Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Commun. Comput. Phys. 1, 1056-1075(2006).

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite-element-based imaginary distance beam propagation solution of complex modes in optical waveguides,” J. Lightwave Technol. 20, 1054-1060(2002).

[CrossRef]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927-933 (2002).

[CrossRef]

J. Yuan and C. W. Shu, “Local discontinuous Galerkin methods for partial differential equations with higher order derivatives,” J. Sci. Comput. 17, 27-47 (2002).

[CrossRef]

J. Yuan and C. W. Shu, “A local discontinuous Galerkin method for KdV type equations,” SIAM J. Numer. Anal. 40, 769-791(2002).

[CrossRef]

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

[CrossRef]

Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618-623 (2000).

[CrossRef]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737-743 (2000).

[CrossRef]

Y. L. Hsueh, M. C. Yang, and H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. 17, 2389-2397 (1999).

[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

J. C. Chen and S. Jungling, “Computation of higher-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, S199-S205 (1994).

[CrossRef]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098-2105 (1994).

[CrossRef]

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

[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).

[CrossRef]

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335-1339 (1990).

[CrossRef]

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

[CrossRef]

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 2^{3}*S* states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).

[CrossRef]

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).

[CrossRef]

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).

[CrossRef]

Y. L. Hsueh, M. C. Yang, and H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. 17, 2389-2397 (1999).

[CrossRef]

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

[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).

[CrossRef]

J. C. Chen and S. Jungling, “Computation of higher-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, S199-S205 (1994).

[CrossRef]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098-2105 (1994).

[CrossRef]

Y. D. Cheng and C. W. Shu, “A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives,” Math. Comput. 77, 699-730 (2008).

[CrossRef]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).

[CrossRef]

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335-1339 (1990).

[CrossRef]

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335-1339 (1990).

[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes in Pascal* (Cambridge University , 1989).

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).

[CrossRef]

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

[CrossRef]

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

[CrossRef]

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

[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).

[CrossRef]

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 2^{3}*S* states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).

[CrossRef]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098-2105 (1994).

[CrossRef]

J. C. Chen and S. Jungling, “Computation of higher-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, S199-S205 (1994).

[CrossRef]

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 2^{3}*S* states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).

[CrossRef]

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Commun. Comput. Phys. 1, 1056-1075(2006).

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

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

[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes in Pascal* (Cambridge University , 1989).

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 2^{3}*S* states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).

[CrossRef]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927-933 (2002).

[CrossRef]

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

[CrossRef]

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).

[CrossRef]

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).

[CrossRef]

R. Shankar, *Principles of Quantum Mechanics*, 2nd ed. (Plenum, 1994), Chap. 21.

Y. D. Cheng and C. W. Shu, “A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives,” Math. Comput. 77, 699-730 (2008).

[CrossRef]

J. Yuan and C. W. Shu, “A local discontinuous Galerkin method for KdV type equations,” SIAM J. Numer. Anal. 40, 769-791(2002).

[CrossRef]

J. Yuan and C. W. Shu, “Local discontinuous Galerkin methods for partial differential equations with higher order derivatives,” J. Sci. Comput. 17, 27-47 (2002).

[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

C. L. Xu, W. P. Huang, M. S. Stern, and 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 waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56-63 (1988).

[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes in Pascal* (Cambridge University , 1989).

H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631-644(1992).

[CrossRef]

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes in Pascal* (Cambridge University , 1989).

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

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

[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).

[CrossRef]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).

[CrossRef]

J. Yuan and C. W. Shu, “Local discontinuous Galerkin methods for partial differential equations with higher order derivatives,” J. Sci. Comput. 17, 27-47 (2002).

[CrossRef]

J. Yuan and C. W. Shu, “A local discontinuous Galerkin method for KdV type equations,” SIAM J. Numer. Anal. 40, 769-791(2002).

[CrossRef]

A. Goldberg, H. M. Schey, and J. L. Schwartz, “Computer-generated motion pictures of one-dimensional quantum-mechanical transmission and reflection phenomena,” Am. J. Phys. 35, 177-186 (1967).

[CrossRef]

Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Commun. Comput. Phys. 1, 1056-1075(2006).

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

[CrossRef]

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

[CrossRef]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. 38, 927-933 (2002).

[CrossRef]

S. Jungling and J. C. Chen, “A study and optimization of eigenmode calculations using the imaginary-distance beam-propagation method,” IEEE J. Quantum Electron. 30, 2098-2105 (1994).

[CrossRef]

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335-1339 (1990).

[CrossRef]

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

[CrossRef]

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 2^{3}*S* states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).

[CrossRef]

C. L. Xu, W. P. Huang, and S. K. Chaudhuri, “Efficient and accurate vector mode calculations by beam propagation method,” J. Lightwave Technol. 11, 1209-1215 (1993).

[CrossRef]

H. Shu and M. Bass, “Analysis and optimization of the numerical calculation in the slowly decaying imaginary distance beam propagation method,” J. Lightwave Technol. 26, 3199-3206 (2008).

[CrossRef]

Y. Tsuji and M. Koshiba, “Guided-mode and leaky-mode analysis by imaginary distance beam propagation method based on finite element scheme,” J. Lightwave Technol. 18, 618-623 (2000).

[CrossRef]

M. Koshiba and Y. Tsuji, “Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. 18, 737-743 (2000).

[CrossRef]

Y. L. Hsueh, M. C. Yang, and H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. 17, 2389-2397 (1999).

[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, K. T. V. Grattan, and H. A. El-Mikati, “Full vectorial finite-element-based imaginary distance beam propagation solution of complex modes in optical waveguides,” J. Lightwave Technol. 20, 1054-1060(2002).

[CrossRef]

P. Chamorro-Posada, “A modified imaginary distance BPM for directly computing arbitrary vector modes of 3-D optical waveguides,” J. Lightwave Technol. 21, 862-867 (2003).

[CrossRef]

H. Shu and M. Bass, “Calculating the guided modes in optical fibers and waveguides,” J. Lightwave Technol. 25, 2693-2699 (2007).

[CrossRef]

J. Yuan and C. W. Shu, “Local discontinuous Galerkin methods for partial differential equations with higher order derivatives,” J. Sci. Comput. 17, 27-47 (2002).

[CrossRef]

Y. D. Cheng and C. W. Shu, “A discontinuous Galerkin finite element method for time dependent partial differential equations with higher order derivatives,” Math. Comput. 77, 699-730 (2008).

[CrossRef]

J. C. Chen and S. Jungling, “Computation of higher-order waveguide modes by imaginary-distance beam propagation method,” Opt. Quantum Electron. 26, S199-S205 (1994).

[CrossRef]

M. M. Spuhler, D. Wiesmann, P. Freuler, and M. Diergardt, “Direct computation of higher-order propagation modes using the imaginary-distance beam propagation method,” Opt. Quantum Electron. 31, 751-761 (1999).

[CrossRef]

D. Blume, M. Lewerenz, P. Niyaz, and K. B. Whaley, “Excited states by quantum Monte Carlo methods: imaginary time evolution with projection operators,” Phys. Rev. E 55, 3664-3375 (1997).

[CrossRef]

K. E. Schmidt, P. Niyaz, A. Vaught, and M. A. Lee, “Green's function Monte Carlo method with exact imaginary-time propagation,” Phys. Rev. E 71, 016707 (2005).

[CrossRef]

P. J. Chiang, C. P. Yu, and H. C. Chang, “Analysis of two-dimensional photonic crystals using a multidomain pseudospectral method,” Phys. Rev. E 75, 026703 (2007).

[CrossRef]

J. Yuan and C. W. Shu, “A local discontinuous Galerkin method for KdV type equations,” SIAM J. Numer. Anal. 40, 769-791(2002).

[CrossRef]

H. A. Van Der Vorst, “BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput. 13, 631-644(1992).

[CrossRef]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, *Numerical Recipes in Pascal* (Cambridge University , 1989).

R. Shankar, *Principles of Quantum Mechanics*, 2nd ed. (Plenum, 1994), Chap. 21.