Abstract

A review and comparison of design procedures for binary-phase and multiphase Fourier gratings used as array generators is presented. Grating structures include one- and two-dimensional binary-phase Dammann gratings, general binary-phase gratings, quarternary-phase Dammann gratings, and kinoforms for which coherent and incoherent designs are considered. One coherent method of design is that due to Dammann, which involves the solution of a set of N nonlinear equations in N unknowns. Although Dammann’s method generates little error, it does not permit the explicit maximization of diffraction efficiency. To increase diffraction efficiency, Dammann’s method is modified such that diffraction efficiency is a design parameter. To ensure the existence of a grating that has high diffraction and generates the desired source array, the number of grating parameters are increased, and an incoherent design is considered. Simulated annealing is applied to the solution of this problem. Examples of grating design using the different techniques are presented.

© 1990 Optical Society of America

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References

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  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]
  2. See, for example, the Session 2 papers on fabrication in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052(1989).
  3. J. A. Davis, S. W. Connely, G. W. Bach, R. A. Lilly, D. M. Contrell, “Programmable optical interconnections with large fan-out capability using the magneto-optic spatial light modulator,” Opt. Lett. 14, 102–104 (1989).
    [Crossref] [PubMed]
  4. M. W. Farn, J. W. Goodman, “Optimal binary phase-only matched filters,” Appl. Opt. 27, 4431–4437 (1988).
    [Crossref] [PubMed]
  5. D. A. Buralli, G. M. Morris, “Design of a wide-field diffractive landscape lens,” Appl. Opt. (to be published).
  6. Other applications were presented at the Symposium on Novel Applications of Diffractive Optics at the 1989 Annual Meeting of the Optical Society of America.
  7. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [Crossref]
  8. H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
    [Crossref]
  9. F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
    [Crossref]
  10. U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
    [Crossref]
  11. W. B. Veldkamp, J. R. Leger, G. J. Swanson, “Coherent summation of laser beams using binary phase gratings,” Opt. Lett. 11, 303–305 (1986).
    [Crossref] [PubMed]
  12. J. N. Mait, K.-H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
    [Crossref] [PubMed]
  13. J. N. Mait, “Design of Dammann gratings for optical symbolic substitution,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 646–652 (1989).
    [Crossref]
  14. R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
    [Crossref]
  15. J. Jahns, N. Streibl, S. J. Walker, “Multilevel phase structures for array generation,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 198–203 (1989).
    [Crossref]
  16. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [Crossref]
  17. M. R. Feldman, C. C. Guest, “High efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [Crossref] [PubMed]
  18. M. S. Kim, C. C. Guest, “Block quantization of an annealed binary phase hologram for interconnects,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151 (to be published).
  19. J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
    [Crossref]
  20. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a new method,” submitted to Opt. Eng.
  21. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [Crossref] [PubMed]
  22. 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 (to be published).
  23. J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).
    [Crossref]
  24. J. N. Mait, “Design of Dammann gratings for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
    [Crossref] [PubMed]
  25. J. Turunen, A. Vasara, J. Westerholm, “Generalized two-dimensional binary phase gratings,” presented at the 1988 Annual Meeting of the Optical Society of America.
  26. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).
  27. Corrected data from Ref. 8 for N= 3 and N= 4 (corrections shown in boldface):Nηze,1ze,2ze,3ze,4337.70.1810.3100.436365.50.1220.3440.395356.70.0380.3270.468466.30.1000.1590.3701.492
  28. U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. (to be published).
  29. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [Crossref]

1989 (6)

J. A. Davis, S. W. Connely, G. W. Bach, R. A. Lilly, D. M. Contrell, “Programmable optical interconnections with large fan-out capability using the magneto-optic spatial light modulator,” Opt. Lett. 14, 102–104 (1989).
[Crossref] [PubMed]

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[Crossref]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

M. R. Feldman, C. C. Guest, “High efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[Crossref] [PubMed]

J. N. Mait, “Design of Dammann gratings for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
[Crossref] [PubMed]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[Crossref]

1988 (3)

1986 (1)

1982 (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
[Crossref]

1977 (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

1973 (1)

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]

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]

Acklin, B.

R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
[Crossref]

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[Crossref]

Bach, G. W.

Brenner, K.-H.

Buralli, D. A.

D. A. Buralli, G. M. Morris, “Design of a wide-field diffractive landscape lens,” Appl. Opt. (to be published).

Connely, S. W.

Contrell, D. M.

Dammann, H.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

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

Dändliker, R.

R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
[Crossref]

Davis, J. A.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

Farn, M. W.

Feldman, M. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Gallagher, N. C.

Goodman, 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, “High efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
[Crossref] [PubMed]

M. S. Kim, C. C. Guest, “Block quantization of an annealed binary phase hologram for interconnects,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151 (to be published).

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]

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

J. Jahns, N. Streibl, S. J. Walker, “Multilevel phase structures for array generation,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 198–203 (1989).
[Crossref]

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (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]

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
[Crossref]

Kim, M. S.

M. S. Kim, C. C. Guest, “Block quantization of an annealed binary phase hologram for interconnects,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151 (to be published).

Klotz, E.

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Krackhardt, U.

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. (to be published).

Leger, J. R.

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]

Lilly, R. A.

Lin, G.

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

Liu, B.

Mait, J. N.

J. N. Mait, “Design of Dammann gratings for two-dimensional, nonseparable, noncentrosymmetric responses,” Opt. Lett. 14, 196–198 (1989).
[Crossref] [PubMed]

J. N. Mait, K.-H. Brenner, “Optical symbolic substitution: system design using phase-only holograms,” Appl. Opt. 27, 1692–1700 (1988).
[Crossref] [PubMed]

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).
[Crossref]

J. N. Mait, “Design of Dammann gratings for optical symbolic substitution,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 646–652 (1989).
[Crossref]

McCormick, F. B.

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[Crossref]

Morris, G. M.

D. A. Buralli, G. M. Morris, “Design of a wide-field diffractive landscape lens,” Appl. Opt. (to be published).

Pedrini, G.

R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
[Crossref]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
[Crossref]

Salin, A.

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

Streibl, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

J. Jahns, N. Streibl, S. J. Walker, “Multilevel phase structures for array generation,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 198–203 (1989).
[Crossref]

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. (to be published).

Swanson, G. J.

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[Crossref]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Thalmann, R.

R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
[Crossref]

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a new method,” submitted to Opt. Eng.

J. Turunen, A. Vasara, J. Westerholm, “Generalized two-dimensional binary phase gratings,” presented at the 1988 Annual Meeting of the Optical Society of America.

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a new method,” submitted to Opt. Eng.

J. Turunen, A. Vasara, J. Westerholm, “Generalized two-dimensional binary phase gratings,” presented at the 1988 Annual Meeting of the Optical Society of America.

Veldkamp, W. B.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

J. Jahns, N. Streibl, S. J. Walker, “Multilevel phase structures for array generation,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 198–203 (1989).
[Crossref]

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a new method,” submitted to Opt. Eng.

J. Turunen, A. Vasara, J. Westerholm, “Generalized two-dimensional binary phase gratings,” presented at the 1988 Annual Meeting of the Optical Society of America.

Wyrowski, F.

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 (to be published).

Appl. Opt. (3)

Fiber Integ. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
[Crossref]

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. Phys. D (1)

J. Turunen, A. Vasara, J. Westerholm, G. Lin, A. Salin, “Optimisation and fabrication of grating beamsplitters,” J. Phys. D 21, S102–S105 (1988).
[Crossref]

Opt. Acta (1)

H. Dammann, E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977).
[Crossref]

Opt. Commun. (1)

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. (3)

F. B. McCormick, “Generation of large spot arrays from a single laser beam by multiple imaging with binary phase gratings,” Opt. Eng. 28, 299–304 (1989).
[Crossref]

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[Crossref]

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[Crossref]

Opt. Lett. (4)

Other (14)

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 (to be published).

J. N. Mait, “Extensions to Dammann’s method of binary-phase grating design,” Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 41–46 (1989).
[Crossref]

M. S. Kim, C. C. Guest, “Block quantization of an annealed binary phase hologram for interconnects,” in Optical Information Processing Systems and Architectures, B. Javidi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1151 (to be published).

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a new method,” submitted to Opt. Eng.

J. Turunen, A. Vasara, J. Westerholm, “Generalized two-dimensional binary phase gratings,” presented at the 1988 Annual Meeting of the Optical Society of America.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, Cambridge, 1986).

Corrected data from Ref. 8 for N= 3 and N= 4 (corrections shown in boldface):Nηze,1ze,2ze,3ze,4337.70.1810.3100.436365.50.1220.3440.395356.70.0380.3270.468466.30.1000.1590.3701.492

U. Krackhardt, N. Streibl, “Design of Dammann-gratings for array generation,” Opt. Commun. (to be published).

D. A. Buralli, G. M. Morris, “Design of a wide-field diffractive landscape lens,” Appl. Opt. (to be published).

Other applications were presented at the Symposium on Novel Applications of Diffractive Optics at the 1989 Annual Meeting of the Optical Society of America.

See, for example, the Session 2 papers on fabrication in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052(1989).

J. N. Mait, “Design of Dammann gratings for optical symbolic substitution,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 646–652 (1989).
[Crossref]

R. Thalmann, G. Pedrini, B. Acklin, R. Dändliker, “Optical symbolic substitution using diffraction gratings,” in Optical Computing 88, J. W. Goodman, P. Chavel, G. Roblin, eds., Proc. Soc. Photo-Opt. Instrum. Eng.963, 635–641 (1989).
[Crossref]

J. Jahns, N. Streibl, S. J. Walker, “Multilevel phase structures for array generation,” in Holographic Optics: Optically and Computer Generated, I. N. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1052, 198–203 (1989).
[Crossref]

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Figures (14)

Fig. 1
Fig. 1

Representation of a single period P ˜ ( e ) ( u ) of an even-ordered Dammann grating P(e)(u).

Fig. 2
Fig. 2

Representation of a single period P ˜ ( o ) ( u ) of an odd Dammann grating P(o)(u).

Fig. 3
Fig. 3

Single-period representation of a kinoform.

Fig. 4
Fig. 4

Single-period representation of a two-dimensional, nonseparable binary-phase grating.

Fig. 5
Fig. 5

Source array for one-dimensional, real design: (a) desired response, (b) even component of desired response, (c) odd component of desired response.

Fig. 6
Fig. 6

Predicted responses for one-dimensional, real Dammann design: (a) total, (b) even, (c) odd.

Fig. 7
Fig. 7

Dammann gratings corresponding to responses in Fig. 6: (a) total, (b) even, (c) odd.

Fig. 8
Fig. 8

(a) Dammann grating and (b) magnitude response corresponding to one-dimensional, complex Dammann design.

Fig. 9
Fig. 9

(a) Kinoform and (b) magnitude response corresponding to one-dimensional, real Dammann design.

Fig. 10
Fig. 10

(a) Kinoform and (b) magnitude response corresponding to one-dimensional, complex Dammann design.

Fig. 11
Fig. 11

(a) Kinoform and (b) magnitude response corresponding to one-dimensional, complex Dammann design with α as a design parameter.

Fig. 12
Fig. 12

(a) Annealed kinoform and (b) magnitude response corresponding to one-dimensional, complex design. Spatial resolution is 1/16, and phase quantization step size is π/4.

Fig. 13
Fig. 13

(a) Annealed kinoform and (b) magnitude response corresponding to one-dimensional, incoherent design. Spatial resolution is 1/16, and phase quantization step size is π/4.

Fig. 14
Fig. 14

(a) Annealed kinoform and (b) magnitude response corresponding to one-dimensional, incoherent design using magnitude error of measure. Spatial resolution is 1/16, and phase quantization step size is π/128.

Tables (9)

Tables Icon

Table 1 One-Dimensional Binary-Phase-Grating Representations and Corresponding Source-Array Symmetry Characteristics

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Table 2 One-Dimensional Kinoform Representations and Corresponding Source-Array Symmetry Characteristics

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Table 3 Dammann Grating Parameters for 2N + 1 Complex Source Array

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Table 4 Kinoform Grating Parameters for 2N + 1 Complex Source Arraya

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Table 5 Dammann Grating Parameters for 2N + 1 Complex Source Array with Scale Factor as a Design Parameter

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Table 6 Kinoform Grating Parameters for 2N + 1 Complex Source Array with Scale Factor as a Design Parametera

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Table 7 Results of Real Grating Designa

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Table 8 Results of Complex Grating Designa

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Table 9 Results of Incoherent Grating Designa

Equations (77)

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P ( u ) = P ˜ ( u ) * comb ( u ) ,
p ( x ) = n = p ˜ ( n ) δ ( x n ) ,
p ˜ ( n ) = 1 / 2 1 / 2 P ˜ ( u ) exp ( j 2 π n u ) d u .
P ˜ ( e ) ( u ) = ( 1 ) N z rect ( u ) 2 k = 1 N z ( 1 ) k rect [ u ( z k + z k ) / 2 z k z k ] .
p ˜ ( e ) ( n ) = ( 1 ) N z sinc ( n ) 2 k = 1 N z ( 1 ) k ( z k z k ) × sinc [ ( z k z k ) n ] exp [ j π ( z k + z k ) n ] = ( 1 ) N z sinc ( n ) 2 k = 1 N z ( 1 ) k { [ z k sinc ( 2 z k n ) z k × sinc ( 2 z k n ) ] + j [ z k cosc ( 2 z k n ) z k cosc ( 2 z k n ) ] } ,
cosc ( x ) = 1 cos π x π x .
p ˜ e ( n ) = ( 1 ) N z sinc ( n ) 2 k = 1 N z ( 1 ) k 2 z k sinc ( 2 z k n ) .
P ˜ ( o ) ( u ) = { ( 1 ) N z rect ( u ) 2 k = 1 N z ( 1 ) k × rect [ u ( z k + z k ) / 2 z k z k ] } sgn ( u z 0 ) ,
p ˜ ( o ) ( n ) = j exp ( j 2 π z 0 n ) ( ( 1 ) N z [ 1 cos ( π n ) exp ( j 2 π z 0 n ) π n ] 2 k = 1 N z ( 1 ) k ( z k z k ) × { 1 cos [ π ( z k z k ) n ] exp [ j π ( z k + z k 2 z 0 ) n ] π ( z k z k ) n } ) = [ 2 z 0 sinc ( 2 z 0 n ) + j 2 z 0 cosc ( 2 z 0 n ) ] + j ( 1 ) N z × cosc ( n ) 2 k = 1 N z ( 1 ) k { [ z k sinc ( 2 z k n ) + z k sinc ( 2 z k n ) ] + j [ z k cosc ( 2 z k n ) + z k cosc ( 2 z k n ) ] } .
p ˜ o ( n ) = j [ ( 1 ) N z cosc ( n ) 2 k = 1 N z ( 1 ) k 2 z k cosc ( 2 z k n ) ] .
P ˜ 2 ϕ ( u ) = exp ( j ϕ 2 ) rect ( u ) + [ exp ( j ϕ 1 ) exp ( j ϕ 2 ) ] ( u ) ,
( e ) ( u ) = ( 1 ) N z k = 1 N z ( 1 ) k rect [ u ( z k + z k ) / 2 z k z k ]
( o ) ( u ) = rect [ u ( z 0 1 / 2 ) / 2 z 0 + 1 / 2 ] + ( 1 ) N z sgn ( u z 0 ) × k = 1 N z ( 1 ) k rect [ u ( z k + z k ) / 2 z k z k ] .
p ˜ 2 ϕ ( n ) = exp ( j ϕ 2 ) sinc ( n ) + [ exp ( j ϕ 1 ) exp ( j ϕ 2 ) ] σ ( n ) = exp ( j ϕ 2 ) sinc ( n ) + 2 j σ ( n ) sin ( ϕ 1 ϕ 2 2 ) × exp ( j ϕ 1 + ϕ 2 2 ) ,
p ˜ 2 ϕ ( c ) ( n ) = exp ( j ϕ 1 ) sinc ( n ) + 2 j sin ( ϕ 1 ) σ ( n ) .
p ˜ 2 ϕ ( s ) ( n ) = exp ( j ϕ 1 ) sinc ( n ) 2 cos ( ϕ 1 ) σ ( n ) .
g ( n ) = ( 1 / 2 ) [ g ( n ) + g * ( n ) ] + ( 1 / 2 ) [ g ( n ) g * ( n ) ] = g s ( n ) + g a ( n ) .
g s ( n ) = g s * ( n ) , g a ( n ) = g a * ( n ) .
G ( u ) = G s ( u ) + G a ( u ) = ( 1 / 2 ) [ G ( u ) + G * ( u ) ] + ( 1 / 2 ) [ G ( u ) G * ( u ) ] = Re { G ( u ) } + j Im { G ( u ) } .
P ˜ ( e e ) ( u ) = 1 2 [ P ˜ 1 ( e ) ( u ) + j P ˜ 2 ( e ) ( u ) ] ,
P ˜ ( e o ) ( u ) = 1 2 [ P ˜ ( e ) ( u ) + j P ˜ ( o ) ( u ) ] ,
P ˜ ( o e ) ( u ) = 1 2 [ P ˜ ( o ) ( u ) + j P ˜ ( e ) ( u ) ] ,
P ˜ ( o o ) ( u ) = 1 2 [ P ˜ 1 ( o ) ( u ) + j P ˜ 2 ( o ) ( u ) ] .
P ˜ κ ( u ) = k = 1 N ϕ exp ( j ϕ k ) rect [ u ( z k + z k 1 ) / 2 ( z k z k 1 ) ] ,
p ˜ κ ( n ) = k = 1 N ϕ cos ϕ k [ z k sinc ( 2 z k n ) z k 1 sinc ( 2 z k 1 n ) ]   sin ϕ k [ z k cosc ( 2 z k n ) z k 1 cosc ( 2 z k 1 n ) ] + j { sin ϕ k [ z k sinc ( 2 z k n ) z k 1 sinc ( 2 z k 1 n ) ] + cos ϕ k × [ z k cosc ( 2 z k n ) z k 1 cosc ( 2 z k 1 n ) ] } ,
p ˜ κ e ( n ) = 2 k = 1 N ϕ / 2 ( z k z k 1 ) sinc [ ( z k z k 1 ) n ] cos [ π ( z k + z k 1 ) n + ( ϕ N ϕ + 1 k ϕ k ) / 2 ] exp [ j ( ϕ N ϕ + 1 k + ϕ k ) / 2 ] ,
p ˜ κ o ( n ) = exp ( j ϕ N ϕ ) ( z ( N ϕ + 1 ) / 2 z ( N ϕ 1 ) / 2 ) × sinc [ ( z ( N ϕ + 1 ) / 2 z ( N ϕ 1 ) / 2 ) n ] + 2 k = 1 ( N ϕ 1 ) / 2 ( z k z k 1 ) × sinc [ ( z k z k 1 ) n ] cos [ π ( z k + z k 1 ) n + ( ϕ N ϕ + 1 k ϕ k ) / 2 ] exp [ j ( ϕ N ϕ + 1 k + ϕ k ) / 2 ] .
P ˜ κ ( u ) = m = 1 M exp [ j Ө 2 π / 2 m ( u ) ] ,
ϕ 1 = 0 ,
ϕ 2 = 2 π / 2 m .
P ( u , υ ) = P ˜ ( u , υ ) * * comb ( u , υ ) .
p ( x , y ) = n = m = p ˜ ( n , m ) δ ( x n , y m ) ,
p ˜ ( n , m ) = 1 / 2 1 / 2 1 / 2 1 / 2 P ˜ ( u , υ ) exp [ j 2 π ( n u + m υ ) ] d u d υ .
P ˜ ( s p ) ( u , υ ) = P ˜ 1 ( u ) P ˜ 2 ( υ ) ,
P ˜ 2 ϕ ( n s ) ( u , υ ) = exp ( j ϕ 2 ) rect ( u , υ ) + [ exp ( j ϕ 1 ) exp ( j ϕ 2 ) ] × k = 1 K rect [ u ( u k , 2 + u k , 1 ) / 2 u k , 2 u k , 1 , υ ( υ k , 2 + υ k , 1 ) / 2 υ k , 2 υ k , 1 ] ,
p ˜ 2 ϕ ( n s ) ( n , m ) = exp ( j ϕ 2 ) sinc ( n , m ) + [ exp ( j ϕ 1 ) exp ( j ϕ 2 ) ] × k = 1 K ( u k , 2 u k , 1 ) ( υ k , 2 υ k , 1 ) sinc [ ( u k , 2 u k , 1 ) n × ( υ k , 2 υ k , 1 ) m ] exp { j π [ ( u k , 2 + u k , 1 ) n + ( υ k , 2 + υ k , 1 ) m ] } ,
P ˜ ( u , υ ) = P ˜ ( u , υ ) ,
P ˜ ( u , υ ) = P ˜ ( u , υ ) ,
p ˜ ( n , m ) = ( 1 / 2 ) [ p ˜ ( n , m ) + p ˜ * ( n , m ) ] + ( 1 / 2 ) [ p ˜ ( n , m ) p ˜ * ( n , m ) ] ,
P ˜ 4 ϕ ( n s ) ( u , υ ) = 1 2 [ P ˜ 1 ( n s ) ( u , υ ) + j P ˜ 2 ( n s ) ( u , υ ) ] ,
P ˜ κ ( n s ) ( u , υ ) = k = 1 K exp ( j ϕ k ) × rect [ u ( u k , 2 + u k , 1 ) / 2 u k , 2 u k , 1 , υ ( υ k , 2 + υ k , 1 ) / 2 υ k , 2 υ k , 1 ] .
p ˜ ( n ) = α q ( n ) , n = [ N , N ] .
η = n = N N | p ( n ) | 2 n = | p ( n ) | 2 = n = N N | p ˜ ( n ) | 2 1 / 2 1 / 2 | P ˜ ( u ) | 2 d u = n = N N | p ˜ ( n ) | 2 .
E coh = n = N N | p ˜ ( n ) / α q ( n ) | 2 .
η ˆ = n = N N | q ( n ) | 2 ,
SNR coh = η ˆ / E coh .
| p ˜ ( n ) | = α | q ( n ) | , n = [ N , N ] .
E inc = n = N N | | p ˜ ( n ) | 2 / α 2 | q ( n ) | 2 | 2 ,
SNR inc = η ˆ / E inc .
q ( n ) = q e ( n ) + q o ( n ) .
q ( 2 ) = 1 , q ( 1 ) = 0 , q ( 0 ) = 0 , q ( 1 ) = 1 , q ( 2 ) = 1.
q e ( 0 ) = 0 , q e ( 1 ) = 1 / 2 , and q e ( 2 ) = 1
q o ( 0 ) = 0 , q o ( 1 ) = 1 / 2 , and q o ( 2 ) = 0.
α q e ( n ) = ( 1 ) N sinc ( n ) 2 k = 1 N ( 1 ) k 2 z e , k sinc ( 2 z e , k n ) , n = [ 0 , N ] .
α q e ( 0 ) = 1 + 4 z e , 1 4 z e , 2 , α q e ( 1 ) = 4 z e , 1 sinc ( 2 z e , 1 ) 4 z e , 2 sinc ( 2 z e , 2 ) , α q e ( 2 ) = 4 z e , 1 sinc ( 4 z e , 1 ) 4 z e , 2 sinc ( 4 z e , 2 ) .
z e , 1 = 0.172309 z e , 2 = 0.422309 α = 0.527393.
α q o ( n ) = ( 1 ) N + 1 cosc ( n ) + 2 k = 1 N ( 1 ) k 2 z o , k cosc ( 2 z o , k n ) , n = [ 1 , N ] .
α q o ( 1 ) = cosc ( 1 ) 4 z o , 1 cosc ( 2 z o , 1 ) + 4 z o , 2 cosc ( 2 z o , 2 ) , α q o ( 2 ) = 4 z o , 1 cosc ( 4 z o , 1 ) + 4 z o , 2 cosc ( 4 z o , 2 ) .
z o , 1 = 0.125000 , z o , 2 = 0.375000.
α q ( n ) = p s ( n ) + p a ( n ) , n = [ 0 , N ] .
q ( 2 ) = exp ( j π / 3 ) , q ( 1 ) = 0 , q ( 0 ) = 0 , q ( 1 ) = 1 , q ( 2 ) = exp ( j π / 4 ) .
q s ( 0 ) = 0 , q s ( 1 ) = 1 / 2 , q s ( 2 ) = cos ( 7 π / 24 ) exp ( j π / 24 )
q a ( 0 ) = 0 , q a ( 1 ) = 1 / 2 , q a ( 2 ) = j sin ( 7 π / 24 ) exp ( j π / 24 ) .
z s , 2 = 0.349650 , z a , 2 = 0.333819 , z s , 1 = 0.301840 , z a , 1 = 0.194581 , z 0 = 0.167468 , z a , 1 = 0.062985 , z s , 1 = 0.172395 , z a , 2 = 0.423748 , z s , 2 = 0.387673 , α = 0.615140.
ϕ 1 = 1.249046 , ϕ 2 = 0.463343 , ϕ 3 = 1.892547 , ϕ 4 = 5.961435 , α = 0.402634.
ϕ 1 = 0.638102 , ϕ 2 = 2.784871 , ϕ 3 = 4.825152 , ϕ 4 = 1.551774 , ϕ 5 = 0.595096 , ϕ 6 = 5.185083 , ϕ 7 = 3.154047 , ϕ 8 = 5.508580 , ϕ 9 = 3.050902 , α = 0.2894945 ,
α max = [ n = N N | q ( n ) | 2 ] 1 / 2 .
z s , 2 = 0.329639 , z a , 2 = 0.474790 , z s , 1 = 0.270245 , z a , 1 = 0.318396 , z s , 0 = 0.152849 , z a , 0 = 0.183162 , z s , 1 = 0.169989 , z a , 1 = 0.065499 , z s , 2 = 0.382231 , z a , 2 = 0.405055.
ϕ 1 = 0.517634 , ϕ 2 = 3.075526 , ϕ 3 = 3.317389 , ϕ 4 = 0.073130 , ϕ 5 = 1.090330 , ϕ 6 = 0.565255 , ϕ 7 = 4.864360 , ϕ 8 = 4.178706 , ϕ 9 = 4.320730 , ϕ 10 = 2.039253.
α q ( n ) = p ˜ ( n ) , n = [ N , N ] = k = 1 N ϕ exp ( j ϕ k ) ( z k z k 1 ) sinc [ ( z k z k 1 ) n ] × exp [ j π ( z k + z k 1 ) n ]
E coh = n = N N | p ˜ ( n ) / α q ( n ) | 2
  ϕ k = random ( l ) 2 π / L ,
α | q ( n ) | = | p ˜ ( n ) | , n = [ N , N ] ,
E inc = n = N N | | p ˜ ( n ) | 2 / α 2 | q ( n ) | 2 | 2 .
| q ( 2 ) | = 1 , | q ( 1 ) | = 0 , | q ( 0 ) | = 0 , | q ( 1 ) | = 1 , | q ( 2 ) | = 1.
E p h = n = N N | | p ˜ ( n ) | 2 / α 2 | q ( n ) | 2 | 2 + c | θ p ˜ ( n ) θ q ( n ) | 2
E mag = n = N N | | p ˜ ( n ) | / α | q ( n ) | | 2 .

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