Abstract

The diffraction efficiency η of analog and quantized amplitude holograms is treated. Possible ways to manipulate η are discussed in terms of utilization of different kinds of freedom in the reconstructed field. Iterative methods to increase η are suggested, and experimental results are presented. The methods are based on iterative Fouriertransform algorithms. Results of iterative and noniterative quantization techniques are compared with respect to their dependence on the original analog hologram.

© 1990 Optical Society of America

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  1. H. Bartelt, S. K. Case, “High-efficient hybrid computergenerated holograms,” Appl. Opt. 21, 2886–2890 (1982).
    [Crossref] [PubMed]
  2. J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
    [Crossref]
  3. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [Crossref]
  4. F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174 (1989).
    [Crossref]
  5. L. B. Lesem, P. M. Hirsch, J. A. Jordan, “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661–674 (1968).
    [Crossref]
  6. T. S. Huang, B. Prasada, “Considerations on the generation and processing of holograms by digital computers,” Quarterly Progress Rep. 81, Research Laboratory of Electronics (Massachusetts Institute of Technology, Cambridge, Mass., 1966), pp. 199–205.
  7. P. M. Hirsch, J. A. Jordan, L. B. Lesem, U.S. PatentNo. 3,619,022 (November9, 1971).
  8. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [Crossref] [PubMed]
  9. B. Liu, N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt. 13, 2470–2471 (1974).
    [Crossref] [PubMed]
  10. B. Liu, N. C. Gallagher, “Optimum Fourier-transform division filters with magnitude constraint,” J. Opt. Soc. Am. 64, 1227–1236 (1974).
    [Crossref]
  11. 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).
  12. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [Crossref]
  13. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]
  14. E. N. Leith, J. Upatnieks, “Wavefronts reconstruction with diffused illumination and three-dimensional objects,” J. Opt. Soc. Am. 54, 1295–1301 (1964).
    [Crossref]
  15. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [Crossref] [PubMed]
  16. W.-H. Lee, “Binary synthetic holograms,” Appl. Opt. 13, 1677–1682 (1974).
    [Crossref] [PubMed]
  17. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [Crossref]
  18. 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]
  19. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [Crossref] [PubMed]
  20. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Display 17, 75–77 (1976).
  21. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
    [Crossref]
  22. F. Wyrowski, “Diffractive optical elements, iterative calculation of quantized blazed phase structures,” J. Opt. Soc. Am. A. (to be published).
  23. F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219.
  24. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [Crossref]

1989 (3)

1988 (2)

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

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[Crossref]

1987 (1)

1984 (1)

1982 (2)

1980 (1)

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

1976 (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

1974 (3)

1973 (1)

1972 (1)

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).

1968 (1)

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

1967 (2)

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[Crossref]

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
[Crossref] [PubMed]

1964 (1)

Allebach, J. P.

Bartelt, H.

Bryngdahl, O.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

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]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[Crossref]

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

Burch, J. J.

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[Crossref]

Case, S. K.

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, 297–305 (1980).
[Crossref]

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

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).

Hauck, R.

Hirsch, P. M.

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, U.S. PatentNo. 3,619,022 (November9, 1971).

Huang, T. S.

T. S. Huang, B. Prasada, “Considerations on the generation and processing of holograms by digital computers,” Quarterly Progress Rep. 81, Research Laboratory of Electronics (Massachusetts Institute of Technology, Cambridge, Mass., 1966), pp. 199–205.

Jordan, J. A.

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, U.S. PatentNo. 3,619,022 (November9, 1971).

Lee, W.-H.

Leith, E. N.

Lesem, L. B.

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, U.S. PatentNo. 3,619,022 (November9, 1971).

Liu, B.

Lohmann, A. W.

Paris, D. P.

Prasada, B.

T. S. Huang, B. Prasada, “Considerations on the generation and processing of holograms by digital computers,” Quarterly Progress Rep. 81, Research Laboratory of Electronics (Massachusetts Institute of Technology, Cambridge, Mass., 1966), pp. 199–205.

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).

Seldowitz, M. A.

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

Sweeney, D. W.

Upatnieks, J.

Weissbach, S.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[Crossref]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[Crossref]

Wyrowski, F.

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[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]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[Crossref]

F. Wyrowski, “Diffractive optical elements, iterative calculation of quantized blazed phase structures,” J. Opt. Soc. Am. A. (to be published).

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219.

Appl. Opt. (8)

Commun. ACM (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (2)

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Quantization noise in pulse density modulated holograms,” Opt. Commun. 67, 167–171 (1988).
[Crossref]

S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
[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]

Optik (1)

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).

Proc. IEEE (1)

J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
[Crossref]

Proc. Soc. Inf. Display (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial grayscale,” Proc. Soc. Inf. Display 17, 75–77 (1976).

Other (4)

T. S. Huang, B. Prasada, “Considerations on the generation and processing of holograms by digital computers,” Quarterly Progress Rep. 81, Research Laboratory of Electronics (Massachusetts Institute of Technology, Cambridge, Mass., 1966), pp. 199–205.

P. M. Hirsch, J. A. Jordan, L. B. Lesem, U.S. PatentNo. 3,619,022 (November9, 1971).

F. Wyrowski, “Diffractive optical elements, iterative calculation of quantized blazed phase structures,” J. Opt. Soc. Am. A. (to be published).

F. Wyrowski, “Coding and quantization techniques in digital phase holography,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 215–219.

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Figures (12)

Fig. 1
Fig. 1

Illustration of the reconstructed distribution and notation used.

Fig. 2
Fig. 2

Illustration of how the PF of a signal is used: a term c(x) is added to the amplitude g(x), resulting in (x) with the same modulus.

Fig. 3
Fig. 3

(a) Central part of basic hologram distribution G(u) of the test signal used and (b) calculated reconstruction (zeroth-order reduced).

Fig. 4
Fig. 4

Change |c(x)| in the signal plane obtained by modifying the bias function.

Fig. 5
Fig. 5

Illustration of the iterative FT algorithm

Fig. 6
Fig. 6

(a) Central part of the hologram distribution after an iterative increase of η is obtained by using the AF and the SF, (b) distribution |cJ(x)| in the signal plane due to the change from Fig. 3(a) to Fig. 6(a).

Fig. 7
Fig. 7

(a) Central part of the hologram distribution after superposition of a random phase onto the signal and (b) after an iteratively introduced phase, (c) The intensity of the distribution c(x) due to the change from (a) to (b).

Fig. 8
Fig. 8

(a) Central part of the hologram distribution with an iteratively increased η obtained by using the AF, the SF, and the PF and (b) the optical reconstruction.

Fig. 9
Fig. 9

Normalized histograms of gray values of different hologram distributions. Kinds of freedom used: (a) no freedom; (b) SF and AF; (c), (d) SF and PF; (e) SF, PF, and AF.

Fig. 10
Fig. 10

Central part of binary holograms obtained from analog holograms by application of ED. In (a) the analog hologram was obtained by using a random phase and in (c), with an iterative procedure. (b), (d) The corresponding calculated reconstructions.

Fig. 11
Fig. 11

(a) Optical reconstruction of the hologram shown in Fig. 10(c), (b) that of the hologram shown in Fig. 12(a).

Fig. 12
Fig. 12

(a) Central part of iteratively binarized hologram, (b) its calculated reconstruction.

Tables (2)

Tables Icon

Table 1 Diffraction Efficiencies of Different Analog Holograms of Test Signal for Different Kinds of Freedom Used in Their Calculation

Tables Icon

Table 2 Diffraction Efficiencies η ¯ and SNR Values of Differently Binarized Holograms in Comparison with Diffraction Efficiency η of Original Analog Hologram

Equations (48)

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F ( u ) = F T [ f ( x ) ]
G ( u ) = α { | F ( u ) | cos [ 2 π u x 0 Φ ( u ) ] + B } .
B = F min = min u { F ( u ) cos [ 2 π u x 0 Φ ( u ) ] } ,
α = 1 / ( F max F min ) ,
g ( x ) = α { 0.5 f ( x x 0 ) + 0.5 f * [ ( x + x 0 ) ] + B δ ( x ) } ,
F = { x | x R 2 Λ x 0 Δ f / 2 x x 0 + Δ f / 2 } ,
( u ) = G ( u ) + C ( u ) ,
( x ) = g ( x ) + c ( x ) ,
c ( x ) = 0 , x F
c ( x ) f ( x x 0 ) , x F .
| f ( x x 0 ) + c ( x ) | 2 i ( x x 0 ) = | f ( x x 0 ) | 2 , x F .
η = F | g ( x ) | 2 d x d y Δ G u Δ G υ ,
η = η t η d ,
η t = D ( G ) | G ( u ) | 2 d u d υ Δ G u Δ G υ
η d = F | g ( x ) | 2 d x d y R 2 | g ( x ) | 2 d x d y .
C ( u ) = ( β α ) H ( u ) + D ( u ) ,
H ( u ) = | F ( u ) | cos [ 2 π u x 0 Φ ( u ) ]
( u ) = [ β H ( u ) α F min ] + D ( u ) .
( x ) = β h ( x ) α F min δ ( x ) + d ( x ) ,
h ( x ) = FT 1 [ H ( u ) ] = 0.5 { f ( x x 0 ) + f * [ ( x + x 0 ) ] } ,
( case 1 ) d ( x ) x F ( AF )
( case 2 ) | f ( x x 0 ) + d ( x ) | 2 = | f ( x x 0 ) | 2 , x F ,
d ( x ) = 0 , x F F * ( PF )
( case 3 ) | f ( x x 0 ) + d ( x ) | 2 = | f ( x x 0 ) | 2 , x F ,
d ( x ) 0 , x F F * ( PF and AF ) .
η ¯ = ( β α ) 2 η .
B ( u ) = | F ( u ) | .
C ( u ) = ( β α ) H ( u ) + α F min + β | F ( u ) | ,
β = 1 | F | max
| F | max = max u ( | F ( u ) | { cos [ 2 π u x 0 Φ ( u ) ] + 1 } ) .
c ( x ) = ( β α ) h ( x ) + d ( x ) ,
d ( x ) = α F min δ ( x ) + β FT 1 [ | F ( u ) | ] .
G 0 ( u ) = β 0 H ( u ) α F min ,
j ( u ) = U [ G j ( u ) ] { 1 G j ( u ) > 1 0 G j ( u ) < 1 G j ( u ) else .
C j ( u ) = A j ( u ) cos [ 2 π u x 0 Φ ( u ) ] ,
A j ( u ) = { G j ( u ) 1 G j ( u ) > 1 G j ( u ) G j ( u ) < 1 0 else .
g j + 1 ( x ) = X [ j ( x ) ] = { β j f ( x x 0 ) x F β j * f * [ ( x x 0 ) ] x F * j ( x ) else ,
β j = F j * ( x ) f ( x x 0 ) + j ( x ) f * ( x x 0 ) d x d y 2 F | f ( x x 0 ) | 2 d x d y .
g 0 ( x ) = β 0 h ( x ) α F min δ ( x ) ,
h ( x ) = 0.5 { | f ( x x 0 ) | exp [ i φ ( x x 0 ) ] + c.c } ,
g j + 1 ( x ) = X [ j ( x ) ] = { β 0 | f ( x x 0 ) | exp [ i γ ¯ j ( x ) ] x F β 0 | f [ ( x + x 0 ) ] | exp [ i γ ¯ j ( x ) ] x F * 0 else
g j + 1 ( x ) = X [ j ( x ) ] = { β j | f ( x x 0 ) | exp [ i γ ¯ j ( x ) ] x F β j | f [ ( x + x 0 ) ] | exp [ i γ ¯ j ( x ) ] x F * j ( x ) else ,
β j = F | j ( x ) | | f ( x x ) | d x d y F | f ( x x 0 ) | 2 d x d y .
Q ( u ) = ( u ) G ( u ) .
( x ) = g ( x ) + q ( x ) .
q ( x ) = 0 , x F ,
| f ( x x 0 ) + q ( x ) | 2 | f ( x x 0 ) | 2 , x F ,
SNR = F | f ( x x 0 ) | 2 d x d y F [ | f ( x x 0 ) | β | ( x ) | ] 2 d x d y ,

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