme ultraviolet (UV) and vacuum ultraviolet
(VUV) spectra.166

13.5. SPECIAL USES FOR GRATINGS

In addition to the “traditional” uses of gratings – in analytical instruments,
lasers and astronomical telescopes – there are a number of other uses for which
diffraction gratings are well-suited.

13.5.1. Gratings as filters

Diffraction gratings may be employed as reflectance filters when working in
the far infrared, in order to remove the unwanted second- and higher- diffraction
orders from the light.167 For this purpose, small plane gratings are used that are
blazed for the wavelength of the unwanted shorter-wavelength radiation. The
grating acts as a mirror for the longer-wavelength light, reflecting the desired

166 D. L. Ederer, ed., Selected Papers on VUV Synchrotron Radiation Instrumentation –
Beam Line and Instrument Development, SPIE Milestone Series vol. MS 152, SPIE
(Bellingham, Washington: 1998).
167J. U. White, “Gratings as broad band filters for the infra-red,” J. Opt. Soc. Am. 37, 713-
717 (1947).

214

light into the instrument, while diffracting shorter wavelengths out of the optical
path. The groove spacing d must be chosen so that

sinβ > 1 for all λ > λC, (13-1)

where λC is a wavelength between the short wavelengths to be diffracted and the
long wavelengths to be reflected (see Eq. (2-1)).
A grating can also be used as a color filter if it is illuminated such that its
zero-order efficiency is highly wavelength-dependent.168
It should be recognized that a diffraction grating by itself cannot serve as a
spectral bandpass filter. The grating provides spectral dispersion but not
spectral resolution, so the analogue of a thin-film filter designed to pass a narrow
spectral band would be the combination of a grating and a slit (see Figure 13-10).
A grating monochromator (as described in Chapter 3) may be thought of as a
tunable filter – rotating the grating tunes the central wavelength in the
transmitted spectral band, and the exit slit serves to narrow this band.

13.5.2. Gratings in fiber-optic telecommunications

In the late 1990s, surface-relief diffraction gratings became widely used in
two types of equipment for fiber-optic telecommunications networks operating in
the 1.3–1.7 µm wavelength range. While other wavelength selective tech-
nologies exist (e.g., interference filters, fiber Bragg gratings and array waveguide
gratings), the cost advantage of surface-relief gratings becomes significant as
the channel count increases, since a system with N channels requires N–1 filters
but only a single grating; that is, the filters must act in series (in a cascade
arrangement) but the grating acts on all channels in parallel. Moreover, as N
increases, the spectral bandpass of the filters must decreases, further increasing
their cost.

168 K. Knop, “Diffract ion gratings for color filtering in the zero diffraction order,” Appl.
Opt. 17, 3598-3603 (1978).

215

XS
G

∆λ

Figure 13-10. Spectral resolution using a grating and a slit. Polychromatic light incident
on and diffracted by the grating G is not spectrally resolved; the grating merely diffracts
each wavelength in the incident beam in a different direction. A spectral narrow band ∆λ
is obtained by using exit slit XS to prevent all wavelengths outside this band from passing
to the detector.

Multiplexers & Demultiplexers.169 A multiplexer (see Figure 13-11(a)) is a
component in a fiber-optic network that combines many input channels into one
output channel; as the input channels have different wavelengths, the
multiplexer can be considered a spectrograph used in reverse. A demultiplexer
(Figure 13-11(b)) separates many wavelengths in a single input channel so that
each is transmitted into a unique output channel (this is functionally equivalent
to a spectrograph). Multiplexers and demultiplexers may be employed together
to produce add-drop routers.

169T. Kita and T. Harada, “Use of aberration-corrected concave gratings in optical
demultiplexers,” Appl. Opt. 22, 819-825 (1983).

216

MUX DEMUX

(a) (b)

Figure 13-11. Fiber-optic network components. (a) Multiplexer: many input beams (each
of a unique wavelength) are combined to propagate down the same output path. (b)
Demultiplexer: the several signals in the (combined) input beam are separated by
wavelength. For simplicity, only four wavelengths are shown.

Optical Spectrum Analyzers. In addition to serving in network components,
gratings are used in optical spectrum analyzers which use a small fraction of the
light in the network to monitor the intensity and stability of each channel. These
systems are essentially spectrographs, and may use plane or concave gratings.

13.5.3 Gratings as beam splitters

Gratings can be used as beam splitters in conjunction with Moiré fringe ap-
plications or interferometers. Under normal illumination (α = 0), a grating with a
symmetric groove profile will diffract both first-order beams with equal intensity.
A diffraction grating used as a beam divider provides higher efficiencies when its
groove profile is rectangular, whereas a grating used for spectroscopic purposes
should have a sinusoidal or triangular groove profile.
Transmission gratings can be used as two-beam splitters (where the zero-
order beam has negligible efficiency or is otherwise trapped), three-beam splitters
(where the groove profile is chosen so that the zero-order beam has the same
intensity as the two first-order beams), or for multiple beam sampling, depending
on the choice of groove profile.170

170 E. G. Loewen, L. B. Mashev and E. K. Popov, “Transmission gratings as 3-way beam
splitters,” Proc. SPIE 815, 66-72 (1987); E. K. Popov, E. G. Loewen and M. Nevière,
"Transmission gratings for beam sampling and beam splitting," Appl. Opt. 35, 3072-3075

217

13.5.4 Gratings as optical couplers

Gratings can be used to couple light into and out of waveguide
structures.171 Generally the groove spacing d is specifically chosen to ensure
that only one diffraction order (other than the zero order) propagates.

13.5.5 Gratings in metrological applications

Diffraction gratings can be employed in a variety of metrological
applications. The precise microscopic surface-relief pattern can be used to
calibrate atomic force microscopes (AFMs). Gratings can also be used in
systems designed to measure displacement172 and strain.173

(1996); E. G. Loewen and E. Popov, Diffraction Gratings and Applications, Marcel Dekker,
Inc. (1997), ch. 5.
171T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235-
254 (1977).
172 J.-A. Kim, K.-C. Kim, E. W. Bae, S. Kim and Y. K. Kwak, “ Six-degree-of-freedom
displacement measurement using a diffraction grating,” Rev. Sci. Instrum. 71, 3214-3219
(2000).
173 B. Zhao and A. Asundi, ”Strain microscope with grating diffraction method,” Opt. Eng.
38, 170-174 (1999).

218

14. ADVICE TO GRATING USERS

.

14.1. CHOOSING A SPECIFIC GRATING

If a diffraction grating is to be used only to disperse light by wavelength
(rather than provide focusing as well), then choosing the proper grating is often
a straightforward matter involving the specification of the blaze angle and groove
spacing. In other instances, the problem is one of deciding on the spectrometric
system itself. The main parameters that must be specified are

• Spectral region (wavelength range)
• Wavelength of peak efficiency
• Speed (focal ratio) or throughput
• Resolution or resolving power
• Dispersion
• Free spectral range
• Output optics
• Size limitations

The spectral region, spectral resolution and size requirements will usually
lead to a choice of plane vs. concave design, as well as the coating (if the grating
is reflecting). The size and weight of the system, the method of receiving output
data, the intensity, polarization and spectral distribution of the energy available,
etc., must also be considered. The nature of the detection system, especially for
array detectors, plays a major role in system design: its size, resolution, and
image field flatness are critical issues in the specification of the optical system,
and the sensitivity (vs. wavelength) of the detector will lead to a grating
efficiency vs. wavelength specification.
Spectral resolution depends on many aspects of the optical system and the
quality of its components. In some cases, the grating may be the limiting

219

component (see Section 8.3). The decision here involves the size of the grating
and the angle at which it is to be used, but not on the number of grooves on the
grating or the groove spacing (see Chapter 2).
Speed (or throughput) determines the focal length as well as the sizes of the
optical elements and of the system itself. Special overcoatings become important
in certain regions of the spectrum, especially the vacuum ultraviolet.
When thermal stability is important, gratings should be made on a low
expansion material, such as ZeroDur™ or ULE® fused silica.
Guidelines for specifying gratings are found in Chapter 16.

14.2. APPEARANCE

In the early days of diffraction grating manufacture, R.W. Wood remarked
that the best gratings were nearly always the worst ones in their cosmetic or
visual appearance. While no one would go so far today, it is important to realize
that a grating with certain types of blemishes may well perform better than one
that appears perfect to the eye.

14.2.1. Ruled gratings

Cosmetic defects on ruled gratings may be caused by small droplets of metal
or oxide that have raised the ruling diamond, or streaks may be caused by
temporary adhesion of aluminum to the sides of the diamond tool. On ruled
concave gratings, one can usually detect by eye a series of concentric rings
called a target pattern. It is caused by minor changes in tool shape as the
diamond swings through the arc required to rule on a curved surface. Every
effort is made to reduce the visibility of target patterns to negligible proportions.
Some ruled master gratings have visible surface defects. The most common
sort of defect is a region of grooves that are burnished too lightly (in relation to
the rest of the grating surface). While readily seen with the eye, such a region
has little effect on spectroscopic performance.

220

14.2.2 Holographic gratings

Holographic gratings are susceptible to a different set of cosmetic defects.
Comets are caused by specks on the substrate; when the substrate is rotated
(spun) as the photoresist is applied, these specks cause the photoresist to flow
around them, leaving comet-like trails. Artifacts created during the recording
process are also defects; these are holograms of the optical components used in
the recording of the grating.

14.3. GRATING M OUNTING

The basic rule of mounting a grating is that for any precise optical element:
its shape should not be changed accidentally through excessive clamping
pressure. This problem can be circumvented by kinematic (three-point)
cementing, using a nonrigid cement, or by supporting the surface opposite the
point where clamping pressure is applied.
If a grating is to be mounted from the rear, the relative orientations of the
front and rear surfaces is more important than if the grating is to be mounted from
the front. Front-mounting a plane grating, by contacting the mount to the front
surface of the grating (near the edge of the grating and outside the free aperture),
allows the cost of the substrate to be lower, since the relative parallelism of the
front and back surfaces need not be so tightly controlled.

14.4. GRATING SIZE

Grating size is usually dictated by the light throughput desired (and, in the
case of concave gratings, imaging and instrument size limitations as well).
Should none of the standard substrate sizes listed in the Newport Diffraction
Grating Catalog be suitable to match an instrument design, these same gratings
can be supplied on special size substrates. Special elongated substrate shapes
are available for echelles and laser tuning gratings.

221

14.5. SUBSTRATE M ATERIAL

The standard material for small and medium-sized grating substrates is
specially annealed borosilicate crown glass (BK-7). Low-expansion material,
such as ZeroDur® or fused silica, can be supplied upon request. For large
gratings (approximately 135 x 265 mm or larger), low-expansion material is
standard; BK-7 can be requested as well. For certain applications, it is possible
to furnish metal substrates (e.g., copper or aluminum) that are good heat sinks.

14.6. GRATING COATINGS

While evaporated aluminum is the standard coating for reflection gratings,
fast-fired aluminum with overcoatings of magnesium fluoride (MgF2) can be used
to enhance efficiency in the spectrum between 120 and 160 nm. For the extreme
ultraviolet (below 50 nm), gold replica gratings are recommended, while platinum
is recommended for 80–110 nm. Gold replicas also have higher reflectivity in
most regions of the infrared spectrum, and are particularly useful for fiber-optic
telecommunications applications in the S, C and L (infrared) transmission bands.†

† The transmission bands used in fiber-optic telecommunications are usually defined as
follows: S (short) band: c. 1435 to 1535 nm, C (conventional) band: c. 1525 to 1565 nm, and
L (long) band: c. 1565 to 1630 nm. These definitions are by no means universal, but between
them they cover the near-IR amplification range of erbium -doped optical fibers.

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15. HANDLING GRATINGS

A diffraction grating is a first surface optic, so its surface cannot be
touched or otherwise come in contact with another object without damaging it
and perhaps affecting its performance. Damage can take the form of
contamination (as in the adherence of finger oils) or distortion of the microscopic
groove profile in the region of contact. This chapter describes the reasons why a
grating must be handled carefully and provides guidelines for doing so.

15.1. THE GRATING SURFACE

Commercially available diffraction gratings are replicated optics comprised of
three layers: a substrate, a resin layer, and (usually) a reflective coating (see
Chapter 5). Each layer serves a different purpose: (1) the metallic layer provides
high reflectivity, (2) the resin layer holds the groove pattern and groove profile,
and (3) the substrate (usually glass) keeps the optical surface rigid.

15.2. PROTECTIVE COATINGS

Since the groove profile is maintained by the resin layer, rather than the
reflective (metallic) coating on top of it, protective coatings such as those that
meet the military specification MIL-M-13508 (regarding first-surface aluminum
mirrors) do not serve their intended purpose. Even if the aluminum coating itself
were to be well-protected against contact damage, it is too thin to protect the
softer resin layer underneath it. “Fully cured" resin is not very hard, resembling
modeling clay in its resistance to contact damage. Consequently gratings are not
provided with contact-protecting coatings.

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aluminum

epoxy

substrate

Figure 15-1. Composition of a replica diffraction grating. A section of a standard blazed
grating with an aluminum coating is shown. Layer thicknesses are not shown to scale:
generally the aluminum film thickness is about 1 µm, and the resin layer is between 10
and 50 µm, depending on groove depth and grating size; the substrate thickness is usually
between 3 and 100 mm.

15.3. GRATING COSMETICS AND PERFORMANCE

Warnings against touching the grating surface notwithstanding, damage to
the surface occasionally occurs. Contact from handling, mounting or packaging
can leave permanent visible marks on the grating surface. Moreover, some
gratings have cosmetic defects that do not adversely impair the optical
performance, or perhaps represent the best available quality for a grating with a
particular set of specifications. For example, some gratings have 'worm tracks'
due to mechanical ruling of the master grating from which the replicated grating
was taken, others have coating defects like spit or spatter, and others have
'pinholes' (tiny voids in the reflective coating), etc. The many possible
classifications of surface defects and the many opportunities to render the
surface permanently damaged conspire to make the surfaces of m any gratings
look less than cosmetically perfect.
While this damage may be apparent upon looking that the grating, it is not
straightforward to determine the effect this damage has on the performance of
the grating. Often the area affected by damage or c ontamination is a small
fraction of the total area of the grating. Therefore, only a small portion of the

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total number of grooves under illumination may be damaged, displaced or
contaminated. A damaged or contaminated region on the surface of a grating
may have little, if any, noticeable effect on the performance of the optical system,
because a diffraction grating is usually used as an integrating optic (meaning
that all light of a given wavelength diffracted from the grating surface is brought
to focus in the spectral order of interest). In contrast, a lens or mirror that does
not focus (say, an eyeglass lens or a bathroom mirror) will show a distortion in its
image corresponding to the damaged region of the optic. This familiar experience
– the annoying effect of a chip on an eyeglass lens or a smudge on a bathroom
mirror – has led many to assume that a similar defect on the surface of a grating
will lead to a similar deficiency in performance. The most appropriate
performance test of a grating with surface damage or cosmetic defects is not
visual inspection but instead to use the grating in its optical system and
determine whether the entire system meets its specifications.
Damage to a region of grooves, or their displacement, will theoretically have
some effect on the efficiency of the light diffracted from that region, as well as
the total resolving power of the grating, but in practice such effects are generally
not noticeable. Of more concern, since it may be measurable, is the effect surface
damage may have on light scattered from the grating, which may decrease the
signal-to-noise (SNR) of the optical system. Most forms of surface damage can
be thought of as creating scattering centers where light that should be diffracted
(according to the grating equation) is scattered into other directions instead.

15.4. UNDOING DAMAGE TO THE GRATING SURFACE

Damage to the microscopic groove profile is, unfortunately, irreversible; the
resin layer, like modeling clay, will retain a permanent imprint. Contamination of
the grating surface with finger oils, moisture, vacuum pump oil, etc. is also often
permanent, particularly if the contaminated grating surface has been irradiated.
Sometimes surface contamination can be partially removed, and once in a
while completely, using a mild unscented dishwashing liquid. Care should be
taken not to apply any pressure (even gentle scrubbing) to the grating surface. If
contaminants remain, try using spectroscopic-grade solvents; the purity of such
solvents should be as certained before use, and only the purest form available

225

used. The use of carbon dioxide (CO2) snow,174 which reaches the grating
surface in a sublimed state and evaporates, carrying with it the contaminants, has
also been used with some success. The key to cleaning a grating surface is to
provide no friction ( .g., scrubbing) that might damage the delicate groove
e
structure.

15.5. GUIDELINES FOR HANDLING GRATINGS

• Never touch the grooved surface of a diffraction grating. Handle a
grating by holding it by its edges. If possible, use powder-free gloves
while handling gratings.
• Never allow any mount or cover to come in contact with the grooved
surface of a diffraction grating. A grating that will be shipped should
have its surface protected with a specially-designed cover that does not
touch the surface itself. Gratings that are not in use, either in the
laboratory or on the manufacturing floor, should be kept in a closed box
when not covered. Keep any oils that may be used to lubricate grating
mount adjustments away from the front surface of the grating.
• Do not talk or breathe over the grooved surface of a diffraction
grating. Wear a nose and face mask when it is required that you talk
over the surface of a grating. Breath spray is particularly bad for
reflection gratings, so one should not speak directly over the grating
surface; instead, either turn away or cover the mouth (with the hand or a
surgical mask).

174 R. R. Zito, “Cleaning large optics with CO2 snow,” Proc. SPIE 1236, 952-972 (1990).

226

16. GUIDELINES FOR SPECIFYING
GRATINGS

Proper technical specifications are needed to ensure that the part supplied
by the manufacturer meets the requirements of the customer. This is especially
true for diffraction gratings, whose complete performance features may not be
fully recognized. Documents that provide guidance in the specification of optical
components, such as the ISO 10110 series ("Optics and optical instruments:
Preparation of drawings for optical elements and systems"), do not clearly lend
themselves to the specification of diffraction gratings. This chapter provide
guidelines for generating clear and complete technical specifications for gratings.
Specifications should meet the following criteria.
• They should refer to measurable properties of the grating.
• They should be as objective as possible (avoiding judgment or
interpretation).
• They should be quantitative where possible.
• They should employ common units where applicable (the SI system is
preferred).
• They should contain tolerances.
A properly written engineering print for a diffraction grating will be clear and
understandable to both the customer and the manufacturer.

16.1. REQUIRED SPECIFICATIONS

All grating prints should contain, at a minimum, the following specifications.

1. Free Aperture. The free aperture, also called the clear aperture, of a
grating is the maximum area of the surface that will be illuminated. The
free aperture is assumed to be centered within the ruled area (see
below) unless otherwise indicated. For configurations in w hich the

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grating will rotate, such as in a monochromator, it is important to specify
the free aperture as the maximum dimensions of the beam on the grating
surface (i.e., when the grating is rotated most obliquely to the incident
beam). Also, it is important to ensure that the free aperture specifies an
area that is completely circumscribed by the ruled area, so that the
illuminated area never includes part of the grating surface that does not
have grooves.
The free aperture of the grating is that portion of the grating surface for
which the optical specifications apply (e.g., Diffraction Efficiency,
Wavefront Flatness or Curvature, Scattered Light – see below).
2. Ruled Area. The ruled area of a grating is the maximum area of the
surface that will be covered by the groove pattern. The ruled area is
assumed to be centered on the substrate face unless otherwise
indicated. By convention, the ruled area of a rectangular grating is
specified as "groove length by ruled width" – that is, the grooves are
parallel to the first dimension; for example, a ruled area of 30 mm x 50 mm
indicates that the grooves are 30 mm long.
Most rectangular gratings have their grooves parallel to the shorter
substrate dimension. For gratings whose grooves are parallel to the
longer dimension, it is helpful to specify "long lines" to ensure that the
grooves are made parallel to the longer dimension.
3. Substrate Dimensions. The substrate dimensions (width, length, and
thickness) should be called out, as should their tolerances. If the
grating is designed to be front-mounted, the substrate specifications
can be somewhat looser than if the grating surface will be positioned or
oriented by the precise placement of the substrate. Front-mounting a
grating generally reduces its cost and production time (see Alignment
below).
A grating substrate should have bevels on its active face, so that it is
easier to produce and to reduce chipping the edges while in use. Bevel
dimensions should be specified explicitly and should be considered in
matching the Ruled Area (above) with the substrate dimensions.† For
custom (special-size) substrates, certain minimum bevel dimensions may

† Newport’s standard bevels have a face width of 1.5 mm and are oriented at 45° to the two
edges.

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be required to ensure that the grating is manufacturable – please
contact us for advice.
4. Substrate Material. The particular substrate material should be
specified. If the material choice is of little consequence, this can be left
to the manufacturer, but especially for applications requiring substrates
with low thermal expansion coefficients, or requiring gratings that can
withstand high heat loads, the substrate material and its grade should
be identified. For transmission gratings, the proper specification of the
substrate material should include reference to the fact that the substrate
will be used in transmission, and may additionally refer to specifications
for refractive index, inclusions, bubbles, striae, etc.
5. Nominal Surface Figure. Plane (flat) gratings should be specified as
being planar; concave gratings should have a radius specified, and the
tolerance in the radius should be indicated in either millimeters or
λ
fringes of red HeNe light ( = 632.8 nm) (a "wave" being a single
wavelength, equaling 632.8 nm, and a "fringe" being a single half-
wavelength, equaling 316.4 nm). Deviations from the nominal surface
figure are specified separately as "wavefront flatness" or "wavefront
curvature" (see below).
6. Wavefront Flatness or Curvature. This specification refers to the
allowable deviation of the optical surface from its Nominal Surface
Figure (see above). Plane gratings should ideally diffract plane
wavefronts when illuminated by collimated incident light. Concave
gratings should ideally diffract spherical wavefronts that converge
toward wavelength-specific foci. In both cases, the ideal radius of the
diffracted wavefront should be specified (it is infinite for a plane
grating) and maximum deviations from the ideal radius should also be
called out (e.g., the tolerance in the radius, higher-power irregularity in
the wavefront). It is important to specify that grating wavefront testing
be done in the diffraction order of use if possible, not in zero order,
since the latter technique does not measure the effect of the groove
pattern on the diffracted wavefronts. Deviations from a perfect
wavefront are most often specified in terms of waves or fringes of red
HeNe light. Generally, wavefront is specified as an allowable deviation
from the nominal focus ("power") and allowable higher-order curvature
("irregularity").

229

7. Groove Spacing or Frequency. The number of grooves per millimeter,
or the spacing between adjacent grooves, should be specified, but not
both (unless one is subjugated to the other by labeling it as
"reference"). For a grating whose groove spacing varies across the
surface (e.g., an aberration-corrected concave holographic grating), the
groove spacing (or frequency) is generally specified at the center of the
grating surface.
8. Groove Alignment. Alignment refers to the angle between the groove
direction and an edge of the grating substrate. Sometimes this angular
tolerance is specified as a linear tolerance by stating the maximum
displacement of one end of a groove (to an edge) relative to the other
end of the groove. Generally a tight alignment specification increases
manufacturing cost; it is often recommended that alignment be allowed
to be somewhat loose and that the grating substrate dimensions not be
considered for precise alignment but that the grating surface be oriented
and positioned optically instead of mechanically (see comments in
Substrate Dimensions above).
9. Diffraction Efficiency. Grating efficiency is generally specified as a
minimum at a particular wavelength; often this is the peak wavelength
(i.e., the wavelength of maximum efficiency). Occasionally efficiency
specifications at more than one wavelength are called out.
Either relative or absolute diffraction efficiency should be specified.
Relative efficiency is specified as the percentage of the power at a given
wavelength that would be reflected by a mirror (of the same coating as
the grating) that is diffracted into a particular order by the grating (that
is, efficiency relative to a mirror). Absolute efficiency is specified as the
percentage of the power incident on the grating that is diffracted into a
particular order by the grating.
In addition to the wavelength and the diffraction order, grating
efficiency depends on the incidence and diffraction angles α and β; if
these angles are not explicitly stated, the standard configuration
(namely the Littrow configuration, in which the incident and diffracted
beams are coincident) will generally be assumed. Unless otherwise
noted on the curves themselves, all Newport efficiency curves are
generated for the near-Littrow conditions of use with eight degrees
between the incident and diffracted beams: α − β = 2K = 8°.
Generally diffraction gratings are polarizing elements, so that the
efficiency in both polarizations should be considered:

230

P-plane TE light polarized parallel to grooves
S-plane TM light polarized perpendicular to grooves

For each wavelength that has an efficiency specification, the following
should be indicated: the wavelength, the efficiency (in percent), whether
the efficiency specification is relative or absolute, the diffraction order,
the polarization of the light, and the angles α and β. In some cases, the
bandwidth of the exit slit in the spectrometer used to measure the
grating efficiency may need to be called out as well.

16.2. SUPPLEMENTAL SPECIFICATIONS

Additional specifications are sometimes required based on the particular
application in which the grating is to be used.

10. Blaze Angle. Although it is better to specify diffraction efficiency,
which is a performance characteristic of the grating, sometimes the blaze
angle is specified instead (or additionally). A blaze angle should be
specified only if it is to be measured and verified (often done by
measuring efficiency anyway), and a tolerance should be noted. In
cases where both the diffraction efficiency and the blaze angle are
specified, the efficiency specification should be controlling and the
blaze angle specification should be for reference only.
11. Coating Material. Generally the Diffraction Efficiency specifications
will dictate the coating material, but sometimes a choice exists and a
particular coating should be specified. Additionally, dielectric
overcoatings may be called out that are not implied by the efficiency
specifications.
12. Scattered Light. Grating scattered light is usually specified by requiring
that the fraction of monochromatic light power incident on the grating
and measured a particular angle away from the diffracted order falls
below a certain upper limit. The proper specification of scattered light
would call out the test configuration, the polarization and wavelength of
the incident light, the incidence angle, the solid angle subtended by the

231

detector aperture, and the dimensions of the exit slit. Grating scatter is
measured at Newport using red HeNe light.
13. Cosmetics. The cosmetic appearance of a diffraction grating does not
correlate strongly w the performance of the grating, and for this
ith
reason specifications limiting the type, number and size of cosmetic
defects are not recommended. Nevertheless, all Newport gratings
undergo a rigorous cosmetic inspection before shipment.
14. Imaging Characteristics. Concave holographic gratings may be
aberration-corrected, in which case they can provide focusing without
the use of auxiliary optics. In these cases, imaging characteristics
should be specified, generally by calling out the full width at half
maximum intensity (FWHM) of the images.
15. Damage Threshold. In some instances, such as pulsed laser
applications, diffracted gratings are subjected to beams of high power
density that may cause damage to the delicate grating surface, in which
case the maximum power per unit area that the grating surface must
withstand should be specified.
16. Other specifications. Other specifications that relate to the functional
performance of the grating should be called out in the print. For
example, if the grating must perform in extreme environments (e.g., a
satellite or space-borne rocket, high heat and/or humidity
environments), this should be noted in the specifications.

16.3. ADDITIONAL REQUIRED SPECIFICATIONS FOR CONCAVE
ABERRATION-REDUCED GRATINGS

Concave aberration-reduced gratings, often used in constant-deviation and
flat-field spectrograph mounts (see Sections 7.5.3 and 7.5.5), have imaging
properties that are tailored to the specific geometry of the spectrometer; that is,
the grating recording coordinates γ, rC, δ and rD depend on the use coordinates α,
r, β and r′ (all of these quantities are defined in Chapter 7). Consequently, a
concave aberration-reduced grating requires additional specifications to be fully
described.
The cases for constant-deviation monochromators and flat-field
spectrographs are given separately below. In all cases, a clear optical schematic

232

showing the quantities defined is highly recommended, especially to ensure that
the definition of angles is understood.
17. Constant-deviation monochromator gratings. Concave holographic
gratings used in constant deviation monochromators should have the
following three parameters specified (see Figure 16-1) to be defined
uniquely:
• the distance r from the entrance slit to the grating center (often
called the entrance arm distance),
• the distance r′ from the exit slit to the grating center (the exit arm
distance), and
• the angle 2K between these two arms (the deviation angle);
alternatively, the half-deviation angle K may be specified provided
it is clear which angle is called out.

entrance slit
exit
slit
r
2K
r′

grating

Figure 16-1. Constant-deviation monochromator geometry. The quantities that should
be specified are the entrance arm distance r, the exit arm distance r′, and the angle 2K
between these arms.

18. Flat-field spectrograph gratings. Concave holographic gratings used
in flat-field spectrographs require four parameters (see Figure 16-2):
• the entrance arm distance r,
• the angle α the entrance arm makes with the grating normal,

233

• the distance r′ S from the grating center to the image (on the
detector) of the shortest wavelength λS in the spectrum, and
• the obliquity angle Φ of the detector (as described in 2.3.2).

entrance grating
slit normal

r Φ
λL
λS
βS
α detector
r′S

grating

Figure 16-2. Flat-field spectrograph geometry. The quantities that should be specified
are the incidence angle α, the entrance arm distance r, the exit arm distance r′S for the
shortest wavelength in the spectrum to reach the detector, and the obliquity angle Φ of
the detector. The detector is shown; the shortest and longest wavelengths λS and λL image
at either end of the detector. [The distance r′L from the grating center to the image of λL is
not shown.]

An alternative set of parameters for defining a flat-field spectrograph is the
set of quantities α, r, lH and βH, where α and r are as above and
• the distance lH is measured from the grating center to the line
defined by the detector, such that these two lines are
perpendicular, and
• the angle βH is the angle the line lH makes with the grating normal
(see Figure 16-3).

234

entrance grating
slit normal

r
detector
βH
lH
α

grating

Figure 16-3. Alternative flat-field spectrograph geometry. A flat-field spectrograph can
also be described uniquely by the following quantities: the incidence angle α, the entrance
arm distance r, the distance lH (the line from the grating center to the line defined by the
detector, such that these two lines are perpendicular), and the angle β H that the line lH
makes with the grating normal.

Converting from the parameter set in Figure 16-3 to that in Figure 16-2 can be
accomplished using the formulas

r′ S = lH sec(βH – βS),
(16-1)
r′ L = lH sec(βH – βL).

235

APPENDIX A. Sources of Error in
Monochromator-Mode Efficiency
Measurements of Plane Diffraction Gratings

Jeffrey L. Olson, Newport Corporation

While simple in principle, measuring the efficiency of diffraction gratings is a
complex process requiring precise methods to achieve acceptable results. Every
optical, mechanical, and electronic component comprising an efficiency
measuring system is a potential source of error. Environmental factors may also
contribute to the overall measurement uncertainty. Each source of error is
identified and its effect on efficiency measurement is discussed in detail.

A.0. INTRODUCTION

In his 1982 book Diffraction Gratings, M.C. Hutley makes the following
statement regarding the measurement of diffraction grating efficiency:

“One seldom requires a very high degree of photometric accuracy in
these measurements as one is usually content to know that a grating is
60% efficient rather than 50% and the distinction between, say, 61% and
60% is of little practical significance.”175

While this statement may have been true at the time it was written, it is no
longer the case today. Certain industries, such as laser tuning and
telecommunications, demand gratings with efficiencies approaching theoretical
limits. The efficiency specifications for these gratings are well defined, and
measurement errors as small as one percent may mean the difference between the
acceptance and rejection of a particular grating.

237

In principle, measuring the efficiency of diffraction gratings is simple. A
ratiometric approach is used in which the energy of a diffracted beam is
compared to the energy of the incident beam. The incident beam may be either
measured directly (absolute measurement) or indirectly (relative measurement, by
reflection from a reference mirror). Conversion from relative to absolute
efficiency can be made easily by mu ltiplying the known reflectance of the
reference mirror by the relative efficiency of the grating. (Exceptions to this rule
have been noted, namely 1800 to 2400 g/mm gold or copper gratings measured at
wavelengths below 600 nm).176
As mentioned in Section 11.2, a monochromator mode efficiency-measuring
instrument, in essence, is a double grating monochromator, with the grating
under test serving as the dispersing element in the second monochromator. The
first monochromator scans through the spectral r ange while the test grating
rotates in order to keep the diffracted beam incident upon a detector that remains
in a fixed position throughout the measurement.
A typical efficiency measuring apparatus (see Figure A-1) consists of a
monochromator, collimator, polarizer, grating rotation stage, grating mount,
detector positioning stage, detector and associated optics, amplifier, and signal
processing hardware. Once the beam exits the monochromator it is collimated,
polarized, and, if necessary, stopped-down to a diameter appropriate for the
grating being tested. The beam is then directed toward the grating to be tested
where it is diffracted toward the detector. The electronic signal generated by the
detector is amplified, filtered, and presented to the user via any number of
devices ranging from a simple analog meter to a computer. In any case, a
comparison is made between a reference signal, obtained by direct or indirect
measurement of the incident beam, and the signal from the grating being tested.
Efficiency measurement results are normally reported on a graph (see Figure
A-2) with wavelength on the X-axis and percent efficiency (absolute or relative)
on the Y-axis. It is very unusual to see a published efficiency curve with error
bars or some other i dication of the measurement uncertainty. It must be
n
understood that these measurements are not exact, and may be in error by
several percent. A complete understanding of the measurement process as well
as the sources of error and how to minimize them would be of great value to the

175 M. C. Hutley, Diffraction Gratings, Academic Press (1982), p.168.
(1997), p. 415.

238

technician or engineer making the measurements as well as those involved in
making decisions to accept or reject gratings based on efficiency.

Light
source
Detector
Rotation stage

Monochromator

Collimator Polarizer Aperture Grating

Figure A-1. Typical monochromator-mode efficiency measuring apparatus.

A.1. OPTICAL SOURCES OF ERROR

A.1.1. Wavelength error

Perhaps the most obvious error of an optical nature is an error in
wavelength. If the monochromator does not accurately select the desired
wavelength, efficiency peaks, anomalies, etc., will appear at the wrong spectral
position on the efficiency curve. If the grating being measured is rotated to the
appropriate incident angle for a given wavelength, the diffracted beam may
partially or totally miss the detector if the wavelength is not correct. This is less
of a problem in manually controlled instruments, since the operator can adjust
the wavelength or grating rotation angle to obtain a maximum reading. On an
automated instrument, however, a significant error may result unless the
instrument has the ability to “hunt” for the efficiency peak.

239

Figure A-2. Typical efficiency curve.

Wavelength errors are usually caused by a failure of the monochromator
indexing mechanism to move the grating to the correct rotation angle. Most
computer based monochromator systems employ correction factors or calibration
tables in firmware to correct systematic wavelength errors. Even so, many
monochromators use open-loop stepper motor drives to position the grating.
Since there is no explicit feedback from the rotation mechanism, the controller
must assume that the grating is in the correct position. If the motor fails to move
the proper number of steps due to binding in the mechanical system or for some
other reason, the wavelength will be in error.
To ensure wavelength accuracy, periodic wavelength calibration should be
done using a calibration lamp or other spectral line source. The author has used
the Schumann-Runge O2 absorption lines effectively for monochromator
wavelength calibration in the far ultraviolet region near 193 nm. The well-defined

240

Schumann-Runge transitions occurring at 192.6 nm are especially useful spectral
features.177

A.1.2. Fluctuation of the light source intensity

One of the drawbacks of using a single detector system is that the light
source intensity can change between the times the reference and sample
measurements are made. With filament lamps, the intensity is proportional to the
power dissipated in the filament. According to Ohm’s law, power P is the
product of current I and voltage E, and voltage is the product of current and
resistance R, therefore:

P = IE = I (IR) = I2R. (A-1)

Typically electrical power is applied to the lamp socket, rather than to the
lamp directly. The contact resistance between the lamp and socket can be
significant and is prone to change over time. If a constant voltage is applied to
the socket and the contact resistance was to increase, the current, power, and
lamp intensity will decrease as a result. If a constant current is applied instead,
no change in power will occur as the result of a change in contact resistance
(provided the filament resistance remains constant). For this reason current-
regulated, rather than voltage-regulated, power supplies are preferred whenever
filament-type lamps are used. A photo-feedback system, in which a detector
monitors and controls the lamp intensity, is also a good choice. Regardless of
the type of light source used, it is always best that the sample and reference
measurements be made in quick succession.

A.1.3. Bandpass

As a rule, the bandpass of the light source should always be narrower than
that of the grating under test. The bandpass B of the grating under test is
estimated by the angular dispersion D of the grating, the distance r from the
grating to the detector aperture, and the width w′ of the exit aperture according
to the equation:

177 R. C. Sze and C. A. Smith, “High-temperature absorption studies of the Schumann-Runge

241

w′
B≈ . (A-2)
rD

[Eq. (A-2) assumed the case where the aberrated image of the entrance slit is not
wider than the exit aperture; see Section 8.3).]
Whenever a grating is measured using a source with a bandpass that is too
broad, some of the outlying wavelengths will be diffracted away from the
detector. In contrast, when the reference measurement is made using a mirror or
by direct measurement of the incident beam, no dispersion occurs.
Consequently, the detector captures all wavelengths contained within the
incident beam during the reference measurement, but not during the grating
measurement (resulting in an artificial decrease in grating efficiency). Another
consequence of measuring gratings using a light source with a broad bandpass
is that sharp efficiency peaks will appear flattened and broadened, and may be
several percent lower than if measured using a spectrally-narrow light source.
Efficiency curves should, but often do not, state the bandpass of the source
used to make the measurement. When using a monochromator, it is generally
best to adjust the slits to obtain the narrowest bandpass that will provide an
acceptable signal-to-noise ratio (SNR). Alternatively, a narrow band spectral
source, such as a laser or calibration lamp, may be used in conjunction with a
monochromator or interference filter to eliminate unwanted wavelengths.

A.1.4. Superposition of diffracted orders

According to the grating equation (see Eq. (2-1)), the first order at
wavelength λ and the second order at wavelength λ/2 will diffract at exactly the
same angle (see Section 2.2.2). Therefore the light emerging from a
monochromator exit slit will contain wavelengths other than those desired. The
unwanted orders must be removed in order to accurately determine the efficiency
at the desired wavelength. “Order-sorting” filters are most commonly used for
this purpose. These are essentially high-pass optical filters that transmit longer
wavelengths while blocking the shorter wavelengths.
Another problematic situation arises when the adjacent diffracted orders are
very closely spaced. In this case, adjacent orders must be prevented from
overlapping at the detector aperture, which would result in a significant error.

band of oxygen at ArF laser wavelengths,” J. Opt. Soc. Am . B7, 3, 462-475 (2000).

242

This situation can be avoided by ensuring that the bandpass of the source is
less than the free spectral range of the grating being tested. As shown in
Section 2.7, the free spectral range Fλ is defined as the range of wavelengths ∆λ
in a given spectral order m that are not overlapped by an adjacent order,
expressed by Eq. (2-28):

λ
Fλ = ∆λ = . (2-28)
m

For an echelle being measured in the m = 100th order at λ = 250 nm, the free
spectral range is 2.5 nm. The detector aperture must also be sufficiently narrow to
prevent adjacent orders from being detected, but not so narrow as to violate the
“rule” regarding the bandpass of the light source and grating under test.

A.1.5. Degradation of the reference mirror

When a mirror is used to determine the incident light energy, its reflectance
as a function of wavelength needs to be well characterized. Mirrors tend to
degrade over time due to atmospheric exposure, and if not re-characterized
periodically, optimistic measurements of grating efficiency will result. At the
National Physical Laboratory (NPL) in England, an aluminum-coated silica flat
was used as a reference mirror. This is nothing new, but in this case the “buried”
surface of the mirror was used as the mirror surface instead of the metal surface
itself. Since the aluminum is never exposed to the atmosphere, its reflectance is
stable, and since the mirror was characterized through the silica substrate, its
influence is automatically taken into account.178 The restriction in using the
buried surface method is that the incident beam must be normal to the mirror
surface to avoid beam separation caused by multiple reflections from the front
and buried surfaces. When an unprotected mirror surface is used as a reference,
absolute measurements of its reflectance should be made on a regular basis.

178 M. C. Hutley, Diffraction Gratings, Academic Press (1982), p.169.

243

A.1.6. Collimation

If the incident beam is not reasonably well collimated, the rays will fall upon
the grating at a variety of angles and will be diffracted at different angles. In the
case of a diverging diffracted beam, the beam will spread, possibly overfilling the
detector. Since the reference beam does not encounter a dispersing element in
its path (but the sample beam does), it is possible that all of the energy will be
collected during the reference measurement but not during the sample
measurement, causing the measured efficiency to be low.
Whenever a monochromator-based light source is used it is difficult, if not
impossible, to perfectly collimate the beam emerging from the exit slit in both
planes. It is important to collimate the beam in the direction perpendicular to the
grooves, but it is not as critical for the beam to be well collimated in the direction
parallel to the grooves, since no diffraction occurs in that direction. A limiting
aperture may be used to restrict the beam size and prevent overfilling the grating
under test.
It should be emphasized that beam collimation is not nearly as important in
an efficiency measuring system as it is in an imaging system, such as a
spectrograph. It is only necessary to ensure that the detector collects all of the
diffracted light. The degree of collimation required largely depends on the
dispersion of the grating under test, but in most cases a beam collimated to
within 0.1o is adequate. For example, an angular spread of 0.1o in a beam incident
st
upon a 1200 g/mm grating measured in the 1 order at 632.8 nm (Littrow
configuration) will produce a corresponding spread in the diffracted beam of less
than 1 mm over a distance of 500 mm.

A.1.7. Stray light or “optical noise”

The influence of stray radiation must always be taken into consideration
when making efficiency measurements. If the level of background radiation is
very high, the detector may be biased enough to result in a significant error.
This is especially true when simple DC detection methods are used. Any bias
introduced by background radiation must be subtracted from both the reference
and sample measurements before the ratio is computed. For example, if the
background radiation equals 2% of the reference beam, and the grating being
tested measures 50% relative efficiency, the actual efficiency is 48/98 or 49%.
This represents an error of 1% of the full-scale measured efficiency. In many

244

cases simply operating the instrument in a dark lab or enclosure is sufficient to
reduce background light to insignificant levels. Averaging is often used to
“smooth out” noisy signals, but unlike other more random noise sources that
tend to be bipolar, stray light-induced noise is always positive. Averaging
several measurements containing a significant level of optical noise may bias the
final measurement. In most cases, it is best to use phase-sensitive detection to
remove the effects of unwanted radiation.

A.1.8. Polarization

Most efficiency curves display the S and P as well as the 45-degree
polarization efficiency vs. wavelength. When making polarized efficiency
measurements using an unpolarized source, it is necessary to use some form of
optical element to separate the two polarization vectors. It is critical that the
polarizer be aligned as closely as possible to be parallel (P plane) or
perpendicular (S plane) to the grooves or a polarization mixing error will result.
To determine the 45-degree polarization efficiency of a grating, it seems easy
enough to set the polarizer to 45 degrees and make the measurement, but unless
the output from the light source is exactly balanced in both S and P planes, an
error will result.
Figure A-3 shows the effect of source polarization on the measured
efficiency of a hypothetical grating having efficiencies of 90% in the P plane and
50% in the S plane at some arbitrary wavelength. In one case the light source
contains equal S and P intensities while the other has a 70:30 S to P ratio. In the
case of the balanced light source, as the polarizer is rotated to 45o, the efficiency
is 70%, exactly the average of the S and P measurements. On the other hand, the
unbalanced source results in a measured efficiency of 60%. This represents an
error of 10% of the full-scale measured value. For that reason it is always
recommended to make separate S and P measurements and then average them to
determine the grating’s 45-degree polarization efficiency.

245

1 Source Polarization
0.9
0.8 50% S
Efficiency (%)

0.7 50% P
0.6
0.5 70% S
0.4
0.3 30% P
0.2
0.1
0
0 10 20 30 40 50 60 70 80 90
Polarizer rotation angle (deg)

Figure A-3. Efficiency comparison with balanced and unbalanced source polarization (0º
= P, 90º = S).

A.1.9. Unequal path length

An error can result in a single detector system purely as the result of the
optical path being different between the reference and sample measurements.
This is especially true at UV wavelengths where the atmospheric absorption is
significant. Different optical path lengths are not as much of a problem in dual
detector systems since the relative calibration of the two detectors can
compensate for atmospheric effects.

A.2. M ECHANICAL SOURCES OF ERROR

A.2.1. Alignment of incident beam to grating rotation axis

It is critical to align the incident beam to the rotation axis of the grating stage
and mount. If not, the beam will “walk” across the grating surface at relatively
low incident angles, and partially miss the grating surface at very high incident
angles. Since the incident and diffracted beams are displaced from their correct

246

location, it is entirely possible that all or part of the diffracted beam will miss the
detector aperture.

A.2.2. Alignment of grating surface to grating rotation axis

The effect of not having the grating surface located exactly over the rotation
axis will be similar to that of not having the beam aligned to the grating rotation
axis. Optimally, a grating mount that references the grating’s front surface, rather
than the sides or back of the grating substrate, may be used to ensure alignment.
This is often not practical since the contact points on the mount may leave an
impression on the grating surface. To avoid this problem, an inclined lip or rail is
sometimes used that makes contact with the grating on the extreme outer edge
only. On beveled grating substrates this can be a source of error since the
dimensional variation of the bevels can be significant. If the grating mount does
not reference the front surface, an adjustment must be provided in order to
accommodate gratings of various thicknesses.

A.2.3. Orientation of the grating grooves (tilt adjustment)

On grating mounts that use the substrate to locate the grating to be tested,
the plane in which the diffracted orders lie will be tilted if the grooves are not
properly aligned with the sides of the grating substrate. This may cause the
diffracted beam to pass above or below the detector aperture. Most gratings do
not have perfect alignment of the grooves to the substrate, so it is necessary to
incorporate a method for rotating the grating a small amount in order to
compensate for groove misalignment.

A.2.4. Orientation of the grating surface (tip adjustment)

Due to some wedge in the grating substrate, for example, the grating surface
may not be parallel to the grating rotation axis. This will cause the diffracted
beam to fall above or below the detector aperture. On most grating mounts an
adjustment is provided to correct this situation. Ideally the grating tip, tilt, and
rotation axes all intersect at a common point on the grating surface, but in fact it
is extremely rare to find a grating mount in which the tip axis does so. In most
cases the tip axis is located behind or below the grating substrate, so when

247

adjusted, the grating surface will no longer lie on the rotation axis. The ideal
situation is one in which the grating is front-surface referenced on the mount so
that no adjustment is needed.

A.2.5. Grating movement

It is essential that the grating being tested be held securely in the mount
during the testing process. Vibration from m otors and stages as well as the
inertia generated by the grating as it is rotated may cause it to slip. Any motion
of the grating relative to the mount will result in alignment errors and invalidate
any measurements taken after the movement occurred.

A.3. ELECTRICAL SOURCES OF ERROR

A.3.1. Detector linearity

In principle, all that is required to make satisfactory ratiometric
measurements is a linear response from the sensing element and associated
electronics. Nearly all detectors have response curves that exhibit non-linearity
near saturation and cut off. It is extremely important to ensure that the detector
is biased such that it is operating within the linear region of its response curve.
In addition, the detector preamplifier and signal processing electronics must also
have a linear response, or at least have the non-linearity well characterized in
order for a correction to be applied. Neutral density filters may be inserted into
the optical path to verify or characterize the detection system linearity. The
following set of five calibrated neutral density filters is sufficient, in most cases,
for verifying the detector response to within ± 1%:

248

Optical Density Transmission*
0.1 79%
0.3 50%
0.6 25%
1.0 10%
2.0 1%

* rounded to nearest whole percent.

Detector non-linearity becomes a major source of error when the reference
and grating signals differ significantly in intensity. Unless the linearity of the
detector and associated electronics has been well established, using a mirror with
a reflectance of 90% or higher as a reference may introduce an error if the grating
being measured has an efficiency of 20%. This is analogous to sighting in a rifle
at 100 yards and using it to shoot at targets 25 yards away. In some situations it
is best to use a well-characterized grating as nearly identical to the grating to be
tested as possible. This method is especially useful for making “go/no-go”
efficiency measurements. If the reference grating is carefully chosen to be one
that is marginally acceptable, then the efficiency measuring instrument will have
its greatest accuracy at the most critical point. All gratings measuring greater
than or equal to 100% relative to the reference grating are assumed to be good
and those below 100% are rejected. Of course this method requires periodic
recharacterization of the reference grating in order to maintain measurement
integrity.

A.3.2. Changes in detector sensitivity

Some efficiency measuring instruments use separate detectors for making
the reference and grating measurements (these are not to be confused with
systems that use secondary detectors to monitor light source fluctuations).
Most, however, use a single detector for both the reference and grating
measurements instead. There are very good reasons for doing this. First of all,
detectors and the associated electronics are expensive, so using a single detector
is far more cost effective. Detector response characteristics change over time, so

249

frequent calibration is necessary in a dual-detector system to ensure that the
photometric accuracy of each detector has not changed relative to the other. By
using the same detector for sample and reference measurements, photometric
accuracy is not an issue, since an error in the reference measurement will also be
present in the grating measurement and consequently nullified.

A.3.3. Sensitivity variation across detector surface

A significant error can result if the reference or diffracted beam is focused
down to form a spot that is much smaller than the detector’s active area. Some
detectors, especially photomultipliers, may exhibit a sensitivity variation
amounting to several percent as the spot moves across the detector surface. It is
often sufficient to place the detector aperture far enough away from the detector
such that the spot is defocused and just under-fills the active area. Alternately a
diffuser or integrating sphere is sometimes used to distribute the light more
uniformly across the detector surface.

A.3.4. Electronic noise

Any form of optical or electronic noise can influence efficiency
measurements. It is desirable to maintain the highest signal-to-noise ratio (SNR)
possible, but often a trade-off must be made between signal strength and
spectral resolution. Decreasing the monochromator slit width in order to narrow
the bandpass of the source results in a reduced detector output signal. Care
must be taken not to limit the intensity to a point where electronic (and optical)
noise becomes a significant factor. In most cases, an SNR value of 200:1 is
adequate.

A.4. ENVIRONMENTAL FACTORS

A.4.1. Temperature

Normally it is not necessary to perform efficiency measurements in an
extremely well regulated environment, but there are a few cases in which
temperature control is needed. Whenever very high spectral resolution

250

measurements are made (c. ∆λ ≤ 1 nm), temperature variation within the
monochromator may cause a significant wavelength drift. Temperature
fluctuations may cause optical mounts to expand or contract resulting in a
displacement of the beam. It is always a good idea to keep heat sources well
away from all optical and mechanical components that may affect the grating
being tested or the beam. It is also wise to allow gratings that are to be tested to
acclimate in the same environment as the test instrument.

A.4.2. Humidity

Humidity is not usually a significant error source, but since it can affect the
system optics and electronics, it merits mentioning. A high humidity level may
influence measurements a wavelengths where atmospheric absorption varies
t
with relative humidity. Low humidity promotes the generation of static electricity
that may threaten sensitive electronic components. In general, the humidity level
should be maintained in a range suitable for optical testing.

A.4.3. Vibration

Vibration becomes an error source when its amplitude is sufficient to cause
the grating under test or any of the optical components to become displaced. If
the vibration is from a source other than the instrument itself, then mounting the
instrument on a vibration isolated optical bench will solve the problem. If the
instrument itself is the vibration source, then the problem becomes a little more
difficult. Stepper-motors are most often used to rotate and translate the grating
being tested, as well as tune the monochromator, select filters, etc. As the
motors ramp up to predetermined velocity, a resonant frequency is often
encountered that will set up an oscillation with one or more mechanical
components in the system. While it is sometimes necessary to pass through
these resonant frequencies, it is never advisable to operate continuously at
those frequencies. Most motion controllers have provisions for tuning the
motion profile to minimize resonance. Some motion controllers allow micro-
stepped operation of the motors, producing a much smoother motion. Although
they are generally more expensive, servo controllers, amplifiers, and motors
provide exceptional accuracy and very smooth motion.

251

A.5. SUMMARY

Many of the error sources identified can be eliminated entirely, but only at
the expense of decreased functionality. Greater accuracy can be obtained using
an instrument that operates at a fixed wavelength in a fixed geometry and is only
used to test gratings that have identical physical properties. When a large
variety of gratings are to be tested, each with a different size, shape, groove
frequency, wavelength range, test geometry, etc., it is not practical to construct a
dedicated instrument for each. In this case, a more complex instrument is
required. In specifying such an instrument, each source of error should be
identified, and if possible, quantified. An error budget can then be generated
that will determine if the instrument is able to perform at the desired level. Most
likely it will not, and then a decision needs to be made regarding which features
can be compromised, eliminated, or implemented on another instrument.
Disagreements often arise between measurements made of the same grating
on different efficiency measuring instruments. Slight differences in test
geometry, bandpass, and beam size can have a surprisingly large effect on
efficiency measurements. What is sometimes difficult to understand is that it is
possible for two instruments to m easure the same grating and get different
results that are valid!
Grating efficiency is largely determined by the groove properties of the
master from which the grating was replicated, and to some degree the coating. It
is very rare for a master, regardless of the process used to create it, to have
perfectly uniform efficiency at every spot along its surface. In some cases the
efficiency may vary by several percent. If a grating is measured using a small
diameter beam, then these efficiency variations are very noticeable compared to
measurements made using a larger beam. If two different instruments are used to
measure the same grating, it is possible that the beams are not exactly the same
size or, in the case of a small beam diameter, are not sampling exactly the same
spot on the grating surface. Both instruments are correct in their measurements,
but still do not agree. For this reason, whenever comparisons between
instruments are made, the differences in their configuration must be taken into
consideration.
The goal of efficiency measurement is to characterize the grating under test,
not the apparatus making the measurements. For this reason, efficiency curves
should report not only the relative or absolute efficiency vs. wavelength, but the
properties of the instrument making the measurement as well. Only then is it
possible to reproduce the results obtained with any degree of accuracy.

252

APPENDIX B. Lie aberration theory for grating
systems

Besides the wavefront aberration theory described in Chapter 7, geometrical
optics can be formulated in a manner in direct analogy with the Hamiltonian
theory of classical dynamics.179 The basis for this analogy is the recognition
that Fermat’s principle, which requires that a physical light path be an extremum,
is equivalent to the requirement that this physical light path follow a trajectory
governed by a Hamiltonian. The coordinates and momenta of a particle in
classical mechanics therefore correspond to the coordinates and direction
cosines of a light ray in optics, and the tools of classical mechanics can be
applied directly to geometrical optics.
The characterization of optical systems using Lie transformations, hitherto
applied to dynamical systems, was first developed by Dragt, who considered
axially symmetric systems.180 Later this formulation was extended by Goto and
Kurosaki to optical systems without axial symmetry (but with a plane of
symmetry).181
Dragt considered the coordinates x and y of a point in the object plane, as
well as their direction cosines p and q, as the object phase space variables and
primed quantities (x′, y′, p′ and q′) as their corresponding image phase space
variables (see Figure B-1), and expressed the transformation of the ray in object
space to the ray in image space (due to the optical system) as

W′ = M w (B-1)

179 A. J. Dragt, E. Forest and K. B. Wolf, “Foundations of a Lie algebraic theory of
geometrical optics,” in Lie Methods in Optics, J. J. Sanchez-Mondragon and K. B. Wolf, eds.
(Springer-Verlag, Berlin, 1984), ch. 4.
180 A. J. Dragt, “Lie algebraic theory of geometric optics and optical aberrations,” J. Opt.
Soc. Am. 72, 372-379 (1982).
181 K. Goto and T. Kurosaki, “Canonical formulation for the geometrical optics of concave
gratings,” J. Opt. Soc. Am. A10, 452-465 (1993).

253

where M is a mapping (or simply map), an operator that transforms coordinates
in object space to corresponding coordinates in image space.
A transformation that maps coordinates in object space into coordinates in
image space according to Fermat’s principle is called symplectic. Dragt and Finn
showed in 1976 that a symplectic map can be expressed in terms of Lie
transformations.182 More specifically, this map can be exp ressed as the product
of Lie transformations, each of which is homogeneous in the object space
coordinates (that is, all terms in each Lie transformation are of the same power in
the independent variables). Furthermore, truncating the product at any power
leaves a symplectic transformation, so lower-order imaging properties can be
examined without considering the higher-order Lie transformations in the map.
Goto and Kurosaki used Dragt’s formalism to derive, using Lie algebraic theory
rather than wavefront aberration theory, equations that are formally identical to
the aberration coefficients F20, F02, F30, etc. seen in Chapter 7.

y
z
φ

x θ y

x

Figure B-1. Definition of the object space variables. The optical ray in object space has
coordinate (x, y) in the object plane and direction defined by angles θ and φ (shown),
whose direction cosines are p and q.

182A. J. Dragt and J. M. Finn, “Lie series and invariant functions for analytical symplectic
maps,” J. Math. Phys. 17, 2215-2227 (1976).

254

The transformation from object phase space to image phase space may be
represented as a sequence of operations; for example, for diffraction by a grating,
these operations are (in sequence)
1. transit from the object plane to the grating, through the distance r,
2. rotation, through the angle α, to a coordinate frame centered at the
grating center and oriented with an axis along the grating normal,
3. diffraction in this coordinate frame,
4. rotation through the angle β, and
5. transit from the grating to the image plane, through the distance r′.
An advantage of the Lie transformation approach over the wavefront
aberration technique is that general points (x, y) in the object plane are naturally
considered; which this is also true of wavefront aberration theory, the algebra is
cumbersome and, as a result, most authors consider only a point source in the
dispersion plane.
Another immediate advantage of the Lie transformation approach is that
systems with more than one optical element can be addressed in a
computationally straightforward manner, simply by appending transformations
(in the sequence in which the optical ray encounters the optical elements).183
Using wavefront aberration theory this is not straightforward, in large part
because the grating surface coordinates appear explicitly in the optical path
difference (see Eq. (7-5)). Many researchers overcame this complication by
imposing intermediate foci between successive elements,184 though Chrisp
introduced the use of toroidal reference surfaces and provided an important
advance in the development of wavefront aberration theory for multielement
systems.185

183 C. Palmer, W. R. McKin ney and B. Wheeler, "Imaging equations for spectroscopic
systems using Lie transformations. Part I - Theoretical foundations," Proc. SPIE 3450, 55-
66 (1998).
184 T. Namioka, H. Noda, K. Goto and T. Katayama, “Design studies of mirror-grating
systems for use with an electron storage ring source at the Photon Factory,” Nucl. Inst. Meth.
208, 215-222 (1983).
185 M. Chrisp, “The theory of holographic toroidal grating systems,” Ph. D. dissertation, U.
London (1981); M. Chrisp, “Aberrations of holographic toroidal gr ating systems,” Appl. Opt.
22, 1508-1518 (1983).

255

As an example of the power of Lie algebraic techniques in the analysis of
multielement optical system, below is the equation for defocus for a system of
two aberration-reduced concave holographic gratings:186

cos β1 cos β 2 cosα 1 cosα 2 cosα1 cos β 2
F20 = r1 + r2′ + D
cosα 1 cosα 2 cosβ 1 cos β 2 cosα 2 cos β1

2 A[1] r1  cos β 2 cosα 2 
− 2
 D+ r2′ 
cosα1 cos β1  cosα 2 cos β 2 
(B-2)
′
2 A[ 2 ]r2  cosα 1 cos β1 
− 2
 D+ r1 
cosα 2 cos β 2  cos β 1 cosα1 

4 A21] A22 ]r1 r2′ D
[ [
+ .
cosα1 cosα 2 cos β1 cos β 2

Here the subscripts on the angles and distances refer to each grating, and we
have defined

D = r1′ + r2 (B-3)

as the distance between the two gratings. The quantity A[ i] depends on the
2
substrate curvature and groove pattern for the ith grating.

186 C. Palmer, B. Wheeler and W. R. McKinney, "Imaging equations for spectroscopic
systems using Lie transformations. Part II - Multi-element systems," Proc. SPIE 3450, 67-
77 (1998).

256

FURTHER READING

ASTM standard E131-02, Standard Terminology Relating to Molecular Spectroscopy,
ASTM International (2003).

Ball, D. W., 2001. The Basics of Spectroscopy, SPIE Press (Bellingham, Washington).

Born, M., and E. Wolf, 1980. Principles of Optics, Pergamon Press (Oxford, England).

Clark, B. J., T. Frost and M. A. Russell, 1993. UV Spectroscopy: Techniques, instru-
mentation, data handling, Chapman and Hall (London, England).

Ederer, D. L., ed., 1998. Selected Papers on VUV Synchrotron Radiation Instrumentation
– Beam Line and Instrument Development, SPIE Milestone Series vol. MS 152,
SPIE (Bellingham, Washington).

Harris, D. A. and C. L. Bashford, eds., 1987. Spectrophotometry and Spectrofluorimetry,
IRL Press (Oxford, England).

Harrison, G. R., 1973. "The diffraction grating – an opinionated appraisal," Appl. Opt.
12, 2039-2049.

Hunter, W. R., 1985. "Diffraction Gratings and Mountings for the Vacuum Ultraviolet
Spectral Region," Spectrometric Techniques, vol. IV., 63-180 (1985). This article
provides a very thorough and detailed review of the use and manufacture of
diffraction gratings.

Hutley, M. C., 1976. "Interference (holographic) diffraction gratings," J. Phys.E9, 513-
520 (1976). This article offers more detail on holographic gratings, and compares
them with ruled gratings.

Hutley, M. C., 1982. Diffraction Gratings, Academic Press (New York, New York).
This book provides a complete and thorough tour of diffraction gratings, their manu-
facture and their application.

James, R. F. and R. S. Sternberg, 1969. The Design of Optical Spectrometers, Chapman
and Hall (London, England).

Loewen. E. G., 1970. “Diffraction Gratings for Spectroscopy,” J. Physics E3, 953-961.

Loewen. E. G., M. Nevière and D. Maystre, 1977. “Grating Efficiency Theory as it
Applies to Blazed and Holographic Gratings,” Appl. Optics 16, 2711-2721 (1977).

257

Loewen, E. G., 1983. "Diffraction Gratings, Ruled and Holographic," in Applied Optics
and Optical Engineering, vol. IX (chapter 2), R. Shannon, ed., Academic Press (New
York, New York). This chapter describes developments in grating efficiency theory
as well as those in concave grating aberration reduction.

Loewen, E. G. and E. Popov, 1997. Diffraction Gratings and Applications, Marcel
Dekker (New York, New York).

Maystre, D., ed., 1993. Selected Papers on Diffraction Gratings, SPIE Milestone Series
vol. MS 83, SPIE (Bellingham, Washington).

Meltzer, R. J., 1969. "Spectrographs and Monochromators," in Applied Optics and
Optical Engineering, vol. V (chapter 3), R. Shannon, ed., Academic Press (New
York, New York).

Petit, R., ed., 1980. Electromagnetic Theory of Gratings, Springer-Verlag (New York,
New York); volume 22 in “Topics in Current Physics” series.

Richardson, D., 1969. "Diffraction Gratings", in Applied Optics and Optical Engineering,
vol. V (chapter 2), R. Kingslake, ed., Academic Press (New York, New York).

Samson, J. A. R., 1967. Techniques of Vacuum Ultraviolet Spectroscopy, John Wiley &
Sons (New York, New York).

Schroeder, D. J., 1987. Astronomical Optics, Academic Press (San Diego, California).
Chapters 12 through 15 serve as an excellent introduction to gratings and their
instruments, with application toward stellar spectrometry. Chapters 2 through 5
form a clear and quite complete introduction to the ideas of geometrical optics used
to design lens, mirror, and grating systems.

Skoog, D. A., F. J. Holler and T. A. Nieman, 1998. Principles of Instrumental Analysis,
fifth edition, Brooks/Cole. The sections on atomic and molecular spectroscopy
serve as excellent introductions to the subject.

Stover, J. C., 1990. Optical Scattering, McGraw-Hill, Inc. (New York, New York).

Stroke, G. W. 1967. “Diffraction Gratings,” in Handbook of Physics 29, S. Flugge, ed.
(Springer, Berlin).

Strong, J., 1960. “The Johns Hopkins University and diffraction gratings,” J. Opt. Soc.
Am. 50, 1148-1152.

Williard, H. H., et al., 1988. Instrumental Methods of Analysis, seventh edition, Wads-
worth. (Belmont, California). The chapters on absorption and emission
spectroscopy, and ultraviolet and visible spectroscopic instrumentation, are of par-
ticular interest.

258

Grating Publications by Newport Corporation
Personnel

Below is a partial list of publications by Newport scientists and engineers
pertaining to diffraction gratings and their uses.

Reference Books
E. G. Loewen and E. Popov, Diffraction Gratings and Applications, 1997 Marcel Dekker,
Inc. (ISBN 0-8247-9923-2).

Gratings in General
D. Richardson, "Diffraction Gratings," chapter 2, volume II of Applied Optics and Optical
Engineering, R. Kingslake, ed. (Academic Press, New York: 1969)
E. G. Loewen, “Diffraction Gratings for Spectroscopy,” J. Physics E3, 953-961 (1970).
E. G. Loewen, Diffraction Grating Handbook, first edition, Bausch & Lomb (1970).
G. R. Harrison and E. G. Loewen, “Ruled gratings and wavelength tables,” Appl. Opt. 15,
1744-1747 (1976).
E. G. Loewen, M. Nevière and D. Maystre, “Optimal design for beam sampling mirror
gratings,” Appl. Opt. 15, 2937-2939 (1976).
E. G. Loewen, "Diffraction Gratings, Ruled and Holographic," chapter 2, volume IX of
Applied Optics and Optical Engineering, R. Shannon, ed. (Academic Press, New
York: 1983), pp. 33-71.
W. R. McKinney, Diffraction gratings: manufacture, specification and application, SPIE
Tutorial T25 (1984).
E. G. Loewen, “What’s new in gratings?” in Instrumentation for Ground-Based Optical
Astronomy, Present and Future, L. B. Robinson, ed. (Springer-Verlag, 1987),
pp. 118-123.
E. G. Loewen, "The Ruling and Replication of Diffraction Gratings," Optics and
Photonics News, May 1991.
C. Palmer, Diffraction Grating Handbook, sixth edition, Newport Corporation (2005).
[E. G. Loewen wrote the first edition, originally published in 1970.]

260

C. Palmer, "Diffraction Gratings: The Crucial Dispersive Component," Spectroscopy 10,
14-15 (1995).

Grating Efficiency
E. G. Loewen, D. Maystre, R. C. McPhedran and I. Wilson, “Correlation between
efficiency of diffraction gratings and theoretical calculations over a wide spectral
range,” Japanese J. Appl. Phys. 14, suppl. 1, 143-152 (1975).
E. G. Loewen, M. Nevière and D. Maystre, "Grating efficiency theory as it applies to
E. G. Loewen and M. Nevière, “Dielectric coated gratings: a curious property,” Appl. Opt.
16, 3009-3011 (1977).
E. G. Loewen, M. Nevière and D. Maystre, “On an asymptotic theory of diffraction
gratings used in the scalar domain,” J. Opt. Soc. Am. 68, 496-502 (1978).
E. G. Loewen and M. Nevière, "Simple selection rules for VUV and XUV diffraction
gratings," Appl. Opt. 17, 1087-1092 (1978).
I. Wilson, B. Brown and E. G. Loewen, “Grazing incidence grating efficiencies,” Appl.
Opt. 18, 426-427 (1979).
E. G. Loewen, M. Nevière and D. Maystre, "Efficiency optimization of rectangular
groove gratings for use in the visible and IR regions," Appl. Opt. 18, 2262-2266
(1979).
E. G. Loewen, W. R. M cKinney and R. McPhedran, "Experimental Investigation of
Surface Plasmon Scatter from Diffraction Gratings," Proc. SPIE 503, 187-197
1984).
L. B. Mashev, E. K. Popov and E. G. Loewen, "Asymmetrical trapezoidal grating
efficiency," Appl. Opt. 26, 2864-2866 (1987).
L. B. Mashev, E. K. Popov and E. G. Loewen, "Optimization of grating efficiency in
grazing incidence," Appl. Opt. 26, 4738-4741 (1987).
L. Mashev, E. Popov and E. G. Loewen, “Total absorption of light by a sinusoidal grating
near grazing incidence,” Appl. Opt. 27, 152-154 (1988).
E. Popov , L. Mashev and E. G. Loewen, “Total absorption of light by gratings in grazing
incidence: a connection in the complex plane with other types of anomaly,”
Appl. Opt. 28, 970-975 (1989).
L. Mashev, E. Popov and E. G. Loewen, “Brewster effect for deep metallic gratings,”
Appl. Opt. 28, 2538-2541 (1989).

261

E. Popov, L. Tsonev, E. Loewen and E. Alipieva, “Spectral behavior of anomalies in deep
metallic grooves,” J. Opt. Soc. Am. A7, 1730-1735 (1990).
E. G. Loewen, E. K. Popov, L. V. Tsonev and J. Hoose, “Experimental study of local and
integral efficiency behavior of a concave holographic diffraction grating,” J. Opt.
Soc. Am. A7, 1764-1769 (1990).
E. K. Popov, L. V. Tsonev and E. G. Loewen, “Scalar theory of transmission relief
V. S. Valdes, W. R. McKinney and C. Palmer, “The differential method for grating
efficiencies implemented in Mathematica,” Nucl. Inst. Meth. A347, 216-219
(1994).
C. Palmer, J. Olson and M. M. Dunn, “Blazed diffraction gratings obtained by ion-milling
sinusoidal photoresist gratings,” Proc. SPIE 2622, 112-121 (1995).

Grating Imaging
W. R. McKinney and C. Palmer, "Numerical design method for aberration-reduced
concave grating spectrometers," Appl. Opt. 26, 3108-3118 (1987).
C. Palmer and W. R. McKinney, "Optimizing Angular Deviation for Fixed-Deviation
Monochromators Modified by Use of Aberration-Reduced Holographic
Gratings," Proc. SPIE 815, 57-65 (1987).
C. Palmer, “Limitations of aberration correction in spectrometer imaging,” Proc. SPIE
1055, 359-369 (1989).
C. Palmer, "Theory of second generation holographic gratings," J. Opt. Soc. Am. A6,
1175-1188 (1989).
C. Palmer, “Absolute astigmatism correction for flat-field spectrographs,” Appl. Opt. 28,
1605-1607 (1989).
W. R. McKinney and C. Palmer, "Design of Grazing Incidence Monochromators
Involving Unconventional Gratings," Proc. SPIE 1055, 332-335 (1989).
C. Palmer and W .R. McKinney, "Equivalence of focusing conditions for holographic and
varied line-space gratings," Appl. Opt. 29, 47-51 (1990).
C. Palmer, “Second-order imaging properties of circular-field spectrographs, Appl. Opt.
29, 1451-1454 (1990); erratum, Appl. Opt. 29, 3990 (1990).
C. Palmer, "Deviation of second-order focal curves in common spectrometer mounts," J.
Opt. Soc. Am. A7, 1770-1778 (1990).
C. Palmer and W. R. McKinney, "Imaging Theory of Plane-symmetric Varied Line-space
Grating Systems," Opt. Eng. 33, 820-829 (1994).

262

W. R. McKinney and C. Palmer, “Derivation of aberration coefficients for single-element
plane-symmetric reflecting systems using Mathematica,” Proc. SPIE 3150, 97-
104 (1997).
C. Palmer, W. R. McKinney and B. Wheeler, "Imaging equations for spectroscopic
systems using Lie transformations. Part I - Theoretical foundations," Proc.
SPIE 3450, 55-66 (1998).
C. Palmer, B. Wheeler and W. R. McKinney, "Imaging equations for spectroscopic
systems using Lie transformations. Part II - Multi-element systems," Proc.
SPIE 3450, 67-77 (1998).
C. Palmer and W. R. McKinney, "Imaging properties of varied line-space (VLS) gratings
with adjustable curvature," Proc. SPIE 3450, 87-102 (1998).

Echelle Gratings
G. R. Harrison, E. G. Loewen and R. S. Wiley, "Echelle gratings: their testing and
improvement," Appl. Opt. 15, 971-976 (1976).
E. G. Loewen, D. Maystre, E. Popov and L. Tsonev, "Echelles: scalar, electromagnetic
and real-groove properties," Appl. Opt. 34, 1707-1727 (1995).
J. Hoose et al., "Grand Gratings: Bigger is Better, Thanks to Mosaic Technology,"
Photonics Spectra (December 1995).
E. G. Loewen et al., "Diffraction efficiency of echelles working in extremely high orders,"
Appl. Opt. 35, 1700-1704 (1996).
E. Popov et al., "Integral method for echelles covered with lossless or absorbing thin
dielectric layers," Appl. Opt. 38, 47-55 (1999).

Transmission Gratings
E. G. Loewen, L. B. Mashev and E. K. Popov, “Transmission gratings as three-way
beamsplitters,” Proc. SPIE 815, 66-72 (1981).
E. K. Popov, L. V. Tsonev and E. G. Loewen, “Scalar theory of transmission relief
E. K. Popov, E. G. Loewen and M. Nevière, "Transmission gratings for beam sampling
and beam splitting," Appl. Opt. 35, 3072-3075 (1996).

263

Mosaic Gratings
J. Hoose et al., "Grand Gratings: Bigger is Better, Thanks to Mosaic Technology,"
Photonics Spectra 29, 118-120 (December 1995).
T. Blasiak and S. Zheleznyak, "History and construction of large mosaic diffraction
gratings," Proc. SPIE 4485, 370-377 (2002).

Grating Replication
W. R. McKinney and L. Bartle, “Development in replicated nickel gratings,” Proc. SPIE
315, 170-172 (1981).
E. G. Loewen, Replication of mirrors and diffraction gratings, SPIE Tutorial T10 (1983).

Grating Ruling
R. S. Wiley, “Diamond’s role in ruling diffraction gratings,” Proc. Int. Diamond Conf.,
249-256 (1967).
E. G. Loewen, R. H. Burns and K. H. Kreckel, “Numerically controlled ruling engine with
1/10 fringe interferometric feedback for grids 600 mm square and scales 760 mm
long,” J. Opt. Soc. Am. 60, 726A (1970).
E. G. Loewen, R. S Wiley and G. Portas, “Large diffraction gratings ruling engine with
nanometer digital control system,” Proc. SPIE 815, 88-95 (1987).

Grating Testing
W. Anderson, G. Griffin, C. F. Mooney and R. Wiley, “Electron microscope method for
measuring diffraction grating groove geometry,” Appl. Opt. 4, 999-1003 (1965).
G. R. Harrison, E. G. Loewen and R. S. Wiley, "Echelle gratings: their testing and
improvement," Appl. Opt. 15, 971-976 (1976).

264

INDEX

classical diffraction, 22
A classical equivalent grating, 53, 86
Coddington equations, 92
aberration, 91
collimator, 76
aberration coefficient, 91
colorimetry, 199
Abney mount, 101
comets, 215
additive dispersion, 81
concave grating, 87
anamorphic magnification, 36
conical diffraction, 22
angle, sign convention for, 19, 90
conservation of energy, 145
angular deviation, 23, 106, 130
constant-deviation monochromator, 106
angular dispersion, 27
constant-scan monochromator, 82
anomalies, 126, 147
curvature, of concave blank, 87
Rayleigh, 147
Czerny-Turner monochromator, 76,
resonance, 147
185
threshold, 148
astigmatism, 91
D
astronomy
ground-based, 204 defocus, 91
space-borne, 208 demultiplexer, 210
atomic spectroscopy, 197 deviation angle, 23, 106
dielectric coatings, 145
B diffraction, 19
angle, 90
'B' engine, 44
classical, 22
bandpass, 33
conical, 22
bandwidth, 119
in-plane, 22
blaze angle, 38
limit, 110
blaze condition, 38
order, 22, 201
blaze wavelength, 39, 125
existence, 24
blazing, 16, 123, 142, 143
existence, 146
overlapping, 25
C propagation, 146
camera, 76 zero, 25
Carpenter prism. - see grism dispersion, 27
cementing, 64 additive, 81

266

angular, 27 Foucault knife-edge test, 173
linear, 28 free spectral range, 37
of light by grating, 22 fringe pattern, 50
plane of, 88 full spectral bandwidth, 119
reciprocal linear, 29 full width at half maximum, 119
subtractive, 81 full width at zero height, 119
double monochromator, 80 FWHM, 119
FWZH, 119
E
G
Eagle mount, 101
Ebert-Fastie monochromator, 77, 185 ghosts, 44
echelle, 37, 45, 191 Lyman, 167, 170
effective spectral bandwidth, 119 Rowland, 43, 167, 168
efficiency, 123 grass, 170
absolute, 123, 172 grating, 14, 19
anomalies, 172 anomalies, 147
conservation of energy, 145 as beam splitter, 211
curve, 124, 172 as filter, 209
measurement of, 172 as optical coupler, 212
errors in, 231 blazing, 46
reciprocity theorem, 145 cementing, 64
relative, 123, 172 classical, 86
emission spectrum (or line), 199 cleaning, 144
errors in efficiency measurements, 231 coating, 63
excitation spectrum, 199 dielectric, 145
special overcoatings, 185
F concave, 85, 186
dielectric coatings, 145
ƒ/number, 35
echelle, 37, 45, 191
facility, 55
efficiency, 56
Fermat's principle, 89
equation, 22
filter (grating), 209
fingerprints on, 144
first generation holographic grating, 53,
for laser tuning, 201
86, 96
grazing incidence, 191
flat-field spectrograph, 59, 103
groove
fluorescence spectroscopy, 199
frequency, 21, 59, 248
focal distance, 92
pattern, 59
focal length, 34
profile, 57
focal ratio, 35

267

handling, 217 substrate shape, 60
holographic, 49 tangent plane, 89
blazing, 143 testing, 46
classification, 52 transmission, 14
first generation, 53, 86 varied line-space (VLS), 86
modulation, 134 VLS, 86
recording process, 55 normal, 88
second generation, 54, 86 grating equation, 21
vs. ruled, 56 for Littrow configuration, 23
in fiber-optic telecommunications, in optical media other than air, 189
209 grating normal, 19
in metrology, 212 grating prism. – see grism
inspection, 66 grazing incidence system, 191
interference (holographic), 18 grism, 189
master, 14, 43, 49 groove frequency, 21, 248
manufacture time, 61 groove spacing, 16, 19
modulation, 134 precision (ruled grating), 17
mosaic, 205 variations in, 168, 170
mount, 75 grooves per millimeter, 21, 248
terminology, 75
with concave grating, 101 H
with plane grating, 75
handling gratings, 217
normal, 19, 88
holographic grating, 49
original, 14
overcoating, 143
parting agent, 64 I
pitch, 21, 248 ICP, 198
plane, 75, 185 imaging spectrograph, 105
reflection, 14 incidence angle, 19, 90
replica, 14, 61, 63 inductively coupled plasma, 198
ruled, 43 in-plane diffraction, 22
scattered light, 57 instrumental stray light, 40, 151
selection, 213 measurement, 181
separation, 65 interferometric control, 44, 45
Sheridon, 52 inter-order scatter, 179
special overcoatings, 185 ion etching, 57, 143
special sizes, 215
specifications for, 221
submaster, 64

268

Czerny-Turner, 76
K Ebert-Fastie, 77
Littrow, 79
knife-edge test, Foucault, 173
Monk-Gillieson, 78
mosaic grating, 205
L multiplexer, 210
laser tuning, 201
Lie aberration theory, 247 O
limit of resolution, 30, – see resolution
obliquity factor, 29
line curvature, 92
optical spectrum analyzer, 211
linear dispersion, 28
order. – see diffraction order
reciprocal, 29
order sorting, 27, 37
linespread function, 111
Littrow blaze condition, 38
P
Littrow configuration, 23
angular dispersion, 28 parting agent, 64
for laser tuning, 201 Paschen-Runge mount, 101
Rowland ghosts in, 169 peak wavelength, 125
Littrow monochromator, 79 photoresist, 51
Lyman ghosts, 170 pitch, 19, 21, 248
plane grating, 75, 87
M plate factor, 29
polarization, 125, 172
magnification
conversion, 149
anamorphic, 36
polychromator, 75
sagittal, 112
principal plane, 88
tangential, 112
prism, 189
Mann engine, 44
pulse compression and stretching, 203
master grating, 14, 43, 49
merit function, 99
R
Michelson engine, 44
modulation, 134 radius, of concave blank, 87
molecular laser tuning, 203 Raman gratings, 170
molecular spectroscopy, 197 Raman spectroscopy, 82, 200
Monk-Gillieson monochromator, 78 Rayleigh anomalies, 147
monochromator, 75 Rayleigh criterion, 30
double, 80 reciprocal linear dispersion, 29
constant-deviation, 106 reciprocity theorem, 145
constant-scan, 82

269

reflectance values, for various metals, spectrograph, 75, 83
251 flat-field, 103
reflection grating, 19 imaging, 105
replica, 14, 16, 63 Rowland circle, 101
replication, 16 Wadsworth, 103
tree, 66 spectrometer, 75
replication process, 63 spectroscopy, 197
resolution, spectral 32 atomic, 197
and resolving power, 33 fluorescence, 199
resolving power, 30, 43, 44, 186 molecular, 197
and spectral resolution, 33 Raman, 200
maximum theoretical, 31 spectrum analyzer, 211
measurement, 177 specular reflection, 22
resonance anomalies, 147 S-plane, 203
Rowland circle spectrograph, 101 spot diagram, 109
Rowland ghosts, 168, – see ghosts SPR, 149
ruling engine, 43 SRE (stray radiant energy), 151
stigmatic image, 91
S stray light, 40, 58, 151
measurement of, 181
sagittal focal distance, 92
stray radiant energy (SRE), 151
sagittal focusing, 96
submaster, 64
sagittal plane, 88
subtractive dispersion, 81
sagittal radius, of blank, 87
surface microroughness, 152
satellites, 170
surface plasmon resonance (SPR), 149
scan angle, 23
scattered light, 40, 151, 152
T
measurement of, 179
second generation holographic grating, tangent plane, of grating, 89
54 tangential focal distance, 92
servo control systems, 16 tangential focusing, 96
Sheridon grating, 52 tangential plane, 88
signal-to-noise ratio (SNR), 40 tangential radius, of blank, 87
slit function, 183 target pattern, 214
Snell's law, 189 threshold anomalies, 148
SNR (signal-to-noise ratio), 40 transfer coating, 64
solvents (for cleaning gratings), 144 transmission grating, 19
specifications for gratings, 221
spectral order. – see diffraction order

270

Wood’s anomalies, 126
V
Z
varied line-space (VLS) grating, 47, 86
vibration, 17 zero order, 22, 25
VLS (varied line-space) grating, 47, 86

W
Wadsworth spectrograph, 103
wavefront testing, 175

271

Diffraction grating

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