Abstract

Efficient nonuniform schemes, based on the generalized Douglas (GD) scheme, are developed for the finite-difference beam propagation method (FD-BPM). For a two-dimensional (2-D) problem, two methods are presented: a computational space method and a physical space method. In the former, the GD scheme is employed, after replacing a nonuniform grid in the physical space with a uniform one in the computational space. In the latter, the GD scheme is directly extended to a nonuniform grid in the physical space. We apply these two methods to paraxial and wide-angle FD-BPM's. The fourth-order accuracy is achieved in the transverse direction, provided that the grid growth factor between two adjacent grids is r=1+O (\Delta x ). For the paraxial BPM, the reduction in the truncation error is demonstrated through modal calculations of a graded-index waveguide using an imaginary distance procedure. For the wide-angle BPM, the propagating field in a tilted waveguide is analyzed to show the effectiveness of the present scheme. As an application of the physical space method, an adaptive grid is introduced into the multistep method.

[IEEE ]

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  1. M. D. Feit and J. A. Fleck, Jr., "Light propagation in graded-index optical fibers," Appl. Opt., vol. 17, pp. 3990-3998, 1978.
  2. D. Yevick and M. Glasner, "Forward wide-angle light propagation in semiconductor rib waveguides," Opt. Lett., vol. 15, pp. 174-176, 1990.
  3. G. R. Hadley, "Wide-angle beam propagation using Pade approximant operators," Opt. Lett., vol. 17, pp. 1426-1428, 1992.
  4. Y. Arai, A. Maruta, and M. Matsuhara, "Transparent boundary for the finite-element beam-propagation method," Opt. Lett., vol. 18, pp. 765-766, 1993.
  5. H. E. Hernandez-Figueroa, "Simple nonparaxial beam-propagation method for integrated optics," J. Lightwave Technol., vol. 12, pp. 644-649, 1994.
  6. F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation," J. Lightwave Technol., vol. 11, pp. 1425-1434, 1993.
  7. J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama, and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol., vol. 14, pp. 2401-2406, 1996.
  8. J. Yamauchi, J. Shibayama, and H. Nakano, "Wide-angle propagating beam analysis based on the generalized Douglas scheme for variable coefficients," Opt. Lett., vol. 20, pp. 7-9, 1995.
  9. C. Vassallo and M. J. Van der Keur, "Highly efficient transparent boundary conditions for finite difference beam propagation method at order four," J. Lightwave Technol., vol. 15, pp. 1958-1965, 1997.
  10. F. Ladouceur, "Boundaryless beam propagation," Opt. Lett., vol. 21, pp. 4-5, 1996.
  11. S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: Scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol., vol. 13, pp. 375-383, 1995.
  12. K. M. Lo and E. H. Li, "Solutions of the quasivector wave equation for optical waveguides in a mapped infinite domains by the Galerkin's method," J. Lightwave Technol., vol. 16, pp. 937-944, 1998.
  13. G. R. Hadley, "Multistep method for wide-angle beam propagation," Opt. Lett., vol. 17, pp. 1743-1745, 1992.
  14. D. Yevick and W. Bardyszewski, "Correspondence of variational finite-difference (relaxation) and imaginary-distance propagation methods for modal analysis," Opt. Lett., vol. 17, pp. 329-330, 1992.
  15. C. L. Xu, W. P. Huang, and S. K. Chaudhuri, "Efficient and accurate vector mode calculations by beam propagation method," J. Lightwave Technol., vol. 11, pp. 1209-1215, 1993.
  16. G. R. Hadley, "Transparent boundary condition for beam propagation," Opt. Lett., vol. 16, pp. 624-626, 1991.

Appl. Opt. (1)

J. Lightwave Technol. (7)

H. E. Hernandez-Figueroa, "Simple nonparaxial beam-propagation method for integrated optics," J. Lightwave Technol., vol. 12, pp. 644-649, 1994.

F. Schmidt, "An adaptive approach to the numerical solution of Fresnel's wave equation," J. Lightwave Technol., vol. 11, pp. 1425-1434, 1993.

J. Yamauchi, J. Shibayama, O. Saito, O. Uchiyama, and H. Nakano, "Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis," J. Lightwave Technol., vol. 14, pp. 2401-2406, 1996.

S. J. Hewlett and F. Ladouceur, "Fourier decomposition method applied to mapped infinite domains: Scalar analysis of dielectric waveguides down to modal cutoff," J. Lightwave Technol., vol. 13, pp. 375-383, 1995.

K. M. Lo and E. H. Li, "Solutions of the quasivector wave equation for optical waveguides in a mapped infinite domains by the Galerkin's method," J. Lightwave Technol., vol. 16, pp. 937-944, 1998.

C. Vassallo and M. J. Van der Keur, "Highly efficient transparent boundary conditions for finite difference beam propagation method at order four," J. Lightwave Technol., vol. 15, pp. 1958-1965, 1997.

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

Opt. Lett. (8)

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