【作者】 陈芝花；

【导师】 朱建新；

【作者基本信息】 浙江大学 ， 计算数学， 2008， 博士

【摘要】 本文主要论述了以声波导、光波导为背景的Helmholtz方程的数值计算问题,包括渐近解的计算及波的传播计算,其中渐近解的计算包括开放波导泄漏模渐近解的计算和使用了完美匹配层(PML)后泄漏模及Berenger模渐近解的计算。高精度的泄漏模在波的传播计算中非常重要,尤其是对于光互联等比较精密的仪器。当声速及折射率为分段常数时,泄漏模的渐近解可以通过一个解析的非线性方程得到;但是对于声速随深度变化的声波导以及折射率随横向变化的光波导,无法得到关于特征值的非线性方程。本文首先给出了开放的平板光波导及海洋声波导中泄漏模的渐近解。在这种波导中,运用W.K.B方法,通过对原方程的特征问题进行分析,得到了关于特征值的一个近似的的非线性方程,再对这个方程进行进一步分析得到了泄漏模的渐近解。数值模拟表明,这些渐近解比较接近于泄漏模的精确解,并且随着模的增大,误差越来越小,因此可以作为Rayleigh商迭代的初值求解更精确的泄漏模。开放的波导结构在数值计算时的空间区域是没有边界的,现在比较有效的方法是采用完美匹配层把开放的波导区域截断为有界的区域。在加了完美匹配层之后,除了传播模和泄漏模之外,还会产生一种模,称之为Berenger模。为了分析完美匹配层对波的各种模式的影响,本文还讨论了加了完美匹配层之后对传播计算比较重要的泄漏模的渐近解,此外,粗略地推导了Berenger模的渐近公式。结果表明,在采用了完美匹配层之后,对于声波导和光波导的TE模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导是一样的;对于TM模,略去高阶无穷小量,泄漏模的渐近公式和开放的波导会有所不同。对于光波导,上下包层的折射率若是一样的话,Berenger模的轨迹只有一条;若不一样的话,Berenger模的轨迹有两条。数值模拟表明,这些泄漏模及Berenger模的渐近解在其精确解附近,因此可以作为Rayleigh商迭代的初值来求解更精确的模。本文最后做了一些把以上所得的渐近解应用于波的传播计算的工作,主要做了两种情况下的波传播的模拟:一种是声波的波数及光波的折射率仅随一个方向变化的情况,另一种是波数及折射率随两个方向而变化的情况。数值模拟表明,把渐近解应用于波的传播计算中得到的结果和以前的方法相比,结果得到明显改善,并且计算速度快很多。对于光波导,本文还比较了未选取Berenger模和选取了Berenger模时波的传播。数值模拟结果表明,Berenger模对波的传播会产生轻微的影响。更多还原

【Abstract】 Leaky modes play an important role in wave propagation computation of optical and acoustic waveguide ,especially in the on-chip optical interconnection proposed re-cently. They can be used to partially represent the wave field related to the continuous spectrum of the radiation and evanescent modes.For two-dimensional step-index waveguides,the propagation constants satisfy an analysis nonlinear equation,from which asymptotic solutions of leaky modes have been given. B-ut when the refractive index n is varied with the transverse variable x in optical waveguide or acoustic velocity c is varied with z in acoustic waveguide respectively,analysis nonlinear equation of the leaky modes can not be obtained. Asymptotic solutions of leaky modes in unbounded waveguide are given first. In order to get asymptotic solutions of the leaky modes, W.K.B. method is implemented to create an approximate nonlinear equa-tion. Numerical simulations show that our asymptotic solutions are very close to exact leaky modes and the errors become smaller as the norm of the modes become larger. Our asymptotic solutions can be used as initial guesses for computing the more accurate leaky modes by Rayleigh Quotient iteration .PML as an efficient artificial absorbing material is widely used to solve unbounded wave propagation problem, in which Berenger modes(the eigenmodes in a waveguide terminated by a finite PML) appear. High order asymptotic solutions for the Leaky modes and Berenger modes are derived in waveguide with PML. Asymptotic solutions of the modes are useful,because they can be used as initial guesses for solving the exact eigenvalues numerically.Wave propagation computation is implemented when Leaky modes are obtained by asymptotic solutions which are used as initial guesses for Rayleigh Quotation itera-tion. Two cases are considered: first, refractive index is varied with transverse variable and acoustic velocity c is varied with depth,second, they are all varied with two directions. Numerical simulations show that results from this method are better than the methods from Lapack with highly computation efficiently.For optical waveguide,wave propagation computation are compared with or without Berenger modes. Numerical simulations show that results from these two ways will have a slight difference.更多还原