On an inverse boundary value problem for a nonlinear time harmonic Maxwell system

TL;DR

This paper investigates an inverse boundary value problem for a class of nonlinear time harmonic Maxwell systems, demonstrating that boundary measurements can determine material properties and nonlinear terms in nonlinear optics models.

Contribution

It establishes uniqueness results for recovering permittivity, permeability, and nonlinear functions from boundary data in nonlinear Maxwell systems.

Findings

Boundary measurements determine material parameters and nonlinear terms.

Unique recovery of permittivity and permeability functions.

Applicable to nonlinear optics models.

Abstract

This paper considers a class of nonlinear time harmonic Maxwell systems at fixed frequency, with nonlinear terms taking the form $\mathscr{X}(x,|\vec E(x)|^2)\vec E(x)$, $\mathscr{Y}(x,|\vec H(x)|^2)\vec H(x)$, such that $\mathscr{X}(x,s)$, $\mathscr{Y}(x,s)$ are both real analytic in $s$. Such nonlinear terms appear in nonlinear optics theoretical models. Under certain regularity conditions, it can be shown that boundary measurements of tangent components of the electric and magnetic fields determine the electric permittivity and magnetic permeability functions as well as the form of the nonlinear terms.