光在无界周期波导中的传播建模与快速计算

In this project, some fast and marching numerical methods are studied for Helmholtz equation on unbounded periodic optical waveguides.(1) For a waveguide with periodic corrugated interfaces, choosing a coordinate orthogonal transform and an equation transform, the interfaces are flatten or stepped , then the equation is reduced to a partial differential equation - an improved Helmholtz equation that does not include one order derivative along the main range. (2) An adaptive PML ( perfectly matched layer ), that is, a complex strengthen coordinate transform, is created, then the improved Helmholtz equation on the unbounded periodic waveguide is turned to a partial differential equation on a bounded region. (3) A marching computational method is developped to realize stably DtN, and the correctness and the reliability of numerical results should be raised. (4) High precision treatment of the characteristic problem for the related complex matrix, and a fast and stable transform formula of local base are studied and created to realize the marching computation with a large step along the main range. (5) For eigenmode expansion method, the validity of the approximate solution approaching the exact one is verified, further the actual wave propagation is obtained on the unbounded periodic waveguide. This project will provide some reliable computational methods and practice software to measure quickly the optical wave propagation and optimize the design of photoelectric devices.

本项目研究Helmholtz 方程在无界周期光波导上求解的快速、步进有效数值方法；（1）对带有周期性皱阶界面的光波导，选用坐标正交变换和方程变换，一方面将界面平坦化或阶梯化，另一方面将方程简化为在主传播方向上不含一阶导数的偏微分方程- 改进的Helmholtz 方程；（2）构造合适的完美匹配层，即复伸展坐标变换，将在无界周期波导上改进Helmholtz 方程的求解转化为在有界区域上复偏微分方程的求解；（3）设计稳定地实现DtN 的步进计算方法，提高计算结果的正确性和可靠性；（4）研究和构造在数值实施步进方法时所涉及到的复矩阵特征问题的高精度处理和快速、稳定的局部基转换计算公式，实现在主传播方向上大步长步进计算；（5）研究用特征模展开方法得到的方程近似解逼近其精确解的有效性，进而得到波在无界周期波导中的真实传播性态。为光波传播的快速检测和光电器件的优化设计提供可靠的计算方法和实施软件。

本项目给出Helmholtz 方程在无界周期光波导上求解的快速、步进有效数值方法；（1）对带有周期性皱阶界面的光波导，选用了坐标正交变换和方程变换，一方面将界面平坦化或阶梯化，另一方面将方程简化为在主传播方向上不含一阶导数的偏微分方程-修正Helmholtz 方程；（2）构造了合适的完美匹配层，即复伸展坐标变换，将在无界周期波导上修正Helmholtz 方程的求解转化为在有界区域上复偏微分方程的求解；（3）设计了稳定地实现DtN 的步进计算方法，提高了计算结果的正确性和可靠性；（4）给出在数值实施步进方法时所涉及到的复矩阵特征问题的高精度处理和快速、稳定的局部基转换计算公式，实现了在主传播方向上大步长步进计算；（5）研究了用特征模展开方法得到的方程近似解逼近其精确解的有效性，进而得到了波在无界周期波导中的真实传播性态。为光波传播的快速检测和光电器件的优化设计提供可靠的数学处理和算法。