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

TL;DR

This work provides a rigorous framework for the PML truncation of scattering by a local perturbation of an open, $2\pi$-periodic waveguide in the half-plane. By combining the curve-deformation technique with the Floquet-Bloch transform, the open problem is reformulated as a coupled family of quasi-periodic problems on a designed contour, and the limiting absorption principle selects the physical solution. The authors establish exponential convergence of the PML truncation by comparing Dirichlet-to-Neumann maps along the deformed contour, and they demonstrate the theory with a hybrid spectral-FD numerical scheme and a concrete example. The results offer a principled, implementable approach for accurate simulations of open periodic waveguides with local defects, including guidance on selecting PML parameters and discretization strategies.

Abstract

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.

Figures (4)

• Figure 1: The set $\mathcal{A}$ (red), $\hat{\alpha}_j$ for $j\in J^+$ (green), $\hat{\alpha}_j$ for $j\in J^-$ (blue)
• Figure 2: The real part of the mode $\phi^+$ (left) and the curve $\Gamma$ (right)
• Figure 3: Real parts of $u^{(1)}$ (i.e. Born), $u^{(2)}$, $u^{(5)}$, and $u^{(9)}$.
• Figure 4: Error as a function of $\rho$ between the 4th iterates of the PML-method and the D2N-boundary condition

Theorems & Definitions (18)

• Definition 2.1
• Lemma 2.2
• Lemma 2.4
• Theorem 3.1
• Corollary 3.2
• Theorem 3.4
• Definition 3.5
• Lemma 3.7
• Theorem 3.8
• Lemma 4.1