-----------------------------------------------------------

Waves on Wednesday Seminar Series

Wednesday, November 16,  2005, 10:00 am
Cullimore 611
New Jersey Institute of Technology

Numerical Solution of the Nonlinear Helmholtz Equation Using Nonorthogonal Expansions

S. Tsynkov

Department of Mathematics

North Carolina State University

Abstract

Previously, we have developed a fourth-order numerical method for solving the nonlinear Helmholtz equation that governs the propagation of time-harmonic focusing electromagnetic waves in Kerr-type media. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC). It has provided for the reflectionless propagation of all the outgoing waves while also fully transmitting the given incoming beam(s) at the boundaries of the computational domain. Altogether, the method has enabled the direct simulation of nonlinear self-focusing in the nonparaxial regime, including quantitative prediction of the important phenomenon of backscattering. To the best of our knowledge, this capacity has never been achieved before in nonlinear optics. In the current work we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the transverse direction, we now introduce the Sommerfeld-type local radiation boundary conditions. They are constructed directly in the discrete framework for the chosen fourth- order scheme that approximates the Helmholtz operator. Numerically, implementation of the Sommerfeld conditions requires evaluation of the eigenvalues and eigenvectors of a non-Hermitian matrix; the latter subsequently render the separation of variables on the upper time level of the iterative solver. It turns out that in spite of the additional effort due to the expansion with respect to the resulting nonorthogonal basis, the new algorithm offers considerable overall benefits, both from the standpoint of its numerical performance and the range of physical phenomena that it is capable of simulating. Joint work with G. Fibich, Tel Aviv University