Abstract

Conventional integral equation methods for diffraction gratings require lattice sum techniques to evaluate quasi-periodic Green’s functions. The boundary integral equation Neumann-to-Dirichlet map (BIE-NtD) method in Wu and Lu [J. Opt. Soc. Am. A 26, 2444 (2009)], [J. Opt. Soc. Am. A 28, 1191 (2011)] is a recently developed integral equation method that avoids the quasi-periodic Green’s functions and is relatively easy to implement. In this paper, we present a number of improvements for this method, including a revised formulation that is more stable numerically, and more accurate methods for computing tangential derivatives along material interfaces and for matching boundary conditions with the homogeneous top and bottom regions. Numerical examples indicate that the improved BIE-NtD map method achieves a high order of accuracy for in-plane and conical diffractions of dielectric gratings.

© 2012 Optical Society of America

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References

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  1. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
  2. G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).
  3. G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
    [CrossRef]
  4. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
    [CrossRef]
  5. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
    [CrossRef]
  6. M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
    [CrossRef]
  7. S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
    [CrossRef]
  8. P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  9. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  10. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
    [CrossRef]
  11. N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788 (2007).
    [CrossRef]
  12. I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
    [CrossRef]
  13. R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
    [CrossRef]
  14. P. Lalanne and J. P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
    [CrossRef]
  15. Y.-P. Chiou, W.-L. Yeh, and N.-Y. Shih, “Analysis of highly conducting lamellar gratings with multidomain pseudospectral method,” J. Lightwave Technol. 27, 5151–5159 (2009).
    [CrossRef]
  16. G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
    [CrossRef]
  17. D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
    [CrossRef]
  18. K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
    [CrossRef]
  19. D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d’Aix-Marseille III, 1974).
  20. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, 1980), Chap. 3.
  21. A. Pomp, “The integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
    [CrossRef]
  22. B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
    [CrossRef]
  23. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  24. E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
    [CrossRef]
  25. T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418 (2005).
    [CrossRef]
  26. A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).
  27. L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597 (2010).
    [CrossRef]
  28. G. Schmidt and B. H. Kleemann, “Integral equation methods from grating theory to photonics: an overview and new approaches for conical diffraction,” J. Mod. Opt. 58, 407–423 (2011).
    [CrossRef]
  29. E. Popov, M. Neviére, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  30. Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A 26, 2444–2451(2009).
    [CrossRef]
  31. Y. Wu and Y. Y. Lu, “Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction,” J. Opt. Soc. Am. A 28, 1191–1196 (2011).
    [CrossRef]
  32. G. R. Hadley, “High-accuracy finite-difference equations for dielectric waveguide analysis II: dielectric corners,” J. Lightwave Technol. 20, 1219–1231 (2002).
    [CrossRef]
  33. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).
  34. W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
    [CrossRef]

2012 (1)

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[CrossRef]

2011 (4)

2010 (3)

2009 (2)

2008 (1)

2007 (1)

2006 (2)

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

2005 (2)

2002 (2)

2000 (1)

1999 (1)

1997 (1)

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

1996 (4)

1995 (1)

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

1991 (1)

A. Pomp, “The integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Andriamanampisoa, L. B.

Armeanu, A. M.

Bao, G.

Botten, L. C.

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Bozhkov, B.

Campbell, S.

Chen, Z. M.

Chiou, Y.-P.

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

de Sterke, C. M.

Edee, K.

Foresti, M.

Goray, L. I.

Gralak, B.

Granet, G.

Guizal, B.

Gushchin, I.

Hadley, G. R.

Hoose, J.

Hugonin, J. P.

Kleemann, B. H.

G. Schmidt and B. H. Kleemann, “Integral equation methods from grating theory to photonics: an overview and new approaches for conical diffraction,” J. Mod. Opt. 58, 407–423 (2011).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Kress, R.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

Lalanne, P.

Li, L.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

Lu, W.

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[CrossRef]

Lu, Y. Y.

Lyndin, N. M.

Magath, T.

Mait, J. N.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

Maystre, D.

E. Popov, B. Bozhkov, D. Maystre, and J. Hoose, “Integral method for echelles covered with lossless or absorbing thin dielectric layers,” Appl. Opt. 38, 47–55 (1999).
[CrossRef]

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d’Aix-Marseille III, 1974).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, 1980), Chap. 3.

McPhedran, R. C.

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Menez, L.

Mirotznik, M. S.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Morf, R. H.

Morris, G. M.

Neviére, M.

Parriaux, O.

Pomp, A.

A. Pomp, “The integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

Popov, E.

Prather, D. W.

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

Raniriharinosy, K.

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

Schmidt, G.

G. Schmidt and B. H. Kleemann, “Integral equation methods from grating theory to photonics: an overview and new approaches for conical diffraction,” J. Mod. Opt. 58, 407–423 (2011).
[CrossRef]

L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597 (2010).
[CrossRef]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

Serebryannikov, A. E.

Shih, N.-Y.

Song, D.

Tayeb, G.

Tishchenko, A. V.

Wu, H. J.

Wu, Y.

Wyrowski, F.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

Yeh, W.-L.

Yuan, L.

Appl. Opt. (1)

Commun. Comput. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

J. Comput. Phys. (1)

W. Lu and Y. Y. Lu, “Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations,” J. Comput. Phys. 231, 1360–1371 (2012).
[CrossRef]

J. Lightwave Technol. (2)

J. Mod. Opt. (4)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

A. Pomp, “The integral method for coated gratings—computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile—theory and experiments,” J. Mod. Opt. 43, 1323–1349 (1996).
[CrossRef]

G. Schmidt and B. H. Kleemann, “Integral equation methods from grating theory to photonics: an overview and new approaches for conical diffraction,” J. Mod. Opt. 58, 407–423 (2011).
[CrossRef]

J. Opt. Soc. Am. A (18)

E. Popov, M. Neviére, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
[CrossRef]

Y. Wu and Y. Y. Lu, “Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method,” J. Opt. Soc. Am. A 26, 2444–2451(2009).
[CrossRef]

Y. Wu and Y. Y. Lu, “Boundary integral equation Neumann-to-Dirichlet map method for gratings in conical diffraction,” J. Opt. Soc. Am. A 28, 1191–1196 (2011).
[CrossRef]

T. Magath and A. E. Serebryannikov, “Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs,” J. Opt. Soc. Am. A 22, 2405–2418 (2005).
[CrossRef]

L. I. Goray and G. Schmidt, “Solving conical diffraction with integral equations,” J. Opt. Soc. Am. A 27, 585–597 (2010).
[CrossRef]

M. Foresti, L. Menez, and A. V. Tishchenko, “Modal method in deep metal-dielectric gratings: the decisive role of hidden modes,” J. Opt. Soc. Am. A 23, 2501–2509 (2006).
[CrossRef]

S. Campbell, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Modal method for classical diffraction by slanted lamellar gratings,” J. Opt. Soc. Am. A 25, 2415–2426 (2008).
[CrossRef]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
[CrossRef]

N. M. Lyndin, O. Parriaux, and A. V. Tishchenko, “Modal analysis and suppression of the Fourier modal method instabilities in highly conductive gratings,” J. Opt. Soc. Am. A 24, 3781–3788 (2007).
[CrossRef]

I. Gushchin and A. V. Tishchenko, “Fourier modal method for relief gratings with oblique boundary conditions,” J. Opt. Soc. Am. A 27, 1575–1583 (2010).
[CrossRef]

R. H. Morf, “Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings,” J. Opt. Soc. Am. A 12, 1043–1056 (1995).
[CrossRef]

P. Lalanne and J. P. Hugonin, “Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings,” J. Opt. Soc. Am. A 17, 1033–1042 (2000).
[CrossRef]

G. Granet, L. B. Andriamanampisoa, K. Raniriharinosy, A. M. Armeanu, and K. Edee, “Modal analysis of lamellar gratings using the moment method with subsectional basis and adaptive spatial resolution,” J. Opt. Soc. Am. A 27, 1303–1310 (2010).
[CrossRef]

D. Song, L. Yuan, and Y. Y. Lu, “Fourier-matching pseudospectral modal method for diffraction gratings,” J. Opt. Soc. Am. A 28, 613–620 (2011).
[CrossRef]

K. Edee, “Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings,” J. Opt. Soc. Am. A 28, 2006–2013 (2011).
[CrossRef]

G. Bao, Z. M. Chen, and H. J. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106–1114 (2005).
[CrossRef]

Optica Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Optica Acta 28, 413–428 (1981).
[CrossRef]

Other (6)

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).

G. Bao, L. Cowsar, and W. Masters, Mathematical Modeling in Optical Sciences (Society for Industrial and Applied Mathematics, 2001).

D. Maystre, “Sur la diffraction et l’absorption par les réseaux utilisés dans l’infrarouge, le visible et l’ultraviolet,” Ph.D. dissertation (Université d’Aix-Marseille III, 1974).

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, 1980), Chap. 3.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd ed. (Springer-Verlag, 1998).

D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

A typical diffraction grating.

Fig. 2.
Fig. 2.

Three dielectric diffraction gratings.

Fig. 3.
Fig. 3.

Example 1: absolute error versus 1/N for the diffraction efficiency of the first transmitted order.

Fig. 4.
Fig. 4.

Example 2: absolute error versus 1/N for the diffraction efficiency of the first transmitted order.

Fig. 5.
Fig. 5.

Example 3: absolute error versus 1/N for the diffraction efficiency of the first transmitted order.

Equations (32)

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

x2u+y2u+(k02εγ02)u=0,
xHx+yHy,1ε(yHxxHy)
αj=α0+2πj/L,βj(l)=k02ε(l)αj2γ02,l=1,2.
u(x+L,y)=eiα0Lu(x,y),
yu(x,0)=B(2)u(x,0).
(B(2)f)(x)=iLj=βj(2)0Lf(x~)eiαj(xx~)dx~,
yu(x,D+)=B(1)u(x,D)2B(1)u(i)(x,D+),
w(t)=sew1p+sbw2pw1p+w2pfortbtte,
w1=(121p)ξ3+ξp+12,w2=1w1,ξ=2t(tb+te)tetb.
Vjφ=uonΩj,
Nj[φj1+φj]=[Nj,11Nj,12Nj,21Nj,22][φj1+φj]=[uj1uj],
Qj+uj=φj+,Qjuj=φj,Yjuj=u0,
φ0=B(2)u0onΓ0,
φm+=B(1)um+gmonΓm,
Zj=(I[Nj,11Nj,11]Qj1+)1[Nj,12Nj,12],
Qj=([Nj,22Nj,22]+[Nj,21Nj,21]Qj1+Zj)1,
Yj=Yj1ZjQj.
φj+=Mjφj+Tjψj,
Mj=[1σjνy2σjνxνyσjνxνy1σjνx2],Tj=σj[νxνyνy2νx2νxνy].
Qj+=MjQj+Tj[wτwτ].
r(s)=(x(s),y(s)),0sLj.
τu(r)=1|r(s)|du(r(s))ds=1w(t)|r(w(t))|du(r(w(t)))dt,
ψ(t)=1|r(ω(t))|du(r(ω(t)))dt.
dh(t)dt=eiα0x(w(t))[iα0dx(w(t))dtu(r(w(t)))+|r(w(t))|ψ(t)].
h(t)l=Nj/2Nj/21h^lexp(i2πlt/Tj)
h^l=1Njk=0Nj1hkei2πlk/Nj,Nj2l<Nj2;
dhdt(tk)l=Nj/2Nj/21i2πlTjh^lei2πltk/Tj,k=0,,Nj1.
[ψ(t0)ψ(t1)ψ(tNj1)]Dj[u0u1uNj1].
(B(2)f)(x)iLj=J0J0βj(2)eiαjx0T0f(w(t~))eiαjw(t~)w(t~)dt~iT0LN0j=J0J0βj(2)eiαjxk=0N01f(xk)eiαjxkw(tk),
u(x,y)=j=cjexp[i(αjxβj(2)y)],y0,
cj=1L0Lu(x,0)eiαjxdxT0LN0k=0N01u(xk,0)w(tk)eiαjxk,
Qj+=ηjQj,

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