Abstract

An efficient numerical method is developed for the modal analysis of two-dimensional photonic crystal waveguides (PCWs). Using the Dirichlet-to-Neumann (DtN) map of the supercell, the waveguide modes are solved from an eigenvalue problem formulated on two boundaries of the supercell, leading to significantly smaller matrices when it is discretized. The eigenvalue problem is linear even when the medium is dispersive. The DtN map of a domain is an operator that maps the wave field on the boundary of the domain to the normal derivative of the field. The DtN map of the supercell can be efficiently calculated by merging the DtN maps of the ordinary and defect unit cells.

© 2007 Optical Society of America

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    [CrossRef]

2007

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 337-349 (2007).

J. H. Yuan and Y. Y. Lu, "Computing photonic band structures by Dirichlet-to-Neumann maps: the triangular lattice," Opt. Commun. 273, 114-120 (2007).
[CrossRef]

S. J. Li and Y. Y. Lu, "Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells," J. Opt. Soc. Am. A 24, 2438-2442 (2007).
[CrossRef]

2006

Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

J. H. Yuan and Y. Y. Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 23, 3217-3222 (2006).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

P. Joly, J.-R. Li, and S. Fliss, "Exact boundary conditions for periodic waveguides containing a local perturbation," Comm. Comp. Phys. 1, 945-973 (2006).

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

2005

K. Yasumoto, H. Jia, and K. Sun, "Rigorous modal analysis of two-dimensional photonic crystal waveguides," Radio Sci. 40, RS6S02 (2005).
[CrossRef]

H. Jia and K. Yasumoto, "Rigorous analysis of guidedmodes of two-dimensional metallic electromagnetic crystal waveguides," J. Electromagn. Waves Appl. 19, 1919-1933 (2005).
[CrossRef]

S. Y. Shi, C. H. Chen, and D. W. Prather, "Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs," Appl. Phys. Lett. 86, 43104 (2005).
[CrossRef]

2004

C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

S. F. Helfert, "Numerical stable determination of Floquet modes and the application to the computation of band structures," Opt. Quantum Electron. 36, 87-107 (2004).
[CrossRef]

C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004).
[CrossRef] [PubMed]

S. Guo, F. Wu, S. Albin, and R. S. Rogowski, "Photonic band gap analysis using finite-difference frequency-domain method," Opt. Express 12, 1741-1746 (2004).
[CrossRef] [PubMed]

2003

2002

2001

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

2000

M. Qiu and S. L. He, "Guided modes in a two-dimensional metallic photonic crystal waveguide," Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

A. Adibi, Y. Xu, R. K. Lee, A. Yariv, and A. Scherer, "Properties of the slab modes in photonic crystal optical waveguides," J. Lightwave Technol. 18, 1554-1564 (2000).
[CrossRef]

J. Tausch and J. Butler, "Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps," J. Comput. Phys. 159, 90-102 (2000).
[CrossRef]

1999

D. C. Dobson, "An efficient method for band structure calculations in 2D photonis crystals," J. Comput. Phys. 149, 363-379 (1999).
[CrossRef]

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

1997

K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
[CrossRef]

1996

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

H. Y. D. Yang, "Finite difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1996).
[CrossRef]

J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

H. Benisty, "Modal analysis of optical guides with two-dimensional photonic band-gap boundaries," J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

1995

M. J. Grote and J. B. Keller, "On nonreflecting boundary conditions," J. Comput. Phys. 122, 231-243 (1995).
[CrossRef]

1993

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

1987

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Adibi, A.

Albin, S.

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Asatryan, A. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Axmann, W.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Bassi, P.

Bellanca, G.

Benisty, H.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

H. Benisty, "Modal analysis of optical guides with two-dimensional photonic band-gap boundaries," J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

Botten, L. C.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Butler, J.

J. Tausch and J. Butler, "Efficient analysis of periodic dielectric waveguides using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A 19, 1120-1128 (2002).
[CrossRef]

J. Tausch and J. Butler, "Floquet multipliers of periodic waveguides via Dirichlet-to-Neumann maps," J. Comput. Phys. 159, 90-102 (2000).
[CrossRef]

Byrne, M. A.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

Castaldini, D.

Chang, H. C.

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

C. P. Yu and H. C. Chang, "Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals," Opt. Express 12, 1397-1408 (2004).
[CrossRef] [PubMed]

C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004).
[CrossRef]

Checoury, X.

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

Chen, C. H.

S. Y. Shi, C. H. Chen, and D. W. Prather, "Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs," Appl. Phys. Lett. 86, 43104 (2005).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Chiang, P. J.

P. J. Chiang, C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E 75, 026703 (2007).
[CrossRef]

Cho, Y. S.

David, A.

A. David, H. Benisty, and C. Weisbuch, "Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape," Phys. Rev. B 73, 075107 (2006).
[CrossRef]

de Sterke, C. M.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Dobson, D. C.

D. C. Dobson, "An efficient method for band structure calculations in 2D photonis crystals," J. Comput. Phys. 149, 363-379 (1999).
[CrossRef]

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, "Finite element computation of grating scattering matrices and application to photonic crystal band calculations," J. Comput. Phys. 219, 120-143 (2006).
[CrossRef]

Erni, D.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Fan, S. H.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Fliss, S.

P. Joly, J.-R. Li, and S. Fliss, "Exact boundary conditions for periodic waveguides containing a local perturbation," Comm. Comp. Phys. 1, 945-973 (2006).

Gnan, M.

Grote, M. J.

M. J. Grote and J. B. Keller, "On nonreflecting boundary conditions," J. Comput. Phys. 122, 231-243 (1995).
[CrossRef]

Guo, S.

Hafner, C.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

He, S. L.

M. Qiu and S. L. He, "Guided modes in a two-dimensional metallic photonic crystal waveguide," Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

Helfert, S. F.

S. F. Helfert, "Numerical stable determination of Floquet modes and the application to the computation of band structures," Opt. Quantum Electron. 36, 87-107 (2004).
[CrossRef]

Hernández-Figueroa, H. E.

Huang, Y. X.

Y. X. Huang and Y. Y. Lu, "Modeling photonic crystals with complex unit cells by Dirichlet-to-Neumann maps," J. Comput. Math. 25, 337-349 (2007).

Y. X. Huang and Y. Y. Lu, "Scattering from periodic arrays of cylinders by Dirichlet-to-Neumann maps," J. Lightwave Technol. 24, 3448-3453 (2006).
[CrossRef]

Im, S.

Jia, H.

K. Yasumoto, H. Jia, and K. Sun, "Rigorous modal analysis of two-dimensional photonic crystal waveguides," Radio Sci. 40, RS6S02 (2005).
[CrossRef]

H. Jia and K. Yasumoto, "Rigorous analysis of guidedmodes of two-dimensional metallic electromagnetic crystal waveguides," J. Electromagn. Waves Appl. 19, 1919-1933 (2005).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Joly, P.

P. Joly, J.-R. Li, and S. Fliss, "Exact boundary conditions for periodic waveguides containing a local perturbation," Comm. Comp. Phys. 1, 945-973 (2006).

Jun, S.

Keller, J. B.

M. J. Grote and J. B. Keller, "On nonreflecting boundary conditions," J. Comput. Phys. 122, 231-243 (1995).
[CrossRef]

Kuchment, P.

W. Axmann and P. Kuchment, "An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I. Scalar case," J. Comput. Phys. 150, 468-481 (1999).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Lee, R. K.

Li, J.-R.

P. Joly, J.-R. Li, and S. Fliss, "Exact boundary conditions for periodic waveguides containing a local perturbation," Comm. Comp. Phys. 1, 945-973 (2006).

Li, S. J.

Lourtioz, J. M.

X. Checoury and J. M. Lourtioz, "Wavelet method for computing band diagrams of 2D photonic crystals," Opt. Commun. 259, 360-365 (2006).
[CrossRef]

Lu, Y. Y.

Marrone, M.

McPhedran, R. C.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Moreno, E.

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B 65, 155120 (2002).
[CrossRef]

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, "Photonic band structure calculations using scattering matrices," Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Ohtaka, K.

K. Sakoda, T. Ueta, and K. Ohtaka, "Numerical analysis of eigenmodes localized at line defects in photonic lattices," Phys. Rev. B 56, 14905-14908 (1997).
[CrossRef]

Pendry, J. B.

J. B. Pendry, "Calculating photonic band structure," J. Phys. Condens. Matter 8, 1085-1108 (1996).
[CrossRef]

Prather, D. W.

S. Y. Shi, C. H. Chen, and D. W. Prather, "Revised plane wave method for dispersive material and its application to band structure calculations of photonic crystal slabs," Appl. Phys. Lett. 86, 43104 (2005).
[CrossRef]

Qiu, M.

M. Qiu and S. L. He, "Guided modes in a two-dimensional metallic photonic crystal waveguide," Phys. Lett. A 266, 425-429 (2000).
[CrossRef]

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, and O. L. Alerhand, "Accurate theoretical analysis of photonic band-gap materials," Phys. Rev. B 48, 8434-8437 (1993).
[CrossRef]

Rodriguez-Esquerre, V. F.

Rogowski, R. S.

Sakoda, K.

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Figures (7)

Fig. 1
Fig. 1

Square lattice of circular rods with a line defect.

Fig. 2
Fig. 2

Dispersion curves (for the E polarization) of a photonic crystal waveguide composed of one missing row in a square lattice of dielectric rods.

Fig. 3
Fig. 3

Electric field patterns of the propagating modes (E polarization) of a photonic crystal waveguide at ω L ( 2 π c ) = 0.8 . The top and the bottom figures are the odd and even modes, respectively, and their Bloch wavenumbers are β L ( 2 π ) = 0.3346 and 0.2773.

Fig. 4
Fig. 4

Relative error E N of the propagation constant β of a photonic crystal waveguide obtained with a reference solution at N = 19 .

Fig. 5
Fig. 5

Photonic crystal waveguide composed of air holes in a finite dielectric slab.

Fig. 6
Fig. 6

Dispersion curves of the photonic crystal waveguide depicted in Fig. 5.

Fig. 7
Fig. 7

Electric field patterns of the propagating modes (E polarization) for the photonic crystal waveguide depicted in Fig. 5 and ω L ( 2 π c ) = 0.23 .

Equations (40)

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2 u x 2 + 2 u y 2 + k 0 2 n 2 u = 0 ,
n ( x , y + L ) = n ( x , y ) .
u ( x , y ) = ϕ ( x , y ) exp ( i β y ) ,
ϕ ( x , y + L ) = ϕ ( x , y ) .
u ( x , y + L ) = ρ u ( x , y ) , ρ = exp ( i β L ) .
u ( x , L ) = ρ u ( x , 0 ) , u y ( x , L ) = ρ u y ( x , 0 ) .
Ω = { ( x , y ) < x < , 0 < y < L } .
2 ϕ x 2 + 2 ϕ y 2 + 2 i β ϕ y + [ k 0 2 n 2 ( x , y ) β 2 ] ϕ = 0 ,
ϕ ( x , L ) = ϕ ( x , 0 ) , ϕ y ( x , L ) = ϕ y ( x , 0 ) .
[ y 1 x 2 + k 0 2 n 2 y ] [ ϕ φ ] = i β [ ϕ φ ] .
S [ u + ( , 0 ) u ( , L ) ] = [ S 11 S 12 S 21 S 22 ] [ u + ( , 0 ) u ( , L ) ] = [ u ( , 0 ) u + ( , L ) ] .
u + ( x , L ) = ρ u + ( x , 0 ) , u ( x , L ) = ρ u ( x , 0 ) .
[ S 11 I S 21 0 ] [ u + ( , 0 ) u ( , 0 ) ] = ρ [ 0 S 12 I S 22 ] [ u + ( , 0 ) u ( , 0 ) ] .
T [ u + ( , 0 ) u ( , 0 ) ] = [ u + ( , L ) u ( , L ) ] .
T [ u + ( , 0 ) u ( , 0 ) ] = ρ [ u + ( , 0 ) u ( , 0 ) ] .
T = [ 0 S 12 I S 22 ] 1 [ S 11 I S 21 0 ] .
M [ u ( , 0 ) u ( , L ) ] = [ M 11 M 12 M 21 M 22 ] [ u ( , 0 ) u ( , L ) ] = [ y u ( , 0 ) y u ( , L ) ] .
[ M 11 I M 21 0 ] [ u ( , 0 ) y u ( , 0 ) ] = ρ [ M 12 0 M 22 I ] [ u ( , 0 ) y u ( , 0 ) ] ,
N [ u ( , 0 ) y u ( , 0 ) ] = [ u ( , L ) y u ( , L ) ] ,
N [ u ( , 0 ) y u ( , 0 ) ] = ρ [ u ( , 0 ) y u ( , 0 ) ] .
Ω ̂ = { ( x , y ) x 0 < x < x J , 0 < y < L } .
u ( x , y ) = 0 , x = x 0 , x J .
Λ ( j ) [ u 0 j v j 1 v j u 1 j ] = [ Λ 11 ( j ) Λ 12 ( j ) Λ 13 ( j ) Λ 14 ( j ) Λ 21 ( j ) Λ 22 ( j ) Λ 23 ( j ) Λ 24 ( j ) Λ 31 ( j ) Λ 32 ( j ) Λ 33 ( j ) Λ 34 ( j ) Λ 41 ( j ) Λ 42 ( j ) Λ 43 ( j ) Λ 44 ( j ) ] [ u 0 j v j 1 v j u 1 j ] = [ y u 0 j x v j 1 x v j y u 1 j ] ,
u ( x , y ) = k = 1 N t c k ϕ k ( x , y ) ,
Λ ( j ) = B A 1 .
ξ k = x j 1 + ( k 0.5 ) x j x j 1 N x , k = 1 , 2 , , N x ,
η l = ( l 0.5 ) L N y , l = 1 , 2 , , N y .
u 0 = [ u 01 u 02 u 0 J ] , u 1 = [ u 11 u 12 u 1 J ] , v = [ v 1 v 2 v J 1 ] .
x v j = Λ 31 ( j ) u 0 j + Λ 32 ( j ) v j 1 + Λ 33 ( j ) v j + Λ 34 ( j ) u 1 j = Λ 21 ( j + 1 ) u 0 , j + 1 + Λ 22 ( j + 1 ) v j + Λ 23 ( j + 1 ) v j + 1 + Λ 24 ( j + 1 ) u 1 , j + 1 .
Λ 32 ( j ) v j 1 + ( Λ 33 ( j ) Λ 22 ( j + 1 ) ) v j Λ 23 ( j + 1 ) v j + 1 = Λ 31 ( j ) u 0 j + Λ 21 ( j + 1 ) u 0 , j + 1 Λ 34 ( j ) u 1 j + Λ 24 ( j + 1 ) u 1 , j + 1 .
C 0 v = C 1 [ u 0 u 1 ] ,
v = C 0 1 C 1 [ u 0 u 1 ] .
y [ u 0 j u 1 j ] = [ Λ 11 ( j ) Λ 14 ( j ) Λ 41 ( j ) Λ 44 ( j ) ] [ u 0 j u 1 j ] + [ Λ 12 ( j ) Λ 13 ( j ) Λ 42 ( j ) Λ 43 ( j ) ] [ v j 1 v j ] .
y [ u 0 u 1 ] = D 0 [ u 0 u 1 ] + D 1 v .
y [ u 0 u 1 ] = ( D 0 + D 1 C 0 1 C 1 ) [ u 0 u 1 ] .
M = D 0 + D 1 C 0 1 C 1 .
u ( x , y ) = e i ( α x + β y ) Φ ( x , y ) ,
E N = β ( N ) β ( 19 ) β ( 19 ) ,
x J = x 0 = ( m + 3 ) L + d 2 + b ,
x j + 1 x j = { b , if j = m , j = m + 8 , d , if j = m + 4 , L , otherwise.

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