We construct a regular isotropic approximate cloak for the Maxwell system of equations. The method of transformation optics has enabled the design of electromagnetic parameters that cloak a region from external observation. However, these constructions are singular and anisotropic, making practical implementation difficult. Thus, regular approximations to these cloaks have been constructed that cloak a given region to any desired degree of accuracy. In this paper, we show how to construct isotropic approximations to these regularized cloaks using homogenization techniques, so that one obtains cloaking of arbitrary accuracy with regular and isotropic parameters.Though the regularized approximate cloak is nonsingular, its material parameters are still anisotropic, which poses another difficulty in practical implementation. Therefore, it would be useful if, as a second step these regularized approximate cloaks were further approximated by regular isotropic cloaks. It is a well-known phenomenon in homogenization theory [1,45,7] that homogenization of isotropic material parameters can lead, in the small scale limit, to anisotropic parameters.This approach, called isotropic transformation optics was introduced and performed in the case of electrostatic, acoustic and quantum cloaking by Greenleaf, Kurylev, Lassas and Uhlmann in [17] and [19]. It was shown in [19] that the for a fixed Dirichlet boundary value, the Neumann data for the approximate cloaking construction converge strongly. Moreover, Faraco, Kurylev and Ruiz [11] showed that the same construction in fact gives us strong convergence of the Dirichlet-to-Neumann maps in the operator norm. More recently, it has been implemented in a quasilinear model [12].In this paper, we will construct an approximate isotropic cloak for Maxwell's equations using similar homogenization techniques. We begin by summarizing the main results of the existence-uniqueness and cloaking theory for the Maxwell equations in Section 2. In Section 3, we present a brief overview of homogenization and present a proof of the homogenization result for the Maxwell equations, under the assumption that both ε(x) and µ(x) have positive imaginary parts. Section 4 will be devoted to an explicit construction of an approximate isotropic cloak using inverse homogenization techniques as in [17]. Since the magnetic permeability of the anisotropic approximate cloak does not have an imaginary part, we need to go through a two step process. In the first step, we alter permittivities and permeabilities by a small positive parameter δ so that the assumptions of Section 3 are satisfied, and then construct isotropic parameters ε n δ and µ n that homogenize to the altered paremetrs. In the second step, we let δ → 0. Finally in Section 5, we will prove our Main Theorem: that as we first let n → ∞ and then δ → 0, the electromagnetic boundary measurements corresponding to our approximate cloaking construction converge strongly to the boundary measurements in empty space. The order in which these limits are taken can...
