Perfectly matched layers in photonics computations: 1D and 2D nonlinear coupled mode equations

Extending the general approach for first-order hyperbolic systems developed in [D. Appelö, T. Hagstrom, G. Kreiss, Perfectly matched layers for hyperbolic systems: general formulation, well-posedness and stability, SIAM J. Appl. Math., 2006, to appear], we construct PML equations for the mixed-type system governing propagation of optical wave packets in both 1D and 2D Bragg resonant photonic waveguides with a cubic nonlinearity, i.e. the coupled mode equations. We prove that in the linear case the layer equations are absorbing and perfectly matched. We also prove they are stable for constant parameters. A number of numerical experiments are performed to assess the layer’s performance in both the linear and nonlinear regimes.

扩展在[D. Appelö, T. Hagstrom, G. Kreiss，《双曲型系统的完全匹配层：一般公式、适定性和稳定性》，SIAM J. Appl. Math.，2006年，即将发表]中所发展的一阶双曲型系统的一般方法，我们针对在具有三次非线性的一维和二维布拉格共振光子波导中光波包传播的混合型系统（即耦合模方程）构建了完全匹配层（PML）方程。我们证明在线性情形下，层方程是吸收性的且完全匹配。我们还证明对于常数参数它们是稳定的。进行了大量数值实验以评估该层在线性和非线性两种情形下的性能。