Abstract

In Fourier and Fresnel holography diffuse illumination is used to smooth the power spectrum of the object wave. A speckle pattern will then disturb the diffraction pattern. We present methods to calculate appropriate diffusers in digital holography that do not introduce speckle. The considerations include deterministic object-independent as well as iteratively calculated object-dependent diffusers. Object-independent diffusers are suited for initial phases to avoid stagnation of the iterative procedure.

© 1991 Optical Society of America

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References

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  1. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, Vol. 41 of Topics in Applied Physics, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.
    [CrossRef]
  2. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  3. M. A. Seldowitz, J. P. Allebach, D. W. Sweeny, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  4. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  5. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  6. E. N. Leith, J. Upatnieks, “Wavefront reconstruction with diffused illumination and three-dimensional objects,”J. Opt. Soc. Am. 54, 1295–1301 (1964).
    [CrossRef]
  7. W. J. Dallas, “Deterministic diffusers for holography,” Appl. Opt. 12, 1179–1187 (1973).
    [CrossRef] [PubMed]
  8. Y. Torii, “Synthesis of deterministic phase codes for phase shifter in holography,” Opt. Commun. 24, 175–180 (1978).
    [CrossRef]
  9. F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
    [CrossRef]
  10. Y. Nakayama, M. Kato, “Diffuser with pseudorandom phase sequence,”J. Opt. Soc. Am. 69, 1367–1372 (1979).
    [CrossRef]
  11. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).
  12. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  13. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  14. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  15. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
    [CrossRef]
  16. F. Wyrowski, “Considerations on convolutions and phase factors,” submitted to Opt. Commun.
  17. H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
    [CrossRef]
  18. 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]
  19. F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
    [CrossRef]

1990 (1)

1989 (2)

1988 (1)

1987 (2)

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
[CrossRef]

M. A. Seldowitz, J. P. Allebach, D. W. Sweeny, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
[CrossRef] [PubMed]

1985 (1)

1984 (1)

1983 (1)

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

1980 (1)

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

1979 (1)

1978 (1)

Y. Torii, “Synthesis of deterministic phase codes for phase shifter in holography,” Opt. Commun. 24, 175–180 (1978).
[CrossRef]

1973 (2)

1964 (1)

Allebach, J. P.

Arsenault, H. A.

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
[CrossRef]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

Chalasinska-Macukow, K.

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Dallas, W. J.

W. J. Dallas, “Deterministic diffusers for holography,” Appl. Opt. 12, 1179–1187 (1973).
[CrossRef] [PubMed]

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

Fiddy, M. A.

Fienup, J. R.

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

Gallagher, N. C.

Hauck, R.

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
[CrossRef]

R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
[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 (November9, 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 (November9, 1971).

Kato, M.

Leith, E. N.

Lesem, L. B.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

Liu, B.

Nakayama, Y.

Scivier, M. S.

Seldowitz, M. A.

Sweeny, D. W.

Torii, Y.

Y. Torii, “Synthesis of deterministic phase codes for phase shifter in holography,” Opt. Commun. 24, 175–180 (1978).
[CrossRef]

Upatnieks, J.

Wyrowski, F.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[CrossRef]

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

F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

F. Wyrowski, “Considerations on convolutions and phase factors,” submitted to Opt. Commun.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

Y. Torii, “Synthesis of deterministic phase codes for phase shifter in holography,” Opt. Commun. 24, 175–180 (1978).
[CrossRef]

F. Wyrowski, R. Hauck, O. Bryngdahl, “Computer holography: object-dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
[CrossRef]

H. A. Arsenault, K. Chalasinska-Macukow, “The solution to the phase retrieval problem using the sampling theorem,” Opt. Commun. 47, 380–386 (1983).
[CrossRef]

Opt. Eng. (1)

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

Other (4)

O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1990), Vol. 28, pp. 1–86.
[CrossRef]

F. Wyrowski, “Considerations on convolutions and phase factors,” submitted to Opt. Commun.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. Patent3,619,022 (November9, 1971).

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

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Figures (9)

Fig. 1
Fig. 1

Regular and intermediate sampling values of a sampling grid.

Fig. 2
Fig. 2

Portion of regular sampling values ○ used in approximation.

Fig. 3
Fig. 3

Diffuser for Δφi = 1.335: a, phase distribution (gray levels correspond to phase values); b, magnitude of spectrum.

Fig. 4
Fig. 4

Diffuser for Δφi = 2.0: a, phase distribution; b, magnitude of spectrum.

Fig. 5
Fig. 5

Flow chart of the iterative Fourier-transform algorithm.

Fig. 6
Fig. 6

Magnitude of e(x) after 50 iteration cycles, when a random initial phase is used.

Fig. 7
Fig. 7

a, Magnitude of e(x) when a diffuser with Δφi 1.335 is used as the initial phase; b, magnitude of spectrum of the optimized diffuser.

Fig. 8
Fig. 8

a, Magnitude of e(x) when a diffuser with Δφi = 2.0 is used as the initial phase; b, magnitude of spectrum of the optimized diffuser.

Fig. 9
Fig. 9

Optically generated diffraction patterns of digital holograms: a, a diffuser with a random phase was introduced; b, an object-dependent diffuser was iteratively calculated.

Equations (14)

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e ( x ) = e ( m ) * sinc ( x , δ x f ) = m , n e ( m ) exp [ i φ ( m ) ] sinc ( m δ x f - x ) sinc ( n δ y f - y ) ,
comb ( x , δ x ) ~ α , β δ ( x - α δ x ) δ ( y - β δ y )
sinc ( x , δ x ) ~ [ sin ( π x / δ x ) / ( π x / δ x ) ] [ sin ( π y / δ y ) / ( π y / δ y ) ] .
comb ( x , δ x i ) = | m , n exp [ i φ ( m ) ] sinc ( m δ x f - x ) sinc ( n δ y f - y ) | 2 × comb ( x , δ x i ) .
Δ φ 1 = φ ( m 2 , n 2 ) - φ ( m 2 , n 1 ) ,
Δ φ 2 = φ ( m 1 , n 2 ) - φ ( m 2 , n 2 ) ,
Δ φ 3 = φ ( m 1 , n 1 ) - φ ( m 1 , n 2 ) ,
Δ φ 4 = φ ( m 2 , n 1 ) - φ ( m 1 , n 1 ) .
1 = 2 sinc 2 ( 0.5 ) ( 1 + cos Δ φ i ) ,             i = 1 , , 4 ,
1 = 2 sinc 2 ( 0.5 ) [ 2 + cos Δ φ 1 + cos Δ φ 2 + cos Δ φ 3 + cos Δ φ 4 + cos ( Δ φ 1 + Δ φ 2 ) + cos ( Δ φ 2 + Δ φ 3 ) ] ,
i = 1 4 Δ φ i = 0
| i = 1 4 Δ φ i | = 2 π .
B [ F j ( zk ) ] = F ¯ j ( zk ) = { F j ( zk ) , - M / 2 zk M / 2 0 , otherwise
X [ f ¯ j ( zm ) ] = f j + 1 ( zm ) = f ( zm ) exp [ i φ j ( zm ) ] ,

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