Abstract

In the generation of computer-generated holograms the phase is in many applications a free parameter that can be manipulated to achieve a high diffraction efficiency, a small space–bandwidth product, and a speckle-free reconstruction. An iterative algorithm to determine such phase distributions is described. Experimental verifications are given.

© 1988 Optical Society of America

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References

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  1. W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
    [CrossRef]
  2. E. N. Leith, J. Upatnieks, “Wavefront reconstruction with diffused illumination and three-dimensional objects,”J. Opt. Soc. Am. 54, 1295–1301 (1964).
    [CrossRef]
  3. R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 194–203.
  4. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
    [CrossRef]
  5. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
    [CrossRef]
  6. H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968).
    [CrossRef] [PubMed]
  7. F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer-generated holograms: hologram repetition and phase manipulations,” J. Opt. Soc. Am. A 4, 694–698 (1987).
    [CrossRef]
  8. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  9. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  10. J. P. Allebach, B. Liu, “Minimax spectrum shaping with a bandwidth constraint,” Appl. Opt. 14, 3062–3072 (1975).
    [CrossRef] [PubMed]
  11. J. P. Allebach, N. C. Gallagher, B. Liu, “Aliasing error in digital holography,” Appl. Opt. 15, 2183–2188 (1976).
    [CrossRef] [PubMed]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  14. M. S. Scivier, M. A. Fiddy, “Phase ambiguities and the zeros of multidimensional band-limited functions,” J. Opt. Soc. Am. A 2, 693–697 (1985).
    [CrossRef]
  15. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  16. Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
    [CrossRef]
  17. R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).
  18. T. R. Crimmins, J. R. Fienup, “Uniqueness of phase retrieval for functions with sufficiently disconnected support,”J. Opt. Soc. Am. 73, 218–221 (1983).
    [CrossRef]
  19. H. V. Deighton, M. S. Scievier, M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985).
    [CrossRef] [PubMed]
  20. H. A. Ferwerda, “Phase retrieval in imaging,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 36–39.
  21. R. Barakat, G. Newsam, “Algorithm for reconstruction of partially known, band-limited Fourier-transform pairs from noisy data,” J. Opt. Soc. Am. A 2, 2027–2039 (1985).
    [CrossRef]
  22. J. R. Fienup, C. C. Wackerman, “Phase-retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  23. W. L. Root, “Ill-posedness and precision in object-field reconstruction problems,” J. Opt. Soc. Am. A 4, 171–179 (1987).
    [CrossRef]
  24. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,”U.S. Patent No.3,619,022 (November9, 1971).
  25. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
    [CrossRef]

1987

1986

1985

1983

1982

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1979

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

1976

1975

1973

1972

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

1968

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

H. J. Gerritsen, W. J. Hannan, E. G. Ramberg, “Elimination of speckle noise in holograms with redundancy,” Appl. Opt. 7, 2301–2311 (1968).
[CrossRef] [PubMed]

1964

1963

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Allebach, J. P.

Barakat, R.

Bates, R. H. T.

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).

Bruck, Y. M.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Bryngdahl, O.

Burckhardt, C. B.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 194–203.

Collier, R. J.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 194–203.

Crimmins, T. R.

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

Deighton, H. V.

Ferwerda, H. A.

H. A. Ferwerda, “Phase retrieval in imaging,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 36–39.

Fiddy, M. A.

Fienup, J. R.

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, 237–246 (1972).

Gerritsen, H. J.

Hannan, W. J.

Hauck, R.

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,”U.S. Patent No.3,619,022 (November9, 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, 150–155 (1969).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,”U.S. Patent No.3,619,022 (November9, 1971).

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Dev. 13, 150–155 (1969).
[CrossRef]

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,”U.S. Patent No.3,619,022 (November9, 1971).

Lin, L. H.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 194–203.

Liu, B.

Newsam, G.

Ramberg, E. G.

Root, W. L.

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, 237–246 (1972).

Scievier, M. S.

Scivier, M. S.

Sodin, L. G.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Upatnieks, J.

Wackerman, C. C.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Wyrowski, F.

Appl. Opt.

Commun. ACM

L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
[CrossRef]

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, 150–155 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Opt. Commun.

Y. M. Bruck, L. G. Sodin, “On the ambiguity of the image reconstruction problem,” Opt. Commun. 30, 304–308 (1979).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Opt. Lett.

Optik

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. H. T. Bates, “Fourier phase problems are uniquely solvable in more than one dimension. I. Underlying theory,” Optik 61, 247–262 (1982).

Other

H. A. Ferwerda, “Phase retrieval in imaging,” in Digest of Topical Meeting on Signal Recovery and Synthesis II (Optical Society of America, Washington, D.C., 1986), pp. 36–39.

R. J. Collier, C. B. Burckhardt, L. H. Lin, Optical Holography (Academic, New York, 1971), pp. 194–203.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980), pp. 291–366.
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,”U.S. Patent No.3,619,022 (November9, 1971).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978), Vol. 16, pp. 119–232.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Block diagram of the error-reduction algorithm.

Fig. 2
Fig. 2

Illustration of the effect of superimposing (a) a linear phase and (b) a quadratic phase onto the samples of an amplitude. Two different complex amplitudes of the same band limit result.

Fig. 3
Fig. 3

Illustration of two phase-retrieval iterations. (a) The space-domain constraint used to recover (b) the original object. (c), (e) The resulting space-domain distributions; (d), (f) the corresponding Fourier-domain distributions. In (c) and (d) (the first-iteration example) a nonnegativity restriction was used.

Fig. 4
Fig. 4

Original intensity I(x, y) used in the experiments, sampled in 128 × 128 pixels.

Fig. 5
Fig. 5

Space distribution after 50 iterations, using a random initial phase.

Fig. 6
Fig. 6

Illustration of phase distributions around a dark spot. The phase in (a), in contrast to the one in (b), leads to iteration stagnation.

Fig. 7
Fig. 7

Modulus of the spectrum of the iterated phase function.

Fig. 8
Fig. 8

Moduli of spectra when introducing (a) a random phase, (b) a constant phase, and (c) an iterated phase.

Fig. 9
Fig. 9

Simulations of normalized reconstructed intensities obtained by using (a) a random phase, (b) a constant phase, and (c) an iterated phase to calculate the spectra.

Equations (4)

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u ( x , y ) = p , q u p q sinc ( Δ ν u x p ) sinc ( Δ ν u y q ) .
u ˜ k l = | u ˜ k l | exp ( i Φ k l ) = p , q u p q exp [ i 2 π ( p k M + q l N ) ] ,
u k l = { 0 for k P / 2 or l Q / 2 and for k < P / 2 and l < Q / 2 u ˜ k l if 0 | u ˜ k l | u ˜ max u ˜ max exp ( i Φ ˜ k l ) if | u ˜ k l | > u ˜ max .
Φ m n = Φ m ( 1 ) + Φ n ( 2 ) ,

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