Abstract

We propose a novel type of multiplexed computer-generated hologram (MCGH) with irregular-shaped polygonal apertures and discrete phase levels. Each elementary cell forming the new MCGH is divided into a central aperture and several peripheral apertures. The new MCGH allows us to exploit the huge space–bandwidth product provided by standard lithography technologies. With use of the Abbe transform, the Fraunhofer diffraction patterns from the polygonal apertures and, therefore, the layout coefficients can be computed with simple algebraic expressions. Several symmetries related to the polygonal apertures also facilitate the layout-coefficient computation. In the novel iterative subhologram design algorithm (ISDA), we consider all subholograms equally and apply the image-plane constraint to the total reconstructed image, which is the coherent addition of the subimages from the subholograms. We designed MCGHs with several billions of pixels per period, which cannot be achieved with the classical iterative Fourier transform algorithm, because of the prohibitive computational cost and memory limitation. MCGHs with irregular polygonal apertures and discrete phases, which were designed by the ISDA, reconstruct a desired image of large size with high diffraction efficiencies and low reconstruction errors.

© 2002 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.
  9. J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.
  10. J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
    [CrossRef]
  11. J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
    [CrossRef]
  12. J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chaps. 1 and 2, pp. 6–62.
  13. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, Vol. 41 of Topics in Applied Research, B. R. Frieden, ed. (Springer-Verlag, Berlin, 1980), pp. 291–366.
  14. M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph 39 and IEE Materials and Devices Series 12 (SPIE Press, Bellingham, Wash., 1997), Chap. 2, pp. 139–250.
  15. 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]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berücksichtigung der Theorie der Beugung in Heliometer (Frommansche, Jena, Germany, 1888).
  19. J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated hologram with polygonal apertures,” Appl. Opt. 41, 298–307 (2002).
    [CrossRef] [PubMed]
  20. J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chap. 3, pp. 63–93.
  21. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.
  22. W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.
  23. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
    [CrossRef]
  24. J.-N. Gillet, Y. Sheng, “Multiplexing of arbitrary-shaped polygonal apertures with discrete phase levels to design computer-generated holograms,” in Diffractive Optics and Micro-Optics, Vol. 75 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 55–57.
  25. J. N. Mait, “Understanding diffractive optical design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]

2002 (1)

2000 (1)

1999 (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

1997 (1)

1995 (1)

1994 (1)

1992 (2)

1990 (1)

1989 (1)

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

1988 (1)

1982 (1)

1980 (1)

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

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1965 (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.

Brede, J.

Brown, D. R.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Bryngdahl, O.

Cook, A.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Cooley, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Dallas, W. J.

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

Delaney, B.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Dial, O.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

Ersoy, O. K.

Fienup, J. R.

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

Flannery, B. P.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Gao, X.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Gillet, J.-N.

J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated hologram with polygonal apertures,” Appl. Opt. 41, 298–307 (2002).
[CrossRef] [PubMed]

J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chaps. 1 and 2, pp. 6–62.

J.-N. Gillet, Y. Sheng, “Multiplexing of arbitrary-shaped polygonal apertures with discrete phase levels to design computer-generated holograms,” in Diffractive Optics and Micro-Optics, Vol. 75 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 55–57.

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chap. 3, pp. 63–93.

Hochmuth, D. H.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Ichikawa, H.

Jaakkola, T.

Johnson, E. G.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Kathman, A. D.

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

Komrska, J.

Kuisma, S.

Lee, C. K.

Liao, H. Z.

Lu, C. Y.

Mait, J. N.

J. N. Mait, “Understanding diffractive optical design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
[CrossRef]

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

McCord, M. A.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph 39 and IEE Materials and Devices Series 12 (SPIE Press, Bellingham, Wash., 1997), Chap. 2, pp. 139–250.

Miller, J. M.

Noponen, E.

Prather, D. W.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

Press, W. H.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Rooks, M. J.

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph 39 and IEE Materials and Devices Series 12 (SPIE Press, Bellingham, Wash., 1997), Chap. 2, pp. 139–250.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Scherer, A.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

Sheng, Y.

J.-N. Gillet, Y. Sheng, “Multiplexed computer-generated hologram with polygonal apertures,” Appl. Opt. 41, 298–307 (2002).
[CrossRef] [PubMed]

J.-N. Gillet, Y. Sheng, “Iterative simulated quenching for designing irregular-spot-array generators,” Appl. Opt. 39, 3456–3465 (2000).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

J.-N. Gillet, Y. Sheng, “Multiplexing of arbitrary-shaped polygonal apertures with discrete phase levels to design computer-generated holograms,” in Diffractive Optics and Micro-Optics, Vol. 75 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 55–57.

Straubel, R.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berücksichtigung der Theorie der Beugung in Heliometer (Frommansche, Jena, Germany, 1888).

Taghizadeh, M. R.

Teukolski, S. A.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Tukey, J. W.

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Turunen, J.

Vasara, A.

Vetterling, W. T.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

Wang, J. S.

Westerholm, J.

Wyrowski, F.

Yatagai, T.

Yoshikawa, N.

Zhuang, J. Y.

Appl. Opt. (6)

J. Opt. Soc. Am. (1)

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

Math. Comput. (1)

J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series,” Math. Comput. 19, 297–301 (1965).
[CrossRef]

Opt. Commun. (1)

J.-N. Gillet, Y. Sheng, “Irregular spot array generator with trapezoidal apertures of varying heights,” Opt. Commun. 166, 1–7 (1999).
[CrossRef]

Opt. Eng. (2)

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

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

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Other (10)

E. G. Johnson, A. D. Kathman, D. H. Hochmuth, A. 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 SPIE Critical Review Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 54–74.

J. N. Mait, D. W. Prather, X. Gao, A. Scherer, O. Dial, “Characterization of a binary subwavelength diffractive lens,” in Diffractive Optics and Micro-Optics, Vol. 41 of Trends in Optics and Photonics Series (Optical Society of America, Washington D.C., 2000), pp. 108–109.

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chaps. 1 and 2, pp. 6–62.

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

M. A. McCord, M. J. Rooks, “Electron beam lithography,” in SPIE Handbook of Microlithography, Micromachining and Microfabrication, Vol. 1: Microlithography, P. Rai-Choudhury, ed., SPIE Press Monograph 39 and IEE Materials and Devices Series 12 (SPIE Press, Bellingham, Wash., 1997), Chap. 2, pp. 139–250.

R. Straubel, Über die Berechnung der Fraunhoferschen Beregungserscheinungen durch Randintegrale mit Besondere Berücksichtigung der Theorie der Beugung in Heliometer (Frommansche, Jena, Germany, 1888).

J.-N. Gillet, Y. Sheng, “Multiplexing of arbitrary-shaped polygonal apertures with discrete phase levels to design computer-generated holograms,” in Diffractive Optics and Micro-Optics, Vol. 75 of Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 55–57.

J.-N. Gillet, “Éléments optiques diffractifs conçus avec des ouvertures trapézoı̈dales et polygonales et de nouveaux algorithmes d’optimisation,” Ph.D. thesis (Université Laval, Québec, PQ, Canada, 2001), Chap. 3, pp. 63–93.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. rev. (McGraw-Hill, New York, 1986), Chap. 18, pp. 356–384.

W. H. Press, S. A. Teukolski, W. T. Vetterling, B. P. Flannery, Numerical Recipe in C: The Art of Scientific Computing (Cambridge U. Press, Cambridge, UK, 1992), Chap. 12, pp. 496–536.

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Figures (6)

Fig. 1
Fig. 1

Polygonal layout of a MCGH in which each elementary cell is divided into a central aperture with an irregular octagonal shape and four peripheral apertures with irregular hexagonal shapes: (a) period of the MCGH with M×N=4×4 square cells, (b) cell (j, r) (the central aperture is circumscribed inside a circle of radius R=0.25/N).

Fig. 2
Fig. 2

Polygonal aperture Ω of pentagonal contour C.

Fig. 3
Fig. 3

Flow chart of the ISDA to design a MCGH with polygonal apertures (S is the number of subholograms; the bold symbols denote M×N complex-valued arrays; ×, /, and exp represent, respectively, element-by-element array multiplication, division, and exponentiation and 2 corresponds to a quadratic double summation).

Fig. 4
Fig. 4

510×510-pixel desired image with 256 gray levels.

Fig. 5
Fig. 5

512×512-pixel Fourier image reconstructed by a MCGH with 512×512 cells per period (M=512, A=510), which is composed of irregular octagonal and hexagonal apertures (layout of Fig. 1) with 16 discrete phase levels.

Fig. 6
Fig. 6

Polygonal apertures s and s that are symmetric with respect to the axis Oy.

Tables (2)

Tables Icon

Table 1 MCGHs with Irregular Octagonal and Hexagonal Apertures versus MCGHs with Triangular Apertures and Conventional CGHs with Square Pixels for Reconstructing the Desired Image of 510 × 510  Pixels Shown in Fig. 4 (A = B = 510) a

Tables Icon

Table 2 MCGHs with Irregular Octagonal and Hexagonal Apertures Taking Z Discrete Phase Levels Designed with the ISDA and an Indirect Phase-Digitization Approach for Reconstructing the Desired Image of 510 × 510 Pixels Shown in Fig. 4 (A = B = 510)

Equations (43)

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

UΩ(p)=fΩΩexp(-i2πpξ)dS,
U(p)=-fΩ(2π)2p2Ω2g dS=-fΩ(2π)2p2Cgnˆ dl,
UΩ(p)=fΩi2πp2cexp(-i2πpξ)pnˆ dl,
UΩ(p)=fΩi2πp2q=1Qpn^qLqsinc(pt^qLq)×exp(-i2πpξMq).
UΩ(p)=fΩexp(-i2πpξO)MN K(p2)q=1Qpn^qLq×sinc(pt^qLq)exp(-i2πpξMq).
K(p2)=iMN/(2πp2)
Us(m, n, j, r)=ψ(m, n)fs(j, r) 1MN×exp-i2πmM j+nN rΘs(m, n),
Θs(p)=K(p2)q=1QΛqs(p),
Λqs(p)=p (n^qLq)ssinc[p (t^qLq)s]×exp[-i2πp (ξMq)s]
Θs(p)=MNΩsexp(-i2πp ξ)dS,
Θs(0)=MNωs,
Gs(m, n)=ψ(m, n)Θs(m, n)Fs(m, n),
Fs(m, n)=1MNj=0M-1r=0N-1fs(j, r)expmM j+nN r,
T(m, n)=s=1SGs(m, n)=ψ(m, n)s=1SΘs(m, n)Fs(m, n).
η=m=-A/2A/2-1n=-B/2B/2-1|T(m, n)|21.
Θ2(m, n)=Θ1(-m, n).
Θ4(m, n)=Θ1(m, -n).
Θ3(m, n)=Θ1*(m, n).
σ1(p)=q=1Q/4Λq,5(p)
Θ5(m, n)=K(m2+n2)[σ1(m, n)+σ1(-m, n)-σ1*(m, n)-σ1*(-m, n)]=K(m2+n2){Im[σ1(m, n)]+Im[σ1(-m, n)]},
Fs(1)(m, n)=H(m, n),
fs(k)(j, r)=FFT-1{Fs(k)(m, n)},
f˜s(k)(j, r)=exp[iφ˜s(k)(j, r)],
T˜(k)(m, n)=s=15Θs(m, n)F˜s(k)(m, n),
T(k)(m, n)=β(k)H(m, n) T˜(k)(m, n)|T˜(k)(m, n)|ifm[-A/2, A/2-1]andn[-B/2, B/2-1]0orT˜(k)(m, n)otherwise.
Gs(k+1)(m, n)=T(k)(m, n)-s=1ssSΘs(m, n)F˜s(k)(m, n),
E(k)=m=-M/2M/2-1n=-N/2N/2-1|T(k)(m, n)-T˜(k)(m, n)|2.
φ˜s(k)(j, r)=DZ{φs(k)(j, r)},
DZ{ϕ}
=0ifϕ<π/ZorΦ2π(Z-0.5)/Z2πz/Zif2π(z-0.5)/Zϕ<2π(z+0.5)/Z,
Θs(m, n)=Θs(-m, n).
Λqs(m, n)=(-mνqx+nνqy)Lqsinc[(-mτqx+nτqy)Lq]×exp[-i2π(-mxq+nyq)]=Λqs(-m, n).
Λqs(m, n)=Λqs(-m, n).
Θs(m, n)=K(m2+n2)q=1QΛqs(m, n)=K[(-m)2+n2]q=1QΛqs(-m, n)=Θs(-m, n).
Θs(m, n)=Θs(m, -n).
Λqs(m, n)=Λqs(m, -n)
Θs(p)=Θs*(p).
Λqs(p)=-pn^qLqsinc(pt^qLq)exp(i2πpξMq)=-Λqs*(p),
Λqs(p)=-Λqs*(p).
Θs(p)=iMN2πp2q=1QΛqs(p)=-iMN2πp2q=1QΛqs*(p)=K*(p2)q=1QΛqs*(p)=Θs*(p).
Θs(p)=Θs(p)exp(-i2πpξt),
Λqs(p)=Λqs(p)exp(-i2πξt).
Λqs(p)=pn^qL1sinc(pt^qLq)×exp[-i2πp(ξMq+ξt)]=Λqs(p)exp(-i2πpξt),

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