Inverse Scattering Problems and Shape Optimization with Respect to Electromagnetic Chirality for Long Tubular Objects

We address two distinct but related problems in the field of time-harmonic electromagnetic scattering from both perfectly conducting and penetrable long tubular objects.
The first problem is the inverse scattering problem of reconstructing an object from knowledge of the corresponding far-field pattern for a single incident field.
The inverse scattering problem can be formulated as a non-linear, ill-posed operator equation, where the operator is the far-field map, which maps the boundary of the scatterer to the far-field pattern of the scattered field exited by one incident field.
The shape of the scatterer is reconstructed using a Gauss--Newton minimization procedure for the regularized relative residual of this equation.
We establish a characterization of the shape derivative of the far-field map for the class of tubular objects under consideration.
The second problem focuses on the shape optimization of scatterers with respect to their electromagnetic chirality properties.
A scatterer is called electromagnetically chiral if the scattering response from fields with one pure helicity cannot be reproduced with fields of the opposite helicity. ... mehrIt is called maximally electromagnetically chiral, if it is invisible to fields of one pure helicity.
A chirality measure usually requires the far-field operator. We present a general Fr\'echet differentiability proof for the far-field operator and establish both a characterization of the derivative for tubular objects and an efficient numerical implementation.
Numerical examples are provided for both the inverse problem and the shape optimization.
The computation of the scattered fields and their domain derivatives is conducted using boundary element methods.