Abstract

Critical modeling issues relating to rigorous boundary element method (BEM) analysis of diffractive optical elements (DOEs) are identified. Electric-field integral equation (EFIE) and combined-field integral equation (CFIE) formulations of the BEM are introduced and implemented. The nonphysical interior resonance phenomenon and thin-shape breakdown are illustrated in the context of a guided-mode resonant subwavelength grating. It is shown that modeling such structures by using an open geometric configuration eliminates these problems that are associated with the EFIE BEM. Necessary precautions in defining the incident fields are also presented for the analysis of multiple-layer DOEs.

© 2001 Optical Society of America

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References

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  1. Feature issue on diffractive optics applications, Appl. Opt. 34, 2399–2559 (1995).
    [CrossRef]
  2. M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.
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  6. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Scalar integral diffraction methods: unification, accuracy, and comparison with a rigorous boundary element method with application to diffractive cylindrical lenses,” J. Opt. Soc. Am. A 15, 1822–1837 (1998).
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  7. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Metallic surface-relief on-axis and off-axis focusing diffractive cylindrical mirrors,” J. Opt. Soc. Am. A 16, 113–130 (1999).
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  8. T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
    [CrossRef]
  9. D. W. Prather, M. S. Mirotznik, 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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    [CrossRef]
  11. C. A. Klein, R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of the interior resonant frequencies,” IEEE Trans. Antennas Propag. 23, 431–435 (1975).
    [CrossRef]
  12. A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
    [CrossRef]
  13. R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am. 90, 2728–2738 (1991).
    [CrossRef]
  14. G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
    [CrossRef]
  15. T. W. Wu, “A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies,” J. Acoust. Soc. Am. 97, 84–91 (1995).
    [CrossRef]
  16. K. Hirayama, K. Igarashi, Y. Hayashi, E. N. Glytsis, T. K. Gaylord, “Finite-substrate-thickness cylindrical diffractive lenses: exact and approximate boundary element methods,” J. Opt. Soc. Am. A 16, 1294–1302 (1999).
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  17. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
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    [CrossRef]
  21. N. Morita, “Analysis of scattering by a dielectric rectangular cylinder by means of integral equation formulation,” Electron. Commun. Jpn. 57-B, 72–80 (1974).
  22. J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektr. Uebertrag. 33, 71–80 (1979).
  23. J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings,” J. Opt. Soc. Am. A18 (to be published).
  24. P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
    [CrossRef]
  25. A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, UK, 1973), Chap. 4.

1999 (2)

1998 (2)

1997 (2)

1996 (2)

1995 (3)

S. S. Wang, R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34, 2414–2420 (1995).
[CrossRef] [PubMed]

Feature issue on diffractive optics applications, Appl. Opt. 34, 2399–2559 (1995).
[CrossRef]

T. W. Wu, “A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies,” J. Acoust. Soc. Am. 97, 84–91 (1995).
[CrossRef]

1994 (1)

G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
[CrossRef]

1993 (1)

1992 (1)

P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
[CrossRef]

1991 (2)

R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am. 90, 2728–2738 (1991).
[CrossRef]

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

1990 (1)

A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
[CrossRef]

1985 (1)

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

1979 (1)

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektr. Uebertrag. 33, 71–80 (1979).

1975 (1)

C. A. Klein, R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of the interior resonant frequencies,” IEEE Trans. Antennas Propag. 23, 431–435 (1975).
[CrossRef]

1974 (1)

N. Morita, “Analysis of scattering by a dielectric rectangular cylinder by means of integral equation formulation,” Electron. Commun. Jpn. 57-B, 72–80 (1974).

1973 (1)

J.-C. Bolomey, W. Tabbara, “Numerical aspects on coupling between complementary boundary value problems,” IEEE Trans. Antennas Propag. AP-21, 356–363 (1973).
[CrossRef]

Bendickson, J. M.

Bolomey, J.-C.

J.-C. Bolomey, W. Tabbara, “Numerical aspects on coupling between complementary boundary value problems,” IEEE Trans. Antennas Propag. AP-21, 356–363 (1973).
[CrossRef]

Engel, H.

Friesem, A. A.

Gaylord, T. K.

Glisson, A. W.

P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
[CrossRef]

Glytsis, E. N.

Goggans, P. M.

P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
[CrossRef]

Harrigan, M. E.

Harrington, R. F.

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektr. Uebertrag. 33, 71–80 (1979).

Hayashi, Y.

Hirayama, K.

Ido, J.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Igarashi, K.

Kishk, A. A.

P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
[CrossRef]

Klein, C. A.

C. A. Klein, R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of the interior resonant frequencies,” IEEE Trans. Antennas Propag. 23, 431–435 (1975).
[CrossRef]

Kojima, T.

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

Koshiba, M.

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

Krishnasamy, G.

G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
[CrossRef]

Liu, Y.

G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
[CrossRef]

Magnusson, R.

Mait, J. N.

Martinez, R.

R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am. 90, 2728–2738 (1991).
[CrossRef]

Mautz, J. R.

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektr. Uebertrag. 33, 71–80 (1979).

Miller, E. K.

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, UK, 1973), Chap. 4.

Mirotznik, M. S.

Mittra, R.

C. A. Klein, R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of the interior resonant frequencies,” IEEE Trans. Antennas Propag. 23, 431–435 (1975).
[CrossRef]

Morita, N.

N. Morita, “Analysis of scattering by a dielectric rectangular cylinder by means of integral equation formulation,” Electron. Commun. Jpn. 57-B, 72–80 (1974).

Ohkawa, S.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Peterson, A. F.

A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
[CrossRef]

Poggio, A. J.

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, UK, 1973), Chap. 4.

Prather, D. W.

Rizzo, F. J.

G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
[CrossRef]

Rosenblatt, D.

Sharon, A.

Steingrueber, R.

Tabbara, W.

J.-C. Bolomey, W. Tabbara, “Numerical aspects on coupling between complementary boundary value problems,” IEEE Trans. Antennas Propag. AP-21, 356–363 (1973).
[CrossRef]

Wang, S. S.

Weber, H. G.

Wu, T. W.

T. W. Wu, “A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies,” J. Acoust. Soc. Am. 97, 84–91 (1995).
[CrossRef]

Yashiro, K.

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Appl. Opt. (4)

Arch. Elektr. Uebertrag. (1)

J. R. Mautz, R. F. Harrington, “Electromagnetic scattering from a homogeneous material body of revolution,” Arch. Elektr. Uebertrag. 33, 71–80 (1979).

Electromagnetics (1)

A. F. Peterson, “The ‘interior resonance’ problem associated with surface integral equations of electromagnetics: numerical consequences and a survey of remedies,” Electromagnetics 10, 293–312 (1990).
[CrossRef]

Electron. Commun. Jpn. (1)

N. Morita, “Analysis of scattering by a dielectric rectangular cylinder by means of integral equation formulation,” Electron. Commun. Jpn. 57-B, 72–80 (1974).

Electron. Commun. Jpn., Part 2: Electron. (1)

T. Kojima, J. Ido, “Boundary-element method analysis of light-beam scattering and the sum and differential signal output by DRAW-type optical disk models,” Electron. Commun. Jpn., Part 2: Electron. 74, 11–20 (1991).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

J.-C. Bolomey, W. Tabbara, “Numerical aspects on coupling between complementary boundary value problems,” IEEE Trans. Antennas Propag. AP-21, 356–363 (1973).
[CrossRef]

C. A. Klein, R. Mittra, “An application of the ‘condition number’ concept to the solution of scattering problems in the presence of the interior resonant frequencies,” IEEE Trans. Antennas Propag. 23, 431–435 (1975).
[CrossRef]

P. M. Goggans, A. A. Kishk, A. W. Glisson, “A systematic treatment of conducting and dielectric bodies with arbitrarily thick or thin features using the method of moments,” IEEE Trans. Antennas Propag. 40, 555–560 (1992).
[CrossRef]

K. Yashiro, S. Ohkawa, “Boundary element method for electromagnetic scattering from cylinders,” IEEE Trans. Antennas Propag. AP-33, 383–389 (1985).
[CrossRef]

Int. J. Numer. Methods Eng. (1)

G. Krishnasamy, F. J. Rizzo, Y. Liu, “Boundary integral equations for thin bodies,” Int. J. Numer. Methods Eng. 37, 107–121 (1994).
[CrossRef]

J. Acoust. Soc. Am. (2)

T. W. Wu, “A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies,” J. Acoust. Soc. Am. 97, 84–91 (1995).
[CrossRef]

R. Martinez, “The thin-shape breakdown (TSB) of the Helmholtz integral equation,” J. Acoust. Soc. Am. 90, 2728–2738 (1991).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Other (3)

M. Koshiba, Optical Waveguide Theory by the Finite Element Method (KTK Scientific, Tokyo, 1992), pp. 43–47.

J. M. Bendickson, E. N. Glytsis, T. K. Gaylord, “Guided-mode resonant subwavelength gratings: effects of finite beams and finite gratings,” J. Opt. Soc. Am. A18 (to be published).

A. J. Poggio, E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, ed. (Pergamon, Oxford, UK, 1973), Chap. 4.

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Figures (10)

Fig. 1
Fig. 1

Normalized magnitude of the boundary fields |Ez|/|E0| versus distance along the boundary for a square dielectric structure modeled by a single closed boundary. From top to bottom, the three plots correspond to TE-polarized illumination by an incident beam with λ0=849, 850, and 851 nm, respectively. The vertical dotted lines correspond to the various corners of the square structure as indicated by the inset at the upper right.

Fig. 2
Fig. 2

Matrix condition number versus free-space wavelength for EFIE and CFIE BEM analysis of a square dielectric structure modeled by a single closed boundary.

Fig. 3
Fig. 3

Normalized magnitude of the boundary fields |Ez|/|E0| versus distance along the boundary for a square dielectric structure modeled by two open boundaries. The structure is illuminated by a TE-polarized incident beam with λ0=850 nm. The vertical dotted lines correspond to the various corners of the square structure as indicated by the inset at the upper right.

Fig. 4
Fig. 4

Normalized magnitude of the boundary fields |Ez|/|E0| versus distance along the boundary for a square dielectric structure modeled by two open boundaries. The upper BEM boundary Γ1 almost entirely circumscribes the square structure, which is illuminated by a TE-polarized incident beam with λ0=850 nm. The vertical dotted lines correspond to the various corners of the square structure as indicated by the inset at the upper right.

Fig. 5
Fig. 5

Normalized magnitude of the boundary fields |Ez|/|E0| versus distance along the boundary for a resonant subwavelength grating modeled by a single closed boundary. From top to bottom, the three plots correspond to TE-polarized illumination by an incident beam with λ0=759.1, 760.1, and 761.1 nm, respectively.

Fig. 6
Fig. 6

EFIE and CFIE matrix condition number versus free-space wavelength for a dielectric rectangular structure having dimensions identical to a single ridge of a resonant subwavelength grating.

Fig. 7
Fig. 7

Normalized magnitude of the boundary fields |Ez|/|E0| versus distance along the boundary for a resonant subwavelength grating modeled by two open boundaries. The plot compares the boundary fields determined by the EFIE and CFIE BEMs for TE-polarized illumination and λ0=760.1 nm.

Fig. 8
Fig. 8

Normalized magnitude of the tangential magnetic boundary fields |Htan|/|H0| versus distance along the boundary as determined by using the EFIE and CFIE BEMs for a resonant subwavelength grating modeled by a single closed boundary. The thickness of the thin connecting strips of the structure is 1 and 0.001 nm in the upper and lower plots, respectively, and λ0=850 nm.

Fig. 9
Fig. 9

Normalized magnitude of the tangential magnetic boundary fields |Htan|/|H0| versus distance along the boundary as determined by using the EFIE and CFIE BEMs for a resonant subwavelength grating modeled by two open boundaries. The thickness of the thin connecting strips of the structure is 0.001 nm, and λ0=850 nm.

Fig. 10
Fig. 10

Geometry used in the multiple-layer integral equation formulation of the diffraction problem. The boundaries Γi divide all space into N-1 homogeneous layers bounded by two open, semi-infinite regions. Each region Si has either real refractive index ni or complex refractive index n˜i (in the case of a lossy material). For each boundary Γi, the unit normal and tangent vectors are represented by n^i and t^i, respectively.

Equations (67)

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ϕ1t(r1)=Γ1ϕΓ1(rΓ1) G1n1 (r1, rΓ1)-p1G1(r1, rΓ1)ψΓ1(rΓ1)dl1+ϕinc(r1),
r1S1,
ϕit(ri)=ΓiϕΓi(rΓi) Gini (ri, rΓi)-piGi(ri, rΓi)ψΓi(rΓi)dli-Γi-1ϕΓi-1(rΓi-1) Gini-1 (ri, rΓi-1)-piGi(ri, rΓi-1)ψΓi-1(rΓi-1)dli-1,
riSi,
ϕN+1t(rN+1)=-ΓNϕΓN(rΓN) GN+1nN (rN+1, rΓN)-pN+1GN+1(rN+1, rΓN)ψΓN(rΓN)dlN,
rN+1  SN+1,
Gi(ri, rΓ)=(-j/4)H0(2)(ki|ri-rΓ|)
(i=1, 2,, N+1),
ϕit(rΓi)=ϕi+1t(rΓi)ϕΓi(rΓi),
1piϕitni (rΓi)=1pi+1ϕi+1tni (rΓi)ψΓi(rΓi).
θΓ12π-1ϕ1t(rΓ1)+Γ1ϕΓ1(rΓ1) G1n1 (rΓ1, rΓ1)-p1G1(rΓ1, rΓ1)ψΓ1(rΓ1)dl1=-ϕinc(rΓ1),
θΓi-12πϕi-1t(rΓi-1)
+Γ1-1ϕΓi-1(rΓi-1) Gini-1 (rΓi-1, rΓi-1)-piGi(rΓi-1, rΓi-1)ψΓi-1(rΓi-1)dli-1-ΓiϕΓi(rΓi) Gini (rΓi-1, rΓi)-piGi(rΓi-1, rΓi)ψΓi(rΓi)dli=0
θΓi2π-1ϕit(rΓi)-Γi-1ϕΓi-1(rΓi-1) Gini-1 (rΓi, rΓi-1)-piGi(rΓi, rΓi-1)ψΓi-1(rΓi-1)dli-1+ΓiϕΓi(rΓi) Gini (rΓi, rΓi)-piGi(rΓi, rΓi)ψΓi(rΓi)dli=0,
θΓN2πϕNt(rΓN)+ΓNϕΓN(rΓN) GN+1nN (rΓN, rΓN)-pN+1GN+1(rΓN, rΓN)ψΓN(rΓN)dlN=0,
ϕΓi={M}T{ϕΓi}e,
ψΓi={M}T{ψΓi}e,
{M}T=[-ξ(1-ξ)/21-ξ2ξ(1+ξ)/2],
Ai,jm(ϕΓjt)=ΓjϕΓjt(rΓj)km4j H1(2)(kmRij)cos φjdlj,
Bi,jm(ψΓjt)=-ΓjpmψΓjt(rΓj)14j H0(2)(kmRij)dlj,
Ci,jm(ϕΓjt)=km2ΓjϕΓjt(rΓj)×14j H0(2)(kmRij)cos(φi-φj)dlj,
Di,jm(ϕΓjt)=-ΓjϕΓjttj (rΓj)km4j H1(2)(kmRij)sin φidlj,
Fi,jm(ψΓjt)=ΓjpmψΓjt(rΓj)km4j H1(2)(kmRij)cos φidlj,
Ki,jm(ϕΓjt, ψΓjt)=Ai,jm(ϕΓjt)+Bi,jm(ψΓjt),
Li,jm(ϕΓjt, ψΓjt)=Ci,jm(ϕΓjt)+Di,jm(ϕΓjt)+Fi,jm(ψΓjt).
ϕΓ1t(rΓ1+)=ϕinc(rΓ1+)+K1,11(ϕΓ1t, ψΓ1t),r=rΓ1+,
ϕΓi-1t(rΓi-1-)=-Ki-1,i-1i(ϕΓi-1t, ψΓi-1t)+Ki-1,ii(ϕΓit, ψΓit),
r=rΓi-1-,
ϕΓit(rΓi+)=Ki,ii(ϕΓit, ψΓit)-Ki,i-1i(ϕΓi-1t, ψΓi-1t),
r=rΓi+,
ϕΓNt(rΓN-)=-KN,NN+1(ϕΓNt, ψΓNt),r=rΓN-,
ψΓ1t(rΓ1+)=ψinc(rΓ1+)+L1,11(ϕΓ1t, ψΓ1t),r=rΓ1+,
ψΓi-1t(rΓi-1-)=-Li-1,i-1i(ϕΓi-1t, ψΓi-1t)+Li-1,ii(ϕΓit, ψΓit),
r=rΓi-1-,
ψΓit(rΓi+)=Li,ii(ϕΓit, ψΓit)-Li,i-1i(ϕΓi-1t, ψΓi-1t),
r=rΓi+,
ψΓNt(rΓN-)=-LN,NN+1(ϕΓNt, ψΓNt),r=rΓN-.
θ12π-1ϕΓ1t(rΓ1)+K1,11(ϕΓ1t, ψΓ1t)=-ϕinc(rΓ1),
θi-12πϕΓi-1t(rΓi-1)+Ki-1,i-1i(ϕΓi-1t, ψΓi-1t)
-Ki-1,ii(ϕΓit, ψΓit)=0,
θi2π-1ϕΓit(rΓi)+Ki,ii(ϕΓit, ψΓit)
-Ki,i-1i(ϕΓi-1t, ψΓi-1t)=0,
θN2πϕΓNt(rΓN)+KN,NN+1(ϕΓNt, ψΓNt)=0.
θ12π-1ψΓ1t(rΓ1)+L1,11(ϕΓ1t, ψΓ1t)=-ψinc(rΓ1),
θi-12πψΓi-1t(rΓi-1)+Li-1,i-1i(ϕΓi-1t, ψΓi-1t)
-Li-1,ii(ϕΓit, ψΓit)=0,
θi2π-1ψΓit(rΓi)+Li,ii(ϕΓit, ψΓit)
-Li,i-1i(ϕΓi-1t, ψΓi-1t)=0,
θN2πψΓNt(rΓN)+LN,NN+1(ϕΓNt, ψΓNt)=0.
-ϕΓ1t(rΓ1)+[K1,11(ϕΓ1t, ψΓ1t)-K1,12(ϕΓ1t, ψΓ1t)]
+K1,22(ϕΓ2t, ψΓ2t)=-ϕinc(rΓ1),
-γ1ψΓ1t(rΓ1)+[L1,11(ϕΓ1t, ψΓ1t)-L1,12(ϕΓ1t, ψΓ1t)]
+L1,22(ϕΓ2t, ψΓ2t)=-p1ψinc(rΓ1),
-ϕΓit(rΓi)+[Ki,ii(ϕΓit, ψΓit)-Ki,ii+1(ϕΓit, ψΓit)]
+Ki,i+1i+1(ϕΓi+1t, ψΓi+1t)
-Ki,i-1i(ϕΓi-1t, ψΓi-1t)
=0,
-γiψΓit(rΓi)+[Li,ii(ϕΓit, ψΓit)-Li,ii+1(ϕΓit, ψΓit)]
+Li,i+1i+1(ϕΓi+1t, ψΓi+1t)
-Li,i-1i(ϕΓi-1t, ψΓi-1t)
=0,
-ϕΓNt(rΓN)+[KN,NN(ϕΓNt, ψΓNt)-KN,NN+1(ϕΓNt, ψΓNt)]
-KN,N-1N(ϕΓN-1t, ψΓN-1t)
=0,
-γNψΓNt(rΓN)+[LN,NN(ϕΓNt, ψΓNt)-LN,NN+1(ϕΓNt, ψΓNt)]
-LN,N-1N(ϕΓN-1t, ψΓN-1t)
=0,

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