Abstract

The design theory presented describes characteristic features of thin diffractive elements. In particular it permits the derivation of upper bounds of the diffraction efficiency of diffractive elements. These bounds are independent of the calculation method used to obtain the diffractive elements. The consequences of the results for diffractive optics are briefly discussed.

© 1993 Optical Society of America

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    [CrossRef] [PubMed]
  2. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  3. J. J. Burch, “A computer algorithm for the synthesis of spatial frequency filters,” Proc. IEEE 55, 599–601 (1967).
    [CrossRef]
  4. 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]
  5. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  6. O. Bryngdahl, F. Wyrowski, “Digital holography—computer-generated holograms,” in Progress in Optics 28, E. Wolf, ed. (North-Holland, New York, 1990), Chap. 1, pp. 1–86.
    [CrossRef]
  7. D. C. Chu, J. R. Fienup, J. W. Goodman, “Multiemulsion on-axis computer generated hologram,” Appl. Opt. 12, 1386–1388 (1973).
    [CrossRef] [PubMed]
  8. W. H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
    [CrossRef] [PubMed]
  9. C. K. Hsueh, A. A. Sawchuk, “Computer-generated double-phase holograms,” Appl. Opt. 17, 3874–3883 (1978).
    [CrossRef] [PubMed]
  10. M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
    [CrossRef] [PubMed]
  11. F. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. 30, 4650–4657 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  14. R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  15. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics 21, E. Wolf, ed. (North-Holland, New York, 1984), Chap. 1, pp. 1–67.
    [CrossRef]
  16. N. C. Gallagher, D. W. Sweeney, “Computer-generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
    [CrossRef]
  17. A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
    [CrossRef]
  18. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).
  19. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  20. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  21. F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065, 1988.
    [CrossRef]
  22. F. Wyrowski, O. Bryngdahl, “Speckle-free reconstruction in digital holography,” J. Opt. Soc. Am. A 6, 1171–1174, 1989.
    [CrossRef]
  23. R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578, 1991.
    [CrossRef]
  24. R. Bräuer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers in optical holography,” Opt. Lett. 16, 1427–1429, 1991.
    [CrossRef]
  25. W. H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics 16, E. Wolf, ed. (North-Holland, New York, 1978), Chap. 3, pp. 119–223.
    [CrossRef]
  26. W. J. Dallas, Computer-Generated Holograms, Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980).
  27. 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]
  28. P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
    [CrossRef]
  29. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
    [CrossRef]
  30. M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Eng. 14, 479–481 (1989).
  31. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
    [CrossRef] [PubMed]
  32. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial greyscale,” Proc. Soc. Inf. Displ. 17, 75–77 (1976).
  33. R. Hauck, O. Bryngdahl, “Computer-generated holograms with pulse-density modulation,” J. Opt. Soc. Am. A 1, 5–10 (1984).
    [CrossRef]
  34. S. Weissbach, F. Wyrowski, O. Bryngdahl, “Digital phase holograms: coding and quantization with an error diffusion concept,” Opt. Commun. 72, 37–41 (1989).
    [CrossRef]
  35. S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
    [CrossRef] [PubMed]
  36. P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971).
  37. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–24 (1972).
  38. N. C. Gallagher, B. Liu, “Method for computing kino-forms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  39. B. Liu, N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt. 13, 2470–2471 (1974).
    [CrossRef] [PubMed]
  40. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  41. F. Wyrowski, “Iterative quantization of digital amplitude holograms,” Appl. Opt. 28, 3864–3870 (1989).
    [CrossRef] [PubMed]
  42. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  43. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  44. F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
    [CrossRef]
  45. H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).
  46. In this paper an equidistant quantization within the interval [0, 1] for DAE’s and [0, 2π] for DPE’s has been assumed. A generalization of the results to [0, a] (a< 1) and [0, θ] (θ< 2π) is possible by making an appropriate change of the projection operators C¯p± or C¯p(Z).
  47. For DAE’s, Hermitian signals are of no concern because of the zeroth order in the diffraction pattern of a DAE. Thus the first row of relation (45) is unimportant for DAE’s.
  48. F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–357 (1991).
    [CrossRef]
  49. H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
    [CrossRef] [PubMed]
  50. T. Peter, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. (to be published).
  51. H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).
  52. D. C. Chu, J. W. Goodmann, “Spectrum shaping with parity sequences,” Appl. Opt. 11, 1716–1724 (1972).
    [CrossRef] [PubMed]
  53. H. Akahori, “Comparison of deterministic phase coding with random phase coding in terms of dynamic range,” Appl. Opt. 12, 2336–2343 (1973).
    [CrossRef] [PubMed]
  54. W. J. Dallas, “Deterministic diffusers for holography,” Appl. Opt. 12, 1179–1187 (1973).
    [CrossRef] [PubMed]
  55. F. Wyrowski, O. Bryngdahl, “Computer holography: object dependent deterministic diffusers,” Opt. Commun. 63, 81–84 (1987).
    [CrossRef]
  56. U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
    [CrossRef] [PubMed]
  57. H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
    [CrossRef]

1992 (6)

1991 (7)

1990 (1)

1989 (8)

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

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Eng. 14, 479–481 (1989).

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

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
[CrossRef] [PubMed]

N. C. Gallagher, D. W. Sweeney, “Computer-generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

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

1988 (1)

1987 (2)

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]

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

1986 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1984 (1)

1980 (1)

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

1978 (1)

1976 (1)

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

1974 (1)

1973 (4)

1972 (2)

D. C. Chu, J. W. Goodmann, “Spectrum shaping with parity sequences,” Appl. Opt. 11, 1716–1724 (1972).
[CrossRef] [PubMed]

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–24 (1972).

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1970 (2)

W. H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

1969 (1)

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]

1967 (2)

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

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

1966 (1)

Aarts, E. H. L.

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
[CrossRef]

Akahori, H.

Allebach, J. P.

Bräuer, R.

Brenner, K.-H.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Brown, B. R.

Bryngdahl, O.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578, 1991.
[CrossRef]

R. Bräuer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers in optical holography,” Opt. Lett. 16, 1427–1429, 1991.
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

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

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

F. Wyrowski, 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]

H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).

T. Peter, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. (to be published).

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

Burch, J. J.

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

Chu, D. C.

Dallas, W. J.

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

W. J. Dallas, Computer-Generated Holograms, Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980).

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Feldman, M. R.

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Eng. 14, 479–481 (1989).

Fienup, J. R.

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

D. C. Chu, J. R. Fienup, J. W. Goodman, “Multiemulsion on-axis computer generated hologram,” Appl. Opt. 12, 1386–1388 (1973).
[CrossRef] [PubMed]

Flavin, M. A.

Floyd, R. W.

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

Gallagher, N. C.

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

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, 237–24 (1972).

Goodman, J. W.

Goodmann, J. W.

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Guest, C. C.

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Eng. 14, 479–481 (1989).

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]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971).

Horner, J. L.

Hsueh, C. K.

Huang, A.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Ichikawa, H.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Jaakkola, T.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Jahns, J.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Jewell, J.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

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]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971).

Krackhardt, U.

Kuisma, S.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Lee, W. H.

W. H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

W. H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics 16, E. Wolf, ed. (North-Holland, New York, 1978), Chap. 3, pp. 119–223.
[CrossRef]

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]

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.

Lohmann, A. W.

Lüpken, H.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).

Mait, J. N.

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics 21, E. Wolf, ed. (North-Holland, New York, 1984), Chap. 1, pp. 1–67.
[CrossRef]

Miller, D. A. B.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Miller, J. M.

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Murdocca, M.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Noponen, E.

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Paris, D. P.

Pauka, T.

H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).

Peter, T.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

T. Peter, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. (to be published).

Prise, M. E.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Sawchuk, A. A.

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, 237–24 (1972).

Seldowitz, M. A.

Sizer, T.

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Steinberg, L.

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

Streibl, N.

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

Sweeney, D. W.

Taghizadeh, M. R.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Turunen, J.

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

van Laarhoven, P. J. M.

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
[CrossRef]

Vasara, A.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Weissbach, S.

S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[CrossRef] [PubMed]

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

Westerholm, J.

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

Wojak, U.

Wyrowski, F.

S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[CrossRef] [PubMed]

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[CrossRef]

F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–357 (1991).
[CrossRef]

F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
[CrossRef] [PubMed]

R. Bräuer, U. Wojak, F. Wyrowski, O. Bryngdahl, “Digital diffusers in optical holography,” Opt. Lett. 16, 1427–1429, 1991.
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

R. Bräuer, F. Wyrowski, O. Bryngdahl, “Diffusers in digital holography,” J. Opt. Soc. Am. A 8, 572–578, 1991.
[CrossRef]

F. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. 30, 4650–4657 (1991).
[CrossRef] [PubMed]

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

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

T. Peter, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. (to be published).

H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).

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

Appl. Opt. (18)

D. C. Chu, J. R. Fienup, J. W. Goodman, “Multiemulsion on-axis computer generated hologram,” Appl. Opt. 12, 1386–1388 (1973).
[CrossRef] [PubMed]

W. H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

M. A. Flavin, J. L. Horner, “Amplitude encoded phase-only filters,” Appl. Opt. 28, 1692–1696 (1989).
[CrossRef] [PubMed]

F. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. 30, 4650–4657 (1991).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

S. Weissbach, F. Wyrowski, “Error diffusion procedure: theory and applications in optical signal processing,” Appl. Opt. 31, 2518–2534 (1992).
[CrossRef] [PubMed]

N. C. Gallagher, B. Liu, “Method for computing kino-forms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
[CrossRef] [PubMed]

B. Liu, N. C. Gallagher, “Convergence of a spectrum shaping algorithm,” Appl. Opt. 13, 2470–2471 (1974).
[CrossRef] [PubMed]

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

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[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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H. Akahori, “Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform,” Appl. Opt. 25, 802–811 (1986).
[CrossRef] [PubMed]

U. Krackhardt, J. N. Mait, N. Streibl, “Upper bound on the diffraction efficiency of phase-only fan-out elements,” Appl. Opt. 31, 27–37 (1992).
[CrossRef] [PubMed]

IBM J. Res. Dev. (1)

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. A (6)

Opt. Commun. (7)

F. Wyrowski, “Efficiency of quantized diffractive phase elements,” Opt. Commun. 92, 119–126 (1992).
[CrossRef]

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

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

F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–357 (1991).
[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]

A. Vasara, E. Noponen, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction analysis of Dammann gratings,” Opt. Commun. 91, 337–342 (1991).
[CrossRef]

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Eng. (4)

N. C. Gallagher, D. W. Sweeney, “Computer-generated microwave kinoforms,” Opt. Eng. 28, 599–604 (1989).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

M. R. Feldman, C. C. Guest, “Iterative encoding of high-efficiency holograms for generation of spot arrays,” Opt. Eng. 14, 479–481 (1989).

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

Opt. Lett. (2)

Optik (2)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–24 (1972).

H. Dammann, “Blazed synthetic phase-only holograms,” Optik 31, 95–104 (1970).

Proc. IEEE (3)

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

N. Streibl, K.-H. Brenner, A. Huang, J. Jahns, J. Jewell, A. W. Lohmann, D. A. B. Miller, M. Murdocca, M. E. Prise, T. Sizer, “Digital optics,” Proc. IEEE 77, 1954–1969 (1989).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Proc. Soc. Inf. Displ. (1)

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

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other (12)

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Rep. TKK-F-A690 (Department of Technical Physics, Helsinki University of Technology, Helsinki, Finland, 1991).

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

R. Petit, ed., Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics 21, E. Wolf, ed. (North-Holland, New York, 1984), Chap. 1, pp. 1–67.
[CrossRef]

P. M. Hirsch, J. A. Jordan, L. B. Lesem, “Method of making an object-dependent diffuser,” U.S. patent3,619,022 (November9, 1971).

W. H. Lee, “Computer generated holograms: techniques and applications,” in Progress in Optics 16, E. Wolf, ed. (North-Holland, New York, 1978), Chap. 3, pp. 119–223.
[CrossRef]

W. J. Dallas, Computer-Generated Holograms, Vol. 41 of Topics in Applied Physics (Springer-Verlag, Berlin, 1980).

P. J. M. van Laarhoven, E. H. L. Aarts, Simulated Annealing: Theory and Applications (Reidel, Dordrecht, The Netherlands, 1987).
[CrossRef]

In this paper an equidistant quantization within the interval [0, 1] for DAE’s and [0, 2π] for DPE’s has been assumed. A generalization of the results to [0, a] (a< 1) and [0, θ] (θ< 2π) is possible by making an appropriate change of the projection operators C¯p± or C¯p(Z).

For DAE’s, Hermitian signals are of no concern because of the zeroth order in the diffraction pattern of a DAE. Thus the first row of relation (45) is unimportant for DAE’s.

T. Peter, F. Wyrowski, O. Bryngdahl, “Importance of initial distribution for iterative calculation of quantized diffractive elements,” J. Mod. Opt. (to be published).

H. Lüpken, T. Pauka, R. Bräuer, F. Wyrowski, O. Bryngdahl, “On the design of Dammann gratings,” Opt. Commun. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Examples of signal waves in digital holography: (a) magnitude of the impulse response of an inverse filter for detecting the letter L, (b) intensity of the diffraction pattern of an array illuminator.

Fig. 2
Fig. 2

Mathematically, the encoding of Fs(u) as a DAE is the mapping of the values of Fs(u) onto the interval [0, 1].

Fig. 3
Fig. 3

Illustration of the mapping of three values of Fs(u) onto the interval [−1, 1]: fine arrows, by the operator C ±; dashed arrows, by a hardclip binarization [see Eq. (36)].

Fig. 4
Fig. 4

Illustration concerning an estimation of the intensity of the function Ĉ±(u).

Fig. 5
Fig. 5

Small picture: a row out of the array depicted in Fig. 1(b). After a hardclip binarization of the corresponding Fs(u), this row is drastically disturbed, as shown in the calculated diffraction pattern |g(x)|2 of the binary DAE.

Fig. 6
Fig. 6

Mathematically, the encoding of Fs(u) as a DPE is a mapping of the values of Fs(u) onto the unit circle: Fine arrow, for an arbitrary coding method; dashed arrow, for the projection operator [see Eq. (43)].

Fig. 7
Fig. 7

Illustration of the encoding of a DPE quantized in Z = 4 phase levels as the mapping of Fs(u) onto the four allowed values (bullets) in the complex plane. Fine arrows, arbitrary mapping; dashed arrows, threshold quantization [see Eq. (41)].

Fig. 8
Fig. 8

Specification of the signal wave f(x) on the basis of the application permits calculation of its spectrum and the corresponding upper bound (formulas of Table 1) in a computer. The bound can be utilized to judge the applicability of digital holography to the application of concern in terms of diffraction efficiency.

Tables (1)

Tables Icon

Table 1 Overview of Upper Bounds of the Diffraction Efficiency of Thin Diffractive Elementsa

Equations (50)

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

F G ( u ) = g ( x ) = α f ( x - x 0 ) ,             x F ;
g ( x ) 2 = α 2 f ( x - x 0 ) 2 ,             x F .
max u F s ( u ) = 1 ,
Δ F < ,
f ( x - x 0 ) 0 ,             x F .
G ( u ) H ( DAE ) = { G ( u ) 0 G ( u ) 1 }
G ( u ) H ( DPE ) = { G ( u ) G ( u ) = 1 } .
G N ( u ) = O I T G 1 ( u ) H ,
G ( u ) = C F s ( u ) H ,
η = g ( x ) 2 F .
C = C + C ± .
G ± ( u ) = C ± F s ( u ) .
G ( u ) = C + G ± ( u ) = [ G ± ( u ) + 1 ] / 2.
G ( u ) = C F s ( u ) = [ C ± F s ( u ) + 1 ] / 2.
η = ( 1 / 4 ) η ± ,
η ± = g ± ( x ) 2 F ,
g ± ( x ) = F G ± ( u ) .
G ± ( u ) = C ± F s ( u ) = F s ( u ) + C ± ( u ) .
g ± ( x ) = f ( x - x 0 ) + c ± ( x ) .
c ± ( x ) = ( α ± - 1 ) f ( x - x 0 ) + c ^ ± ( x ) ,             x F ,
c ^ ± ( x ) 0 ,             x F ;
g ± ( x ) = α ± f ( x - x 0 ) + c ^ ± ( x ) .
g ± ( x ) 2 = α ± 2 f ( x - x 0 ) 2 + c ^ ± ( x ) 2 .
η ± = α ± 2 η 0
η 0 = f ( x - x 0 ) 2 F F s ( u ) 2 .
G ± ( u ) 2 = α ± 2 F s ( u ) 2 + C ^ ± ( u ) 2
sgn { Re [ G ± ( u ) ] } 1 sgn { Re [ F s ( u ) ] } 1 ,
C ± ( u ) 2 = α ± 2 Im [ F s ( u ) ] 2 + α ± Re [ F s ( u ) ] + G ± ( u ) 2 ;
sgn { Re [ G ± ( u ) ] } 2 = sgn { Re [ F s ( u ) ] } 2 ,
C ± ( u ) 2 = α ± 2 Im [ F s ( u ) ] 2 + α ± Re [ F s ( u ) ] - G ± ( u ) 2 .
C ± ( u ) 2 α ± 2 Im [ F s ( u ) ] 2 + α ± Re [ F s ( u ) ] - G ± ( u ) 2 ,
G ± ( u ) 2 α ± 2 F s ( u ) 2 + α ± 2 Im [ F s ( u ) ] 2 + α ± 2 Re [ F s ( u ) ] 2 + G ± ( u ) 2 - 2 α ± G ± ( u ) Re [ F s ( u ) ] ,
α ± G ± ( u ) Re [ F s ( u ) ] F s ( u ) 2 ,
η G ± ( u ) Re [ F s ( u ) ] 2 4 F s ( u ) 2 .
η l = Re [ F s ( u ) ] 2 4 F s ( u ) 2 .
C ¯ p ± ( 2 ) F s ( u ) = { + 1 Re [ F s ( u ) ] 0 - 1 Re [ F s ( u ) ] < 0 .
η < η l
η ¯ ( Z ) η ¯ l ( Z ) = F s ( u ) cos [ Δ Φ ¯ p ( u ; Z ) ] 2 F s ( u ) 2 ,
Δ Φ ¯ p ( u ; Z ) = { 2 π / Z sign [ Φ s ( u ) ] - mod [ Φ s ( u ) , 2 π / Z ] τ ( Φ s ) 0 - 2 π / Z mod [ Φ s ( u ) , 2 π / Z ] otherwise ,
τ ( Φ s ) = mod [ Φ s ( u ) , 2 π / Z ] - mod [ Φ s ( u ) , π / Z ] .
C ¯ p ( Z ) F s ( u ) = exp ( i { arg [ F s ( u ) ] + Δ Φ ¯ p ( u ; Z ) } ) ,
η η l = F s ( u ) 2 F s ( u ) 2
C p F s ( u ) = exp { i arg [ F s ( u ) ] } .
η ¯ l ( Z ) = sinc 2 [ 1 / Z ] η l ,
η ¯ ( Z ) η ¯ l ( Z ) = F s ( u ) 2 F s ( u ) 2 × { 1 Hermitian signal , Z = 2 sinc 2 [ 1 / Z ] otherwise ,
[ η l ] DAE = ( 1 / 4 ) [ η ¯ l ( 2 ) ] DPE .
[ η ¯ l ] DAE = F s ( u ) 2 π 2 F s ( u ) 2 .
β = F s ( u ) 2 F s ( u ) 2 .
G J ( u ) = C i t F s ( u ) = C p ( S p C p ) J F s ( u ) .
G J ( u ) = C i t G 1 ( u ) ,

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