Design of broadband omnidirectional antireflection coatings using ant colony algorithm X. Guo,* H. Y. Zhou, S. Guo, X. X. Luan, W. K. Cui, Y. F. Ma and L. Shi Photonic Device Research Laboratory, School of Electronic Information and Control Engineering, Beijing University of Technology, Beijing 100124, China * [email protected]
Abstract: Optimization method which is based on the ant colony algorithm (ACA) is described to optimize antireflection (AR) coating system with broadband omnidirectional characteristics for silicon solar cells incorporated with the solar spectrum (AM1.5 radiation). It’s the first time to use ACA method for optimizing the AR coating system. In this paper, for the wavelength range from 400 nm to 1100 nm, the optimized threelayer AR coating system could provide an average reflectance of 2.98% for θ+ to 80° and 6.56% for incident angles from 0° to incident angles from Rave 90° . ©2014 Optical Society of America OCIS codes: (310.1210) Antireflection coatings; (310.0310) Thin films; (350.6050) Solar energy.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13.
T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for highperformance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008). D. J. Poxson, M. F. Schubert, F. W. Mont, E. F. Schubert, and J. K. Kim, “Broadband omnidirectional antireflection coatings optimized by genetic algorithm,” Opt. Lett. 34(6), 728–730 (2009). Y. Liu, S. H. Sun, J. Xu, L. Zhao, H. C. Sun, J. Li, W. W. Mu, L. Xu, and K. J. Chen, “Broadband antireflection and absorption enhancement by forming nano-patterned Si structures for solar cells,” Opt. Express 19(S5 Suppl 5), A1051–A1056 (2011). K. Choi, S. H. Park, Y. M. Song, Y. T. Lee, C. K. Hwangbo, H. Yang, and H. S. Lee, “Nano-tailoring the surface structure for the monolithic high-performance antireflection polymer film,” Adv. Mater. 22(33), 3713– 3718 (2010). E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). M. L. Kuo, D. J. Poxson, Y. S. Kim, F. W. Mont, J. K. Kim, E. F. Schubert, and S. Y. Lin, “Realization of a near-perfect antireflection coating for silicon solar energy utilization,” Opt. Lett. 33(21), 2527–2529 (2008). Y. M. Song, S. J. Jang, J. S. Yu, and Y. T. Lee, “Bioinspired parabola subwavelength structures for improved broadband antireflection,” Small 6(9), 984–987 (2010). H. Park, D. Shin, G. Kang, S. Baek, K. Kim, and W. J. Padilla, “Broadband optical antireflection enhancement by integrating antireflective nanoislands with silicon nanoconical-frustum arrays,” Adv. Mater. 23(48), 5796– 5800 (2011). P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012). J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thinlm materials with low refractive index for broadband elimination of fresnel reection,” Nat. Photonics 1, 176 (2007). M. F. Schubert, F. W. Mont, S. Chhajed, D. J. Poxson, J. K. Kim, and E. F. Schubert, “Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm,” Opt. Express 16(8), 5290–5298 (2008). H. Greiner, “Robust optical coating design with evolutionary strategies,” Appl. Opt. 35(28), 5477–5483 (1996). S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34(13), 2247–2254 (1995).
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1137
14. Y. J. Chang and Y. T. Chen, “Broadband omnidirectional antireflection coatings for metal-backed solar cells optimized using simulated annealing algorithm incorporated with solar spectrum,” Opt. Express 19(S4 Suppl 4), A875–A887 (2011). 15. M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73– 81 (1997). 16. M. Dorigo and L. M. Gambardella, “Ant colony system: a cooperative learning approach to the traveling salesman problem,” IEEE T Evolut. Comput. 1, 53 (1997). 17. S. N. Kuan, H. L. Ong, and K. M. Ng, “Solving the feeder bus network design problem by genetic algorithms and ant colony optimization,” Adv. Eng. Softw. 37(6), 351–359 (2006). 18. W. Wang, S. Guo, N. Chang, and W. Yang, “Optimum buckling design of composite stiffened panels using ant colony algorithm,” Compos. Struct. 92(3), 712–719 (2010). 19. H. A. Macleod, Thin-Film Optical Filters, (CRC, Bristol, 2001, Chap. 4). 20. S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008). 21. E. D. Palik, “Doped n-Type Silicon (n-Si),” in Handbook of Optical Constants of Solids, (Academic, 1998). 22. H. B. Duan, The Theory and Application of Ant Colony Algorithm (Science, 2005), Chap. 4.
1. Introduction With the increasing urgent demands in clean energy, solar cells have been gaining much attention in recent years. One of the key problems for solar cells is how to decrease the surface reflection due to the large refractive index discontinuity between the semiconductor and the air. On the basis of destructive interference between the incident and reflected light, the reflection loss of the antireflection (AR) coating system, which was applied on the crystalline silicon solar cells, can be achieved less than 5% for one specific wavelength under the normal incidence [1–3]. Broadband AR coating system over a wide incident angle range are highly desirable for solar cells, which can increase light absorption in the active region up to a factor of 4n2 in a relative wide wavelength range, where n is the refractive index of the material [4, 5]. By now, various approaches for broadband and omnidirectional AR coating system design have been reported. They include the use of multilayer porous films [6], the biomimetic moth’s eye structure [7, 8], subwavelength surface Mie resonators [9], and etc.. Recently, a step-graded graded-refractive-index (GRIN) AR coating system with a refractive index as low as 1.05 has been demonstrated which could eliminate Fresnel reflection [10]. However, it is difficult to optimize the GRIN profiles, because the parameter space generally includes many local minima, which makes it unsuitable to find the local minima for deterministic optimization schemes. To meet this challenge, computational genetic algorithm (GA) [2, 11–13] and simulated annealing algorithm (SA) [14] methods have been applied in order to design optimized GRIN profiles for AR coating system. The ant colony algorithm (ACA) is a heuristic optimization method, which was developed to solve traveling salesman problem (TSP) by Dorigo [15, 16]. The searching mechanism of ACA is based on the ants’ capability of finding the shortest path from a food source to their nest. The global optimum found by ACA is insensitive to the initial values which are often critical in conventional optimization algorithms. ACA has been proved to be a useful technique to solve optimization problems in feeder bus network design [17]. In this paper, according to the demands for the broadband and omnidirectional AR coating system, the iterative method of ACA was applied to optimize the AR coating system for silicon-based solar cells, with the objective of minimizing the average reflectivity over the 400 nm to 1100 nm which can be absorbed by silicon [14] from 0° to 90° of all the incident angle ranges. 2. Optimization algorithm Figure 1 depicts a multilayer structure used in this paper. Each layer in this structure is assumed to be homogeneous and is characterized by its thickness d i with the refractive index
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1138
ni , i = {1, 2, …, N}. A plane wave is incident from the semi-infinite air region with
refractive index n0 . For simplicity, the entire absorption layer is assumed by the bottom silicon substrate with the refractive index nSi .
Fig. 1. Schematic cross section of AR coating system on a silicon substrate for the reflectance calculation by ACA-based method.
Assuming the possible maximal thickness of each layer is d max , and the refractive index range is [ nmin , nmax ], the refractive index of the ith layer can be calculated by [18]: s
ni = ( nmax − nmin )
c
j
i
j =1
2 j −1 + nmin ,
2s − 1
(1)
in which the s-bit binary number Ci = ci1ci2 cis , cij is 0 or 1, j = 1, 2, ..., s. Similarly, the thickness of the ith layer Di the m-bit binary number, can be written as: Di = d i1d i2 d im ,
(2)
in which d i j is 0 or 1, j = 1,2, ..., m. The thickness of the ith layer can be calculated by: m
d i = d max
d j =1
j i
2m −1
2m − 1
.
(3)
Then the n-layer AR coating system can be expressed by a string of binary number: L = C1 D1C2 D2 CN DN .
(4)
As a result, we can use a (Ns + Nm)-bit binary number to describe a film structure as shown in Fig. 1. Assuming s = m = g, the film structure could be simplified to be 2Ng-bit binary number. Then, the L was reordered as follow: L = c11d11c12 d12 ...c1g d1g c21 d 21c22 d 22 ...cij d i j ...cNg d Ng ,
(5)
the value of cij d i j which can be {00, 01, 10, 11}, was expressed by {A, B, C, D}.
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1139
Fig. 2. Illustration of the Ng city-layer system. Each city-layer has A, B, C, D four cities.
According to our ACA coding, the optimization process for the AR coating system is illustrated as Fig. 2, according to the multi city-layer TSP (MCLTSP) model [18]. In Fig. 2, the four cities of A, B, C, D in one column is called one city-layer, and the Ng city-layers form a city-matrix. In the MCLTSP model, the traveler must start the tour from the first city layer to the next city-layer one by one until to the last city-layer. In such an open loop tour, one and only one city could be visited in each city-layer. When the traveler reached the last city, the tour was completed. Each completed tour produced a solution by the MCLTSP. For example, the tour shown in Fig. 2 could be expressed by {A C … B D} to represent an AR coating system { c11 = 0, d11 = 0, c12 = 1, d12 = 0, …, cNg −1 = 0, d Ng −1 = 1, cNg = 1, d Ng = 1}. The reflectance of AR coating system could be changed to the traveled distance in the MCLTSP model. The total traveled distance of one tour was defined by the average θ reflectance, Rave [2], θ Rave =
1 2 λ2 − λ1 π
λ2
θ2
1
1
λ θ
RTE + RTM dθ dλ , 2
(6)
where RTE and RTM were the angle- and the wavelength-dependent reflection coefficients for TE- and TM-polarized light modes [19]. Under such situation, the optimizing process of the minimum reflectance of the AR coating system was changed to search the shortest path among all the travelers. The intensity of trail information, which was used to simulate the pheromone of ants, was denoted between city i in the d th city-layer and city j in the ( d + 1) th layer as τ (d,i, j) , where d = {1, 2, ..., Ng-1}, i = {1, 2, 3, 4}, j = {1, 2, 3, 4}. Since the prior trail information was not available, the intensity matrix of trail information was first initiated as a fixed number τ 0 . All the ants with the number of K were randomly placed in the cities in the first city-layer. For any ant in the city i of the d th city-layer, the probability to visit the city j in the ( d + 1) th city-layer can be written in a formula as follows [18]: arg j=
max [τ ( d , i, j )] P
if q ≤ q 0
otherwise
,
(7)
where q was a random number within [0,1], q0 was an experienced parameter, and P was a random variable selected according to the following probability distribution [18]: P(d , i, j ) =
τ (d , i, j ) . j τ ( d , i, j )
(8)
The pheromone trail now could be expressed by:
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1140
τ (i, j ) = (1 − ρ ) ⋅τ (i, j ) + Δτ (i, j ) + e ⋅ Δτ e (i, j ),
(9)
where ρ was a random number within [0,1], e was the number of elite ants, and the updated amount of pheromone Δτ was K
Δτ (i, j ) = Δτ k (i, j ).
(10)
k =1
For the k th ant, θ Q / Rave Δτ k (i, j ) = 0 θ+ Q / Rave Δτ e (i, j ) = 0
if (i,j) ∈ tour otherwise
if (i,j) ∈ the shortest tour otherwise
(11)
,
,
(12)
θ stood for the traveled distance of the tour of where Q was an experienced parameter, Rave θ+ was for the shortest traveled distance of all the tours. According to the each ant, while Rave local update principle, each ant updated the amount of pheromone on the visited path in its tour while the other pheromone was volatilized to disappear gradually. The updated amount of pheromone Δτ deposited on each visited path by one ant was inversely proportional to the traveled distance of its tour. According to the global update principle, the Δτ e was enhanced amount of pheromone on the shortest path. The calculation procedure using ACA can be described as following with its implementation of the AR coating system optimization, as illustrated in Fig. 3.
Fig. 3. Flow chart for ACA-based broadband omnidirectional AR coating system optimization method.
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1141
The main six steps were: 1. Setting up the parameters and initializing the pheromone trails, 2. Putting the ants to the first city-layer, 3. Each ant must go to the next city through a chosen path in available paths depending on the probability given in Eq. (7), 4. Calculating the thickness and refractive index of the n-layer AR coating system by the θ traveled distance of all ant paths, and getting the value of Rave , 5. According to the local and global update principles, updating the pheromone according to Eqs. (9)-(12), to calculate the shorter travelled distance of the tours for θ the smaller Rave , 6. If the iteration cyclecounter reached the maximum value, stop the process, otherwise θ+ repeat steps 2-5. Rave was obtained when the iteration process finished. 3. Numerical results Before the extensive optimizations of AR coating system for typical crystalline silicon solar cells, comparisons were made between the GA, SA and ACA in AR coating applications with the published theoretical and experimental results, by using the same AR coating system and corresponding refractive indices optimized in Ref [14]. and [20] under the same range of wavelength and incident angle. The refractive index of a bulk crystalline silicon in [21]. was taken into account here. There is no a rigorous theory about how to select the parameters used in ACA-based method. Considering the calculate speed and stability, the number K of θ+ was close to the published results in [14]. and [20] by ants was set by 50 [22]. The Rave θ+ was minimized by fine adjusting the roughly adjusting the parameters, and then the Rave parameters several times. The parameters used in this paper for ACA-based method were θ+ shown in Table 1.The optimized average reflectance Rave using ACA, as shown in Fig. 4(c), was 1.89% for λ = [400, 750] nm and θ = [ 40° , 80° ], as opposed to 4.90% for the same wavelength and incident angle ranges reported in [20]. and 3.54% in [14], as shown in Fig. 4(a) and 4(b), respectively. The detailed layer thickness and performance comparisons were given in Table 2. The thickness of each layer was changed a lot compared with other two calculation results by GA and SA methods. Further, both of the thickness and refractive index of three-layer AR coating system were optimized by ACA method at the same time. The index domain was defined by some practically realizable refractive indices ranging from θ+ was further decreased to 1.05 to 2.66 [2, 10, 14]. The optimized average reflectance Rave 1.68% for λ = [400,750] nm and θ = [ 40° , 80° ], with the detailed structure parameters as shown in Table 3. The calculated reflectance performance was shown in Fig. 4(d). It can be seen from Fig. 4(a) and 4(b) that the AR coating system designed by GA has a high reflectance if the incident angle was larger than 75° , and SA optimized AR coating system has a high reflectance for λ
Abstract: Optimization method which is based on the ant colony algorithm (ACA) is described to optimize antireflection (AR) coating system with broadband omnidirectional characteristics for silicon solar cells incorporated with the solar spectrum (AM1.5 radiation). It’s the first time to use ACA method for optimizing the AR coating system. In this paper, for the wavelength range from 400 nm to 1100 nm, the optimized threelayer AR coating system could provide an average reflectance of 2.98% for θ+ to 80° and 6.56% for incident angles from 0° to incident angles from Rave 90° . ©2014 Optical Society of America OCIS codes: (310.1210) Antireflection coatings; (310.0310) Thin films; (350.6050) Solar energy.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12. 13.
T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for highperformance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008). D. J. Poxson, M. F. Schubert, F. W. Mont, E. F. Schubert, and J. K. Kim, “Broadband omnidirectional antireflection coatings optimized by genetic algorithm,” Opt. Lett. 34(6), 728–730 (2009). Y. Liu, S. H. Sun, J. Xu, L. Zhao, H. C. Sun, J. Li, W. W. Mu, L. Xu, and K. J. Chen, “Broadband antireflection and absorption enhancement by forming nano-patterned Si structures for solar cells,” Opt. Express 19(S5 Suppl 5), A1051–A1056 (2011). K. Choi, S. H. Park, Y. M. Song, Y. T. Lee, C. K. Hwangbo, H. Yang, and H. S. Lee, “Nano-tailoring the surface structure for the monolithic high-performance antireflection polymer film,” Adv. Mater. 22(33), 3713– 3718 (2010). E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982). M. L. Kuo, D. J. Poxson, Y. S. Kim, F. W. Mont, J. K. Kim, E. F. Schubert, and S. Y. Lin, “Realization of a near-perfect antireflection coating for silicon solar energy utilization,” Opt. Lett. 33(21), 2527–2529 (2008). Y. M. Song, S. J. Jang, J. S. Yu, and Y. T. Lee, “Bioinspired parabola subwavelength structures for improved broadband antireflection,” Small 6(9), 984–987 (2010). H. Park, D. Shin, G. Kang, S. Baek, K. Kim, and W. J. Padilla, “Broadband optical antireflection enhancement by integrating antireflective nanoislands with silicon nanoconical-frustum arrays,” Adv. Mater. 23(48), 5796– 5800 (2011). P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012). J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thinlm materials with low refractive index for broadband elimination of fresnel reection,” Nat. Photonics 1, 176 (2007). M. F. Schubert, F. W. Mont, S. Chhajed, D. J. Poxson, J. K. Kim, and E. F. Schubert, “Design of multilayer antireflection coatings made from co-sputtered and low-refractive-index materials by genetic algorithm,” Opt. Express 16(8), 5290–5298 (2008). H. Greiner, “Robust optical coating design with evolutionary strategies,” Appl. Opt. 35(28), 5477–5483 (1996). S. Martin, J. Rivory, and M. Schoenauer, “Synthesis of optical multilayer systems using genetic algorithms,” Appl. Opt. 34(13), 2247–2254 (1995).
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1137
14. Y. J. Chang and Y. T. Chen, “Broadband omnidirectional antireflection coatings for metal-backed solar cells optimized using simulated annealing algorithm incorporated with solar spectrum,” Opt. Express 19(S4 Suppl 4), A875–A887 (2011). 15. M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73– 81 (1997). 16. M. Dorigo and L. M. Gambardella, “Ant colony system: a cooperative learning approach to the traveling salesman problem,” IEEE T Evolut. Comput. 1, 53 (1997). 17. S. N. Kuan, H. L. Ong, and K. M. Ng, “Solving the feeder bus network design problem by genetic algorithms and ant colony optimization,” Adv. Eng. Softw. 37(6), 351–359 (2006). 18. W. Wang, S. Guo, N. Chang, and W. Yang, “Optimum buckling design of composite stiffened panels using ant colony algorithm,” Compos. Struct. 92(3), 712–719 (2010). 19. H. A. Macleod, Thin-Film Optical Filters, (CRC, Bristol, 2001, Chap. 4). 20. S. Chhajed, M. F. Schubert, J. K. Kim, and E. F. Schubert, “Nanostructured multilayer graded-index antireflection coating for Si solar cells with broadband and omnidirectional characteristics,” Appl. Phys. Lett. 93(25), 251108 (2008). 21. E. D. Palik, “Doped n-Type Silicon (n-Si),” in Handbook of Optical Constants of Solids, (Academic, 1998). 22. H. B. Duan, The Theory and Application of Ant Colony Algorithm (Science, 2005), Chap. 4.
1. Introduction With the increasing urgent demands in clean energy, solar cells have been gaining much attention in recent years. One of the key problems for solar cells is how to decrease the surface reflection due to the large refractive index discontinuity between the semiconductor and the air. On the basis of destructive interference between the incident and reflected light, the reflection loss of the antireflection (AR) coating system, which was applied on the crystalline silicon solar cells, can be achieved less than 5% for one specific wavelength under the normal incidence [1–3]. Broadband AR coating system over a wide incident angle range are highly desirable for solar cells, which can increase light absorption in the active region up to a factor of 4n2 in a relative wide wavelength range, where n is the refractive index of the material [4, 5]. By now, various approaches for broadband and omnidirectional AR coating system design have been reported. They include the use of multilayer porous films [6], the biomimetic moth’s eye structure [7, 8], subwavelength surface Mie resonators [9], and etc.. Recently, a step-graded graded-refractive-index (GRIN) AR coating system with a refractive index as low as 1.05 has been demonstrated which could eliminate Fresnel reflection [10]. However, it is difficult to optimize the GRIN profiles, because the parameter space generally includes many local minima, which makes it unsuitable to find the local minima for deterministic optimization schemes. To meet this challenge, computational genetic algorithm (GA) [2, 11–13] and simulated annealing algorithm (SA) [14] methods have been applied in order to design optimized GRIN profiles for AR coating system. The ant colony algorithm (ACA) is a heuristic optimization method, which was developed to solve traveling salesman problem (TSP) by Dorigo [15, 16]. The searching mechanism of ACA is based on the ants’ capability of finding the shortest path from a food source to their nest. The global optimum found by ACA is insensitive to the initial values which are often critical in conventional optimization algorithms. ACA has been proved to be a useful technique to solve optimization problems in feeder bus network design [17]. In this paper, according to the demands for the broadband and omnidirectional AR coating system, the iterative method of ACA was applied to optimize the AR coating system for silicon-based solar cells, with the objective of minimizing the average reflectivity over the 400 nm to 1100 nm which can be absorbed by silicon [14] from 0° to 90° of all the incident angle ranges. 2. Optimization algorithm Figure 1 depicts a multilayer structure used in this paper. Each layer in this structure is assumed to be homogeneous and is characterized by its thickness d i with the refractive index
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1138
ni , i = {1, 2, …, N}. A plane wave is incident from the semi-infinite air region with
refractive index n0 . For simplicity, the entire absorption layer is assumed by the bottom silicon substrate with the refractive index nSi .
Fig. 1. Schematic cross section of AR coating system on a silicon substrate for the reflectance calculation by ACA-based method.
Assuming the possible maximal thickness of each layer is d max , and the refractive index range is [ nmin , nmax ], the refractive index of the ith layer can be calculated by [18]: s
ni = ( nmax − nmin )
c
j
i
j =1
2 j −1 + nmin ,
2s − 1
(1)
in which the s-bit binary number Ci = ci1ci2 cis , cij is 0 or 1, j = 1, 2, ..., s. Similarly, the thickness of the ith layer Di the m-bit binary number, can be written as: Di = d i1d i2 d im ,
(2)
in which d i j is 0 or 1, j = 1,2, ..., m. The thickness of the ith layer can be calculated by: m
d i = d max
d j =1
j i
2m −1
2m − 1
.
(3)
Then the n-layer AR coating system can be expressed by a string of binary number: L = C1 D1C2 D2 CN DN .
(4)
As a result, we can use a (Ns + Nm)-bit binary number to describe a film structure as shown in Fig. 1. Assuming s = m = g, the film structure could be simplified to be 2Ng-bit binary number. Then, the L was reordered as follow: L = c11d11c12 d12 ...c1g d1g c21 d 21c22 d 22 ...cij d i j ...cNg d Ng ,
(5)
the value of cij d i j which can be {00, 01, 10, 11}, was expressed by {A, B, C, D}.
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1139
Fig. 2. Illustration of the Ng city-layer system. Each city-layer has A, B, C, D four cities.
According to our ACA coding, the optimization process for the AR coating system is illustrated as Fig. 2, according to the multi city-layer TSP (MCLTSP) model [18]. In Fig. 2, the four cities of A, B, C, D in one column is called one city-layer, and the Ng city-layers form a city-matrix. In the MCLTSP model, the traveler must start the tour from the first city layer to the next city-layer one by one until to the last city-layer. In such an open loop tour, one and only one city could be visited in each city-layer. When the traveler reached the last city, the tour was completed. Each completed tour produced a solution by the MCLTSP. For example, the tour shown in Fig. 2 could be expressed by {A C … B D} to represent an AR coating system { c11 = 0, d11 = 0, c12 = 1, d12 = 0, …, cNg −1 = 0, d Ng −1 = 1, cNg = 1, d Ng = 1}. The reflectance of AR coating system could be changed to the traveled distance in the MCLTSP model. The total traveled distance of one tour was defined by the average θ reflectance, Rave [2], θ Rave =
1 2 λ2 − λ1 π
λ2
θ2
1
1
λ θ
RTE + RTM dθ dλ , 2
(6)
where RTE and RTM were the angle- and the wavelength-dependent reflection coefficients for TE- and TM-polarized light modes [19]. Under such situation, the optimizing process of the minimum reflectance of the AR coating system was changed to search the shortest path among all the travelers. The intensity of trail information, which was used to simulate the pheromone of ants, was denoted between city i in the d th city-layer and city j in the ( d + 1) th layer as τ (d,i, j) , where d = {1, 2, ..., Ng-1}, i = {1, 2, 3, 4}, j = {1, 2, 3, 4}. Since the prior trail information was not available, the intensity matrix of trail information was first initiated as a fixed number τ 0 . All the ants with the number of K were randomly placed in the cities in the first city-layer. For any ant in the city i of the d th city-layer, the probability to visit the city j in the ( d + 1) th city-layer can be written in a formula as follows [18]: arg j=
max [τ ( d , i, j )] P
if q ≤ q 0
otherwise
,
(7)
where q was a random number within [0,1], q0 was an experienced parameter, and P was a random variable selected according to the following probability distribution [18]: P(d , i, j ) =
τ (d , i, j ) . j τ ( d , i, j )
(8)
The pheromone trail now could be expressed by:
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1140
τ (i, j ) = (1 − ρ ) ⋅τ (i, j ) + Δτ (i, j ) + e ⋅ Δτ e (i, j ),
(9)
where ρ was a random number within [0,1], e was the number of elite ants, and the updated amount of pheromone Δτ was K
Δτ (i, j ) = Δτ k (i, j ).
(10)
k =1
For the k th ant, θ Q / Rave Δτ k (i, j ) = 0 θ+ Q / Rave Δτ e (i, j ) = 0
if (i,j) ∈ tour otherwise
if (i,j) ∈ the shortest tour otherwise
(11)
,
,
(12)
θ stood for the traveled distance of the tour of where Q was an experienced parameter, Rave θ+ was for the shortest traveled distance of all the tours. According to the each ant, while Rave local update principle, each ant updated the amount of pheromone on the visited path in its tour while the other pheromone was volatilized to disappear gradually. The updated amount of pheromone Δτ deposited on each visited path by one ant was inversely proportional to the traveled distance of its tour. According to the global update principle, the Δτ e was enhanced amount of pheromone on the shortest path. The calculation procedure using ACA can be described as following with its implementation of the AR coating system optimization, as illustrated in Fig. 3.
Fig. 3. Flow chart for ACA-based broadband omnidirectional AR coating system optimization method.
#205071 - $15.00 USD (C) 2014 OSA
Received 10 Feb 2014; revised 22 May 2014; accepted 23 May 2014; published 6 Jun 2014 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1137 | OPTICS EXPRESS A1141
The main six steps were: 1. Setting up the parameters and initializing the pheromone trails, 2. Putting the ants to the first city-layer, 3. Each ant must go to the next city through a chosen path in available paths depending on the probability given in Eq. (7), 4. Calculating the thickness and refractive index of the n-layer AR coating system by the θ traveled distance of all ant paths, and getting the value of Rave , 5. According to the local and global update principles, updating the pheromone according to Eqs. (9)-(12), to calculate the shorter travelled distance of the tours for θ the smaller Rave , 6. If the iteration cyclecounter reached the maximum value, stop the process, otherwise θ+ repeat steps 2-5. Rave was obtained when the iteration process finished. 3. Numerical results Before the extensive optimizations of AR coating system for typical crystalline silicon solar cells, comparisons were made between the GA, SA and ACA in AR coating applications with the published theoretical and experimental results, by using the same AR coating system and corresponding refractive indices optimized in Ref [14]. and [20] under the same range of wavelength and incident angle. The refractive index of a bulk crystalline silicon in [21]. was taken into account here. There is no a rigorous theory about how to select the parameters used in ACA-based method. Considering the calculate speed and stability, the number K of θ+ was close to the published results in [14]. and [20] by ants was set by 50 [22]. The Rave θ+ was minimized by fine adjusting the roughly adjusting the parameters, and then the Rave parameters several times. The parameters used in this paper for ACA-based method were θ+ shown in Table 1.The optimized average reflectance Rave using ACA, as shown in Fig. 4(c), was 1.89% for λ = [400, 750] nm and θ = [ 40° , 80° ], as opposed to 4.90% for the same wavelength and incident angle ranges reported in [20]. and 3.54% in [14], as shown in Fig. 4(a) and 4(b), respectively. The detailed layer thickness and performance comparisons were given in Table 2. The thickness of each layer was changed a lot compared with other two calculation results by GA and SA methods. Further, both of the thickness and refractive index of three-layer AR coating system were optimized by ACA method at the same time. The index domain was defined by some practically realizable refractive indices ranging from θ+ was further decreased to 1.05 to 2.66 [2, 10, 14]. The optimized average reflectance Rave 1.68% for λ = [400,750] nm and θ = [ 40° , 80° ], with the detailed structure parameters as shown in Table 3. The calculated reflectance performance was shown in Fig. 4(d). It can be seen from Fig. 4(a) and 4(b) that the AR coating system designed by GA has a high reflectance if the incident angle was larger than 75° , and SA optimized AR coating system has a high reflectance for λ
