A full leap frog Fourier method for integrating the Korteweg-de Vries (KdV) equation ut + uux-eUxxx 0 results in an O(N-3) stability constraint on the time step, where N is the number of Fourier modes used. This stability limit is much more restrictive than the accuracy limit for many applications.In this paper we propose a method for which the stability limit is extended by treating the linear dispersive ux term implicitly. Thus timesteps can be taken up to an accuracy limit larger than the explicit stability limit. The implicit method is implemented without solving linear systems by integrating in time in the Fourier space and discretizing the nonlinear uu term by leap frog. A second method we propose uses basis functions which solve the linear part of the KdV equation and leap frog for time integration. A linearized stability analysis of the proposed schemes proves that a version of the first scheme possesses a certain kind of unconditional stability and that the second scheme has an O(N-1) stability limit. The accuracy of the schemes for soliton propagation is analyzed by examining the truncation error. In addition, we analyze a linearization of a nonlinear finite element scheme proposed by Winther that treats the ux term by Crank-Nicolson. Numerical experiments on soliton solutions show that the linearized stability analysis gives accurate predictions for all the nonlinear schemes, that the truncation errors give an estimate for the accuracies and that the Fourier methods are more accurate than the finite element method.
We discuss in this paper some of the methods for 2-D self-consistent simulation of semiconductor quantum well lasers. First, the electronic and optical parts are each treated separately, and then the coupling of the two problems is addressed. We briefly dicuss the evolution of self-consistent laser simulation up to its present level, and afterwards some of the electronic transport concerns are discussed in greater detail. Transport in bulk regions and at heterojunctions, and the coupling of classical and quantum regions are each presented separately. Then, we review the approaches to the solution of the eigenvalue problem for the optical field. Finally, to illustrate the issues involved in coupling electronic and optical solutions, we introduce a model laser structure in which the optical field is poorly confined in the lateral direction, and the different confinement mechanisms are discussed. We then present calculations showing how gain can contribute to the lateral confinement of the fundamental cavity mode.
Maintaining cladding layer material composition uniformity is inherently difficult during high temperature MBE growth of GaAs/AlGaAs laser structures. These non-uniformities can lead to asymmetrical waveguiding structures with distorted optical output characteristics of the laser. Distortions in optical characteristics can greatly affect the alignment and the coupling efficiency between laser diodes and optical fiber or other electro-optical systems in integrated opto-electronic applications. A two-dimensional dielectric waveguide simulator has been used to analyze the optical properties of GRINSCII GaAs/A1GaAs lasers with asymmetrical cladding structures. Through this analysis, we have demonstrated an optimal laser device structure which has the desired optical characteristics and is less sensitive to cladding composition asymmetries arising in typical growth conditions.
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