Abstract

The design of diffractive optical elements that incorporate several optical functions in a single element is discussed. The technique used involves iterative optimization. A previous paper is continued, in which initial results with few sampling points were reported. Here new results that involve a large number of sampling points are reported. Because the algorithm is computationally intensive with a large number of data points, the parallel implementation of the algorithm on a MASPAR machine is described. MASPAR is a single-instruction multiple-data machine with 16,384 processors. The computer simulations discussed involve simultaneous wavelength demultiplexing, focusing, and the filtering out of a particular wavelength component. It is shown that satisfactory designs of diffractive optical elements can be achieved by the assignment of only a small number of sampling points on the output plane that adequately specify what is required at each wavelength.

© 1995 Optical Society of America

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References

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  1. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Iterative techniques to integrate different optical functions in a diffractive phase element,” Appl. Opt. 30, 4629–4635 (1991).
    [CrossRef] [PubMed]
  2. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  3. P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
    [CrossRef]
  4. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  5. D. Prongue, H. P. Herzig, R. Dandliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
    [CrossRef] [PubMed]
  6. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
    [CrossRef]
  7. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  8. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  9. M. Kato, K. Sakuda, “Computer-generated holograms: application to intensity variable and wavelength demultiplexing holograms,” Appl. Opt. 31, 630–635 (1992).
    [CrossRef] [PubMed]
  10. Y. Ishii, T. Kubota, “Wavelength demultiplexer in multi-mode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
    [CrossRef] [PubMed]
  11. Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
    [CrossRef]
  12. A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
    [CrossRef]
  13. 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).
  14. G. Yang, B. Gu, X. Tan, M.-P. Chang, B. Dong, O. K. Ersoy, “An iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1639 (1994).
    [CrossRef]
  15. K. E. Batcher, “Design of a massively parallel processor,” IEEE Trans. Comput. C-29, 836–840 (1980).
    [CrossRef]

1994 (1)

1993 (5)

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Y. Ishii, T. Kubota, “Wavelength demultiplexer in multi-mode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
[CrossRef] [PubMed]

Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
[CrossRef]

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

1992 (2)

1991 (1)

1990 (1)

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1988 (1)

1980 (1)

K. E. Batcher, “Design of a massively parallel processor,” IEEE Trans. Comput. C-29, 836–840 (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).

1967 (1)

Amitai, Y.

Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
[CrossRef]

Batcher, K. E.

K. E. Batcher, “Design of a massively parallel processor,” IEEE Trans. Comput. C-29, 836–840 (1980).
[CrossRef]

Bernhardt, M.

Bryngdahl, O.

Chang, M.-P.

Dandliker, R.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

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

Dong, B.

Ehbets, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Ersoy, O. K.

Gale, M. T.

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).

Gu, B.

Herzig, H. P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

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

Ishii, Y.

Kato, M.

Kewitsch, A.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Kjelberg, I.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Kubota, T.

Lohmann, A. W.

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Paris, D. P.

Prongue, D.

Regnaul, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Sakuda, K.

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).

Segev, M.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Tan, X.

Turunen, J.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Wyrowski, F.

Yang, G.

Yariv, A.

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[CrossRef]

Appl. Opt. (5)

IEEE Trans. Comput. (1)

K. E. Batcher, “Design of a massively parallel processor,” IEEE Trans. Comput. C-29, 836–840 (1980).
[CrossRef]

J. Mod. Opt. (3)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode arrays by continuous surface-relief elements,” J. Mod. Opt. 40, 737–645 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

Y. Amitai, “Design of wavelength-division multiplexing/demultiplexing using substrate-mode holographic elements,” Opt. Commun. 98, 24–28 (1993).
[CrossRef]

Opt. Lett. (1)

A. Kewitsch, M. Segev, A. Yariv, “Electric-field multiplexing/demultiplexing of volume holograms in photorefractive media,” Opt. Lett. 7, 534–536 (1993).
[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).

Prog. Opt. (1)

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Schematic diagram of a diffractive optical system.

Fig. 2
Fig. 2

Wavelength-dependent output intensities in the 1-D case with N1 = 16 and Nλ = N2s = 4.

Fig. 3
Fig. 3

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, and N2s = 32.

Fig. 4
Fig. 4

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, and N2s = 64.

Fig. 5
Fig. 5

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, and N2s = 128.

Fig. 6
Fig. 6

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, and N2s = 256.

Fig. 7
Fig. 7

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, and N2s = 8.

Fig. 8
Fig. 8

Total output intensity versus spatial position index n in the 1-D case with N1 = 1024, Nλ = 4, N2s = 8, and λ2 filtered out.

Fig. 9
Fig. 9

Total output intensity versus spatial position index n in the 1-D case with N1 = 256, Nλ = 4, and N2s = 16.

Fig. 10
Fig. 10

Total output intensity versus spatial position index n in the 1-D case with N1 = 256, Nλ = 4, and N2s = 32.

Fig. 11
Fig. 11

Total output intensity versus spatial position index n in the 1-D case with N1 = 256, Nλ = 4, and N2s = 64.

Fig. 12
Fig. 12

Total output intensity versus spatial position index n in the 1-D case with N1 = 256, Nλ = 4, and N2s = 8.

Fig. 13
Fig. 13

Total output intensity versus spatial position index n in the 1-D case with N1 = 256, Nλ = 4, and N2s = 8.

Fig. 14
Fig. 14

Total output intensity versus spatial position indices nx, ny in the 2-D four-beam generator design with N1x = 1024, N1y = 256, Nλ = 4, N2xs = 8, N2y = 256.

Fig. 15
Fig. 15

Total output intensity versus spatial position indices nx, ny in the 2-D four-beam generator design with N1x = 512, N1y = 64, Nλ = 4, N2xs = 8, and N2y = 64.

Fig. 16
Fig. 16

Total output intensity versus spatial position indices nx, ny in the 2-D eight-beam generator design with N1x = 512, N1y = 64, Nλ = 4, N2xs = 8, and N2y = 64.

Tables (2)

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Table 1 SNR in 1-D Simulations

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Table 2 SNR in 2-D Simulations

Equations (15)

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U 1 α = U 1 ( X 1 , λ α ) = ρ 1 ( X 1 , λ α ) exp [ i ϕ 1 ( X 1 , λ α ) ] ,
U 2 α = U 2 ( X 2 , λ α ) = ρ 2 ( X 2 , λ α ) exp [ i ϕ 2 ( X 2 , λ α ) ] ,
U 2 ( X 2 , λ α ) = G ( X 2 , X 1 , λ α ) U 1 ( X 1 , λ α ) d X 1 .
U 2 ( X 2 , λ α ) = G ˆ ( λ α ) U 1 ( X 1 , λ α ) ,
U 1 n ( λ α ) = ρ 1 n ( λ α ) exp ( i ϕ 1 n ) , n = 1 , 2 , 3 , , N 1 ,
U 2 m α = ρ 2 m α exp ( i ϕ 2 m α ) ,
U 2 m α = n = 1 N 1 G m n ( λ α ) U 1 n ( λ α ) , m = 1 , 2 , 3 , , N 2 s , α = 1 , 2 , 3 , , N λ .
D 2 = α [ U 2 α G ˆ ( λ α ) U 1 ( λ α ) ] 2 = α Tr [ U 2 α + + U 2 α U 2 α + G ˆ ( λ α ) U 1 ( λ α ) U 1 + ( λ α ) G ˆ + ( λ α ) U 2 α + U 1 + ( λ α ) G ˆ + ( λ α ) G ˆ ( λ α ) U 1 ( λ α ) ] .
δ ϕ 1 D 2 = 0 , δ ϕ 2 γ D 2 = 0 ,
ϕ 1 n = arg [ j , α G j n * ( λ α ) ρ 2 j α exp ( i ϕ 2 j α ) j n A k j ( λ α , λ α ) ρ 1 j ( λ α ) exp ( i ϕ 1 j ) ρ 1 n ( λ α ) ] for n = 1 , 2 , 3 , , N 1 ,
ϕ 2 n γ = arg [ j G n j ( λ γ ) ρ 1 j ( λ γ ) exp ( i ϕ 1 j ) ] ,
j | ϕ ˜ 1 j ( 0 , m ) ϕ ˜ 1 j ( 0 , m + 1 ) | ε 1 ,
SSE = D 2 γ U 2 γ 2 = γ, n ρ 2 n γ exp [ i ϕ ˜ 2 n γ ( n ) ] j G n j ( λ γ ) ρ 1 j × exp [ i ϕ ˜ 1 j ( m , 0 ) ] 2 / γ, n ρ 2 n γ 2 ,
SSE = γ, n { ρ 2 n γ j G n j ( λ γ ) ρ 1 j exp [ i ϕ ˜ 1 j ( m , 0 ) ] } 2 / γ , n ρ 2 n γ 2 .
SNR = ( 1 / N λ ) α I i d ( α ) ( 1 / N t ) k α I k n ( α ) ,

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