Abstract

A hybrid-algorithm has been developed that uses the concept of a rapid recursive mean-squared-error (RMSE) function combined with the annealed, multiphase, on-axis capabilities of iterative discrete on-axis (IDO) encoding. High diffraction efficiency and computational speed are obtained through the use of the RMSE algorithm with a constant weighting coefficient in the error function and an iterative initial process for determining phase codes in the output plane. Results for large even spot arrays are presented, and comparisons are made for diffraction efficiency, spot uniformity, and computation speed between the original IDO and the RMSE-based IDO encoding.

© 1992 Optical Society of America

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References

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  1. M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  2. M. R. Feldman, C. C. Guest, “Iterative discrete on-axis encoding for computer-generated holograms,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), pp. 110–111.
  3. M. R. Feldman, C. C. Guest, “Encoding method for high-efficiency multiple-beam holograms using simulated annealing,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 120.
  4. M. R. Feldman, C. C. Guest, “Interconnect density limitations for free space optical interconnections of electronic circuits,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 89.
  5. I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
    [CrossRef]
  6. M. S. Kim, C. C. Guest, “Simulated annealing algorithm for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
    [CrossRef] [PubMed]
  7. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
  8. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798(1987).
    [CrossRef] [PubMed]
  9. B. K. Jennison, J. P. Allebach, “Analysis of the leakage from computer-generated holograms synthesized by direct binary search,” J. Opt. Soc. Am. A 6, 234–243 (1989).
    [CrossRef]
  10. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A 8, 652–660 (1991).
    [CrossRef]
  11. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
  12. R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 111.

1992 (1)

1991 (1)

1990 (1)

1989 (3)

1987 (1)

1986 (1)

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Allebach, J. P.

Bohachevsky, I. O.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Feldman, M. R.

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Iterative discrete on-axis encoding for computer-generated holograms,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), pp. 110–111.

M. R. Feldman, C. C. Guest, “Encoding method for high-efficiency multiple-beam holograms using simulated annealing,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 120.

M. R. Feldman, C. C. Guest, “Interconnect density limitations for free space optical interconnections of electronic circuits,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 89.

Guest, C. C.

M. S. Kim, C. C. Guest, “Simulated annealing algorithm for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[CrossRef] [PubMed]

M. R. Feldman, C. C. Guest, “Iterative discrete on-axis encoding for computer-generated holograms,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), pp. 110–111.

M. R. Feldman, C. C. Guest, “Interconnect density limitations for free space optical interconnections of electronic circuits,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 89.

M. R. Feldman, C. C. Guest, “Encoding method for high-efficiency multiple-beam holograms using simulated annealing,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 120.

Jennison, B. K.

Johnson, M. E.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Kim, M. S.

Krackhardt, U.

Mait, J. N.

Morrison, R. L.

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 111.

Seldowitz, M. A.

Stein, M. L.

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Streibl, N.

Sweeney, D. W.

Walker, S. L.

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 111.

Appl. Opt. (3)

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

Opt. Eng. (1)

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).

Opt. Lett. (1)

Technometrics (1)

I. O. Bohachevsky, M. E. Johnson, M. L. Stein, “Generalized simulated annealing for function optimization,” Technometrics 28, 209–217 (1986).
[CrossRef]

Other (4)

R. L. Morrison, S. L. Walker, “Binary phase gratings generating even numbered spot arrays,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), p. 111.

M. R. Feldman, C. C. Guest, “Iterative discrete on-axis encoding for computer-generated holograms,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1989), pp. 110–111.

M. R. Feldman, C. C. Guest, “Encoding method for high-efficiency multiple-beam holograms using simulated annealing,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1988), p. 120.

M. R. Feldman, C. C. Guest, “Interconnect density limitations for free space optical interconnections of electronic circuits,” in Digest of Optical Society of America Annual Meeting (Optical Society of America, Washington, D.C., 1987), p. 89.

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Figures (7)

Fig. 1
Fig. 1

(a) 32 × 32 cell, binary-amplitude CGH designed from variable scaling factor RMSE-based DBS with random object phase codes. (b) 4 × 4 spot array with 0.8% diffraction efficiency generated from the CGH in (a) replicated twice in each dimension.

Fig. 2
Fig. 2

(a) 32 × 32 cell, 8-phase-level CGH designed from variable scaling factor RMSE-based IDO with random object phase codes. (b) 4 × 4 spot array with 5.63% diffraction efficiency generated from the CGH in (a) replicated twice in each dimension.

Fig. 3
Fig. 3

(a) 32 × 32 cell, 8-phase-level CGH designed from constant scaling factor RMSE-based IDO with iteratively determined object phase codes. (b) 4 × 4 spot array with 82.4% diffraction efficiency generated from the CGH in (a) replicated twice in each dimension.

Fig. 4
Fig. 4

Number of cells changed and efficiency versus the number of iterations for IDO encoding for a 16 × 16 spot array produced from a 128 × 128 cell CGH with 8 phase levels.

Fig. 5
Fig. 5

(a) 64 × 64 cell, 8-phase-level CGH designed from constant scaling factor RMSE-based IDO with iteratively determined object phase codes. (b) 10 × 10 spot array with 79.0% diffraction efficiency generated from the CGH in (a) replicated twice in each dimension.

Fig. 6
Fig. 6

16 × 16 spot array with 77.1% diffraction efficiency generated from a 128 × 128 cell CGH designed by constant scaling factor RMSE-based IDO.

Fig. 7
Fig. 7

32 × 32 spot array with 75.7% diffraction efficiency generated from a 256 × 256 cell CGH designed by constant scaling factor RMSE-based IDO.

Tables (3)

Tables Icon

Table 1 Error Function Comparison

Tables Icon

Table 2 Object Phase Generation Comparison

Tables Icon

Table 3 IDO Versus RMSE for 8-Phase-Level Even Spot Array Generation

Equations (16)

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H ( x , y ) = k = - K / 2 K / 2 - 1 l = L / 2 L / 2 - 1 H k l rect ( x - k δ x δ x , y - l δ y δ y ) ,
h ( u , v ) δ x 2 δ y 2 K L sinc ( u δ x , v δ y ) × comb ( K u δ x , L v δ y ) k = - K / 2 K / 2 - 1 l = - L / 2 L / 2 - 1 H k l × exp [ i 2 π ( k u δ x + l v δ y ) ] .
h ( u , v ) [ δ x δ y K L sinc ( m K , n L ) h m n ] δ ( u - m K δ x , v - n L δ y ) ,
h m n = 1 K L k = - K / 2 K / 2 - 1 l = - L / 2 L / 2 - 1 H k l exp [ i 2 π ( m k K , n l L ) ] .
h m n i + 1 = h m n i + p K L exp [ i 2 π ( m k K + n l L ) ] .
e = 1 A B ( m , n ) R f m n - λ h m n 2
λ = ( m , n ) R f m n h m n * ( m , n ) R h m n 2 .
e = 1 A B [ ( m , n ) R f m n 2 - | ( m , n ) R f m n h m n * | 2 ( m , n ) R h m n 2 ] .
( m , n ) R f m n h m n i + 1 * = ( m , n ) R f m n h m n i * + p K L ( m , n ) R f m n × exp [ - i 2 π ( m k K + n l L ) ] ,
( m , n ) R h m n i + 1 2 = ( m , n ) R h m n i 2 + A B ( K L ) 2 + 2 p K L Re [ DFT ( r m n h m n i - S ) ( k , l ) + 1 K L s = 1 S p s R k - k s , l - l s ]
h m n i + 1 = h m n i + p K L exp [ i 2 π ( m k K + n l L ) ] ,
( m , n ) R f m n h m n i + 1 * = ( m , n ) R f m n h m n i * + p K L ( m , n ) R f m n × exp [ - i 2 π ( m k K + n l L ) ] ,
( m , n ) R h m n i + 1 2 = ( m , n ) R h m n i 2 + A B ( K L ) 2 p 2 + 2 K l × Re { p * [ DFT ( r m n h m n i - S ) ( k . l ) + 1 K L s = 1 S p s R k - k s , l - l s ] } .
e = 1 A B ( m , n ) R f m n - h m n 2 = 1 A B { ( m , n ) R f m n 2 + λ 2 ( m , n ) R h m n 2 - 2 λ Re [ ( m , n ) R f m n h m n * ] } .
Δ P = p max - p min 2 P avg ,
e = k = - K / 2 K / 2 l = - L / 2 L / 2 ( 1 - H k l 2 ) 2 ,

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