Abstract

The computationally efficient DuFort-Frankel beam-propagation method (BPM) is ideally suited to parallel computing. Although the scheme is conditionally stable for structures with lossless materials, in the presence of material loss it can become unstable. It is shown that the use of perfectly matched layer (PML) boundary conditions can also cause instability, especially in the three-dimensional (3-D) case. These instabilities are characterized and a stabilized DuFort-Frankel scheme is presented that extends the scope of this powerful method to cover these practically important scenarios.

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  2. H. M. Masoudi and J. M. Arnold, "Spurious modes in the DuFort-Frankel finite-difference beam propagation method", IEEE Photon. Technol. Lett., vol. 9, pp. 1382-1384, 1997.
  3. E. C. DuFort and S. P. Frankel, "Stability conditions in the numerical treatment of parabolic differential equations", Math Other Aid Comp., vol. 7, pp. 135-153, 1953.
  4. F. Xiang and G. L. Yip, "An explicit and stable finite-difference 2-D vector beam propagation method", IEEE Photon. Technol. Lett., vol. 6, pp. 1248-1250, 1994.
  5. H. M. Masoudi, M. A. Al-Sunaidi and J. M. Arnold, "Efficient time-domain beam-propagation method for modeling integrated optical devices", J. Lightw. Technol. , vol. 19, pp. 759-771, 2001.
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  15. T. B. Koch, J. B. Davies and D. Wickramasinghe, "Finite element/finite difference propagation algorithm for integrated optical device", Electron. Lett., vol. 25, pp. 514-516, 1989.
  16. Y. Tsuji and M. Koshiba, "A finite element beam propagation method for strongly guiding and longitudinally varying optical waveguides", J. Lightw. Technol., vol. 14, pp. 217-222, 1996.
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  18. G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators", Opt. Lett., vol. 17, p. 1426, 1992.
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  25. H. Rao, R. Scarmozzino and R. M. Osgood Jr., "A bidirectional beam propagation method for multiple dielectric interfaces", IEEE Photon. Technol. Lett., vol. 11, pp. 830-832, Jul. 1999.
  26. H. El-Refaei, D. Yevick and I. Betty, "Stable and noniterative bi-directional beam propagation method", IEEE Photon. Technol. Lett., vol. 12, pp. 389-391, Apr. 2000.
  27. H. Rao, M. J. Steel, R. Scarmozzino and R. M. Osgood Jr., "Complex propagators for evanescent waves in bidirectional beam propagation method", J. Lightw. Technol., vol. 18, pp. 1155-1160, Aug. 2000.
  28. P. L. Ho and Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators", IEEE Photon. Technol. Lett., vol. 13, pp. 1316-1318, Dec. 2001.
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  30. H. Derudder, F. Olyslager, D. De Zutter and S. Van den Berghe, "Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers", IEEE Trans. Antennas Propag., vol. 49, pp. 185 -195, 2001.
  31. W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates", Microwave Opt. Technol. Lett., no. 7, pp. 599-604, 1994.

Other (31)

H. M. Masoudi and J. M. Arnold, "Parallel beam propagation methods", IEEE Photon. Technol. Lett., vol. 6, pp. 848-850, 1994.

H. M. Masoudi and J. M. Arnold, "Spurious modes in the DuFort-Frankel finite-difference beam propagation method", IEEE Photon. Technol. Lett., vol. 9, pp. 1382-1384, 1997.

E. C. DuFort and S. P. Frankel, "Stability conditions in the numerical treatment of parabolic differential equations", Math Other Aid Comp., vol. 7, pp. 135-153, 1953.

F. Xiang and G. L. Yip, "An explicit and stable finite-difference 2-D vector beam propagation method", IEEE Photon. Technol. Lett., vol. 6, pp. 1248-1250, 1994.

H. M. Masoudi, M. A. Al-Sunaidi and J. M. Arnold, "Efficient time-domain beam-propagation method for modeling integrated optical devices", J. Lightw. Technol. , vol. 19, pp. 759-771, 2001.

Y. Chung and N. Dagli, "Explicit finite difference beam propagation method: Application to semiconductor rib waveguide Y-junction analysis", Electron. Lett., vol. 26, pp. 711-713, 1990.

Y. Chung and N. Dagli, "An assessment of finite difference beam propagation method", IEEE J. Quantum Eectron., vol. 26, pp. 1335-1339, 1990.

Y. Chung and N. Dagli, "Analysis of Z -invariant and Z -variant semiconductor rib waveguides by explicit finite difference beam propagation method with nonuniform mesh configuration", IEEE J. Quantum Eectron., vol. 27, pp. 2296-2305, 1991.

D. Yevick and B. Hermansson, "Efficient beam propagation techniques", IEEE J. Quantum Eectron., vol. 26, pp. 109-112, 1990.

P. C. Lee, D. Shulz and E. Voges, "Three dimensional finite difference beam propagation algorithms for photonics devices", IEEE J. Lightw. Technol., vol. 10, pp. 1832-1838, 1992.

W. P. Huang and C. L. Xu, "A wide-angle vector beam propagation method", IEEE Photon. Technol. Lett., vol. 4, pp. 1118-1120, 1992.

G. R. Hadley, "Wide-angle beam propagation with a transparent boundary condition", Proc. Int. Photon. Res., pp. 296-297, 1992.

J. P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves", J. Comput. Phys., vol. 114, pp. 185-200, 1994.

W. P. Huang, C. L. Xu, W. Lui and K. Yokoyama, "The perfectly matched layer (PML) boundary condition for the beam propagation method", IEEE Photon. Technol. Lett., vol. 8, pp. 649-651, 1996.

T. B. Koch, J. B. Davies and D. Wickramasinghe, "Finite element/finite difference propagation algorithm for integrated optical device", Electron. Lett., vol. 25, pp. 514-516, 1989.

Y. Tsuji and M. Koshiba, "A finite element beam propagation method for strongly guiding and longitudinally varying optical waveguides", J. Lightw. Technol., vol. 14, pp. 217-222, 1996.

J. Yamauchi, J. Shibayama and H. Nakano, "Modified finite-difference beam propagation method based on the generalized douglas scheme for variable coefficients", IEEE Photon. Technol. Lett., vol. 7, p. 661, 1995.

G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators", Opt. Lett., vol. 17, p. 1426, 1992.

M. D. Feit and J. A. Fleck, "Computation of mode properties in optical fiber waveguides by a propagating beam method", Appl. Opt., vol. 19, p. 1154, 1980.

J. Gerdes and R. Pregla, "Beam-propagation algorithm based on the method of lines", J. Opt. Soc. Amer. B, vol. 8, no. 2, pp. 389-394, 1991.

G. R. Hadley, "Multistep method for wide-angle beam propagation", Opt. Lett., vol. 17, p. 1743, 1992.

R. Scarmozzino, A. Gopinath, R. Pregla and S. Helfert, "Numerical techniques for modeling guided-wave photonic devices", IEEE J. Sel. Topics Quantum Electon. , vol. 6, pp. 150-162, 2001.

P. Kaczmarski and P. E. Lagasse, "Bidirectional beam propagation method", Electron. Lett., vol. 24, pp. 675-676, May 1988.

K. Hayashi, M. Koshiba, Y. Tsuji, S. Yoneta and R. Kaji, "Combination of beam propagation method and mode expansion propagation method for bidirectional optical beam propagation analysis", J. Lightw. Technol., vol. 16, pp. 2040-2045, Nov. 1998.

H. Rao, R. Scarmozzino and R. M. Osgood Jr., "A bidirectional beam propagation method for multiple dielectric interfaces", IEEE Photon. Technol. Lett., vol. 11, pp. 830-832, Jul. 1999.

H. El-Refaei, D. Yevick and I. Betty, "Stable and noniterative bi-directional beam propagation method", IEEE Photon. Technol. Lett., vol. 12, pp. 389-391, Apr. 2000.

H. Rao, M. J. Steel, R. Scarmozzino and R. M. Osgood Jr., "Complex propagators for evanescent waves in bidirectional beam propagation method", J. Lightw. Technol., vol. 18, pp. 1155-1160, Aug. 2000.

P. L. Ho and Y. Y. Lu, "A stable bidirectional propagation method based on scattering operators", IEEE Photon. Technol. Lett., vol. 13, pp. 1316-1318, Dec. 2001.

R. M. Knox and P. P. Toulios, "Integrated circuits for the millimeter through optical frequency range", in Proc MRI Symposium on Submillimeter Waves, Brooklyn, NY, 1970, pp. 497- 516.

H. Derudder, F. Olyslager, D. De Zutter and S. Van den Berghe, "Efficient mode-matching analysis of discontinuities in finite planar substrates using perfectly matched layers", IEEE Trans. Antennas Propag., vol. 49, pp. 185 -195, 2001.

W. C. Chew and W. H. Weedon, "A 3-D perfectly matched medium from modified Maxwell's equations with stretched coordinates", Microwave Opt. Technol. Lett., no. 7, pp. 599-604, 1994.

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