Particle Swarm Optimization for the Design of High Diffraction Efficient Holographic Grating

Transcription

1 Original Artile Partile Swarm Optimization for the Design of High Diffration Effiient Holographi Grating A.K. Tripathy 1, S.K. Das, M. Sundaray 3 and S.K. Tripathy* 4 1, Department of Computer Siene, Berhampur University, Berhampur, Ganjam , Odisha, India 3 Department of Physis, Berhampur University, Berhampur, Ganjam, , Odisha, India 4 National institute of siene and Tehnology, Pallur hills, Berhampur, Ganjam, , Odisha, India Address for Correspondene National institute of siene and Tehnology, Pallur hills, Berhampur, Ganjam, , Odisha, India. ABSTRACT We present a Partile Swarm Optimization (PSO) algorithm for the design of holographi gratings using oupled mode theory. The algorithm is based on a new objetive funtion that has been proposed for ahieving desired diffration effiieny. The performane of the algorithm is determined by designing a grating with speifi appliation to solar onentrator. The algorithm ould predit the different parameters like thikness of grating, modulation index and angular deviation for ahieving maximum diffration effiieny. Keywords: Diffration effiieny, Holograms, Holographi grating, Partile swarm optimization. INTRODUCTION In reent years, different stohasti global optimization algorithms like Geneti Algorithm, Simulated Annealing, have been applied to solve different physial problems. The drawbak of Geneti Algorithm is its expensive omputational ost 1. In simulated annealing, repeatedly annealing with a 1/ log k shedule is very slow, espeially if the ost funtion is expensive to ompute. Reently PSO has been proposed as an alternative tool for solving optimization problems. Due to its simpliity, this algorithm is used to optimize almost every physial problem. We present a Partile Swarm Optimization (PSO) algorithm 3 for the design of holographi gratings using oupled mode theory 4. The algorithm is based on a new objetive funtion that has been proposed for ahieving desired diffration effiieny. The performane of the algorithm is determined by designing a grating with speifi appliation to solar onentrator. The algorithm ould predit the different parameters like thikness of grating, modulation index and angular Amerian Journal of Computer Siene and Engineering Survey

2 S.K. Tripathy et al ISSN deviation for ahieving maximum diffration effiieny. Holographi grating The hologram grating, whih we have onsidered for this investigation is a transmission hologram as shown in fig (1) and is haraterized by -Grating period, K -Grating vetor (perpendiular to the fringe planes), θ- the angular deviation from the Bragg s angle, - slant angle, -Propagation vetors of diffrated waves, -propagation vetors of referene wave, n -Spatial modulation amplitudes of the refrative index, d -Grating thikness. The diffration effiieny for suh a holographi grating using Kogelnik oupling theory [5] an be expressed as: s S S R (1) Where, 1 1 v R i S j e sin 1 s 1 R os K os s os n v () (3) (4) Using fig (1) the diffration effiieny of holographi grating an be shown to be: 1 v sin Where, d s 1 v (5) K K os 4 n (6) (7) k (8) v n d 1 1 R s (9) Equation (5) is the basi equation whih is used for optimization of diffration effiieny. OBJECTIVE FUNCTION First we set 9 different points on the desired flat urve and then minimize the distane between desired urve and objetive plot as shown in fig () by using the fitness funtion in equation (10). n fitness k k i ki 1 (10) Where, k i The desired value of effiieny at k i, where i 1,...9 k It is the diffration effiieny of holographi grating as given in equation (5). Partile swarm optimization (PSO) Partile Swarm Optimization is a population based optimization tehnique. It emulates some aspets of soial behavior of a flok of birds and a shool of fish. The swarm initially has a population of random solutions. Eah potential solution, alled a partile, is given a random veloity and is flown through the problem spae. The partiles have memory and eah partile keeps trak of its previous best position (pbest) and the orresponding fitness value. The swarm has another value alled gbest, whih is the best value of the entire partile s pbests in the swarm. The PSO algorithm 6,7 used in our paper is as summarized below: Step-1: Initialize a population of partiles (30 in our ase) with random positions (s k i ) and veloities (v k k i ).where s i and v k i represent the position of partile i at AJCSES[3][1][015]

3 S.K. Tripathy et al ISSN iteration k and veloity of partile i at iteration k with a number of dimension respetively within the lower and upper bounds of index modulation (n 1 ), thikness of hologram (d) and angle of defletion from Bragg s angle (θ). Evaluate the fitness of eah partile using equation (10) and assign the partile position as pbest. The best among the pbests is global best (gbest). Step-: Plot graph between wave length versus diffration effiieny for different optimum values of modulation index (n 1 ), thikness of hologram (d) and angle of defletion from Bragg s angle (θ) and angle of deviation from Bragg s angle (θ) versus diffration effiieny for different optimum values of wavelength, n 1 modulation index, thikness of hologram d. Step-3: k 1 k k k V i wv i 1 * rand 1*(pbest is i ) * rand *(gbest s i ) (11) w w Max [(wmax wmin) iter]/ maxiter (1) k 1 k k 1 si si Vi (13) Change the veloity and position of the partile aording to equations (11) and k k (13), respetively. V i & si represents the veloity of partile i at iteration k and urrent position of partile i at iteration k with a no. of dimensions respetively. rand1 rand & are two uniform random funtions between 0 and 1, w is the inertia weight alulated aording to equation (1) and 1 & are the aeleration onstants. Step-4: For eah partile, evaluate the fitness using equation (10) if all variables are within the searh range 8. Step-5: For eah partile, if the urrent fitness value is better than pbest, then update pbest. If the urrent best fitness is better than the gbest, update gbest to the urrent best position and fitness value. Step-6: Repeat steps 4 & 5 until the maximum number of funtion evaluation is ompleted. SIMULATION RESULT AND ANALYSIS For an effetive holographi grating to be used in photovoltai appliations, its diffration effiieny should not vary muh with respet to wavelength and deviation from Bragg s angle of inidene. In pratial situation, both wavelength and deviation from Bragg s angle of inidene are not onstant, but vary in a given range. For example the position of sun with respet to onentrator hanges with time. As the diffration effiieny of the holographi grating depends on other parameters like thikness and modulation index, it is possible to optimize the diffration effiieny suh that it does not vary muh with the hange of wavelength or deviation from Bragg s angle of inidene. To ahieve this, one has to searh for the values of thikness of hologram d, angle of deviation from Bragg s angle, modulation index n 1 for whih diffration effiieny beomes maximum and does not hange muh with respet to hange of wavelength. In other words, the variation of diffration effiieny with respet to wavelength and Bragg angle variation should be almost flat and maximum. We use the algorithm presented in the setion IV, to d, n find these values of 1,. Simulation are made with the following values of 0.996, 0.997, 0.998, 0.999, 0.1, 0.999, 0.998, 0.997, k1 k k3 k4 k5 k6 k7 k8 k9 The wavelength range of operation is nm. For example the proposed algorithm is implemented for photovoltai appliation and the different parameters for maximum and flat diffration effiieny are found to be , n , d m as shown AJCSES[3][1][015]

4 S.K. Tripathy et al ISSN in fig (3) and m, n , d m as shown in fig (4). CONCLUSION In this paper we proposed an algorithm based on Partile Swarm Optimization to optimize diffration effiieny of a holographi grating. A new fitness funtion has been proposed and the algorithm is applied for a holographi grating whih an be used as a solar onentrator to ahieve maximum and flat diffration effiieny. REFERENCES 1. K. F. Man, K.S. Tang, S. Kwong: Geneti Algorithms: Conepts and Appliations, IEEE Transation on Industrial Eletronis, Vol. 43, No. 5, pages , P.J.M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and appliations, Kluwer Aademi Publisher, J. Robinson and Y. Rahmat-Samii, Partile Swarm Optimization in eletromagneti, IEEE Antennas and propagation, 5(004) M.G. Moharam and T. K. Gaylord, Rigorous oupled-wave analysis of planar- grating diffration, J. Opt. So. Am. 71 (1981) H. Kogelnik, Coupled wave analysis for thik hologram gratings, The Bell System Tehnial Journal 48(1969) Y. Shi and R.C. Eberhart, A modified partile swarm optimizer, In P. Congress on Evolutionary Computation, (1998), P.N. Suganthan, Partile swarm optimizer with neighbourhood operator, in P. Congress on Evolutionary Computation, Washington, DC, 3(1999), D. Gies and Y. Rahmat-samii, Partile swarm optimization for reonfigurable phase differentiated array design, mirow. opt. Tehnol. Lett., 38(003), Figure 1. Shemati diagram showing the basi model for diffration from a volume hologram AJCSES[3][1][015]

5 S.K. Tripathy et al ISSN Figure. Setting of the desired funtion Figure 3. Variation of diffration effiieny with wavelength at Bragg s angle illumination at fixed fringe spaing m, the angular deviation from the Bragg s angle 6.671, n modulation index 1, and thikness of the grating 6 d m AJCSES[3][1][015]

6 S.K. Tripathy et al ISSN Figure 4. Variation of diffration effiieny with angular deviation from the Bragg s angle at fixed fringe 7 spaing m, wavelength m, modulation n index 1 6, and thikness of the grating d m AJCSES[3][1][015]