Fabry–pérot interferometer picoseconds dispersive properties

International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME
274
FABRY–PÉROT INTERFEROMETER PICOSECONDS DISPERSIVE
PROPERTIES
Elham Jasim Mohammad
Physics Department,Collage of Sciences/Al-Mustansiriyah University, Iraq,
ABSTRACT
Fabry-Pérot interferometers are used in optical modems, spectroscopy, lasers, and
astronomy. In this paper we used the coupled mode equation to design the Fabry–Pérot
interferometers and study the picosecond dispersion. Coupled mode analysis is widely used
in the field of integrated optoelectronics for the description of two coupled waves traveling in
the same direction. The program is written in MATLAB to simulate and analysis the Fabry–
Pérot properties.
Keywords: Coupled Mode Theory, Fabry–Pérot Interferometer, Finesse.
I. INTRODUCTION
The Fabry-Perot interferometer (FPI) is a simple device that relies on the interference
of multiple beams. The interferometer consists of two parallel semi-transparent reflective
surfaces that are well aligned to form an optical Fabry-Perot cavity with cavity length L and
refractive index n. When a monochromatic input light enters the Fabry-Perot cavity, two
reflections at the two surfaces with amplitudes of A1 and A2 are generated respectively as in
Figure 1 below:
Figure 1: Basic structure of a Fabry-Perot Interferometer [1].
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ISSN 0976 – 6545(Print)
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Volume 4, Issue 2, March – April (2013), pp. 274-282
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275
Thus, the two reflections interfere with each other to produce an interference pattern
consisting peaks and valleys as some light constructively interferes and some destructively.
The total reflected light intensity can be written as follows for low finesse [1,2]:
)
4
cos(2)cos(2 21
2
2
2
121
2
2
2
1
λ
π
φ
nL
AAAAAAAAI ++=∆++= (1)
φ∆ is the relative phase shift between the two light signals. λ denotes the wavelength of the
interrogation light source. The basic principle of Fabry-Perot interferometric is quite clear.
Changes in the FP cavity length produce a cosine modulation of the output intensity signal.
The change of the physical parameter under measurement is converted into a change in the
cavity length L and subsequently modifies I. Therefore, those physical parameters changes
could be obtained by examining I [1].
The solid Fabry-Perot interferometer, also known as a single-cavity coating, is formed
by separating two thin-film reflectors with a thin-film spacer. In an all-dielectric cavity, the
thin-film reflectors are quarter-wave stack reflectors made of dielectric materials. The spacer,
which is a single layer of dielectric material having an optical thickness corresponding to an
integral-half of the principal wavelength, induces transmission rather than reflection at the
principal wavelength. Light with wavelengths longer or shorter than the principal wavelength
undergoes a phase condition that maximizes reflectivity and minimizes transmission. In a
metal-dielectric-metal (MDM) cavity, the reflectors of the solid Fabry-Perot interferometer
are thin-films of metal and the spacer is a layer of dielectric material with an integral half-
wave thickness. These are commonly used to filter UV light that would be absorbed by all-
dielectric coatings [3].
II. PARAMETERS WHICH DEFINE FPI AND DISPERSION COMPUTING
USING COUPLED MODE THEORY
The Fabry-Perot is a simple interferometer, which relies on the interference of
multiple reflected beams [4]. The accompanying Figure 2 shows a schematic Fabry-Perot
cavity. Incident light undergoes multiple reflections between coated surfaces which define the
cavity.
Figure 2: Schematic of a Fabry-Perot interferometer [4].

276
Each transmitted wavefront has undergone an even number of reflections (0, 2, 4, . .).
Whenever there is no phase difference between emerging wavefronts, interference between
these wavefronts produces a transmission maximum. This occurs when the optical path
difference is an integral number of whole wavelengths, i.e., when [4]:
θλ cos2 optm = (2)
where m is an integer, often termed the order, opt is the optical thickness, and θ is the angle
of incidence. The phase difference between each succeeding reflection is given by δ [5]:
θ
λ
π
δ cos2
2
ln





= (3)
If both surfaces have a reflectance R, the transmittance function of the etalon is given by [5]:
( )
)2/(sin1
1
cos21
1
22
2
δδ FRR
R
Te
+
=
−+
−
= (4)
where:
2
)1(
4
R
R
F
−
= , is the coefficient of finesse. Maximum transmission 1=eT occurs when the
optical path length difference θcos2nl between each transmitted beam is an integer multiple
of the wavelength. In the absence of absorption, the reflectance of the etalon eR is the
complement of the transmittance, such that 1=+ ee RT . The maximum reflectivity is given by
[5]:
2max
)1(
4
1
1
1
R
R
F
R
+
=
+
−= (5)
and this occurs when the path-length difference is equal to half an odd multiple of the
wavelength. The wavelength separation between adjacent transmission peaks is called the
free spectral range (FSR) of the etalon, λ∆ , and is given by [5]:
θ
λ
λθ
λ
λ
cos2cos2
2
0
0
2
0
ll nn
≈
+
=∆ (6)
Where, 0λ is the central wavelength of the nearest transmission peak. The FSR is related to
the full-width half-maximum (FWHM), of any one transmission band by a quantity known as
the finesse [5]:
R
RF
f
−
=≈
12
2
1
ππ
. Etalons with high finesse show sharper transmission
peaks with lower minimum transmission. At other wavelengths, destructive interference of
transmitted wavefronts reduces transmitted intensity toward zero (i.e., most, or all, of the
light is reflected back toward the source).
Transmission peaks can be made very sharp by increasing the reflectivity of the mirror
surfaces. In a simple Fabry-Perot interferometer transmission curve (see Figure 3), the ratio
of successive peak separation to FWHM transmission peak is termed finesse [4].

277
Figure 3: Transmission pattern showing the free spectral range (FSR) of a simple Fabry-
Perot interferometer [4].
High reflectance results in high finesse (i.e., high resolution). In most Fabry-Perot
interferometers, air is the medium between high reflectors; therefore, the optical thickness,
opt , is essentially equal to d, the physical thickness. The air gap may vary from a fraction of a
millimeter to several centimeters. The Fabry-Perot is a useful spectroscopic tool. It provided
much of the early motivation to develop quality thin films for the high-reflectance mirrors
needed for high finesse [4].
Assuming no absorption, conservation of energy requires 1=+ RT . The total amplitude
of both beams will be the sum of the amplitudes of the two beams measured along a line
perpendicular to the direction of the beam.
Thus: 00cos/32
1
ll ikiki
RTet −+
= θπ
, where 0l is: 00 sintan2 θθll = and k = wavenumber. Neglecting the
π2 phase change due to the two reflections, the phase difference between the two beams is:
00
cos
2
l
l
k
k
−=
θ
δ . The relationship between θ and 0θ is given by Snell's law: 00 sinsin θθ nn = .
So that the amplitude can be rewritten as: δi
T
t
Re1 −
= .
The intensity of the beam will be just t times its complex conjugate. Since the
incident beam was assumed to have an intensity of one, this will also give the transmission
function [5,6]:
δcos21 2
2
*
RR
T
ttTe
−+
== (7)
In this study we used the Coupled mode theory to show the Fabry-Perot dispersive
properties. Coupled mode analysis is widely used for the design of optical filters and mirrors,
which are composed of discrete layers with large differences in the refractive indices (e.g.,
dielectric multilayer coatings), the coupled-mode approach is hardly considered. Its
applicability seems to be questionable because the assumption of a small perturbation is
violated in the case of large index discontinuities. Additionally, a lot of powerful analytical
design tools based on the coupled mode equations have been developed [7]. In coupled mode
equations,
λ
π
κ
2
n∆
= defines the coupling coefficient for the first order refractive-index variation
n∆ andλ is the design wavelength [8]. The group delay (GD) is defined as the negative of
the derivative of the phase response with respect to frequency [9]. In physics and in particular

278
in optics, the study of waves and digital signal processing, the term delay meaning: the rate of
change of the total phase shift with respect to angular frequency [10,11]:
ω
φ
d
d
GD −= . Through
a device or transmission medium, where φ is the total phase shift in radians, and ω is the
angular frequency in radians. The group delay dispersion (GDD) can be determined by
[10,11]:
ωd
dGD
GDD = .
Fabry-Perot interferometers can be constructed from purely metallic coatings, but high
absorption losses limit performance [4]. Furthermore, the Fabry-Perot filter ideally covers a
whole communication band, which is typically tens of nanometers large [12].
III. SIMULATION RESULT AND DISCUSSION
From all results below, it got after following these steps:
1. Calculate the transmittance function, finesse and contrast factor of FPI.
2. Implementation of the Transfer Matrix method for solution of Coupled Mode
equations.
3. Found the phase difference to calculate the amplitude and power transmission
coefficient of FPI.
4. Calculate the delay and dispersion of FPI in picoseconds units.
5. Found delay and dispersion analytical results.
The dispersive and analysis results for the mean, median, mode and the standard
deviation (STD) are tablets in table 1. There are direct relationship among the reflectance,
resolving power and the finesse and as they are shown in the plots that have been shown
below. Figure 4 is about the transmitted intensity versus the interference order. It shows the
transmittance function for different values of F . Instead of δ , the corresponding interference
order
π
δ
2
is noted. Figure 5 is about the finesse and the mirror reflectivity. The finesse is an
important parameter that determines the performance of a FPI. Conceptually, finesse can be
thought of as the number of beams interfering within the FP cavity to form the standing
wave. The primary factor that affects finesse is the reflectance R of the FP mirrors, which
directly affects the number of beams circulating inside the cavity. In Figure 6 we found
another important factor in the design of FPI is the contrast factor which is defined primarily
as the ratio of the maximum to minimum transmission. Figure 7 show finesse against contrast
factor. Figure 8 represents the relationship between the amplitude transmission and the
wavelength. Finally, Figure 9 and Figure 10 show the delay and dispersion versus the
wavelength after using the transfer function and coupled mode equation. The theoretically
designed delay has a small oscillations are visible. Of course, the same behavior can be found
for the dispersion. Figure 11 and Figure 12 show the delay and dispersion versus the
wavelength after using the transfer function, coupled mode equation and then POLYFIT
function. Table 2 show the reflectance and resolving power values for deferent interference
order.

279
-15 -10 -5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Interference Order
Transmittance
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0
50
100
150
200
250
300
350
Mirror Reflectivity
Finesse
Table 1 The dispersive and statistical analysis: mean, median, mode and the standard
deviation.
Index Mean Median Mode STD
1st
order Transmittance
Function
0.5468 0.4639 0.2868 0.2401
2nd
order Transmittance
Function
0.417 0.2981 0.1648 0.2689
3rd
order Transmittance
Function
0.2893 0.1548 0.0784 0.2722
Finesse 24.89 10.63 42.97 4.441
Contrast Factor 247.9 11.12 1.778 1118
Amplitude Transmission -0.0003402
-
0.002776
-0.9718 0.3977
Power Transmission 0.3157 0.1858 0.0006045 0.3178
Delay ps -0.01603 -0.01602 -0.02106 0.000854
Dispersion 2
ps -2.137E-5
-2.132E-
5
-0.0001241 1.738E-5
Fit Delay ps -0.01603 -0.01605 -0.0161 6.181E-5
Fit Dispersion 2
ps -2.137E-5 -2.14E-5 -2.146E-5 9.194E-8
Figure 4 Figure 5
Figure 4: Shows the transmitted intensity versus the interference order for various values of
transmittance of the coatings. Figure 5: Finesse versus the mirror reflectivity. Not that the
transmitted intensity peaks get narrower and the coefficient of finesse increases. When peaks
are very narrow in Figure. 3, light can be transmitted only if the plate separationl , refractive
index n , and the wavelength λ satisfy the precise relation: λθπδ /cos2 ln= .

280
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Mirror Reflectivity
ContrastFactor
0 50 100 150 200 250 300 350
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Finesse
ContrastFactor
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Wavelength (nm)
AmplitudeTransmissin(p.u)
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Wavelength (nm)
PowerTransmisson(p.u)
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
-0.022
-0.02
-0.018
-0.016
-0.014
-0.012
-0.01
Wavelength (nm)
Delay(ps)
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-4
Wavelength (nm)
Dispersion(ps)
Figure 6 Figure 7
Figure 6: Contrast factor and the mirror reflectivity. Figure 7: Finesse against contrast factor.
Very high finesse factors require highly contrast factor. These mean, when finesse increase,
contrast factor increase also.
Figure 8: The relationship between the Figure 9: Power transmissions versus
amplitude transmission and the wavelength the wavelength.
Figure 10: The relationship between the Figure 11: The relationship between the
delay and the wavelength dispersion and the wavelength

281
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
-0.0161
-0.016
-0.016
-0.0159
-0.0159
-0.0159
-0.0158
Wavelength (nm)
Delay(ps)
1498.5 1499 1499.5 1500 1500.5 1501 1501.5
-2.145
-2.14
-2.135
-2.13
-2.125
-2.12
-2.115
-2.11
-2.105
-2.1
x 10
-5
Wavelength (nm)
Dispersion(ps)
Figure 12: The relationship between Figure 13: The relationship between the
the fit delay and the wavelength. fit dispersion and the wavelength.
Table 2 The reflectance and resolving power for deferent interference order.
Reflectanc
e
Resolving
Power
Reflectan
ce
Resolving
Power
56 0.989924 42 0.999842
55 0.979927 41 0.989853
54 0.989929 40 0.998564
53 0.999930 39 0.997873
52 0.989932 38 0.998881
51 0.998933 37 0.996888
50 0.998934 36 0.997894
49 0.999736 35 0.999989
48 0.995937 34 0.999005
47 0.996938 33 0.999098
46 0.999938 32 0.993913
45 0.997939 31 0.997916
44 0.999434 30 0.998919
43 0.988994 29 0.999322
IV. CONCLUSION
The general theory behind interferometry still applies to the Fabry–Perot model,
however, these multiple reflection reinforce the areas where constructive and destructive
effects occur making the resulting fringes much more clearly defined . This paper has
presented a theoretical design of Fabry-Perot interferometer. This theoretical design study
including dispersion, FSR, finesse and contrast, used to assess the performance of the FPI
were discussed. An attempt is made to analyze the factors that control and affect the
performance and the design of the FPI versus the parameter that control those factors. Very
high finesse factors require highly reflective mirrors. A higher finesse value indicates a

282
greater number of interfering beams within the cavity, and hence a more complete
interference process. The figure show that the linear increase in finesse with respect to
contrast increase. The equation and the plots also show that a linear increase in finesse,
translates into a quadratic to each other and the average fit delay and dispersion has small
oscillations around the design wavelength. The Finesse is the most important parameter, its
value depends on the reflectivity of coating parallelism of the etalon mirror and the shape and
size of the field stop.
REFERENCES
[1] X. Zhao, Study of Multimode Extrinsic Fabry-Perot Interferometric Fiber Optic Sensor
on Biosensing, Ms. C. Thesis, Virginia Polytechnic Institute and State University,
Blacksburg, Virginia, 2006.
[2] R. Fowles, Introduction to Modern Optics, (Dover Publication: New York, 1989).
[3] Optical Filter Design Application Note.
[4] Interference Filters: www.mellesgriot.com.
[5] Macleod H. A., Thin-Film Optical Filters: 3rd
Edition, (Published by Institute of Physics
Publishing, wholly owned by The Institute of Physics, London, UK, 2001).
[6] S. Tamir, Fabry-Perot Filter Analysis and Simulation Using MATLAB, 2010.
[7] M. Wiemer, Double Chirped Mirrors for Optical Pulse Compression, 2007.
[8] B. Cakmak, T. Karacali and S. Yu, Theoretical Investigation of Chirped Mirrors in
Semiconductor Lasers, Appl. Phys., 2005, 33-37.
[9] Adobe PDF-View as html, Definition of Group Delay, 2008:
http://www.dsprelated.com/blogimages/AndorBariska/NGD/ngdblog.pdf.
[10] T. Imran, K. H. Hong, T. J. Yu and C. H. Nam, Measurement of the group-delay
dispersion of femtosecond optics using white-light interferometry, American Institute of
Physics, Review of Scientific Instruments, vol. 75, 2004, pp. 2266-2270.
[11] M. Kitano, T. Nakanishi and K. Sugiyama, Negative Group Delay and Superluminal
Propagation: an Electronic Circuit Approach, IEEE J. Select. Topics Quantum
Electronics, vol. 9, no. 1, 2003.
[12] Enabling Technologies, chapter 2.
[13] Gaillan H. Abdullah and Elham Jasim Mohammad, “Analyzing Numerically Study the
Effect of Add a Spacer Layer in Gires-Tournois Interferometer Design”, International
Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4,
Issue 2, 2013, pp. 85 - 91.

Fabry–pérot interferometer picoseconds dispersive properties

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Fabry–pérot interferometer picoseconds dispersive properties