Direct and inverse time-harmonic scattering by Dirichlet periodic curves with local perturbations

Abstract

This is a continuation of the authors' previous work (A. Kirsch, Math. Meth. Appl. Sci., 45 (2022): 5737-5773.) on well-posedness of time-harmonic scattering by locally perturbed periodic curves of Dirichlet kind. The scattering interface is supposed to be given by a non-self-intersecting Lipschitz curve. We study properties of the Green's function and prove new well-posedness results for scattering of plane waves at a propagative wave number. In such a case there exist guided waves to the unperturbed problem, which are also known as Bounded States in the Continuity (BICs) in physics. In this paper uniqueness of the forward scattering follows from an orthogonal constraint condition enforcing on the total field to the unperturbed scattering problem. This constraint condition, which is also valid under the Neumann boundary condition, is derived from the singular perturbation arguments and also from the approach of approximating a plane wave by point source waves. For the inverse problem of determining the defect, we prove several uniqueness results using a finite or infinite number of point source and plane waves, depending on whether a priori information on the size and height of the defect is available.

Figures (3)

• Figure 1: Illustration of wave scattering from a perfectly reflecting periodic curve with a local perturbation in $(0, 2\pi)$. The red area denotes the perturbed domain. The scattering interface is supposed to be a non-self-intersecting curve.
• Figure 2: Illustration of the gap domain $D^*\subset \tilde{D}_1\backslash\overline{\tilde{D}}_2$ between two local perturbations. Here $\tilde{\Gamma}_1=\Gamma$ is identical with the unperturbed grating curve.
• Figure 3: Illustration of $\Gamma$ and its local perturbation $\tilde{\Gamma}$ which generate identical wave fields with $\theta=0$ and $k=2$. Here $D^*=D\backslash\overline{\tilde{D}}$ represents the difference domain between $D$ and $\tilde{D}$.

Theorems & Definitions (45)

• Definition 2.1
• Remark 2.2
• Definition 2.5
• Definition 2.7
• Definition 2.8
• Lemma 2.9
• Proposition 2.10: Well-posedness for point source waves
• Proposition 2.11: Well-posedness for plane waves
• Theorem 3.1
• Remark 3.2