The perfectly matched layers method is a well known truncation technique for its efficiency and convenience in numerical implementations of wave scattering problems in unbounded domains. In this paper, we study the convergence of the perfectly matched layers (PML) for wave scattering from a local perturbation of an open waveguide in the half space above the real line, where the refractive index is a function which is periodic along the axis of the waveguide and equals to one above a finite height. The problem is challenging due to the existence of guided waves, and a typical way to deal with the difficulty is to apply the limiting absorption principle. Based on the Floquet-Bloch transform and a curve deformation theory, the solution from the limiting absorption principle is rewritten as the integral of a coupled family of quasi-periodic problems with respect to the quasi-periodicity parameter on a particularly designed curve. By comparing the Dirichlet-to-Neumann maps on a straight line above the locally perturbed periodic layer, we finally show that the PML method converges exponentially with respect to the PML parameter. Finally, the numerical examples are shown to illustrate the theoretical results.

翻译：摘要：完美匹配层方法因其在无界域波散射问题数值实现中的高效性和便捷性而成为一种广为人知的截断技术。本文研究了完美匹配层（PML）在实线上方半空间中开放波导局部扰动引起的波散射问题中的收敛性，其中折射率是沿波导轴周期变化的函数，且在有限高度之上等于1。由于导波的存在，该问题具有挑战性，处理该难题的典型方法是应用极限吸收原理。基于Floquet-Bloch变换和曲线变形理论，将极限吸收原理的解重新表示为在特定设计曲线上关于拟周期参数对耦合拟周期问题族的积分。通过比较局部扰动周期层上方直线上的Dirichlet-to-Neumann映射，最终证明PML方法关于PML参数呈指数收敛。最后，通过数值算例验证理论结果。