Abstract

A detailed analysis of the polarization characteristics (birefringence) of a silicon-on-insulator (SOI) rib waveguide is given. The fundamental TE- and TM-polarized modes of the SOI rib waveguide are calculated by a semivectorial finite-difference method. The rib width and the slab height of the SOI rib waveguide are normalized with respect to the total height of the silicon layer. A general relation between the two normalized parameters for a nonbirefringent SOI rib waveguide is obtained. According to this relation a nonbirefringent SOI rib waveguide can easily be designed. The fabrication tolerance for a nonbirefringent SOI rib waveguide is also analyzed, revealing that the tolerance can be increased by use of a larger total height of the silicon layer.

© 2004 Optical Society of America

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References

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  1. P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
    [CrossRef]
  2. Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
    [CrossRef]
  3. R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
    [CrossRef]
  4. M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
    [CrossRef]
  5. W. P. Wong, K. S. Chiang, “Design of optical strip-loaded waveguides with zero modal birefringence,” J. Lightwave Technol. 16, 1240–1248 (1998).
    [CrossRef]
  6. B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
    [CrossRef]
  7. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
    [CrossRef]
  8. P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
    [CrossRef]
  9. P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
    [CrossRef]
  10. C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
    [CrossRef]
  11. W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
    [CrossRef]
  12. N.-N. Feng, G.-R. Zhou, W. P. Huang, “Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers,” J. Lightwave Technol. 20, 1976–1980 (2002).
    [CrossRef]
  13. M. S. Stern, “Semivectorial polarized finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
    [CrossRef]
  14. S. P. Pogossian, L. Vescan, A. Vonsovici, “The single-mode condition for semiconductor rib waveguides with large cross section,” J. Lightwave Technol. 16, 1851–1853 (1998).
    [CrossRef]
  15. O. Powell, “Single-mode condition for silicon rib waveguides,” J. Lightwave Technol. 20, 1851–1855 (2002).
    [CrossRef]

2002 (3)

2000 (1)

B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
[CrossRef]

1998 (3)

1997 (1)

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

1996 (2)

P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
[CrossRef]

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

1994 (3)

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
[CrossRef]

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

1990 (1)

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

1988 (1)

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

Albrecht, P.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Ando, S.

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

Chaudhuri, S. K.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

Chiang, K. S.

Coppinger, F.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

El-Mikati, H. A.

B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
[CrossRef]

Feng, N.-N.

Hamacher, M.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Heidrich, H.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Hibino, Y.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Hida, Y.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Hoffmann, D.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Huang, W. P.

N.-N. Feng, G.-R. Zhou, W. P. Huang, “Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers,” J. Lightwave Technol. 20, 1976–1980 (2002).
[CrossRef]

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

Inoue, Y.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

Itoh, M.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Jalali, B.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
[CrossRef]

Jinguji, K.

M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
[CrossRef]

Kasahara, R.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Kawachi, M.

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
[CrossRef]

Lui, W.

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Nolting, H.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Obayya, S. S. A.

B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
[CrossRef]

Ohmori, Y.

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

Okuno, M.

M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
[CrossRef]

Pogossian, S. P.

Powell, O.

Rahman, B. M. A.

B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
[CrossRef]

Rendina, I.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

Saida, T.

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

Sawada, T.

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

Stern, M. S.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

Sugita, A.

M. Okuno, A. Sugita, K. Jinguji, M. Kawachi, “Birefringence control of silica waveguides on Si and its application to a polarization-beam splitter/switch,” J. Lightwave Technol. 12, 625–633 (1994).
[CrossRef]

Takahashi, H.

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

Trinh, P. D.

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
[CrossRef]

Vescan, L.

Vonsovici, A.

Weinert, C. M.

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Wong, W. P.

Xu, C. I.

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Xu, C. L.

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

Yegnanarayanan, S.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
[CrossRef]

Yokoyama, K.

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

Yoon, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

Yoshimoto, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

Zhou, G.-R.

Electron. Lett. (1)

R. Kasahara, M. Itoh, Y. Hida, T. Saida, Y. Inoue, Y. Hibino, “Birefringence compensated silica-based waveguide with undercladding ridge,” Electron. Lett. 38, 1178–1179 (2002).
[CrossRef]

IEE Proc. Optoelectron. (3)

B. M. A. Rahman, S. S. A. Obayya, H. A. El-Mikati, “Minisation of modal birefringence in semiconductor optical guided-wave devices,” IEE Proc. Optoelectron. 147, 151–156 (2000).
[CrossRef]

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguide with arbitrary index profiles,” IEE Proc. Optoelectron. 135, 56–63 (1988).
[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, S. K. Chaudhuri, “Full-vectorial mode calculations by finite difference method,” IEE Proc. Optoelectron. 141, 281–286 (1994).
[CrossRef]

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

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, F. Coppinger, “Advances in silicon-on-insulator optoelectronics,” IEEE J. Sel. Top. Quantum Electron. 4, 938–947 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

P. D. Trinh, S. Yegnanarayanan, F. Coppinger, B. Jalali, “Silicon-on-insulator (SOI) phased-array wavelength multi/multiplexer with extremely low-polarization sensitivity,” IEEE Photon. Technol. Lett. 9, 940–942 (1997).
[CrossRef]

P. D. Trinh, S. Yegnanarayanan, B. Jalali, “5 × 9 integrated optical star coupler in silicon-on-insulator technology,” IEEE Photon. Technol. Lett. 8, 940–942 (1996).
[CrossRef]

P. Albrecht, M. Hamacher, H. Heidrich, D. Hoffmann, H. Nolting, C. M. Weinert, “TE/TM mode splitters on InGaAsP/InP,” IEEE Photon. Technol. Lett. 2, 114–115 (1990).
[CrossRef]

Y. Inoue, Y. Ohmori, M. Kawachi, S. Ando, T. Sawada, H. Takahashi, “Polarization mode converter with polyimide half waveplate in silica-based planar lightwave circuits,” IEEE Photon. Technol. Lett. 6, 626–628 (1994).
[CrossRef]

W. P. Huang, C. I. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer boundary condition for modal analysis of optical waveguides: leaky mode calculations,” IEEE Photon. Technol. Lett. 8, 652–654 (1996).
[CrossRef]

J. Lightwave Technol. (5)

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Figures (6)

Fig. 1
Fig. 1

Cross section of a SOI rib waveguide and the corresponding single-mode condition.

Fig. 2
Fig. 2

Birefringence Δn of a SOI rib waveguide as t varies (for different values of r).

Fig. 3
Fig. 3

Birefringence Δn of a SOI rib waveguide as r varies (for different values of t).

Fig. 4
Fig. 4

Normalized rib width t for a nonbirefringent SOI rib waveguide as r varies.

Fig. 5
Fig. 5

Birefringence of a SOI rib waveguide when H = 5 μm and H = 8 μm.

Fig. 6
Fig. 6

Fabrication tolerance for a nonbirefringent SOI rib waveguide: (a) H = 5 μm, (b) H = 8 μm.

Equations (10)

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

P xx P xy P yx P yy E x E y = 2 z 2 E x E y ,
P xx E x = x ˜ 1 n 2 n 2 E x x ˜ + 2 E x y ˜ 2 + n 2 k 0 2 E x , P xy E y = x ˜ 1 n 2 n 2 E y y ˜ - 2 E y x ˜ y ˜ , P yy E y = y ˜ 1 n 2 n 2 E y y ˜ + x ˜ E y x ˜ + n 2 k 0 2 E y , P yx E x = y ˜ 1 n 2 n 2 E x x ˜ - x ˜ E x y ˜ ,
ζ ˜ = 0 ζ   s ζ ζ d ζ     ζ = x ,   y ,
s ζ ζ = 1 in the non-PML region 1 - j   σ ζ max ω ε 0 | ζ - ζ 0 | 2 d 2 in the PML region ,
P xx Ē x = β 2 Ē x ,
P yy Ē y = β 2 Ē y .
a mn E x m - 1 ,   n + b mn E x m ,   n + c mn E x m + 1 ,   n + d mn E x m ,   n - 1 + e mn E x m ,   n + 1 = β 2 E x m ,   n ,
a mn = T m - 1 Δ x ˜ m Δ x ˜ m - 1 / 2 , b mn = - 1 Δ x ˜ m 2 - T m + 1 Δ x ˜ m + 1 / 2 + 2 - T m - 1 Δ x ˜ m - 1 / 2 - 1 Δ y ˜ n + 1 / 2 + 1 Δ y ˜ n - 1 / 2 1 Δ y ˜ n + n 2 k 0 2 , c mn = 1 Δ x ˜ m Δ x ˜ m + 1 / 2   T m + 1 , d mn = 1 Δ y ˜ n Δ y ˜ n - 1 / 2 , e mn = 1 Δ y ˜ n Δ y ˜ n + 1 / 2 ,
T m ± 1 = 2 n m ± 1 2 / n m ± 1 2 + n m 2 , Δ x ˜ m + 1 / 2 = x ˜ m + 1 - x ˜ m , Δ x ˜ m - 1 / 2 = x ˜ m - x ˜ m - 1 , Δ x ˜ m = x ˜ m + 1 / 2 - x ˜ m - 1 / 2 .
Δ n = n TE - n TM .

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