6545(Print), ISSN (Online) Volume 4, Issue 2, March April (2013), IAEME & TECHNOLOGY (IJEET)

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1 INTERNATIONAL International Journal of JOURNAL Electrical Engineering OF ELECTRICAL and Technology (IJEET), ENGINEERING ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME & TECHNOLOGY (IJEET) ISSN (Print) ISSN (Online) Volume 4, Issue, March April (3), pp IAEME: Journal Impact Factor (3): 5.58 (Calculated by GISI) IJEET I A E M E 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 A and A are generated respectively as in Figure below: Figure : Basic structure of a Fabry-Perot Interferometer []. 74

2 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME 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 [,]: 4π nl I = A + A + A A cos( φ ) = A + A + A A 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 []. 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 halfwave thickness. These are commonly used to filter UV light that would be absorbed by alldielectric 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 shows a schematic Fabry-Perot cavity. Incident light undergoes multiple reflections between coated surfaces which define the cavity. Figure : Schematic of a Fabry-Perot interferometer [4]. 75

3 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME Each transmitted wavefront has undergone an even number of reflections (,, 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 = t op cos θ () 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]: π δ = nl cosθ (3) λ If both surfaces have a reflectance R, the transmittance function of the etalon is given by [5]: where: F 4R ( R) ( R) = = (4) T e + R R cosδ + F sin ( δ / ) =, is the coefficient of finesse. Maximum transmission T e = occurs when the optical path length difference nl cosθ between each transmitted beam is an integer multiple of the wavelength. In the absence of absorption, the reflectance of the etalon R is the e complement of the transmittance, such that T e + R e =. The maximum reflectivity is given by [5]: 4R R = = (5) max + F 76 ( + R) 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]: λ λ λ = (6) nl cosθ + λ nl cosθ Where, λ 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 π F πr the finesse [5]: f =. Etalons with high finesse show sharper transmission 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].

4 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME 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,, is essentially equal to d, the physical thickness. The air gap may vary from a fraction of a t op 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 =. 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. π i+ 3ikl / cos θ ik Thus: l t = RTe, where l is: l = l tanθ sinθ and k = wavenumber. Neglecting the π phase change due to the two reflections, the phase difference between the two beams is: kl = kl cosθ δ. The relationship between θ and θ is given by Snell's law: sinθ n sinθ n =. So that the amplitude can be rewritten as: T t =. iδ Re 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]: T e * T = tt = (7) + R R cosδ 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, π n κ = defines the coupling coefficient for the first order refractive-index variation λ 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 77

5 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME 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 [,]: dφ GD =. 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 [,]: dgd GDD =. 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 []. III. SIMULATION RESULT AND DISCUSSION From all results below, it got after following these steps:. Calculate the transmittance function, finesse and contrast factor of FPI.. 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. 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 π 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 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 and Figure show the delay and dispersion versus the wavelength after using the transfer function, coupled mode equation and then POLYFIT function. Table show the reflectance and resolving power values for deferent interference order. 78

6 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME Table The dispersive and statistical analysis: mean, median, mode and the standard deviation. Index Mean Median Mode STD st order Transmittance Function nd order Transmittance Function rd order Transmittance Function Finesse Contrast Factor Amplitude Transmission Power Transmission Delay ps Dispersion ps -.37E-5 -.3E E-5 Fit Delay ps E-5 Fit Dispersion ps -.37E-5 -.4E E E-8 Transmittance Interference Order Finesse 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 separationl, refractive index n, and the wavelength λ satisfy the precise relation: δ = πnlcosθ / λ. 79

7 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME Contrast Factor Mirror Reflectivity Contrast Factor 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. Amplitude Transmissin (p.u) Power Transmisson (p.u) Figure 8: The relationship between the amplitude transmission and the wavelength Figure 9: Power transmissions versus the wavelength x Delay (ps) Dispersion (ps) Figure : The relationship between the delay and the wavelength Figure : The relationship between the dispersion and the wavelength 8

8 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), IAEME Delay (ps) Dispersion (ps) -. x Figure : The relationship between the fit delay and the wavelength. Figure 3: The relationship between the fit dispersion and the wavelength. Table The reflectance and resolving power for deferent interference order. Reflectanc e Resolving Power Reflectan ce Resolving Power 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 8

9 International Journal of Electrical Engineering and Technology (IJEET), ISSN (Print), ISSN (Online) Volume 4, Issue, March April (3), 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 [] 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, 6. [] R. Fowles, Introduction to Modern Optics, (Dover Publication: New York, 989). [3] Optical Filter Design Application Note. [4] Interference Filters: [5] Macleod H. A., Thin-Film Optical Filters: 3 rd Edition, (Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London, UK, ). [6] S. Tamir, Fabry-Perot Filter Analysis and Simulation Using MATLAB,. [7] M. Wiemer, Double Chirped Mirrors for Optical Pulse Compression, 7. [8] B. Cakmak, T. Karacali and S. Yu, Theoretical Investigation of Chirped Mirrors in Semiconductor Lasers, Appl. Phys., 5, [9] Adobe PDF-View as html, Definition of Group Delay, 8: [] 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, 4, pp [] 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., 3. [] Enabling Technologies, chapter. [3] 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, 3, pp

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