Abstract

We propose a new multicriteria method for the determination of computer-generated holograms (CGH’s). For this purpose, the direct binary search (DBS) algorithm for computing CGH’s has been modified to converge on a new error function that defines the optimal trade-off among different criteria. This approach allows us to control the trade-off between the amplitude error and the diffraction efficiency and to provide a rigorous figure of merit. Comparisons among different encoding methods show that the best results are obtained with a modified version of the DBS method combined with the iterative Fourier transform algorithm.

© 1997 Optical Society of America

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References

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  1. 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]
  2. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 6, 1058–1065 (1988).
    [CrossRef]
  3. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  4. B. K. Jennison, J. P. Allebach, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A 8, 652–660 (1991).
    [CrossRef]
  5. Ph. Réfrégier, “Optimal trade-off filters for noise robustness, sharpness of the correlation peak, and Horner efficiency,”, Opt. Lett. 16, 829–831 (1991).
    [CrossRef] [PubMed]
  6. H. Kogelnik, “Reconstructing response and efficiency of hologram gratings,” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Polytechnic Press, Brooklyn, N.Y., 1967), pp. 605–617.
  7. 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).
  8. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]

1991 (2)

1989 (1)

1988 (1)

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 6, 1058–1065 (1988).
[CrossRef]

1987 (1)

1982 (1)

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).

Allebach, J. P.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 6, 1058–1065 (1988).
[CrossRef]

Fienup, J. R.

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).

Jennison, B. K.

Kogelnik, H.

H. Kogelnik, “Reconstructing response and efficiency of hologram gratings,” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Polytechnic Press, Brooklyn, N.Y., 1967), pp. 605–617.

Réfrégier, Ph.

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).

Seldowitz, M. A.

Sweeney, D. W.

Wyrowski, F.

F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
[CrossRef] [PubMed]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 6, 1058–1065 (1988).
[CrossRef]

Appl. Opt. (3)

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

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 6, 1058–1065 (1988).
[CrossRef]

B. K. Jennison, J. P. Allebach, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A 8, 652–660 (1991).
[CrossRef]

Opt. Lett. (1)

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).

Other (1)

H. Kogelnik, “Reconstructing response and efficiency of hologram gratings,” in Proceedings of the Symposium on Modern Optics, J. Fox, ed. (Polytechnic Press, Brooklyn, N.Y., 1967), pp. 605–617.

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Figures (6)

Fig. 1
Fig. 1

Normalized amplitude error versus the diffraction efficiency of binary amplitude Fourier holograms as computed with the DBS algorithm converging on a multicriteria error for values of μ varying from 0 to 1.

Fig. 2
Fig. 2

Close-up of Fig. 1.

Fig. 3
Fig. 3

IFTA according to Wyrowski3 for computing binary amplitude holograms. FT-1, inverse Fourier transform.

Fig. 4
Fig. 4

Comparison of the normalized amplitude error versus the diffraction efficiency of binary amplitude Fourier holograms, as computed with different algorithms.

Fig. 5
Fig. 5

Close-up of Fig. 4.

Fig. 6
Fig. 6

Comparison of the normalized amplitude error versus the diffraction efficiency of four-phase-level Fourier holograms, as computed with different algorithms.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

ERRA=q,rIfq, r-cagq, r2q,rIfq, r2,
ca=q,rIfq, r·gq, rq,rIgq, r2,
η=q,rIgq, r2N·M,
E1h1E1h2, E2h1E2h2
E1hE1ho, E2hE2ho
C=1-μE1h+μE2h,
C=1-μERRA+μ1η,
E1h1E1h2,E2h1E2h2
E1hE1ho,E2hE2ho
Eμ, h=1-μE1h+μE2h,
E1hμ1E1hμ2,
E2hμ2E2hμ1.
E1hμ2E1hμ1,E2hμ2<E2hμ1,

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