非线性薛定谔方程求解.pdfVIP

CHAPTER IV NUMERICAL SOLUTIONS TO THE NONLINEAR SCHRÖDINGER EQUATION 4.1 Introduction In general, analytical solutions to the full Maxwell wave equation for a nonlinear optical system do not exist. Even numerical solutions to the wave equation are extremely difficult to implement due to the dimensionality of the problem. The vector form of the wave equation is a four-dimensional (three spatial, one temporal), second-order partial differential equation. Thus, approximations based on propagation conditions and experimental results are needed in order to solve an approximate scalar form of the wave equation, i.e. the nonlinear Schrödinger equation. However, the approximations listed in the previous chapter do limit the generality and validity of the solutions. For example, the condition extreme nonlinearity, as for the case in supercontinuum generation, is a propagation regime where slowly varying envelope approximation may be violated. The purpose of this chapter is to provide an introduction to a very powerful method in numerically solving the NLSE, known as the split-step Fourier method (SSFM) [15]. The chapter will begin with a list pointing the advantages of the SSFM 134 compared to finite-difference methods. Then, the SSFM the symmetric SSFM will be introduced. The chapter will then detail the inclusion of the Raman effect in the numerical solution. 4.2 Why use the Split-Step Fourier Method? The SSFM is the technique of choice for solving the NLSE due to its easy implementation and speed compared to other methods, notably time-domain finite- difference methods [73]. The finite difference method solves the Maxwell’s wave equ