Ruming Zhang

A Convergent Numerical Scheme for Scattering of Aperiodic Waves from Periodic Surfaces Based on the Floquet--Bloch Transform

Periodic surface structures are nowadays standard building blocks of optical devices. If such structures are illuminated by aperiodic time-harmonic incident waves as, e.g., Gaussian beams, the resulting surface scattering problem must be formulated in an unbounded layer including the periodic surface structure. An obvious recipe to avoid the need to discretize this problem in an unbounded domain is to set up an equivalent system of quasiperiodic scattering problems in a single (bounded) periodicity cell via the Floquet-Bloch transform. The solution to the original surface scattering problem then equals the inverse Floquet-Bloch transform applied to the family of solutions to the quasiperiodic problems, which simply requires to integrate these solutions in the quasiperiodicity parameter. A numerical scheme derived from this representation hence completely avoids the need to tackle differential equations on unbounded domains. In this paper, we provide rigorous convergence analysis and error bounds for such a scheme when applied to a two-dimensional model problem, relying upon a quadrature-based approximation to the inverse Floquet-Bloch transform and finite element approximations to quasiperiodic scattering problems. Our analysis essentially relies upon regularity results for the family of solutions to the quasiperiodic scattering problems in suitable mixed Sobolev spaces. We illustrate our error bounds as well as efficiency of the numerical scheme via several numerical examples.

A Floquet--Bloch Transform Based Numerical Method for Scattering from Locally Perturbed Periodic Surfaces

Scattering problems for periodic structures have been studied a lot in the past few years. A main idea for numerical solution methods is to reduce such problems to one periodicity cell. In contrast to periodic settings, scattering from locally perturbed periodic surfaces is way more challenging. In this paper, we introduce and analyze a new numerical method to simulate scattering from locally perturbed periodic structures based on the Bloch transform. As this transform is applied only in periodic domains, we first rewrite the scattering problem artificially in a periodic domain. With the help of the Bloch transform, we next transform this problem into a coupled family of quasi-periodic problems posed in the periodicity cell. A numerical scheme then approximates the family of quasi-periodic solutions (we rely on the finite element method) and backtransformation provides the solution to the original scattering problem. In this paper, we give convergence analysis and error bounds for a Galerkin discretizatio...

Non-periodic acoustic and electromagnetic, scattering from periodic structures in 3D

Abstract Scattering of non-periodic waves from unbounded structures is difficult to treat, as one typically formulates the problem in an unbounded domain covering the unbounded periodic structure. The Floquet–Bloch transform reduces the latter problem to a family of decoupled periodic scattering problems. This reduction is in particular interesting from the point of view of numerical computations. We analyze a corresponding fully discrete numerical solution algorithm for three-dimensional scattering problems in acoustics and electromagnetics, proving in particular convergence rates under suitable assumptions on the geometry and the material coefficients. A crucial part of our analysis relies on the continuous dependence of the family of solutions to the quasiperiodic scattering problems on the quasiperiodicity. The latter part is actually more difficult to establish than for corresponding two-dimensional problems. We further provide a numerical example in 3D acoustics that illustrates feasibility of the proposed algorithm.

Near-field imaging of periodic inhomogeneous media

This paper is concerned with the inverse electromagnetic scattering problem by a periodic structure in the two-dimensional transverse electric (TE) polarization case. The structure is assumed to separate the whole space into three parts: the medium above and below the structure is assumed to be homogeneous, and the medium inside the structure is assumed to be inhomogeneous with the refractive index. The inverse problem recovers the refractive index from knowledge of the scattered field measured on a straight line above and below the periodic structure, corresponding to several incident plane waves with different incident angles. This inverse problem is reformulated as a regularized optimization problem, and a quasi-Newton method is developed to numerically solve this optimization problem. To overcome the severe ill-posedness of the inverse problem and to get high-resolution imaging of the refractive index, we utilize a continuation technique with multiple wave numbers, taking the Born approximation as the initial guess. In each iteration step, a fast integral equation method is used to solve the direct problem. Several numerical examples are presented to show the effectiveness of the inversion algorithm.

The numerical solution of scattering by infinite rough interfaces based on the integral equation method

In this paper, we describe a Nystrom integration method for the integral operator T which is the normal derivative of the double-layer potential arising in problems of two-dimensional acoustic scattering by infinite rough interfaces. The hypersingular kernel and unbounded integral interval of T are the key difficulties. By using a mollifier, we separately deal with these two difficulties and propose its Nystrom integration method. Furthermore, we establish convergence of the method. Finally, we apply the method to the scattering problem by infinite rough interfaces and carry out some numerical experiments to show the validity.

Reconstruction of Local Perturbations in Periodic Surfaces

This paper concerns the inverse scattering problem to reconstruct a local perturbation in a periodic structure. Unlike the periodic problems, the periodicity for the scattered field no longer holds, thus classical methods, which reduce quasi-periodic fields in one periodic cell, are no longer available. Based on the Floquet-Bloch transform, a numerical method has been developed to solve the direct problem, that leads to a possibility to design an algorithm for the inverse problem. The numerical method introduced in this paper contains two steps. The first step is initialization, that is to locate the support of the perturbation by a simple method. This step reduces the inverse problem in an infinite domain into one periodic cell. The second step is to apply Newton-CG method to solve the associated optimization problem. The perturbation is then approximated by a finite spline basis. Numerical examples are given at the end of this paper, shows the efficiency of the numerical method.

A sampling method for the reconstruction of a periodic interface in a layered medium

In this paper, we consider the inverse problem of reconstructing periodic interfaces in a two-layered medium with TM-mode. We propose a sampling-type method to recover the top periodic interface from the near-field data measured on a straight line above the total structure. Finally, numerical experiments are illustrated to show the effectiveness of the method.