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40220140501002
1. International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), INTERNATIONAL JOURNAL OF ELECTRICAL ENGINEERING & ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME TECHNOLOGY (IJEET) ISSN 0976 – 6545(Print) ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), pp. 10-16 © IAEME: www.iaeme.com/ijeet.asp Journal Impact Factor (2013): 5.5028 (Calculated by GISI) www.jifactor.com IJEET ©IAEME EFFICIENT EVOLUTIONARY ALGORITHM FOR THE THIN-FILM SYNTHESIS USING THE EFFECTS OF MERIT FUNCTION M.F.A. Alias1, M.A. Hussain2 1 Department of Physics, College of Science, University of Baghdad P.O. Box Jadiriyah, Baghdad, Iraq 2 University of Technology, Department of Laser and Optoelectronics Engineering Baghdad, Iraq ABSTRACT We propose a new method for designing of multilayer coatings which makes use of a specified reflectance or transmittance values over a range of wavelengths by using the evolutionary algorithms method. The crossover and mutation processes are adapted to depend directly on the value of merit function. The proposed method offers explicit values for the thickness and the refractive index of each layer. Comparing our theoretical evolutionary algorithm with other theoretical results which using different mathematical formulas for crossover, mutation and selection led to obtain acceptable solutions for different types of coating designs. Keywords: Evolutionary Algorithm, Multilayer System Design. I. INTRODUCTION The basic task of the approaches to the evolutionary algorithms for designing multilayer systems is to find the values of the construction parameters (refractive indices, and thicknesses) that bring the computed optical performance close to the target specification over the desired wavelength band. The mathematical framework for this paper assumes normal incidence of light, and the best solution for designing depends on the basis of minimizing merit function. The merit function (MF) employed here is the basic one that is used to represent the difference between the target reflectance and the computed reflectance over a range of wavelengths [1]. 10
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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME MF = 1 W 2 1/ 2 W Rcomp ( n , d , λ j ) − Rtar ( λ j ) ∑ δR j =1 ( ) (1) where Rcomp and Rtar are the computed and the target reflectance at the wavelength λj, n and d are the refractive index and the thickness of all layers of coating system, and δR is the tolerance at the wavelength λj. In general δR is set to 0.01, W is the number of interesting points used to compute the merit function. Rcomp can be calculated using the most used method based on a matrix formulation [2]. The evolution process starting with an initial population of a number of chromosomes which each one of them is having a certain number of genes. A few statistical processes are acting on populations and species. These processes are selection, crossover, mutation and reproduction which are the principle of modern biological thought of Darwinian evolutionary theory. Starting with an initial population of chromosomes, we evaluate each of its genes by a certain fitness function. This work uses the evolutionary algorithms to the design of a thin film multilayer coating to achieve a particular reflectance or transmittance over a defined wavelength region. Also we propose a method that used merit function to determine the probability percentage of crossover and mutation processes. II. PROBLEM DEFINITION Function optimization is a rather theoretical application of a genetic algorithm. In general an optimization problem requires finding a setting of parameters of the system under consideration, such that a certain fitness function is maximized or equivalently minimized [3-5]. For a function consist of six unknown parameters: ݂ ሺ݊ଵ , ݀ଵ , ݊ଶ , ݀ଶ , ݊ଷ , ݀ଷ , … ሻ ൌ ݉݅݊݅݉݁ݑ݈ܽݒ ݉ݑ (2) The above equation can represent three layer problems for antireflection (n) is the refractive index and d is the thickness of each layer. We can choose any possible solution of the unknown parameters, each possible parameter represent a gene while the whole set of parameters represent a chromosome. For the above function each chromosome will contain six genes, while for more complex function with more unknown parameters such as the thin-film multilayer function, the genes will exceeded to be equivalent to the layers number. The thin-film multilayer system consists of M layers of different materials with different thicknesses. This system can achieve a spectral reflectance (or transmittance) which can be compared with the desired or target reflectance via merit function. The basic operations for evolutionary algorithm are: 1- Cross-over operation The original version of this operation produced a child that inherits genes from each parent with equal probability, but in general the child inherits genes with different probability from each parents, some researchers considered that each gene of the child has an 80% probability of coming from the father and a 20% probability of coming from the mother [6], while other ones considered equal probability (50%) for each two chromosomes. Let the gene from the father be X f , and the gene from the mother or the other parent be X m , then the gene of the child Xc equal to: 11
3.
International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME X =P X + P X c f f m m (3) where Pf and Pm are the inheriting probability of the father and the mother, respectively. The first chromosome or the father chromosome will share with a percentage equal to: ( MF ) m P = f ( MF ) + ( MF ) m f (4) while the mother chromosome has a sharing percentage equal to: ( MF ) f P = m ( MF ) + ( MF ) m f (5) The child chromosome that comes from the cross-over of the two parents will be: X = c X .( MF ) + X .( MF ) f m m f ( MF ) + ( MF ) m f (6) Xf , Xm, Xc are the father, mother and child chromosomes, respectively. This new procedure will allow the child chromosome to inherit the best performance of the two parent chromosomes. 2- Mutation operation In this research a new formula for mutation operation has been produced, the mutation operation is assumed to have a probability of 0.2% of the total number of genes, these mutant genes are chosen randomly. The value of each mutant gene will change to another value according to the relation: X ′ = X 1+τ c c (7) τ is the variation in the value of the mutant gene, it is assumed to be equal to: τ = Rand . ( MF ) (8) c Rand is a random number between -0.5 to 0.5. Obviously, the change in the value of the mutant gene depends on its merit function value, the variation τ approaches zero as the merit function value (the perfect solution) approaches zero. 3- Replacement operation At the end of each generation of the evolutionary algorithm method, we apply the replacement operation. The replacement operation tends to remove the weakest chromosome which has maximum merit function value with another one coming from the stronger one which has minimum merit function. 12
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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME If the genes of the stronger chromosome are (Xc)s , then the genes of the weakest chromosome (Xc)w will change to: ( X ) = ( X ) . Rand c w c s (9) (Rand) is a random number between 0.75 to 1.25. The gene of the weakest chromosome takes any value between 75% and 125% of the gene of the stronger chromosome. This operation has an important effect for elimination of any chromosome with high merit function value; this will increase the speed of reaching the best solution. III. RESULTS We will compare our theoretical evolutionary algorithm with other theoretical results which using different mathematical formulas for crossover, mutation and selection. The major results for comparison are the value of merit function and the shape of transmittance (or reflectance) curves. A. Antireflection Coating We have applied this analysis to two design problems. The first problem was treated in the work of Yang and Kao who used the family competition evolutionary algorithm and has also been the focus of several other studies [7-10], this problem is a wideband antireflection coating for germanium in the infrared region. The target design is for zero reflectance in the wavelength region between 7.7 and 12.3 µm. The incident medium (the zero layer) is air with refractive index 1.0 and the last layer is germanium substrate with refractive index 4.0, the first layer has the low refractive index (zinc sulfide with refractive index 2.2), and the second layer has the high refractive index (germanium with refractive index 4.2) . The coating materials are assumed to be non-dispersive and non-absorbing in the above wavelength region, which means that the extinction coefficient of the whole layers equals to zero then all layers will expressed with the refractive indices and thicknesses only. The total number of layers was 15. Figure (1) shows the convergence of merit function values with generation number. 1.0 15-Layer Film 0.9 0.8 Mi Fnto et uc n r i 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 2 4 6 8 10 12 14 16 18 20 Generation Number Figure 1. Convergence of merit function for M=15 The result for the reflectance as a function of wavelength is shown in figure (2), the merit function value was 0.0916 and the values of ∑ ni di was 35.1025 µm. The mean value of reflectance along the required wavelength range (7.7-12.3) µm is 0.8421%. The construction parameters of the layers are shown in Table (1). 13
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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME 4.0 3.8 15-Layer Film 3.6 3.4 3.2 3.0 % e c ne R fle ta c 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 Wavelength (um) Figure 2. Antireflection coating design for M=15 B. Beam Splitter The second design was treated in the work of Yang and Kao [11] to design a beam splitter in the wavelength range (0.5-1.0) µm, the target design is the reflectance equal to 50% and transmittance 50%. The incident media (the zero layer) was air, and the substrate (the last layer) was the glass (ns=1.52), while the beams were incident normal to the plane of the splitter. The design consists of two materials with low refractive index 1.35 (NaAlF2) and high refractive index 2.35 (TiO2) arranged in sequences for 16 layers with first layer has the low refractive index. All extinction coefficients are assumed to have zero value. The result for the reflectance as a function of wavelength is shown in figure (3), the merit function value was 0.0349 and the value of ∑ ni di was 2.0920 µm. The resulting reflectance is (50±0.23) % with average value of 50.0% along the whole desired wavelength range. The construction parameters of the layers are shown in Table (1). 55.0 54.5 54.0 16-Layer Film 53.5 53.0 52.5 52.0 R eflec ce (% tan ) 51.5 51.0 50.5 50.0 49.5 49.0 48.5 48.0 47.5 47.0 46.5 46.0 45.5 45.0 0.5 0.6 0.7 0.8 0.9 Wavelengh (um) Figure 3. Splitter coating design for M=16 14 1.0
6.
International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME Table(1): Construction parameters of the antireflection and beam splitter coating Antireflection Coating Layer 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Beam Splitter Coating Refractive Thickness Index µm 1.00 -----------2.20 1.119305 4.20 0.719271 2.20 0.324698 4.20 0.387474 2.20 0.556538 4.20 0.340919 2.20 2.488036 4.20 1.495034 2.20 0.341780 4.20 0.387282 2.20 0.827030 4.20 0.216319 2.20 0.596713 4.20 1.471448 2.20 0.122277 4.00 ------------ Refractive Index 1.00 1.35 2.35 1.35 2.35 1.35 2.35 1.35 2.35 1.35 2.35 1.35 2.35 1.35 2.35 1.35 2.35 Thickness µm ---------0.10811 0.06555 0.00765 0.05713 0.03912 0.12353 0.16354 0.11681 0.02721 0.08659 0.05567 0.09766 0.04615 0.04442 0.05719 0.00859 1.52 ---------- 17 MF 0.0916 0.03495 IV. CONCLUSIONS These results have demonstrated that the new formula of crossover and mutation is a synthesis approach for optical thin-film designs. These formulas can closely cooperate with any other method to improve the overall search performance. The results of two optical coating designs verify that the proposed approach, although some-what slower, is competitive with comparable algorithm to obtain acceptable solutions for different types of coating designs. We believe that the crossover and mutation operations can depend on the value of merit function to produce designs with high quality comparing with other methods. REFERENCES [1] [2] [3] J.Dobrowolski, and R.Kemp, Refinement of optical multilayer systems with different optimization procedures, Applied Optics, 29, 1990,2876-2893. K.Wang, and T. Woo, Genetic algorithms and adaptive optimization, Lecture Notes, Department of Industrial Engineering, University of Washington, USA., 2001. Ioannis G. Tsoulos , Solving constrained optimization problems using a novel genetic algorithm, Applied Mathematics and Computation, 208(1), 2009,273-283. 15
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International Journal of Electrical Engineering and Technology (IJEET), ISSN 0976 – 6545(Print), ISSN 0976 – 6553(Online) Volume 5, Issue 1, January (2014), © IAEME [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] P. Bumroongsri, S.Kheawhom, Simultaneous estimation of thin film thickness and optical properties using two-stage optimization, Journal of Global Optimization 54(3), 2012, 583-597. S. Patel and V. Kheraj, Determination of refractive index and thickness of thin-film from reflectivity spectrum using genetic algorithm, AIP Conference. Proceeding, 1536, 509-510, 2013. P. Berning, Theory and Calculations of Optical Thin Films, In: G. Hass (ed.) Physics of Thin Films, Academic Press, New York , 1963,69-121. J-M Yang and C-Y Kao, A robust evolutionary algorithm for optical thin-film designs, Proceedings of the 2000 Congress on Evolutionary Computation, (CEC00).La Jolla, California, UAS,2000, 978-985. J.A. Dobrowolski, Numerical Methods for Optical Thin Films, Optics and Photonics News, 8 (6), 1997,24-33. S. Martin, J.rivory, and M. Schoeannauer, Synthesis of optical multiplayer systems using genetic algorithms, Applied Optics, 34(13) 1995, 2247-2254. J.Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J.Dobrowolski, F. Goldstein, D. Gustafson, and R. Kemp, Antireflection coatings for Germanium IR Optics: A Comparison of Numerical Design Methods, Applied Optics, 27, 1988, 2832-2840. J-M Yang, and C-Y Kao, Efficient evolutionary algorithm for the thin-film synthesis of inhomogeneous optical coatings, Applied Optics, 40 (19), 2001, 3256-3267. Anwar A. Alsagaf, Yousef Y. Holba, Anwar H. Jarndal and Gamil R. Salman, “The Hybrid Evolutionary Algorithm For Optimal Planning Of Hybrid Woban”, International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 3, Issue 3, 2012, pp. 122 - 138, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. D Kathirvel, N Suriyanarayanan, S Prabahar, S Srikanth and P Rajasekarana, “Optical and AFM Studies of Vacuum Evaporated CDS Thin Films”, International Journal of Electronics and Communication Engineering & Technology (IJECET), Volume 2, Issue 2, 2011, pp. 16 - 22, ISSN Print: 0976- 6464, ISSN Online: 0976 –6472. 16
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