Abstract

This paper describes a new iterative algorithm for synthesizing the kinoform so that the Fourier spectrum of an object is leveled by adjusting the information of a dummy area introduced into the object's domain, spatially isolated from the signal area. Theoretical consideration of the effect of the dummy area derives the required size of the dummy area, and computer simulations prove it to be valid, although restricted to an object composed of binary numbers. Also, it is shown that highly efficient use of the incident light is possible to achieve. Experimental results verify the proposed scheme.

© 1986 Optical Society of America

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References

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  1. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  2. C. B. Burckhardt, “Use of a Random Phase Mask for the Recording of Fourier Transform Holograms of Data Masks,” Appl. Opt. 9, 695 (1970).
    [CrossRef] [PubMed]
  3. Y. Takeda, Y. Oshida, Y. Miyamura, “Random Phase Shifters for Fourier Transformed Holograms,” Appl. Opt. 11, 818 (1972).
    [CrossRef] [PubMed]
  4. R. H. Katyl, “Use of Pseudorandom Sequences in the Synthesis of Kinoforms,” Appl. Opt. 11, 198 (1972).
    [CrossRef] [PubMed]
  5. W. C. Stewart, A. H. Firester, E. C. Fox, “Random Phase Data Masks: Fabrication Tolerances and Advantages of Four Phase Level Masks,” Appl. Opt. 11, 604 (1972).
    [CrossRef] [PubMed]
  6. W. J. Dallas, “Deterministic Diffusers for Holography,” Appl. Opt. 12, 1179 (1973).
    [CrossRef] [PubMed]
  7. H. Akahori, “Comparison of Deterministic Phase Coding with Random Phase Coding in Terms of Dynamic Range,” Appl. Opt. 12, 2336 (1973).
    [CrossRef] [PubMed]
  8. P.M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object Dependent Diffuser,” U.S. Patent3,619,022 (1971).
  9. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that reduces Image Reconstruction Error,” Appl. Opt. 12, 2328 (1973).
    [CrossRef] [PubMed]
  10. J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
    [CrossRef]
  11. D. C. Chu, J. W. Goodman, “Spectrum Shaping with Parity Sequences,” Appl. Opt. 11, 1716 (1972).
    [CrossRef] [PubMed]
  12. D. C. Chu, “Spectrum Shaping for Computer Generated Holograms,” Ph.D. Dissertation, Department of Electrical Engineering, Stanford U. (1974).
  13. B. Liu, N. C. Gallagher, “Convergence of a Spectrum Shaping Algorithm,” Appl. Opt. 13, 2470 (1974).
    [CrossRef] [PubMed]
  14. R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).
  15. J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  16. S. D. Conte, C. de Boor, Elementary Numerical Analysis, and Algorithmic Approach (McGraw-Hill, Japanese translation).
  17. Fuji Film Co., Ltd. of Japan.
  18. A display processor for the research of digital image processing that was developed at the Electrotechnical Laboratory of Japan.

1982

1980

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

1974

1973

1972

1970

1969

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Akahori, H.

Burckhardt, C. B.

Chu, D. C.

D. C. Chu, J. W. Goodman, “Spectrum Shaping with Parity Sequences,” Appl. Opt. 11, 1716 (1972).
[CrossRef] [PubMed]

D. C. Chu, “Spectrum Shaping for Computer Generated Holograms,” Ph.D. Dissertation, Department of Electrical Engineering, Stanford U. (1974).

Conte, S. D.

S. D. Conte, C. de Boor, Elementary Numerical Analysis, and Algorithmic Approach (McGraw-Hill, Japanese translation).

Dallas, W. J.

de Boor, C.

S. D. Conte, C. de Boor, Elementary Numerical Analysis, and Algorithmic Approach (McGraw-Hill, Japanese translation).

Fienup, J. R.

J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

Firester, A. H.

Fox, E. C.

Gallagher, N. C.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Goodman, J. W.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Hirsch, P.M.

P.M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object Dependent Diffuser,” U.S. Patent3,619,022 (1971).

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

P.M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object Dependent Diffuser,” U.S. Patent3,619,022 (1971).

Katyl, R. H.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

P.M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object Dependent Diffuser,” U.S. Patent3,619,022 (1971).

Liu, B.

Miyamura, Y.

Oshida, Y.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Stewart, W. C.

Takeda, Y.

Appl. Opt.

IBM J. Res. Dev.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297 (1980).
[CrossRef]

Optik

R. W. Gerchberg, W. O. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik 35, 237 (1972).

Other

S. D. Conte, C. de Boor, Elementary Numerical Analysis, and Algorithmic Approach (McGraw-Hill, Japanese translation).

Fuji Film Co., Ltd. of Japan.

A display processor for the research of digital image processing that was developed at the Electrotechnical Laboratory of Japan.

D. C. Chu, “Spectrum Shaping for Computer Generated Holograms,” Ph.D. Dissertation, Department of Electrical Engineering, Stanford U. (1974).

P.M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object Dependent Diffuser,” U.S. Patent3,619,022 (1971).

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Figures (13)

Fig. 1
Fig. 1

Arrangement of the dummy area: (a) example of a 1-D object; (b) example of a 2-D object.

Fig. 2
Fig. 2

Flow diagram illustrating the iterative algorithm with a dummy area.

Fig. 3
Fig. 3

Four original sequences of one hundred binary numbers. The number of zeros which DATA A, DATA B, DATA C, and DATA D contain is 0, 25, 50, 75, respectively.

Fig. 4
Fig. 4

The rms intensity error for DATA A.

Fig. 5
Fig. 5

The rms intensity error for DATA B.

Fig. 6
Fig. 6

The rms intensity error for DATA C.

Fig. 7
Fig. 7

The rms intensity error for DATA D.

Fig. 8
Fig. 8

Variation in the efficiency η when ε0 = 0.01 and DATA C of Fig. 3 is used as the original sequence. A circle denotes the average of one hundred values of η.

Fig. 9
Fig. 9

Efficiency η gained by the revised algorithm when ε0 = 0.01 and I = 50. A plot of any specific figure represents the average of ten values of η for the corresponding data sequence and number of dummy points.

Fig. 10
Fig. 10

Original object of 16 × 16 binary numbers.

Fig. 11
Fig. 11

The rms intensity error of reconstructed image vs the number of iterations.

Fig. 12
Fig. 12

Samples of the digitally reconstructed images relative to the two points labeled a and b in Fig. 11: (a) after thirty iterations of the case without the dummy area (Q = 0); (b) after thirty iterations of the case with the dummy area (Q = 5); (c) a key to the grid numbers used in Figs. 12(a) and (b), where x denotes |f′|2, the intensity of a sample.

Fig. 13
Fig. 13

Optically reconstructed images: (a) the image for Q = 0 (without the dummy area) corresponding to Fig. 12(a); (b) the image for Q = 5 corresponding to Fig. 12(b).

Equations (31)

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a n , 0 = { s n n = 0 , 1 , , N 1 0 n = N , N + 1 , , L 1
F m = n = 0 L 1 f n exp ( i 2 π m n / L ) = | F m | exp ( i ϕ m ) , m = 0 , 1 , , L 1
f n = 1 L m = 0 L 1 F m exp ( i 2 π m n / L ) = a n exp ( i α n ) , n = 0 , 1 , , L 1
ɛ 2 = n = 0 N 1 ( | f n | 2 | f n , 0 | 2 ) 2 n = 0 N 1 | f n , 0 | 4 .
f n = { s n exp ( i α n ) n = 0 , 1 , , N 1 a n exp ( i α n ) . n = N , N + 1 , , L 1
f n , k = { s n exp ( i α n , k ) n = 0 , 1 , , N 1 d n , k exp ( i α n , k ) , n = N , N + 1 , , L 1
f n , K = { s n exp ( i α n , K ) n = 0 , 1 , , N 1 d n , K exp ( i α n , K ) , n = N , N + 1 , , L 1
α n , K = α n , k + Δ α n , d n , K = d n , k + Δ d n .
F m , K = n = 0 N 1 s n exp i ( α n , K 2 π m n / L ) + n = N N 1 d n , K exp i ( α n , K 2 π m n / L ) . m = 0 , 1 , , L 1
| F m , K | 2 n = 0 L 1 | f n , K | 2 + 2 X m , m = 0 , 1 , , L 1
X m = n = 0 N 2 l > n N 1 s n s l { cos θ m , n , l ( Δ α n Δ α l ) sin θ m , n , l } + n = 0 N 1 l = N L 1 s n { ( d l , k + Δ d l ) cos θ m , n , l d l , k ( Δ α n Δ α l ) sin θ m , n , l } + n = N L 2 l > n L 1 { ( d n , k d l , k + d l , k Δ d n + d n , k Δ d l ) cos θ m , n , l ( Δ α n Δ α l ) d n , k d l , k sin θ m , n , l } ,
θ m , n , l = α n , k α l , k 2 π m ( n l ) / L .
1 L m = 0 L 1 | F m , K | 2 = n = 0 L 1 | f n , K | 2 ,
X m = 0 , for m = 0 , 1 , , L 1 .
N e N υ .
M N 0 .
ξ ( k ) = n = 0 N 1 | f n , k | 2 n = 0 L 1 | f n , k | 2 ,
η = ξ ( k 0 ) ,
f n = { s n exp ( i α n ) n = 0 , 1 , , N 1 0 . n = N , N + 1 , , L 1
τ m n ( r ) = { τ m n ; signal area τ m n ( r 1 ) ; dummy area .
( 1 N ) 2 m , n = 0 N 1 | τ m n ( r + 1 ) exp [ i ϕ m n ( r + 1 ) ] τ m n ( r ) exp [ i ϕ m n ( r + 1 ) ] | 2 = p , q = 0 N 1 | A p q ( r + 1 ) exp [ i ψ p q ( r + 1 ) ] α A p q exp [ i ψ p q ( r ) ] | 2 p , q = 0 N 1 | A p q ( r + 1 ) exp [ i ψ p q ( r + 1 ) ] α A p q exp [ i ψ p q ( r + 1 ) ] | 2 .
τ m n ( r + 1 ) = { τ m n ( r ) = τ m n ; signal area τ m n ( r ) ; dummy area ,
m , n = 0 N 1 | τ m n ( r ) τ m n ( r ) | 2 m , n = 0 N 1 | τ m n ( r + 1 ) τ m n ( r ) | 2 .
| F m , K | 2 = F m , K × F m , K * = n = 0 N 1 l = 0 N 1 s n s l exp ( i ψ m , n , l ) + n = 0 N 1 l = N L 1 s n d l , K exp ( i ψ m , n , l ) + l = 0 N 1 n = N L 1 s l d n , K exp ( i ψ m , n , l ) + n = N L 1 l = N L 1 d n , K d l , K exp ( i ψ m , n , l ) ,
ψ m , n , l = α n , K α l , K 2 π m ( n l ) / L .
n = 0 N 1 s n 2 + 2 n = 0 N 2 l > n N 1 s n s l cos ψ m , n , l ,
2 n = 0 N 2 l = n L 1 s n d l , K cos ψ m , n , l ,
n = N L 1 ( d n , K ) 2 + 2 n = N L 2 l > n L 1 d n , K d l , K cos ψ m , n , l .
θ m , n , l = α n , k α l , k 2 π m ( n l ) / L
| F m , K | 2 = n = 0 L 1 | f n , K | 2 + 2 n = 0 N 2 l > 0 N 1 s n s l cos ( θ m , n , l + Δ α n Δ α l ) + 2 n = 0 N 1 l = N L 1 s n ( d l , k + Δ d l ) × cos ( θ m , n , l + Δ α n Δ α l ) + 2 n = N L 2 l > n L 1 ( d n , k + Δ d n ) ( d l , k + Δ d l ) × cos ( θ m , n , l + Δ α n Δ α l ) .
cos ( θ m , n , l + Δ α n Δ α l ) cos θ m , n , l ( Δ α n Δ α l ) sin θ m , n , l , ( d l , k + Δ d l ) ( Δ α n Δ α l ) d l , k ( Δ α n Δ α l ) , ( d n , k + Δ d n ) ( d l , k + Δ d l ) d n , k d l , k + d l , k Δ d n + d n , k Δ d l , ( d n , k + Δ d n ) ( d l , k + Δ d l ) ( Δ α n Δ α l ) d n , k d l , k ( Δ α n Δ α l ) ,

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