变系数光波导中特征模求解及其在传播计算中的应用

发布时间：2018-04-30 17:22

本文选题：无界区域 + 模式　；参考：《浙江大学》2017年博士论文

【摘要】：谱元方法(Spectral Element Method (SEM)),类似于谱方法(Spectral Method (SM)),把解展开成一组基函数,这些基函数大多是高阶多项式;同时类似于有限元方法(Finite Element Method (FEM)),把计算域分解成多个更加简单的区域,在这些区域中定义局部基函数。而网格适应算法,在自适应有限元中具有至关重要的地位,它主要通过研究解的收敛即“光滑性”以及相应的误差估计即“残差”来进行的。在光波导结构中,特征模式分析具有重要的地位。本论文中将给出一种带网格适应的谱元方法(即改进的谱元方法MSEM),以此来计算变折射率的无界光波导结构中的特征模问题;另一方面,对于三维无界光波导中波的传播问题,本文给出了一个三维的算子步进算法,并验证其有效性,这里主要关注于三维标量的Helmholtz方程,其中特征模的求解也用到了上述的MSEM。具体地来说:本文给出了一种基于网格适应的改进的谱元方法MSEM,对折射率面是不断变化的无界光波导进行特征模求解,这里光波导的折射率面对应于一个二维函数,不再是仅仅依赖于一个方向。首先,对一个一维无界光波导的特征模问题进行研究,在这里波导的横向折射率面是一个连续函数。对于无界区域,用完美匹配层(Perfectly Matched Layer (PML))进行截断,这样就把一个无界特征问题转化为有界特征问题。通过MSEM,有效给出了这种一维问题的特征模分布(包括波导中传播模、泄漏模以及Berenger模)。然后MSEM对折射率面是二位函数的波导结构也给出了有效的特征模解法。事实上,MSEM可以应用于折射率面剧烈变化的波导结构,不再局限于缓慢变化或者均匀的折射率面。文中为了验证MSEM的有效性,与其他经典的求解偏微分方程的数值方法:切比雪夫谱配点法(Chebyshev Spectral Collocation Method (CSCM)),FEM以及SEM进行比较,MSEM做到了利用更少的插值点而得到相近的精度。对于三维无界光波导中的传播问题,本文将以前只关注于二维介质的算子步进算法推广到三维无界区域。这里首先关注一个带散射边界条件的三维标量的Helmholtz方程。三维的算子步进算法基于二维的方法,假定波在传播方向是缓慢变化的,然后运用MSEM求解其二维区域Helmholtz算子的特征值和特征向量,通过局部特征截断,分片进行算子步进。这种三维的算子步进算法一方面利用了MSEM在求解特征模方面的良好性质,另一方面又保持了算子步进方法的优点:大步长和快速计算。
[Abstract]:The spectral element method is similar to the spectral method, Spectral Method, which expands the solution into a set of basis functions, which are mostly higher-order polynomials, and similar to the finite element method (FEM), which decomposes the computational domain into several simpler domains. Local basis functions are defined in these regions. The mesh adaptation algorithm plays an important role in the adaptive finite element method. It is mainly carried out by studying the convergence of the solution, that is, "smoothness", and the corresponding error estimation, that is, "residual error". Characteristic mode analysis plays an important role in optical waveguide structures. In this paper, we present a mesh-adaptive spectral element method (the improved spectral element method, MSEMN) to calculate the eigenmode problem in an unbounded optical waveguide structure with variable refractive index, on the other hand, for the propagation of waves in a three-dimensional unbounded optical waveguide. In this paper, we give a three dimensional operator step algorithm and verify its validity. Here we mainly focus on the Helmholtz equation of the three dimensional scalar, in which the above MSEM is also used to solve the eigenmode. Specifically, an improved spectral element method based on mesh adaptation, MSEMM, is presented. The eigenmode of an unbounded optical waveguide whose refractive index surface is constantly changing is solved, where the refractive index face of the optical waveguide should be a two-dimensional function. Is no longer dependent on only one direction. Firstly, the eigenmode problem of a one-dimensional unbounded optical waveguide is studied, where the transverse refractive index plane of the waveguide is a continuous function. For the unbounded region, the perfectly matched Matched Layer is used to truncate the unbounded region, so that an unbounded feature problem is transformed into a bounded feature problem. The characteristic mode distribution (including propagating mode, leaky mode and Berenger mode) of this one-dimensional problem is given. Then MSEM gives an effective eigenmode method for waveguide structures whose refractive index plane is a two-position function. In fact, MSEM can be applied to waveguide structures with dramatically changing refractive index surface, and is no longer limited to slowly varying or even refractive index surface. In order to verify the validity of MSEM, compared with other classical numerical methods for solving partial differential equations: Chebyshev Spectral Collocation Method and SEM, the Chebyshev spectrum collocation method has achieved similar accuracy by using fewer interpolation points. For the problem of propagation in 3D unbounded optical waveguides, this paper extends the operator step algorithm, which was only concerned with two-dimensional media, to the three-dimensional unbounded region. Here we first focus on a three-dimensional scalar Helmholtz equation with scattering boundary conditions. The three-dimensional operator step algorithm is based on the two-dimensional method, assuming that the wave is slowly changing in the propagation direction. Then the eigenvalue and eigenvector of the Helmholtz operator in its two-dimensional domain are solved by using MSEM, and then the local feature truncation is used to step the operator piecewise. On the one hand, this three-dimensional operator step algorithm takes advantage of the good properties of MSEM in solving characteristic modules, on the other hand, it preserves the advantages of the operator step method: large step size and fast computation.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.8

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