Abstract

The governing equation of the slowly decaying imaginary distance beam propagation method (SD-ID-BPM) is further modified, for calculating the eigenmodes in optical fibers and waveguides. Its convergence is analyzed in detail and compared to the earlier version of SD-ID-BPM and other methods. It is demonstrated that the method described here can converge to the same desired accuracy within fewer propagation steps than the earlier version of SD-ID-BPM and other methods. Since the governing equation of the SD-ID-BPM is a partial differential equation with higher order derivatives, it might be interesting if the discretization in the transverse xy plane is performed by applying the numerical techniques for partial differential equations with higher order derivatives.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. D. Feit and J. A. Fleck Jr., “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154-1164 (1980).
    [CrossRef] [PubMed]
  2. D. Yevick and W. Bardyszewski, “Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis,” Opt. Lett. 17, 329-330 (1992).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. H. Shu and M. Bass, “Calculating the guided modes in optical fibers and waveguides,” J. Lightwave Technol. 25, 2693-2699 (2007).
    [CrossRef]
  10. 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]
  11. R. Shankar, Principles of Quantum Mechanics, 2nd ed. (Plenum, 1994), Chap. 21.
  12. S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
    [CrossRef]
  13. 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]
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. 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]
  19. 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]
  20. 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]
  21. J. Yuan and C. W. Shu, “A local discontinuous Galerkin method for KdV type equations,” SIAM J. Numer. Anal. 40, 769-791(2002).
    [CrossRef]
  22. C. C. Huang, “Simulation of optical waveguides by novel full vectorial pseudospectral-based imaginary distance beam propagation method,” Opt. Express 16, 17915-17934 (2008).
    [CrossRef] [PubMed]
  23. 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]
  24. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335-1339 (1990).
    [CrossRef]
  25. 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]
  26. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Pascal (Cambridge University , 1989).
  27. 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]
  28. M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56-63 (1988).
    [CrossRef]
  29. 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]
  30. 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]
  31. Y. Y. Lu, “Some techniques for computing wave propagation in optical waveguides,” Commun. Comput. Phys. 1, 1056-1075(2006).

2008

2007

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]

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

2006

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

2005

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]

2003

2002

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. 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]

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]

2000

1999

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]

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]

1997

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]

1994

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]

1993

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]

1992

D. Yevick and W. Bardyszewski, “Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis,” Opt. Lett. 17, 329-330 (1992).
[CrossRef] [PubMed]

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]

1990

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

1988

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

1980

1971

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
[CrossRef]

1967

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]

Bardyszewski, W.

Bass, M.

Blume, D.

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]

Chamorro-Posada, P.

Chang, H. C.

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]

Chaudhuri, S. K.

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]

Chen, J. C.

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]

Cheng, Y. D.

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]

Chiang, P. J.

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]

Chung, Y.

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

Dagli, N.

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

Diergardt, M.

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]

El-Mikati, H. A.

Feit, M. D.

Flannery, B. P.

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

Fleck, J. A.

Freuler, P.

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]

Goldberg, A.

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]

Gopinath, A.

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]

Grattan, K. T. V.

Helfert, S.

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]

Hsueh, Y. L.

Huang, C. C.

Huang, W. P.

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]

Jensen, C. A.

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
[CrossRef]

Jungling, S.

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]

Koshiba, M.

Lawande, S. V.

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
[CrossRef]

Lee, M. A.

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]

Lewerenz, M.

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]

Lu, Y. Y.

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

Niyaz, P.

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]

Obayya, S. S. A.

Pregla, R.

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]

Press, W. H.

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

Rahman, B. M. A.

Sahlin, H. L.

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
[CrossRef]

Saitoh, K.

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]

Scarmozzino, R.

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]

Schey, H. M.

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]

Schmidt, K. E.

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]

Schwartz, J. L.

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]

Shankar, R.

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

Shu, C. W.

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]

Shu, H.

Spuhler, M. M.

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]

Stern, M. S.

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]

Teukolsky, S. A.

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

Tsuji, Y.

Van Der Vorst, H. A.

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]

Vaught, A.

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]

Vetterling, W. T.

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

Whaley, K. B.

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]

Wiesmann, D.

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]

Xu, C. L.

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]

Yang, M. C.

Yevick, D.

Yu, C. P.

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]

Yuan, J.

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]

Am. J. Phys.

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]

Appl. Opt.

Commun. Comput. Phys.

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

IEE Proc. J. Optoelectron.

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

IEE Proc. Optoelectron.

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]

IEEE J. Quantum Electron.

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]

IEEE J. Sel. Top. Quantum Electron.

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]

J. Chem. Phys.

S. V. Lawande, C. A. Jensen, and H. L. Sahlin, “He and H−11S and 23S states computed from Feynman path integrals in imaginary time,” J. Chem. Phys. 54, 445-452 (1971).
[CrossRef]

J. Lightwave Technol.

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. 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]

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]

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. 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]

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. 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]

J. Sci. Comput.

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]

Math. Comput.

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]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

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]

Phys. Rev. E

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]

SIAM J. Numer. Anal.

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

SIAM J. Sci. Stat. Comput.

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]

Other

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.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Plot of the modulus of the amplification factor as a function of b = ( Δ z / δ ) ( λ m α ) 4 .

Fig. 2
Fig. 2

Plot of the number of propagation steps needed to converge to a specific accuracy as a function of the parameter δ for the considered step index fiber. The specific accuracy is for the effective index converging to within 10 5 from the eventual solution. (a) LP 01 mode with α set to be α = ( k 0 2 n core 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) . (b) LP 02 mode with α set to be α = ( k 0 2 · 1.4615 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) .

Fig. 3
Fig. 3

Plots of the convergence of β / k 0 for the LP 01 mode as a function of the propagation step number for the considered step index fiber, with α set to be α = ( k 0 2 n core 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) . The open circles represent the calculation using the quartic SD-ID-BPM with δ = 10 8 m 3 ; the solid triangles represent the calculation using the quadratic SD-ID-BPM [10] with c = 10 2 m 1 . The stars represent the convergence of the calculated effective index n eff as a function of number of numerical solutions using Eq. (14), with α set to be α = ( k 0 2 n core 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) .

Fig. 4
Fig. 4

Plots of the convergence of β + / k 0 for the LP 02 mode as a function of the propagation step number for the considered step index fiber, with α set to be α = ( k 0 2 · 1.4615 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) . The open circles represent the calculation using the quartic SD-ID-BPM with δ = 10 8 m 3 ; the solid triangles represent the calculation using the quadratic SD-ID-BPM [10] with c = 10 2 m 1 . The stars represent the convergence of the calculated effective index n eff as a function of number of numerical solutions using Eq. (14), with α set to be α = ( k 0 2 · 1.4615 2 k 0 2 n 0 2 ) / ( 2 k 0 n 0 ) .

Equations (15)

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

A z = j · ( H α ) · ( α H ) c · A
H = 1 2 k 0 n 0 · [ 2 x 2 + 2 y 2 + k 0 2 · ( n 2 n 0 2 ) ]
A z = j · ( H α ) 4 δ · A ,
A ( x , y , z ) = e j · ( H α ) 4 δ · z · A ( x , y , z = 0 ) .
A ( x , y , 0 ) = m a m ϕ m ( x , y ) ,
H · ϕ m ( x , y ) = λ m ϕ m ( x , y ) ,
A ( x , y , z ) = m a m e ( λ m α ) 4 δ · z ϕ m ( x , y ) .
A ( x , y , z + Δ z ) = 1 1 + Δ z ( H α ) 4 δ · A ( x , y , z ) ,
A ( x , y , z ) = m a m ( z ) ϕ m ( x , y ) , A ( x , y , z + Δ z ) = m a m ( z + Δ z ) ϕ m ( x , y ) .
m a m ( z + Δ z ) ϕ m ( x , y ) = m a m ( z ) 1 1 + Δ z ( λ m α ) 4 δ ϕ m ( x , y ) .
a m ( z + Δ z ) = 1 1 + Δ z ( λ m α ) 4 δ a m ( z ) .
A l + 1 = I I + Δ z ( H ( n ) α I ) 4 δ · A l ,
λ i = α ± { [ A ( x 0 , y 0 , z ) A ( x 0 , y 0 , z + Δ z ) 1 ] · δ Δ z } 1 4 ,
A l + 1 = I H ( n ) α I A l .
A z = j · ( H α ) 2 N S 2 N 1 · A ,

Metrics