Noniterative vectorial beam propagation method with a smoothing digital filter

The alternating direction implicit (ADI) technique is used in the numerical solution scheme for each longitudinal step, in the well-known vectorial beam propagation method (VBPM). It is used to reduce the linear-system matrix into a set of tridiagonal ones, which can then be solved by Thomas algorithm. Solutions using standard (Crank-Nicolson) VBPM for step-index waveguides contain small oscillations, which can significantly reduce the accuracy of the solutions. We show that this "noise" can be removed by a properly tailored low-pass digital filter in the spatial frequency domain. Results become as accurate as those obtained by iterative techniques. Examples deal first with the scaler LP/sub 01/ mode and a Gaussian field in a single-mode fiber converging to the vector HE/sub 11/ mode. Second, with an optimized Ti:LiNbO/sub 3/ bent waveguide using MgO. Finally, with a rib waveguide where advantages related to the use of the digital filter become very evident.

[1]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[2]  C.L. Xu,et al.  A wide-angle vector beam propagation method , 1992, IEEE Photonics Technology Letters.

[3]  Wei-Ping Huang,et al.  Simulation of three-dimensional optical waveguides by a full-vector beam propagation method , 1993 .

[4]  H. H. Rachford,et al.  The Numerical Solution of Parabolic and Elliptic Differential Equations , 1955 .

[5]  Junji Yamauchi,et al.  Beam-propagation analysis of optical fibres by alternating direction implicit method , 1991 .

[6]  Paul Lagasse,et al.  Application of propagating beam methods to electromagnetic and acoustic wave propagation problems: A review , 1987 .

[7]  G. R. Hadley,et al.  Transparent boundary condition for the beam propagation method , 1992 .

[8]  G. Yip,et al.  An improved finite-difference vector beam propagation formulation for graded-index waveguides , 1994, IEEE Photonics Technology Letters.

[9]  J. Chrostowski,et al.  A full-vectorial beam propagation method for anisotropic waveguides , 1994 .

[10]  Lars Thylén,et al.  The beam propagation method: an analysis of its applicability , 1983 .

[11]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[12]  Wei-Ping Huang,et al.  The finite-difference vector beam propagation method: analysis and assessment , 1992 .

[13]  Charles Vassallo,et al.  Reformulation for the beam-propagation method , 1993 .

[14]  P.K.W. Vinsome,et al.  Block iterative methods for fully implicit reservoir simulation , 1982 .

[15]  Pao-Lo Liu,et al.  Semivectorial beam-propagation method for analyzing polarized modes of rib waveguides , 1992 .

[16]  Paul Lagasse,et al.  Beam-propagation method: analysis and assessment , 1981 .

[17]  C.L. Xu,et al.  A finite-difference vector beam propagation method for three-dimensional waveguide structures , 1992, IEEE Photonics Technology Letters.