Abstract

A general procedure is presented for the design of diffractive optical elements, and scalar diffraction theory is used to apply it to the design of three common diffractive elements: a diffractive lens, an array generator, and a correlation filter. The procedure reveals that most common design techniques can be classified as either direct or indirect optimizations. The key feature of the design procedure is the specification of a suitable performance measure based on the designer’s understanding of the optical system and the fabrication technology used to realize the diffractive element. The use of complex-wave amplitude, scale, and phase freedoms in design is also emphasized.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
    [CrossRef]
  2. N. C. Gallagher, B. Liu, “Method for computing kinoforms that reduces image reconstruction error,” Appl. Opt. 12, 2328–2335 (1973).
    [CrossRef] [PubMed]
  3. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
    [CrossRef]
  4. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
    [CrossRef]
  5. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
    [CrossRef]
  6. M. R. Feldman, C. C. Guest, “High-efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  7. F. Wyrowski, “Digital holography as part of diffractive optics,” Rep. Prog. Phys.1481–1571 (1991).
    [CrossRef]
  8. D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.
  9. D. C. Dobson, J. A. Cox, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 32–53.
  10. G. Bao, D. C. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.
  11. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  12. J. N. Mait, J. D. Vaaler, “Necessary and sufficient conditions for bipolar incoherent spatial filtering,” J. Opt. Soc. Am. A 6, 147–149 (1989).
    [CrossRef]
  13. O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).
  14. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  15. J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.
  16. D. A. Buralli, G. M. Morris, “Design of diffractive singlets for monochromatic imaging,” Appl. Opt. 30, 2151–2158 (1991).
    [CrossRef] [PubMed]
  17. G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Lincoln Laboratory Tech. Rep. 914 (MIT Lincoln Laboratory, Lexington, Mass., 1991).
  18. Data supplied by Michael Feldman, Department of Electrical Engineering, University of North Carolina at Charlotte, Charlotte, N.C. 28223 (personal communication, February1994).
  19. W. H. Welch, J. E. Morris, M. R. Feldman, “Design and fabrication of radially symmetric computer generated holograms,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 97.
  20. U. Killat, G. Rabe, W. Rave, “Binary phase gratings for star couplers with high splitting ratio,” Fiber Integ. Opt. 4, 159–167 (1982).
    [CrossRef]
  21. R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.
  22. 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]
  23. J. R. Leger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26, 4391–4399 (1987).
    [CrossRef] [PubMed]
  24. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  25. 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]
  26. 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]
  27. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  28. 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]
  29. J. N. Mait, “Upper bound on the diffraction efficiency of phase-only array generators,” in Holographic Optics: Computer and Optically Generated, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 53–62 (1991).
    [CrossRef]
  30. H. Lüpken, F. Wyrowski, “General design concept for periodic and non-periodic diffractive phase elements,” in Diffractive and Holographic Optics Technology, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2152, 84–94 (1994).
    [CrossRef]
  31. J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
    [CrossRef]
  32. B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
    [CrossRef]
  33. G. Gheen, F. Dickey, J. DeLaurentis, “Examination of metrics and assumptions used in correlation filter design,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Miceli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 107–118 (1993).
    [CrossRef]
  34. A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
    [CrossRef] [PubMed]
  35. F. Wyrowski, “Digital phase-encoded inverse filter for optical pattern recognition,” Appl. Opt. 30, 4650–4657 (1991).
    [CrossRef] [PubMed]
  36. J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLMs in pattern recognition systems,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–261 (1993).
    [CrossRef]
  37. J. A. Davis, D. M. Cottrell, J. E. Davis, R. A. Lilly, “Fresnel lens-encoded binary phase-only filters for optical pattern recognition,” Opt. Lett. 14, 659–661 (1989).
    [CrossRef] [PubMed]
  38. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
    [CrossRef] [PubMed]
  39. V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters,” Appl. Opt. 33, 4465–4471 (1994).
    [CrossRef] [PubMed]
  40. M. W. Farn, J. W. Goodman, “Optimal maximum correlation filters for arbitrarily constrained devices,” Appl. Opt. 28, 4865–4869 (1989).
  41. A. Khan, P. K. Rajan, “Design of SLM-constrained MACE filters using simulated annealing optimization,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 284–292 (1993).
    [CrossRef]
  42. W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.
  43. T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Scale-invariant matched-filter generation by simulated annealing,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.
  44. M. S. Kim, C. C. Guest, “Simulated annealing algorithm for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
    [CrossRef] [PubMed]
  45. J. Rosen, J. Shamir, “Application of the projection-onto-constraint-sets algorithm for optical pattern recognition,” Opt. Lett. 16, 752–754 (1991).
    [CrossRef] [PubMed]

1994

1993

1992

1991

1990

1989

1987

1986

1982

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

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

1973

1971

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

Bao, G.

G. Bao, D. C. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

Bowen, T.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

Brown, D. R.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Buralli, D. A.

Casasent, D.

Cloonan, T. J.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.

Cook, A. L.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Cottrell, D. M.

Cox, J. A.

D. C. Dobson, J. A. Cox, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 32–53.

G. Bao, D. C. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

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]

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.

Davis, J. E.

Delaney, B.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

DeLaurentis, J.

G. Gheen, F. Dickey, J. DeLaurentis, “Examination of metrics and assumptions used in correlation filter design,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Miceli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 107–118 (1993).
[CrossRef]

Dickey, F.

G. Gheen, F. Dickey, J. DeLaurentis, “Examination of metrics and assumptions used in correlation filter design,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Miceli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 107–118 (1993).
[CrossRef]

Dobson, D. C.

D. C. Dobson, J. A. Cox, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 32–53.

G. Bao, D. C. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

Falkenstörfer, O.

O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).

Farn, M. W.

Feldman, M.

W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Feldman, M. R.

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

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

W. H. Welch, J. E. Morris, M. R. Feldman, “Design and fabrication of radially symmetric computer generated holograms,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 97.

Feldman, Michael

Data supplied by Michael Feldman, Department of Electrical Engineering, University of North Carolina at Charlotte, Charlotte, N.C. 28223 (personal communication, February1994).

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

Gallagher, N. C.

Gheen, G.

G. Gheen, F. Dickey, J. DeLaurentis, “Examination of metrics and assumptions used in correlation filter design,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Miceli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 107–118 (1993).
[CrossRef]

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.

Heyward, W.

W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Hochmuth, D. H.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

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]

Johnson, D. E. G.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Juday, R. D.

Kathman, A. D.

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

Keinonen, T.

O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).

Khan, A.

A. Khan, P. K. Rajan, “Design of SLM-constrained MACE filters using simulated annealing optimization,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 284–292 (1993).
[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.

Krackhardt, U.

Laude, V.

Leger, J. R.

Lilly, R. A.

Lindlein, N.

O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).

Liu, B.

Lüpken, H.

H. Lüpken, F. Wyrowski, “General design concept for periodic and non-periodic diffractive phase elements,” in Diffractive and Holographic Optics Technology, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2152, 84–94 (1994).
[CrossRef]

Mahalanobis, A.

Mait, J. 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]

J. N. Mait, J. D. Vaaler, “Necessary and sufficient conditions for bipolar incoherent spatial filtering,” J. Opt. Soc. Am. A 6, 147–149 (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]

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLMs in pattern recognition systems,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–261 (1993).
[CrossRef]

J. N. Mait, “Upper bound on the diffraction efficiency of phase-only array generators,” in Holographic Optics: Computer and Optically Generated, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 53–62 (1991).
[CrossRef]

McCormick, F. B.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.

Mears, R. J.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Scale-invariant matched-filter generation by simulated annealing,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Morris, G. M.

Morris, J. E.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

W. H. Welch, J. E. Morris, M. R. Feldman, “Design and fabrication of radially symmetric computer generated holograms,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 97.

Morrison, R. L.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.

O’Brien, D. C.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Scale-invariant matched-filter generation by simulated annealing,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

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]

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]

Rajan, P. K.

A. Khan, P. K. Rajan, “Design of SLM-constrained MACE filters using simulated annealing optimization,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 284–292 (1993).
[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]

Réfrégier, Ph.

Roff, R.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

Rosen, J.

Rowlette, J. R.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

Sarama, S. D.

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLMs in pattern recognition systems,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–261 (1993).
[CrossRef]

Schwider, J.

O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).

Shamir, J.

Stack, J.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

Streibl, N.

Swanson, G. J.

J. R. Leger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26, 4391–4399 (1987).
[CrossRef] [PubMed]

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]

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Lincoln Laboratory Tech. Rep. 914 (MIT Lincoln Laboratory, Lexington, Mass., 1991).

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]

Tam, E.

W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

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, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[CrossRef]

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

Vaaler, J. D.

van der Gracht, J.

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLMs in pattern recognition systems,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–261 (1993).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[CrossRef]

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

Veldkamp, W. B.

Vijaya Kumar, B. V. K.

Walker, S. L.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Welch, W. H.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

W. H. Welch, J. E. Morris, M. R. Feldman, “Design and fabrication of radially symmetric computer generated holograms,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 97.

Westerholm, J.

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

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[CrossRef]

Wilkinson, T. D.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Scale-invariant matched-filter generation by simulated annealing,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Wyrowski, F.

F. Wyrowski, “Design theory of diffractive elements in the paraxial domain,” J. Opt. Soc. Am. A 10, 1553–1561 (1993).
[CrossRef]

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

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

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

H. Lüpken, F. Wyrowski, “General design concept for periodic and non-periodic diffractive phase elements,” in Diffractive and Holographic Optics Technology, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2152, 84–94 (1994).
[CrossRef]

Yang, H.

W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Appl. Opt.

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

D. A. Buralli, G. M. Morris, “Design of diffractive singlets for monochromatic imaging,” Appl. Opt. 30, 2151–2158 (1991).
[CrossRef] [PubMed]

J. R. Leger, G. J. Swanson, W. B. Veldkamp, “Coherent laser addition using binary phase gratings,” Appl. Opt. 26, 4391–4399 (1987).
[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]

B. V. K. Vijaya Kumar, “Tutorial survey of composite filter designs for optical correlators,” Appl. Opt. 31, 4773–4801 (1992).
[CrossRef]

A. Mahalanobis, B. V. K. Vijaya Kumar, D. Casasent, “Minimum average correlation energy filters,” Appl. Opt. 26, 3633–3640 (1987).
[CrossRef] [PubMed]

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

R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100–5111 (1993).
[CrossRef] [PubMed]

V. Laude, Ph. Réfrégier, “Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters,” Appl. Opt. 33, 4465–4471 (1994).
[CrossRef] [PubMed]

M. W. Farn, J. W. Goodman, “Optimal maximum correlation filters for arbitrarily constrained devices,” Appl. Opt. 28, 4865–4869 (1989).

M. S. Kim, C. C. Guest, “Simulated annealing algorithm for binary phase only filters in pattern classification,” Appl. Opt. 29, 1203–1208 (1990).
[CrossRef] [PubMed]

Fiber Integ. Opt.

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

IEEE Trans. Med. Imaging

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part I—Theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

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

J. Turunen, A. Vasara, J. Westerholm, “Stripe-geometry two-dimensional Dammann gratings,” Opt. Commun. 74, 245–252 (1989).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
[CrossRef]

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

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

Opt. Lett.

Rep. Prog. Phys.

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

Other

D. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. L. Cook, D. R. Brown, B. Delaney, “Advantages of genetic algorithm optimization methods in diffractive optic design,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 54–74.

D. C. Dobson, J. A. Cox, “Mathematical modeling for diffractive optics,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 32–53.

G. Bao, D. C. Dobson, J. A. Cox, “Mathematical issues in the electromagnetic theory of gratings,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 8–11.

O. Falkenstörfer, T. Keinonen, N. Lindlein, J. Schwider, “Iterative correction of holographic lenses,” in 16th Congress of the International Commission for Optics, G. Lupkovics, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1983, 649–650 (1993).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

J. R. Rowlette, T. Bowen, R. Roff, J. Stack, J. E. Morris, W. H. Welch, M. R. Feldman, “Achromatic holographic optical elements for coupling laser diode to single mode fiber,” Proceedings of the Annual Meeting IEEE–LEOS (93CH3297-9) (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 474–475.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Lincoln Laboratory Tech. Rep. 914 (MIT Lincoln Laboratory, Lexington, Mass., 1991).

Data supplied by Michael Feldman, Department of Electrical Engineering, University of North Carolina at Charlotte, Charlotte, N.C. 28223 (personal communication, February1994).

W. H. Welch, J. E. Morris, M. R. Feldman, “Design and fabrication of radially symmetric computer generated holograms,” in Annual Meeting, Vol. 17 of 1991 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), p. 97.

R. L. Morrison, S. L. Walker, F. B. McCormick, T. J. Cloonan, “Practical applications of diffractive optics in free-space photonic switching,” in Diffractive and Miniaturized Optics, S.-H. Lee, ed., Vol. CR49 of Critical Reviews (SPIE, Bellingham, Wash., 1993), pp. 265–289.

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. 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]

J. N. Mait, “Upper bound on the diffraction efficiency of phase-only array generators,” in Holographic Optics: Computer and Optically Generated, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 53–62 (1991).
[CrossRef]

H. Lüpken, F. Wyrowski, “General design concept for periodic and non-periodic diffractive phase elements,” in Diffractive and Holographic Optics Technology, I. N. Cindrich, S.-H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2152, 84–94 (1994).
[CrossRef]

J. N. Mait, J. van der Gracht, S. D. Sarama, “Diffractive filter design for SLMs in pattern recognition systems,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 250–261 (1993).
[CrossRef]

G. Gheen, F. Dickey, J. DeLaurentis, “Examination of metrics and assumptions used in correlation filter design,” in Photonics for Processors, Neural Networks, and Memories, B. Javidi, J. L. Horner, W. J. Miceli, S. T. Kowel, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2026, 107–118 (1993).
[CrossRef]

A. Khan, P. K. Rajan, “Design of SLM-constrained MACE filters using simulated annealing optimization,” in Optical Pattern Recognition IV, D. P. Casasent, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1959, 284–292 (1993).
[CrossRef]

W. Heyward, H. Yang, E. Tam, M. Feldman, “Multilevel phase-only filter for inspection of printed circuit boards,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

T. D. Wilkinson, D. C. O’Brien, R. J. Mears, “Scale-invariant matched-filter generation by simulated annealing,” in Annual Meeting, Vol. 16 of 1993 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1993), p. 221.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Flowchart representation of design steps 4 and 5 that highlights the distinction between direct and indirect designs.

Fig. 2
Fig. 2

Optimization algorithms used for design: (a) bidirectional, (b) unidirectional.

Fig. 3
Fig. 3

Schematic representation of the function of a focusing lens.

Fig. 4
Fig. 4

Quarternary quantization of a quadratic phase function to realize a diffractive lens: (a) quadratic phase, (b) quadratic-phase modulo 2π, (c) quantized-phase function.

Fig. 5
Fig. 5

Complex-plane representation of diffractive lens values: (a) unconstrained, (b) constrained.

Fig. 6
Fig. 6

Coherent optical processor.

Fig. 7
Fig. 7

Representation of 3 × 3 fan-out. (a) Array intensity, (b) Fourier magnitude, (c) Fourier phase. Two periods in the (u, v) plane are shown. (d) Complex-plane representation.

Fig. 8
Fig. 8

Complex-plane representation of Fourier array generator values determined by IFTA for a quarternary-phase element. (a) Initial distribution, (b)–(d) intermediate distributions, (e) final distribution before final quantization. The unit circle is indicated in (a)–(c).

Fig. 9
Fig. 9

Design of multilevel-phase array generators. Binary-phase array generators and corresponding spot array intensities resulting from (a) quantization design, (b) IFTA, and (c) simulated annealing. (d)–(f) As in (a)–(c) except that the array generators are 16-phase elements. Two periods of the array generator in the (u, v) plane are shown.

Fig. 10
Fig. 10

Letters A to be recognized and classified with use of a MACE filter.

Fig. 11
Fig. 11

Representation of the MACE filter to recognize and classify the letters A in Fig. 10: (a) intensity response, (b) Fourier magnitude, (c) Fourier phase, (d) complex-plane representation of Fourier values.

Fig. 12
Fig. 12

Representation of the binary-phase DOE’s designed to function as MACE filters and their corresponding responses: (a) minimum eMSE (quantized), (b) minimum eMSE (iterated), (c) minimum eave + epeak (quantized), (d) minimum eave + epeak (annealed).

Tables (7)

Tables Icon

Table 1 Examples of Basic Physical Understanding Necessary for Diffractive Optical Design

Tables Icon

Table 2 Examples of Basic Mathematical Understanding Necessary for Diffractive Optical Design

Tables Icon

Table 3 Summary of Physics Information Used in Designs

Tables Icon

Table 4 Summary of Mathematical Information Used in Designs

Tables Icon

Table 5 Performance of Diffractive f/1 Lenses with 100-μm Diameter

Tables Icon

Table 6 Performance of Multilevel Quantized Phase Array Generators Designed to Generate a 3 × 3 Fan-Out

Tables Icon

Table 7 Performance of Binary-Phase Diffractive Elements Designed to Perform As Correlation Filters

Equations (22)

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

i ( x , y ) = - - O ( u , v ) P ( u , v ) exp { j ( π / λ z 0 ) [ ( u - x ) 2 + ( v - y ) 2 ] } d u d v .
η = i ( x 0 , y 0 ) 2 = | - - P ( u , v ) exp { j ( π / λ z 0 ) [ ( u - x 0 ) 2 + ( v - y 0 ) 2 ] } d u d v | .
Q lens ( u , v ) = exp { j - π / λ z 0 ) [ ( u - x 0 ) 2 + ( v - y 0 ) 2 ] } ,
e q = P ( u , v ) - Q lens ( u , v ) 2 d u d v .
p ( x , y ) = P ( u , v ) exp [ j 2 π ( u x + v y ) ] d u d v .
e int = ( x , y ) X | p ( x , y ) 2 - α ub 2 q ( x , y ) 2 | 2 d x d y ,
q ( x , y ) = n = 1 N q n δ ( x - x n , y - y n )
α ub = η u b / N ,
η ub = [ - 1 / 2 1 / 2 - 1 / 2 1 / 2 Q ( u , v ) d u d v ] 2 - 1 / 2 1 / 2 - 1 / 2 1 / 2 Q ( u , v ) 2 d u d v ,
arg { q ub ( x , y ) } = { 2.079287 , 2.821833 , - 2.079287 , - 2.821833 , 0.000000 , - 2.821833 , - 2.079287 , 2.821833 , 2.079287 } .
i ( x , y ) = o ( ζ , η ) p * ( - x - ζ , - y - η ) d ζ d η = o ( x , y ) p ( x , y ) .
c i ( x , y ) = f i ( x , y ) p ( x , y ) ,             i = [ 1 , N ] ,
E ave = 1 N i = 1 N E i ,
E i = F i ( u , v ) P * ( u , v ) 2 d u d v = c i ( x , y ) 2 d x d y .
Q MACE = D ^ - 1 F ^ ( F ^ + D ^ - 1 F ^ ) - 1 c ^ * .
V = c 1 , max - c 2 , max c 1 , max + c 2 , max ,
e MSE = X p ( x , y ) - α q MACE ( x - x c , y - y c ) 2 d x d y .
W X , x > W o , x + W q , x ,             W X , y > W o , y + W q , y ,
e ave = 1 N i = 1 N e i ,
e i = X c i ( x - x c , y - y c ) 2 d x d y ,
e peak = 1 N i = 1 N ( c i ( x c , y c ) - γ ) 2 .
β i = { F i ( u , v ) exp [ j ψ i ( u , v ) ] } exp [ - j ψ i ( u , v ) ] d u d v = F i ( u , v ) d u d v .

Metrics