Abstract

An efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the unit cells is developed for accurate simulations of two-dimensional photonic crystal (PhC) devices in the frequency domain. The DtN map of a unit cell is an operator that maps the wave field on the boundary of the cell to its normal derivative and it can be approximated by a small matrix. Using the DtN maps of the regular and defect unit cells, we can avoid computations in the interiors of the unit cells and calculate the wave field only on the edges. This gives rise to a significant reduction in the total number of unknowns. Reasonably accurate solutions can be obtained using 10 to 15 unknowns for each unit cell. In contrast, standard finite element, finite difference or plane wave expansion methods may require a few hundreds unknowns for each unit cell at the same level of accuracy. We illustrate our method by a number of examples, including waveguide bends, branches, microcavities coupled with waveguides, waveguides with stubs, etc.

© 2008 Optical Society of America

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2008 (2)

J. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys.227, 4617-3629 (2008).
[CrossRef]

Y. Wu and Y. Y. Lu, "Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice," J. Opt. Soc. Am. B25, 1466-1473 (2008).
[CrossRef]

2007 (8)

2006 (2)

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

J. Yuan and Y. Y.Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A23, 3217-3222 (2006).
[CrossRef]

2005 (2)

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, 043104 (2005).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[CrossRef]

2004 (2)

2003 (2)

2002 (2)

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B65, 155120 (2002).
[CrossRef]

M. Marrone, V. F. Rodriguez-Esquerre, and H. E. Hernandez-Figueroa, "Novel numerical method for the analysis of 2D photonic crystals: the cell method," Opt. Express10, 1299-1304 (2002).
[PubMed]

2001 (2)

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. E64, 046603 (2001).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express8, 173-190 (2001).
[CrossRef] [PubMed]

2000 (1)

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightw. Technol.18, 102-110 (2000).
[CrossRef]

1999 (1)

J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightw. Technol.17, 1500-1508 (1999).
[CrossRef]

1996 (1)

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]

1994 (1)

1987 (2)

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]

Albin, S.

Antoine, X.

J. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys.227, 4617-3629 (2008).
[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. E64, 046603 (2001).
[CrossRef]

Baba, T.

J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightw. Technol.17, 1500-1508 (1999).
[CrossRef]

Botten, L. C.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[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. E64, 046603 (2001).
[CrossRef]

Chang, H. C.

P. J. Chiang and C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E75, 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. Express12, 1397-1408 (2004).
[CrossRef] [PubMed]

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, 043104 (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 and C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E75, 026703 (2007).
[CrossRef]

Cho, Y. S.

de Sterke, C. M.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[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. E64, 046603 (2001).
[CrossRef]

Erni, D.

J. Smajic, C. Hafner, and D. Erni, "Design and optimization of an achromatic photonic crystal bend," Opt. Express11, 1378-1384 (2003).
[CrossRef] [PubMed]

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B65, 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]

Felbacq, D.

Guo, S.

Hafner, C.

J. Smajic, C. Hafner, and D. Erni, "Design and optimization of an achromatic photonic crystal bend," Opt. Express11, 1378-1384 (2003).
[CrossRef] [PubMed]

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B65, 155120 (2002).
[CrossRef]

Hernandez-Figueroa, H. E.

Hikari, M.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightw. Technol.18, 102-110 (2000).
[CrossRef]

Huang, Y.

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

Y. Huang, Y. Y. Lu, and S. Li, "Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps," J. Opt. Soc. Am. B24, 2860-2867 (2007).
[CrossRef]

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

Ikeda, M.

J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightw. Technol.17, 1500-1508 (1999).
[CrossRef]

Im, S.

Jao, R. F.

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwells equations in a planewave basis," Opt. Express8, 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]

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.

Jun, S.

Koshiba, M.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightw. Technol.18, 102-110 (2000).
[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]

Li, S.

Lin, M. C.

Lu, Y. Y.

J. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys.227, 4617-3629 (2008).
[CrossRef]

Y. Wu and Y. Y. Lu, "Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice," J. Opt. Soc. Am. B25, 1466-1473 (2008).
[CrossRef]

S. Li and Y. Y. Lu, "Multipole Dirichlet-to-Neumann map method for photonic crystals with complex unit cells," J. Opt. Soc. Am. A24, 2438-2442 (2007).
[CrossRef]

S. Li and Y. Y. Lu, "Computing photonic crystal defect modes by Dirichlet-to-Neumann maps," Opt. Express15, 14454-14466 (2007).
[CrossRef] [PubMed]

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

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

Y. Huang, Y. Y. Lu, and S. Li, "Analyzing photonic crystal waveguides by Dirichlet-to-Neumann maps," J. Opt. Soc. Am. B24, 2860-2867 (2007).
[CrossRef]

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

J. Yuan and Y. Y.Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A23, 3217-3222 (2006).
[CrossRef]

Marrone, M.

Maystre, D.

McPhedran, R. C.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[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. E64, 046603 (2001).
[CrossRef]

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. B65, 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. E64, 046603 (2001).
[CrossRef]

Ogusu, K.

Poulton, C. G.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[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, 043104 (2005).
[CrossRef]

Rodriguez-Esquerre, V. F.

Rogowski, R. S.

Shi, S. Y.

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, 043104 (2005).
[CrossRef]

Smajic, J.

Takayama, K.

Tayeb, G.

Tsuji, Y.

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightw. Technol.18, 102-110 (2000).
[CrossRef]

Villeneuve, P. R.

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]

Wilcox, S.

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[CrossRef]

Wu, F.

Wu, Y.

Yablonovitch, E.

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

Yonekura, J.

J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightw. Technol.17, 1500-1508 (1999).
[CrossRef]

Yu, C. P.

P. J. Chiang and C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E75, 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. Express12, 1397-1408 (2004).
[CrossRef] [PubMed]

Yuan, J.

J. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys.227, 4617-3629 (2008).
[CrossRef]

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

J. Yuan and Y. Y.Lu, "Photonic bandgap calculations using Dirichlet-to-Neumann maps," J. Opt. Soc. Am. A23, 3217-3222 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

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, 043104 (2005).
[CrossRef]

J. Comput. Math. (1)

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

J. Comput. Phys. (1)

J. Yuan, Y. Y. Lu, and X. Antoine, "Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps," J. Comput. Phys.227, 4617-3629 (2008).
[CrossRef]

J. Lightw. Technol. (3)

J. Yonekura, M. Ikeda, and T. Baba, "Analysis of finite 2-D photonic crystals of columns and lightwave devices using the scattering matrix method," J. Lightw. Technol.17, 1500-1508 (1999).
[CrossRef]

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

M. Koshiba, Y. Tsuji, and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits," J. Lightw. Technol.18, 102-110 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

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

Opt. Express (8)

Opt. Lett. (1)

Phys. Rev. B (1)

E. Moreno, D. Erni, and C. Hafner, "Band structure computations of metallic photonic crystals with the multiple multipole method," Phys. Rev. B65, 155120 (2002).
[CrossRef]

Phys. Rev. E (3)

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. E64, 046603 (2001).
[CrossRef]

S. Wilcox, L. C. Botten, R. C. McPhedran, C. G. Poulton, and C. M. de Sterke, "Modeling of defect modes in photonic crystals using the fictitious source superposition method," Phys. Rev. E71, 056606 (2005).
[CrossRef]

P. J. Chiang and C. P. Yu, and H. C. Chang, "Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method," Phys. Rev. E75, 026703 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

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]

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]

Other (4)

H. Ikuno and Y. Naka, "Finite-difference time-domain method applied to photonic crystals," in Electromagnetic Theory and Appications for Photonic Crystals, ed., K. Yasumoto, (CRC Press, Taylor & Francis Group, 2006).

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

T. Fujisawa and M. Koshiba, "Finite-element modeling of nonlinear interferometers based on photonic-crystal waveguides for all-optical signal processing," J. Lightw. Technol.24, 617-623 (1006).
[CrossRef]

P. A. Martin, Multiple Scattering (Cambridge University Press, Cambridge, UK, 2006).
[CrossRef]

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Figures (14)

Fig. 1.
Fig. 1.

Two neighboring square unit cells.

Fig. 2.
Fig. 2.

One period of a line defect waveguide in a bulk photonic crystal composed of cylinders in a square lattice.

Fig. 3.
Fig. 3.

Fig. 3. One period of a line defect waveguide in a bulk photonic crystal composed of cylinders in a triangular lattice.

Fig. 4.
Fig. 4.

Left: computation domain for a 90° photonic crystal waveguide bend. Right: transmission and reflection spectra of the 90° waveguide bend.

Fig. 5.
Fig. 5.

Electric field patterns of a 90° waveguide bend at ωa/(2πc)=0.353 (left) and ωa/(2πc)=0.42 (right).

Fig. 6.
Fig. 6.

Left: A microcavity coupled to waveguides. Right: Transmission spectrum of the microcavity coupled to waveguides.

Fig. 7.
Fig. 7.

Left: A double microcavity coupled to waveguides. Right: Transmission spectrum of the double microcavity coupled to waveguides.

Fig. 8.
Fig. 8.

Left: A photonic crystal waveguide Y-branch. Ports 1, 2 and 3 correspond to the waveguide in the left and the two waveguides in the right. Right: Transmission and reflection spectra of the Y-branch.

Fig. 9.
Fig. 9.

Left: A photonic crystal waveguide T-branch. Ports 1, 2 and 3 correspond to the waveguide in the left and the two waveguides in the right. Right: Transmission and reflection spectra of the T-branch.

Fig. 10.
Fig. 10.

Photonic crystal waveguides with one stub (left) and two stubs (right). A stub consists of two rods of radius r s . The three rods between the two stubs has a radius r e .

Fig. 11.
Fig. 11.

Transmission spectra of a PhC waveguide with a stub for different values of r s (the radius of the stub rods).

Fig. 12.
Fig. 12.

Transmission spectra of a PhC waveguide with two stubs for different values of r e (the radius of the three rods between the stubs).

Fig. 13.
Fig. 13.

Left: computation domain for a 60° waveguide bend with a microcavity and a special rod at the center. Right: Transmission spectra of the 60° waveguide bend for different values of the dielectric constant of the special rod.

Fig. 14.
Fig. 14.

Magnitude of the electric field in a 60° photonic crystal waveguide bend (with a microcavity) at a resonant frequency.

Equations (19)

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ρ x ( 1 ρ u x ) + ρ y ( 1 ρ u y ) + k 0 2 n 2 u = 0 ,
Λ u Γ = u ν Γ ,
u ( x ) j = 1 K c j ϕ j ( x ) ,
Λ ( 1 ) u = [ Λ 11 ( 1 ) Λ 12 ( 1 ) Λ 13 ( 1 ) Λ 14 ( 1 ) Λ 21 ( 1 ) Λ 22 ( 1 ) Λ 23 ( 1 ) Λ 24 ( 1 ) Λ 31 ( 1 ) Λ 32 ( 1 ) Λ 33 ( 1 ) Λ 34 ( 1 ) Λ 41 ( 1 ) Λ 42 ( 1 ) Λ 43 ( 1 ) Λ 44 ( 1 ) ] [ u 01 v 01 u 11 v 11 ] = [ y u 01 x v 01 y u 11 x v 11 ] .
1 ρ 1 ( Λ 41 ( 1 ) u 01 + Λ 42 ( 1 ) v 01 + Λ 43 ( 1 ) u 11 + Λ 44 ( 1 ) v 11 )
= 1 ρ 2 ( Λ 21 ( 2 ) u 02 + Λ 22 ( 2 ) v 11 + Λ 23 ( 2 ) u 12 + Λ 24 ( 2 ) v 21 ) ,
M [ u 0 u 1 ] = y [ u 0 u 1 ] ,
A 1 v 1 = A 2 [ u 0 u 1 ] ,
y [ u 0 u 1 ] = B 1 [ u 0 u 1 ] + B 2 v 1 ,
M = B 1 + B 2 A 1 1 A 2 .
w ( x , y ) = Φ ( x , y ) e i β y ,
[ M 11 I M 21 0 ] [ w 0 y w 0 ] = μ [ M 12 0 M 22 I ] [ w 0 y w 0 ] ,
u + ( x , y ) = j = 1 c j Φ j ( x , y ) e i β j y , u ( x , y ) = j = 1 d j Φ ˜ j ( x , y ) e i β j y .
y u 0 + = + u 0 + , for + = M 11 + M 12 T .
y u 0 = u 0 , for = M 21 T ˜ + M 22 .
y u = + u + ( + ) u , y = 0 + .
y u = u + ( + ) u + , y = 0 ,
ν u = + u + ( + ) u on Γ 0 + ,
v u = u + ( + ) u + on Γ 0 ,

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