Field sensitivity to L^p variations of a scatterer

TL;DR

This paper investigates how the scattered electromagnetic or acoustic fields from a periodic slab are sensitive to changes in material properties, providing a mathematical framework for derivatives with respect to these properties in L^p norms.

Contribution

It establishes the variational Frechet derivative of the scattered field and transmitted energy with respect to material coefficients in L^p norms, under resonance-free conditions.

Findings

Derivatives are Lipschitz continuous.

Valid for coefficients bounded above and below.

Applicable for 2<p<infinity.

Abstract

For the problem of diffraction of harmonic scalar waves by a lossless periodic slab scatterer, we analyze field sensitivity with respect to the material coefficients of the slab. The governing equation is the Helmholtz equation, which describes acoustic or electromagnetic fields. The main theorem establishes the variational Frechet derivative of the scattered field measured in the H^1 (root-mean-square-gradient) norm as a function of the material coefficients measured in an L^p (p-power integral) norm, with 2<p<infinity, as long as these coefficients are bounded above and below by positive constants and do not admit resonance. The derivative is Lipschitz continuous. We also establish the variational derivative of the transmitted energy with respect to the material coefficients in L^p.