Abstract

A new two-step design algorithm for the calculation of a diffractive phase element (DPE) for use with partially coherent laser beams is presented. The optical reconstruction of the DPE is modeled by the convolution of a coherent diffraction pattern and the far-field intensity distribution of a partially coherent laser beam. Numerical deconvolution is applied to derive a suitable amplitude pattern as signal input to a standard iterative Fourier transform algorithm (IFTA). Theory and numerical results are presented. Compared with a single-step IFTA design, this new approach yields nearly equal diffraction efficiencies and a relative improvement of 15% in signal reconstruction error.

© 2001 Optical Society of America

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  1. D. Basting, ed., Excimer Laser Technology: Laser Sources, Optics, Systems and Applications (Lambda Physik Göttingen, Göttingen, Germany, to be published).
  2. S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
  4. J. Bernges, L. Unnebrink, T. Henning, “Mask adapted beam shaping for material processing with excimer laser radiation,” in Optika 98: 5th Congress on Modern Optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 108–111 (1998).
  5. T. Henning, M. Scholl, “Beamshaping by multifaceted integrator mirrors: effects of partial coherence,” in Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds. (Festkörper-Laser-Institut, Berlin, 1994), pp. 117–128.
  6. J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
    [CrossRef]
  7. M. Beyerlein, S. Schuberth, T. Dresel, “Design and numerical analysis of diffractive optical elements for beam-shaping of partially coherent light,” in Diffractive Optics 99, Vol. 22 of EOS Topical Meeting Digest Series (European Optical Society, Orsay, France, 1999), pp. 157–158.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  12. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London, Ser. A 208, 263–277 (1951).
    [CrossRef]
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  15. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995).
  16. P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).
  17. P. C. Hansen, “Numerical aspects of deconvolution,” in Lecture Notes of the Department of Informatics and Mathematical Modelling of the Technical University of Denmark, Technical University of Denmark, Vol. 1, P. C. Hansen, ed. (Lyngby, Denmark, 2000), http://www.imm.dtu.dk/~pch/regularization/deconv.html .
  18. G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md.1996).
  19. F. Wyrowski, “Diffractive optical elements: iterative calcu-lation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  20. M. Bernhardt, F. Wyrowski, O. Bryngdahl, “Iterative techniques to integrate different optical functions in a diffractive phase element,” Appl. Opt. 30, 4629–4635 (1991).
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  21. M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
    [CrossRef]
  22. D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.
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2000 (2)

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

1996 (2)

1995 (2)

1991 (1)

1990 (1)

1987 (1)

1977 (1)

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London, Ser. A 208, 263–277 (1951).
[CrossRef]

Allebach, J. P.

Bengtsson, J.

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

Bernges, J.

J. Bernges, L. Unnebrink, T. Henning, “Mask adapted beam shaping for material processing with excimer laser radiation,” in Optika 98: 5th Congress on Modern Optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 108–111 (1998).

Bernhardt, M.

Beyerlein, M.

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
[CrossRef] [PubMed]

M. Beyerlein, S. Schuberth, T. Dresel, “Design and numerical analysis of diffractive optical elements for beam-shaping of partially coherent light,” in Diffractive Optics 99, Vol. 22 of EOS Topical Meeting Digest Series (European Optical Society, Orsay, France, 1999), pp. 157–158.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

Bryngdahl, O.

Burghardt, B.

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

Carter, W. H.

Dresel, T.

T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
[CrossRef] [PubMed]

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

M. Beyerlein, S. Schuberth, T. Dresel, “Design and numerical analysis of diffractive optical elements for beam-shaping of partially coherent light,” in Diffractive Optics 99, Vol. 22 of EOS Topical Meeting Digest Series (European Optical Society, Orsay, France, 1999), pp. 157–158.

Friberg, A. T.

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md.1996).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Hansen, P. C.

P. C. Hansen, “Numerical aspects of deconvolution,” in Lecture Notes of the Department of Informatics and Mathematical Modelling of the Technical University of Denmark, Technical University of Denmark, Vol. 1, P. C. Hansen, ed. (Lyngby, Denmark, 2000), http://www.imm.dtu.dk/~pch/regularization/deconv.html .

Hård, S.

Henning, T.

J. Bernges, L. Unnebrink, T. Henning, “Mask adapted beam shaping for material processing with excimer laser radiation,” in Optika 98: 5th Congress on Modern Optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 108–111 (1998).

T. Henning, M. Scholl, “Beamshaping by multifaceted integrator mirrors: effects of partial coherence,” in Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds. (Festkörper-Laser-Institut, Berlin, 1994), pp. 117–128.

Holmer, A.-K.

Hopkins, H. H.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London, Ser. A 208, 263–277 (1951).
[CrossRef]

Ihlemann, J.

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

Jansson, P. A.

P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).

Johansson, M.

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

Kaivola, M.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Kajava, T.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Kuittinen, M.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Laakkonen, P.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995).

Marowsky, G.

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

Marsden, P. J.

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

Pääkkönen, P.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Roberts, N. C.

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

Schäfer, D.

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

Scholl, M.

T. Henning, M. Scholl, “Beamshaping by multifaceted integrator mirrors: effects of partial coherence,” in Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds. (Festkörper-Laser-Institut, Berlin, 1994), pp. 117–128.

Schuberth, S.

M. Beyerlein, S. Schuberth, T. Dresel, “Design and numerical analysis of diffractive optical elements for beam-shaping of partially coherent light,” in Diffractive Optics 99, Vol. 22 of EOS Topical Meeting Digest Series (European Optical Society, Orsay, France, 1999), pp. 157–158.

Schwider, J.

Seldowitz, M. A.

Sidhu, J.

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

Simonen, J.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Sweeney, D. W.

Timm, M.

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

Turunen, J.

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

Unnebrink, L.

J. Bernges, L. Unnebrink, T. Henning, “Mask adapted beam shaping for material processing with excimer laser radiation,” in Optika 98: 5th Congress on Modern Optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 108–111 (1998).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md.1996).

Venables, M. A.

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

Williams, S. W.

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

Wolf, E.

Wyrowski, F.

Appl. Opt. (5)

J. Mod. Opt. (2)

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

J. Turunen, P. Pääkkönen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Proc. R. Soc. London, Ser. A (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London, Ser. A 208, 263–277 (1951).
[CrossRef]

Other (12)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, Cambridge, 1995).

P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).

P. C. Hansen, “Numerical aspects of deconvolution,” in Lecture Notes of the Department of Informatics and Mathematical Modelling of the Technical University of Denmark, Technical University of Denmark, Vol. 1, P. C. Hansen, ed. (Lyngby, Denmark, 2000), http://www.imm.dtu.dk/~pch/regularization/deconv.html .

G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md.1996).

D. Basting, ed., Excimer Laser Technology: Laser Sources, Optics, Systems and Applications (Lambda Physik Göttingen, Göttingen, Germany, to be published).

S. W. Williams, P. J. Marsden, N. C. Roberts, J. Sidhu, M. A. Venables, “Excimer laser beam shaping and material processing using diffractive optics,” in High-Power Laser Ablation, C. R. Phipps, ed., Proc. SPIE3343, 205–211 (1998).
[CrossRef]

M. Beyerlein, S. Schuberth, T. Dresel, “Design and numerical analysis of diffractive optical elements for beam-shaping of partially coherent light,” in Diffractive Optics 99, Vol. 22 of EOS Topical Meeting Digest Series (European Optical Society, Orsay, France, 1999), pp. 157–158.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, 1999).

D. Schäfer, J. Ihlemann, G. Marowsky, B. Burghardt, M. Timm, “Multifacet kinoforms for KrF excimer laser,” in Diffractive Optics and Micro-Optics, 2000 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 2000), pp. 189–191.

J. Bernges, L. Unnebrink, T. Henning, “Mask adapted beam shaping for material processing with excimer laser radiation,” in Optika 98: 5th Congress on Modern Optics, G. Akos, G. Lupkovics, A. Podmaniczky, Proc. SPIE3573, 108–111 (1998).

T. Henning, M. Scholl, “Beamshaping by multifaceted integrator mirrors: effects of partial coherence,” in Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejias, eds. (Festkörper-Laser-Institut, Berlin, 1994), pp. 117–128.

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Figures (6)

Fig. 1
Fig. 1

Influence of the SepOp operator and of simulated noise on a measured excimer laser far-field intensity pattern. Focus length z=1250 mm; scales in millimeters.

Fig. 2
Fig. 2

Scaled rms error and efficiency η of the TSVD solutions Ipck.

Fig. 3
Fig. 3

(a) Simulated intensity Iconv=I0 * Iraw of an ideal flat-topped profile I0 smoothed by the partially coherent beam corresponding to Iraw (rmss, 17.0%; efficiency, 79.2%). (b) Simulated intensity Ipc140=Icoh140 * Iraw of the solution Icoh140 (cf. Fig. 4) obtained by numerical deconvolution (rmss, 14.4%; efficiency, 79.5%). For nearly equal efficiencies the deconvolved solution yields a relative improvement in homogeneity of (0.170-0.144)/0.170=15.3%.

Fig. 4
Fig. 4

Regularized solution Icoh140 of Eq. (18): (a) gray-scale image, (b) mesh plot. In section 5 this intensity distribution is used for calculation of a DPE phase function as the input signal of an IFTA design.

Fig. 5
Fig. 5

Test of the stability of the reconstructions Ipcns if the incident beam characterized by the far-field intensity Iraws differs from the TSVD input far-field intensity Irawn. The dependence of the increase of reconstruction error Δrmss on the difference ΔIraw of the far-field intensities is shown.

Fig. 6
Fig. 6

(a) Simulated intensity FpcI0 reconstructed by a four-level DPE calculated with I0 as the IFTA input signal [cf. Fig. 3(a)] (rmss, 17.1%; efficiency, 56.5%). (b) Simulated intensity Fpc140 reconstructed by a four-level DPE calculated with Icoh140 as the IFTA input signal [cf. Fig. 3(b)] (rmss=14.6%; efficiency, 54.0%). For a slightly reduced diffraction efficiency a relative improvement in the homogeneity of (0.171-0.146)/0.171=14.6% is achieved.

Equations (45)

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Δλ/λ01,
|ΔS|=|s2-s1|=cτc 1Δν=λ02Δλ.
Γ12(τ)J12 exp(2πiν0τ).
Γ12(τ)J12.
I(x, y)=- K(x, y; u1, v1)K*(x, y; u2, v2)×t(u1, v1)t*(u2, v2)×J(u1, v1; u2, v2)du1dv1du2dv2,
K(x, y; u, v)=exp[i(2π/λ0)z]iλ0z expi πλ0z(x2+y2)×exp-2πixλ0zu+yλ0zv,
I(x, y)=1λ02z2 - t(u1, v1)t*(u2, v2)×J(u1, v1; u2, v2)×exp-2πixλ0zu1+yλ0zv1×exp2πixλ0zu2+yλ0zv2du1dv1du2dv2.
J(u1, v1; u2, v2)=I0j(Δu, Δv),
Δu=u1-u2,u¯=1/2(u1+u2),
Δv=v1-v2,v¯=1/2(v1+v2).
I(x, y)=I0λ02z2 - tu¯+Δu2, v¯+Δv2×t*u¯-Δu2, v¯-Δv2j(Δu, Δv)×exp-2πixλ0zΔu+yλ0zΔv×dΔudΔvdu¯dv¯.
(tt)(Δu, Δv)=-tu¯+Δu2, v¯+Δv2×t*u¯-Δu2, v¯-Δv2du¯dv¯.
I(x, y)=I0λ02z2 -(tt)(Δu, Δv)j(Δu, Δv)×exp-2πixλ0zΔu+yλ0zΔvdΔudΔv.
I(x, y)FT[(tt)j](x, y)
 convolutiontheorem
=FT[(tt)](x, y) * FT[j](x, y)
 correlationtheorem
=|FT[t](x, y)|2 * FT[j](x, y)
=Icoh(x, y) * FT[j](x, y),
Iraw(x, y)=C×FT[j(Δu, Δv)](x, y).
I(x, y)=Icoh(x, y) * Iraw(x, y).
v=T¯¯(u)wn,
T¯¯(u)=un/2+1un/2u100un/2+2un/2+1u2u10un+1unun/2+1un/2u20un+1un/2+2un/2+1u300unun-1un/2+1.
I0(x, y)=Irawx(x-x)×Irawy(y-y)Icoh(x, y)dxdy.
Iraw¯¯=IrxIryT.
Irx(i)=SepOpx(Iraw¯¯)=1S l=1mIraw¯¯(i, l),i=1,,n,
Iry(i)=SepOpy(Iraw¯¯)=1S k=1nIraw¯¯(k, i),i=1,,m,
I0¯¯=T¯¯(Irx)Icoh¯¯T¯¯T(Iry).
T¯¯(Irx)=Ux¯¯x¯¯Vx¯¯T=i=1nuxiσxivxiT,
T¯¯(Iry)=Uy¯¯y¯¯Vy¯¯T=i=1muyiσyivyiT.
Icoh k¯¯=(i, j) uxiTI0¯¯uyjσxiσyj vxivyjT.
I¯¯=P1(I¯¯)=I¯¯-min(I¯¯),
I¯¯=P2(I¯¯)=Inm(n, m)S0(n, m)S,
I¯¯=P3(I¯¯)=I¯¯n,m I0nmn,m Inm.
Icoh¯¯ kP(Icoh¯¯ k).
IpckIcoh k * Iraw.
mean[rmss(I0¯¯, Iconv¯¯)-rmss(I0¯¯, Ipck¯¯)]=4.0%±0.7%,
mean[η(I0¯¯, Iconv¯¯)-η(I0¯¯, Ipck¯¯)]=0.49%±0.57%.
Δrmss=rmss(I0, Ipcnn)-rmss(I0, Ipcns),
ΔIraw=rmss(Irawn, Iraws).
rms(S¯¯, A¯¯)1NM n=1Nm=1M(Snm-Anm)21/2.
rmsn(S¯¯, A¯¯)1max(A) 1NM n=1Nm=1M(Snm-Anm)21/2.
rmss(S¯¯, A¯¯)1N (n, m)S(Snm-γ×Anm)21/2,
γ=(n,m)S Snm×Anm(n,m)S Anm2.
η(S¯¯, A¯¯)(n,m)S Anm(n,m) Snm.

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