Abstract
In the current paper we consider an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium. We show the unique determination of the electromagnetic material parameters and the nonlinear susceptibility parameters of the medium by making electromagnetic measurements on the boundary. We are interested in the case of the time-harmonic Maxwell equations.
Highlights
Let (M, g) be a compact 3-dimensional Riemannian manifold with smooth boundary and let E(·, t) and H(·, t) be the time-dependent 1-forms on M representing electric and magnetic fields
The electric and magnetic fields E and H are said to be time-harmonic with frequency ω > 0 if
MNL(x, H(x, t)) = χm(x, |H|2g)H(x, t), where χm is the scalar susceptibilities depending only on the time-average of the intensity of H. Such nonlinear magnetizations appear in the study of metamaterials built by combining an array of wires and split-ring resonators embedded into a Kerr-type dielectric [32]
Summary
Let (M, g) be a compact 3-dimensional Riemannian manifold with smooth boundary and let E(·, t) and H(·, t) be the time-dependent 1-forms on M representing electric and magnetic fields. MNL(x, H(x, t)) = χm(x, |H|2g)H(x, t), where χm is the scalar susceptibilities depending only on the time-average of the intensity of H Such nonlinear magnetizations appear in the study of metamaterials built by combining an array of wires and split-ring resonators embedded into a Kerr-type dielectric [32]. These metamaterials have complicated form of nonlinear magnetization. If the intensity |H|2g is sufficiently small, relatively to the resonant frequency, the nonlinear magnetization can be assumed to be of the. This assumption has successful numerical implementation [17]. The complex functions μ and ε represent the material parameters (permettivity and permeability, respectively)
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