Fabry–pérot interferometer picoseconds dispersive properties

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  • 1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME274FABRY–PÉROT INTERFEROMETER PICOSECONDS DISPERSIVEPROPERTIESElham Jasim MohammadPhysics Department,Collage of Sciences/Al-Mustansiriyah University, Iraq,ABSTRACTFabry-Pérot interferometers are used in optical modems, spectroscopy, lasers, andastronomy. In this paper we used the coupled mode equation to design the Fabry–Pérotinterferometers and study the picosecond dispersion. Coupled mode analysis is widely usedin the field of integrated optoelectronics for the description of two coupled waves traveling inthe 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. INTRODUCTIONThe Fabry-Perot interferometer (FPI) is a simple device that relies on the interferenceof multiple beams. The interferometer consists of two parallel semi-transparent reflectivesurfaces that are well aligned to form an optical Fabry-Perot cavity with cavity length L andrefractive index n. When a monochromatic input light enters the Fabry-Perot cavity, tworeflections at the two surfaces with amplitudes of A1 and A2 are generated respectively as inFigure 1 below:Figure 1: Basic structure of a Fabry-Perot Interferometer [1].INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING& TECHNOLOGY (IJEET)ISSN 0976 – 6545(Print)ISSN 0976 – 6553(Online)Volume 4, Issue 2, March – April (2013), pp. 274-282© IAEME: www.iaeme.com/ijeet.aspJournal Impact Factor (2013): 5.5028 (Calculated by GISI)www.jifactor.comIJEET© I A E M E
  • 2. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME275Thus, the two reflections interfere with each other to produce an interference patternconsisting 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]:)4cos(2)cos(2 212221212221λπφnLAAAAAAAAI ++=∆++= (1)φ∆ is the relative phase shift between the two light signals. λ denotes the wavelength of theinterrogation 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 thecavity length L and subsequently modifies I. Therefore, those physical parameters changescould be obtained by examining I [1].The solid Fabry-Perot interferometer, also known as a single-cavity coating, is formedby separating two thin-film reflectors with a thin-film spacer. In an all-dielectric cavity, thethin-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 anintegral-half of the principal wavelength, induces transmission rather than reflection at theprincipal wavelength. Light with wavelengths longer or shorter than the principal wavelengthundergoes a phase condition that maximizes reflectivity and minimizes transmission. In ametal-dielectric-metal (MDM) cavity, the reflectors of the solid Fabry-Perot interferometerare 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 COMPUTINGUSING COUPLED MODE THEORYThe Fabry-Perot is a simple interferometer, which relies on the interference ofmultiple reflected beams [4]. The accompanying Figure 2 shows a schematic Fabry-Perotcavity. Incident light undergoes multiple reflections between coated surfaces which define thecavity.Figure 2: Schematic of a Fabry-Perot interferometer [4].
  • 3. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME276Each transmitted wavefront has undergone an even number of reflections (0, 2, 4, . .).Whenever there is no phase difference between emerging wavefronts, interference betweenthese wavefronts produces a transmission maximum. This occurs when the optical pathdifference 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 angleof incidence. The phase difference between each succeeding reflection is given by δ [5]:θλπδ cos22ln= (3)If both surfaces have a reflectance R, the transmittance function of the etalon is given by [5]:( ))2/(sin11cos211222δδ FRRRTe+=−+−= (4)where:2)1(4RRF−= , is the coefficient of finesse. Maximum transmission 1=eT occurs when theoptical path length difference θcos2nl between each transmitted beam is an integer multipleof the wavelength. In the absence of absorption, the reflectance of the etalon eR is thecomplement of the transmittance, such that 1=+ ee RT . The maximum reflectivity is given by[5]:2max)1(4111RRFR+=+−= (5)and this occurs when the path-length difference is equal to half an odd multiple of thewavelength. The wavelength separation between adjacent transmission peaks is called thefree spectral range (FSR) of the etalon, λ∆ , and is given by [5]:θλλθλλcos2cos220020ll nn≈+=∆ (6)Where, 0λ is the central wavelength of the nearest transmission peak. The FSR is related tothe full-width half-maximum (FWHM), of any one transmission band by a quantity known asthe finesse [5]:RRFf−=≈1221ππ. Etalons with high finesse show sharper transmissionpeaks with lower minimum transmission. At other wavelengths, destructive interference oftransmitted wavefronts reduces transmitted intensity toward zero (i.e., most, or all, of thelight is reflected back toward the source).Transmission peaks can be made very sharp by increasing the reflectivity of the mirrorsurfaces. In a simple Fabry-Perot interferometer transmission curve (see Figure 3), the ratioof successive peak separation to FWHM transmission peak is termed finesse [4].
  • 4. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME277Figure 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-Perotinterferometers, 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 amillimeter to several centimeters. The Fabry-Perot is a useful spectroscopic tool. It providedmuch of the early motivation to develop quality thin films for the high-reflectance mirrorsneeded for high finesse [4].Assuming no absorption, conservation of energy requires 1=+ RT . The total amplitudeof both beams will be the sum of the amplitudes of the two beams measured along a lineperpendicular to the direction of the beam.Thus: 00cos/321ll ikikiRTet −+= θπ, where 0l is: 00 sintan2 θθll = and k = wavenumber. Neglecting theπ2 phase change due to the two reflections, the phase difference between the two beams is:00cos2llkk−=θδ . The relationship between θ and 0θ is given by Snells law: 00 sinsin θθ nn = .So that the amplitude can be rewritten as: δiTtRe1 −= .The intensity of the beam will be just t times its complex conjugate. Since theincident beam was assumed to have an intensity of one, this will also give the transmissionfunction [5,6]:δcos21 22*RRTttTe−+== (7)In this study we used the Coupled mode theory to show the Fabry-Perot dispersiveproperties. 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. Itsapplicability seems to be questionable because the assumption of a small perturbation isviolated in the case of large index discontinuities. Additionally, a lot of powerful analyticaldesign tools based on the coupled mode equations have been developed [7]. In coupled modeequations,λπκ2n∆= defines the coupling coefficient for the first order refractive-index variationn∆ andλ is the design wavelength [8]. The group delay (GD) is defined as the negative ofthe derivative of the phase response with respect to frequency [9]. In physics and in particular
  • 5. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME278in optics, the study of waves and digital signal processing, the term delay meaning: the rate ofchange of the total phase shift with respect to angular frequency [10,11]:ωφddGD −= . Througha device or transmission medium, where φ is the total phase shift in radians, and ω is theangular frequency in radians. The group delay dispersion (GDD) can be determined by[10,11]:ωddGDGDD = .Fabry-Perot interferometers can be constructed from purely metallic coatings, but highabsorption losses limit performance [4]. Furthermore, the Fabry-Perot filter ideally covers awhole communication band, which is typically tens of nanometers large [12].III. SIMULATION RESULT AND DISCUSSIONFrom 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 Modeequations.3. Found the phase difference to calculate the amplitude and power transmissioncoefficient 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 standarddeviation (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 shownbelow. Figure 4 is about the transmitted intensity versus the interference order. It shows thetransmittance function for different values of F . Instead of δ , the corresponding interferenceorderπδ2is noted. Figure 5 is about the finesse and the mirror reflectivity. The finesse is animportant parameter that determines the performance of a FPI. Conceptually, finesse can bethought of as the number of beams interfering within the FP cavity to form the standingwave. The primary factor that affects finesse is the reflectance R of the FP mirrors, whichdirectly affects the number of beams circulating inside the cavity. In Figure 6 we foundanother important factor in the design of FPI is the contrast factor which is defined primarilyas the ratio of the maximum to minimum transmission. Figure 7 show finesse against contrastfactor. Figure 8 represents the relationship between the amplitude transmission and thewavelength. Finally, Figure 9 and Figure 10 show the delay and dispersion versus thewavelength after using the transfer function and coupled mode equation. The theoreticallydesigned delay has a small oscillations are visible. Of course, the same behavior can be foundfor the dispersion. Figure 11 and Figure 12 show the delay and dispersion versus thewavelength after using the transfer function, coupled mode equation and then POLYFITfunction. Table 2 show the reflectance and resolving power values for deferent interferenceorder.
  • 6. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME279-15 -10 -5 0 5 10 1500. OrderTransmittance0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1050100150200250300350Mirror ReflectivityFinesseTable 1 The dispersive and statistical analysis: mean, median, mode and the standarddeviation.Index Mean Median Mode STD1storder TransmittanceFunction0.5468 0.4639 0.2868 0.24012ndorder TransmittanceFunction0.417 0.2981 0.1648 0.26893rdorder TransmittanceFunction0.2893 0.1548 0.0784 0.2722Finesse 24.89 10.63 42.97 4.441Contrast Factor 247.9 11.12 1.778 1118Amplitude Transmission -0.0003402-0.002776-0.9718 0.3977Power Transmission 0.3157 0.1858 0.0006045 0.3178Delay ps -0.01603 -0.01602 -0.02106 0.000854Dispersion 2ps -2.137E-5-2.132E-5-0.0001241 1.738E-5Fit Delay ps -0.01603 -0.01605 -0.0161 6.181E-5Fit Dispersion 2ps -2.137E-5 -2.14E-5 -2.146E-5 9.194E-8Figure 4 Figure 5Figure 4: Shows the transmitted intensity versus the interference order for various values oftransmittance of the coatings. Figure 5: Finesse versus the mirror reflectivity. Not that thetransmitted intensity peaks get narrower and the coefficient of finesse increases. When peaksare very narrow in Figure. 3, light can be transmitted only if the plate separationl , refractiveindex n , and the wavelength λ satisfy the precise relation: λθπδ /cos2 ln= .
  • 7. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME2800.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1010002000300040005000600070008000900010000Mirror ReflectivityContrastFactor0 50 100 150 200 250 300 350010002000300040005000600070008000900010000FinesseContrastFactor1498.5 1499 1499.5 1500 1500.5 1501 1501.5-1-0.8-0.6-0.4- (nm)AmplitudeTransmissin(p.u)1498.5 1499 1499.5 1500 1500.5 1501 1501.500. (nm)PowerTransmisson(p.u)1498.5 1499 1499.5 1500 1500.5 1501 1501.5-0.022-0.02-0.018-0.016-0.014-0.012-0.01Wavelength (nm)Delay(ps)1498.5 1499 1499.5 1500 1500.5 1501 1501.5-1.5-1-0.500.511.5x 10-4Wavelength (nm)Dispersion(ps)Figure 6 Figure 7Figure 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 versusamplitude transmission and the wavelength the wavelength.Figure 10: The relationship between the Figure 11: The relationship between thedelay and the wavelength dispersion and the wavelength
  • 8. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME2811498.5 1499 1499.5 1500 1500.5 1501 1501.5-0.0161-0.016-0.016-0.0159-0.0159-0.0159-0.0158Wavelength (nm)Delay(ps)1498.5 1499 1499.5 1500 1500.5 1501 1501.5-2.145-2.14-2.135-2.13-2.125-2.12-2.115-2.11-2.105-2.1x 10-5Wavelength (nm)Dispersion(ps)Figure 12: The relationship between Figure 13: The relationship between thethe fit delay and the wavelength. fit dispersion and the wavelength.Table 2 The reflectance and resolving power for deferent interference order.ReflectanceResolvingPowerReflectanceResolvingPower56 0.989924 42 0.99984255 0.979927 41 0.98985354 0.989929 40 0.99856453 0.999930 39 0.99787352 0.989932 38 0.99888151 0.998933 37 0.99688850 0.998934 36 0.99789449 0.999736 35 0.99998948 0.995937 34 0.99900547 0.996938 33 0.99909846 0.999938 32 0.99391345 0.997939 31 0.99791644 0.999434 30 0.99891943 0.988994 29 0.999322IV. CONCLUSIONThe general theory behind interferometry still applies to the Fabry–Perot model,however, these multiple reflection reinforce the areas where constructive and destructiveeffects occur making the resulting fringes much more clearly defined . This paper haspresented a theoretical design of Fabry-Perot interferometer. This theoretical design studyincluding dispersion, FSR, finesse and contrast, used to assess the performance of the FPIwere discussed. An attempt is made to analyze the factors that control and affect theperformance and the design of the FPI versus the parameter that control those factors. Veryhigh finesse factors require highly reflective mirrors. A higher finesse value indicates a
  • 9. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 –6545(Print), ISSN 0976 – 6553(Online) Volume 4, Issue 2, March – April (2013), © IAEME282greater number of interfering beams within the cavity, and hence a more completeinterference process. The figure show that the linear increase in finesse with respect tocontrast 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 smalloscillations around the design wavelength. The Finesse is the most important parameter, itsvalue depends on the reflectivity of coating parallelism of the etalon mirror and the shape andsize of the field stop.REFERENCES[1] X. Zhao, Study of Multimode Extrinsic Fabry-Perot Interferometric Fiber Optic Sensoron 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: 3rdEdition, (Published by Institute of PhysicsPublishing, 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 inSemiconductor 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-delaydispersion of femtosecond optics using white-light interferometry, American Institute ofPhysics, Review of Scientific Instruments, vol. 75, 2004, pp. 2266-2270.[11] M. Kitano, T. Nakanishi and K. Sugiyama, Negative Group Delay and SuperluminalPropagation: an Electronic Circuit Approach, IEEE J. Select. Topics QuantumElectronics, vol. 9, no. 1, 2003.[12] Enabling Technologies, chapter 2.[13] Gaillan H. Abdullah and Elham Jasim Mohammad, “Analyzing Numerically Study theEffect of Add a Spacer Layer in Gires-Tournois Interferometer Design”, InternationalJournal of Advanced Research in Engineering & Technology (IJARET), Volume 4,Issue 2, 2013, pp. 85 - 91.