Propagation-based classification of linear magnetoelectric response in dielectrics

We study electromagnetic wave propagation in homogeneous dielectrics endowed with a linear magnetoelectric (ME) response in the geometric-optics regime. Assuming isotropic permittivity and permeability while keeping a generic $3\times 3$ ME matrix $\alpha_{ij}$, we derive the eikonal (Fresnel) eigenvalue problem for the polarization vector and obtain a compact quartic dispersion relation for the normalized phase speed $r=v/v_d$, where $v_d=(\mu\varepsilon)^{-1/2}$ is the phase speed of the underlying dielectric. We then classify the propagation effects of $\alpha_{ij}$ by decomposing it into trace, symmetric-traceless, and antisymmetric sectors. We show that (i) the pure-trace sector is propagation-silent at leading geometric-optics order; (ii) the antisymmetric sector yields a factorized quartic and produces two branches with closed-form phase speeds, including regimes where $|v|>v_d$; and (iii) the symmetric-traceless sector encodes the richest directional dependence through algebraic invariants that control the Fresnel wave surface and polarization mixing. Finally, we discuss how the predicted phase-speed shifts can be accessed by phase-sensitive transmission and resonant techniques, and we outline numerical workflows to validate the analytic dispersion and map polarization signatures in bulk and finite geometries.