Abstract

The microgenetic algorithm is utilized in the design of dielectric grating structures. To our knowledge, this is the first time that a genetic algorithm has been demonstrated with a small population size, a mixed variable type, and an adaptive mutation scheme. A wide range of grating devices is optimized with the microgenetic algorithm to demonstrate its usefulness and efficiency in diffractive-optic components. Specific examples include fan-out phase gratings, antireflection grating surfaces, resonant fan-out devices, and cascaded grating structures. Conclusions and recommendations for additional efforts are also addressed for increasing the applicability and the convergence rates of the algorithm.

© 1995 Optical Society of America

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
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    [Crossref] [PubMed]
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  4. E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.
  5. K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1196, 289–296 (1989).
    [Crossref]
  6. D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1988), Chap. 3, pp. 59–88.
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    [Crossref]
  8. R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check for the validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
    [Crossref]
  9. M. Moharam, T. Gaylord, “Diffractive analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [Crossref]
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    [Crossref] [PubMed]

1994 (1)

1993 (1)

1992 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

1982 (1)

Abushagar, M. A. G.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

Brown, D. R.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

Cook, A.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

Delaney, W.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

Feldman, M. R.

Gaylord, T.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Goldberg, D.

D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1988), Chap. 3, pp. 59–88.

Hochmuth, D. H.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

Johnson, E.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

Kathman, A.

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

Kathman, A. D.

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

Krishnakumar, K.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1196, 289–296 (1989).
[Crossref]

Magnusson, R.

Michalewicz, Z.

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992), Chap. 2, pp. 31–42.
[Crossref]

Moharam, M.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Morris, G. M.

Petit, R.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check for the validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[Crossref]

Raguin, D. H.

Stack, J.

Tayeb, G.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check for the validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[Crossref]

Wang, S. S.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Other (6)

E. Johnson, A. D. Kathman, D. H. Hochmuth, A. Cook, D. R. Brown, W. Delaney, “Advantages of Genetic Algorithm Optimization Methods in Diffractive Optic Design,” in Diffractive and Miniaturized Optics, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.CR49, 54–74 (1993).

E. Johnson, M. A. G. Abushagar, A. Kathman, “Phase grating optimization using genetic algorithms,” in Optical Design for Photonics, Vol. 9 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), pp. 71–73.

K. Krishnakumar, “Micro-genetic algorithms for stationary and non-stationary function optimization,” in Intelligent Control and Adaptive Systems, G. Rodriguez, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1196, 289–296 (1989).
[Crossref]

D. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, Reading, Mass., 1988), Chap. 3, pp. 59–88.

Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution Programs (Springer-Verlag, Berlin, 1992), Chap. 2, pp. 31–42.
[Crossref]

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check for the validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[Crossref]

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Figures (17)

Fig. 1
Fig. 1

Crossover comparisons for single-, two-, and multiple-point operations.

Fig. 2
Fig. 2

Optical interconnection by means of a fan-out phase grating in a 4f coherent imaging system.

Fig. 3
Fig. 3

Unit-cell chromosome encoding. Each subpixel is encoded into a gene location governed by its position in the unit cell.

Fig. 4
Fig. 4

Resonance gratings for diffraction optimization: (a) single-surface corrugation, (b) dual-surface corrugation.

Fig. 5
Fig. 5

Multilayer slab with dielectric subregions for coupled-mode analysis. The E mode is orthogonal to the plane of incidence, and the H mode is in the plane. The thickness of each modulated sublayer is denoted by ti for the ith sublayer.

Fig. 6
Fig. 6

Diffraction grating chromosome encoding. Three layers are used with five subregions in each modulated sublayer. The variables are subregion dielectric constant (epsilon 1 or 2), thickness of each subregion, and period of the unit cell.

Fig. 7
Fig. 7

Convergence results for the binary-phase-level case (solid curve) and the 16-phase-level case (dashed curve). The resulting phase functions are illustrated in Figs. 8(a) and 8(b).

Fig. 8
Fig. 8

1:16 fan-out phase grating unit cells: (a) binary-phase device, (b) 16-phase-level device. Both structures were optimized with 32 × 32 subcells.

Fig. 9
Fig. 9

Actual 9 × 9 diffraction pattern for the 1:16 fan-out design.

Fig. 10
Fig. 10

Arbitrary 8-phase-level fan-out device: (a) 32 × 32 unit-cell phase function, (b) 9 × 9 diffraction pattern in pixelated form.

Fig. 11
Fig. 11

Arbitrary 8-phase-level fan-out device: (a) 32 × 32 unit-cell phase function, (b) 9 × 9 diffraction pattern in pixelated form.

Fig. 12
Fig. 12

Optimized unit-cell profile for an antireflection structure with three modulated sublayers at normal incidence with n = 4.0.

Fig. 13
Fig. 13

Transmission results for an optimized antireflection structure: (a) spectral comparison between optimized design (solid curve) and effective-medium theory (dashed curve), (b) angle-of-incidence performance for optimized three-layer grating design.

Fig. 14
Fig. 14

Resonance 1:5 fan-out optimization results: (a) resulting unit cell (n = 1.5), (b) convergence comparison between average beam intensity (solid curve) and beam uniformity (dashed curve).

Fig. 15
Fig. 15

Optimized resonance reflector: (a) unit-cell description, (b) spectral response for zero-order reflection at normal incidence.

Fig. 16
Fig. 16

Optimized polarization beam splitter with a cascaded grating structure separated by a glass substrate (n = 1.5).

Fig. 17
Fig. 17

Angle-of-incidence effects for the cascaded grating structure: −1-order DE for E-mode formulation (solid curve) and zero-order DE for H-mode formulation (dashed curve).

Equations (7)

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T ( x , y ) = n = m = δ ( x n N , y m N ) p = N / 2 ( N / 2 ) 1 q = N / 2 ( N / 2 ) 1 rect ( x p , y q ) exp ( j ϕ p q ) ,
T ( m , n ) = sinc ( n N , m N ) p = N / 2 ( N / 2 ) 1 q = N / 2 ( N / 2 ) 1 exp ( j ϕ p q ) × exp [ j 2 π ( n p N + m q N ) ] .
C ( n , m ) = 2 D FFT [ exp ( j ϕ p q ) ] .
DE ( n , m ) = sinc 2 ( n N , m N ) | C ( n , m ) | 2 .
MSE = m n | I ( n , m ) γ DE ( n , m ) | 2 ,
E 1 ( x , y ) = E inc ( x , y ) + n R n exp [ j ( α n x + β n 1 y ) ] , E 2 ( x , y ) = n T n exp [ j ( α n x + β n 2 y ) ] .
MSE = m | I ( m ) DE ( m ) | 2 ,

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