一类变系数亥姆霍兹算子的特征模计算及应用研究

发布时间：2018-03-17 14:29

本文选题：光波导　切入点：亥姆霍兹方程　出处：《浙江大学》2016年博士论文　论文类型：学位论文

【摘要】：以物理中的光波导为背景,数学上,本文分别对无界变系数亥姆霍兹方程的模式求解问题及反散射问题的数值算法两类问题进行了研究。第一类问题,对于缓变光波导,本文推导得到了使用完美匹配层(perfectly matched layer,简称PML)边界的TM模式下传播常数满足的三角矩阵近似的色散方程；进一步地,对一般的光波导,本文得到了使用PML边界的TE和TM模式下全矩阵近似的高精度的色散方程。另外,本文给出了泄漏模和Berenger模的解析的渐近解公式,当求解这两种模在大模(|β|较大)时的高精度解时,它们可以作为使用牛顿迭代法高精度求解色散方程时的初始值。对于另一类问题,反介质的重建问题,本文给出了较好的数值反演算法。具体地说,对于无界的光波导,本文使用PML边界条件将无界问题有界化。由于讨论的是波导非均匀的情况,所以使用了微分转移矩阵来推导芯层的转移矩阵。本文首先推导了TM模式的微分转移矩阵,并对该微分转移矩阵进行积分,对积分后的形式进行矩阵指数运算。但是因为指数运算难以给出简单明确的表达式,所以对波导缓变的情况,本文使用上、下三角矩阵近似微分转移矩阵,最终推导得到了PML边界下TM模式的两个三角矩阵近似的色散关系,并对波导的大模情况,本文给出了泄漏模和Berenger模的解析的渐近解公式。因为在波导模式的分析中,相对于小模,高精度求解大模要困难得多,所以使用牛顿法求解泄漏模和Berenger模的高精度解时,它们的渐近解可以作为迭代初始值。在另一个研究工作中,为了进一步提高特征值的精度,本文不对微分转移矩阵进行任何近似,直接对其积分,并对积分后的形式进行矩阵变换,然后近似地进行矩阵指数的计算,最终结合PML边界条件,推导得到了特征模式(传播模、泄漏模、Berenger模)满足的全矩阵近似的高精度的解析的色散关系。这个色散关系对一般折射率连续变化且可导的波导都适用,不在局限于缓变波导的情况。而且,在一定的条件下,可以精确地计算矩阵指数,此时上面得到的高精度的色散关系变为精确的色散关系。对于波导中的反介质重建问题,基于已有的理论方法,本文给出了一种有效的反演数值算法。首先对于每一个固定的入射频率,使用数值搜索和牛顿迭代,得到了一系列高精度的特征值。其次使用多频信息,结合理论表达式的特点,建立了规模相对小的线性系统。最后通过分析奇异值的分布情况,主要使用奇异值分解,迭代正则化等技术对线性系统进行正则化反演求解。数值实验部分对复杂程度不同的杂质(含有低频到高频傅里叶模式的杂质)进行数值验证,结果表明本文提出了一个较好的反演数值算法。
[Abstract]:The optical waveguide in the physics background of mathematics, numerical algorithms to solve the problem of unbounded model Helmholtz equations with variable coefficients and the inverse scattering problem of the two kinds of problems are studied. First, the slow light waveguide, is presented in this paper using the perfect matching layer (perfectly matched layer, referred to as PML) the dispersion equation of triangular matrix constants satisfy the approximate boundary of the TM communication mode; further, the optical waveguide in general, this paper obtained the high precision dispersion using the PML boundary of the TE and TM mode full matrix approximation process. In addition, this paper gives the asymptotic analysis of leakage mode and Berenger mode solution formula and when solving these two kinds of modules in the module (| beta | larger) high precision solutions, they can be used as the initial use of the Newton iterative method with high accuracy for solving dispersion equation of value. For another kind of problem, inverse medium heavy This paper gives the construction problem, numerical inversion algorithm better. Specifically, the optical waveguide is unbounded, this paper use the PML boundary conditions will bounded unbounded problem. Because the discussion is non uniform waveguide, transfer matrix are used to derive the differential transfer matrix of the core layer. The paper deduced the differential TM mode transfer matrix, and the integral of the differential transfer matrix, the matrix calculation of the integral form of the index. But because the expression index is difficult to give a clear and simple operation, so the waveguide slowly changed, this paper use, lower triangular matrix approximate differential transfer matrix, finally deduced dispersion relation two triangular matrix TM model under PML boundary approximation, and mode of the waveguide, this paper gives the asymptotic analysis of the leakage mode and Berenger mode solution formula. Because in the analysis of waveguide mode in, Yu Xiaomo, high precision solution mode is much more difficult, so the high accuracy using Newton method to solve the leakage mode and Berenger mode solution, the asymptotic solutions can be used as the initial value of iteration. In another study, in order to further improve the accuracy of the eigenvalues, the transfer matrix of any differential approximation. The direct integral, and matrix transformation of the integral of the form, and then calculate approximate matrix index, finally combined with PML boundary condition, derived characteristic patterns (propagation mode, leaky mode, Berenger mode) can satisfy the dispersion relation of full matrix approximation with high accuracy. The analytical dispersion relation of the general index are applicable and can guide the continuous variation of rate of waveguide, is not confined to a slowly varying waveguide. Moreover, under certain conditions, we can accurately calculate the index matrix, the above obtained high precision The dispersion relation becomes the exact dispersion relation for inverse medium waveguide reconstruction problem, based on the existing theories and methods, this paper presents an effective numerical inversion algorithm. Firstly, for each fixed incident frequency, using numerical search and Newton iteration, we obtained a series of high precision eigenvalues. Second use multi frequency information, combined with the characteristics of the theoretical formula, a linear system is relatively small. Finally, through the analysis of the distribution of the singular value, using singular value decomposition, iterative regularization techniques such as regularization to solve the linear system. The numerical experiments of impurities of different complexity (with low frequency to high frequency Fourier model the impurity) was verified, the results show that the proposed a better numerical inversion algorithm.

【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.8

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