Abstract

Using a novel genetic algorithm (GA) with a Lamarckian search we optimize the polygonal layout of a new type of multiplexed computer-generated hologram (MCGH) with polygonal apertures. A period of the MCGH is divided into cells, and the cell is further divided into polygonal apertures according to a polygonal layout, which is to be optimized. Among an ensemble of 1.21 × 1024 possible polygonal layouts, we take a population of 102 solutions, which are coded as chromosomes of bits, and find the optimal solution with our GA. We introduce rank-based selection with cumulative normal distribution fitness, double crossover, exponentially decreasing mutation probability and Lamarckian downhill search with a small number of offspring chromosomes into our GA, which shows a rapid convergence to the global minimum of the cost function. In a second step of optimization the phase distributions over the subholograms in the MCGH are determined with our iterative subhologram design algorithm. Our MCGH designs show large-size reconstructed images with high diffraction efficiency and low reconstruction error.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2002 (2)

2000 (1)

1999 (3)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

E. E. Agoston, R. Hinterding, Z. Michalewicz, “Parameter control in evolutionary algorithms,” IEEE Trans. Comput. 3, 124–141 (1999).
[CrossRef]

G. Zhou, Y. Chen, Z. Wang, H. Song, “Genetic local search algorithm for optimization design of diffractive optical elements,” Appl. Opt. 38, 4281–4290 (1999).
[CrossRef]

1997 (1)

1995 (1)

P. Charbonneau, “Genetic algorithms in astronomy and astrophysics,” The Astrophysical Journal Supplement Series 101, 309–334 (1995).
[CrossRef]

1993 (1)

1992 (2)

1990 (1)

1988 (1)

1982 (1)

1980 (1)

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

1972 (1)

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

Agoston, E. E.

E. E. Agoston, R. Hinterding, Z. Michalewicz, “Parameter control in evolutionary algorithms,” IEEE Trans. Comput. 3, 124–141 (1999).
[CrossRef]

Brede, J.

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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Bryngdahl, O.

Chang, M. P.

Charbonneau, P.

P. Charbonneau, “Genetic algorithms in astronomy and astrophysics,” The Astrophysical Journal Supplement Series 101, 309–334 (1995).
[CrossRef]

Chen, Y.

Cheng, R.

M. Gen, R. Cheng, Genetic Algorithms and Engineering Design (Wiley, New York, 1997).

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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Cummings, M. R.

W. S. Klug, M. R. Cummings, Concepts of Genetics, 6th ed. (Prentice Hall, Upper Saddle River, N.J., 2000).

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, Vol. 41 of Topics in Applied Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.

Darwin, C.

C. Darwin, The Origin of Species, 6th ed., 1872 (The New American Library of World Literature, New York, 1958; with a special Introduction by J. Huxley).

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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Dial, O.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

Ersoy, O. K.

Fienup, J. R.

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

Gao, X.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

Gen, M.

M. Gen, R. Cheng, Genetic Algorithms and Engineering Design (Wiley, New York, 1997).

Gerchberg, R. W.

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

Gillet, J.-N.

J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated holograms with irregular-shaped polygonal apertures and discrete phase levels,” J. Opt. Soc. Am. A 19, 2403–2413 (2002).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated hologram with polygonal apertures,” Appl. Opt. 41, 298–307 (2002).
[CrossRef] [PubMed]

J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoïdales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. dissertation (Université Laval, Québec, PQ, Canada, 2001), Chap. 1 and 2, pp. 6–62.

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoïdales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. dissertation (Université Laval, Québec, PQ, Canada, 2001), Chap. 3, pp. 63–93.

Goldberg, D. E.

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

Hancock, P. J. B.

P. J. B. Hancock, “An empirical comparison of selection methods in evolutionary algorithms,” in Evolutionary Computing, Lecture Notes in Computer Science 865, T. C. Fogarty, ed. (Springer-Verlag, Berlin, 1994), pp. 80–94.
[CrossRef]

Haupt, R. L.

R. L. Haupt, S. E. Haupt, Practical Genetic Algorithms (Wiley, New York, 1998).

Haupt, S. E.

R. L. Haupt, S. E. Haupt, Practical Genetic Algorithms (Wiley, New York, 1998).

Hinterding, R.

E. E. Agoston, R. Hinterding, Z. Michalewicz, “Parameter control in evolutionary algorithms,” IEEE Trans. Comput. 3, 124–141 (1999).
[CrossRef]

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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Holland, J. H.

J. H. Holland, “Genetic algorithms,” Scientific American, July1992, pp. 66–72.

J. H. Holland, Adaptation in Natural and Artificial System (Univ. of Michigan Press, Ann Arbor, Mich., 1975).

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., CR49 of SPIE Critical Review Series (SPIE, 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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Klug, W. S.

W. S. Klug, M. R. Cummings, Concepts of Genetics, 6th ed. (Prentice Hall, Upper Saddle River, N.J., 2000).

Komrska, J.

Lamarck, J.-B.-P.-A.

J.-B.-P.-A. Lamarck, Philosophie Zoologique, 1809 (GF-Flammarion, Paris, 1994; with Presentation and Notes by A. Pichot).

Lee, C. K.

Liao, H. Z.

Lu, C. Y.

Mait, J. N.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

McCord, M. A.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph39 and IEE Materials and Devices Series12 (SPIE Press, Bellingham, WA, 1997), Chap. 2, pp. 139–250.

Michalewicz, Z.

E. E. Agoston, R. Hinterding, Z. Michalewicz, “Parameter control in evolutionary algorithms,” IEEE Trans. Comput. 3, 124–141 (1999).
[CrossRef]

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992).
[CrossRef]

Prather, D. W.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

Rooks, M. J.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph39 and IEE Materials and Devices Series12 (SPIE Press, Bellingham, WA, 1997), Chap. 2, pp. 139–250.

Saxton, W. O.

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

Scherer, A.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

Sheng, Y.

Song, H.

Straubel, R.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berücksichtigung der Theorie der Beugung in Heliometer (Frommansche, Jena, Germany, 1888).

Wang, J. S.

Wang, Z.

Wyrowski, F.

Zhou, G.

Zhuang, J. Y.

Appl. Opt. (6)

IEEE Trans. Comput. (1)

E. E. Agoston, R. Hinterding, Z. Michalewicz, “Parameter control in evolutionary algorithms,” IEEE Trans. Comput. 3, 124–141 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Opt. Eng. (1)

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

Optik (Stuttgart) (1)

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

Scientific American (1)

J. H. Holland, “Genetic algorithms,” Scientific American, July1992, pp. 66–72.

The Astrophysical Journal Supplement Series (1)

P. Charbonneau, “Genetic algorithms in astronomy and astrophysics,” The Astrophysical Journal Supplement Series 101, 309–334 (1995).
[CrossRef]

Other (16)

C. Darwin, The Origin of Species, 6th ed., 1872 (The New American Library of World Literature, New York, 1958; with a special Introduction by J. Huxley).

J.-B.-P.-A. Lamarck, Philosophie Zoologique, 1809 (GF-Flammarion, Paris, 1994; with Presentation and Notes by A. Pichot).

W. S. Klug, M. R. Cummings, Concepts of Genetics, 6th ed. (Prentice Hall, Upper Saddle River, N.J., 2000).

M. Gen, R. Cheng, Genetic Algorithms and Engineering Design (Wiley, New York, 1997).

R. L. Haupt, S. E. Haupt, Practical Genetic Algorithms (Wiley, New York, 1998).

P. J. B. Hancock, “An empirical comparison of selection methods in evolutionary algorithms,” in Evolutionary Computing, Lecture Notes in Computer Science 865, T. C. Fogarty, ed. (Springer-Verlag, Berlin, 1994), pp. 80–94.
[CrossRef]

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., CR49 of SPIE Critical Review Series (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992).
[CrossRef]

J. H. Holland, Adaptation in Natural and Artificial System (Univ. of Michigan Press, Ann Arbor, Mich., 1975).

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

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, Vol. 41 of Topics in Applied Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berücksichtigung der Theorie der Beugung in Heliometer (Frommansche, Jena, Germany, 1888).

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoïdales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. dissertation (Université Laval, Québec, PQ, Canada, 2001), Chap. 3, pp. 63–93.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series OSA (Optical Society of America, Washington, D.C., 2000), pp. 108–109.

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoïdales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. dissertation (Université Laval, Québec, PQ, Canada, 2001), Chap. 1 and 2, pp. 6–62.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph39 and IEE Materials and Devices Series12 (SPIE Press, Bellingham, WA, 1997), Chap. 2, pp. 139–250.

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Figures (7)

Fig. 1
Fig. 1

One period of an MCGH with M × N = 4 × 4 cells divided into five polygonal apertures assigned to five subholograms. The polygonal layout for the MCGH cells is given in Fig. 2(a).

Fig. 2
Fig. 2

Two polygonal layouts of the MCGH cells: (a) for 5 subholograms, (b) for 8 subholograms. The polygonal layouts were optimized by our GA.

Fig. 3
Fig. 3

Sixteen candidate stripe patterns used to fill a MCGH cell during optimization with our GA.

Fig. 4
Fig. 4

Rank-based fitness functions with a cumulative Gaussian distribution, linear distribution, and exponential distribution for a number of offspring chromosomes N 0 = 100.

Fig. 5
Fig. 5

Evolution of the trapezoidal-aperture layout in a cell of a MCGH with S = 5 subholograms after generations (a) k = 0, (b) 30, (c) 90, (d) 129 of our GA.

Fig. 6
Fig. 6

640 × 640 pixel desired image with 256 gray levels.

Fig. 7
Fig. 7

642 × 642 pixel Fourier image reconstructed by a MCGH with 642 × 642 cells per period (M = N = 642, A = B = 640), consisting of 8 subholograms, the optimal polygonal layout shown in Fig. 2(b) and apertures taking 8 discrete phase levels.

Tables (2)

Tables Icon

Table 1 Cost Function Values and Probability for the Global Minimum in the Proposed GA Versus Other GAs

Tables Icon

Table 2 MCGHs with Optimal Polygonal Layoutsa

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

Tm, n=s=1S Θsm, nFsm, n,
η=m=-A/2A/2-1n=-B/2B/2-1 |Tm, n|2,
|Θsm, n|Θs0, 0
η=m=-A/2A/2-1n=-B/2B/2-1s=1S Θsm, nFsm, n2m=-M/2M/2-1n=-M/2N/2-1s=1S |Θsm, n|2s=1S |Fsm, n|2m=-M/2M/2-1n=-M/2N/2-1s=1S |Θs0, 0|2s=1S |Fsm, n|2=ηmax,
γm, n=s=1SΘs0, 0-|Θsm, n|=1-s=1S |Θsm, n|,
Γ=MN-1m=-M/2M/2-1n=-N/2N/2-1 γm, n=1-s=1SMN-1m=-M/2M/2-1n=-N/2N/2-1 |Θsm, n|1-s=1S |Θsm, n|,
w=w1w2wΔG,
CΦp-1<rCΦp,
ϕp=1-Cgp=1-2πσ-1-pexpy-μ2σ2dy,
Φp=ϕpp=1N0 ϕp,
wα=w1pwxp wx+1qwyq wy+1pwΔpwβ=w1qwxq wx+1pwyp wy+1qwΔq,

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