Bounds on the Uniaxial Effective Complex Permittivity of Two-phase Composites and Optimal or Near Optimal Microstructures

Electromagnetic materials with a uniaxial effective permittivity tensor, characterized by its transverse ($\epsilon_\perp$) and axial ($\epsilon_\parallel$) components, play a central role in the design of advanced photonic and electromagnetic materials including hyperbolic metamaterials, and biological imaging platforms. Tight bounds on the complex effective permittivity of such metamaterials are critical for predicting and optimizing their macroscopic electromagnetic response. While rigorous tight bounds exist for isotropic two-phase composites, corresponding results for uniaxial composites remain relatively unexplored. In this work, we systematically investigate the attainable range of $\epsilon_\perp$ and $\epsilon_\parallel$ in the quasistatic regime for two-phase metamaterials with isotropic homogeneous phases. By analyzing known microgeometries and constructing hierarchical laminates (HLs), we demonstrate that the classical bounds on $\epsilon_\perp$ are not optimal. We conjecture improved bounds based on numerically fitted circular arcs derived from convex hulls of $\epsilon_\perp$ values obtained from HLs, and we identify optimal rank-4 HL structures that achieve all points on the conjectured bounds. Additionally, we quantify the correlation between $\epsilon_\perp$ and $\epsilon_\parallel$ for fixed volume fractions, and propose a design algorithm to construct HL microstructures achieving prescribed values of $\epsilon_\perp$. Leveraging the Cherkaev-Gibiansky transformation and the translation method, we extend recent techniques developed for isotropic composites by Kern-Miller-Milton to derive translation bounds on the uniaxial complex effective permittivity tensor. Finally, bounds on the sensitivity of the effective permittivity tensor of low-loss composites are obtained and their optimality is shown in two-dimensions.