Abstract

We present numerical implementation and verification of a rigorous full-vector, integral-equation formulation suitable for analyzing modal characteristics of complex, two-dimensional (2D) rectangular-like dielectric waveguides. By dividing the waveguide into vertical slices, a system of integral equations we call vector-coupled transverse-mode integral equations (VCTMIE) is derived. The entire electromagnetic mode fields are completely determined by one-dimensional unknown field functions on the slice interfaces. To further reduce numerical computation, we expand these functions in terms of the guiding modes of a slab waveguide with a large normalized frequency. Through orthogonal projection the resulting nonlinear eigenvalue and eigenvector matrix formulation enables us to obtain the effective mode index with 107 precision and to compute with high resolution the 2D vectorial mode field solutions of an open dielectric waveguide. We show stable and speedy convergence of our method as well as techniques to overcome the Gibbs phenomenon in the reconstruction of the transverse fields.

© 2006 Optical Society of America

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  1. H. W. Chang, T. L. Wu, and M. H. Sheng, "Vectorial modal analysis of dielectric waveguides based on a coupled transverse-mode integral equation. I. Mathematical formulation," J. Opt. Soc. Am. A 23, 1468-1477 (2006).
    [CrossRef]
  2. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
    [CrossRef]
  3. E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).
  4. J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).
  5. S. F. Tsao, S. M. Chang, H. W. Chang, "Vectorial modal analysis of 2D dielectric waveguides by simple orthogonal basis," in Proceedings of Optics and Photonics Taiwan 2003 (National Taipei University of Technology, 2003), pp. 94-96.
  6. C. C. Huang, C.-C. Huang, and J. Y. Yang, "An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition," J. Lightwave Technol. 21, 2284-2296 (2003).
    [CrossRef]
  7. P. C. Waterman, "New formulations of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
    [CrossRef]
  8. M. H. Sheng and H. W. Chang, "Accurate first-order leaky-wave analysis of antiresonant reflecting optical waveguides," Appl. Opt. 44, 751-764 (2005).
    [CrossRef] [PubMed]
  9. N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
    [CrossRef]
  10. M. Kohtoku, T. Hirono, S. Oku, Y. Kadota, Y. Shibata, and Y. Yoshikuni, "Control of higher order leaky modes in deep-ridged waveguides and application to low-crosstalk arrayed waveguide gratings," J. Lightwave Technol. 22, 499-508 (2004).
    [CrossRef]
  11. T. L. Wu and H. W. Chang, "Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination," J. Opt. Soc. Am. A 18, 2823-2832 (2001).
    [CrossRef]
  12. A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993).
    [CrossRef]
  13. D. U. Li and H. C. Chang, "Full-vectorial finite element modal analysis of bounded and unbounded waveguides," in Proceedings of Asia Pacific Microwave Conference 2001 (National Taiwan University, 2001), pp. 376-379.
  14. Y. C. Chiang, Y. P. Chiou, and H. C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
    [CrossRef]
  15. G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis II: Dielectric corners," J. Lightwave Technol. 20, 1219-1231 (2002).
    [CrossRef]
  16. N. Thomas, "Finite-difference methods for the modal analysis of dielectric waveguides with rectangular corners," Ph.D thesis(University of Nottingham, UK, 2004).
  17. A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
    [CrossRef]
  18. W. W. Lui, C.-L. Xu, W.-P. Huang, K. Yokoyama, and S. Seki, "Full-vectorial mode analysis with considerations of field singularities at corners of optical waveguides," J. Lightwave Technol. 17, 1509-1513 (1999).
    [CrossRef]
  19. T. Lu and D. Yevick, "Comparative evaluation of a novel series approximation for electromagnetic fields at dielectric corners with boundary element method applications," J. Lightwave Technol. 22, 1426-1432 (2004).
    [CrossRef]

2006 (1)

2005 (1)

2004 (3)

2003 (1)

2002 (2)

2001 (1)

2000 (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

1999 (1)

1993 (1)

A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993).
[CrossRef]

1992 (1)

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

1969 (3)

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

P. C. Waterman, "New formulations of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

Chang, H. C.

Y. C. Chiang, Y. P. Chiou, and H. C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightwave Technol. 20, 1609-1618 (2002).
[CrossRef]

D. U. Li and H. C. Chang, "Full-vectorial finite element modal analysis of bounded and unbounded waveguides," in Proceedings of Asia Pacific Microwave Conference 2001 (National Taiwan University, 2001), pp. 376-379.

Chang, H. W.

Chang, S. M.

S. F. Tsao, S. M. Chang, H. W. Chang, "Vectorial modal analysis of 2D dielectric waveguides by simple orthogonal basis," in Proceedings of Optics and Photonics Taiwan 2003 (National Taipei University of Technology, 2003), pp. 94-96.

Chiang, Y. C.

Chiou, Y. P.

Goell, J. E.

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

Gopinath, A.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Hadley, G. R.

Helfert, S.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Hirono, T.

Huang, C. C.

Huang, C.-C.

Huang, W.-P.

Kadota, Y.

Kanaguchi, Y.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

Kikuchi, N.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

Kohtoku, M.

Kondo, Y.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

Li, D. U.

D. U. Li and H. C. Chang, "Full-vectorial finite element modal analysis of bounded and unbounded waveguides," in Proceedings of Asia Pacific Microwave Conference 2001 (National Taiwan University, 2001), pp. 376-379.

Lu, T.

Lui, W. W.

Marcatili, E. A.

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

Okamoto, H.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

Oku, S.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

M. Kohtoku, T. Hirono, S. Oku, Y. Kadota, Y. Shibata, and Y. Yoshikuni, "Control of higher order leaky modes in deep-ridged waveguides and application to low-crosstalk arrayed waveguide gratings," J. Lightwave Technol. 22, 499-508 (2004).
[CrossRef]

Pregla, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Scarmozzino, R.

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

Seki, S.

Sheng, M. H.

Shibata, Y.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

M. Kohtoku, T. Hirono, S. Oku, Y. Kadota, Y. Shibata, and Y. Yoshikuni, "Control of higher order leaky modes in deep-ridged waveguides and application to low-crosstalk arrayed waveguide gratings," J. Lightwave Technol. 22, 499-508 (2004).
[CrossRef]

Sudbo, A. S.

A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993).
[CrossRef]

A. S. Sudbo, "Why are accurate computations of mode fields in rectangular dielectric waveguides difficult?" J. Lightwave Technol. 10, 418-419 (1992).
[CrossRef]

Thomas, N.

N. Thomas, "Finite-difference methods for the modal analysis of dielectric waveguides with rectangular corners," Ph.D thesis(University of Nottingham, UK, 2004).

Tohmori, Y.

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

Tsao, S. F.

S. F. Tsao, S. M. Chang, H. W. Chang, "Vectorial modal analysis of 2D dielectric waveguides by simple orthogonal basis," in Proceedings of Optics and Photonics Taiwan 2003 (National Taipei University of Technology, 2003), pp. 94-96.

Waterman, P. C.

P. C. Waterman, "New formulations of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

Wu, T. L.

Xu, C.-L.

Yang, J. Y.

Yevick, D.

Yokoyama, K.

Yoshikuni, Y.

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

E. A. Marcatili, "Dielectric rectangular waveguide and dielectric coupler for integrated optics," Bell Syst. Tech. J. 48, 2071-2103 (1969).

J. E. Goell, "A circular-harmonic computer analysis of rectangular dielectric waveguides," Bell Syst. Tech. J. 48, 2133-2160 (1969).

IEEE J. Sel. Top. Quantum Electron. (1)

R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, "Numerical techniques for modeling guided wave photonic devices," IEEE J. Sel. Top. Quantum Electron. 6, 150-162 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

N. Kikuchi, Y. Shibata, H. Okamoto, Y. Kanaguchi, S. Oku, Y. Kondo, and Y. Tohmori, "Monolithically integrated 100-channel WDM channel selector employing low-crosstalk AWG," IEEE Photon. Technol. Lett. 16, 2481-2483 (2004).
[CrossRef]

J. Acoust. Soc. Am. (1)

P. C. Waterman, "New formulations of acoustic scattering," J. Acoust. Soc. Am. 45, 1417-1429 (1969).
[CrossRef]

J. Lightwave Technol. (7)

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

Pure Appl. Opt. (1)

A. S. Sudbo, "Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides," Pure Appl. Opt. 2, 211-233 (1993).
[CrossRef]

Other (3)

D. U. Li and H. C. Chang, "Full-vectorial finite element modal analysis of bounded and unbounded waveguides," in Proceedings of Asia Pacific Microwave Conference 2001 (National Taiwan University, 2001), pp. 376-379.

N. Thomas, "Finite-difference methods for the modal analysis of dielectric waveguides with rectangular corners," Ph.D thesis(University of Nottingham, UK, 2004).

S. F. Tsao, S. M. Chang, H. W. Chang, "Vectorial modal analysis of 2D dielectric waveguides by simple orthogonal basis," in Proceedings of Optics and Photonics Taiwan 2003 (National Taipei University of Technology, 2003), pp. 94-96.

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Figures (7)

Fig. 1
Fig. 1

Simple rectangular dielectric waveguide of dimensions a by b.

Fig. 2
Fig. 2

Enclosed ridged dielectric waveguide with (a) one interface with two slices, (b) two interfaces with three slices. The two configurations are physically identical.

Fig. 3
Fig. 3

Propagation curves for (a) E 11 x mode with a b = 1 , (b) E 11 x with a b = 2 for rectangular waveguide exposed in air. Here Δ n r = 0.5 .

Fig. 4
Fig. 4

Propagation curves for (a) E 11 x (b) E 11 y modes of rectangular waveguide with different a b ratios and Δ n r = 0.5 .

Fig. 5
Fig. 5

Transverse vector field plots of the E 11 x and E 11 y modes for a square dielectric waveguide with a full width n 1 = 1.5 , n 0 = 1.0 , and a wavelength λ = 1.5 μ m corresponding to a normalized frequency B = 2 . The waveguide boundaries are outlined in solid lines. Note that these two modes are degenerate due to the square symmetric.

Fig. 6
Fig. 6

Transverse vector field plots of higher-order modes E 21 y and E 12 y of a square dielectric waveguide as in Fig. 5.

Fig. 7
Fig. 7

Effective index and two adjacent slice fields E y ( y ) at the interface x = a 2 ( = λ 4 ) for rectangular waveguide with a b = 1 , n 1 = 1.5 , n 0 = 1.0 , and a = λ 2 . PCW at x = 3.01 λ and y = 2 λ . Data in the left column are simulated FMM results using 10 to 80 slice modes from each polarization. Data on the right column are VCTMIE results with 10 to 80 vertical 1D mode functions from a large-V-basis waveguide.

Tables (1)

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Table 1 First Ten Effective Indices of a Ridged Waveguide a

Equations (47)

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[ H z ( 1 ) ( x 1 δ , y ) E z ( 1 ) ( x 1 δ , y ) ] = d y [ R h e ( 1 ) ( y , y ) R h h ( 1 ) ( y , y ) R e e ( 1 ) ( y , y ) R e h ( 1 ) ( y , y ) ] [ E 1 ( y ) H 1 ( y ) ] ,
[ H x ( 2 ) ( x 1 + δ , y ) E z ( 2 ) ( x 1 + δ , y ) ] = d y [ Q h e ( 2 ) ( y , y ) Q h h ( 2 ) ( y , y ) Q e e ( 2 ) ( y , y ) Q e h ( 2 ) ( y , y ) ] [ E 1 ( y ) H 1 ( y ) ] ,
Q h e ( 2 ) ( y , y ) = n j ω [ k c e , n ( 2 ) ] 2 Ψ D , n ( 2 , l ) ( x 1 ) ϕ D , n ( 2 ) ( y ) [ ϕ D , n ( 2 ) ( y ) ] * ,
Q h h ( 2 ) ( y , y ) = n j β [ k c h , n ( 2 ) ] 2 ψ B , n ( 2 , l ) ( x 1 ) Φ B , n ( 2 ) ( y ) [ ϕ B , n ( 2 ) ( y ) ] * ,
Q e e ( 2 ) ( y , y ) = n j β [ k c e , n ( 2 ) ] 2 Ψ D , n ( 2 , l ) ( x 1 ) ϕ D , n ( 2 ) ( y ) [ ϕ D , n ( 2 ) ( y ) ] * ,
Q e h ( 2 ) ( y , y ) = n j ω [ k c h , n ( 2 ) ] 2 Ψ B , n ( 2 , l ) ( x 1 ) ϕ B , n ( 2 ) ( y ) [ ϕ B , n ( 2 ) ( y ) ] * .
Ψ F , n ( 1 , r ) ( x 1 ) = k x F , n ( 1 ) , Ψ F , n ( 2 , l ) ( x 1 ) = k x F , n ( 2 ) ,
Ψ F , n ( 1 , r ) ( x 1 ) = k x F , n ( 1 ) cot ( k x F , n ( 1 ) Δ x 1 ) ,
Ψ F , n ( 2 , l ) ( x 1 ) = k x F , n ( 2 ) cot ( k x F , n ( 2 ) Δ x 2 ) ,
Ψ F , n ( 1 , r ) ( x 1 ) = k x F , n ( 1 ) tan ( k x F , n ( 1 ) Δ x 1 ) ,
Ψ F , n ( 2 , l ) ( x 1 ) = k x F , n ( 2 ) tan ( k x F , n ( 2 ) Δ x 2 ) ,
d y R ( 1 ) F 1 = d y Q ( 2 ) F 1 ,
E 1 ( y ) = n c e , n ( 1 ) ϕ e , n ( 1 ) ( y ) , H 1 ( y ) = n c h , n ( 1 ) ϕ h , n ( 1 ) ( y ) ,
F 1 = [ n = 1 N b ( 1 ) c e , n ( 1 ) ϕ e , n ( 1 ) ( y ) n = 1 N b ( 1 ) c h , n ( 1 ) ϕ h , n ( 1 ) ( y ) ] 2 , 1 = [ [ ϕ ¯ e ( 1 ) ( y ) ] T 0 0 [ ϕ ¯ h ( 1 ) ( y ) ] T ] 2 , 2 N b ( 1 ) C 1 .
{ [ T E , D ( 1 , 1 ) R h e ( 1 ) T D , E ( 1 , 1 ) T E , d B ( 1 , 1 ) R h h ( 1 ) T B , H ( 1 , 1 ) T H , d D ( 1 , 1 ) R e e ( 1 ) T D , E ( 1 , 1 ) T H , D ( 1 , 1 ) R e h ( 1 ) T B , H ( 1 , 1 ) ] [ T E , D ( 1 , 2 ) Q h e ( 2 ) T D , E ( 2 , 1 ) T E , d B ( 1 , 2 ) Q h h ( 2 ) T B , H ( 2 , 1 ) T H , d D ( 1 , 2 ) Q e e ( 2 ) T D , E ( 2 , 1 ) T H , B ( 1 , 2 ) Q e h ( 2 ) T B , H ( 2 , 1 ) ] } C 1 = 0 .
[ T u , v ( m , n ) ] i , j = ϕ u , j ( m ) ϕ v , j ( n ) ,
[ R h e ( 1 ) ] i , j = j ω [ k c e , i ( 1 ) ] 2 Ψ D , i ( 1 , r ) ( x 1 ) δ i , j , [ R h h ( 1 ) ] i , j = j β [ k c h , i ( 1 ) ] 2 δ i , j ,
[ R e e ( 1 ) ] i , j = j β [ k c e , i ( 1 ) ] 2 δ i , j , [ R e h ( 1 ) ] i , j = j ω [ k c h , i ( 1 ) ] 2 Ψ B , i ( 1 , r ) ( x 1 ) δ i , j ,
[ Q h e ( 2 ) ] i , j = j ω [ k c e , i ( 1 ) ] 2 Ψ D , i ( 2 , l ) ( x 1 ) δ i , j , [ Q h h ( 2 ) ] i , j = j β [ k c h , i ( 2 ) ] 2 δ i , j ,
[ Q e e ( 2 ) ] i , j = j β [ k c e , i ( 2 ) ] 2 δ i , j , [ Q e h ( 2 ) ] i , j = j ω [ k c h , i ( 2 ) ] 2 Ψ B , i ( 2 , l ) ( x 1 ) δ i , j .
[ A ( β ) ] 2 N b , 2 N b C = 0 .
{ [ R h e ( 1 ) T E , d B ( 1 , 1 ) R h h ( 1 ) T H , d D ( 1 , 1 ) R e e ( 1 ) R e h ( 1 ) ] [ T E , D ( 1 , 2 ) Q h e ( 2 ) T D , E ( 2 , 1 ) T E , d B ( 1 , 2 ) Q h h ( 2 ) T B , H ( 2 , 1 ) T H , d D ( 1 , 2 ) Q e e ( 2 ) T D , E ( 2 , 1 ) T H , B ( 1 , 2 ) Q e h ( 2 ) T B , H ( 2 , 1 ) ] } C = 0 .
{ [ T E , D ( 2 , 1 ) R h e ( 1 ) T D , E ( 1 , 2 ) T E , d B ( 2 , 1 ) R h h ( 1 ) T B , H ( 1 , 2 ) T H , d D ( 2 , 1 ) R e e ( 1 ) T D , E ( 1 , 2 ) T H , B ( 2 , 1 ) R e h ( 1 ) T B , H ( 1 , 2 ) ] [ Q h e ( 2 ) T E , d B ( 2 , 2 ) Q h h ( 2 ) T H , d D ( 2 , 2 ) Q e e ( 2 ) Q e h ( 2 ) ] } C = 0
d y [ R ( 1 ) Q ( 2 ) S ( 2 ) P ( 2 ) R ( 2 ) Q ( 3 ) ] [ F 1 F 2 ] = 0 .
[ A 11 A 12 A 21 A 22 ] [ C 1 C 2 ] = 0 ,
A 11 = [ T E , D ( 1 , 1 ) R h e ( 1 ) T D , E ( 1 , 1 ) T E , d B ( 1 , 1 ) R h h ( 1 ) T B , H ( 1 , 1 ) T H , d D ( 1 , 1 ) R e e ( 1 ) T D , E ( 1 , 1 ) T H , B ( 1 , 1 ) R e h ( 1 ) T B , H ( 1 , 1 ) ] [ T E , D ( 1 , 2 ) Q h e ( 2 ) T D , E ( 2 , 1 ) T E , d B ( 1 , 2 ) Q h h ( 2 ) T B , H ( 2 , 1 ) T H , d D ( 1 , 2 ) Q e e ( 2 ) T D , E ( 2 , 1 ) T H , B ( 1 , 2 ) Q e h ( 2 ) T B , H ( 2 , 1 ) ] ,
A 12 = [ T E , D ( 1 , 2 ) S h e ( 2 ) T D , E ( 2 , 2 ) 0 0 T H , B ( 1 , 2 ) S e h ( 2 ) T B , H ( 2 , 2 ) ] ,
A 21 = [ T E , D ( 2 , 2 ) P h e ( 2 ) T D , E ( 2 , 1 ) 0 0 T H , B ( 2 , 2 ) P e h ( 2 ) T B , H ( 2 , 1 ) ] ,
A 22 = [ T E , D ( 2 , 2 ) R h e ( 2 ) T D , E ( 2 , 2 ) T E , d B ( 2 , 2 ) R h h ( 2 ) T B , H ( 2 , 2 ) T H , d D ( 2 , 2 ) R e e ( 2 ) T D , E ( 2 , 2 ) T H , B ( 2 , 2 ) R e h ( 2 ) T B , H ( 2 , 2 ) ] [ T E , D ( 2 , 3 ) Q h e ( 3 ) T D , E ( 3 , 2 ) T E , d B ( 2 , 3 ) Q h h ( 3 ) T B , H ( 3 , 2 ) T H , d D ( 2 , 3 ) Q e e ( 3 ) T D , E ( 3 , 2 ) T H , B ( 2 , 3 ) Q e h ( 3 ) T B , H ( 3 , 2 ) ] .
[ R h e ( 2 ) ] i , j = j ω [ k c e , i ( 2 ) ] 2 Ψ D , i ( 2 , r ) ( x 2 ) δ i , j , [ R h h ( 2 ) ] i , j = j β [ k c h , i ( 2 ) ] 2 δ i , j ,
[ R e e ( 2 ) ] i , j = j β [ k c e , i ( 2 ) ] 2 δ i , j , [ R e h ( 2 ) ] i , j = j ω [ k c h , i ( 2 ) ] 2 Ψ B , i ( 2 , r ) ( x 2 ) δ i , j ,
[ S h e ( 2 ) ] i , j = j ω [ k c e , i ( 2 ) ] 2 Ψ D , i ( 2 , r ) ( x 1 ) δ i , j ,
[ S e h ( 2 ) ] i , j = j ω [ k c h , i ( 2 ) ] 2 Ψ B , i ( 2 , r ) ( x 1 ) δ i , j ,
[ P h e ( 2 ) ] i , j = j ω [ k c e , i ( 2 ) ] 2 Ψ D , i ( 2 , l ) ( x 2 ) δ i , j ,
[ P e h ( 2 ) ] i , j = j ω [ k c h , i ( 2 ) ] 2 Ψ B , i ( 2 , l ) ( x 2 ) δ i , j ,
[ Q h e ( 2 ) ] i , j = j ω [ k c e , i ( 2 ) ] 2 Ψ D , i ( 2 , l ) ( x 1 ) δ i , j , [ Q h h ( 2 ) ] i , j = j β [ k c h , i ( 2 ) ] 2 δ i , j ,
[ Q e e ( 2 ) ] i , j = j β [ k c e , i ( 2 ) ] 2 δ i , j , [ Q e h ( 2 ) ] i , j = j ω [ k c h , i ( 2 ) ] 2 Ψ B , i ( 2 , l ) ( x 1 ) δ i , j .
Ψ F , n ( m , l ) ( x m 1 ) = k x F , n ( m ) cot ( k x F , n ( m ) Δ x m ) ,
Ψ F , n ( m , l ) ( x m ) = k x F , n ( m ) csc ( k x F , n ( m ) Δ x m ) ,
Ψ F , n ( m , r ) ( x m 1 ) = k x F , n ( m ) csc ( k x F , n ( m ) Δ x m ) ,
Ψ F , n ( m , r ) ( x m ) = k x F , n ( m ) cot ( k x F , n ( m ) Δ x m ) .
[ A 11 A 12 0 0 A 21 A 22 0 0 0 0 A M 1 , M 1 A M 1 , M 0 0 A M , M 1 A M , M ] [ C 1 C 2 C M 1 C M ] = 0 ,
P 2 = ( β k 0 ) 2 1 ( n 1 n 0 ) 2 1 , B = 2 b λ 0 ( n 1 n 0 ) 2 1 .
FLOPs [ N n ( m ) ] 3 × M + N b 2 × N ( m ) × M .
T D , E ( m , n ) = [ ϕ D , 1 ( m ) ( y 1 ) ϕ D , 1 ( m ) ( y N g ) ϕ D , N ( m ) ( y 1 ) ϕ D , N ( m ) ( y N g ) ] N × N g [ w 1 0 0 w N g ] [ ϕ E , 1 ( m ) ( y 1 ) ϕ E , N b ( m ) ( y 1 ) ϕ E , 1 ( m ) ( y N g ) ϕ E , N b ( m ) ( y N g ) ] N g × N b ,
T E , D ( 1 , 1 ) R h e ( 1 ) T E , E ( 1 , 1 ) = [ ϕ E , i ( 1 ) ϕ D , j ( 1 ) ] N b × N [ R h e ( 1 ) ] N × N [ ϕ D , i ( 1 ) ϕ E , j ( 1 ) ] N × N b = [ n = 1 N t i , n r n τ n , j ] N b × N b .
Δ [ T E , D ( 1 , 1 ) R h e ( 1 ) T D , E ( 1 , 1 ) ] i , j = t i , n r n τ n , j .

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