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Fourier Transform Spectroscopy 1st Edition Sumner P.
Davis Digital Instant Download
Author(s): Sumner P. Davis, Mark C. Abrams, James W. Brault
ISBN(s): 9780120425105, 0120425106
Edition: 1
File Details: PDF, 10.41 MB
Year: 2001
Language: english

PREFACE
Fourier transform spectroscopy has evolved over several decades into an analytic
spectroscopic method with applications throughout the physical, chemical, and bi-
ological sciences. As the instruments have become automated and computerized
the user has been able to focus on the experiment and not on the operation of the
instrument. However, in many applications where source conditions are not ideal
or the desired signal is weak, the success of an experiment can depend critically on
an understanding of the instrument and the data-processing algorithms that extract
the spectrum from the interferogram. Fourier Transform Spectrometry provides the
essential background in Fourier analysis, systematically develops the fundamental
concepts governing the design and operation of Fourier transform spectrometers,
and illustrates each concept pictorially. The methods of transforming the interfero-
gram and phase correcting the resulting spectrum are presented, and are focused on
understanding the capabilities and limitations of the algorithms. Techniques of com-
puterized spectrum analysis are discussed with the intention of allowing individual
spectroscopists to understand the numerical processing algorithms without becom-
ing computer programmers. Methods for determining the accuracy of numerical
algorithms are discussed and compared pictorially and quantitatively. Algorithms
for line finding, fitting spectra to voigt profiles, filtering, Fourier transforming, and
spectrum synthesis form a basis of spectrum analysis tools from which complex
signal-processing procedures can be constructed.
This book should be of immediate use to those who use Fourier transform spec-
trometers in their research or are considering their use, especially in astronomy,
atmospheric physics and chemistry, and high-resolution laboratory spectroscopy.
We give the mathematical and physical background for understanding the operation
of an ideal interferometer, illustrate these ideas with examples of interferograms that
are obtained with ideal and nonideal interferometers, and show how the maximum
amount of information can be extracted from the interferograms. Next, we show
how practical considerations of sampling and noise affect the spectrum.
This book evolved out of 20 years of conversations about the methods andpractice of
Fourier transform spectroscopy. Brault is the author of several papers incorporated
into this text, as well as author of a seminal set of lecture notes on the subject. He
is the prime mover in establishing the Fourier transform spectrometer (FTS) as the
instrument of choice for high-resolution atomic and molecular spectroscopy. The
xiii

xiv Preface
content of this book is taken mainly from his work in optics and instrumentation
over a period of many years. Some of the text was initially written up as a part of
Abrams*s doctoral dissertation in order to clarify and quantify many rules of thumb
that were developed in thefieldby Brault and others. Davis and his graduate students
were early users of the Kitt Peak and Los Alamos instruments and have continually
pushed for ever-greater simplicity, accuracy, and flexibility of the data-taking and
data-processing procedures.
Sumner P. Davis is a Professor of Physics at the University of California at Berke-
ley. His research focuses on laboratory spectroscopy of diatomic molecules of
astrophysical interest. Since the late 1950s, Davis and his graduate students pushed
the limits of high-resolution molecular spectroscopy — initially with 1- to 12-m
grating spectrometers and echelle gratings, then with high-resolution Fabry-Perot
interferometers (crossed with high-resolution gratings), and since 1976 with Fourier
transform spectrometers. Most recently, he has returned to echelle grating spec-
trometers — bringing full circle a five decade adventure in spectroscopy.
Mark C. Abrams is Manager of Advanced Programs for ITT Industries Aerospace
/Communications Division in Fort Wayne, Indiana. He was a staff member at the
Jet Propulsion Laboratory, California Institute of Technology, where he was the
instrument scientist for the Atmospheric Trace Molecule Spectroscopy (ATMOS)
experiment (a Space Shuttle-borne Fourier transform spectrometer used for Earth
remote sensing). His research focuses on remote sensing from space and instrument
design.
James W. Brault is a physicist and was a staff scientist of the National Solar Ob-
servatory, Kitt Peak, with an appointment at the University of Colorado in Boulder.
He is the designer of the one-meter instrument at the Observatory, and a codesigner
of the 2.5-meter spectrometer formerly at the Los Alamos National Laboratory
and now at the National Institute for Standards and Technology (NIST). He has
also pioneered numerical methods for transforming and reducing Fourier transform
spectra. His other areas of research are atomic and molecular spectroscopy.
We owe much to our contemporaries and predecessors in the field. We gratefully
acknowledge several kind colleagues and reviewers. Professor Luc Delbouille, Dr.
Ginette Roland, Professor Anne P. Thome, and Dr. Brenda Winnewisser, who gave
their time generously and shepherded the book through many necessary changes.
The inevitable mistakes are ours (with apologies, but enjoy them), and we hope that
these pages will inspire a new generation of researchers to push beyond the current
state-of-the-art and take the community forward with enthusiasm.
S. P D., M. C. A., and J. W. B. — Spring 2001

1
INTRODUCTION
Electromagnetic radiation in the classical picture is a traveling wave of orthog-
onal electric and magnetic fields whose amplitudes vary in time. The propagation
of the wave is described by the wave equation, which is derivable from Maxwell's
equations. We will consider only the electric field, since the magnetic field ampli-
tude and phase are linearly related to the electric field, and the observable effects
on our detectors are largely due to the electric rather than the magnetic fields.
As we shall see, this wave carries information about the source that generates
it. We can make a model of the source when we know the frequencies and their
amplitudes and phases that make up the time-varying wave. Think of how a prism
separates light into its constituent colors and how we can use it as a tool for ex-
amining the spectra of different light sources. It disperses the light by changing
its direction of propagation proportionally to the frequency. Sunlight is spread into
an almost uniformly intense visible spectrum from violet to red, while light from
a high-pressure mercury lamp is concentrated in the violet and blue. A diffraction
grating also disperses the light, but into several spectra rather than a single one, and
through angles proportional to the wavelength rather than the frequency. Because
we know how to describe the action of a dispersing element on an electromagnetic
wave, we like to think that there is an underlying physical principle and mathemat-
ical process that describes the resolution of a time-varying wave into its constituent
frequencies or colors, independent of what form the dispersing instrument takes.
The process is called Fourier decomposition of the waves. The amplitude vs. time
function is transformed into an amplitude v^. frequency spectrum. The decomposi-
tion technique is not restricted to electromagnetic radiation, but applies to all waves.

2 1. Introduction
including sound and water waves. Often the term Fourier transformation is used
as a general term encompassing both Fourier synthesis and Fourier decomposition.
Let*s pick an example to show what this transformation does. We will add
three sinusoids of different frequencies and then decompose the combination into
its three components. Strike a tuning fork lightly and we get a sinusoidal wave, in
this case sound in air, say, at a frequency of 300 Hz. An amplitude vs, time plot
is shown in Fig. 1.1. Strike a second fork and get a frequency of, say, 600 Hz, as
shown in Fig. 1.1. A third fork, with a frequency of 1200 Hz, as shown in Fig. 1.1.
Now strike them simultaneously and observe the beat frequency (Fig. 1.1).
Time (sec)
Time (sec)
Time (sec)
Time (sec)
Frequency (Hz)
Fig. 1.1 Sinusoidal waves of frequency 300 Hz, 600 Hz, and 1200 Hz. All three waves added together.
Fourier decomposition of the combined wave, showing the presence of three frequencies.

1. Introduction 3
The complex wave looks very different from any of the three sinusoids. We
have synthesized it from individual sinusoidal waves. If we see only the complex
wave itself, we might not realize that it is three distinct frequencies combined. But
when we execute a Fourier transformation, we get back a plot of amplitude vs.
frequency as show in Fig. 1.1. This plot tells us that we have three sinusoids at
300, 600, and 1200 Hz. The transform does not plot out three sine waves for us,
but only specifies the relative amplitudes of the frequency components illustrated
in Figs. 1.1. When we listen to the sound given off by the three tuning forks struck
simultaneously, we can actually hear the three frequencies making up the wave.
The ear performs a Fourier transform.
In optics, there are many such devices, so many that we do not even think of them
as instruments performing Fourier transformations on radiation. For examples, a
thin film of oil on the ocean splits sunlight into its constituent colors, and we see these
colors floating on the surface. A diffraction grating disperses light into different
directions depending on the frequencies present. The amplitudes and frequencies
tell us about processes in the source from which the radiation comes. In the cases
we are interested in here, the sources are solids, liquids, or gases and the radiation
emitters are atoms or molecules.
The observation and measurement of radiation emitted from or absorbed by
atoms and molecules provides information about their identity and structure and
their physical environment. A spectrum may contain discrete lines and bands and
perhaps a continuum. Each feature can be identified and associated with a particular
atom or molecule and its environment. For examples, spectrochemical analysis uses
spectroscopic features to determine the chemical composition of the source of the
radiation. Spectroscopists use the frequency distribution to measure the energy level
structure within an atom or molecule. This structure dictates the unique interaction
between radiation and emitting or absorbing matter. For a gas, measurements of
spectral features can be interpreted in terms of abundances, temperatures, pressures,
velocities, and radiative transfer in the material.
Spectrometry, as we are going to discuss it, is the detection and measurement of
radiation and its analysis in terms of frequency and energy distribution, called the
spectrum. The measurement of the spectrum involves the determination of spectral
line positions, intensities, line shapes, and areas, Fourier transform spectrometry is
a method of obtaining high spectral resolution and accurate photometry so that the
measured intensity is an accurate representation of the radiation. We will use the ab-
breviation FTS to mean either Fourier transform spectrometry or Fourier transform

4 7. Introduction
spectrometer, as the occasion requires. What we measure ideally should represent
the frequency spectrum of the radiation unmodified by any optical instrument or
computational artifacts. In practice, accurate spectrometry is a result of understand-
ing how the observed spectrum is modified by the observational process. To find
the frequencies present in the radiation we could in principle record the intensity
of the radiation as a function of time and take a Fourier transform to obtain the
frequency spectrum. In practice, we cannot do this because we have no detectors
and associated electronic circuitry that respond to optical frequencies. Instead, we
use a scanning Michelson interferometer to transform the incoming radiation into
amplitude-modulated radiation whose modulation frequencies are the scaled-down
optical frequencies in the original signal and typically fall in the range of 0 to 50
KHz. We detect these modulation frequencies and record them as the interferogram.
Our task is then to take the transform of this interferogram to recover the original
frequency spectrum of the radiation, interferometer
Why go through these complicated transformations when there are optical
instruments such as the diffraction grating spectrometer that transform incoming
radiation directly into its individual spectral components? The FTS is the most
accurate general-purpose passive spectrometer available. Even if not always the
simplest or most convenient, it can be used to provide working standards for testing
and calibrating more rapid but less precise techniques. In addition to its inherent pre-
cision, because it is an interferometer with a large path difference and a well-known
instrumental function, the FTS is noted for high optical efficiency, no diffraction
losses to higher-order spectra, high throughput, simultaneous observation of all
frequencies/wavelengths, precise photometry, easily adjustable free spectral range,
wide spectral coverage, and two-dimensional stigmatic imaging. Where spectral
line profiles are important, the FTS offers the most thoroughly understood and
simply characterized apparatus function of any passive spectrometer, because all
radiation falls on a single detector and instrumental distortions are often accurately
calculable and correctable.
Far more than most instruments, the Fourier transform spectrometer is a useful
and practical realization in metal and glass of a simple and elegant mathematical
idea, in this case Fourier's theorem. It is true that all of our instruments derive
their usefulness from physical phenomena that can be described by equations, but
their basis is usually well hidden by the time the instrument is in the hands of an
experimenter. The complex processes involved are normally summarized in some
kind of working equation, for example, the grating equation, which maps the desired

1.1 Spectra and Spectroscopic Measurement 5
variable, wavelength, onto the directly observed variable, angle or position at which
the intensity is observed.
In Fourier spectroscopy, the observations are made in a conjugate space, and
only produce recognizable results after a mathematical transformation. We observe
on the other side of the equation, so to speak. Now, this transformation poses no
difficulties for the instrument. The needed measurements are clearly defined and
readily obtained, and the computer programs for making the transformation are
simple and efficient. The real practical difficulty with this technique is that all of
the instincts and experience of the user have come on the one side of the equation, so
we must either operate blindly, and hence often foolishly, or find some way to carry
our experience across the transform sign. The discussion that follows is aimed at
developing some modes of thought and mathematical tools to make the latter path
easier to follow. The treatment is brief for topics that are already well covered
in articles or texts, and more complete for areas that are less well understood in
practice.
Several extensive bibliographies are collected at the end of the book. The
first is a chapter-by-chapter bibliography in which the references are presented by
topic within each chapter. The next is a chronological bibliography; the third is
an applications bibliography, covering laboratory work and applications to remote
sensing. The last is an author bibliography.
1.1 Spectra and Spectroscopic Measurement
The decomposition of electromagnetic radiation into a spectrum separates the
radiation into waves with various frequencies and corresponding wavelengths: v =
c, where v is the frequency in hertz (Hz) and A is the wavelength, with c the speed
of light. For optical spectroscopy in the infrared (IR), visible, and ultraviolet (uv)
regions of the spectrum, it is customary to work with the wavelength A measured
in a length unit of nanometers (10~^ m), abbreviated nm, or in a length unit of
micrometers (10~^ m), abbreviated pm, and with the wavenumber a = i//c = 1/A
measured in waves/centimeter in vacuum, abbreviated cm~^ and termed reciprocal
centimeters. However, the latest recommendation by an international commission
is to use a for the wavenumber in a medium, and i> for vacuum wavenumber or
without the tilde "when there is no chance of confusion with frequency." We will
consistently use the first definition of a in this book and note that it is sometimes
useful to consider cr as a spatial frequency.

6 1. Introduction
The spectrum is partitioned into broad regions as illustrated in Fig. 1.2. Which
of the preceding terms is used in any given case depends on the spectral region, the
observational techniques, and the particular phenomenon being investigated.
Fourier transform spectrometry covers a significant portion of this spectrum,
from 5 cm~^ in the far infrared to 75 000 cm~^ in the vacuum ultraviolet. FTS
instruments are distinguished by an ability to cover broad spectral ranges with high
resolution. A single scan can collect spectral data over ranges as large as 10 000-30
000 cm~^ simultaneously. While other spectrometers can cover similar spectral
ranges, the range of a single set of measurements is typically limited to 10% or less
of the potential range of the instrument class.
I — I — ——h- H- H h H h H h
Wavelength 1 km
Frequency
Wavenumber
Energy
Im
33 MHz
1 mm
33 GHz
1 cm
1 m
33THz
1 nm
• 100 cm' 100,000 cm"
1.2 eV 1.2 KeV
Name
I 1 h -
Radio Wave
Medium I Short
Micro
wave
Infrared
iFarlRINear
I IR
j<iUltraviolet
Vacuum
UV
X-Ray
1 pm
1.2 MeV
H 1
-Ray
Physical
Phenomena
Free Electrons Outer Electron Transitions Nuclear Effects
Molecular Rotation
Hyperfine Transitions
Inner Electron Transitions
H 1 -
Molecular
Vibration and Rotation
^ — — — — h H — h H 1
Instrument Tunable Sources and Detectors Crystal Transmission
Technology Ruled Diffraction Gratings
Tunable Lasers Crystal Gratings
Fourier Transform Spectrometers
Fig. 1.2 Frequency, wavelength, and wavenumber ranges for the measurement of electromagnetic radi-
ation.
Fundamentally, this book is about methods developed for measuring the best
possible spectra of laboratory, terrestrial, and astrophysical objects. Each of us
migrated to FTS as a means of obtaining the best spectra, rather than developing
a technique and seeking applications for that technology. It is spectra that have
captured our attention, and consequently various spectra will appear throughout
this book.
These sample spectra are intended to be representative rather than comprehen-
sive, and they reflect our personal biases. Because they are spectra we obtained in

LI Spectra and Spectroscopic Measurement
our research, they also have the convenience of proximity and accessibility. With
the advent of the Internet, many of the historic Ubraries of spectra that were locked
away in archives are now accessible to the general public. The electronic archive
at the National Solar Observatory (NSO), Kitt Peak, in which the McMath-Pierce
FTS spectra are archived, is a unique record of the atomic, molecular, solar, and
stellar spectra that have provided a major contribution to modem spectroscopy
in the last 25 years (perhaps 25,000 spectra). Similarly, the Atmospheric Trace
Molecule Spectroscopy (ATMOS) experiment (a Space Shuttle-based FTS), used
for atmospheric profiling and solar observations from space, provided some 80,000
high-resolution spectra in the 2- to 16-micrometer spectral region. Indeed, the two
laboratories supported each other, with the HITRAN (high-resolution transmission)
molecular spectroscopy database summarizing the results of years of measurements
at NSO that provided the theoretical basis for modeling the Earth's atmosphere.
20
L
15
I
10
I
8 Lim
500
800
814 815
Wavenumber (cm'')
816
Fig. 1.3 A solar absorption spectrum as observed from the Space Shuttle. The Atmospheric Trace
Molecule Spectroscopy (ATMOS) experiment obtained high-resolution solar absorption spectra between
(2-16 micrometers) with signal-to-noise ratios of 300-400, as illustrated in the
625 and 5000 cm
traces.
- 1

8 1. Introduction
High-resolution infrared solar spectra have remained a major research topic
throughout the past 30 years. Of the 16 000 detected solar features, the majority
are lines of vibration-rotation bands of the diatomic molecular constituents of the
photosphere: CO, CH, OH, and NH. The Av = I and At; = 2 bands of ^^c^^O
and its isotopic variants dominate the spectrum, about 55% of the lines. About
11% of the lines are due to atomic transitions in neutral atoms of Fe, Si, Mg, C,
Ca, and Al. But even after 15 years of examination, 24%, or 3700 lines, remain
unidentified.
In the long-wavelength infrared spectral region (10-15 micrometers) the solar
spectrum shown in Fig. 1.3 is dominated by the rotational bands of the hydroxyl
radical and the Mg emission lines. Since their discovery, the Mg emission lines,
which are Zeeman-split by the magnetic fields present on the surface of the Sun,
have been extensively studied for magnetic field mapping.
800
jpY---K|jW^
900
I I I I I
nrY^nhni
920 940
941
Wavenumber (cm"')
942
Fig. 1.4 Solar absorption spectra of the Earth's atmosphere illustrating the absorption features of various
trace gases: (a) CO2, (b) CFC-11, (c) HNO3, (d) CFC-12, (e) CO2, (f) O3.

1J Spectra and Spectroscopic Measurement
Solar absorption spectra of the Earth's limb during sunrise and sunset using
the Sun as a source, provide long-path observations from which the composition
and state of the Earth's atmosphere can be measured. They are shown in Fig. 1.4.
The low-dispersion spectrum illustrates absorption features due to carbon dioxide,
ozone, and nitric acid, which are essential components of the atmosphere, with
concentrations of parts per thousand, parts per million, and parts per billion, respec-
tively. In addition, the absorption features of the anthropogenic refrigerants CFC-11
(CCI3F) and CFC-12 (CCI2F2) are clearly visible in this tropospheric spectrum at
10-km tangent altitude. The noise level is approximately the vertical excursions of
the trace between spectral lines in the high-dispersion lower panel. The periodic
ripple apparent in this panel is molecular absorption of HNO3. Theoretical calcula-
tions do a remarkable job of simulating the infrared spectrum, as evidenced by the
work summarized in the HITRAN database.
p- 1
LUlMU
6600
. 1 , , , , 1 ,
1 1 1 1 I 1 1
6400
1 . 1 ,
1 7 — 1 1 I I I r
A
1 1 1
I I I -
1 -:
15000 15500 16000
15450
K.h^K.M
15420 15425
Wavenumber (cm')
15430
Fig. 1.5 Emission spectrum of ZrO in a furnace. The traces shown are all plotted from the same
observational broadband run and illustrate the fact that a broad spectral region can be covered at high
resolution in a single scan.

10 7. Introduction
Simulation of stellar environments with a high-temperature (T > 2000 K) fur-
nace is a method of observing molecular emission and absorption under controlled
conditions. The visible spectrum of zirconium oxide shown in Fig. 1.5 is observed
in the spectra of cool carbon stars. Laboratory measurements of line positions,
amplitudes, and equivalent widths permit accurate simulation of spectra in these
stars and the determination of the structure, composition, and evolution of the stars.
The FTS enables accurate and consistent broadband high-resolution spectrometry
covering large intensity ranges. With a properly calibrated FTS and spectral mea-
surement software, such measurements are routine and performed on a regular basis.
The catalog of atomic and molecular parameters measured with the NSO FTS is
remarkable in regard to both its sheer volume and the duration of this instrument as
a standard for laboratory measurements over its 25 years of operation.
1.2 The Classical Michelson Interferometer
We start with a Michelson interferometer, as shown in Fig. 1.6. Light from a
source at the object plane is coUimated and then divided at a beam splitter into two
beams of equal amplitude.
Fixed Mirror
Moving
Mirror
Image Plane
Fig. 1.6 Sketch of Michelson interferometer with an extended source.
These beams are reflected back on themselves by two separate mirrors, onefixedand
the other movable. Each single beam strikes the beam splitter again, where they are
recombined and split again. Consider only the recombined beam that is directed to
the image plane. The two components in the recombined beam interfere with each

12 The Classical Michelson Interferometer 11
other and form a spot whose intensity depends upon the different paths traversed by
the two beams before recombination. As one mirror moves, the path length of one
beam changes and the spot on the screen in the image plane becomes brighter and
dimmer successively, in synchronization with the mirror position. Circular fringes
rather than spots are formed when the source is an extended one.
For an input beam of monochromatic light of wavenumber Go and intensity
B{GO), the intensity of the interferogram as a function of the optical path difference
X between the two beams is given by the familiar two-beam interference relation
for the intensity,
Io{x) = B{ao)[l + cos {27raox)] (1.1)
where as noted earlier the wavenumber a is defined by cr = 1/A = u/c, measured
in reciprocal centimeters. When x is changed by scanning one of the mirrors, the
interferogram is a cosine of wavenumber, or spatial frequency, ao.
When the source contains more than one frequency, the detector sees a super-
position of such cosines,
/•OO
Io{x) ^ / B{a)[l + cos {27rax)] da. (1.2)
Jo
We can subtract the mean value of the interferogram and form an expression for the
intensity as a function of JC:
/•OO
I{x) = Io{x) - I{x) = B(a) COS {27rax) da. (1.3)
Jo
o (cm)
Fig. 1.7 Symmetric interferogram and the spectrum derived from its Fourier transform.
The right-hand side contains all the spectral information in the light and is the cosine
Fourier transform of the source distribution B{a). The distribution can therefore
be recovered by the inverse Fourier transform,
/•OO
B{a) = I{x)cos{27rax)dx. (1.4)
Jo

12 1. Introduction
A schematic representation of an interferogram and the resulting spectrum are
shown in Fig. 1.7.
1.3 Precision, Accuracy, and Dynamic Range
How precise are the measurements? Often we mean how many decimals in the
frequencies are significant, but we can also apply the question to intensities, line
shapes, and widths. In most current applications it is not enough to know only the
peak position and the peak intensity. Line shapes and widths are important, and they
influence the position and intensity. These parameters are accurately determined
only when the lines are fitted to an appropriate profile, something easy to determine
for the FTS. Fitting is also required because the point of maximum intensity almost
never occurs at a sampling point. Our fitting codes assume a voigtian profile, and
also account for the instrumental profile. With proper sampling and fitting we can
measure a single unblended line position to a precision of the line width divided by
the S/N ratio or even better. For example, a line at 2000 cm~^ with a width of 0.015
cm~^ and a S/N ratio of 100 can be measured to 0.00015 cm~ approximately 1.3
parts in 10 million.
How accurate are the measurements? When we assign a wavenumber value,
how good is it on an absolute scale? Two considerations are important, the precision
just discussed and the accuracy of our standards. At the present time our standards
are provided by N2, O2, CO, Ar, and a few other spectrum lines, so these set the
ultimate accuracy. Not every source operates with the same internal conditions nor
do they radiate exactly the same frequencies, so source variations always affect the
wavenumber accuracy. Current practice yields results accurate to one part in 10^,
while state of the art is an order of magnitude better. All this may change in the
next few years because standards are continually improving and being extended to
very large and very small wavenumbers.
It is in the realm of measuring line intensities, both absolute and relative, that
the FTS is without a serious competitor. Atomic spectrum lines vary from line to
line by orders of magnitude in intensity, and it is always the weak lines that verify
the correctness of an energy level analysis. The ratio of the strongest lines to the
weakest detectable ones is called the dynamic range of detection.
Lines from typical sources have dynamic ranges of 1000 or less, but for low-
noise-emission sources, dynamic ranges approaching 1 million are occasionally
encountered. Observing a dynamic range of 1000 is routine for FTS measurements,
and with care a dynamic range of 30,000 can be achieved. Reaching this limit with
other spectrometric techniques is challenging at best.

14 Units 13
1.4 Units
The diversity of units in a given field is related to the longevity of the discipline,
and optics and spectroscopy span nearly the entire period between Newton's discov-
ery of the dispersive properties of prisms in 1672 to the present day. Consequently,
there are more units than necessary.
Spectral plots are two-dimensional representations of energy as a function of
wavelength or frequency. The wavelength and wavenumber scales and their units
have been discussed in Section 1.1. Recall that the appropriate scale for Fourier
transform spectrometry is wavenumbers measured in reciprocal centimeters or
inverse centimeters, written as cm~^. Two equivalents are 1 cm~^ = 2.9979 x 10^^
Hz - 30 GHz.
The intensity scale (j-axis) can be even more confusing. Once upon a time the
scale was quantified in terms of spectral radiance, which was measured in units of
erg sec~^ cm~^ Hz~^ steradian"^. The spectral radiance can be integrated over
all frequencies to yield a total radiance in units of erg sec~^cm~^. In SI units we
have joule sec~^ m~^ Hz~^ steradian"^ and watt m~^. More often we use the
term spectral intensity, measured in watt/nm for wavelength-dispersing instruments
and watt/cm~^ for wavenumber-dispersing instruments, such as the prism and the
FTS. In the case of the FTS with photometric (photon) detectors, we often use
photons/cm~^ as the appropriate measure of radiation.
The difference between W/nm and photons/cm~^ is significant, because there
are two types of detector. One type is a photon detector, which puts out a signal
proportional to the photon density in the radiation striking it. The photomultiplier
is an example. The other puts out a signal proportional to the energy density of
the incident radiation. A bolometer is a detector of this type. For a given energy
density, the photon density is much larger in the infrared than in the ultraviolet;
and conversely, for a given photon density, the energy density is much smaller
in the infrared than in the ultraviolet. In practical terms, we can see that for a
given energy per unit interval (wavenumber or wavelength), a signal may saturate a
photon detector if the radiation is in the infrared, while it will hardly register in the
ultraviolet because there are so few photons.
To illustrate the differences between the three representations W/cm~^, W/nm,
and photons/cm~^, we can examine the emission spectrum of a blackbody source
at the temperature of the photosphere of the Sun, shown in Fig. 1.8.

14 1. Introduction
The thermal radiation profile (Planck curve, given in W/nm) is illustrated in Fig.
1.8. It is familiar from radiative transfer theory as the Planck curve, and describes
the energy flux as a function of wavelength:
B{T)
2hc^ 1
;^5 ^hc/XkT _ I
The radiometric flux as a function of frequency is
(Wm-^sr-^nm-^). (1-5)
B{a,T) = 2hca'^^^J^_^ (W m"^ sr"^ [cm-^]-^). (1.6)
The photometric profile or Planck curve in photons/ cm ^ is even less familiar
1
B{a,T) = 27rca'^^^^^^^_^ (photonsm"^ sr"^ [cm"^]-^) (1.7)
5.
I
0.5 0.33 mm
I
B
<
10000 20000
Wavenumber (cm"')
30000
- 1
Fig. 1.8 (a) Blackbody source at 5500 K, maximum of 0.41 watts/nm at 527 nm, or 18 985cm" (b)
Maximum of 0.30 watts/cm~^ at927nm,or 10790cm~^. (c) Maximum of 1.8 X 10^^ photons/cm~^;
maximum at 1642 nm, or 6089 cm"" .
In these equations, h and k are Planck's and Boltzmann's constants, respectively,
and T is the absolute temperature. The marked differences in shapes are due to the

1.5 A Glance Ahead and to the Side 15
differences in frequency dependence: The first equation goes as cr^, the second as
a^, and the third as cr^.
Standard pressure and temperature (STP) provide a reference point for all
measurements of gases. They define a standard density of 1 amagat. Standard
temperature is 273.15 K, and standard pressure is 1013.25 mbar, or 1.01325x 10^
N/m~^, or 760 mmHg. Unfortunately, the relevant quantity in much of spectrometry
is how much of a given species is present in a sample, which is measured in terms
of the total number of particles in the line of sight, called the column density.
The units are particles/cm^, which can be translated into centimeter amagats, a
more convenient unit for measuring column densities in a cell of known length and
pressure. The conversion factor is 1 cm amagat = 2.68675 x 10^^ particles/cm^.
For absorption spectra, the spectral intensity and the column density are linked
by Beers's law, which relates the absorption to the column density. For emission
spectra, the amount of radiation for thin sources is linearly dependent on the number
of radiating particles. Column densities can be translated into fractional abundances
in parts per million by mass or volume, abbreviated ppmM or ppmV, or more
typically into a concentration times a length such as parts per million meter or ppm
m, although the conversion is dependent on the local atmospheric density. At sea
level, 1 ppm m ~ 2.37 x 10^^ particles cm~^).
1.5 A Glance Ahead and to the Side
We hope this text serves two purposes and two communities of readers: students
in chemistry and physics who are preparing for research using spectrometry, and
practitioners in the field who are interested in the best methods used in the field. We
first derive some basic equations that describe an idealized interferometer and then
extend the description to include some unavoidable physical limitations, such as
maximum path difference and finite input aperture. The next step takes us into the
world of digital signal processing: sampling theory, discrete Fourier transforms, etc.
Then we move into the non ideal world to look at noise and ghosts, into the practical
problems involved in operating an FTS. And finally we present some applications.
We refer to our experience with the development and use of the McMath-Pierce
FTS in the broader context of how to get the most from your instrument and how to
understand the inevitable modifications of spectra by the instrument. The Fourier
transform spectrometry community is quite large and is growing rapidly, although
the high-resolution portion is small. There are many facts or rules of the game
that are known to only a few individuals in the field but unknown to most of the

16 1. Introduction
community and consequently go unheeded. The result is that many FTS users
throw away much of the information inherent in their data. Each rule is based
on a mathematical expression that defines a limitation of the method, which has
practical consequences in the laboratory in terms of how the data are acquired and
interpreted. Look for the following rules and examples as you peruse this book.
They delineate the essential steps required to set up an experiment, in the order that
one ought to address them.
1. Using resolution beyond that required to determine the expected line shape
adds high-frequency noise to the interferogram.
2. The size of the entrance pupil should be matched to the resolution required.
3. Three to five samples per FWHM are necessary to keep the instrumental dis-
tortion (ringing) below 0.1 % of the central intensity, depending on the shape of
the line (three for gaussian lines, five for lorentzian).
4. Violating the sampling theorem at any stage in the data processing introduces
nonphysical features into the data. This admonition may sound obvious, but
fitting minimally sampled data often violates the sampling theorem, because
the algorithms compute derivatives of the spectrum, which require twice as
many sampling points.
5. The use of apodizing functions with discontinuous derivatives creates manifold
problems, so don't use them.
6. Excessive apodization may make Fourier transform spectra look like grating
spectra, smooth with no ringing, but it only wastes information when used in
data-processing algorithms, particularly with fitting routines.
This book has its origin in Brault's seminal paper (1985), which captured many
of the lessons learned in the first decade of work on the 1-meter FTS at the McMath-
Pierce Solar telescope at Kitt Peak. It is a collection of experiences and lessons
learned, motivated by the desire to obtain the best possible measurements of spectral
distributions of electromagnetic radiation.

2
WHY CHOOSE A FOURIER TRANSFORM
SPECTROMETER?
What follows is a discussion of the merits of the Fourier transform spectrometer
(FTS) and why we, the authors, each separately made it our instrument of choice.
To put our work into perspective, together we have measured or supervised mea-
surements of a few thousand spectrum lines produced by prisms, a few more thou-
sands from Fabry-Perot interferometers, and several million produced by diffraction
gratings, and have ourselves measured tens of millions produced by Fourier trans-
form spectrometers.
An evaluation of the usefulness of any tool must begin with an understanding
of the task it is expected to perform. Our area of interest is passive spectrometry
— we expect to set up a source of light and analyze its output without disturbing
the source. We are practitioners of spectrometry in the region between 500 and 50
000 cm~^ (200 and 20 000 nm), with an emphasis on obtaining high-resolution,
broadband, and low-noise spectra.
Every spectrometer has an entrance aperture, focusing optics, a dispersing
element, and one or more detectors. Their comparative usefulness is characterized
by the throughput (how much light passes through), chromatic resolving power
(how close in energy two spectral features can be before they are indistinguishable),
and free spectral range (how wide a spectral range can be viewed before two features
of different wavelengths overlap in the spectral display). A block diagram might
look like Fig. 2.1.
17

18 2. Why Choose an FTS?
Source
Aperture Spectrometer
or >
S l i | / ^
l
Dispersive
Element:
Prism
Grating
Fabry-Perot
Michelson
Collimating Lens
Focal
7KT
JK'
Camera Lens
Prism
Spectrometer
Grating
Spectrometer
Fabry-Perot
Interferometer
Michelson
Interferometer
Fig. 2.1. Block diagram of spectrometer.
In this block diagram we have shown only lenses as the focusing elements,
although in practice mirrors are used for almost all grating spectrometers and
Michelson interferometers, and the optical path is folded back almost on itself.
The FTS uses spherical mirrors at f-numbers typically between f/16 and f/50.
The optical principles and practices are the same for both lenses and mirrors. In
simplified terms, slice a simple positive lens in half and put a reflecting coating on
the plane surface, and you have the equivalent of a concave (positive) mirror.
The job of the passive spectrometer is to gather spectral information from a
source as rapidly and accurately as possible. We will consider in turn three aspects
of information flow: the quantity of information per unit time, the quality of that
information, and some vague sense of the cost of the information.
2.1 Quantity
The magnitude of information flow through a spectrometer may be thought of
as the product of two quantities, one determined by the spectrometer optics and the
other by the detector:
information flow = (optical throughput) x (detector acceptance).
The optical throughput may be defined as the product of the area A of the entrance
aperture and the solid angle fi subtended there by the collimator, further multiplied
by the optical efficiency rjo of the system:
optical throughput = AQrjo (2.1)

2.1 Quantity 19
Because of its axis of symmetry, the FTS interferometer has a large entrance aperture
and, consequently, a large A^ product. A typical interferometer might have a 5-
mm-diameter circular aperture.
Another aspect of the quantity of data obtained is the fact that the FTS records
data at all frequencies simultaneously, a process called multiplexing. There is a
great saving in observation time when we wish to look at many frequencies, as
compared with scanning each frequency separately with a dispersive instrument
such as a diffraction grating.
To determine the role of the detector on the throughput, we need to consider the
mode of detection as well as the intrinsic properties of the detector. Let us combine
the effects of detector quantum efficiency and the noise into a useful hybrid, the
effective quantum sensitivity q, defined by:
q = [(5/iV)observed/(5/iV)ideal] ' (2.2)
where {S/N)i^ea. is the signal-to-noise ratio that would result from a perfect de-
tector, one with unit quantum efficiency and no noise. With this concept, we define
the detector acceptance as
detector acceptance = (quantum sensitivity) x (number of detectors) = qn.
The quantum sensitivity can be more usefully written as
^ NQ-^Nd'
where Q is the actual quantum efficiency of the detector, A^ is the number of
photons per measurement interval incident on the detector, and N^ is the number
of detected photons per measurement interval that it would take to produce the
observed detector noise (noise doesn't always come from photons!). For large
signals, NQ > A^^ and we obtain q ^ Q, while for small signals with NQ < Nd
we have instead q « {NQ/Nd)Q, and this effective quantum efficiency depends on
all three quantities, but especially strongly on the real quantum efficiency, which is
not usually specified by detector manufacturers.
Finally, there are the separate but related topics of spectral coverage and free
spectral range as touched upon earlier. Some spectroscopic problems can be solved
by observing only a fraction of a wavenumber, while others require broad coverage,
up to tens of thousands of wavenumbers. In the latter case, the amount of spectrum

that can be covered without readjusting or changing components becomes a factor
in the information flow. The FTS spectral coverage is Umited by the beamspUtter
material, beamsplitter coatings, substrate transmission, and detector sensitivity.
Wavelength ratios of 5 to 1 are achievable in a single scan, and ratios of 100 to 1
are possible by switching beamsplitters or detectors or both, although the switching
may not be trivial.
2.2 Quality
2.2.1 Resolution and Line Shape
Here we are concerned with the resolution and cleanness of the apparatus func-
tion, the precision of the intensity and wavenumber scales, and any possible sources
of excess noise. The instrumental resolution is determined by the maximum path
difference in the interfering beams. For major research instruments, this effective
maximum path difference is typically 1 to 5 m, corresponding to a resolution of 0.01
to 0.002 cm~^. The absolute wavenumber accuracy of any spectrum can be made
to the same degree as the precision, providing there is a single standard line with
which to set the wavenumber scale. Standard lines nearly equally spaced through-
out the spectral region are not required to set up an accurate scale. The subject of
calibration is discussed further in Chapter 9. On the other hand, many problems do
not require the full resolution of such instruments. For these problems, it is useful
to have variable resolution, because excess resolution reduces the signal-to-noise
ratio. The FTS is especially flexible in this regard and has no equal in the ease of
setting the instrumental resolution to the required value.
The accuracy in determining intensities ideally is limited only by photon statis-
tics, but in practice there are many systematic effects that degrade performance.
Some of these are apparatus function-smearing effects, which distort the shapes of
spectral lines, and nonlinearity and crosstalk in detectors, which create artifacts.
One of our main concerns is with line shapes. In the past, spectroscopy
has treated its two main variables very differently, being highly quantitative on the
wavenumber axis but only quahtative on the intensity axis, largely because intensity
measurements were difficult and unreliable. But accurate intensity information is
increasingly important in many areas: modeling stellar atmospheres, unraveling
complex hyperfine structure patterns, ratioing or differencing spectra to see small
differential effects in the presence of large systematic effects, understanding non-
voigtian line shapes, and so forth.

22 Quality 21
In measuring intensities it is necessary to take into account the apparatus or
instrument function of the spectrometer, defined as the output response to a purely
monochromatic input.
A major part of the value of FTS data is that a broadband interval of the spectrum
can be observed in single or multiple scans with the same instrument settings and
that the dispersion and the instrument line shape function are nominally the same
for every spectral line no matter where it lies in the range. The FTS has an
instrument function whose frequency response is essentially flat out to the end of
the interferogram, where it drops suddenly to zero. All other instruments have an
instrument function that changes markedly with wavelength or wavenumber. We
will discuss this function in a later chapter. In the meantime, to illustrate one
problem, line shape errors quantified as the decrease in peak intensity as a result of
the instrument function are plotted in Fig. 2.2 as a function of resolution for both
the grating spectrometer and the FTS.
If 1% line shape distortion is necessary, then an FTS with an optimum aperture
as defined in Section 5.2 will require five resolution elements across a line width.
In contrast, the grating with an optimum slit width will require 30 elements across a
line width. The factor of 6 in required resolving power is a large part of the practical
advantage of an FTS.
Wavenumber accuracy can be a large and nettlesome subject, although in the
best of all possible worlds it is limited only by photon noise. Under these conditions,
the uncertainty in position of a spectral line is roughly the line width divided by the
product of the signal-to-noise ratio in the line and the square root of the number
of samples in the line width. For example, a spectrum of N2O taken at NSO with
the 1-m FTS shows line widths of 0.01 cm~^ and SjN ratios of several thousand,
resulting in wavenumbers with a root mean square (r.m.s.) scatter of 2 x 10~^
cm~^ when compared with values calculated from fitted molecular parameters.
Such precision is possible though not common in modem FTS work.
22 2 Fixed and Variable Quantities in Experiments
There is yet another way to assess spectrometer performance, in terms of the
obtainable signal-to-noise ratio. Practically, there is a trade-off among signal-to-
noise ratio, spectral and spatial resolution, and measurement time, given the best
electronics, detectors, and optics available.

10.0
1.0
c
OH
0.1
0.01
l

N
1

I
I (a)
(h)
(c)
5 10 20
Resolution Elements per Line Width
50 100
Fig. 2.2 The amplitude distortion of a gaussian line by the FTS and a grating. Curve (a) gives the
limiting error for the FTS due tofinitepath difference alone when the aperture contribution is negligible;
(b) shows the FTS error when the optimum aperture is used. Curve (c) is for a grating with an optimum
slit.
All radiometric devices, including radiometers and interferometers, have com-
mon elements: an aperture of area A and solid angle fi, and optics to channel
radiation to the detector. The devices differ in their methods of spectral separation
and may be compared based on the signal-to-noise ratio within a narrow spectral
interval ACT that is the filter bandwidth for a radiometer or the spectral resolution
width for a spectrometer.
The noise equivalent power (NEP) is the signal power for a signal-to-noise ratio
of unity and is the inverse of the detectivity D
NEP{W) = D-W-^) =
D*
1 An.
T ''
(2.4)
where Ad is the detector area, Af ^ 1/T is the effective bandwidth, which
is determined by the dwell or integration time T at each point, and D* is the

22 Quality 23
detectivity in the detector-noise-limited regime. See Section 8.2.3 for comments on
the usefulness of D*.
The noise equivalent spectral radiance (NESR) describes the overall efficiency
and throughput of the instrument:
_ NEP((7) _ 1 lAp
where r/i is the system efficiency and r/2 is the optical efficiency, including the
transmission properties of the optical components. The spectral bandwidth is the
spectral resolution ACT of the instrument, and the etendue (throughput) A^ is the
product of the collecting area and the solid angle describing the field of view.
The signal-to-noise ratio of the observation is
S_ _ l{a) _ J((T)r;i7^2(^)A(7An
N ~ NESR(a) ~ NEP(a)
(2.6)
or, including the detector characteristics (appropriate for the infrared in the detector-
noise-limited regime),
S _ Iia)mV2ia)^aAnD*Vf
Equation (2.7) leads to the conclusion that the best observations are obtained when
the best detector is used (high D*), the integration time is long, the condensing optics
are fast (large Q), and the bandwidth (spectral resolution) ACT is large (minimum
spectral resolution).
By rearranging Eq. (2.7) we can partition the instrument performance into
terms that are largely fixed and into those that are variable in the measurement
design:
S/N _ rj,v,{a)AI{a)D*_ ^^.8)
AanVf y/A^
The right-hand side is essentially constant. System and optical efficiencies are
always optimized and constrained by material properties, the aperture is as large as
physically possible, the specific intensity is determined by the source and the spectral
resolution required, and the detector performance is determined by its inherent
properties. To gain a factor of 2 improvement requires significant investments of
time and money.

On the other hand, the left-hand side is flexible in trading off one property
for another. The required signal-to-noise ratio can be achieved by many different
combinations of the three parameters in the denominator. These parameters are
the spectral resolution, the spatial resolution or solid angle or field of view, and
the observing time. We can view these parameters as axes of a three-dimensional
space, as shown in Fig. 2.3.
Photometer
(S/N)/T*'^
Narrow Beam
Radiometer (camera)
1/Q
Fig. 2.3. A three-dimensional space representing the trade-off space in which instrument designs are
optimized. The product of the three coordinates representing any instrument must have a fixed value
determined by the right-hand side of Eq. (2.8).
Narrow-beam radiometers using camera systems emphasize angular resolu-
tion at the expense of spectral resolution and signal-to-noise ratio. In contrast,
photometers trade off spectral resolution and solid angle to obtain the best possi-
ble signal-to-noise ratio in a given time interval. Finally, high spectral resolution
requires compromises on the angular resolution and signal-to-noise ratio. As ex-
perimenters largely interested in high-quality spectra, we have had the luxury of
practically infinite integration times and correspondingly have designed instruments
with high spectral resolution and small field of view.
2.3 Cost
One concern is with the resources required to perform useful spectrometry,
including not just capital outlay, but the time used in understanding and becoming
familiar with the equipment, maintaining and extending it, and handling the data
that justify the whole apparatus in the first place. There is a widespread feeling
that grating instruments are cheap and simple and that an FTS is complex and
expensive, and to some extent this is true. But the instruments being visualized
when such comparisons are made are usually vastly different in power. The most

2.4 Summary 25
challenging problems are handled not with, say, a 1-m Ebert-Fastie spectrograph
with photographic recording, but with 10-m-class multiple-passed scanning gratings
or an echelle crossed with a grating and having a two-dimensional spectral display,
and not with a simple single Fabry-Perot etalon, but with multiple-etalon systems.
However it is accomplished, high-precision spectrometry is expensive in the time
of experts as well as in capital.
It is easy to ignore the cost of data reduction, but this can be a real mistake. An
instrument is built once but used many times to obtain data. The natural output of
an FTS after a straightforward numerical transform is a set of numbers representing
the intensities on a linear scale, at a set of points equidistant in wavenumber.
Computer programs exist that operate directly on such records, producing plots and
lists of spectral line parameters almost automatically and making it possible to deal
with spectra of quite remarkable complexity. The importance and value of such
capability cannot be overemphasized. Furthermore, the required computational
power, including that needed to perform the numerical transform, is readily available
on personal computers.
2.4 Summary
To put these comparisons in perspective, we can take several practical cases of
spectra we wish to measure and discuss which instrument we might choose.
Consider fluorescent lamps, which come with several different colors as seen
by the eye — white, blue, red, etc., with no radiation outside the visible spectrum.
Suppose you wanted to make a quick comparison of the color content of each.
Simply look at the lamp with a hand-held prism spectroscope. To get a more
precise evaluation, try a spectrometer with a 60-degree prism of base size 75 mm, a
dispersion index of 50, with f/16 optics. The resolving power is 7500 (0.1 nm). It
produces a single spectrum, with the visible region covering about 20 mm and no
overlapping of spectral regions.
To look at the same lamps with a resolution large enough to resolve the mercury
yellow lines at 577 and 579 nm, try a grating of 50-mm width used in a Littrow
mounting (equal angles of incidence and diffraction) with f/5 optics - a 1/4 meter
scanning monochromator, available commercially. It has a maximum theoretical
resolving power of 200 000 at 500 nm. Since resolving power is most often used
as the basis for comparison, remember that it is expressed as
R = (order of interference) x (number of grating grooves) = mN

or using the grating, because R = mN = (dsin 6/X){W/d) = l^sin 0/X, the
number of wavelengths that will fit into the maximum path difference between
rays diffracted from opposite ends of the grating. In practice the resolving power
is far less than that theoretically possible because the spectrum is observed in
the first order for simplicity of data reduction, and the range of groove spacings
available is limited - representative values are 300/mm, 600/mm, 1200/mm. The
slit width also affects the resolving power. To put in some numbers, consider the
instrument just mentioned, used in the first order with a 50-mm-wide grating having
600 grooves/mm. The theoretical resolving power is 30 000, but with a typical 5-
micron sUt it is reduced to 20 000. The yellow lines are easily resolved. There is
no overlapping of orders because of the restricted range of visible radiation. The
width of the visible spectrum is 50 mm, and a typical scan might take 3 minutes.
Now try observing the mercury green line (also present in a fluorescent lamp)
with a Fabry-Perot interferometer for the purpose of examining the central line
structure in detail, where a resolving power of 800 000 is needed. A Fabry-Perot
interferometer with a spacing of 7 mm, a reflectance of 90%, and f/16 optics has a
resolving power of 800 000 and a free spectral range of 1.5 cm" ^, or 0.05 nm. Here
the resolving power R = (order of interference)(equivalent number of interfering
beams) = {2t/X)NR, where NR is the finesse, about 30 for a reflectance of 90%. In
this case a narrow band filter of width 1.5 cm"^ is required to isolate the line from
the background radiation. An auxiliary dispersing spectrometer (grating or prism)
is often used for this purpose, such as the 1/4 meter monochromator described
earlier.
When we wish to observe the entire lamp spectrum in great detail, including
the hyperfine structure in the green line, we can use an FTS with a maximum
path difference of 200 mm, which gives a resolving power of 800 000. The
path difference of 200 mm is 30 times the plate separation in the Fabry-Perot
interferometer, but in return there is not the same limitation on the free spectral
range. The limit depends on the sampling frequency of the electronics and the speed
of the moving mirror. A typical value of spectral range is 10 000 cm~^, or 250
nm. A single scan might take 2 minutes. The resolution can be changed by simply
changing the value of the maximum path difference. The same FTS can be changed
from a low-resolution to a high-resolution spectrometer on demand, from a "quick
look" instrument to observe changes in spectra with changing source conditions
almost in real time to a high-resolution maximum signal-to-noise instrument. Its
flexibility in this regard is unequaled.

2.4 Summary 27
Each of the three systems — grating, Fabry-Perot, and FTS — occupies a useful
niche in the overall scheme of spectroscopy.
Broadband spectra of modest quality are most simply and cheaply obtained by
the grating with photographic or CCD recording, at least in the visible and UV. This
system is also the most tolerant of source intensity variation. Echelle spectrographs
with array detectors bring at least an order of magnitude improvement in quantity
of data gathered with a diffraction grating and in digital data processing. However,
at high resolution they reproduce line shapes and positions with only modest accu-
racy, owing to optical aberrations and nonlinearities in dispersion. Data reduction
and analysis initially require a minimum of computation to get a first look at the
spectrum, but the extra computations required to convert wavelengths to wavenum-
bers, fit the spectral lines, and construct atlases are time consuming and full of
pitfalls. A typical spectrum might consist of 20 successive echelle orders, each with
a variable dispersion within an order, and a changing dispersion from order to order.
The data are in wavelengths rather than wavenumbers and consequently require an
extra computation to get the energies of levels. The number of samples in each
spectral line must be much larger than for FTS data to get accurate fits for position,
intensity, shape, width, and area, and even then the lines are always asymmetrical
in shape. When constructing atlases, each order must be interpolated to the same
dispersion linear in wavenumber, and then the orders must be trimmed and shifted
to match each one with the preceding and succeeding ones. These computations
are all doable, but not trivial.
High-resolution and compact size are the strong points of the Fabry-Perot inter-
ferometer, though it is restricted to problems that need only a small free spectral
range and are tolerant of apparatus function smearing. The FTS is the system of
choice in the infrared under almost any conditions (with or without a multiplex
advantage) and in the visible and UV when high accuracy is required in intensity,
line shape, or wavenumber.

3
THEORY OF THE IDEAL INSTRUMENT
The essential problem of spectrometry is the measurement of the intensity of
light as a function of frequency or wavelength. The Fourier transform spectrometer
is a multiplex instrument, meaning that the spectral information is encoded in such a
manner that the intensity distribution at all frequencies is measured simultaneously
by a single detector, producing an interferogram, as we have noticed in the previous
chapters.
A simplified optical arrangement has already been shown in Fig. 1.6. A
more sophisticated optical configuration is shown in Fig. 3.1. In the plane mirror
configuration, one-half of the output signal is returned to the source and is lost.
With retro-reflectors instead of plane mirrors, the two outputs are separable. Using
the second output as well as the first doubles the system efficiency. On a more
subtle level, a dual-output system provides a direct measure of the constant (DC)
flux incident on the detector, which is critical for nonlinearity corrections and the
removal of time-domain intensity variations. The choice of retro-reflectors stems
from the fact that it is difficult to keep two plane mirrors exactly perpendicular to
the optical axis as they are scanning. Optical alignment errors produce errors in
the spectrum line profiles. Errors in perpendicularity needfirst-ordercorrections,
while retro-reflectors require only second-order corrections.
In an interferometer, the incident light is focused onto an entrance aperture, a
circular opening typically 1 to 10 mm in diameter, as shown in Fig. 1.6. Optically
speaking, this aperture is the entrance pupil. The light is then coUimated into
plane waves, which are divided by a beamsplitter (ideally 50% transmitting and
50% reflecting) so that the two beams can travel separately through the two arms
29

30 3. Theory of the Ideal Instrument
Corner Reflector Ci
A = e'^^ pi{Lut—2naxi^2)
0 a:i,2
Unbalanced Output (B) ^
kr.
^
Recombiner
-^
Splitter
f ^2
Balanced Output (A)
Fig. 3.1 Optical configuration for a Michelson interferometer. The symbol r^ is the external amplitude
reflection coefficient at the beamsplitter and recombiner, t is the transmission, and r^ is the overall
reflection coefficient of the comer reflector.
of the interferometer. The beams are reflected by mirrors and recombined by a
second beamsplitter unit, the recombiner, into a single beam, which is focused onto
a detector placed at the balanced output position in the figure.
In the plane of the detector the interference pattern is a set of focused circular
rings calledyrm^^^". How many fringes are detected depends on the size of the exit
pupil (the image of the entrance aperture) and on the wavenumber and difference
in path length of the two interfering beams. The size of the entrance aperture is
adjusted so that the central fringe at maximum path difference just fills the exit pupil
where the detector is placed. The exit pupil is an image of the entrance aperture. A
separate physical aperture is not needed.
With monochromatic light incident on the instrument, and when the optical
pathlengths and beamsplitter phase shifts in the two arms of the interferometer are
equal, then the beams interfere constructively at the detector, and the central fringe
is bright at the balanced output. If either or both of the mirrors are moved so
that the path lengths differ, then the field is bright to an extent determined by the

3.1 Equationfor the Balanced Output 31
degree of constructive or destructive interference as a result of the total optical path
difference. When the mirrors are moved at constant speed, the signal at the detector
alternates between light and dark in a sinusoidal fashion.
Several assumptions are implicit within the previous paragraphs that dictate
the form of the following material. We have said nothing yet about how we are
going to record the interferogram and take its transform. Some early instruments
used bandpass filters to perform the Fourier analysis, but often the signal strength
was too small to detect easily, especially in the infrared. To solve this problem the
interferometer was scanned only once in a step-by-step movement of the mirrors.
The interferogram was sampled at small, uniform intervals of path difference for
times long enough to average out the noise. However, stepwise movement of
mirrors in a Michelson interferometer is complicated to control, gives less accurate
positioning, and introduces additional errors as compared to continuous scanning,
as was shown by Harrison in the 1950s with his grating ruling engines. The more
accurate method of data taking is to sample the interferogram while the mirrors are
moving smoothly and continuously and to scan repeatedly to reduce the noise to
an acceptable level, as demonstrated by the Kitt Peak instrument constructed in the
1970s.
As we shall see, the mathematical expression for the interference of the com-
bined beams consists of two terms, a constant term and an interference term that
contains all of the desired information. In some instruments the constant term is
simply removed by a high-pass filter. However, there are always two recombined
beams in which interference occurs as shown in Fig. 3.1, and it is possible to have
two detectors and two output signals that can be combined to eliminate the constant
term and double the signal amplitude, as we shall now discuss.
3.1 Equation for the Balanced Output
The balanced output is so called because both beams undergo one single exter-
nal reflection at the beamsplitter or recombiner, and therefore produce constructive
interference at the detector when the total path difference is zero. Conversely, in
the unbalanced outipuU which is imaged back onto the source in a classical Michel-
son interferometer, one beam has a single external reflection while the other has
none, and their sum has zero intensity because of a phase change of n on external
reflection.
We begin by writing down the fundamental equation that describes what hap-
pens when a plane wave of monochromatic light is incident on the interferometer.

Let the incident light-wave amplitude be represented by e^^^ then the amplitude of
the emergent wave at the balanced output is
A = e*'^Verct(e-*^^^^i + g-^^^^^^j. (3.1)
The emergent time-averaged intensity is the square of the amplitude
/ = |Ap = 2ReRcT{l + cos [27ra(xi - ^2)]}, (3.2)
where Re = rl and Re = r^ are ordinary intensity reflection coefficients and
T = ^^ is the intensity transmission coefficient. The emergent intensity is modified
by three different aspects of the interferometer. We define
77^ = optical efficiency = Re
rjt = beamsplitter efficiency = AR^T
X — path difference = X2 — xi
and rewrite the equation as
^/ X r 1 + COS (27rcrx) 1 ,^ „,
i{x) = rjom [ ^ ^J. (3.3)
The first term, the optical efficiency, is a simple multiplier with a maximum value
of 100% when the mirrors are perfectly reflecting. The beamsplitter efficiency has
a maximum value of 100% when there is no absorption and exactly half the light
is reflected and half transmitted. If we use a dielectric coating so that there are no
losses to absorption or scattering, i? -h T = 1 and rn, = 4i?(l - R), Even a coating
as unbalanced as iZ = 0.15 and T = 0.85 has an efficiency greater than 50%, and a
ratio of 0.25:0.75 results in 75% efficiency.
3.2 The Unbalanced Output
What about the second output? We can simply note that if no energy is lost
in the beamsplitter or recombiner, the outputs are complementary, and energy not
appearing at the balanced output must be at the unbalanced output. For unit input.
IA-^ IB = constant = rio
' 1 + cos {2'Kax)'
IA = VoVb [- (3.4)

3J From Monochromatic Light to Broadband Light 33
IB=VO-IA= rjoVb [ ^ ^J + Vo{l - m)' (3.5)
Since both outputs contain the desired information (half the photons go in each
path), we combine them appropriately by taking their difference. This has the dual
advantage of doubling the signal strength and eliminating most of the constant term
that introduces additive noise (see Section 8.2). There is yet another advantage to
having two outputs. Their sum is a measure of the total intensity of the source, which
may vary slowly in time. Using this information, the interferogram amplitude can
be partially corrected for intensity variations as the interferogram is being recorded.
It should be noted that even though the preceding appears mathematically
and scientifically sensible, many instruments use only one output and remove the
constant term with a high-pass filter between the detector and the preamplifier. This
is equivalent to subtracting the mean signal value from the interferogram, which,
while normally effective, underutilizes the interferometer to avoid the complexity
of a second detector. It degrades by /2 the signal-to-noise ratio the instrument is
capable of achieving.
From now on, for simplicity we shall ignore the constant term and take as our
measure of the output the modulation term
I{x) = IA{X) — IB{X) ~ VoVb cos(27rcra:) = r;cos (27rc7x), (3.6)
where we have combined the optical and beamsplitter efficiencies into a single
overall efficiency rj = ryo^6. Since it is a simple multiplier, it will be left out of the
equations altogether.
3.3 From Monochromatic Light to Broadband Light
In our progression from monochromatic to broadband light, we will use the
concepts and techniques of Fourier analysis, which is the process of representing an
arbitrary function by a superposition of sinusoids. We will introduce each concept
as required, and summarize them all in Chapter 4 with more mathematical rigor for
reference purposes.
In general a source radiates more than one frequency of light, and in such cases
the detector records a superposition of cosines, each one weighted according to the
intensity at its given spectral frequency. For example, a spectrum consisting of two
close, narrow lines of similar intensity, as shown in Fig. 3.2, produces an interfer-

ogram that looks like the familiar beat frequency phenomenon. The carrier fre-
quency is (cTi + o-2)/2 and the beat frequency is (ai — 0-2).
Fig. 3.2 Sodium doublet and interferogram.
Many emission sources consist of strong, apparently randomly spaced spectral
lines. The interferogram of such a source displays constructive interference only
near the position of zero path difference, which results in a bright central fringe,
often called the white light fringe, with an amplitude proportional to A^, the number
of lines in the spectrum. Away from zero, the cosine wave amplitudes decrease
rapidly to a value proportional to ^/N, as illustrated in Fig. 3.3.
Fig. 3.3 Emission spectrum and interferogram.
Other sources emit a near-continuum of radiation with few or many absorption
lines. The continuum produces only a single intense peak in the interferogram at
zero path difference. Information about the absorption lines comes from the small
ripples in the interferogram that extend out to large path differences. They almost
look like noise, as shown in Fig. 3.4.

3.3 From Monochromatic Light to Broadband Light 35
l^.s-v •^^"
Fig. 3.4 Absorption spectrum and interferogram.
In practice, therefore, we always have a polychromatic wave rather than a
monochromatic one, and we need the techniques of Fourier analysis to sort out the
various frequencies. The final interferogram is a superposition of the individual
interferograms for each different frequency, as we shall see.
3.3.1 Generalization to Polychromatic Light
Thus far we have assumed that the input light wave was a monochromatic
wave (Eq. 3.6) of unit amplitude. We may generalize to polychromatic waves with
realistic intensities by letting B{a)da be the energy in a spectral interval da at the
frequency a and the corresponding interferogram dl{x) be the energy detected at
the optical path difference x:
dl{x) = B{a)da cos {27rax). (3.7)
Integrating over the frequency variable yields the energy detected at a path difference
x:
I{x) = B{a) COS {27rax) da. (3.8)
Jo
The quantity we want to recover from this equation is the spectral distribution B{a),
which we can do by taking the inverse transform of the interferogram
/»oo
B{a) = / I{x)cos{27rax) dx.
Jo
(3.9)
3.3.2 Extension to Plus and Minus Infinity
Our arguments leading up to the Fourier transform in Eq. (3.9) make sense phys-
ically, but the mathematics actually produce not only the spectrum B{a) but also

its mirror image, B(—a), at negative frequencies. Remember that cos (27rcrx) =
cos (—27rcra:) and therefore B{a) and B{—a) produce identical interferograms.
The negative frequencies are physically unreal. But when we consider discretely
sampling the interferogram and transforming this representation of the true inter-
ferogram, the mirror spectrum at negative frequencies plays an important role. For
complete symmetry in transforming back and forth from the interferogram domain
to the spectral domain, we would like the integral to extend over all frequencies
from minus to plus infinity. We want the interferogram and the spectrum to be
symmetrical (even functions), so we need to have an expression for B that is also
symmetrical. We can construct such a spectral function B^ from B, as shown in Fig.
3.5 and Eqs. (3.11) to (3.13), and change our definitions to include all frequencies:
Be{a) = ^[B{a) + B{-a)] (3.10)
Be{cr) COS {27rax) da (3.11)
-co
/
+00
I{x)cos{27Tax)dx. (3.12)
-oo
We now have symmetric sets of functions to work with mathematically, and
that will reproduce the true spectrum properly.
Fig. 3.5 Construction of a symmetric function Be from an asymmetric function B.
3.4 The Fourier TVansform Spectrometer as a Modulator
Now imagine that the reflectors are moved in such a way that the path difference
varies linearly in time so that x = vt. Then the dependence of intensity on time is
/ ( ^ ) a cosi27ravt). (3.13)

3.5 Summary 37
Since we assume the input beam is stationary (steady in spectral content and average
amplitude), we can think of the FTS as a modulator that produces a frequency
(typically audio) f = av from that steady beam. Or we may think of it as a
frequency multiplier that maps the light frequency ac to an audio frequency av.
The envelope of the interferogram has exactly the same shape as that of the original
wave, but the "carrier" frequencies are reduced by the factor v/c and hence are
easily detectable and the correct frequency distribution is still recoverable.
3.5 Summary
The heart of Fourier transform spectroscopy is the recognition that polychro-
matic spectral distributions can be determined by measuring the interferogram
produced in an amplitude-division (Michelson) interferometer and then calculating
the Fourier transform of the interferogram. We measure I{x) and then perform
mathematical operations to obtain Be {o) and construct B{a), the desired spectrum.
Figure 3.6 illustrates the process with a spectrum consisting of three lines with
different widths and profiles.
While the spectroscopist's experience is largely in the spectral domain, with
some practice a considerable amount of information can be inferred from the inter-
ferograms themselves.
Figure 3.6 presents three individual interferograms, their sum, and the resulting
spectrum. The quasi-monochromatic frequency of each individual interferogram is
indicated by the oscillation frequency in the wave train. The interferogram in (a)
has the smallest frequency of the three, with a corresponding spectral line of the
lowest frequency, as shown in (e). A larger number of oscillations in a unit interval
of the interferogram corresponds to a higher frequency in the spectrum, as shown
in the interferograms (b) and (c) and in (e).
The spectral line width determines the length of the interferogram. The smaller
the line width, the longer the interferogram. The central spectral line in (e) has the
longest interferogram (b).
The line shape determines the shape of the interferogram envelope. The inter-
ferogram in (a) is produced by a spectrum line with a voigtian profile having equal
gaussian and lorentzian components. The interferogram in (b) is produced by a
gaussian profile, and that in (c) by a lorentzian profile. Notice that the lorentzian
profile interferogram has a cusp at the central maximum; in contrast, the gaussian is
smooth across the central maximum, while the voigtian (a convolution of gaussian

38 3, Theory of the Ideal Instrument
(a) Voigtian
Optical Path Difference x (cm)
(b) Gaussian
(c) Lorentzian
^^^^'^m^^^^m^
optical Path Difference x (cm)
(d) Superposition (a-c)
i (e) Power Spectrum (FT of d)
Wavenumbera (cm )
Fig. 3.6 From signal to spectrum. Individual frequency components radiated by the source (a ~ c). (d)
Sum of the wave trains. The Fourier transform of (d) into (e) completes the cycle from wave trains to
interferogram to spectrum lines.

3.5 Summary 39
and lorentzian profiles with the same full widths at half maximum) displays some
of both constituent shapes. Also note that the lorentzian line profile (c) displays
characteristic exponential wings that decay very slowly, whereas the voigtian profile
decays more rapidly due to the contribution of the gaussian profile.
Of course the combined interferogram shown in (d) does not display all the
foregoing features quite so distinctly, but the principles are clear and the transform
of the combined individual interferograms reproduces the spectrum as exactly as if
we had separate interferograms, by the principle of superposition. In many spectra
all lines have roughly the same shapes and often the same widths. Then the envelope
of the combined interferogram does show clearly the characteristic shape and width
of the lines.

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Title: The prince of space
Author: Jack Williamson
Illustrator: Leo Morey
Release date: June 1, 2024 [eBook #73750]
Language: English
Original publication: Jamaica, NY: Experimenter Publications Inc,
1931
Credits: Greg Weeks, Mary Meehan and the Online Distributed
Proofreading Team at http://www.pgdp.net
*** START OF THE PROJECT GUTENBERG EBOOK THE PRINCE OF
SPACE ***

The Prince of Space
By Jack Williamson
Author of "The Metal Man," "The Green Girl," etc.
Illustrated by MOREY
Even the Lick Observatory,
which was built at the summit
of Mount Wilson, 5885 feet
high, at tremendous expense,
cannot satisfy the
astronomers. An observatory
that would reach about twice
that height, such as the one
built by the scientist in this
story, would be more likely to
hit the mark. Certainly, the
views obtained of the Moon,
and even of Mars, through our
present apparently gigantic

telescopes, undoubtedly call
for a higher observatory,
fitted with a more enormous
telescope, which will some
day be established. What may
be seen then cannot be
foretold with certainty. But
that's where the imagination
—with scientific visualizations
—enters. Mr. Williamson's
writing is not new to our
readers. At that, this story is
sure to make stronger friends
for him, and add many new
ones to his ever fast-growing
list of admirers.
[Transcriber's Note: This etext was produced from
Amazing Stories January 1931.
Extensive research did not uncover any evidence that
the U.S. copyright on this publication was renewed.]

CHAPTER I
Ten Million Eagles Reward!
"Space Flier Found Drifting with Two Hundred Dead!
Notorious Interplanetary Pirate—Prince of Space—
Believed to Have Committed Ghastly Outrage!"
Mr. William Windsor, a hard-headed, grim-visaged newspaperman of
forty, stood nonchalantly on the moving walk that swept him briskly
down Fifth Avenue. He smiled with pardonable pride as he listened
to the raucous magnetic speakers shouting out the phrases that
drew excited mobs to the robot vending machines which sold the
yet-damp news strips of printed shorthand. Bill had written the
account of the outrage; he had risked his life in a mad flight upon a
hurtling sunship to get his concise story to New York in time to beat
his competitors. Discovering the inmost details of whatever was
puzzling or important or exciting in this day of 2131, regardless of
risk to life or limb, and elucidating those details to the ten million
avid readers of the great daily newspaper, The Herald-Sun, was the
prime passion of Bill's life.
Incidentally, the reader might be warned at this point that Bill is not,
properly speaking, a character in this narrative; he is only an
observer. The real hero is that amazing person who has chosen to
call himself "The Prince of Space." This history is drawn from Bill's
diary, which he kept conscientiously, expecting to write a book of the
great adventure.
Bill stepped off the moving sidewalk by the corner vending machine,
dropped a coin in the slot, and received a copy of the damp
shorthand strip delivered fresh from the presses by magnetic tube.
He read his story, standing in a busy street that rustled quietly with
the whir of moving walks and the barely audible drone of the
thousands of electrically driven heliocars which spun smoothly along

on rubber-tired wheels, or easily lifted themselves to skimming flight
upon whirling helicopters.
Heliographic advices from the Moon Patrol flier
Avenger state that the sunship Helicon was found
today, at 16:19, Universal Time, drifting two thousand
miles off the lunar lane. The locks were open, air had
escaped, all on board were frozen and dead. Casualties
include Captain Stormburg, the crew of 71 officers and
men, and 132 passengers, of whom 41 were women.
The Helicon was bound to Los Angeles from the
lunarium health resorts at Tycho on the Moon. It is
stated that the bodies were barbarously torn and
mutilated, as if the most frightful excesses had been
perpetrated upon them. The cargo of the sunship had
been looted. The most serious loss is some thousands
of tubes of the new radioactive metal, vitalium, said to
have been worth nearly a million eagles.
A crew was put aboard the Helicon from the Avenger,
her valves were closed, and she will be brought under
her own motor tubes to the interplanetary base at
Miami, Florida, where a more complete official
examination will be made. No attempt has been made
to identify the bodies of the dead. The passenger list is
printed below.
Military officials are inclined to place blame for the
outrage upon the notorious interplanetary outlaw, who
calls himself "The Prince of Space." On several
occasions the "Prince" has robbed sunships of cargoes
of vitalium, though he has never before committed so
atrocious a deed as the murder of scores of innocent
passengers. It is stated that the engraved calling card,
which the "Prince" is said always to present to the
captain of a captured sunship, was not found on the
wreck.

Further details will be given the public as soon as it is
possible to obtain them.
The rewards offered for the "Prince of Space," taken
dead or alive, have been materially increased since the
outrage. The total offered by the International
Confederation, Interplanetary Transport, Lunar Mining
Corporation, Sunship Corporation, Vitalium Power
Company, and various other societies, corporations,
newspapers, and individuals, is now ten million eagles.
"Ten million eagles!" Bill exclaimed. "That would mean a private
heliocar, and a long, long vacation in the South Seas!"
He snorted, folded up the little sheet and thrust it into his green silk
tunic, as he sprang nimbly upon the moving sidewalk.
"What chance have I to see the Prince of Space?"
About him, the slender spires of widely spaced buildings rose two
hundred stories into a blue sky free from dust or smoke. The white
sun glinted upon thousands of darting heliocars, driven by silent
electricity. He threw back his head, gazed longingly up at an
amazing structure that rose beside him—at a building that was the
architectural wonder of the twenty-second century.
Begun in 2125, Trainor's Tower had been finished hardly a year. A
slender white finger of aluminum and steel alloy, it rose twelve
thousand feet above the canyons of the metropolis. Architects had
laughed, six years ago, when Dr. Trainor, who had been an obscure
western college professor, had returned from a vacation trip to the
moon and announced his plans for a tower high enough to carry an
astronomical observatory giving mountain conditions. A building five
times as high as any in existence! It was folly, they said. And certain
skeptics inquired how an impecunious professor would get funds to
put it up. The world had been mildly astonished when the work

began. It was astounded when it was known that the slender tower
had safely reached its full height of nearly two and a half miles. A
beautiful thing it was, in its slim strength—girder-work of glistening
white metal near the ground, and but a slender white cylinder for
the upper thousands of feet of its amazing height.
The world developed a hungry curiosity about the persons who had
the privilege of ascending in a swift elevator to the queer, many-
storied cylindrical building atop the astounding tower. Bill had spent
many hours in the little waiting room before the locked door of the
elevator shaft—bribes to the guard had been a heavy drain upon a
generous expense account. But not even bribery had won him into
the sacred elevator.
He had given his paper something, however, of the persons who
passed sometimes through the waiting room. There was Dr. Trainor,
of course, a mild, bald man, with kindly blue eyes and a slow,
patient smile. And Paula, his vivaciously beautiful daughter, a slim,
small girl, with amazingly expressive eyes. She had been with her
father on the voyage to the moon. Scores of others had passed
through; they ranged from janitors and caretakers to some of the
world's most distinguished astronomers and solar engineers—but
they were uniformly reticent about what went on in Trainor's Tower.
And there was Mr. Cain—"The mysterious Mr. Cain," as Bill had
termed him. He had seen him twice, a slender man, tall and wiry,
lean of face, with dark, quizzical eyes. The reporter had been able to
learn nothing about him—and what Bill could not unearth was a very
deep secret. It seemed that sometimes Cain was about Trainor's
Tower and that more often he was not. It was rumored that he had
advanced funds for building it and for carrying on the astronomical
research for which it was evidently intended.
Impelled by habit, Bill sprang off the moving walk as he glided past
Trainor's Tower. He was standing, watching the impassive guard,
when a man came past into the street. The man was Mr. Cain, with a
slight smile upon the thin, dark face that was handsome in a stern,
masculine sort of way. Bill started, pricked up his ears, so to speak,

and resolved not to let this mysterious young man out of sight until
he knew something about him.
To Bill's vast astonishment, Mr. Cain advanced toward him, with a
quick, decisive step, and a speculative gleam lurking humorously in
his dark eyes. He spoke without preamble.
"I believe you are Mr. William Windsor, a leading representative of
the Herald-Sun."
"True. And you are Mr. Cain—the mysterious Mr. Cain!"
The tall young man smiled pleasantly.
"Yes. In fact, I think the 'mysterious' is due to you. But Mr. Windsor
——"
"Just call me Bill."
"——I believe that you are desirous of admission to the Tower."
"I've done my best to get in."
"I am going to offer you the facts you want about it, provided you
will publish them only with my permission."
"Thanks!" Bill agreed. "You can trust me."
"I have a reason. Trainor's Tower was built for a purpose. That
purpose is going to require some publicity very shortly. You are
better able to supply that publicity than any other man in the world."
"I can do it—provided——"
"I am sure that our cause is one that will enlist your enthusiastic
support. You will be asked to do nothing dishonorable."
Mr. Cain took a thin white card from his pocket, scrawled rapidly
upon it, and handed it to Bill, who read the words, "Admit bearer.
Cain."
"Present that at the elevator, at eight tonight. Ask to be taken to Dr.
Trainor."

Mr. Cain walked rapidly away, with his lithe, springy step, leaving Bill
standing, looking at the card, rather astounded.
At eight that night, a surprised guard let Bill into the waiting room.
The elevator attendant looked at the card.
"Yes. Dr. Trainor is up in the observatory."
The car shot up, carrying Bill on the longest vertical trip on earth. It
was minutes before the lights on the many floors of the cylindrical
building atop the tower were flashing past them. The elevator
stopped. The door swung open, and Bill stepped out beneath the
crystal dome of an astronomical observatory.
He was on the very top of Trainor's Tower.
The hot stars shone, hard and clear, through a metal-ribbed dome of
polished vitrolite. Through the lower panels of the transparent wall,
Bill could see the city spread below him—a mosaic of fine points of
light, scattered with the colored winking eyes of electric signs; it was
so far below that it seemed a city in miniature.
Slanting through the crystal dome was the huge black barrel of a
telescope, with ponderous equatorial mounting. Electric motors
whirred silently in its mechanism, and little lights winked about it. A
man was seated at the eyepiece—he was Dr. Trainor, Bill saw—he
was dwarfed by the huge size of the instrument.
There was no other person in the room, no other instrument of
importance. The massive bulk of the telescope dominated it.
Trainor rose and came to meet Bill. A friendly smile spread over his
placid face. Blue eyes twinkled with mild kindliness. The subdued
light in the room glistened on the bald dome of his head.
"Mr. Windsor, of the Herald-Sun, I suppose?" Bill nodded, and
produced a notebook. "I am very glad you came. I have something
interesting to show you. Something on the planet Mars."
"What——"

"No. No questions, please. They can wait until you see Mr. Cain
again."
Reluctantly, Bill closed his notebook. Trainor seated himself at the
telescope, and Bill waited while he peered into the tube, and pressed
buttons and moved bright levers. Motors whirred, and the great
barrel swung about.
"Now look," Trainor commanded.
Bill took the seat, and peered into the eyepiece. He saw a little circle
of a curious luminous blue-blackness, with a smaller disk of light
hanging in it, slightly swaying. The disk was an ocherous red, with
darker splotches and brilliantly white polar markings.
"That is Mars—as the ordinary astronomer sees it," Trainor said.
"Now I will change eyepieces, and you will see it as no man has ever
seen it except through this telescope."
Rapidly he adjusted the great instrument, and Bill looked again.
The red disk had expanded enormously, with great increase of
detail. It had become a huge red globe, with low mountains and
irregularities of surface plainly visible. The prismatic polar caps stood
out with glaring whiteness. Dark, green-gray patches, splotched
burned orange deserts, and thin, green-black lines—the controversial
"canals" of Mars—ran straight across the planet, from white caps
toward the darker equatorial zone, intersecting at little round
greenish dots.
"Look carefully," Trainor said. "What do you see in the edge of the
upper right quadrant, near the center of the disk and just above the
equator?"
Bill peered, saw a tiny round dot of blue—it was very small, but
sharply edged, perfectly round, bright against the dull red of the
planet.
"I see a little blue spot."
"I'm afraid you see the death-sentence of humanity!"

Ordinarily Bill might have snorted—newspapermen are apt to
become exceedingly skeptical. But there was something in the
gravity of Trainor's words, and in the strangeness of what he had
seen through the giant telescope in the tower observatory, that
made him pause.
"There's been a lot of fiction," Bill finally remarked, "in the last
couple of hundred years. Wells' old book, 'The War of the Worlds,'
for example. General theory seems to be that the Martians are
drying up and want to steal water. But I never really——"
"I don't know what the motive may be," Trainor said. "But we know
that Mars has intelligent life—the canals are proof of that. And we
have excellent reason to believe that that life knows of us, and
intends us no good. You remember the Envers Expedition?"
"Yes. In 2099. Envers was a fool who thought that if a sunship could
go to the moon, it might go to Mars just as well. He must have been
struck by meteorites."
"There is no reason why Envers might not have reached Mars in
2100," said Trainor. "The heliographic dispatches continued until he
was well over half way. There was no trouble then. We have very
good reason to think that he landed, that his return was prevented
by intelligent beings on Mars. We know that they are using what
they learned from his captured sunship to launch an interplanetary
expedition of their own!"
"And that blue spot has something to do with it?"
"We think so. But I have authority to tell you nothing more. As the
situation advances, we will have need for newspaper publicity. We
want you to take charge of that. Mr. Cain, of course, is in supreme
charge. You will remember your word to await his permission to
publish anything."
Trainor turned again to the telescope.

With a little clatter, the elevator stopped again at the entrance door
of the observatory. A slender girl ran from it across to the man at the
telescope.
"My daughter Paula, Mr. Windsor," said Trainor.
Paula Trainor was an exquisite being. Her large eyes glowed with a
peculiar shade of changing brown. Black hair was shingled close to
her shapely head. Her face was small, elfinly beautiful, the skin
almost transparent. But it was the eyes that were remarkable. In
their lustrous depths sparkled mingled essence of childish innocence,
intuitive, age-old wisdom, and quick intelligence—intellect that was
not coldly reasonable, but effervescent, flashing to instinctively
correct conclusions. It was an oddly baffling face, revealing only the
mood of the moment. One could not look at it and say that its owner
was good or bad, indulgent or stern, gentle or hard. It could be, if
she willed, the perfect mirror of the moment's thought—but the
deep stream of her character flowed unrevealed behind it.
Bill looked at her keenly, noted all that, engraved the girl in the
notebook of his memory. But in her he saw only an interesting
feature story.
"Dad's been telling you about the threatened invasion from Mars,
eh?" she inquired in a low, husky voice, liquid and delicious. "The
most thrilling thing, isn't it? Aren't we lucky to know about it, and to
be in the fight against it!—instead of going on like all the rest of the
world, not dreaming there is danger?"
Bill agreed with her.
"Think of it! We may even go to Mars, to fight 'em on their own
ground!"
"Remember, Paula," Trainor cautioned. "Don't tell Mr. Windsor too
much."
"All right, Dad."
Again the little clatter of the elevator. Mr. Cain had come into the
observatory, a tall, slender young man, with a quizzical smile, and

eyes dark and almost as enigmatic as Paula's.
Bill, watching the vivacious girl, saw her smile at Cain. He saw her
quick flush, her unconscious tremor. He guessed that she had some
deep feeling for the man. But he seemed unaware of it. He merely
nodded to the girl, glanced at Dr. Trainor, and spoke briskly to Bill.
"Excuse me, Mr. Win—er, Bill, but I wish to see Dr. Trainor alone. We
will communicate with you when it seems necessary. In the
meanwhile, I trust you to forget what you have seen here tonight,
and what the Doctor has told you. Good evening."
Bill, of necessity, stepped upon the elevator. Five minutes later he
left Trainor's Tower. Glancing up from the vividly bright, bustling
street, with its moving ways and darting heliocars, he instinctively
expected to see the starry heavens that had been in view from the
observatory.
But a heavy cloud, like a canopy of yellow silk in the light that shone
upon it from the city, hung a mile above. The upper thousands of
feet of the slender tower were out of sight above the clouds.
After breakfast next morning Bill bought a shorthand news strip from
a robot purveyor. In amazement and some consternation he read:
Prince of Space Raids Trainor's Tower
Last night, hidden by the clouds that hung above the
city, the daring interplanetary outlaw, the self-styled
Prince of Space, suspected of the Helicon outrage,
raided Trainor's Tower. Dr. Trainor, his daughter Paula,
and a certain Mr. Cain are thought to have been
abducted, since they are reported to be missing this
morning.
It is thought that the raiding ship drew herself against
the Tower, and used her repulsion rays to cut through
the walls. Openings sufficiently large to admit the body

of a man were found this morning in the metal outer
wall, it is said.
There can be no doubt that the raider was the "Prince
of Space" since a card engraved with that title was left
upon a table. This is the first time the pirate has been
known to make a raid on the surface of the earth—or
so near it as the top of Trainor's Tower.
Considerable alarm is being felt as a result of this and
the Helicon outrage of yesterday. Stimulated by the
reward of ten million eagles, energetic efforts will be
made on the part of the Moon Patrol to run down this
notorious character.
CHAPTER II
Bloodhounds of Space
Two days later Bill jumped from a landing heliocar, presented his
credentials as special correspondent, and was admitted to the
Lakehurst base of the Moon Patrol. Nine slender sunships lay at the
side of the wide, high-fenced field, just in front of their sheds. In the
brilliant morning sunlight they scintillated like nine huge octagonal
ingots of polished silver.
These war-fliers of the Moon Patrol were eight-sided, about twenty
feet in diameter and a hundred long. Built of steel and the new
aluminum bronzes, with broad vision panels of heavy vitrolite, each
carried sixteen huge positive ray tubes. These mammoth vacuum
tubes, operated at enormous voltages from vitalium batteries, were
little different in principle from the "canal ray" apparatus of some
centuries before. Their "positive rays," or streams of atoms which

had lost one or more electrons, served to drive the sunship by
reaction—by the well-known principle of the rocket motor.
And the sixteen tubes mounted in twin rings about each vessel
served equally well as weapons. When focused on a point, the
impact-pressure of their rays equaled that of the projectile from an
ancient cannon. Metal in the positive ray is heated to fusion, living
matter carbonized and burned away. And the positive charge carried
by the ray is sufficient to electrocute any living being in contact with
it.
This Moon Patrol fleet of nine sunships was setting out in pursuit of
the Prince of Space, the interplanetary buccaneer who had abducted
Paula Trainor and her father, and the enigmatic Mr. Cain. Bill was
going aboard as special correspondent for the Herald-Sun.
On the night before the Helicon, the sunship which had been
attacked in space, had been docked at Miami by the rescue crew put
aboard from the Avenger. The world had been thrown into a frenzy
by the report of the men who had examined the two hundred dead
on board.
"Blood sucked from Helicon victims!" the loud speakers were
croaking. "Mystery of lost sunship upsets world! Medical examination
of the two hundred corpses found on the wrecked space flier show
that the blood had been drawn from the bodies, apparently through
curious circular wounds about the throat and trunk. Every victim
bore scores of these inexplicable scars. Medical men will not attempt
to explain how the wounds might have been made.
"In a more superstitious age, it might be feared that the Prince of
Space is not man at all, but a weird vampire out of the void. And, in
fact, it has been seriously suggested that, since the wounds
observed could have been made by no animal known on earth, the
fiend may be a different form of life, from another planet."
Bill found Captain Brand, leader of the expedition, just going on
board the slender, silver Fury, flagship of the fleet of nine war-fliers.

He had sailed before with this bluff, hard-fighting guardsman of the
space lanes; he was given a hearty welcome.
"Hunting down the Prince is a good-sized undertaking, from all
appearances," Bill observed.
"Rather," big, red-faced Captain Brand agreed. "We have been after
him seven or eight times in the past few years—but I think his ship
has never been seen. He must have captured a dozen commercial
sunships."
"You know, I rather admire the Prince—" Bill said, "or did until that
Helicon affair. But the way those passengers were treated is simply
unspeakable. Blood sucked out!"
"It is hard to believe that the Prince is responsible for that. He has
never needlessly murdered anyone before—for all the supplies and
money and millions worth of vitalium he has taken. And he has
always left his engraved card—except on the Helicon.
"But anyhow, we blow him to eternity on sight!"
The air-lock was open before them, and they walked through, and
made their way along the ladder (now horizontal, since the ship lay
on her side) to the bridge in the bow. Bill looked alertly around the
odd little room, with its vitrolite dome and glistening instruments,
while Captain Brand flashed signals to the rest of the fleet for sealing
the locks and tuning the motor ray generators.
A red rocket flared from the Fury. White lances of flame darted from
the down-turned vacuum tubes. As one, the nine ships lifted
themselves from the level field. Deliberately they upturned from
horizontal to vertical positions. Upward they flashed through the air,
with slender white rays of light shooting back from the eight rear
tubes of each.
Bill, standing beneath the crystal dome, felt the turning of the ship.
He felt the pressure of his feet against the floor, caused by
acceleration, and sat down in a convenient padded chair. He
watched the earth become a great bowl, with sapphire sea on the

one hand and green-brown land and diminishing, smokeless city on
the other. He watched the hazy blue sky become deepest azure,
then black, with a million still stars bursting out in pure colors of
yellow and red and blue. He looked down again, and saw the earth
become convex, an enormous bright globe, mistily visible through
haze or air and cloud.
Swiftly the globe drew away. And a tiny ball of silver, half black, half
rimmed with blinding flame, sharply marked with innumerable round
craters, swam into view beyond the misty edge of the globe—it was
the moon.
Beyond them flamed the sun—a ball of blinding light, winged with a
crimson sheet of fire—hurling quivering lances of white heat through
the vitrolite panels. Blinding it was to look upon it, unless one wore
heavily tinted goggles.
Before them hung the abysmal blackness of space, with the canopy
of cold hard stars blazing as tiny scintillant points of light, at an
infinite distance away. The Galaxy was a broad belt of silvery
radiance about them, set with ten thousand many-colored jewels of
fire. Somewhere in the vastness of that void they sought a daring
man, who laughed at society, and called himself the Prince of Space.
The nine ships spread out, a thousand miles apart. Flickering
heliographs—swinging mirrors that reflected the light of the sun—
kept them in communication with bluff Captain Brand, while many
men at telescopes scanned the black, star-studded sweep of space
for the pirate of the void.
Days went by, measured only by chronometer, for the winged, white
sun burned ceaselessly. The earth had shrunk to a little ball of
luminous green, bright on the sunward side, splotched with the
dazzling white of cloud patches and polar caps.
Sometimes the black vitalium wings were spread, to catch the
energy of the sun. The sunship draws its name from the fact that it
is driven by solar power. It utilizes the remarkable properties of the
rare radioactive metal, vitalium, which is believed to be the very

basis of life, since it was first discovered to exist in minute traces in
those complex substances so necessary to all life, the vitamins.
Large deposits were discovered at Kepler and elsewhere on the
moon during the twenty-first century. Under the sun's rays vitalium
undergoes a change to triatomic form, storing up the vast energy of
sunlight. The vitalium plates from the sunshine are built into
batteries with alternate sheets of copper, from which the solar
energy may be drawn in the form of electric current. As the battery
discharges, the vitalium reverts to its stabler allotropic form, and
may be used again and again. The Vitalium Power Company's plants
in Arizona, Chili, Australia, the Sahara, and the Gobi now furnish
most of the earth's power. The sunship, recharging its vitalium
batteries in space, can cruise indefinitely.
It was on the fifth day out from Lakehurst. The Fury, with her sister
ships spread out some thousands of miles to right and left, was
cruising at five thousand miles per hour, at heliocentric elevation
93.243546, ecliptic declination 7°, 18' 46" north, right ascension XIX
hours, 20 min., 31 sec. The earth was a little green globe beside her,
and the moon a thin silver crescent beyond.
"Object ahead!" called a lookout in the domed pilot-house of the
Fury, turning from his telescope to where Captain Brand and Bill
stood smoking, comfortably held to the floor by the ship's
acceleration. "In Scorpio, about five degrees above Antares. Distance
fifteen thousand miles. It seems to be round and blue."
"The Prince, at last!" Brand chuckled, an eager grin on his square
chinned face, light of battle flashing in his blue eyes.
He gave orders that set the heliographic mirrors flickering signals for
all nine of the Moon Patrol fliers to converge about the strange
object, in a great crescent. The black fins that carried the charging
vitalium plates were drawn in, and the full power of the motor ray

tubes thrown on, to drive ahead each slender silver flier at the limit
of her acceleration.
Four telescopes from the Fury were turned upon the strange object.
Captain Brand and Bill took turns peering through one of them.
When Bill looked, he saw the infinite black gulf of space, silvered
with star-dust of distant nebulae. Hanging in the blackness was an
azure sphere, gleaming bright as a great globe cut from turquoise.
Bill was reminded of a similar blue globe he had seen—when he had
stood at the enormous telescope on Trainor's Tower, and watched a
little blue circle against the red deserts of Mars.
Brand took two or three observations, figured swiftly.
"It's moving," he said. "About fourteen thousand miles per hour.
Funny! It is moving directly toward the earth, almost from the
direction of the planet Mars. I wonder——" He seized the pencil,
figured again. "Queer. That thing seems headed for the earth, from
a point on the orbit of Mars, where that planet was about forty days
ago. Do you suppose the Martians are paying us a visit?"
"Then it's not the Prince of Space?"
"I don't know. Its direction might be just a coincidence. And the
Prince might be a Martian, for all I know. Anyhow, we're going to
find what that blue globe is!"
Two hours later the nine sunships were drawn up in the form of a
great half circle, closing swiftly on the blue globe, which had been
calculated to be about one hundred feet in diameter. The sunships
were nearly a thousand miles from the globe, and scattered along a
curved line two thousand miles in length. Captain Brand gave orders
for eight forward tubes on each flier to be made ready for use as
weapons. From his own ship he flashed a heliographic signal.
"The Fury, of the Moon Patrol, demands that you show ship's
papers, identification tags for all passengers, and submit to search
for contraband."

The message was three times repeated, but no reply came from the
azure globe. It continued on its course. The slender white sunships
came plunging swiftly toward it, until the crescent they formed was
not two hundred miles between the points, the blue globe not a
hundred miles from the war-fliers.
Then Bill, with his eye at a telescope, saw a little spark of purple
light appear beside the blue globe. A tiny, bright point of violet-red
fire, with a white line running from it, back to the center of the
sphere. The purple spark grew, the white line lengthened. Abruptly,
the newspaperman realized that the purple was an object hurtling
toward him with incredible speed.
Even as the realization burst upon him, the spark became visible as
a little red-blue sphere, brightly luminous. A white beam shone
behind it, seemed to push it with ever-increasing velocity. The purple
globe shot past, vanished. The white ray snapped out.
"A weapon!" he exclaimed.
"A weapon and a warning!" said Brand, still peering through another
eyepiece. "And we reply!"
"Heliograph!" he shouted into a speaking tube. "Each ship will open
with one forward tube, operating one second twelve times per
minute. Increase power of rear tubes to compensate repulsion."
White shields flickered. Blindingly brilliant rays, straight bars of
dazzling opalescence, burst intermittently from each of the nine
ships, striking across a hundred miles of space to batter the blue
globe with a hail of charged atoms.
Again a purple spark appeared from the sapphire globe, with a beam
of white fire behind it. A tiny purple globe, hurtling at an
inconceivable velocity before a lance of white flame. It reached out,

with a certain deliberation, yet too quickly for a man to do more
than see it.
It struck a sunship, at one tip of the crescent formation.
A dazzling flash of violet flame burst out. The tiny globe seemed to
explode into a huge flare of red-blue light. And where the slim,
eight-sided ship had been was a crushed and twisted mass of metal.
"A solid projectile!" Brand cried. "And driven on the positive ray! Our
experts have tried it, but the ray always exploded the shell. And that
was some explosion! I don't know what—unless atomic energy!"
The eight sunships that remained were closing swiftly upon the blue
globe. The dazzling white rays flashed intermittently from them.
They struck the blue globe squarely—the fighting crews of the Moon
Patrol are trained until their rays are directed with deadly accuracy.
The azure sphere, unharmed, shone with bright radiance—it seemed
that a thin mist of glittering blue particles was gathering about it,
like a dust of powdered sapphires.
Another purple spark leapt from the turquoise globe.
In the time that it took a man's eyes to move from globe to slim,
glistening sunship, the white ray had driven the purple spark across
the distance. Another vivid flash of violet light. And another sunship
became a hurtling mass of twisted wreckage.
"We are seven!" Brand quoted grimly.
"Heliograph!" he shouted into the mouthpiece. "Fire all forward
tubes one second twenty times a minute. Increase rear power to
maximum."
White rays burst from the seven darting sunships, flashing off and
on. That sapphire globe grew bright, with a strange luminosity. The
thin mist of sparkling blue particles seemed to grow more dense
about it.
"Our rays don't seem to be doing any good," Brand muttered,
puzzled. "The blue about that globe must be some sort of vibratory

screen."
Another purple spark, with the narrow white line of fire behind it,
swept across to the flier from the opposite horn of the crescent,
burst into a sheet of blinding red-violet light. Another ship was a
twisted mass of metal.
"Seven no longer!" Brand called grimly to Bill.
"Looks as if the Prince has got us beaten!" the reporter cried.
"Not while a ship can fight!" exclaimed the Captain. "This is the
Moon Patrol!"
Another tiny purple globe traced its line of light across the black,
star-misted sky. Another sunship crumpled in a violet flash.
"They're picking 'em off the ends," Bill observed. "We're in the
middle, so I guess we're last."
"Then," said Captain Brand, "we've got time to ram 'em."
"Control!" he shouted into the speaking tube. "Cut off forward tubes
and make all speed for the enemy. Heliograph! Fight to the end! I
am going to ram them!"
Another red-blue spark moved with its quick deliberation. A purple
flash left another ship in twisted ruin.
Bill took his eye from the telescope. The blue globe, bright under the
rays, with the sapphire mist sparkling about it, was only twenty
miles away. He could see it with his naked eye, drifting swiftly
among the familiar stars of Scorpio.
It grew larger very swiftly.
With the quickness of thought, the purple sparks moved out
alternately to right and to left. They never missed. Each one
exploded in purple flame, crushed a sunship.
"Three fliers left," Bill counted, eyes on the growing blue globe
before them. "Two left. Good-by, Brand." He grasped the bluff
Captain's hand. "One left. Will we have time?"

He looked forward. The blue globe, with the dancing, sparkling haze
of sapphire swirling about it, was swiftly expanding.
"The last one! Our turn now!"
He saw a tiny fleck of purple light dart out of the expanding azure
sphere that they had hoped to ram. Then red-violet flame seemed to
envelope him. He felt the floor of the bridge tremble beneath his
feet. He heard the beginning of a shivering crash like that of
shattering glass. Then the world was mercifully dark and still.
CHAPTER III
The City of Space
Bill lay on an Alpine glacier, a painful broken leg inextricably wedged
in a crevasse. It was dark, frightfully cold. In vain he struggled to
move, to seek light and warmth, while the grim grip of the ice held
him, while bitter wind howled about him and the piercing cold of the
blizzard crept numbingly up his limbs.
He came to with a start, realized that it was a dream. But he was
none the less freezing, gasping for thin, frigid air, that somehow
would not come into his lungs. All about was darkness. He lay on
cold metal.
"In the wreck of the Fury!" he thought. "The air is leaking out. And
the cold of space! A frozen tomb!"
He must have made a sound, for a groan came from beside him. He
fought to draw breath, tried to speak. He choked, and his voice was
oddly high and thin.
"Who are——"
He ended in a fit of coughing, felt warm blood spraying from his
mouth. Faintly he heard a whisper beside him.

"I'm Brand. The Moon Patrol—fought to the last!"
Bill could speak no more, and evidently the redoubtable captain
could not. For a long time they lay in freezing silence. Bill had no
hope of life, he felt only very grim satisfaction in the fact that he and
Brand had not been killed outright.
But suddenly he was thrilled with hope. He heard a crash of hammer
blows upon metal, sharp as the sound of snapping glass in the thin
air. Then he heard the thin hiss of an oxygen lance.
Someone was cutting a way to them through the wreckage. Only a
moment later, it seemed, a vivid bar of light cleft the darkness,
searched the wrecked bridge, settled upon the two limp figures. Bill
saw grotesque figures in cumbrous metal space suits clambering
through a hole they had cut. He felt an oxygen helmet being
fastened about his head, heard the thin hiss of the escaping gas,
and was once more able to breathe.
Again he slipped into oblivion.
He awoke with the sensation that infinite time had passed. He sat up
quickly, feeling strong, alert, fully recovered in every faculty, a clear
memory of every detail of the disastrous encounter with the strange
blue globe-ship springing instantly to his mind.
He was in a clean bed in a little white-walled room. Captain Brand, a
surprised grin on his bluff, rough-hewn features, was sitting upon
another bed beside him. Two attendants in white uniform stood just
inside the door; and a nervous little man in black suit, evidently a
doctor, was hastily replacing gleaming instruments in a leather bag.
A tall man appeared suddenly in the door, clad in a striking uniform
of black, scarlet, and gold—black trousers, scarlet military coat and
cap, gold buttons and decorations. He carried in his hand a glittering
positive ray pistol.
"Gentlemen," he said in a crisp, gruff voice, "you may consider
yourselves prisoners of the Prince of Space."
"How come?" Brand demanded.

"The Prince was kind enough to have you removed from the wreck
of your ship, and brought aboard the Red Rover, his own sunship.
You have been kept unconscious until your recovery was complete."
"And what do you want with us now?" Brand was rather aggressive.
The man with the pistol smiled. "That, gentlemen, I am happy to
say, rests largely with yourselves."
"I am an officer in the Moon Patrol," said Brand. "I prefer death to
anything——"
"Wait, Captain. You need have none but the kindest feelings for my
master, the Prince of Space. I now ask you nothing but your word as
an officer and a gentleman that you will act as becomes a guest of
the Prince. Your promise will lose you nothing and win you much."
"Very good, I promise," Brand agreed after a moment. "——for
twenty-four hours."
He pulled out his watch, looked at it. The man in the door lowered
his pistol, smiling, and walked across to shake hands with Brand.
"Call me Smith," he introduced himself. "Captain of the Prince's
cruiser, Red Rover."
Still smiling, he beckoned toward the door.
"And if you like, gentlemen, you may come with me to the bridge.
The Red Rover is to land in an hour."
Brand sprang nimbly to the floor, and Bill followed. The flier was
maintaining a moderate acceleration—they felt light, but were able
to walk without difficulty. Beyond the door was a round shaft, with a
ladder through its length. Captain Smith clambered up the ladder.
Brand and Bill swung up behind him.
After an easy climb of fifty feet or so, they entered a domed pilot-
house, with vitrolite observation panels, telescopes, maps and
charts, and speaking tube—an arrangement similiar to that of the
Fury.

Black, star-strewn heavens lay before them. Bill looked for the earth,
found it visible in the periscopic screens, almost behind them. It was
a little green disk; the moon but a white dot beside it.
"We land in an hour!" he exclaimed.
"I didn't say where," said Captain Smith, smiling. "Our landing place
is a million miles from the earth."
"Not on earth! Then where——"
"At the City of Space."
"The City of Space!"
"The capital of the Prince of Space. It is not a thousand miles before
us."
Bill peered ahead, through the vitrolite dome, distinguished the
bright constellation of Sagittarius with the luminous clouds of the
Galaxy behind it.
"I don't see anything——"
"The Prince does not care to advertise his city. The outside of the
City of Space is covered with black vitalium—which furnishes us with
power. Reflecting none of the sun's rays, it cannot be seen by
reflected light. Against the black background of space it is invisible,
except when it occults a star."
Captain Smith busied himself with giving orders for the landing. Bill
and Brand stood for many minutes looking forward through the
vitrolite dome, while the motor ray tubes retarded the flier. Presently
a little black point came against the silver haze of the Milky Way. It
grew, stars vanishing behind its rim, until a huge section of the
heavens was utterly black before them.
"The City of Space is in a cylinder," Captain Smith said. "Roughly five
thousand feet in diameter, and about that high. It is built largely of

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