Abstract

An iterative method for generating holograms on spatial light modulators is based on measuring the reconstructed complex image and on-line correction of the hologram. Apart from recording complex amplitude distributions, the new procedure may become a useful tool for applications such as adaptive optics and reconfigurable interconnection networks.

© 1992 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  2. J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnections and their limitations,” Appl. Opt. 28, 311–324 (1989).
    [CrossRef]
  3. W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
    [CrossRef]
  4. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” Prog. Opt. 28, 1–86 (1990).
    [CrossRef]
  5. 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]
  6. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [CrossRef]
  7. U. Mahlab, J. Rosen, J. Shamir, “Iterative generation of holograms on spatial light modulators,” Opt. Lett. 15, 556–558 (1990).
    [CrossRef] [PubMed]
  8. C. S. Weaver, J. W. Goodman, “A technique for optically convolving two functions,” Appl. Opt. 5, 1248–1249 (1966).
    [CrossRef] [PubMed]
  9. J. L. Horner, P. D. Gianino, “Phase-only matched filtering,” Appl. Opt. 23, 812–816 (1984).
    [CrossRef] [PubMed]
  10. B. V. K. Vijaya Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef]
  11. M. B. Reid, P. W. Ma, J. D. Downie, E. Ochoa, “Experimental verification of modified synthetic discriminant function filters for rotation invariance,” Appl. Opt. 29, 1209–1214 (1990).
    [CrossRef] [PubMed]
  12. U. Mahlab, J. Shamir, “Iterative optimization algorithms for filter generation in optical correlators: a comparison,” Appl. Opt. 31, 1117–1125 (1992).
    [CrossRef] [PubMed]
  13. J. A. Nelder, R. Mead, “A simpler method for function minimization,” Comput. J. 8, 308–313 (1965).
    [CrossRef]
  14. The CUE-2 is an image-processing system based on a personal computer manufactured by GALAI Laboratories, Industrial Zone, Migdal Haemek, Israel.

1992 (1)

1990 (4)

1989 (2)

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

J. Shamir, H. J. Caulfield, R. B. Johnson, “Massive holographic interconnections and their limitations,” Appl. Opt. 28, 311–324 (1989).
[CrossRef]

1987 (1)

1984 (1)

1978 (1)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

1967 (1)

1966 (1)

1965 (1)

J. A. Nelder, R. Mead, “A simpler method for function minimization,” Comput. J. 8, 308–313 (1965).
[CrossRef]

Allebach, J. P.

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

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]

Bryngdahl, O.

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

Caulfield, H. J.

Downie, J. D.

Gianino, P. D.

Goodman, J. W.

Hassebrook, L.

Horner, J. L.

Jennison, B. K.

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

Johnson, R. B.

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

Lohmann, A. W.

Ma, P. W.

Mahlab, U.

Mead, R.

J. A. Nelder, R. Mead, “A simpler method for function minimization,” Comput. J. 8, 308–313 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder, R. Mead, “A simpler method for function minimization,” Comput. J. 8, 308–313 (1965).
[CrossRef]

Ochoa, E.

Paris, D. P.

Reid, M. B.

Rosen, J.

Seldowitz, M. A.

Shamir, J.

Sweeney, D. W.

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

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]

Vijaya Kumar, B. V. K.

Weaver, C. S.

Wyrowski, F.

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

Appl. Opt. (8)

Comput. J. (1)

J. A. Nelder, R. Mead, “A simpler method for function minimization,” Comput. J. 8, 308–313 (1965).
[CrossRef]

Opt. Eng. (1)

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

Opt. Lett. (1)

Prog. Opt. (2)

W. H. Lee, “Computer-generated holograms: techniques and applications,” Prog. Opt. 16, 119–232 (1978).
[CrossRef]

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

Other (1)

The CUE-2 is an image-processing system based on a personal computer manufactured by GALAI Laboratories, Industrial Zone, Migdal Haemek, Israel.

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Figures (13)

Fig. 1
Fig. 1

Experimental system. The pattern generated on the SLM is transformed by the linear system and observed by the CCD.

Fig. 2
Fig. 2

Experimental system for calculating a Fourier hologram with a control on the complex distribution of the reconstructed image. BS, beam splitter.

Fig. 3
Fig. 3

Schematic illustration of the system for calculating any kind of hologram. BS’s, beam splitters.

Fig. 4
Fig. 4

Final BCGH of the reconstructed intensity distribution image shown in Fig. 5.

Fig. 5
Fig. 5

Reconstructed image of the letter F. containing magnitude information only.

Fig. 6
Fig. 6

Two examples of sequences of reconstructed images with random phase distribution. The numbers of iterations between every picture are (a) 500 and (b) 1000.

Fig. 7
Fig. 7

Error, defined in Eq. (6), versus the number of iterations. The reconstructed images are shown in Fig. 6(a).

Fig. 8
Fig. 8

Final BCGH of the reconstructed complex amplitude distribution shown in Fig. 9.

Fig. 9
Fig. 9

Reconstructed image of the letter F containing all the complex information.

Fig. 10
Fig. 10

Output distribution of the correlation channel.

Fig. 11
Fig. 11

Sequence of reconstructed images from two-channel process. The number of iterations between the pictures is 2000.

Fig. 12
Fig. 12

Final output distribution of the correlation channel when it was measured alone, after 25,000 iterations.

Fig. 13
Fig. 13

Reconstructed object from the hologram that was created by maximization of the inner product value alone.

Equations (18)

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h ( x , y ) = - + - + H ( u , v ) S ( u , v , x , y ) d u d v .
e = A f ( x , y ) - γ f ˜ ( x , y ) 2 d x d y ,
f ˜ ( x , y ) = w ( x , y ) h ( x , y ) ,
w ( x , y ) = { 1 if x A 0 otherwise .
I ˜ A ( x , y ) = A 2 + f ˜ ( x , y ) 2 + A f ˜ ( x , y ) × cos [ x k sin θ + φ ˜ ( x , y ) ] ,
I A ( x , y ) = A 2 + f ( x , y ) 2 + A f ( x , y ) × cos [ x k sin θ + φ ( x , y ) ] .
e I = A I A ( x , y ) - γ I ˜ A ( x , y ) 2 d x d y ,
e = A f ( x , y ) 2 d x d y - 2 γ cos ϕ | A f * ( x , y ) f ˜ ( x , y ) d x d y | + γ 2 A f ˜ ( x , y ) 2 d x d y ,
γ 0 = cos ϕ | A f ˜ ( x , y ) f * ( x , y ) d x d y | A f ˜ ( x , y ) 2 d x d y ,
e 0 = A f ( x , y ) 2 d x d y - cos 2 ϕ | A f ˜ ( x , y ) f * ( x , y ) d x d y | 2 A f ˜ ( x , y ) 2 d x d y .
e c = A f ( x , y ) 2 d x d y - | A f ˜ ( x , y ) f * ( x , y ) d x d y | 2 A f ˜ ( x , y ) 2 d x d y .
e a = A f ˜ ( x , y ) 2 d x d y | A f ˜ ( x , y ) f * ( x , y ) d x d y | 2 .
arg { g ( x , y ) } = arg { f ( x , y ) } , g ( x , y ) 0 ,             for all x , y where f ( x , y ) 0 ,
e g = A | f ( x , y ) 2 - γ f ˜ ( x , y ) 2 | 2 d x d y | A f ˜ ( x , y ) g * ( x , y ) d x d y | 2 .
e p = | A f ˜ ( x , y ) g * ( x , y ) d x d y | - 2 ,
F ˜ ( u , v ) = G ( u , v ) / G ( u , v ) ,
e i = A | A f ˜ ( x , y ) g * ( x - x , y - y ) d x d y | 2 d x d y | A f ˜ ( x , y ) g * ( x , y ) d x d y | 2 ,
F ˜ ( u , v ) = [ G ( u , v ) ] - 1 .

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