Abstract

A vector mode solver for bending waveguides by using a modified finite-difference (FD) method is developed in a local cylindrical coordinate system, where the perfectly matched layer absorbing boundary conditions are incorporated. Utilizing Taylor series expansion technique and continuity condition of the longitudinal field components, a standard matrix eigenvalue equation without the averaged index approximation approach for dealing with the discrete points neighboring the dielectric interfaces is obtained. Complex effective indexes and field distributions of leaky modes for a typical rib bending waveguide and a silicon wire bend are presented, and solutions accord well with those from the film mode matching method, which shows the validity and utility of the established method.

© 2012 OSA

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  1. P. J. Harshman and T. K. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4(8), 498–499 (2010).
    [CrossRef]
  2. D. F. Welch, A. K. Kish, R. Nagarajan, C. H. Joyner, R. P. Schneider, V. G. Dominic, M. L. Mitchell, S. G. Grubb, T. K. Chiang, D. D. Perkins, and A. C. Nilsson, “The realization of large-scale photonic integrated circuits and the associated impact on fiber-optic communication systems,” J. Lightwave Technol.24(12), 4674–4683 (2006).
  3. N. Somasiri, B. M. A. Rahman, and S. S. A. Obayya, “Fabrication tolerance study of a compact passive polarization rotator,” J. Lightwave Technol.20(4), 751–757 (2002).
    [CrossRef]
  4. C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
    [CrossRef]
  5. J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
    [CrossRef]
  6. W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett.8(5), 649–651 (1996).
    [CrossRef]
  7. J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
    [CrossRef]
  8. R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol.14(4), 634–639 (1996).
    [CrossRef]
  9. L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
    [CrossRef]
  10. L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
    [CrossRef]
  11. K. Kakihara, N. Kono, K. Saitoh, and M. Koshiba, “Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends,” Opt. Express14(23), 11128–11141 (2006).
    [CrossRef] [PubMed]
  12. B. M. A. Rahman, D. M. H. Leung, S. S. A. Obayya, and K. T. V. Grattan, “Numerical analysis of bent waveguides: bending loss, transmission loss, mode coupling, and polarization coupling,” Appl. Opt.47(16), 2961–2970 (2008).
    [CrossRef] [PubMed]
  13. J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett.33(16), 1848–1850 (2008).
    [CrossRef] [PubMed]
  14. J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
    [CrossRef]
  15. M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13(2), 233–238 (1995).
    [CrossRef]
  16. S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol.14(9), 2085–2092 (1996).
    [CrossRef]
  17. W. W. Liu, C. L. Xu, T. Hirono, K. Yokoyama, and W. P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximation,” J. Lightwave Technol.16(5), 910–914 (1998).
    [CrossRef]
  18. N. N. Feng, G. R. Zhou, C. L. Xu, and W. P. Huang, “Computation of full-vector modes for bending waveguides using cylindrical perfectly matched layers,” J. Lightwave Technol.20(11), 1976–1980 (2002).
    [CrossRef]
  19. J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
    [CrossRef]
  20. M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29(5), 691–699 (2011).
    [CrossRef]
  21. F. L. Texeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett.7(9), 285–287 (1997).
    [CrossRef]
  22. P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
    [CrossRef]
  23. Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express12(8), 1622–1631 (2004).
    [CrossRef] [PubMed]
  24. H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
    [CrossRef]
  25. T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
    [CrossRef]
  26. C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
    [CrossRef]

2011 (3)

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29(5), 691–699 (2011).
[CrossRef]

2010 (1)

P. J. Harshman and T. K. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4(8), 498–499 (2010).
[CrossRef]

2009 (2)

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

2008 (2)

2006 (3)

2005 (1)

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
[CrossRef]

2004 (2)

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
[CrossRef]

Y. A. Vlasov and S. J. McNab, “Losses in single-mode silicon-on-insulator strip waveguides and bends,” Opt. Express12(8), 1622–1631 (2004).
[CrossRef] [PubMed]

2002 (2)

1998 (1)

1997 (1)

F. L. Texeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett.7(9), 285–287 (1997).
[CrossRef]

1996 (4)

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol.14(9), 2085–2092 (1996).
[CrossRef]

R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol.14(4), 634–639 (1996).
[CrossRef]

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett.8(5), 649–651 (1996).
[CrossRef]

1995 (1)

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13(2), 233–238 (1995).
[CrossRef]

1994 (2)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

1991 (1)

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
[CrossRef]

Arakawa, Y.

H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
[CrossRef]

Bai, N.

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

Bakir, B. B.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

Bernabe, S.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Besse, P. A.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
[CrossRef]

Chew, W. C.

F. L. Texeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett.7(9), 285–287 (1997).
[CrossRef]

Chiang, T. K.

Chu, T.

H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
[CrossRef]

Ctyroky, J.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
[CrossRef]

Dominic, V. G.

Fedeli, J. M.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Feng, N. N.

Gopinath, A.

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol.14(9), 2085–2092 (1996).
[CrossRef]

Grattan, K. T. V.

Groen, G. H.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Grubb, S. G.

Gu, J. S.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
[CrossRef]

Hamacher, M.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Harshman, P. J.

P. J. Harshman and T. K. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4(8), 498–499 (2010).
[CrossRef]

Heidrich, H.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Hirono, T.

Huang, W. P.

Hubalek, M.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
[CrossRef]

Ishida, S.

H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
[CrossRef]

Itabashi, S.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Joyner, C. H.

Kakihara, K.

Kim, S.

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol.14(9), 2085–2092 (1996).
[CrossRef]

Kish, A. K.

Kono, N.

Kopp, C.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Koshiba, M.

Kou, R.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Krause, M.

Leung, D. M. H.

Liu, W. W.

Liu, X.

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett.8(5), 649–651 (1996).
[CrossRef]

Lusse, P.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

Ma, H.

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

McNab, S. J.

Melchior, H.

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
[CrossRef]

Mitchell, M. L.

Moerman, I.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Nagarajan, R.

Ni, H.

Nilsson, A. C.

Nishi, H.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Obayya, S. S. A.

Orobtchouk, R.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Paniccia, T. K. M.

P. J. Harshman and T. K. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4(8), 498–499 (2010).
[CrossRef]

Park, S.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Pascher, W. W.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Perkins, D. D.

Porte, H.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Pregla, R.

R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol.14(4), 634–639 (1996).
[CrossRef]

Prkna, L.

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
[CrossRef]

Rahman, B. M. A.

Rivera, M.

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13(2), 233–238 (1995).
[CrossRef]

Saitoh, K.

Schneider, R. P.

Schrank, F.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Schule, J.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

Shinojima, H.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Smit, M. K.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Somasiri, N.

Spiekman, L. H.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Stuwe, P.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

Sun, H.

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

Sun, X.

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett.33(16), 1848–1850 (2008).
[CrossRef] [PubMed]

Tekin, T.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Texeira, F. L.

F. L. Texeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett.7(9), 285–287 (1997).
[CrossRef]

Tsuchizawa, T.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Unger, H. G.

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

van Dam, C.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

van der Tol, J. J. G. M.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

van Ham, F. P. G. M.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Vlasov, Y. A.

Watanabe, T.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Weinert, C. M.

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

Welch, D. F.

Xiao, J.

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

J. Xiao, H. Ni, and X. Sun, “Full-vector mode solver for bending waveguides based on the finite-difference frequency-domain method in cylindrical coordinate systems,” Opt. Lett.33(16), 1848–1850 (2008).
[CrossRef] [PubMed]

Xu, C. L.

Yamada, H.

H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
[CrossRef]

Yamada, K.

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

Yokoyama, K.

W. W. Liu, C. L. Xu, T. Hirono, K. Yokoyama, and W. P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximation,” J. Lightwave Technol.16(5), 910–914 (1998).
[CrossRef]

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett.8(5), 649–651 (1996).
[CrossRef]

Zhou, G. R.

Zimmermann, L.

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

J. S. Gu, P. A. Besse, and H. Melchior, “Method of lines for the analysis of the propagation characteristics of curved optical rib wave-guides,” IEEE J. Quantum Electron.27(3), 531–537 (1991).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (4)

L. Prkna, M. Hubalek, and J. Ctyroky, “Field modeling of circular microresonators by film mode matching,” IEEE J. Sel. Top. Quantum Electron.11(1), 217–223 (2005).
[CrossRef]

H. Yamada, T. Chu, S. Ishida, and Y. Arakawa, “Si photonic wire waveguide devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1371–1379 (2006).
[CrossRef]

T. Tsuchizawa, K. Yamada, T. Watanabe, S. Park, H. Nishi, R. Kou, H. Shinojima, and S. Itabashi, “Monolithic integration of silicon-, germanium-, and silica-based optical devices for telecommunications applications,” IEEE J. Sel. Top. Quantum Electron.17(3), 516–525 (2011).
[CrossRef]

C. Kopp, S. Bernabe, B. B. Bakir, J. M. Fedeli, R. Orobtchouk, F. Schrank, H. Porte, L. Zimmermann, and T. Tekin, “Silicon photonic circuits: on-CMOS integration, fiber optical coupling, and packaging,” IEEE J. Sel. Top. Quantum Electron.17(3), 498–509 (2011).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

F. L. Texeira and W. C. Chew, “PML-FDTD in cylindrical and spherical grids,” IEEE Microwave Guided Wave Lett.7(9), 285–287 (1997).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

W. P. Huang, C. L. Xu, W. Lui, and K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett.8(5), 649–651 (1996).
[CrossRef]

L. Prkna, M. Hubalek, and J. Ctyroky, “Vectorial eigenmode solver for bent waveguides based on mode matching,” IEEE Photon. Technol. Lett.16(9), 2057–2059 (2004).
[CrossRef]

J. Xiao, H. Ma, N. Bai, X. Liu, and H. Sun, “Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system,” IEEE Photon. Technol. Lett.21(23), 1779–1781 (2009).
[CrossRef]

C. van Dam, L. H. Spiekman, F. P. G. M. van Ham, G. H. Groen, J. J. G. M. van der Tol, I. Moerman, W. W. Pascher, M. Hamacher, H. Heidrich, C. M. Weinert, and M. K. Smit, “Novel compact polarization converters based on ultra short bends,” IEEE Photon. Technol. Lett.8(10), 1346–1348 (1996).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114(2), 185–200 (1994).
[CrossRef]

J. Lightwave Technol. (9)

R. Pregla, “The method of lines for the analysis of dielectric waveguide bends,” J. Lightwave Technol.14(4), 634–639 (1996).
[CrossRef]

M. Rivera, “A finite difference BPM analysis of bent dielectric waveguides,” J. Lightwave Technol.13(2), 233–238 (1995).
[CrossRef]

S. Kim and A. Gopinath, “Vector analysis of optical dielectric waveguide bends using finite-difference method,” J. Lightwave Technol.14(9), 2085–2092 (1996).
[CrossRef]

W. W. Liu, C. L. Xu, T. Hirono, K. Yokoyama, and W. P. Huang, “Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximation,” J. Lightwave Technol.16(5), 910–914 (1998).
[CrossRef]

N. Somasiri, B. M. A. Rahman, and S. S. A. Obayya, “Fabrication tolerance study of a compact passive polarization rotator,” J. Lightwave Technol.20(4), 751–757 (2002).
[CrossRef]

N. N. Feng, G. R. Zhou, C. L. Xu, and W. P. Huang, “Computation of full-vector modes for bending waveguides using cylindrical perfectly matched layers,” J. Lightwave Technol.20(11), 1976–1980 (2002).
[CrossRef]

P. Lusse, P. Stuwe, J. Schule, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol.12(3), 487–494 (1994).
[CrossRef]

D. F. Welch, A. K. Kish, R. Nagarajan, C. H. Joyner, R. P. Schneider, V. G. Dominic, M. L. Mitchell, S. G. Grubb, T. K. Chiang, D. D. Perkins, and A. C. Nilsson, “The realization of large-scale photonic integrated circuits and the associated impact on fiber-optic communication systems,” J. Lightwave Technol.24(12), 4674–4683 (2006).

M. Krause, “Finite-difference mode solver for curved waveguides with angled and curved dielectric interfaces,” J. Lightwave Technol.29(5), 691–699 (2011).
[CrossRef]

Nat. Photonics (1)

P. J. Harshman and T. K. M. Paniccia, “Integrating silicon photonics,” Nat. Photonics4(8), 498–499 (2010).
[CrossRef]

Opt. Commun. (1)

J. Xiao, H. Ma, N. Bai, X. Liu, and X. Sun, “Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems,” Opt. Commun.282(13), 2511–2515 (2009).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

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Figures (9)

Fig. 1
Fig. 1

Cross section of a typical bending waveguide, where local cylindrical coordinate system is used and the radius of curvature R is defines as the radius of the center of the rib.

Fig. 2
Fig. 2

Point P and its neighboring points and sub-regions.

Fig. 3
Fig. 3

Horizontal (a) and vertical (b) dielectric interfaces.

Fig. 4
Fig. 4

Effective indexes of the fundamental modes for a typical bending rib waveguide as a function of the mesh grid size by different FD method: real (a) and imaginary (b) parts.

Fig. 5
Fig. 5

Effective indexes of the fundamental modes for a typical bending rib waveguide as a function of the bending radius: real (a) and imaginary (b) parts.

Fig. 6
Fig. 6

Distributions of field intensity of the real parts of the fundamental modes for a typical bending waveguide: (a) Hx and (b) Hy for TE-like modes; (c) Hx and (d) Hy for TM-like modes.

Fig. 7
Fig. 7

Effective indexes of the fundamental modes for a silicon wire bend as a function of the bending radius: real (a) and imaginary (b) parts.

Fig. 8
Fig. 8

Effective indexes of the fundamental modes for a silicon wire bend as a function of the operating wavelength: real (a) and imaginary (b) parts.

Fig. 9
Fig. 9

Distributions of field intensity of the real parts of the fundamental modes for a silicon wire bend: (a) Hx and (b) Hy for TE-like modes; (c) Hx and (d) Hy for TM-like modes.

Tables (1)

Tables Icon

Table 1 Effective Indexes of the Fundamental TE-like Modes for a Silicon Wire Bend Computed by Different Methods

Equations (53)

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ς ˜ = 0 ς s ς ( ς )d ς ,
s ς (ς)={ 1, in the non-PML region 1 1 50πΔςω ε 0 | ς ς 0 | 2 d 2 , in the PML region ,
˜ 2 H+( ˜ logε)× ˜ ×H+ k 0 2 εH=0,
˜ r ˜ 0 r ˜ + θ ˜ 0 1 r ˜ θ ˜ + z ˜ 0 z ˜ ,
˜ 2 1 r ˜ r ˜ ( r ˜ r ˜ )+ 1 r ˜ 2 2 θ ˜ 2 + 2 z ˜ 2 ,
[ P xx P xy P yx P yy ][ H x H y ]= β 2 [ H x H y ],
P xx H x = x ˜ [ t ˜ x ( t ˜ x H x ) x ˜ ]+ t ˜ x 2 n 2 y ˜ ( 1 n 2 H x y ˜ )+ t ˜ x 2 k 0 2 n 2 H x ,
P xy H y = x ˜ ( t ˜ x 2 H y y ˜ ) t ˜ x 2 n 2 y ˜ ( 1 n 2 H y x ˜ ),
P yy H y = t ˜ x n 2 x ˜ ( t ˜ x n 2 H y x ˜ )+ t ˜ x 2 2 H y y ˜ 2 + t ˜ x 2 k 0 2 n 2 H y ,
P yx H x = t ˜ x 2 2 H x x ˜ y ˜ t ˜ x 2 n 2 x ˜ ( 1 n 2 H x y ˜ ),
r ˜ 2 2 H x x ˜ 2 +3 r ˜ H x x ˜ + r ˜ 2 2 H x y ˜ 2 + H x + r ˜ 2 k 0 2 ε H x +2 r ˜ H y y ˜ = R 2 β 2 H x ,
r ˜ 2 2 H y x ˜ 2 + r ˜ H y x ˜ + r ˜ 2 2 H y y ˜ 2 + r ˜ 2 k 0 2 ε H y = R 2 β 2 H y ,
r ˜ P 2 ( 2 H x W w 2 2 H x P w 2 + 2 w H x x ˜ | w )+3 r ˜ P H x x | w + r ˜ P 2 ( 2 H x N n 2 2 H x P n 2 2 n H x y ˜ | n )+ H x P + r ˜ P 2 k 0 2 ε 1 H x P + 2 r ˜ P H y y | n = R 2 β 2 H x P ,
r ˜ P 2 ( 2 H x W w 2 2 H x P w 2 + 2 w H x x ˜ | w )+3 r ˜ P H x x | w + r ˜ P 2 ( 2 H x S s 2 2 H x P s 2 + 2 s H x y ˜ | s )+ H x P + r ˜ P 2 k 0 2 ε 2 H x P + 2 r ˜ P H y y | s = R 2 β 2 H x P ,
r ˜ P 2 ( 2 H x E e 2 2 H x P e 2 2 e H x x ˜ | e )+3 r ˜ P H x x | e + r ˜ P 2 ( 2 H x S s 2 2 H x P s 2 + 2 s H x y ˜ | s )+ H x P + r ˜ P 2 k 0 2 ε 3 H x P + 2 r ˜ P H y y | s = R 2 β 2 H x P ,
r ˜ P 2 ( 2 H x E e 2 2 H x P e 2 2 e H x x ˜ | e )+3 r ˜ P H x x | e + r ˜ P 2 ( 2 H x S n 2 2 H x P n 2 2 n H x y ˜ | n )+ H x P + r ˜ P 2 k 0 2 ε 4 H x P + 2 r ˜ P H y y | n = R 2 β 2 H x P ,
r ˜ P 2 ( 2 H y W w 2 2 H y P w 2 + 2 w H y x ˜ | w )+ r ˜ P H y x ˜ | w + r ˜ P 2 ( 2 H y N n 2 2 H y P n 2 2 n H y y ˜ | n )+ r ˜ P 2 k 0 2 ε 1 H y P = R 2 β 2 H y P ,
r ˜ P 2 ( 2 H y W w 2 2 H y P w 2 + 2 w H y x ˜ | w )+ r ˜ P H y x ˜ | w + r ˜ P 2 ( 2 H y S s 2 2 H y P s 2 + 2 s H y y ˜ | s )+ r ˜ P 2 k 0 2 ε 2 H y P = R 2 β 2 H y P ,
r ˜ P 2 ( 2 H y E e 2 2 H y P e 2 2 e H y x ˜ | e )+ r P H y x ˜ | e + r ˜ P 2 ( 2 H y S s 2 2 H y P s 2 + 2 s H y y ˜ | s )+ r ˜ P 2 k 0 2 ε 3 H y P = R 2 β 2 H y P ,
r ˜ P 2 ( 2 H y E e 2 2 H y P e 2 2 e H y x ˜ | e )+ r ˜ P H y x | e + r P 2 ( 2 H y N n 2 2 H y P n 2 2 n H y y ˜ | n )+ r ˜ P 2 k 0 2 ε 4 H y P = R 2 β 2 H y P ,
ε n H x y ˜ | s ε s H x y ˜ | n =( ε n ε s ) H y x ˜ ,
H y y ˜ | n = H y y ˜ | s ,
ε e H y x ˜ | w ε w H y x ˜ | e =( ε e ε w ) H x y ˜ ,
H x x ˜ | e = H x x ˜ | w ,
a xx N H x N + a xx S H x S + a xx E H x E + a xx W H x W + a xx P H x P + a xy N H y N + a xy S H y S + a xy E H y E + a xy W H y W + a xy P H y P = β 2 H x P ,
a yx N H x N + a yx S H x S + a yx P H x P + a yy N H y N + a yy S H y S + a yy E H y E + a yy W H y W + a yy P H y P = β 2 H y P ,
[ [ A xx ] [ A xy ] [ A yx ] [ A yy ] ][ { H x } { H y } ]= β 2 [ { H x } { H y } ],
a xx N = r ˜ P 2 R 2 2 n( e+w ) ( w ε 2 n ε 2 +s ε 1 + e ε 3 s ε 4 +n ε 3 ),
a xx S = r ˜ P 2 R 2 2 s( e+w ) ( w ε 1 n ε 2 +s ε 1 + e ε 4 s ε 4 +n ε 3 ),
a xx E = r ˜ P 2 R 2 1 e+w ( 2 e + 3 r ˜ P ),
a xx W = r ˜ P 2 R 2 1 e+w ( 2 w 3 r ˜ P ),
a xx P = r ˜ P 2 R 2 [ e( c 1 + c 2 ) ( e+w )( s ε 4 +n ε 3 ) + w( c 3 + c 4 ) ( e+w )( n ε 2 +s ε 1 ) ],
c 1 =s ε 4 ( k 0 2 ε 3 2 e 2 2 s 2 + 1 r ˜ P 2 3 e r ˜ P ),
c 2 =n ε 3 ( k 0 2 ε 4 2 e 2 2 n 2 + 1 r ˜ P 2 3 e r ˜ P ),
c 3 =n ε 2 ( k 0 2 ε 1 2 w 2 2 n 2 + 1 r ˜ p 2 + 3 w r ˜ P ),
c 4 =s ε 1 ( k 0 2 ε 2 2 w 2 2 s 2 + 1 r ˜ P 2 + 3 w r ˜ P ),
a xy N = r ˜ P R 2 2 e+w ( w ε 2 n ε 2 +s ε 1 + e ε 3 s ε 4 +n ε 3 ),
a xy S = r ˜ P R 2 2 e+w ( w ε 1 n ε 2 +s ε 1 + e ε 4 s ε 4 +n ε 3 ),
a xy E = r ˜ P 2 R 2 2w e ( e+w ) 2 [ w( ε 1 ε 2 ) n ε 2 +s ε 1 + e( ε 4 ε 3 ) s ε 4 +n ε 3 ],
a xy W = r ˜ P 2 R 2 2e w ( e+w ) 2 [ w( ε 1 ε 2 ) n ε 2 +s ε 1 + e( ε 4 ε 3 ) s ε 4 +n ε 3 ],
a xy P = r ˜ P 2 R 2 2 e+w ( ew ew + 1 r P )[ w( ε 1 ε 2 ) n ε 2 +s ε 1 + e( ε 4 ε 3 ) s ε 4 +n ε 3 ],
a yx N = r ˜ P 2 R 2 2s n ( n+s ) 2 [ s( ε 3 ε 2 ) w ε 3 +e ε 2 + n( ε 4 ε 1 ) w ε 4 +e ε 1 ],
a yx S = r ˜ P 2 R 2 2n s ( n+s ) 2 [ s( ε 3 ε 2 ) w ε 3 +e ε 2 + n( ε 4 ε 1 ) w ε 4 +e ε 1 ],
a yx P = r ˜ P 2 R 2 2( ns ) ns( n+s ) [ s( ε 3 ε 2 ) w ε 3 +e ε 2 + n( ε 4 ε 1 ) w ε 4 +e ε 1 ],
a yy N = r ˜ P 2 R 2 2 n( n+s ) ,
a yy S = r ˜ P 2 R 2 2 s( n+s ) ,
a yy E = r ˜ P 2 R 2 1 n+s ( 2 e + 1 r ˜ P )( s ε 2 w ε 3 +e ε 2 + n ε 1 w ε 4 +e ε 1 ),
a yy W = r ˜ P 2 R 2 1 n+s ( 2 w 1 r ˜ P )( s ε 3 w ε 3 +e ε 2 + n ε 4 w ε 4 +e ε 1 ),
a yy P = r ˜ P 2 R 2 [ s( d 1 + d 2 ) ( n+s )( w ε 3 +e ε 2 ) + n( d 3 + d 4 ) ( n+s )( w ε 4 +e ε 1 ) ],
d 1 =w ε 3 ( k 0 2 ε 2 2 w 2 2 s 2 + 1 w r ˜ P ),
d 2 =e ε 2 ( k 0 2 ε 3 2 e 2 2 s 2 1 e r ˜ P ),
d 3 =w ε 4 ( k 0 2 ε 1 2 w 2 2 n 2 + 1 w r ˜ P ),
d 4 =e ε 1 ( k 0 2 ε 4 2 e 2 2 n 2 1 e r ˜ P ),

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