Genetic algorithm approach for phase extraction in interferometric fiber optic sensor

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1 Genetic alorithm approach for phase etraction in interferometric fiber optic sensor * S.K. Ghorai, Dilip Kumar and P. Mondal Department of Electronics and Communication Enineerin, Birla Institute of Technolo, Mesra, Ranchi-855 * skhorai@rediffmail.com ABSTRACT A phase etraction method based on enetic alorithm (GA) in a Mach-Zehnder tpe fiber interferometer is presented. An application has been attempted for aial strain measurement. Results have been compared with Fourier Transform method (FTM) and Data Dependent Sstem (DDS) method. It is found that GA technique provides ood result without an unwrappin process. KEYWORD: Fiber optic interferometric sensor, enetic alorithm, phase etraction.. INTRODUCTION Interferometric fiber sensors have drawn sinificant attention in industrial application over the past few ears as the offer etremel hih resolution to different measurands. The rane of measurands includes strain, temperature, pressure, surface deformation etc. Different tpes of interferometers have been used dependin upon the sensor applications. However, in each case a frine pattern modulated b the measurand is captured and the accurac of the measurement depends upon the etraction of phase from the interferoram. Different methods have been demonstrated for phase retrieval from the interferoram. A three-step phase shiftin technique was used for etractin surface toporaph information from the interferometric frines []. Other analtical methods such as four- step, Hariharan five-step etc. [] were reported for phase measurement of intensities of interferomtric pattern at man phase shifts. Fourier transform profilometr [] is a popular technique for determination of phase distribution of object field. In Fourier transform method, the frine pattern is Fourier transformed and filtered in the spatial frequenc domain. After filterin the fundamental component of the spectrum inverse Fourier transform is applied. The obtained phase is unwrapped with the suitable alorithm to obtain the phase distribution of the object. Other methods like snchronous method [], Quasi-one-frame alorithm method [5] were also reported to determine the phase from the frine pattern. However, in all the above techniques the common problem is the wrapped phase which provides the principal values ranin from p to + p. It is necessar to correct it b usin an unwrappin alorithm. Reliable phase wrappin alorithms are the critical need of the above techniques. Various factors such as noises, under samplin, frine discontinuities, irreular surface brihtness make the unwrappin process more complicated. Reularization technique based on low pass filterin and binar threshold operation has also been reported for phase retrieval in recent times [6]. But it is reatl affected durin demodulation of nois frine pattern. In the present paper, we report a phase etraction technique based on enetic alorithm (GA) in a Mach-Zehnder tpe fiber optic interferometer. Earlier GA technique has been used for phase recoverin in moiré pattern and optical testin [7]. Recentl we have reported a phase etraction technique based on Data Dependent sstem (DDS) method for strain measurement in interferometric fiber optic sensor. In the present contet we etract the phase information from the interferoram of the fiber optic sensor with the GA technique where no unwrappin method is required. The phase data of interference pattern is approimated b a parametric estimation of n th order polnomial. The enetic alorithm is used to fit

2 the best nonlinear function to the phase. A population of chromosome is codified with the parameters of the function that estimate the phase. An application has been made for aial strain measurement in Mach-Zehnder tpe interferometer.. THEORETICAL BACKGROUND In Mach-Zehnder tpe fiber interferometer dark and briht frines are produced b the interference of liht from two arms. When measurand field is applied in sensor arm it chanes the dispersion properties of the fiber. This causes the chane in phase between the two liht beams which is turn modulates the interferoram. The intensit of the interferoram enerated ma be written as I = I + I + I I γ ( τ) cos( ωτ ) () Ψ where I and I are the intensities of the two arms, γ is the normalized coherence function, τ is the time difference of the interferin beams and Ψ is the phase difference between the beams. When lonitudinal strain is applied in the sensin arm, the phase imbalance due to measurand is iven b Ψ = Ψε + Ψ εt () where the first term is the imbalance due to strain and second term due to cross-sensitivit between strain and temperature. The applied strain causes the chane in refractive inde of the fiber in the sensor arm (strain optic effect). Assumin neliible diameter chane, the phase imbalance due to strain is described as Ψε π = L ε λ [ n / n ( )] P + P where P and P are the elements of the strain optic tensor. Thus when a lenth of one arm of the interferometer is subjected to a measurand, the interference pattern will chane. The interferoram is captured b a CCD and its intensit distribution can be written in one dimensional form as * I( ) = I ( ) + Γ( ) + Γ ( ) () Γ denotes the self coherence function and Γ * ( ) is its conjuate, I ( ) where () is the backround intensit and is the piel number. The required phase information due to measurand is the arument of self coherence function which is iven b Im{ Γ ( )} ar Γ( ) = tan (5) Re{ Γ ( )} ω, it ma be epressed as, When the phase is epanded around a central optical frequenc () Φ( ω ) = L [ β ( ω ) + β( ω )( ω ω ) + / β ( ω )( ω ω ) + / 6β ( ω )( ω ω ) +...] d m where β =, m =,,,... (6) m dω β m ω = ω represent the arm imbalances in the propaation constant β and its derivatives. Usin enetic alorithm method, the phase is recovered from the interferoram and the values of the coefficients of equation (6) are determined. The enetic alorithm is used to etract the phase from the interferoram and then the values of the element of β are determined. These would provide the values of the measurand applied in the sensor arm.

3 . FORMULATION FOR PHASE EXTRACTION USING GA The intensit distribution of an intrfroram can be written in two-dimensional form as I(, = I (, + a(, cos{ + ω + Φ(, n(, } where ω (7) +, are the inteer values representin indees of piel location in the frine pattern, a(, is the amplitude modulation, Q(, is the phase information related to the phsical quantit bein measured, ω and ω are anular career is the noise present in interferoram To recover the phase from the frequenc in the and directions. n(, interferoram enetic alorithm has been used. The phase data of the interferoram has been approimated b the th order polnomial fittin function as follows: P (, + = (8) where,,,, are the coefficients to be estimated b GA. The function has been selected based on the backround eperimental information. To etract phase from the interferoram, we have developed an objective function usin equation (8) and polnomial fittin function, which is iven b [8] O( ) = {[ I (, cos( ω + { P (, P (, ) } ]} m(, where Y X = = N + ω + P (, )] + w [{ P (, P (, }, coordinates represent the piel location in the interference pattern. (, piels in the frine pattern. X Y is the imae resolution and w is the weihtin factor. (, I N ( 9) is the normalized value of the m is the concerned area in the frine pattern. The first term within the summation is used to match the frine model to the observed pattern and the second term is kept for the smoothness and continuit of the phase concerned. Further we define a fitness function as f() = O ma - O() () where O ma is the maimum values of O (). Genetic alorithm has been used to compute the values of the coefficient,,,, in the phase function usin optimization of the fitness function. GA is a stochastic search method based on mechanics of natural selection and natural enetics [9]. The advantae of GA is just to obtain the lobal optimal solution fairl. In GA search spaces are irreular and it avoids ettin stuck at local optimal solution. At first a set of random solutions,,,,... ) is enerated b a set of n bit binar strins called ( chromosomes. The solutions are codified accordin to their limits ( ma, min ) and the desired resolution. The different steps of GA are as follows: Step-I Selection: A random population of m members (m was set to ) is selected. The values of each coefficient are initialized randoml usin a random number enerator. The fitness of each chromosome is evaluated and the correspondin averae fitness of the initial population is computed. Step-II Reproduction: The primar aim of reproduction is to eliminate bad solutions from the current population to prevent further replication in the net enerations. The ood solutions from the current population were selected usin roulette wheel selection. In the process the best chromosomes et more copies, the averae sta even and the worst die off.

4 Step-III Crossover operation: Crossover operation is used to create new and better solutions from eistin solutions. In this operation GA will echane some portion of bits from the two parent chromosomes with a cross over probabilit P c. In our work P c was taken as.8, if a random enerated number is less than the crossover probabilit, then GA will crossover the parent chromosomes otherwise no crossover will take place. In this work a two-point crossover was used to increase diversit of the population and to enerate new solutions. Step-IV Mutation: A mutation operator is used to maintain chromosome population variet and to prevent converence to local optimum. The mutation probabilit ives us the epected number of chromosomes, which undero the mutation operation. A mutation probabilit of.8 was taken in our simulation. Finall, after the computation of the new eneration each candidate solution is checked for the desired accurac. If the condition is satisfied the proram is terminated acceptin the correspondin solution, otherwise the above processes are repeated until the desired accurac is met.. EXPERIMENT SETUP Fi. shows the eperimental set up for aial strain measurement in a Mach-Zehnder fiber interferometer. Liht from a He- Ne laser source is split into two components b means of a directional coupler (DC). In the sensor arm two rods have been used to fi a portion of the fiber (~cm) b epo. One rod has been fied on an aluminium base plate and another on a translation stae (Newport made, model - -s). The fiber has been strained usin micro-movement. The reference arm has been kept isolated from eternal perturbations. The output beams of the two arms have been made collimated b GRIN rod lenses. The two beams overlap at the center of the beam combiner and from interference pattern. The intensit patterns have been captured usin CCD camera for different strain (5-µs). The captured patterns have been converted into bmp format and divided into desired piel matri (56 56). An imae processin alorithm has been developed to diitize the interferoram data and thus to obtain the ra level values of selected piels of the interference pattern. The piel values have been utilized to etract the phase chane due to measurand in the interferoram usin enetic alorithm. 5. RESULTS AND DISCUSSION Lonitudinal strains were applied in one arm of the Mach-Zehnder fiber interferometer. The phase modulated frine patterns were captured usin CCD camera and the ra level values of the piel of a selected size were determined usin a imae processin alorithm. Net GA technique was used to estimate the phase term from eperimental frine pattern. We used to a chromosome population size equal to, a cross over probabilit of.8 and mutation probabilities of.8for each allele in the chromosome. In each chromosome 5 coefficients were coded inside population. The search rane for each allele or polnomial cofficient in chromosome was [-,]. The alorithm was terminated after a specific number of enerations or iterations so that the desired accurac was met. The time taken to achieve this condition was sec on Pentium IV-.5GHz. computer. Fi. shows the interference pattern obtained from the Mach-Zehnder fiber interferometer and Fi. shows the correspondin phase contour maps obtained usin the polnomial coefficient under different strain conditions. Reference Arm LS CP BC CM CL TS PC Fi : Eperimental setup, LS - Laser Source, CP - Directional Coupler, CL - Clamp, TS - Translational Stae, BC - Beam Combiner, CM - Camera

5 (a) Fi : Interferorams for different applied strain: 5µ strain for (a), µ strain for (b). (b) (a) (b) Fi.: Correspondin phase distribution of frine patterns of fi. (a) & (b). Phase in radians Phase in radians Fi. (a) Wrapped phase usin FTM Normalized Phase Normalized Phase Z (Radians) Z (Radians) Y (piels) X (piels) Y (piels) X (piels) Fi. (b) Unwrapped phase usin FTM Fi. (c) Wrapped phase usin DDS Fi. (d) Unwrapped phase usin DDS 5

6 Normalized Phase Group Dela (fs) Fi. (e) Phase recovered usin GA Fi. (a)-(e) show the recovered phases usin FTM, DDS and GA methods. Talor coefficient of the phase field was calculated usin Matlab and those provide the roup dela and dispersion due to the measurand. The results were compared with those obtained usin data dependent sstem (DDS) and Fourier transform method (FTM) as shown in fi.5. A better result has been obtained usin DDS method as compared to FTM. Similar results have been achieved usin GA without an additional unwrappin process. The difficult in GA was the proper selection of enetic parameters for the required converence. 6. CONCLUSION Genetic alorithm method has been used to recover the phase distribution in a Mach-Zehnder tpe fiber interferometer. No unwrappin process is needed as in Fourier transform and data dependent sstem method. Results have been compared in case of aial strain applied in the fiber interferometer. The method can be etended to low coherence interferometer and other measurands. ACKNOWLEDGMENT: The authors deepl acknowlede the ISRO (VSSC) for providin the necessar financial support. Relative strain (micro strain) Fi. 5. Dependences of roup dela with measurand. REFERENCES:. Three dimensional surface metrolo of manetic recordin materials throuh direct phase detectin microscopic Interferometr: D. M. Perr, P. J. Moran and G. M. Robinson, J. Inst. Electron. Radio En., Vol No.55, 5 (985).. Phase-measurement interferometric techniques: K. Creath, Proress in Optics, Wolf E. ed. Elsevier, Amsterdam Vol.No. 6, 5 (988).. Fourier transform profilometr a review: X. Su and W. Chen, Opt. Laser En.,Vol No. 5, 6 ().. Interferometric phase measurement usin spatial snchronous detection : K. H. Womack, Opt En.,Vol. No., 9 (98). 5. A quasi one-frame phase unwrappin alorithm throuh zone-switchin and zone-shiftin hbrid implementation: M. J. Huan, Opt. Commu., Vol. No., 87 (). 6. Frine-follower reularized phase tracker for demodulation of close-frine interferoram : M. Servin, J.L. Marroquin and F. J. Cuevas, J.Opt.Soc.Am A, Vol.No. 8, 689 (). 7. A parametric method applied to phase recover a frine pattern based on a enetic alorithm: F. J. Cuevas, J. H. Sossa-Azuela and M. A. Servin, Opt. Commun.,Vol. No., (). 8. Data dependent sstem approach for strain measurment in interferometric fiber optic sensor: S. K. Ghorai, S. Naraan, Opt. Laser Technol., Vol.No. 6/7, 57 (). 9. D. E. Goldber, Genetic Alorithm in Search, Optimization and Machine Learnin, Addition Wesle, New York,