Direct and inverse electromagnetic scattering problems for bi-anisotropic periodic media

Abstract

This work focuses on electromagnetic direct and inverse scattering problems for bi-anisotropic periodic media in three dimensions. The scattering phenomenon of interest occurs when electromagnetic waves interact with a bi-anisotropic periodic scattering object and is governed by the time-harmonic Maxwell’s equations with Rayleigh radiation condition. Bi-anisotropy means the object is characterized by three quantities: permittivity, permeability and magnetoelectric coupling coefficient. These are 3 × 3 matrix-valued functions of the spatial variable and make up two constitutive relations in addition to the Maxwell’s equations. In the equations, the incident fields which we send towards the object are known. For the direct problem, we also know the three quantities characterizing the object, and the goal is to solve for the scattered fields. The inverse problem, on the other hand, aims to find the shape of the scattering object from the measurements of the scattered fields on two planes, one above and one below the object. Both problems have many applications in real life, especially in the study of photonic crystals. Regarding the direct problem, we formulate the Maxwell’s equations and constitutive relations with Rayleigh radiation condition into an equivalent integro-differential equation. We then show, under certain constraints on the material parameters of the scattering object, existence and uniqueness of solution. To numerically solve the problem, we utilize a periodization technique and fast Fourier Transform to switch to solving for the Fourier coefficients of the unknown. This helps avoid direct computation of the Green’s function which has singularities. Then, we analyze convergence of a Galerkin scheme and implement it to obtain a numerical solution to the problem. The results regarding the direct problem are taken from the published paper [22] of myself and my advisor. For the inverse problem, we rigorously justify the Factorization method to reconstruct the shape of the scattering medium from scattered field data. To be precise, we prove a unique determination of the geometry of the scattering object. This determination can also be easily implemented numerically, thus, gives a fast imaging algorithm to find the shape of the scattering object. We also provide numerical results to show how the method performs with different types of bi-anisotropic periodic objects, from simple to more complex geometries. The results regarding the inverse problem are taken from the published paper [23] of myself and my advisor.