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FABRY–PÉROT INTERFEROMETER PICOSECONDS DISPERSIVE PROPERTIES

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FABRY–PÉROT INTERFEROMETER PICOSECONDS DISPERSIVE PROPERTIES Powered By Docstoc
					INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING
 International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
 6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME
                            & TECHNOLOGY (IJEET)
ISSN 0976 – 6545(Print)
ISSN 0976 – 6553(Online)
Volume 4, Issue 2, March – April (2013), pp. 274-282
                                                                              IJEET
© IAEME: www.iaeme.com/ijeet.asp
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      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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 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 π nL       (1)
             I = A12 + A 22 + 2 A1 A 2 cos( ∆ φ ) = A12 + A 22 + 2 A1 A 2 cos(            )
                                                                                   λ

  ∆ 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].



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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME

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]:

                                                    m λ = 2 t op cos θ                            (2)

where m is an integer, often termed the order, t op is the optical thickness, and θ is the angle
of incidence. The phase difference between each succeeding reflection is given by δ [5]:

                                                            2π                                  (3)
                                                    δ =        2nl cosθ
                                                            λ 

If both surfaces have a reflectance R , the transmittance function of the etalon is given by [5]:

                                       Te =
                                                      (1 − R )2         =
                                                                                    1             (4)
                                                       2
                                              1 + R − 2 R cos δ             1 + F sin 2 (δ / 2)


where: F =      4R       , is the coefficient of finesse. Maximum transmission Te = 1 occurs when the
             (1 − R) 2
optical path length difference 2nl cosθ between each transmitted beam is an integer multiple
of the wavelength. In the absence of absorption, the reflectance of the etalon Re is the
complement of the transmittance, such that Te + Re = 1 . The maximum reflectivity is given by
[5]:
                                              1   4R                                      (5)
                                   Rmax = 1 −   =
                                                              1+ F    (1 + R) 2

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]:

                                                      λ2
                                                       0          λ20
                                    ∆λ =                      ≈                                   (6)
                                                2nl cos θ + λ0 2nl cos θ

Where, λ 0 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
                                            1
                             π F       πR       2
the finesse [5]: f ≈               =  . Etalons with high finesse show sharper transmission
                          2     1− R
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].

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  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,
t op , 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 T + R = 1. 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: t1 = RTe 2πi +3ikl / cos θ −ik l , where l 0 is: l 0 = 2l tan θ sin θ 0 and k = wavenumber. Neglecting the
                          0 0




2π phase change due to the two reflections, the phase difference between the two beams is:
    2kl
δ =     − k 0 l 0 . The relationship between θ and θ 0 is given by Snell's law: n sin θ = n0 sin θ 0 .
    cos θ
So that the amplitude can be rewritten as: t =            T       .
                                                      1 − Re iδ
       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]:

                                                        T2
                                  Te = tt * =                                                               (7)
                                                1 + R 2 − 2 R cos δ

        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, κ = π∆n defines the coupling coefficient for the first order refractive-index variation
                  2λ
 ∆n and λ 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

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   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]: GD = − dφ . Through
                                                                                     dω
   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]: GDD = dGD .
                   dω
   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 δ is noted. Figure 5 is about the finesse and the mirror reflectivity. The finesse is an
          2π
   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.




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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
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Table 1 The dispersive and statistical analysis: mean, median, mode and the standard
deviation.

                                Index                                          Mean                     Median                         Mode                            STD
                            st
                      1 order Transmittance
                                                                               0.5468                        0.4639                   0.2868                       0.2401
                              Function
                      2nd order Transmittance
                                                                               0.417                         0.2981                   0.1648                       0.2689
                              Function
                      3rd order Transmittance
                                                                               0.2893                        0.1548                   0.0784                       0.2722
                              Function
                               Finesse                                         24.89                     10.63                         42.97                           4.441
                           Contrast Factor                                     247.9                     11.12                         1.778                           1118
                                                                                                           -
                      Amplitude Transmission                              -0.0003402                                                 -0.9718                       0.3977
                                                                                                       0.002776
                            Power Transmission                                 0.3157                   0.1858                     0.0006045                      0.3178
                                Delay ps                                      -0.01603                 -0.01602                     -0.02106                     0.000854
                                                                                                       -2.132E-
                                  Dispersion ps 2                             -2.137E-5                                            -0.0001241                    1.738E-5
                                                                                                           5
                                    Fit Delay ps                               -0.01603                -0.01605                     -0.0161                      6.181E-5
                                 Fit Dispersion ps 2                          -2.137E-5                -2.14E-5                    -2.146E-5                     9.194E-8

                       1                                                                               350

                      0.9
                                                                                                       300
                      0.8

                      0.7                                                                              250
      Transmittance




                      0.6
                                                                                                       200
                                                                                             Finesse




                      0.5
                                                                                                       150
                      0.4

                      0.3                                                                              100

                      0.2
                                                                                                       50
                      0.1

                       0                                                                                0
                       -15          -10    -5             0          5   10      15                     0.5    0.55   0.6   0.65    0.7    0.75     0.8   0.85   0.9    0.95   1
                                                Interference Order                                                                  Mirror Reflectivity




                                          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 separation l , refractive
index n , and the wavelength λ satisfy the precise relation: δ = 2πnl cosθ / λ .




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                                                                10000
                                                                                                                                                                                      10000
                                                                    9000
                                                                                                                                                                                              9000
                                                                    8000
                                                                                                                                                                                              8000
                                                                    7000
                                                                                                                                                                                              7000
                                              Contrast Factor




                                                                    6000




                                                                                                                                                              Contrast Factor
                                                                                                                                                                                              6000
                                                                    5000
                                                                                                                                                                                              5000
                                                                    4000
                                                                                                                                                                                              4000
                                                                    3000
                                                                                                                                                                                              3000
                                                                    2000
                                                                                                                                                                                              2000
                                                                    1000
                                                                                                                                                                                              1000

                                                                      0
                                                                      0.5     0.55   0.6      0.65   0.7    0.75     0.8   0.85   0.9     0.95      1                                                                       0
                                                                                                                                                                                                                                0         50          100        150      200          250     300    350
                                                                                                     Mirror Reflectivity
                                                                                                                                                                                                                                                                   Finesse



                                                                                              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.

                                                                1                                                                                                                                                           1.4

                                                     0.8
                                                                                                                                                                                                                            1.2
                                                     0.6
                Amplitude Transmissin (p.u)




                                                     0.4                                                                                                                                                                         1
                                                                                                                                                                                                  Power Transmisson (p.u)




                                                     0.2
                                                                                                                                                                                                                            0.8
                                                                0
                                                                                                                                                                                                                            0.6
                                                -0.2

                                                -0.4                                                                                                                                                                        0.4

                                                -0.6
                                                                                                                                                                                                                            0.2
                                                -0.8

                                                          -1                                                                                                                                                                  0
                                                         1498.5               1499      1499.5        1500       1500.5       1501         1501.5                                                                            1498.5            1499         1499.5        1500       1500.5    1501    1501.5
                                                                                                 Wavelength (nm)                                                                                                                                                     Wavelength (nm)




   Figure 8: The relationship between the                                                                                                                         Figure 9: Power transmissions versus
 amplitude transmission and the wavelength                                                                                                                                  the wavelength.
                                                                                                                                                                                                                                     -4
                                                                                                                                                                                                                                 x 10
               -0.01                                                                                                                                                                                              1.5



              -0.012                                                                                                                                                                                                        1



              -0.014                                                                                                                                                                                              0.5
                                                                                                                                                                                Dispersion (ps)
 Delay (ps)




              -0.016                                                                                                                                                                                                        0



              -0.018                                                                                                                                                                                       -0.5



               -0.02                                                                                                                                                                                                        -1



              -0.022                                                                                                                                                                                       -1.5
                  1498.5                                                   1499      1499.5        1500       1500.5       1501         1501.5                                                              1498.5                        1499          1499.5        1500       1500.5       1501    1501.5
                                                                                              Wavelength (nm)                                                                                                                                                    Wavelength (nm)




 Figure 10: The relationship between the                                                                                                                                    Figure 11: The relationship between the
      delay and the wavelength                                                                                                                                                   dispersion and the wavelength


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                                                                                                                          -5
                                                                                                                       x 10
                   -0.0158                                                                                      -2.1

                                                                                                              -2.105
                   -0.0159
                                                                                                               -2.11

                                                                                                              -2.115
                   -0.0159




                                                                                            Dispersion (ps)
                                                                                                               -2.12
      Delay (ps)




                   -0.0159                                                                                    -2.125

                                                                                                               -2.13
                    -0.016
                                                                                                              -2.135

                                                                                                               -2.14
                    -0.016
                                                                                                              -2.145

                   -0.0161
                        1498.5   1499   1499.5        1500       1500.5   1501   1501.5                          1498.5        1499   1499.5        1500       1500.5   1501   1501.5
                                                 Wavelength (nm)                                                                               Wavelength (nm)




                    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                        Resolving                 Reflectan                            Resolving
                                             e                             Power                       ce                                 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

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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –
6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME

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
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[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
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[6] S. Tamir, Fabry-Perot Filter Analysis and Simulation Using MATLAB, 2010.
[7] M. Wiemer, Double Chirped Mirrors for Optical Pulse Compression, 2007.
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[9] Adobe PDF-View as html, Definition of Group Delay, 2008:
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[10] T. Imran, K. H. Hong, T. J. Yu and C. H. Nam, Measurement of the group-delay
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[11] M. Kitano, T. Nakanishi and K. Sugiyama, Negative Group Delay and Superluminal
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[12] Enabling Technologies, chapter 2.
[13] Gaillan H. Abdullah and Elham Jasim Mohammad, “Analyzing Numerically Study the
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     Issue 2, 2013, pp. 85 - 91.




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