Abstract

The performance of the recently developed time-domain beam-propagation methods (TD-BPMs) is compared with that of the finite-difference time-domain (FDTD) method. For the TD-BPMs, we investigate full-band (FB), wide-band (WB), and narrow-band (NB) methods based on the implicit finite-difference (FD) schemes. Owing to the use of the slowly varying envelope, a time step of the TD-BPM can be chosen to be larger than that of the FDTD. Although the numerical results of a waveguide grating obtained from the FB-and WB-TD-BPMs agree well with that from the FDTD, the CPU times are longer than that of the FDTD due to the solution of broadly banded matrices. Introducing the alternating-direction implicit method (ADIM) into the WB-and NB-TD-BPMs contributes to a reduction in the CPU time. To make the methods more efficient, a fourth-order accurate FD formula is applied to the ADIM-based WB-and NB-TD-BPMs, leading to reduced CPU times to 40% and 6% of that of the FDTD, respectively.

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  1. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Norwood, MA: Artech House, 2000.
  2. S. T. Chu, W. P. Huang and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits", Comput. Phys. Commun., vol. 68, no. 1-3, pp. 451-484, Nov. 1991.
  3. W. P. Huang, S. T. Chu, A. Goss and S. K. Chaudhuri, "A scalar finite-difference time-domain approach to guided-wave optics", IEEE Photon. Technol. Lett., vol. 3, no. 6, pp. 524-526, Sep. 1991.
  4. W. P. Huang, S. T. Chu and S. K. Chaudhuri, "A semivectorial finite-difference time-domain method", IEEE Photon. Technol. Lett., vol. 3, no. 9, pp. 803-806, Sep. 1991.
  5. C. Conti, A. Di Falco and G. Assanto, "Parametric oscillations in photonic crystal slabs 3-D time-domain analysis", IEEE Photon. Technol. Lett., vol. 16, no. 6, pp. 1495-1497, Jun. 2004.
  6. T. Matsumoto and T. Baba, "Photonic crystal k-vector superprism", J. Lightw. Technol., vol. 22, no. 3, pp. 917-922, Mar. 2004.
  7. Y. Ohtera, Y. Sasaki and S. Kawakami, "Postprocessing of FDTD solutions for precise calculations of eigenfrequencies of photonic periodic structures utilizing the variational expression", J. Lightw. Technol., vol. 22, no. 6, pp. 1628-1636, Jun. 2004.
  8. P. L. Liu, Q. Zhao and F. S. Choa, "Slow-wave finite-difference beam propagation method", IEEE Photon. Technol. Lett., vol. 7, no. 8, pp. 890-892, Aug. 1995.
  9. G. H. Jin, J. Harari, J. P. Vilcot and D. Decoster, "An improved time domain beam propagation method for integrated optics components", IEEE Photon. Technol. Lett., vol. 9, no. 3, pp. 348-350, Mar. 1997.
  10. F. Ma, "Slowly varying envelope simulation of optical waves in time domain with transparent and absorbing boundary conditions", J. Lightw. Technol., vol. 15, no. 10, pp. 1974-1985, Oct. 1997.
  11. J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Finite-difference time-domain beam propagation method for analysis of three-dimensional optical waveguides", Electron. Lett., vol. 35, no. 18, pp. 1548-1549, Sep. 1999.
  12. J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Time-domain finite-difference BPM with Padé approximants in time axis for analysis of circularly symmetric fields", Electron. Lett., vol. 36, no. 4, pp. 319-321, Feb. 2000.
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  14. J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Comparative study of absorbing boundary conditions for the time-domain beam propagation method", IEEE Photon. Technol. Lett., vol. 13, no. 4, pp. 314-316, Apr. 2001.
  15. J. Shibayama, A. Yamahira, T. Mugita, J. Yamauchi and H. Nakano, "A finite-difference time-domain beam-propagation method for TE-and TM-wave analyses", J. Lightw. Technol., vol. 21, no. 7, pp. 1709-1715, Jul. 2003.
  16. M. Koshiba, Y. Tsuji and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits", J. Lightw. Technol., vol. 18, no. 1, pp. 102-110, Jan. 2000.
  17. J. J. Lim, T. M. Benson, E. C. Larkins and P. Sewell, "Wideband finite-difference-time-domain beam propagation method", Microw. Opt. Technol. Lett., vol. 34, no. 4, pp. 243-247, Jul. 2002.
  18. T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits", J. Lightw. Technol., vol. 22, no. 2, pp. 684-691, Feb. 2004.
  19. V. F. Rodriguez-Esquerre and H. E. Hernandez-Figueroa, "Novel time-domain step-by-step scheme for integrated optical applications", IEEE Photon. Technol. Lett., vol. 13, no. 4, pp. 311-313, Apr. 2001.
  20. V. F. Rodriguez-Esquerre, M. Koshiba and H. E. Hernandez-Figueroa, "Finite-element time-domain analysis of 2-D photonic crystal resonant cavities", IEEE Photon. Technol. Lett., vol. 16, no. 3, pp. 816-818, Mar. 2004.
  21. S. S. A. Obayya, "Efficient finite-element-based time-domain beam propagation analysis of optical integrated circuits", IEEE J. Quantum Electron., vol. 40, no. 5, pp. 591-595, May 2004.
  22. J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama and H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides", IEEE Photon. Technol. Lett., vol. 9, no. 7, pp. 961-963, Jul. 1997.
  23. J. Yamauchi, G. Takahashi and H. Nakano, "Modified finite-difference formula for semivectorial H-field solutions of optical waveguides", IEEE Photon. Technol. Lett., vol. 10, no. 8, pp. 1127-1129, Aug. 1998.
  24. N. M. Newmark, "A method of computation for structural dynamics", J. Eng. Mech. Div., ASCE, vol. 85, no. EM-3, pp. 67-94, Jul. 1959.
  25. H. A. Van Der Vorst, "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear system", SIAM J. Sci. Stat. Comput., vol. 13, no. 2, pp. 631-644, Mar. 1992.
  26. D. W. Peaceman and H. H. Rachford Jr., "The numerical solution of parabolic and elliptic differential equations", J. Soc. Ind. Appl. Math., vol. 3, no. 1, pp. 28-41, Mar. 1955.
  27. J. Douglas Jr. and H. H. Rachford Jr., "On the numerical solution of the heat conduction problem in two and three variables", Trans. Amer. Math. Soc., vol. 82, no. 2, pp. 421-439, Jul. 1956.
  28. H. Yokota, M. Hira and S. Kurazono, "Iterative finite difference beam propagation method analysis of nonlinear optical waveguide excitation problem", IEICE Trans., vol. J77-C-I, no. 10, pp. 529-535, Oct. 1994.
  29. H. Rao, R. Scarmozzino and R. M. Osgood Jr., "An improved ADI-FDTD method and its application to photonic simulations", IEEE Photon. Technol. Lett., vol. 14, no. 4, pp. 477-479, Apr. 2002.
  30. Y. Y. Lu, "New unconditionally stable ADI schemes for the wave equation", presented at the 5th Pacific Rim Conf. Laser Electro-Optics, Taipei, Taiwan,W1A-(14)-2, 2003.
  31. N. N. Feng and W. P. Huang, "Time-domain reflective beam propagation method", IEEE J. Quantum Electron., vol. 40, no. 6, pp. 778-783, Jun. 2004.
  32. J. B. Schneider, C. L. Wagner and O. M. Ramahi, "Implementation of transparent sources in FDTD simulations", IEEE Trans. Antennas Propag., vol. 46, no. 8, pp. 1159-1168, Aug. 1998.
  33. S. M. Wang, "On the current source implementation for the ADI-FDTD method", IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 513-515, Nov. 2004.
  34. J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves", J. Comput. Phys., vol. 114, no. 2, pp. 185-200, Oct. 1994.
  35. D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang and B. Chen, "The perfectly matched layer boundary condition for scalar finite-difference time-domain method", IEEE Photon. Technol. Lett., vol. 13, no. 5, pp. 454-456, May 2001.
  36. C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space", IEEE Microw. Guided Wave Lett., vol. 5, no. 3, pp. 90-92, Mar. 1995.
  37. K. H. Dridi, J. S. Hesthaven and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries", IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 749-756, May 2001.
  38. T. Ando, H. Nakayama, S. Numata, J. Yamauchi and H. Nakano, "Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface", J. Lightw. Technol., vol. 20, no. 8, pp. 1627-1634, Aug. 2002.
  39. C. P. Yu and H. C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers", Opt. Express, vol. 12, no. 25, pp. 6165-6177, Dec. 2004.
  40. G. Marrocco, M. Sabbadini and F. Bardati, "FDTD improvement by dielectric subgrid resolution", IEEE Trans. Microw. Theor. Tech., vol. 46, no. 12, pp. 2166-2169, Dec. 1998.
  41. Y. P. Chiou, Y. C. Chiang and H. C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices", J. Lightw. Technol., vol. 18, no. 2, pp. 243-251, Feb. 2000.
  42. J. Yamauchi, T. Murata and H. Nakano, "Semivectorial H -field analysis of rib waveguides by a modified beam-propagation method based on the generalized Douglas scheme", Opt. Lett., vol. 25, no. 24, pp. 1771-1773, Dec. 2000.
  43. J. L. Young, D. Gaitonde and J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach", IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1573-1580, Nov. 1997.
  44. T. Hirono, W. W. Lui, K. Yokoyama and S. Seki, "Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation", J. Lightw. Technol., vol. 16, no. 10, pp. 1915-1920, Oct. 1998.

Other (44)

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Norwood, MA: Artech House, 2000.

S. T. Chu, W. P. Huang and S. K. Chaudhuri, "Simulation and analysis of waveguide based optical integrated circuits", Comput. Phys. Commun., vol. 68, no. 1-3, pp. 451-484, Nov. 1991.

W. P. Huang, S. T. Chu, A. Goss and S. K. Chaudhuri, "A scalar finite-difference time-domain approach to guided-wave optics", IEEE Photon. Technol. Lett., vol. 3, no. 6, pp. 524-526, Sep. 1991.

W. P. Huang, S. T. Chu and S. K. Chaudhuri, "A semivectorial finite-difference time-domain method", IEEE Photon. Technol. Lett., vol. 3, no. 9, pp. 803-806, Sep. 1991.

C. Conti, A. Di Falco and G. Assanto, "Parametric oscillations in photonic crystal slabs 3-D time-domain analysis", IEEE Photon. Technol. Lett., vol. 16, no. 6, pp. 1495-1497, Jun. 2004.

T. Matsumoto and T. Baba, "Photonic crystal k-vector superprism", J. Lightw. Technol., vol. 22, no. 3, pp. 917-922, Mar. 2004.

Y. Ohtera, Y. Sasaki and S. Kawakami, "Postprocessing of FDTD solutions for precise calculations of eigenfrequencies of photonic periodic structures utilizing the variational expression", J. Lightw. Technol., vol. 22, no. 6, pp. 1628-1636, Jun. 2004.

P. L. Liu, Q. Zhao and F. S. Choa, "Slow-wave finite-difference beam propagation method", IEEE Photon. Technol. Lett., vol. 7, no. 8, pp. 890-892, Aug. 1995.

G. H. Jin, J. Harari, J. P. Vilcot and D. Decoster, "An improved time domain beam propagation method for integrated optics components", IEEE Photon. Technol. Lett., vol. 9, no. 3, pp. 348-350, Mar. 1997.

F. Ma, "Slowly varying envelope simulation of optical waves in time domain with transparent and absorbing boundary conditions", J. Lightw. Technol., vol. 15, no. 10, pp. 1974-1985, Oct. 1997.

J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Finite-difference time-domain beam propagation method for analysis of three-dimensional optical waveguides", Electron. Lett., vol. 35, no. 18, pp. 1548-1549, Sep. 1999.

J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Time-domain finite-difference BPM with Padé approximants in time axis for analysis of circularly symmetric fields", Electron. Lett., vol. 36, no. 4, pp. 319-321, Feb. 2000.

J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Efficient time-domain finite-difference beam propagation methods for the analysis of slab and circularly symmetric waveguides", J. Lightw. Technol., vol. 18, no. 3, pp. 437-442, Mar. 2000.

J. Shibayama, T. Takahashi, J. Yamauchi and H. Nakano, "Comparative study of absorbing boundary conditions for the time-domain beam propagation method", IEEE Photon. Technol. Lett., vol. 13, no. 4, pp. 314-316, Apr. 2001.

J. Shibayama, A. Yamahira, T. Mugita, J. Yamauchi and H. Nakano, "A finite-difference time-domain beam-propagation method for TE-and TM-wave analyses", J. Lightw. Technol., vol. 21, no. 7, pp. 1709-1715, Jul. 2003.

M. Koshiba, Y. Tsuji and M. Hikari, "Time-domain beam propagation method and its application to photonic crystal circuits", J. Lightw. Technol., vol. 18, no. 1, pp. 102-110, Jan. 2000.

J. J. Lim, T. M. Benson, E. C. Larkins and P. Sewell, "Wideband finite-difference-time-domain beam propagation method", Microw. Opt. Technol. Lett., vol. 34, no. 4, pp. 243-247, Jul. 2002.

T. Fujisawa and M. Koshiba, "Time-domain beam propagation method for nonlinear optical propagation analysis and its application to photonic crystal circuits", J. Lightw. Technol., vol. 22, no. 2, pp. 684-691, Feb. 2004.

V. F. Rodriguez-Esquerre and H. E. Hernandez-Figueroa, "Novel time-domain step-by-step scheme for integrated optical applications", IEEE Photon. Technol. Lett., vol. 13, no. 4, pp. 311-313, Apr. 2001.

V. F. Rodriguez-Esquerre, M. Koshiba and H. E. Hernandez-Figueroa, "Finite-element time-domain analysis of 2-D photonic crystal resonant cavities", IEEE Photon. Technol. Lett., vol. 16, no. 3, pp. 816-818, Mar. 2004.

S. S. A. Obayya, "Efficient finite-element-based time-domain beam propagation analysis of optical integrated circuits", IEEE J. Quantum Electron., vol. 40, no. 5, pp. 591-595, May 2004.

J. Yamauchi, M. Sekiguchi, O. Uchiyama, J. Shibayama and H. Nakano, "Modified finite-difference formula for the analysis of semivectorial modes in step-index optical waveguides", IEEE Photon. Technol. Lett., vol. 9, no. 7, pp. 961-963, Jul. 1997.

J. Yamauchi, G. Takahashi and H. Nakano, "Modified finite-difference formula for semivectorial H-field solutions of optical waveguides", IEEE Photon. Technol. Lett., vol. 10, no. 8, pp. 1127-1129, Aug. 1998.

N. M. Newmark, "A method of computation for structural dynamics", J. Eng. Mech. Div., ASCE, vol. 85, no. EM-3, pp. 67-94, Jul. 1959.

H. A. Van Der Vorst, "Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear system", SIAM J. Sci. Stat. Comput., vol. 13, no. 2, pp. 631-644, Mar. 1992.

D. W. Peaceman and H. H. Rachford Jr., "The numerical solution of parabolic and elliptic differential equations", J. Soc. Ind. Appl. Math., vol. 3, no. 1, pp. 28-41, Mar. 1955.

J. Douglas Jr. and H. H. Rachford Jr., "On the numerical solution of the heat conduction problem in two and three variables", Trans. Amer. Math. Soc., vol. 82, no. 2, pp. 421-439, Jul. 1956.

H. Yokota, M. Hira and S. Kurazono, "Iterative finite difference beam propagation method analysis of nonlinear optical waveguide excitation problem", IEICE Trans., vol. J77-C-I, no. 10, pp. 529-535, Oct. 1994.

H. Rao, R. Scarmozzino and R. M. Osgood Jr., "An improved ADI-FDTD method and its application to photonic simulations", IEEE Photon. Technol. Lett., vol. 14, no. 4, pp. 477-479, Apr. 2002.

Y. Y. Lu, "New unconditionally stable ADI schemes for the wave equation", presented at the 5th Pacific Rim Conf. Laser Electro-Optics, Taipei, Taiwan,W1A-(14)-2, 2003.

N. N. Feng and W. P. Huang, "Time-domain reflective beam propagation method", IEEE J. Quantum Electron., vol. 40, no. 6, pp. 778-783, Jun. 2004.

J. B. Schneider, C. L. Wagner and O. M. Ramahi, "Implementation of transparent sources in FDTD simulations", IEEE Trans. Antennas Propag., vol. 46, no. 8, pp. 1159-1168, Aug. 1998.

S. M. Wang, "On the current source implementation for the ADI-FDTD method", IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 513-515, Nov. 2004.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves", J. Comput. Phys., vol. 114, no. 2, pp. 185-200, Oct. 1994.

D. Zhou, W. P. Huang, C. L. Xu, D. G. Fang and B. Chen, "The perfectly matched layer boundary condition for scalar finite-difference time-domain method", IEEE Photon. Technol. Lett., vol. 13, no. 5, pp. 454-456, May 2001.

C. M. Rappaport, "Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space", IEEE Microw. Guided Wave Lett., vol. 5, no. 3, pp. 90-92, Mar. 1995.

K. H. Dridi, J. S. Hesthaven and A. Ditkowski, "Staircase-free finite-difference time-domain formulation for general materials in complex geometries", IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 749-756, May 2001.

T. Ando, H. Nakayama, S. Numata, J. Yamauchi and H. Nakano, "Eigenmode analysis of optical waveguides by a Yee-mesh-based imaginary-distance propagation method for an arbitrary dielectric interface", J. Lightw. Technol., vol. 20, no. 8, pp. 1627-1634, Aug. 2002.

C. P. Yu and H. C. Chang, "Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers", Opt. Express, vol. 12, no. 25, pp. 6165-6177, Dec. 2004.

G. Marrocco, M. Sabbadini and F. Bardati, "FDTD improvement by dielectric subgrid resolution", IEEE Trans. Microw. Theor. Tech., vol. 46, no. 12, pp. 2166-2169, Dec. 1998.

Y. P. Chiou, Y. C. Chiang and H. C. Chang, "Improved three-point formulas considering the interface conditions in the finite-difference analysis of step-index optical devices", J. Lightw. Technol., vol. 18, no. 2, pp. 243-251, Feb. 2000.

J. Yamauchi, T. Murata and H. Nakano, "Semivectorial H -field analysis of rib waveguides by a modified beam-propagation method based on the generalized Douglas scheme", Opt. Lett., vol. 25, no. 24, pp. 1771-1773, Dec. 2000.

J. L. Young, D. Gaitonde and J. S. Shang, "Toward the construction of a fourth-order difference scheme for transient EM wave simulation: Staggered grid approach", IEEE Trans. Antennas Propag., vol. 45, no. 11, pp. 1573-1580, Nov. 1997.

T. Hirono, W. W. Lui, K. Yokoyama and S. Seki, "Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation", J. Lightw. Technol., vol. 16, no. 10, pp. 1915-1920, Oct. 1998.

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