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 three-layer AR coating system could provide an average reflectance of 2.98% for incident angles from Raveθ+ to 80° and 6.56% for incident angles from 0° to 90°.

© 2014 Optical Society of America

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  1. T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
    [CrossRef] [PubMed]
  2. 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).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. 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).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  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).
    [CrossRef]
  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).
    [CrossRef]
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    [CrossRef]
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  22. H. B. Duan, The Theory and Application of Ant Colony Algorithm (Science, 2005), Chap. 4.

2012

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012).
[CrossRef] [PubMed]

2011

2010

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).
[CrossRef]

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

2009

2008

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

2007

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

2006

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).
[CrossRef]

1997

M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73–81 (1997).
[CrossRef] [PubMed]

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).

1996

1995

1982

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982).
[CrossRef]

Baek, S.

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).
[CrossRef] [PubMed]

Brunner, R.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Chang, N.

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).
[CrossRef]

Chang, Y. J.

Chen, K. J.

Chen, M.

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Chen, Y. T.

Chhajed, S.

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).
[CrossRef]

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).
[CrossRef] [PubMed]

Choi, K.

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).
[CrossRef] [PubMed]

Cody, G. D.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982).
[CrossRef]

Dorigo, M.

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).

M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73–81 (1997).
[CrossRef] [PubMed]

Gambardella, L. M.

M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73–81 (1997).
[CrossRef] [PubMed]

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).

Greiner, H.

Guo, S.

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).
[CrossRef]

Helgert, M.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Hwangbo, C. K.

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).
[CrossRef] [PubMed]

Jang, S. J.

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).
[CrossRef] [PubMed]

Kang, G.

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).
[CrossRef] [PubMed]

Kim, J. K.

Kim, K.

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).
[CrossRef] [PubMed]

Kim, Y. S.

Kuan, S. N.

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).
[CrossRef]

Kuo, M. L.

Lee, H. S.

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).
[CrossRef] [PubMed]

Lee, Y. T.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Li, J.

Lin, S. Y.

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).
[CrossRef] [PubMed]

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Liu, W.

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Liu, Y.

Lohmüller, T.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Martin, S.

Mont, F. W.

Mu, W. W.

Ng, K. M.

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).
[CrossRef]

Ong, H. L.

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).
[CrossRef]

Padilla, W. J.

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).
[CrossRef] [PubMed]

Park, H.

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).
[CrossRef] [PubMed]

Park, S. H.

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).
[CrossRef] [PubMed]

Polman, A.

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012).
[CrossRef] [PubMed]

Poxson, D. J.

Rivory, J.

Schoenauer, M.

Schubert, E. F.

Schubert, M. F.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

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).
[CrossRef]

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Shin, D.

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).
[CrossRef] [PubMed]

Smart, J. A.

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Song, Y. M.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Spatz, J. P.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Spinelli, P.

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012).
[CrossRef] [PubMed]

Sun, H. C.

Sun, S. H.

Sundermann, M.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Verschuuren, M. A.

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012).
[CrossRef] [PubMed]

Wang, W.

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).
[CrossRef]

Xi, J. Q.

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Xu, J.

Xu, L.

Yablonovitch, E.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982).
[CrossRef]

Yang, H.

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).
[CrossRef] [PubMed]

Yang, W.

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).
[CrossRef]

Yu, J. S.

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).
[CrossRef] [PubMed]

Zhao, L.

Adv. Eng. Softw.

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).
[CrossRef]

Adv. Mater.

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).
[CrossRef] [PubMed]

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).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. Lett.

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).
[CrossRef]

Biosystems

M. Dorigo and L. M. Gambardella, “Ant colonies for the travelling salesman problem,” Biosystems 43(2), 73–81 (1997).
[CrossRef] [PubMed]

Compos. Struct.

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).
[CrossRef]

IEEE T Evolut. Comput.

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).

IEEE Trans. Electron. Dev.

E. Yablonovitch and G. D. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE Trans. Electron. Dev. 29(2), 300–305 (1982).
[CrossRef]

Nano Lett.

T. Lohmüller, M. Helgert, M. Sundermann, R. Brunner, and J. P. Spatz, “Biomimetic interfaces for high-performance optics in the deep-UV light range,” Nano Lett. 8(5), 1429–1433 (2008).
[CrossRef] [PubMed]

Nat Commun

P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface mie resonators,” Nat Commun 3, 692 (2012).
[CrossRef] [PubMed]

Nat. Photonics

J. Q. Xi, M. F. Schubert, J. K. Kim, E. F. Schubert, M. Chen, S. Y. Lin, W. Liu, and J. A. Smart, “Optical thin-film materials with low refractive index for broadband elimination of fresnel reflection,” Nat. Photonics 1, 176 (2007).

Opt. Express

Opt. Lett.

Small

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).
[CrossRef] [PubMed]

Other

H. A. Macleod, Thin-Film Optical Filters, (CRC, Bristol, 2001, Chap. 4).

E. D. Palik, “Doped n-Type Silicon (n-Si),” in Handbook of Optical Constants of Solids, (Academic, 1998).

H. B. Duan, The Theory and Application of Ant Colony Algorithm (Science, 2005), Chap. 4.

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Figures (5)

Fig. 1
Fig. 1

Schematic cross section of AR coating system on a silicon substrate for the reflectance calculation by ACA-based method.

Fig. 2
Fig. 2

Illustration of the Ng city-layer system. Each city-layer has A, B, C, D four cities.

Fig. 3
Fig. 3

Flow chart for ACA-based broadband omnidirectional AR coating system optimization method.

Fig. 4
Fig. 4

Simulation results of the reflectance characteristics of theAR coating system designed by (a) GA (b) SA (c) ACA using the same parameters as in [20, 15], (d) ACA (both the thickness and refractive index were optimized) as a function of wavelength from 400nm to 750nm and incident angle from 40 ° to 80 ° .

Fig. 5
Fig. 5

Simulation results of the reflectance performance of optimized AR coating system designed by ACA method as a function of wavelength from 400nm to 1100nm and incident angle from 0 ° to 90 ° .

Tables (4)

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Table 1 Parameters chosen in the optimization using ACA-based method

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Table 2 Thickness (nm) of individual layers of the three-layer AR coating system on silicon designed by different algorithms and the related average reflectivity (incident angle 4 0 ° - 8 0 ° wavelength 400-750 nm)

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Table 3 Structure parameters and average reflectances of the three-layer AR coating system on silicon designed by ACA method (incident angle 4 0 ° - 8 0 ° wavelength 400-750 nm)

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Table 4 Structure parameters and average reflectances of the three-layer AR coating system on silicon designed by ACA method for wavelength from 400 nm to 1100 nm

Equations (11)

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n i =( n max n min ) j=1 s c i j 2 j1 2 s 1 + n min ,
D i = d i 1 d i 2 d i m ,
d i = d max j=1 m d i j 2 m1 2 m 1 .
L= C 1 D 1 C 2 D 2 C N D N .
L= c 1 1 d 1 1 c 1 2 d 1 2 ... c 1 g d 1 g c 2 1 d 2 1 c 2 2 d 2 2 ... c i j d i j ... c N g d N g ,
R ave θ = 1 λ 2 λ 1 2 π λ 1 λ 2 θ 1 θ 2 R TE + R TM 2 dθdλ,
j={ arg max[ τ(d,i,j) ] if q q 0 P otherwise ,
P(d,i,j)= τ(d,i,j) j τ(d,i,j) .
Δτ(i,j)= k=1 K Δ τ k (i,j) .
Δ τ k (i,j)={ Q/ R ave θ if (i,j)tour 0 otherwise ,
Δ τ e (i,j)={ Q/ R ave θ+ if (i,j)the shortest tour 0 otherwise ,

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