Abstract

We propose a pseudospectral mode solver for optical waveguide mode analysis formulated by the frequency-domain Maxwell equations. Special attention is paid upon identifying the required boundary operator for the formulation and the relationships between the derived operator and the physical boundary conditions. These theoretical results are adopted into a Legendre pseudospectral multidomain computational framework to compute the propagation characteristics of optical waveguides. Numerical experiments are conducted, and the expected spectral convergence of the scheme is observed for smooth problems and for problems having field jumps at material interfaces. For dielectric waveguides with sharp corners, the spectral convergence is lost due to the singular nature of fields at the corner. Nevertheless, compared with other methods, the present formulation remains as an efficient approach to obtain waveguide modes.

© 2012 IEEE

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  1. K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Tech. 34, 1104-1113 (1986).
  2. M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," IEE Proc. J. Optoelectron. 135, 56-63 (1988).
  3. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," IEE Proc. J. Optoelectron. 139, 137-142 (1992).
  4. P. Lüsse, P. Stuwe, J. Schüle, H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightw. Technol. 12, 487-494 (1994).
  5. G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis: II. Dielectric corners," J. Lightw. Technol. 20, 1219-1231 (2002).
  6. N. Thomas, P. Sewell, T. M. Benson, "A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides," J. Lightw. Technol. 25, 2563-2570 (2007).
  7. Y. P. Chiou, Y. P. Chiang, H. C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).
  8. S. Zhao, "Full-vectorial matched interface and boundary (MIB) method for the modal analysis of dielectric waveguides," J. Lightw. Technol. 26, 2251-2259 (2008).
  9. S. Zhao, "High-order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces," J. Comput. Phys. 229, 3155-3170 (2010).
  10. M. Koshiba, K. Inoue, "Simple and efficient finite-element analysis of microwave and optical waveguides," IEEE Trans. Microw. Theory Tech. 40, 371-377 (1992).
  11. J. F. Lee, D. K. Sun, Z. J. Cendes, "Tangential vector finite element for electromagnetic field computation," IEEE Trans. Magn. 27, 4032-4035 (1991).
  12. M. Koshiba, S. Maruyama, K. Hirayama, "A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems," J. Lightw. Technol. 12, 495-502 (1994).
  13. D. U. Li, H. C. Chang, "An efficient full-vectorial finite-element modal analysis of dielectric waveguides incorporating inhomogeneous elements across dielectric discontinuities," IEEE J. Quantum Electron. 36, 1251-1261 (2000).
  14. P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, H.-C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).
  15. C.-C. Huang, C.-C. Huang, J.-Y. Yang, "An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition," J. Lightw. Technol. 21, 2284-2296 (2003).
  16. C.-C. Huang, C.-C. Huang, J.-Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Quantum Electron. 11, 457-465 (2005).
  17. J. Xiao, X. Sun, "Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method," Opt. Commun. 283, 2835-2840 (2010).
  18. J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge Univ. Press, 2007).
  19. D. Funaro, D. Gottlieb, "A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations," Math. Comput. 51, 599-613 (1988).
  20. C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).
  21. B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, Y. L. Wang, "Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation," Opt. Exp. 17, 14211-14228 (2009).
  22. B. Y. Lin, C. H. Teng, H. C. Chang, H. H. Hsiao, J. K. Wang, Y. L. Wang, "Pseudospectral modeling of nano-optics in Ag sphere arrays," J. Sci. Comput. 45, 429-446 (2010).
  23. S. F. Chiang, B. Y. Lin, C. H. Teng, H. C. Chang, "Improved analysis of rectangular dielectric waveguides based on a Legendre pseudospectral penalty scheme," presented at the Integr. Photon. Res., Silicon and Nano Photon. WashingtonDC (2010) Paper IWH8.
  24. W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).
  25. R. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).
  26. W. Lu, Y. Y. Lu, "Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations," J. Comput. Phys. 231, 1360-1371 (2012).

2012 (1)

W. Lu, Y. Y. Lu, "Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations," J. Comput. Phys. 231, 1360-1371 (2012).

2010 (3)

B. Y. Lin, C. H. Teng, H. C. Chang, H. H. Hsiao, J. K. Wang, Y. L. Wang, "Pseudospectral modeling of nano-optics in Ag sphere arrays," J. Sci. Comput. 45, 429-446 (2010).

S. Zhao, "High-order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces," J. Comput. Phys. 229, 3155-3170 (2010).

J. Xiao, X. Sun, "Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method," Opt. Commun. 283, 2835-2840 (2010).

2009 (1)

B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, Y. L. Wang, "Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation," Opt. Exp. 17, 14211-14228 (2009).

2008 (3)

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, H.-C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).

S. Zhao, "Full-vectorial matched interface and boundary (MIB) method for the modal analysis of dielectric waveguides," J. Lightw. Technol. 26, 2251-2259 (2008).

2007 (1)

N. Thomas, P. Sewell, T. M. Benson, "A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides," J. Lightw. Technol. 25, 2563-2570 (2007).

2005 (1)

C.-C. Huang, C.-C. Huang, J.-Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Quantum Electron. 11, 457-465 (2005).

2003 (1)

C.-C. Huang, C.-C. Huang, J.-Y. Yang, "An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition," J. Lightw. Technol. 21, 2284-2296 (2003).

2002 (1)

G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis: II. Dielectric corners," J. Lightw. Technol. 20, 1219-1231 (2002).

2000 (2)

Y. P. Chiou, Y. P. Chiang, H. C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).

D. U. Li, H. C. Chang, "An efficient full-vectorial finite-element modal analysis of dielectric waveguides incorporating inhomogeneous elements across dielectric discontinuities," IEEE J. Quantum Electron. 36, 1251-1261 (2000).

1994 (2)

M. Koshiba, S. Maruyama, K. Hirayama, "A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems," J. Lightw. Technol. 12, 495-502 (1994).

P. Lüsse, P. Stuwe, J. Schüle, H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightw. Technol. 12, 487-494 (1994).

1992 (2)

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," IEE Proc. J. Optoelectron. 139, 137-142 (1992).

M. Koshiba, K. Inoue, "Simple and efficient finite-element analysis of microwave and optical waveguides," IEEE Trans. Microw. Theory Tech. 40, 371-377 (1992).

1991 (1)

J. F. Lee, D. K. Sun, Z. J. Cendes, "Tangential vector finite element for electromagnetic field computation," IEEE Trans. Magn. 27, 4032-4035 (1991).

1988 (2)

D. Funaro, D. Gottlieb, "A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations," Math. Comput. 51, 599-613 (1988).

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," IEE Proc. J. Optoelectron. 135, 56-63 (1988).

1986 (1)

K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Tech. 34, 1104-1113 (1986).

1973 (1)

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

IEE Proc. J. Optoelectron. (1)

C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," IEE Proc. J. Optoelectron. 139, 137-142 (1992).

IEE Proc. J. Optoelectron. (1)

M. S. Stern, "Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles," IEE Proc. J. Optoelectron. 135, 56-63 (1988).

IEEE J. Quantum Electron. (1)

D. U. Li, H. C. Chang, "An efficient full-vectorial finite-element modal analysis of dielectric waveguides incorporating inhomogeneous elements across dielectric discontinuities," IEEE J. Quantum Electron. 36, 1251-1261 (2000).

IEEE Trans. Magn. (1)

J. F. Lee, D. K. Sun, Z. J. Cendes, "Tangential vector finite element for electromagnetic field computation," IEEE Trans. Magn. 27, 4032-4035 (1991).

IEEE J. Quantum Electron. (2)

P.-J. Chiang, C.-L. Wu, C.-H. Teng, C.-S. Yang, H.-C. Chang, "Full-vectorial optical waveguide mode solvers using multidomain pseudospectral frequency-domain (PSFD) formulations," IEEE J. Quantum Electron. 44, 56-66 (2008).

C.-C. Huang, C.-C. Huang, J.-Y. Yang, "A full-vectorial pseudospectral modal analysis of dielectric optical waveguides with stepped refractive index profiles," IEEE J. Quantum Electron. 11, 457-465 (2005).

IEEE Trans. Microw. Theory Tech. (2)

K. Bierwirth, N. Schulz, F. Arndt, "Finite-difference analysis of rectangular dielectric waveguide structures," IEEE Trans. Microw. Theory Tech. 34, 1104-1113 (1986).

M. Koshiba, K. Inoue, "Simple and efficient finite-element analysis of microwave and optical waveguides," IEEE Trans. Microw. Theory Tech. 40, 371-377 (1992).

J. Comput. Phys. (1)

W. Lu, Y. Y. Lu, "Waveguide mode solver based on Neumann-to-Dirichlet operators and boundary integral equations," J. Comput. Phys. 231, 1360-1371 (2012).

J. Lightw. Technol. (2)

P. Lüsse, P. Stuwe, J. Schüle, H.-G. Unger, "Analysis of vectorial mode fields in optical waveguides by a new finite difference method," J. Lightw. Technol. 12, 487-494 (1994).

C.-C. Huang, C.-C. Huang, J.-Y. Yang, "An efficient method for computing optical waveguides with discontinuous refractive index profiles using spectral collocation method with domain decomposition," J. Lightw. Technol. 21, 2284-2296 (2003).

J. Sci. Comput. (1)

C. H. Teng, B. Y. Lin, H. C. Chang, H. C. Hsu, C. N. Lin, K. A. Feng, "A Legendre pseudospectral penalty scheme for solving time-domain Maxwell's equations," J. Sci. Comput. 36, 351-390 (2008).

J. Comput. Phys. (1)

S. Zhao, "High-order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces," J. Comput. Phys. 229, 3155-3170 (2010).

J. Lightw. Technol. (5)

G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis: II. Dielectric corners," J. Lightw. Technol. 20, 1219-1231 (2002).

N. Thomas, P. Sewell, T. M. Benson, "A new full-vectorial higher order finite-difference scheme for the modal analysis of rectangular dielectric waveguides," J. Lightw. Technol. 25, 2563-2570 (2007).

Y. P. Chiou, Y. P. Chiang, H. C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices," J. Lightw. Technol. 18, 243-251 (2000).

S. Zhao, "Full-vectorial matched interface and boundary (MIB) method for the modal analysis of dielectric waveguides," J. Lightw. Technol. 26, 2251-2259 (2008).

M. Koshiba, S. Maruyama, K. Hirayama, "A vector finite element method with the high-order mixed-interpolation-type triangular elements for optical waveguiding problems," J. Lightw. Technol. 12, 495-502 (1994).

J. Sci. Comput. (1)

B. Y. Lin, C. H. Teng, H. C. Chang, H. H. Hsiao, J. K. Wang, Y. L. Wang, "Pseudospectral modeling of nano-optics in Ag sphere arrays," J. Sci. Comput. 45, 429-446 (2010).

Math. Comput. (1)

D. Funaro, D. Gottlieb, "A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations," Math. Comput. 51, 599-613 (1988).

Numer. Math. (1)

W. J. Gordon, C. A. Hall, "Transfinite element methods: Blending-function interpolation over arbitrary curved element domains," Numer. Math. 21, 109-129 (1973).

Opt. Commun. (1)

J. Xiao, X. Sun, "Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method," Opt. Commun. 283, 2835-2840 (2010).

Opt. Exp. (1)

B. Y. Lin, H. C. Hsu, C. H. Teng, H. C. Chang, J. K. Wang, Y. L. Wang, "Unraveling near-field origin of electromagnetic waves scattered from silver nanorod arrays using pseudo-spectral time-domain calculation," Opt. Exp. 17, 14211-14228 (2009).

Other (3)

J. S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge Univ. Press, 2007).

R. Collin, Field Theory of Guided Waves (McGraw-Hill, 1960).

S. F. Chiang, B. Y. Lin, C. H. Teng, H. C. Chang, "Improved analysis of rectangular dielectric waveguides based on a Legendre pseudospectral penalty scheme," presented at the Integr. Photon. Res., Silicon and Nano Photon. WashingtonDC (2010) Paper IWH8.

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