Target Signatures for Anisotropic Screens in Electromagnetic Scattering

TL;DR

The paper addresses detecting changes in anisotropic ultra-thin screens (metasurfaces) in electromagnetic scattering by formulating a Maxwell forward problem with a surface tensor $\Sigma$ and proving well-posedness under strengthened passivity. It then introduces a target-signature framework where a modified far-field operator $\mathcal{F}=F-F^{(\lambda)}$ exposes interior $\Sigma$-Steklov eigenvalues as indicators of screen properties, and proves a uniqueness result for the inverse problem. The key contribution is linking these interior eigenvalues to observable far-field data and providing both a variational eigenproblem and practical numerical methods (FEM, sampling/Tikhonov) to compute them from scattering data. This approach offers a non-invasive means to monitor isotropic or anisotropic thin films from far-field measurements, with preliminary numerical demonstrations for isotropic and anisotropic cases and clearly identified avenues for extending to complex-valued $\Sigma$.

Abstract

Anisotropic thin sheets of materials possess intriguing properties because of their ability to modify the phase, amplitude and polarization of incident waves. Such sheets are usually modeled by imposing transmission conditions of resistive or conductive type on a surface called a screen. We start by analyzing this model, and show that the standard passivity conditions can be slightly strengthened to provide conditions under which the forward scattering problem has a unique solution. We then turn to the inverse problem and suggest a target signature for monitoring such films. The target signature is based on a modified far field equation obtained by subtracting an artificial far field operator for scattering by a closed surface containing the thin sheet and parametrized by an artificial impedance. We show that this impedance gives rise to an interior eigenvalue problem, and these eigenvalues can be determined from the far field pattern, so functioning as target signatures. We prove uniqueness for the inverse problem, and give preliminary numerical examples illustrating our theory.

Figures (4)

• Figure 1: Target signatures for the full unit sphere at $\kappa=1.9$ and $\Sigma=(0.5i)I$. We show results computed from the far field pattern as the curve of the average norm of ${\bf g}$ against the auxiliary parameter $\eta$. We also show the first three $\Sigma$-Steklov eigenvaues marked as $*$. Peaks of the avergae norm of ${\bf g}$ correspond well to $\Sigma$-Steklov eigenvalues.
• Figure 2: A contour map of the real part of the third component of the scattered electric field in the plane $z=0$. Creeping waves along the screen are clearly visible. These waves have a shorter wavelength than the field in the bulk, so imposing an additional computational burden on the forward solver.
• Figure 3: Predicted target signatures and computed $\Sigma$-Steklov eigenvalues for the hemisphere at $\kappa=1.9$. Left: scalar $\Sigma=0.5iI$. Right: anisotropic $\Sigma$ with $\sigma_1=0.5$ and $\sigma_3=0.4$. In each panel the curve shows the average of the norm of ${\bf g}$ as the parameter $\lambda$ varies, and the $*$ mark eigenvalues computed by FEM.
• Figure 4: Results of changing parameters in an anisotropic choice of $\Sigma$ for the hemispherical screen. We show changes in the smallest (in magnitude) target signatures as the parameters defining $\Sigma$ given by (\ref{['Sig_proj']})) vary. Left panel: we set $\sigma_{3,3}=0.5$ and vary $\sigma_{1,1}$. Right panel: we set $\sigma_{1,1}=0.5$ and vary $\sigma_{3,3}$. Eigenvalues for different parameter values are shown as $*$.

Theorems & Definitions (15)

• Lemma 1
• Theorem 1
• Definition 1: Inverse Problem
• Theorem 2
• Definition 2: $\Sigma$-Steklov Eigenvalues
• Theorem 3
• Lemma 2
• Theorem 4
• Proposition 1
• Lemma 3