Abstract

There is a trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements. It is caused by the inherent ill-posedness of the design problem itself. For the optimal design, the optimum trade-off needs to be obtained. The trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements is theoretically investigated based on the Tikhonov regularization theory. A novel scheme of an iterative Fourier transform algorithm with regularization to obtain the optimum trade-off is proposed.

© 2004 Optical Society of America

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References

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  1. J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, New York, 1997).
  2. B. Kress, P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, New York, 2000).
  3. J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.
  4. B. Dong, Y. Zhang, B. Gu, G. Yang, “Numerical investigation of phase retrieval in a fractional Fourier transform,” J. Opt. Soc. Am. A 14, 2709–2714 (1997).
    [CrossRef]
  5. V. A. Soifer, V. V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997).
  6. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).
  7. F. Wyrowski, “Upper bound of the diffraction efficiency of diffractive phase elements,” Opt. Lett. 16, 1915–1917 (1991).
    [CrossRef] [PubMed]
  8. F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
    [CrossRef]
  9. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized blazed phase structure,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  10. J. R. Fienup, “Phase-retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  11. J. R. Fienup, “Phase-retrieval algorithms for a complicated optical system,” Appl. Opt. 32, 1737–1746 (1993).
    [CrossRef] [PubMed]
  12. S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
    [CrossRef]
  13. V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
    [CrossRef]
  14. H. Kim, B. Lee, “Iterative Fourier transform algorithmwith adaptive regularization parameter distribution for the optimal design of diffractive optical elements,” Jpn. J. Appl. Phys., Part 1 43, L702–L705 (2004).
    [CrossRef]
  15. S. Rudnaya, “Analysis and optimal design of diffractive optical elements,” Ph.D. thesis (University of Minnesota, Minneapolis, Minn., 1999).
  16. E. G. Johnson, M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12, 1152–1160 (1995).
    [CrossRef]
  17. H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Boston, Mass., 1996).
  18. J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  19. H. Kim, B. Lee, “Calculation of the transmittance function of a multilevel diffractive optical element considering multiple internal reflections,” Opt. Eng. (to be published).
  20. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  21. H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  22. P. Senthilkumaran, F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
    [CrossRef]

2004

H. Kim, B. Lee, “Iterative Fourier transform algorithmwith adaptive regularization parameter distribution for the optimal design of diffractive optical elements,” Jpn. J. Appl. Phys., Part 1 43, L702–L705 (2004).
[CrossRef]

2002

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

P. Senthilkumaran, F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

1998

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
[CrossRef]

1997

1995

1993

1991

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

F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
[CrossRef]

1990

1982

Abushagur, M. A. G.

Backlund, J.

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

Bengtsson, J.

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

Buhling, S.

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

Dong, B.

Doskolovich, L.

V. A. Soifer, V. V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997).

Engl, H. W.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Boston, Mass., 1996).

Fienup, J. R.

Goncharsky, A. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Gu, B.

Hanke, M.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Boston, Mass., 1996).

Johnson, E. G.

Kim, H.

H. Kim, B. Lee, “Iterative Fourier transform algorithmwith adaptive regularization parameter distribution for the optimal design of diffractive optical elements,” Jpn. J. Appl. Phys., Part 1 43, L702–L705 (2004).
[CrossRef]

H. Kim, B. Lee, “Calculation of the transmittance function of a multilevel diffractive optical element considering multiple internal reflections,” Opt. Eng. (to be published).

Kotlyar, V. V.

V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
[CrossRef]

V. A. Soifer, V. V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997).

Kress, B.

B. Kress, P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, New York, 2000).

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Larsson, A.

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

Lee, B.

H. Kim, B. Lee, “Iterative Fourier transform algorithmwith adaptive regularization parameter distribution for the optimal design of diffractive optical elements,” Jpn. J. Appl. Phys., Part 1 43, L702–L705 (2004).
[CrossRef]

H. Kim, B. Lee, “Calculation of the transmittance function of a multilevel diffractive optical element considering multiple internal reflections,” Opt. Eng. (to be published).

Lindberg, H.

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

Mendlovic, D.

Meyrueis, P.

B. Kress, P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, New York, 2000).

Modh, P.

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

Neubauer, A.

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Boston, Mass., 1996).

Ozaktas, H. M.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Rudnaya, S.

S. Rudnaya, “Analysis and optimal design of diffractive optical elements,” Ph.D. thesis (University of Minnesota, Minneapolis, Minn., 1999).

Sahin, A.

Senthilkumaran, P.

P. Senthilkumaran, F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

Seraphimovich, P. G.

V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
[CrossRef]

V. A. Soifer, V. V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997).

Stepanov, V. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).

Turunen, J.

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, New York, 1997).

Wyrowski, F.

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

P. Senthilkumaran, F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

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

F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
[CrossRef]

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

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, New York, 1997).

Yagola, A. G.

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).

Yang, G.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

Zhang, Y.

Appl. Opt.

J. Mod. Opt.

P. Senthilkumaran, F. Wyrowski, “Phase synthesis in wave-optical engineering: mapping- and diffuser-type approaches,” J. Mod. Opt. 49, 1831–1850 (2002).
[CrossRef]

S. Buhling, F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49, 1871–1892 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys., Part 1

H. Kim, B. Lee, “Iterative Fourier transform algorithmwith adaptive regularization parameter distribution for the optimal design of diffractive optical elements,” Jpn. J. Appl. Phys., Part 1 43, L702–L705 (2004).
[CrossRef]

Opt. Commun.

F. Wyrowski, “Considerations on convolutions and phase factors,” Opt. Commun. 81, 353–358 (1991).
[CrossRef]

Opt. Lasers Eng.

V. V. Kotlyar, P. G. Seraphimovich, V. A. Soifer, “An iterative algorithm for designing diffractive optical elements with regularization,” Opt. Lasers Eng. 29, 261–268 (1998).
[CrossRef]

Opt. Lett.

Other

S. Rudnaya, “Analysis and optimal design of diffractive optical elements,” Ph.D. thesis (University of Minnesota, Minneapolis, Minn., 1999).

H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, Boston, Mass., 1996).

J. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

H. Kim, B. Lee, “Calculation of the transmittance function of a multilevel diffractive optical element considering multiple internal reflections,” Opt. Eng. (to be published).

V. A. Soifer, V. V. Kotlyar, L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, London, 1997).

A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer Academic, Boston, Mass., 1995).

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Wiley, New York, 1997).

B. Kress, P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, New York, 2000).

J. Bengtsson, P. Modh, J. Backlund, H. Lindberg, A. Larsson, “Progress in diffractive integrated optics,” in Diffractive Optics and Micro-Optics, R. Magnusson, ed., Vol. 75 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2002), pp. 17–19.

H. M. Ozaktas, Z. Zalevsky, M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).

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Figures (6)

Fig. 1
Fig. 1

Schematic of a paraxial optical system with a DOE.

Fig. 2
Fig. 2

Computation grid used for the DOE design. The signal area is equal to the target image of nonzero magnitude. Three regularization parameters are indicated in the corresponding regions.

Fig. 3
Fig. 3

Comparison of the trade-off curves of the four types of IFTA: conventional IFTA, IFTA with the ARPD, IFTA with the first-order Tikhonov regularization, and IFTA with the ARPD and the first-order Tikhonov regularization for (a) design A with full freedom in the DOE plane and (b) design B with decreased freedom. For a specific value of diffraction efficiency, the obtained uniformity is different for each case. The lowest value of uniformity indicates the best solution.

Fig. 4
Fig. 4

Comparative illustration of the design results with use of the conventional IFTA and the proposed IFTA with the ARPD and the first-order Tikhonov regularization in the case of design A with full freedom in the DOE plane. (a) Intensity distribution of the resulting diffraction image and (b) phase profile of the DOE obtained by the conventional IFTA. (c) Intensity distribution of the resulting diffraction image and (d) phase profile of the DOE obtained by the proposed IFTA with the ARPD and the first-order Tikhonov regularization. For both IFTAs, the iteration number is 200.

Fig. 5
Fig. 5

Same as Fig. 4 but for the case of design B with decreased freedom in the DOE plane.

Fig. 6
Fig. 6

Comparison of the changes in (a) uniformity and (b) diffraction efficiency for several values of the regularization parameter αD (in the range 0–0.2) with use of the IFTA with the first-order Tikhonov regularization and of the IFTA with the ARPD and the first-order Tikhonov regularization in the case of design A with full freedom in the DOE plane.  

Equations (47)

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Uniformity=|F|max-|F|min|F|max+|F|min,
Diffractionefficiency=S|F|2dxdy--|F|2dxdy×100(%),
MSE=S||F|-F0|2dxdy,
F(x2, y2)=Fr[G(x1, y1)]=--h(x2, y2, x1, y1)G(x1, y1)dx1dy1,
h(x2, y2, x1, y1)
=-jλ(d1+d2)-λd1d2fexpjπλ(d1+d2)-λd1d2f  ×1-d1f(x22+y22)-2(x2x1+y2y1)+1-d2f(x12+y12).
G(x1, y1)=Fr-1[F(x2, y2)]=--h-1(x1, y1, x2, y2)×F(x2, y2)dx2dy2,
h-1(x1, y1, x2, y2)
=jλ(d1+d2)-λd1d2fexp-jπλ(d1+d2)-λd1d2f  ×1-d2f(x12+y12)-2(x2x1+y2y1)+1-d1f(x22+y22).
F(x2, y2)
=-jλ(d1+d2)-λd1d2fexpjπ(1-d1/f)(x22+y22)λ(d1+d2)-λd1d2f  ×--expjπ(1-d2/f)(x12+y12)λ(d1+d2)-λd1d2f  ×G(x1, y1)exp-j2π(x2x1+y2y1)λ(d1+d2)-λd1d2f  dx1dy1.
ωM(x1)=x1π(1-d2/f)(x12+y12)λ(d1+d2)-λd1d2f  =2π(1-d2/f)x1λ(d1+d2)-λd1d2f  .
ωS>2|ωM(x1)|=4π(1-d2/f)x1λ(d1+d2)-d1d2f,
Δx1=2πωS<λ[(d1+d2)-d1d2/f]2(1-d2/f)x1.
Lx1R<λ(d1+d2-d1d2/f)2(1-d2/f)N,RLx12.
J(F)=-|DS|F|-F0|2dx2dy2+αSS|F|2dx2dy2+αNN|F|2dx2dy2,
DSF=Ffor(x2, y2)S0for(x2, y2)S,
-|F|2dx2dy2=S|F|2dx2dy2+N|F|2dx2dy2=-|F0|2dx2dy2.
δJ(F)=2-Re{[-|F0|exp(iψ)+DSF+DSαSF+(1-DS)αNF]δF*}dx2dy2,
δF=F¯-F=-τ[DS(1+αS)F+(1-DS)αNF-F0exp(jψ)],
F¯n=Fn-τ[DS(1+αS)Fn+(1-DS)αNFn-F0exp(jψn)].
Fn+1=Fr DDOE Fr-1(F¯n),
DDOEG=A0exp[j arg(G)]for(x1, y1)Ω0for(x1, y1)Ω,
F¯n=τF0exp(jψn)+(1-τ-ταS)Fnfor(x2, y2)S(1-ταN)Fnfor(x2, y2)S.
F¯n=τF0exp(jψn)+(1-τ)Fnfor(x2, y2)SβFnfor(x2, y2)S,
τ=1-ταS,
β=1-ταN1-ταS.
J(F)=(αS+1)S|F|-F0αS+12dx2dy2+αNN|F|2dx2dy2+αSαS+1SF02dx2dy2.
αS(x2, y2)=2γπtan-1|F(x2, y2)|-F0(x2, y2)F0(x2, y2)+γ-1,
F¯n=τF0exp(jψn)+1-τ-τ2γπtan-1|F(x, y)|-F0(x, y)F0(x, y)+γ-1Fnfor(x, y)SFnfor(x, y)S.
J(F)=-|DS|F|-F0|2dx2dy2+αSS|F|2dx2dy2+αNN|F|2dx2dy2+αDS[(x2|F|)2+(y2|F|)2]dx2dy2.
J(F)=T(F)+αDS[(x2|F|)2+(y2|F|)2]dx2dy2,
δJ(F)=J(F+δF)-J(F)=δT(F)+αDS[(x|F+δF|)2+(y|F+δF|)2]-[(x|F|)2+(y|F|)2]dxdy,
δT(F)=2-Re{[-F0exp(iψ)+DSF+DSαSF+(1-DS)αNF]δF*}dxdy.
(x|F+δF|)2+(y|F+δF|)2-(x|F|)2-(y|F|)2=(x|F|+x|δF|)2+(y|F|y+y|δF|y)2-(x|F|)2-(y|F|)2=2(x|F|x|δF|+y|F|y|δF|).
S[(x|F+δF|)2+(y|F+δF|)2]-[(x|F|)2+(y|F|)2]dxdy=S2(x|F|x|δF|+y|F|y|δF|)dxdy=2Sx|F|x|δF|dxdy+2Sx|F|x|δF|dxdy=Sy[2(x|F|)|δF|]-Sx+Sx-2Sx(xx|F|)|δFs|dxdy+Sx0[2(y|F|)|δF|]-Sy+Sy-2Sy(yy|F|)|δF|dydx=-2S(xx|F|+yy|F|)|δF|dxdy=-2S2|F|exp(iψ)exp(-iψ)|δF|dxdy=-2S2|F|exp(iψ)δF*dxdy=-2SRe[2|F|exp(iψ)δF*]dxdy,
δJ(F)=2-Re{[-F0exp(iψ)+DSF+DSαSF+(1-DS)αNF-DSαD2|F|exp(iψ)]δF*}dxdy.
δF=F¯-F=-τ[DS(1+αS)F+(1-DS)αNF-F0×exp(jψ)-DSαD2|F|exp(iψ)],
F¯n=Fn-τ[DS(1+αS)Fn+(1-DS)αNFn-F0×exp(jψn)-DSαD2|Fn|exp(iψn)].
Fn+1=Fr DDOE Fr-1(F¯n).
F¯n=τF0exp(jψn)+(1-τ-ταS)Fn+ταD2|Fn|exp(iψn)for(x2, y2)S,(1-ταN)Fnfor(x2, y2)S,
F¯n=τF0exp(jψn)+1-τ-τ2γπtan-1|F(x2, y2)|-F0(x2, y2)F0(x2, y2)+γ-1Fn+ταD2|Fn|exp(iψn)for(x2, y2)S(1-ταN)Fnfor(x2, y2)S.
(2|Fn|)(k, l)=1Δ2 [|Fn(k+1, l)|+|Fn(k, l+1)|+|Fn(k-1, l)|+|Fn(k, l-1)|-4|Fn(k, l)|],
k, l=0, 1, 2,, N,
(2|Fn|)(k, l)=1Δ2 [|Fn(k, l)|+|Fn(k, l+1)|+|Fn(k, l)|+|Fn(k, l-1)|-4|Fn(k, l)|],
F¯n(k, l)=τF0(k, l)exp[jψn(k, l)]+(1-τ-ταS)Fn(k, l)for(x2(k), y2(l))S+ταD1Δ2 [|Fn(k, l)|+|Fn(k, l+1)|+|Fn(k, l)|+|Fn(k, l-1)|-4|Fn(k, l)|]exp[iψn(k, l)](1-ταN)Fn(k, l)for(x2(k), y2(l))S,
Fn+1=Fr DDOE Fr-1(F¯n).

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