Fresnel Fibers of Skewon-Free Linear Media: Duality, Polarization, and Interface Response
I-AIMS preprint sh-184 · 2026
The Fresnel polynomial of a local linear electromagnetic medium is a homogeneous quartic in the wave covector. For a generic skewon-free medium the constitutive tensor and the axion form a quadratic line complex, and the Fresnel surface is its focal Kummer surface. We use this to determine exactly what the wave surface forgets. The Maxwell symbol at a wave covector is the conic of complex lines through it, so the polarization of a mode is the vertex of a singular conic, and polarization degenerates exactly at the sixteen nodes. For every scalar m the law χₘ=ϵ(χ−mϵ)⁻¹ϵ satisfies ℱ_(χₘ)=det(χ−mϵ)⁻¹ℱᵪ, and for generic χ these laws, together with coordinate change, scale, and axion, exhaust the media with a projectively equivalent wave surface. For a biaxial dielectric the members are explicit magnetoelectric media D=E−mH, B=−mE+(ε+m²)H with positive energy for every real m; their coupling is of Tellegen type, skewon-free but not Lorentz reciprocal. The duality carries plane waves of the crystal to plane waves of these media with the same wave vector and Poynting vector, so they have the transmitted modes of the crystal at every planar interface. Their reflection differs: at a principal interface with vacuum the duality acts on the surface impedance by Z↦ mI−Z⁻¹, and the reflection matrix has cross-polarized entry 2m/(m²+c), where c>0 depends only on the crystal and the incidence; the crystal itself has none. Wave-surface invariants are complete for incidence and incomplete for interface response; in fixed laboratory units the missing coordinates are the duality parameter m, the constitutive scale, and the axion.
MSC classes: Primary 78A25; Secondary 14J28, 78A05