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The perfectly matched layers method is a well known truncation technique forits efficiency and convenience in numerical implementations of wave scatteringproblems in unbounded domains. In this paper, we study the convergence of theperfectly matched layers (PML) for wave scattering from a local perturbation ofan open waveguide in the half space above the real line, where the refractiveindex is a function which is periodic along the axis of the waveguide andequals to one above a finite height. The problem is challenging due to theexistence of guided waves, and a typical way to deal with the difficulty is toapply the limiting absorption principle. Based on the Floquet-Bloch transformand a curve deformation theory, the solution from the limiting absorptionprinciple is rewritten as the integral of a coupled family of quasi-periodicproblems with respect to the quasi-periodicity parameter on a particularlydesigned curve. By comparing the Dirichlet-to-Neumann maps on a straight lineabove the locally perturbed periodic layer, we finally show that the PML methodconverges exponentially with respect to the PML parameter. Finally, thenumerical examples are shown to illustrate the theoretical results.

The PML-Method for a Scattering Problem for a Local Perturbation of an Open Periodic Waveguide