Abstract

A finite-difference scheme for accurately analyzing step-index dielectric waveguides with angled and curved dielectric interfaces has been proposed by Chiang Here, this formulation is generalized to take into account a possible curvature of the entire waveguide along the propagation direction, and it is shown how Perfectly Matched Layers can be incorporated in order to accurately compute the bending or curvature loss of dielectric waveguides with step-index profiles of arbitrary shape. Furthermore, the formulation of the interface equations that relate the fields and their derivatives on both sides of a dielectric interface has been simplified, so that their implementation no longer requires any further explicit conversions between coordinate systems.

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  1. J. Saijonmaa, D. Yevick, "Beam-propagation analysis of loss in bent optical waveguides and fibers," J. Opt. Soc. Amer. 73, 1785-1791 (1983).
  2. J.-S. Gu, P.-A. Besse, H. Melchior, "Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides," IEEE J. Quantum Electron. 27, 531-537 (1991).
  3. M. Rivera, "Accurate full vector beam propagation method for bent channel waveguides," Integr. Photon. Res. OSA Tech. Dig.6, (1996) paper IMB3.
  4. R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightw. Technol. 14, 634-639 (1996).
  5. S. Kim, A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightw. Technol. 14, 2085-2092 (1996).
  6. W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightw. Technol. 16, 910-914 (1998).
  7. P. D. Sewell, T. M. Benson, "Efficient curvature analysis of buried waveguides," J. Lightw. Technol. 18, 1321-1329 (2000).
  8. G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces," J. Lightw. Technol. 20, 1210-1218 (2002).
  9. N.-N. Feng, G.-R. Zhou, C. Xu, W.-P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightw. Technol. 20, 1976-1980 (2002).
  10. Y. Tsuji, M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).
  11. L. Prkna, M. Hubálek, J. Čtyroký, "Field modeling of circular microresonators by film mode matching," IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).
  12. K. Kakihara, N. Kono, K. Saitoh, M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Exp. 14, 11 128-11 141 (2006).
  13. R. Jedidi, R. Pierre, "High-order finite-element methods for the computation of bending loss in opticalwaveguides," J. Lightw. Technol. 25, 2618-2630 (2007).
  14. M. Oxborrow, "Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).
  15. J. Xiao, H. Ni, X. Sun, "Full-vector mode solver for bending waveguides based on the finite-difference frequency domain method in cylindrical coordinate systems," Opt. Lett. 33, 1848-1850 (2008).
  16. J. Xiao, H. Ma, N. Bai, X. Liu, X. Sun, "Full-vectorial mode solver for bending waveguides using multidomain pseudospectral method in a cylindrical coordinate system," IEEE Photon. Technol. Lett. 21, 1779-1781 (2009).
  17. S. Zhao, "High order vectorial analysis of waveguides with curved dielectric interfaces," IEEE Microw. Wireless Compon. Lett. 19, 266-268 (2009).
  18. J. Xiao, H. Ma, N. Bai, X. Liu, X. Sun, "Full-vectorial analysis of bending waveguides using finite difference method based on H-fields in cylindrical coordinate systems," Opt. Commun. 282, 2511-2515 (2009).
  19. J. Xiao, X. Sun, "Full-vector analysis of optical dielectric waveguide bends using improved finite difference method based on E fields in cylindrical coordinate systems," J. Opt. 12, 055404 (2010).
  20. V. P. Tzolov, M. Fontaine, N. Godbout, S. Lacroix, "Nonlinear self-phase-modulation effects: A vectorial first-order perturbation approach," Opt. Lett. 20, 456-458 (1995).
  21. M. Krause, H. Renner, E. Brinkmeyer, "Polarization-dependent curvature loss in silicon rib waveguides," IEEE J. Sel. Top. Quantum Electron. 12, II1359-1362 (2006).
  22. T. G. Nguyen, R. S. Tummidi, T. L. Koch, A. Mitchell, "Lateral leakage of TM-like mode in thin-ridge silicon-on-insulator bent waveguides and ring resonators," Opt. Exp. 18, 7243-7252 (2010).
  23. M. Heiblum, J. H. Harris, "Analysis of curved optical waveguides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).
  24. D. M. Shyroki, "Exact equivalent straight waveguide model for bent and twisted waveguides," IEEE Trans. Microw. Theory Tech. 56, 414-419 (2008).
  25. A. Säynätjoki, B. Bai, A. Tervonen, J. Turunen, S. Honkanen, "Enhanced vertical confinement in angled-wall slot waveguides," Opt. Rev. 17, 181-186 (2010).
  26. E. J. Teo, A. A. Bettiol, B. Xiong, M. B. H. Breese, P. T. Shuvan, "An all-silicon, single-mode Bragg cladding rib waveguide," Opt. Exp. 18, 8816-8823 (2010).
  27. J. Cardenas, C. B. Poitras, J. T. Robinson, K. Preston, L. Chen, M. Lipson, "Low loss etchless silicon photonic waveguides," Opt. Exp. 17, 4752-4757 (2009).
  28. O. Horn, J. Amthor, T. Lipka, J. Müller, "Tunable silicon photonic wires fabricated by contact lithography and thermal oxidation," Proc. SPIE Photonic Materials, Devices, and Applications III (2009) pp. 73660C.
  29. M. Krause, H. Renner, A. Harke, J. Müller, E. Brinkmeyer, "Leakage loss in trench-bulge waveguides," J. Lightw. Technol. 23, 1890-1895 (2005).
  30. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).
  31. Y.-C. Chiang, Y.-P. Chiou, H.-C. Chang, "Improved full-vectorial finite-difference mode solver for optical waveguides with step-index profiles," J. Lightw. Technol. 20, 1609-1618 (2002).
  32. 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).
  33. R. B. Lehoucq, D. C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (1998) Society for Industrial & Applied Mathematics.
  34. J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, "A supernodal approach to sparse partial pivoting," SIAM J. Matrix Anal. Applic. 20, 720-755 (1999).
  35. C. Vassallo, "Improvement of finite difference methods for step-index optical waveguides," Proc. Inst. Elect. Eng.—J 139, 137-142 (1992).
  36. W. C. Chew, J. M. Jin, E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).
  37. F. L. Teixeira, W. C. Chew, "Systematic derivation of anisotropric PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guided Wave Lett. 7, 371-373 (1997).
  38. Z. S. Sacks, D. M. Kingsland, R. Lee, J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).
  39. S. D. Gedney, A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  40. A. F. Oskooi, L. Zhang, Y. Avniel, S. G. Johnson, "The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers," Opt. Exp. 16, 11 376-11 392 (2008).
  41. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

2010 (5)

T. G. Nguyen, R. S. Tummidi, T. L. Koch, A. Mitchell, "Lateral leakage of TM-like mode in thin-ridge silicon-on-insulator bent waveguides and ring resonators," Opt. Exp. 18, 7243-7252 (2010).

A. Säynätjoki, B. Bai, A. Tervonen, J. Turunen, S. Honkanen, "Enhanced vertical confinement in angled-wall slot waveguides," Opt. Rev. 17, 181-186 (2010).

E. J. Teo, A. A. Bettiol, B. Xiong, M. B. H. Breese, P. T. Shuvan, "An all-silicon, single-mode Bragg cladding rib waveguide," Opt. Exp. 18, 8816-8823 (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-vector analysis of optical dielectric waveguide bends using improved finite difference method based on E fields in cylindrical coordinate systems," J. Opt. 12, 055404 (2010).

2009 (4)

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

S. Zhao, "High order vectorial analysis of waveguides with curved dielectric interfaces," IEEE Microw. Wireless Compon. Lett. 19, 266-268 (2009).

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

J. Cardenas, C. B. Poitras, J. T. Robinson, K. Preston, L. Chen, M. Lipson, "Low loss etchless silicon photonic waveguides," Opt. Exp. 17, 4752-4757 (2009).

2008 (3)

D. M. Shyroki, "Exact equivalent straight waveguide model for bent and twisted waveguides," IEEE Trans. Microw. Theory Tech. 56, 414-419 (2008).

A. F. Oskooi, L. Zhang, Y. Avniel, S. G. Johnson, "The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers," Opt. Exp. 16, 11 376-11 392 (2008).

J. Xiao, H. Ni, X. Sun, "Full-vector mode solver for bending waveguides based on the finite-difference frequency domain method in cylindrical coordinate systems," Opt. Lett. 33, 1848-1850 (2008).

2007 (2)

R. Jedidi, R. Pierre, "High-order finite-element methods for the computation of bending loss in opticalwaveguides," J. Lightw. Technol. 25, 2618-2630 (2007).

M. Oxborrow, "Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).

2006 (3)

K. Kakihara, N. Kono, K. Saitoh, M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Exp. 14, 11 128-11 141 (2006).

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).

M. Krause, H. Renner, E. Brinkmeyer, "Polarization-dependent curvature loss in silicon rib waveguides," IEEE J. Sel. Top. Quantum Electron. 12, II1359-1362 (2006).

2005 (2)

M. Krause, H. Renner, A. Harke, J. Müller, E. Brinkmeyer, "Leakage loss in trench-bulge waveguides," J. Lightw. Technol. 23, 1890-1895 (2005).

L. Prkna, M. Hubálek, J. Čtyroký, "Field modeling of circular microresonators by film mode matching," IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).

2004 (1)

Y. Tsuji, M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).

2002 (3)

G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces," J. Lightw. Technol. 20, 1210-1218 (2002).

N.-N. Feng, G.-R. Zhou, C. Xu, W.-P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightw. Technol. 20, 1976-1980 (2002).

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

2000 (1)

P. D. Sewell, T. M. Benson, "Efficient curvature analysis of buried waveguides," J. Lightw. Technol. 18, 1321-1329 (2000).

1999 (1)

J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, "A supernodal approach to sparse partial pivoting," SIAM J. Matrix Anal. Applic. 20, 720-755 (1999).

1998 (1)

W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightw. Technol. 16, 910-914 (1998).

1997 (2)

W. C. Chew, J. M. Jin, E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).

F. L. Teixeira, W. C. Chew, "Systematic derivation of anisotropric PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guided Wave Lett. 7, 371-373 (1997).

1996 (3)

M. Rivera, "Accurate full vector beam propagation method for bent channel waveguides," Integr. Photon. Res. OSA Tech. Dig.6, (1996) paper IMB3.

R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightw. Technol. 14, 634-639 (1996).

S. Kim, A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightw. Technol. 14, 2085-2092 (1996).

1995 (2)

V. P. Tzolov, M. Fontaine, N. Godbout, S. Lacroix, "Nonlinear self-phase-modulation effects: A vectorial first-order perturbation approach," Opt. Lett. 20, 456-458 (1995).

Z. S. Sacks, D. M. Kingsland, R. Lee, J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).

1992 (1)

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

1991 (1)

J.-S. Gu, P.-A. Besse, H. Melchior, "Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides," IEEE J. Quantum Electron. 27, 531-537 (1991).

1983 (1)

J. Saijonmaa, D. Yevick, "Beam-propagation analysis of loss in bent optical waveguides and fibers," J. Opt. Soc. Amer. 73, 1785-1791 (1983).

1975 (1)

M. Heiblum, J. H. Harris, "Analysis of curved optical waveguides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).

Electromagnetics (1)

Y. Tsuji, M. Koshiba, "Complex modal analysis of curved optical waveguides using a full-vectorial finite element method with perfectly matched layer boundary conditions," Electromagnetics 24, 39-48 (2004).

IEEE J. Quantum Electron. (1)

M. Heiblum, J. H. Harris, "Analysis of curved optical waveguides by conformal transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).

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

L. Prkna, M. Hubálek, J. Čtyroký, "Field modeling of circular microresonators by film mode matching," IEEE J. Sel. Top. Quantum Electron. 11, 217-223 (2005).

IEEE Microw. Wireless Compon. Lett. (1)

S. Zhao, "High order vectorial analysis of waveguides with curved dielectric interfaces," IEEE Microw. Wireless Compon. Lett. 19, 266-268 (2009).

IEEE Trans. Antennas Propag. (1)

Z. S. Sacks, D. M. Kingsland, R. Lee, J.-F. Lee, "A perfectly matched anisotropic absorber for use as an absorbing boundary condition," IEEE Trans. Antennas Propag. 43, 1460-1463 (1995).

IEEE Trans. Microw. Theory Tech. (1)

D. M. Shyroki, "Exact equivalent straight waveguide model for bent and twisted waveguides," IEEE Trans. Microw. Theory Tech. 56, 414-419 (2008).

IEEE J. Quantum Electron. (1)

J.-S. Gu, P.-A. Besse, H. Melchior, "Method of lines for the analysis of the propagation characteristics of curved optical rib waveguides," IEEE J. Quantum Electron. 27, 531-537 (1991).

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

M. Krause, H. Renner, E. Brinkmeyer, "Polarization-dependent curvature loss in silicon rib waveguides," IEEE J. Sel. Top. Quantum Electron. 12, II1359-1362 (2006).

IEEE Microw. Guided Wave Lett. (1)

F. L. Teixeira, W. C. Chew, "Systematic derivation of anisotropric PML absorbing media in cylindrical and spherical coordinates," IEEE Microw. Guided Wave Lett. 7, 371-373 (1997).

IEEE Photon. Technol. Lett. (1)

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

IEEE Trans. Microw. Theory Tech. (1)

M. Oxborrow, "Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators," IEEE Trans. Microw. Theory Tech. 55, 1209-1218 (2007).

Integr. Photon. Res. (1)

M. Rivera, "Accurate full vector beam propagation method for bent channel waveguides," Integr. Photon. Res. OSA Tech. Dig.6, (1996) paper IMB3.

J. Lightw. Technol. (2)

R. Pregla, "The method of lines for the analysis of dielectric waveguide bends," J. Lightw. Technol. 14, 634-639 (1996).

N.-N. Feng, G.-R. Zhou, C. Xu, W.-P. Huang, "Computation of full-vector modes for bending waveguide using cylindrical perfectly matched layers," J. Lightw. Technol. 20, 1976-1980 (2002).

J. Opt. Soc. Amer. (1)

J. Saijonmaa, D. Yevick, "Beam-propagation analysis of loss in bent optical waveguides and fibers," J. Opt. Soc. Amer. 73, 1785-1791 (1983).

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. (7)

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

M. Krause, H. Renner, A. Harke, J. Müller, E. Brinkmeyer, "Leakage loss in trench-bulge waveguides," J. Lightw. Technol. 23, 1890-1895 (2005).

S. Kim, A. Gopinath, "Vector analysis of optical dielectric waveguide bends using finite-difference method," J. Lightw. Technol. 14, 2085-2092 (1996).

W. W. Lui, C.-L. Xu, T. Hirono, K. Yokoyama, W.-P. Huang, "Full-vectorial wave propagation in semiconductor optical bending waveguides and equivalent straight waveguide approximations," J. Lightw. Technol. 16, 910-914 (1998).

P. D. Sewell, T. M. Benson, "Efficient curvature analysis of buried waveguides," J. Lightw. Technol. 18, 1321-1329 (2000).

G. R. Hadley, "High-accuracy finite-difference equations for dielectric waveguide analysis I: Uniform regions and dielectric interfaces," J. Lightw. Technol. 20, 1210-1218 (2002).

R. Jedidi, R. Pierre, "High-order finite-element methods for the computation of bending loss in opticalwaveguides," J. Lightw. Technol. 25, 2618-2630 (2007).

J. Opt. (1)

J. Xiao, X. Sun, "Full-vector analysis of optical dielectric waveguide bends using improved finite difference method based on E fields in cylindrical coordinate systems," J. Opt. 12, 055404 (2010).

Microw. Opt. Technol. Lett. (1)

W. C. Chew, J. M. Jin, E. Michielssen, "Complex coordinate stretching as a generalized absorbing boundary condition," Microw. Opt. Technol. Lett. 15, 363-369 (1997).

Opt. Exp. (1)

A. F. Oskooi, L. Zhang, Y. Avniel, S. G. Johnson, "The failure of perfectly matched layers, and towards their redemption by adiabatic absorbers," Opt. Exp. 16, 11 376-11 392 (2008).

Opt. Lett. (1)

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, "Improving accuracy by subpixel smoothing in the finite-difference time domain," Opt. Lett. 31, 2972-2974 (2006).

Opt. Commun. (1)

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

Opt. Exp. (4)

K. Kakihara, N. Kono, K. Saitoh, M. Koshiba, "Full-vectorial finite element method in a cylindrical coordinate system for loss analysis of photonic wire bends," Opt. Exp. 14, 11 128-11 141 (2006).

E. J. Teo, A. A. Bettiol, B. Xiong, M. B. H. Breese, P. T. Shuvan, "An all-silicon, single-mode Bragg cladding rib waveguide," Opt. Exp. 18, 8816-8823 (2010).

J. Cardenas, C. B. Poitras, J. T. Robinson, K. Preston, L. Chen, M. Lipson, "Low loss etchless silicon photonic waveguides," Opt. Exp. 17, 4752-4757 (2009).

T. G. Nguyen, R. S. Tummidi, T. L. Koch, A. Mitchell, "Lateral leakage of TM-like mode in thin-ridge silicon-on-insulator bent waveguides and ring resonators," Opt. Exp. 18, 7243-7252 (2010).

Opt. Lett. (2)

Opt. Rev. (1)

A. Säynätjoki, B. Bai, A. Tervonen, J. Turunen, S. Honkanen, "Enhanced vertical confinement in angled-wall slot waveguides," Opt. Rev. 17, 181-186 (2010).

Proc. Inst. Elect. Eng.—J (1)

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

SIAM J. Matrix Anal. Applic. (1)

J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, J. W. H. Liu, "A supernodal approach to sparse partial pivoting," SIAM J. Matrix Anal. Applic. 20, 720-755 (1999).

Other (4)

R. B. Lehoucq, D. C. Sorensen, C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods (1998) Society for Industrial & Applied Mathematics.

S. D. Gedney, A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

O. Horn, J. Amthor, T. Lipka, J. Müller, "Tunable silicon photonic wires fabricated by contact lithography and thermal oxidation," Proc. SPIE Photonic Materials, Devices, and Applications III (2009) pp. 73660C.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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