Abstract

We have developed a phase optimization method of a quantized kinoform by a genetic algorithm. Because the genetic algorithm inherently deals with discrete values, the quantized phase of the kinoform can be easily estimated. The two-dimensional Fourier kinoform can utilize effectively the periodicity of the discrete Fourier transform in the genetic algorithm. This condition enables us to perform the crossover process that is one of the processes in genetic algorithm without a spatial bandwidth of the kinoform. The optimization has been performed successfully in computer simulation. The optically reconstructed image agrees well with the theoretical one.

© 1995 Optical Society of America

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  14. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).
  15. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

1994 (1)

1991 (1)

1990 (1)

1989 (2)

J. Turunen, A. Vasara, J. Westerholm, Opt. Eng. 28, 1162 (1989).

M. S. Kim, M. R. Feldman, C. C. Guest, Opt. Lett. 14, 545 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

1983 (1)

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

1980 (1)

J. R. Fienup, Opt. Eng. 19, (1980).

1975 (1)

T. Yatagai, M. Takeda, Optik 43, 337 (1975).

1973 (1)

1969 (1)

L. P. Lesem, P. M. Kirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Allebach, J. P.

Amako, J.

Brown, D. R.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Bryngdahl, O.

Cook, A.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Delaney, B.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Feldman, M. R.

Fienup, J. R.

J. R. Fienup, Opt. Eng. 19, (1980).

Gallagher, M. C.

Gellatt, C. D.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Goldberg, D. E.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

Guest, C. C.

Hochmuth, D. H.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Johnson, E. G.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Jordan, J. A.

L. P. Lesem, P. M. Kirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Kathman, A. D.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

Kim, M. S.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Kirsch, P. M.

L. P. Lesem, P. M. Kirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lesem, L. P.

L. P. Lesem, P. M. Kirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Liu, B.

Seldowitz, M. A.

Sonehara, T.

Sweeney, D. W.

Takeda, M.

T. Yatagai, M. Takeda, Optik 43, 337 (1975).

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, Opt. Eng. 28, 1162 (1989).

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, Opt. Eng. 28, 1162 (1989).

Vecchi, M. P.

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, Opt. Eng. 28, 1162 (1989).

Wyrowski, F.

Yatagai, T.

Yoshikawa, N.

Appl. Opt. (6)

IBM J. Res. Dev. (1)

L. P. Lesem, P. M. Kirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Eng. (2)

J. Turunen, A. Vasara, J. Westerholm, Opt. Eng. 28, 1162 (1989).

J. R. Fienup, Opt. Eng. 19, (1980).

Opt. Lett. (1)

Optik (1)

T. Yatagai, M. Takeda, Optik 43, 337 (1975).

Science (1)

S. Kirkpatrick, C. D. Gellatt, M. P. Vecchi, Science 220, 671 (1983).
[CrossRef] [PubMed]

Other (2)

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1989).

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, B. Delaney, in Diffractive and Miniaturized Optics, S. H. Lee, ed., Vol. CR49 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54– 74.

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Figures (4)

Fig. 1
Fig. 1

Arrangement of the crossover process in the two-dimensional representation. A kinoform is the grid area. Areas A, B, C, and D are the selected areas for the crossover process.

Fig. 2
Fig. 2

Value of the cost function E as a function of the iteration number t in three different representations: the lexicographic representation (dotted curve), the two-dimensional representation without periodicity (shaded dashed curve), and the two-dimensional representation with periodicity (solid curve).

Fig. 3
Fig. 3

Reconstructed images of the kinoform in the tth iteration number during the optimization in computer simulation: (a) t = 0, (b) t = 100, (c) t = 1000, (d) t = 5000.

Fig. 4
Fig. 4

Reconstructed images of the kinoform in the tth iteration number corresponding to Fig. 3: (a) t = 0, (b) t = 100, (c) t = 1000, (d) t = 5000.

Equations (3)

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U ( ν , μ ) = A ( ν , μ ) exp [ i θ ( ν , μ ) ] = u ( x , y ) exp [ 2 π i λ f ( ν x + μ y ) ] d x d y ,
I ^ ( x , y ) = | u ^ ( x , y ) | 2 = | A exp [ i θ ^ ( ν , μ ) ] exp [ 2 π i ( ν x + μ y ) d ν d μ ] | 2 .
E = | I 0 ( x , y ) I ^ ( x , y ) | 2 d x d y | I 0 ( x , y ) | 2 d x d y ,

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