Abstract

We compare the performance of very fast simulated quenching; generalized simulated quenching, which unifies classical Boltzmann simulated quenching and Cauchy fast simulated quenching; and variable step size simulated quenching. The comparison is carried out by applying these algorithms to the design of diffractive optical elements for beam shaping of monochromatic, spatially incoherent light to a tightly focused image spot, whose central lobe should be smaller than the geometrical-optics limit. For generalized simulated quenching we choose values of visiting and acceptance shape parameters recommended by other investigators and use both a one-dimensional and a multidimensional Tsallis random number generator. We find that, under our test conditions, variable step size simulated quenching, which generates each parameter's new states based on the acceptance ratio instead of a certain theoretical probability distribution, produces the best results. Finally, we demonstrate experimentally a tightly focused image spot, with a central lobe 0.22–0.68 times the geometrical-optics limit and a relative sidelobe intensity 55%–60% that of the central maximum intensity.

© 2008 Optical Society of America

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  1. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788-2798 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. M. J. Thomson and M. R. Taghizadeh, “Diffractive elements for high-power fibre coupling applications,” J. Mod. Opt. 50, 1691-1699 (2003).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. L. Ingber, “Very fast simulated reannealing,” Math. Comput. Modell. 12, 967-973 (1989).
    [CrossRef]
  9. L. Ingber, “Simulated annealing--practice versus theory,” Math. Comput. Modell. 18, 29-57 (1993).
    [CrossRef]
  10. L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 12, 967-973 (1995).
  11. I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
    [CrossRef]
  12. J. M. Sutter and J. H. Kalivas, “Convergence of generalized simulated annealing with variable step size with application toward parameter estimations of linear and nonlinear models,” Anal. Chem. 63, 2383-2386 (1991).
    [CrossRef]
  13. C. Tsallis and D. A. Stariolo, “Generalized simulated annealing,” Physica A 233, 395-406 (1996).
    [CrossRef]
  14. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Arial. Mach. Intell. 6, 721-741 (1984).
    [CrossRef]
  15. J. N. Gillet and Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456-3465 (2000).
    [CrossRef]
  16. T. Schanze “An exact D-dimensional Tsallis random number generator for generalized simulated annealing,” Comput. Phys. Commun. 175, 708-712 (2006).
    [CrossRef]
  17. J. S. Liu, A. J. Caley, and M. R. Taghizadeh, “Diffractive optical elements for beam shaping of monochromatic spatially incoherent light,” Appl. Opt. 45, 8440-8447 (2006).
    [CrossRef] [PubMed]
  18. M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
    [CrossRef]
  19. E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
    [CrossRef]
  20. C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
    [CrossRef]
  21. Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
    [CrossRef]
  22. E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
    [CrossRef]
  23. A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
    [CrossRef]
  24. D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).
  25. J. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31, 1181-1183 (2006).
    [CrossRef] [PubMed]
  26. H. Nishihara and I. Suhara, “Micro Fresnel Lenses,” Prog. Opt. 24, 1-37 (1987).
    [CrossRef]
  27. T. J. Suleski and D. C. O'Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. 34, 7507-7517 (1995).
    [CrossRef] [PubMed]
  28. S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747-1754 (2003).
    [CrossRef]
  29. Z. Zhai and B. Zhao, “Diffraction intensity distribution of an axicon illuminated by polychromatic light,” J. Opt. A , Pure Appl. Opt. 9, 862-867 (2007).
    [CrossRef]

2007 (2)

2006 (4)

J. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31, 1181-1183 (2006).
[CrossRef] [PubMed]

T. Schanze “An exact D-dimensional Tsallis random number generator for generalized simulated annealing,” Comput. Phys. Commun. 175, 708-712 (2006).
[CrossRef]

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, “Diffractive optical elements for beam shaping of monochromatic spatially incoherent light,” Appl. Opt. 45, 8440-8447 (2006).
[CrossRef] [PubMed]

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

2005 (2)

C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
[CrossRef]

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

2004 (2)

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).

2003 (2)

S. Teng, L. Liu, J. Zu, Z. Luan, and D. Liu, “Uniform theory of the Talbot effect with partially coherent light illumination,” J. Opt. Soc. Am. A 20, 1747-1754 (2003).
[CrossRef]

M. J. Thomson and M. R. Taghizadeh, “Diffractive elements for high-power fibre coupling applications,” J. Mod. Opt. 50, 1691-1699 (2003).

2000 (2)

1997 (1)

Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
[CrossRef]

1996 (2)

M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
[CrossRef]

C. Tsallis and D. A. Stariolo, “Generalized simulated annealing,” Physica A 233, 395-406 (1996).
[CrossRef]

1995 (3)

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 12, 967-973 (1995).

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

T. J. Suleski and D. C. O'Shea, “Gray-scale masks for diffractive-optics fabrication: I. Commercial slide imagers,” Appl. Opt. 34, 7507-7517 (1995).
[CrossRef] [PubMed]

1993 (1)

L. Ingber, “Simulated annealing--practice versus theory,” Math. Comput. Modell. 18, 29-57 (1993).
[CrossRef]

1991 (1)

J. M. Sutter and J. H. Kalivas, “Convergence of generalized simulated annealing with variable step size with application toward parameter estimations of linear and nonlinear models,” Anal. Chem. 63, 2383-2386 (1991).
[CrossRef]

1989 (1)

L. Ingber, “Very fast simulated reannealing,” Math. Comput. Modell. 12, 967-973 (1989).
[CrossRef]

1987 (3)

H. Szu and R. Hartley, “Fast simulated annealing,” Phys. Lett. A 122, 157-162 (1987).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788-2798 (1987).
[CrossRef] [PubMed]

H. Nishihara and I. Suhara, “Micro Fresnel Lenses,” Prog. Opt. 24, 1-37 (1987).
[CrossRef]

1984 (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Arial. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

1983 (1)

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

Allebach, J. P.

Burke, G. J.

Caley, A. J.

Castilho, C. M. C. D.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

Chang, C. W.

Chen, Z. G.

M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
[CrossRef]

Correia, E. R.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

D. Carvalho, V. E.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

Dall'Igna, A.

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

Dardenne, L. E.

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

Dawson, M. D.

C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
[CrossRef]

Esperidiao, A. S. C.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

Fan, W.

Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
[CrossRef]

Gelatt, C. D.

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

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Arial. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Arial. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

Gillet, J. N.

Gong, X. G.

Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
[CrossRef]

Gu, E.

C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
[CrossRef]

H. Park, C.

D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).

Hartley, R.

H. Szu and R. Hartley, “Fast simulated annealing,” Phys. Lett. A 122, 157-162 (1987).
[CrossRef]

Herzig, H. P.

H. P. Herzig, Micro-optics. Elements, Systems and Applications (Taylor & Francis, 1997).

Ingber, L.

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 12, 967-973 (1995).

L. Ingber, “Simulated annealing--practice versus theory,” Math. Comput. Modell. 18, 29-57 (1993).
[CrossRef]

L. Ingber, “Very fast simulated reannealing,” Math. Comput. Modell. 12, 967-973 (1989).
[CrossRef]

Jeon, C. W.

C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
[CrossRef]

Kalivas, J. H.

J. M. Sutter and J. H. Kalivas, “Convergence of generalized simulated annealing with variable step size with application toward parameter estimations of linear and nonlinear models,” Anal. Chem. 63, 2383-2386 (1991).
[CrossRef]

Kip, D.

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

Kirkpatrick, S.

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

Lam, K. S.

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

Lane, R. G.

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

Langer, M.

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

Lee, J. S.

D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).

Li, J.

Liew, S. C.

Liu, D.

Liu, J. S.

Liu, L.

Luan, Z.

Mitchell, M.

M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
[CrossRef]

Morrill, S. M.

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

Mundim, K. C.

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

Nam, D.

D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).

Nascimento, V. B.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

Nishihara, H.

H. Nishihara and I. Suhara, “Micro Fresnel Lenses,” Prog. Opt. 24, 1-37 (1987).
[CrossRef]

O'Shea, D. C.

Rosen, I. I.

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

Salinas, S. V.

Schanze, T.

T. Schanze “An exact D-dimensional Tsallis random number generator for generalized simulated annealing,” Comput. Phys. Commun. 175, 708-712 (2006).
[CrossRef]

Segev, M.

M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
[CrossRef]

Seldowitz, M. A.

Shandarov, V.

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

Sheng, Y.

Shih, M. F.

M. Mitchell, Z. G. Chen, M. F. Shih, and M. Segev “Self-trapping of partially spatially incoherent light,” Phy. Review. Lett. 77, 490-493 (1996).
[CrossRef]

Silva, R. S.

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

Smirnov, E.

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

Soares, E. A.

E. R. Correia, V. B. Nascimento, C. M. C. D. Castilho, A. S. C. Esperidiao, E. A. Soares, and V. E. D. Carvalho, “The generalized simulated annealing algorithm in the low energy electron diffraction search problem,” J. Phys.: Condens. Matter 17, 1-16 (2005).
[CrossRef]

Stariolo, D. A.

C. Tsallis and D. A. Stariolo, “Generalized simulated annealing,” Physica A 233, 395-406 (1996).
[CrossRef]

Stepic, M.

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

Suhara, I.

H. Nishihara and I. Suhara, “Micro Fresnel Lenses,” Prog. Opt. 24, 1-37 (1987).
[CrossRef]

Suleski, T. J.

Sun, D. Y.

Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
[CrossRef]

Sutter, J. M.

J. M. Sutter and J. H. Kalivas, “Convergence of generalized simulated annealing with variable step size with application toward parameter estimations of linear and nonlinear models,” Anal. Chem. 63, 2383-2386 (1991).
[CrossRef]

Sweeney, D. W.

Szu, H.

H. Szu and R. Hartley, “Fast simulated annealing,” Phys. Lett. A 122, 157-162 (1987).
[CrossRef]

Taghizadeh, M. R.

J. S. Liu, A. J. Caley, and M. R. Taghizadeh, “Diffractive optical elements for beam shaping of monochromatic spatially incoherent light,” Appl. Opt. 45, 8440-8447 (2006).
[CrossRef] [PubMed]

M. J. Thomson and M. R. Taghizadeh, “Diffractive elements for high-power fibre coupling applications,” J. Mod. Opt. 50, 1691-1699 (2003).

Teng, S.

Thompson, C. A.

Thomson, M. J.

M. J. Thomson and M. R. Taghizadeh, “Diffractive elements for high-power fibre coupling applications,” J. Mod. Opt. 50, 1691-1699 (2003).

Tsallis, C.

C. Tsallis and D. A. Stariolo, “Generalized simulated annealing,” Physica A 233, 395-406 (1996).
[CrossRef]

Vecchi, M. P.

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

Webb, K. J.

White, D. A.

Xiang, Y.

Y. Xiang, D. Y. Sun, W. Fan, and X. G. Gong, “Generalized simulated annealing and its application to the Thomson model,” Phys. Lett. A 233, 216-220 (1997).
[CrossRef]

Zhai, Z.

Z. Zhai and B. Zhao, “Diffraction intensity distribution of an axicon illuminated by polychromatic light,” J. Opt. A , Pure Appl. Opt. 9, 862-867 (2007).
[CrossRef]

Zhao, B.

Z. Zhai and B. Zhao, “Diffraction intensity distribution of an axicon illuminated by polychromatic light,” J. Opt. A , Pure Appl. Opt. 9, 862-867 (2007).
[CrossRef]

Zu, J.

Anal. Chem. (1)

J. M. Sutter and J. H. Kalivas, “Convergence of generalized simulated annealing with variable step size with application toward parameter estimations of linear and nonlinear models,” Anal. Chem. 63, 2383-2386 (1991).
[CrossRef]

Appl. Opt. (6)

Appl. Phy. B (1)

E. Smirnov, M. Stepic, V. Shandarov, and D. Kip “Pattern formation by spatially incoherent light in a nonlinear ring cavity,” Appl. Phy. B 85, 135-138 (2006).
[CrossRef]

Appl. Phy. Lett. (1)

C. W. Jeon, E. Gu, and M. D. Dawson, “Mask-free photolightographic exposure using a matrix-addressable micropixellated AlInGaN ultraviolet light-emitting diode,” Appl. Phy. Lett. 86, 221105 (2005).
[CrossRef]

Comput. Phys. Commun. (1)

T. Schanze “An exact D-dimensional Tsallis random number generator for generalized simulated annealing,” Comput. Phys. Commun. 175, 708-712 (2006).
[CrossRef]

Contr. Cybernet. (1)

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 12, 967-973 (1995).

Genet. Mol. Biol. (1)

A. Dall'Igna, R. S. Silva, K. C. Mundim, and L. E. Dardenne, “Performance and parametrization of the algorithmn Simplified Generalized simulated annealing,” Genet. Mol. Biol. 27, 616-622 (2004).
[CrossRef]

IEEE Trans. Pattern. Arial. Mach. Intell. (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern. Arial. Mach. Intell. 6, 721-741 (1984).
[CrossRef]

IEICE Trans. Inf. Syst. (1)

D. Nam, J. S. Lee, and C. H. Park, “n-dimensional Cauchy neighbor generation for the fast simulated annealing,” IEICE Trans. Inf. Syst. E87-D, 2499-2502 (2004).

Int. J. Radiat. Oncol. (1)

I. I. Rosen, K. S. Lam, R. G. Lane, M. Langer, and S. M. Morrill, “Comparison of simulated annealing algorithms for conformal therapy treatment planning,” Int. J. Radiat. Oncol. , Biol., Phy. 33, 1091-1099 (1995).
[CrossRef]

J. Mod. Opt. (1)

M. J. Thomson and M. R. Taghizadeh, “Diffractive elements for high-power fibre coupling applications,” J. Mod. Opt. 50, 1691-1699 (2003).

J. Opt. A (1)

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Figures (8)

Fig. 1
Fig. 1

Normalized exponential cooling scheme and GSQ's cooling scheme. The initial temperature at the first time step of all cooling schemes has been normalized to 1. Exponential cooling allows the temperature to reduce more slowly at the beginning of the iteration process compared with GSQ's cooling scheme.

Fig. 2
Fig. 2

Test problem used for comparison of SQs: Fraunhofer-type configuration for beam shaping of monochromatic, spatially incoherent light to a tightly focused image spot whose central lobe should be smaller than the geometrical optics limit.

Fig. 3
Fig. 3

(Color online) Initial temperature profiles used for VFSA. (a) Uniform temperature profile, (b) lower temperatures correspond to more sensitive pixel, (c) higher temperatures correspond to more sensitive pixels.

Fig. 4
Fig. 4

Relative sidelobe intensity produced by DOEs designed by VSSQ with different m values.

Fig. 5
Fig. 5

(Color online) (a) Change in the step size factor h of the most sensitive pixels when different m values are used. (b) Corresponding change in the cost function, i.e., relative sidelobe intensity K.

Fig. 6
Fig. 6

(Color online) Cross-sectional structures of the designed and fabricated DOEs: (a) designed by S-VSSQ, (b) magnified structure of (a), (c) designed by M-VSSQ, (d) magnified structure of (c), (e) measured depth profile of a DOE shown in (d).

Fig. 7
Fig. 7

(Color online) Spectrum of the LED light used in the test. The peak wavelength of the LED light is 637.9   nm .

Fig. 8
Fig. 8

(Color online) Simulated (solid curves) and experimental (dashed curves) intensity profiles generated by the DOE designed by M-VSSQ, whose profiles are shown in Figs. 6(c)–6(e). (a) Results with diffused 633   nm He–Ne laser beam, (b) results with LED light.

Tables (1)

Tables Icon

Table 1 Parameters Used in Tsallis's GSQ

Equations (17)

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T ( k ) = T ( 1 ) ln ( 2 ) ln ( 1 + k ) ,
T ( k ) = T ( 1 ) 1 k ,
T ( k ) = T ( 1 ) 2 q 1 1 ( 1 + k ) q 1 1 ,
T ( k ) i = T ( 1 ) i exp ( c i k 1 / D ) ,
T ( k ) = T ( 1 ) exp ( cr k ) ,
I out ( y ) = [ U out ( y ) ] 2 = β = w / 2 β = w / 2 | { U 2 _ ind ( y , z in ) exp [ j ϕ DOE ] } | 2 ,
U 2 _ ind ( y , z in ) = C exp { j k z in 1 + ( y β z i n ) 2 } z i n { 1 + ( y β z i n ) 2 } 5 / 4 ,
K = max [ I o u t ( y y signal ) ] max [ I o u t ( y y signal ) ] ,
G = size signal size geometrical   image = size signal size object × z out / z in .
P ( ψ k i ψ k + 1 i ) = { 1 , i f   E ( ψ k + 1 i ) < E ( ψ k i ) exp { [ E ( ψ k + 1 i ) E ( ψ k i ) ] / T a i } , if   E ( ψ k + 1 i ) E ( ψ k i ) ,
ψ k + 1 i = ψ k i + c × y i × 2 π ,
y i = sgn ( u i 1 2 ) T v i [ ( 1 + 1 / T v i ) | 2 u i 1 | 1 ] ,
ψ k + 1 i = ψ k i + h j i × u i × 2 π ,
ψ k + 1 = ψ k + Δ x k × 2 π ,
g q v ( Δ x k ) = ( q v 1 π ) D / 2 Γ ( 1 q v 1 + D 1 2 ) Γ ( 1 q v 1 + D 1 2 ) × [ T v , q v ] D / ( 3 q v ) [ 1 + ( q v 1 ) ( Δ x ) 2 [ T v , q v ] 2 / ( 3 q v ) ] 1 / ( q v 1 ) + ( D 1 ) / 2 .
P q a ( x k x k + 1 ) = { 1 if   E ( x k + 1 ) < E ( x k ) 1 [ 1 + ( q a 1 ) ( E ( x k + 1 ) E ( x k ) ) / T a , q a ] 1 ^ / ( q a 1 ) if   E ( x k + 1 ) E ( x k ) .
ϕ ( y ) lens = 2 π λ f 2 + y 2 ,

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