Abstract

An iterative concept is suggested to quantize digital amplitude holograms. It is based on an iterative Fourier transform algorithm. A stepwise introduction of the quantization constraint results in a convergent algorithm. The production of holograms is described and their optical reconstructions are presented.

© 1989 Optical Society of America

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References

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  1. W.-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1978), pp. 119–232.
    [CrossRef]
  2. B. R. Brown, A. W. Lohmann, “Complex Spatial Filtering with Binary Masks,” Appl. Opt. 5, 967–969 (1966).
    [CrossRef] [PubMed]
  3. W. -H. Lee, “Binary Synthetic Holograms,” Appl. Opt 13, 1677–1682 (1974).
    [CrossRef] [PubMed]
  4. R. Hauck, O. Bryngdahl, “Computer-Generated Holograms with Pulse-Density Modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  5. D. Just, R. Hauck, O. Bryngdahl, “Computer-Generated Holograms: Structure Manipulation,” J. Opt. Soc. Am. A 2, 644–648 (1985).
    [CrossRef]
  6. 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]
  7. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of Making an Object-Dependent Diffuser,” U.S. Patent3,619,022 (1971).
  8. R. A. Gabel, B. Liu, “Minimization of Reconstruction Errors with Computer Generated Binary Holograms,” Appl. Opt. 9, 1180–1190 (1970).
    [CrossRef] [PubMed]
  9. N. C. Gallagher, B. Liu, “Method for Computing Kinoforms that Reduces Image Reconstruction Error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  10. J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–000 (1980).
    [CrossRef]
  11. H. Akahori, “Spectrum Leveling by an Iterative Algorithm with a Dummy Area for Synthesizing the Kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef] [PubMed]
  12. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A, 5, 1058–1065 (1988).
    [CrossRef]
  13. J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  14. M. Broja, F. Wyrowski, O. Bryngdahl, “Digital Halftoning by Iterative Procedure,” Opt. Commun. 69, 205–000 (1988).
    [CrossRef]

1988

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A, 5, 1058–1065 (1988).
[CrossRef]

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital Halftoning by Iterative Procedure,” Opt. Commun. 69, 205–000 (1988).
[CrossRef]

1987

1986

1985

1984

1982

1980

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–000 (1980).
[CrossRef]

1974

W. -H. Lee, “Binary Synthetic Holograms,” Appl. Opt 13, 1677–1682 (1974).
[CrossRef] [PubMed]

1973

1970

1966

Akahori, H.

Allebach, J. P.

Broja, M.

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital Halftoning by Iterative Procedure,” Opt. Commun. 69, 205–000 (1988).
[CrossRef]

Brown, B. R.

Bryngdahl, O.

Fienup, J. R.

J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–000 (1980).
[CrossRef]

Gabel, R. A.

Gallagher, N. C.

Hauck, R.

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.

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

Just, D.

Lee, W. -H.

W. -H. Lee, “Binary Synthetic Holograms,” Appl. Opt 13, 1677–1682 (1974).
[CrossRef] [PubMed]

Lee, W.-H.

W.-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1978), pp. 119–232.
[CrossRef]

Lesem, L. B.

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

Liu, B.

Lohmann, A. W.

Seldowitz, M. A.

Sweeney, D. W.

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-Transform Algorithm Applied to Computer Holography,” J. Opt. Soc. Am. A, 5, 1058–1065 (1988).
[CrossRef]

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital Halftoning by Iterative Procedure,” Opt. Commun. 69, 205–000 (1988).
[CrossRef]

Appl. Opt

W. -H. Lee, “Binary Synthetic Holograms,” Appl. Opt 13, 1677–1682 (1974).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

M. Broja, F. Wyrowski, O. Bryngdahl, “Digital Halftoning by Iterative Procedure,” Opt. Commun. 69, 205–000 (1988).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative Method Applied to Image Reconstruction and to Computer-Generated Holograms,” Opt. Eng. 19, 297–000 (1980).
[CrossRef]

Other

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

W.-H. Lee, “Computer-Generated Holograms: Techniques and Applications,” Progress in Optics, E. Wolf, Ed. (North-Holland, Amsterdam, 1978), pp. 119–232.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Simulated reconstruction of analog hologram distribution with dc peak suppressed.

Fig. 2
Fig. 2

Illustration of the hardclip operator Q 2 ¯.

Fig. 3
Fig. 3

Hologram binarized with Q 2 ¯ is shown in (a) and in (b) the simulated reconstruction with dc peak suppressed.

Fig. 4
Fig. 4

Illustration of the iterative Fourier transform algorithm. The constraints are executed by the operator U ¯ in the hologram plane and by X ¯ in the signal plane.

Fig. 5
Fig. 5

Histograms of Gj(u) binarized with U ¯ = Q 2 ¯ after the iteration cycle (a) j = 0, (b) j = 1, (c) j = 5, and (d) j = 10.

Fig. 6
Fig. 6

Illustration of the Q 2 ¯ p operator used to avoid stagnation of the iterative procedure.

Fig. 7
Fig. 7

Illustration of the stepwise change of the Q 2 ¯ p operator during the iteration.

Fig. 8
Fig. 8

Histogram of Gj(u) binarized with U ¯ = Q 2 ¯ p after the last iteration cycle of (a) step 2 and (b) step 4.

Fig. 9
Fig. 9

(a) Hologram binarized with U ¯ = Q 2 ¯ p and (b) the simulated reconstructed intensity with dc peak suppressed.

Fig. 10
Fig. 10

Optical reconstructions from holograms calculated by iteration. (a) and (b) are quantized with Z = 3 and (c) and (d) with Z = 2. Parameters: M = N = 256, Mf = Nf = 64, and m0 = (−64,64).

Equations (21)

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G ( u ) = | F ( u ) | cos [ 2 π u x 0 Φ ( u ) ] + B ,
g ( x ) = ½ f ( x x 0 ) + ½ f * [ ( x + x 0 ) ] + B · δ ( x ) ,
G ̅ ( u ) = Q Z ¯ [ G ( u ) ] = { 0 , G ( u ) Δ z · 2 Δ , ( 2 z 1 ) Δ < G ( u ) ( 2 z + 1 ) Δ , 1 , ( 2 Z 3 ) Δ < G ( u )
Q ( u ) = G ̅ ( u ) G ( u ) ,
g ̅ ( x ) = g ( x ) + q ( x ) = ½ f ( x x 0 ) + ½ f * [ ( x + x 0 ) ] + B δ ( x ) + q ( x ) .
| f ( x x 0 ) + q ( x ) | 2 i ( x x 0 ) , x .
σ j 2 = | c j | f ( x x 0 ) | g ̅ j ( x ) | 2 dxdy ,
c j = | f ( x x 0 ) | | g ̅ j ( x ) | dxdy D ( f ) | f ( x ) | 2 dxdy .
g 0 ( x ) = | f ( x x 0 ) | exp [ i φ 0 ( x x 0 ) ] + | f [ ( x + x 0 ) ] | exp { i φ 0 [ ( x + x 0 ) ] }
X ¯ [ g ̅ j ( x ) ] = g j + 1 ( x ) ] = { c j | f ( x x 0 ) | exp [ i γ ̅ j ( x ) ] , x c j | f [ ( x + x 0 ) ] | exp [ i γ ̅ j ( x ) ] , x * , g ̅ j ( x ) , otherwise
Q ̅ max Δ = 0.5 .
G ̅ j ( u ) = U ¯ [ G j ( u ) ] = Q Z ¯ p [ G j ( u ) ] = { 0 , G j ( u ) ( p ) Δ z · 2 Δ , [ 2 z ( p ) ] Δ < G j ( u ) [ 2 z + ( p ) ] Δ 1 , [ 2 ( z 1 ) ( p ) ] Δ < G j ( u ) G j ( u ) , otherwise
0 < ( 1 ) < ( 2 ) < ( p ) < ( P ) = 1 ,
f ( m f ) = f ( x ) comb ( x , δ x ) ,
comb ( x , δ x ) α , β δ ( x α · δ x ) δ ( y β · δ y ) ,
g 0 ( m ) = f ( m m 0 ) + f * [ ( m + m 0 ) ] ,
G ̅ J ( k ) = G ̅ J ( u ) comb ( u , δ u ) ,
rect ( u , δ u ) rect ( u , δ u ) rect ( υ , δ υ ) ,
rect ( u , δ u ) = { 1 , | u | δ u / 2 0 , otherwise ,
H ( u ) = G ̅ J ( k ) * rect ( u , δ u ) ,
SNR = D ( f ) | f ( x ) | 2 dxdy σ J 2 ,

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