Physical Limits and Optimal Synthesis of Beyond Diagonal Anomalous Scatterers

TL;DR

The paper addresses fundamental limits on anomalous scattering from passive metasurfaces by formulating the problem as a quadratically constrained quadratic program over induced currents under passivity, and solves the dual to obtain tight bounds. It provides explicit synthesis methods for non-local (beyond-diagonal) matching networks and non-local material distributions that achieve these bounds, and extends the analysis to arbitrary design regions. Numerically, it demonstrates a typical $-6\mathrm{\,dB}$ reduction in anomalous bistatic scattering relative to forward directions for representative configurations, and derives asymptotic relations for plane-wave excitation. These results illuminate the trade-offs and costs in RIS and metasurface design, offering analytical insight and practical guidance for achieving targeted non-specular scattering while respecting power conservation and material constraints.

Abstract

Realizing metasurfaces for anomalous scattering is fundamental to designing reflector arrays, reconfigurable intelligent surfaces, and metasurface antennas. However, the basic cost of steering scattering into non-specular directions is not fully understood. This paper derives tight physical bounds on anomalous scattering using antenna array systems equipped with non-local matching networks. The matching networks are explicitly synthesized based on the solutions of the optimization problems that define these bounds. Furthermore, we analyze fundamental limits for metasurface antennas implemented with metallic and dielectric materials exhibiting minimal loss within a finite design region. The results reveal a typical 6dB reduction in bistatic radar cross section (RCS) in anomalous directions compared to the forward direction. Numerical examples complement the theory and illustrate the inherent cost of achieving anomalous scattering relative to forward or specular scattering for canonical configurations.

Figures (9)

• Figure 1: Illustration of an antenna scatterer consisting of $5\times 7$ dual polarized patch elements.
• Figure 2: The antenna in Fig. \ref{['fig:antennascatt']} is contained with in a design rectangular region $\varOmega$ modeled by the complex resistivity $\boldsymbol\rho(\boldsymbol r)$ with side lengths $\ell_\mathrm{x}$ and $\ell_\mathrm{y}$.
• Figure 3: Examples of scattering configurations (side view) analyzed in this paper. (a) single meta surface. (b) multiple layers. (cd) ground plane below the ab) structures. The asymptotic forward scattering is indicated, with $A$ denoting the shadow area. In this paper, rectangular regions with side lengths $\ell_\mathrm{x}=2\ell_\mathrm{y}$ are used as illustrated in Fig. \ref{['fig:antennascattreg']}.
• Figure 4: Illustration of the scattering setup with an illuminating plane wave from in the $\hat{\boldsymbol k}$ direction and maximization of the scattered field in the $\hat{\boldsymbol r}$ direction. The directions $\hat{\boldsymbol k}$ and $\hat{\boldsymbol r}$ are represented in a spherical coordinate system $(\theta,\phi)$ as indicated in the figure. In the presented numerical examples, a rectangular resistive sheet with surface resistance $R_\mathrm{s}=0.01\mathrm{\,\Omega/\square}$ and side lengths $\ell_\mathrm{x}=2\ell_\mathrm{y}$ is used in the configurations depicted in Fig. \ref{['fig:ScattConfig']}.
• Figure 5: Upper bound on the bi-static scattering $\sigma_\mathrm{b}$ in dB for the four configurations in Fig. \ref{['fig:ScattConfig']} for a 2:1 rectangular region with longest side length $\ell_\mathrm{x}=10\lambda$, see Fig. \ref{['fig:scattgeo']}, modeled with surface resistivity $R_\mathrm{s}=0.01\mathrm{\,\Omega/\square}$. Plotted by projecting the upper hemisphere in Fig. \ref{['fig:scattgeo']} on the xy-plane.