Abstract

We propose a genetic local search algorithm (GLSA) for the optimization design of diffractive optical elements (DOE’s). This hybrid algorithm incorporates advantages of both genetic algorithm (GA) and local search techniques. It appears better able to locate the global minimum compared with a canonical GA. Sample cases investigated here include the optimization design of binary-phase Dammann gratings, continuous surface-relief grating array generators, and a uniform top-hat focal plane intensity profile generator. Two GLSA’s whose incorporated local search techniques are the hill-climbing method and the simulated annealing algorithm are investigated. Numerical experimental results demonstrate that the proposed algorithm is highly efficient and robust. DOE’s that have high diffraction efficiency and excellent uniformity can be achieved by use of the algorithm we propose.

© 1999 Optical Society of America

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  1. V. V. Wong, G. J. Swanson, “Design and fabrication of a Gaussian fan-out optical interconnect,” Appl. Opt. 32, 2502–2511 (1993).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. T. Stone, N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  7. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  8. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  9. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
    [CrossRef]
  10. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [CrossRef]
  11. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.
  12. G. Yang, “Genetic algorithm to the optimal design of diffractive optical elements and its comparison with simulated annealing algorithm,” Acta Opt. Sin. 13, 577–584 (1993).
  13. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  14. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
    [CrossRef]
  15. X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
    [CrossRef]
  16. Z. B. Xu, Y. Gao, “The characteristic analysis and prevention of premature convergence of genetic algorithms,” Science in China E26, 364–375 (1996) (in Chinese).
  17. N. N. Schraudolph, R. K. Belew, “Dynamic parameter encoding for genetic algorithms,” Mach. Learning 9, 9–21 (1992).
    [CrossRef]
  18. A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Simulated-annealing-based genetic algorithm for modeling the optical constants of solids,” Appl. Opt. 36, 7097–7103 (1997).
    [CrossRef]
  19. A. M. S. Zalzala, P. J. Fleming, eds., Genetic Algorithms in Engineering Systems (Institution of Electrical Engineers, London, 1997), Chap. 1, pp. 10–12.
  20. J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
    [CrossRef]
  21. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  22. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
  23. R. L. Morrison, S. L. Walker, T. J. Cloonan, “Beam array generation and holographic interconnections in a free-space optical switching network,” Appl. Opt. 32, 2512–2518 (1993).
    [CrossRef] [PubMed]
  24. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
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    [CrossRef]
  26. P. Ehbets, H. P. Herzig, D. Prongué, M. T. Gale, “High-efficiency continuous surface-relief gratings for two-dimensional array generation,” Opt. Lett. 17, 908–910 (1992).
    [CrossRef] [PubMed]
  27. D. Prongué, H. P. Herzig, R. D. Dändliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
    [CrossRef] [PubMed]
  28. E. Sidick, A. Knoesen, J. N. Mait, “Design and rigorous analysis of high-efficiency array generators,” Appl. Opt. 32, 2599–2605 (1993).
    [CrossRef] [PubMed]
  29. Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
    [CrossRef]
  30. S. N. Dixit, I. M. Thomas, B. W. Woods, A. J. Morgan, M. A. Henesian, P. J. Wegner, H. T. Powell, “Random phase plates for beam smoothing on the Nova laser,” Appl. Opt. 32, 2543–2554 (1993).
    [CrossRef] [PubMed]
  31. X. Deng, X. Liang, Z. Chen, W. Yu, R. Ma, “Uniform illumination of large targets using a lens arrays,” Appl. Opt. 25, 377–381 (1986).
    [CrossRef] [PubMed]
  32. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.

1997

J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
[CrossRef]

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Simulated-annealing-based genetic algorithm for modeling the optical constants of solids,” Appl. Opt. 36, 7097–7103 (1997).
[CrossRef]

1996

Z. B. Xu, Y. Gao, “The characteristic analysis and prevention of premature convergence of genetic algorithms,” Science in China E26, 364–375 (1996) (in Chinese).

1995

1994

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

1993

1992

1991

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1989

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

1988

1986

1984

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

1983

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1953

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Arinaga, S.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Belew, R. K.

N. N. Schraudolph, R. K. Belew, “Dynamic parameter encoding for genetic algorithms,” Mach. Learning 9, 9–21 (1992).
[CrossRef]

Brown, D. R.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Chen, Z.

Cloonan, T. J.

Cook, A.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Dändliker, R. D.

Delaney, B.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Deng, X.

Dixit, S. N.

Djurišic, A. B.

Dong, B. Z.

Ehbets, P.

Elazar, J. M.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Gale, M. T.

Gao, Y.

Z. B. Xu, Y. Gao, “The characteristic analysis and prevention of premature convergence of genetic algorithms,” Science in China E26, 364–375 (1996) (in Chinese).

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

George, N.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Gu, B. Y.

Henesian, M. A.

Herzig, H. P.

Hochmuth, D. H.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Johnson, E. G.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Kathman, A. D.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Kato, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Kitagawa, Y.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Knoesen, A.

Krackhardt, U.

Leger, J. R.

Leung, Y.

J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
[CrossRef]

Liang, X.

Ma, R.

Mait, J. N.

Metropolis, N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Mima, K.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Miyanaga, N.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Morgan, A. J.

Morrison, R. L.

Nakatsuka, M.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Palmieri, F.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

Powell, H. T.

Prongué, D.

Qi, X.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

Rakic, A. D.

Rosenbluth, A. W.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Rosenbluth, M. N.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Schraudolph, N. N.

N. N. Schraudolph, R. K. Belew, “Dynamic parameter encoding for genetic algorithms,” Mach. Learning 9, 9–21 (1992).
[CrossRef]

Sidick, E.

Stone, T.

Streibl, N.

Swanson, G. J.

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Tan, X.

Teller, A. H.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Teller, E.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

Thomas, I. M.

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Veldkamp, W. B.

Walker, S. L.

Wegner, P. J.

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

Wong, V. V.

Woods, B. W.

Wyrowski, F.

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Xu, Z.

J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
[CrossRef]

Xu, Z. B.

Z. B. Xu, Y. Gao, “The characteristic analysis and prevention of premature convergence of genetic algorithms,” Science in China E26, 364–375 (1996) (in Chinese).

Yamanaka, C.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Yang, G.

G. Yang, “Genetic algorithm to the optimal design of diffractive optical elements and its comparison with simulated annealing algorithm,” Acta Opt. Sin. 13, 577–584 (1993).

Yang, G. Z.

Yu, W.

Zhang, J.

J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
[CrossRef]

Acta Opt. Sin.

G. Yang, “Genetic algorithm to the optimal design of diffractive optical elements and its comparison with simulated annealing algorithm,” Acta Opt. Sin. 13, 577–584 (1993).

Appl. Opt.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fanout elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

V. V. Wong, G. J. Swanson, “Design and fabrication of a Gaussian fan-out optical interconnect,” Appl. Opt. 32, 2502–2511 (1993).
[CrossRef] [PubMed]

R. L. Morrison, S. L. Walker, T. J. Cloonan, “Beam array generation and holographic interconnections in a free-space optical switching network,” Appl. Opt. 32, 2512–2518 (1993).
[CrossRef] [PubMed]

S. N. Dixit, I. M. Thomas, B. W. Woods, A. J. Morgan, M. A. Henesian, P. J. Wegner, H. T. Powell, “Random phase plates for beam smoothing on the Nova laser,” Appl. Opt. 32, 2543–2554 (1993).
[CrossRef] [PubMed]

A. B. Djurišić, J. M. Elazar, A. D. Rakić, “Simulated-annealing-based genetic algorithm for modeling the optical constants of solids,” Appl. Opt. 36, 7097–7103 (1997).
[CrossRef]

X. Tan, B. Y. Gu, G. Z. Yang, B. Z. Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
[CrossRef] [PubMed]

X. Deng, X. Liang, Z. Chen, W. Yu, R. Ma, “Uniform illumination of large targets using a lens arrays,” Appl. Opt. 25, 377–381 (1986).
[CrossRef] [PubMed]

T. Stone, N. George, “Hybrid diffractive-refractive lenses and achromats,” Appl. Opt. 27, 2960–2971 (1988).
[CrossRef] [PubMed]

D. Prongué, H. P. Herzig, R. D. Dändliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
[CrossRef] [PubMed]

E. Sidick, A. Knoesen, J. N. Mait, “Design and rigorous analysis of high-efficiency array generators,” Appl. Opt. 32, 2599–2605 (1993).
[CrossRef] [PubMed]

IEEE Trans. Neural Netw.

X. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space: Part I. Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

J. Chem. Phys.

N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys. 21, 1087–1092 (1953).
[CrossRef]

J. Opt. Soc. Am. A

Mach. Learning

N. N. Schraudolph, R. K. Belew, “Dynamic parameter encoding for genetic algorithms,” Mach. Learning 9, 9–21 (1992).
[CrossRef]

Opt. Commun.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1176 (1989).
[CrossRef]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. Lett.

Y. Kato, K. Mima, N. Miyanaga, S. Arinaga, Y. Kitagawa, M. Nakatsuka, C. Yamanaka, “Random phasing of lasers for uniform target acceleration and plasma instability suppression,” Phys. Rev. Lett. 53, 1057–1060 (1984).
[CrossRef]

Rep. Prog. Phys.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Science

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Science in China

Z. B. Xu, Y. Gao, “The characteristic analysis and prevention of premature convergence of genetic algorithms,” Science in China E26, 364–375 (1996) (in Chinese).

J. Zhang, Z. Xu, Y. Leung, “Global annealing genetic algorithm and its convergence analysis,” Science in China E40, 414–424 (1997).
[CrossRef]

Other

A. M. S. Zalzala, P. J. Fleming, eds., Genetic Algorithms in Engineering Systems (Institution of Electrical Engineers, London, 1997), Chap. 1, pp. 10–12.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 5, pp. 85–90.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

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Figures (12)

Fig. 1
Fig. 1

Block diagram of an IFTA.

Fig. 2
Fig. 2

Chromosome encoding for a parameter set that describes the DOE.

Fig. 3
Fig. 3

Flowchart of a GLSA.

Fig. 4
Fig. 4

Flowchart of a local search technique associated with the GLSA for DOE design.

Fig. 5
Fig. 5

One normalized grating period of a binary-phase Dammann grating.

Fig. 6
Fig. 6

Convergence properties of a canonical GA, GLSA1, and GLSA2 for the optimization design of 1 × 15 fan-out binary-phase Dammann gratings, when C = 10 and η d = 0.8. The population size N s , crossover probability p c , and mutation probability p m of the three algorithms are the same: N s = 50, p c = 0.65, p m = 0.01. In GLSA1 α = 0.9, ε = 0.005, N l = 16, and in GLSA2 α = 0.9, ε = 0.005, β = 0.9, χ0 = 0.85, N l = 32.

Fig. 7
Fig. 7

Convergence properties of a canonical GA, GLSA1, and GLSA2 for the optimization design of a 1 × 15 fan-out CSR grating array generator, when C = 10 and η d = 0.95. The population size N s , crossover probability p c , and mutation probability p m of the three algorithms are the same: N s = 50, p c = 0.65, p m = 0.01. In GLSA1 α = 0.94, ε = 0.02, N l = 32, and in GLSA2 α = 0.94, ε = 0.02, β = 0.9, χ0 = 0.85, N l = 32.

Fig. 8
Fig. 8

Phase profile of a 1 × 15 fan-out CSR grating array generator within the length of one grating period.

Fig. 9
Fig. 9

Schematic illustration of a DOE-based uniform target illumination system.

Fig. 10
Fig. 10

Convergence properties of a canonical GA, GLSA1, and GLSA2 for a uniform focal intensity profile generator, when C = 5 and η d = 0.85. The population size N s , crossover probability p c , and mutation probability p m of the three algorithms are the same: N s = 200, p c = 0.65, p m = 0.01. In GLSA1 α = 0.94, ε = 0.05, N l = 128, and in GLSA2 α = 0.94, ε = 0.05, β = 0.9, χ0 = 0.8,N l = 256.

Fig. 11
Fig. 11

Relative focal plane intensity profile generated by a DOE, which is designed by use of a GLSA; the uniformity error within the desired region (0–200 µm) is ∼1%, and the efficiency is 85%.

Fig. 12
Fig. 12

Phase profile of a uniform focal plane intensity generator designed with the GLSA.

Tables (2)

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Table 1 Optimization Design Results of 1 × 15 Fan-Out Binary-Phase Dammann Gratings

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Table 2 Optimization Design Results of CSR Grating Array Generators with Different Spot Numbersa

Equations (26)

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i=x-ab-a 2l.
pi=Fii Fi,
yj=xj+rΔk,
yi=xi, i=1, 2,, N,  ij,
Cost=CRη+σ,
Rη=0,ηηdηd-η,η<ηd,
ηηl=- |Fu|du2- |Fu|2du,
An=1+2 m=1M+1-1m+1Xm, n=01iπnm=1M+1-1m exp-i2πnXm, n0,
In=|An|2,
η=k=-NN Ik,  σ=k=-NNIˆ-Ik2Iˆ22N+11/2,
Iˆ=k=-NN Ik/2N+1.
minX1,X2,...,XMCostX1, X2,, XM,  0=X0<X1<<XM<XM+1=1.
Xk=j=1k tjj=1M+1 tj,  k=1, 2,, M.
Tx=n=-N1N2 an expjϕnexpj2πnx=rxexpjϕx,
An=01expjϕxexp-j2πnxdx,
In=|An|2,  -N1nN2.
η=n=-N1N2 In,  σ=n=-N1N2Iˆ-In2Iˆ2N1+N2+11/2,
Iˆ=k=-N1N2 Ik/N1+N2+1.
0.5an1.5,  0ϕn2π,  n=0, 1,, N.
ϕn=ϕn+2kπ,
Ur=2πiλfexpi2πf/λexpiπr2λf0d/2×J02πρrλfexpiϕρρdρ,
Um=Urm=2πiλfexpi2πf/λexpiπrm2λf×n=1Nexpiϕnρn-1ρn J02πρrmλfρdρ,
Im=Irm=|Um|2,  m=1, 2,, M,
η=2π 0δ/2 Irrdrπd24,
σ=m=1MIˆ-Im2Iˆ2M1/2.
Iˆ=m=1M Im/M.

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