Generic Recovery of Permittivity and Permeability in Anisotropic Maxwell Systems

Abstract

We study the inverse problem of recovering the constitutive tensors of a homogeneous anisotropic electromagnetic medium without magnetoelectric coupling (non-chiral) from its Fresnel surface, the characteristic variety of Maxwell's equations governing electromagnetic wave propagation. For known isotropic permeability, normalized to $\mu =I$, we prove that the Fresnel surface uniquely determines the permittivity tensor $\epsilon$, and that the associated Fresnel polynomial is reducible precisely when $\epsilon$ has a repeated eigenvalue. For general, positive-definite symmetric tensors $(\epsilon ,\mu )$, we prove that the Fresnel polynomial is generically irreducible over $\mathbb{C}$ and we identify the natural gauge symmetry under which it is invariant. Using geometric invariant theory, a powerful tool of modern algebraic geometry, we construct an affine quotient of the parameter space by this gauge action and prove that the induced Fresnel-polynomial map is birational onto its image. We deduce that, outside a proper real algebraic exceptional set, the real Fresnel surface determines $(\epsilon ,\mu )$ up to gauge. This establishes generic uniqueness for the inverse problem and, to our knowledge, provides a new application of affine geometric invariant theory to gauge freedom in a PDE inverse problem.

23 pages, 2 magma scripts supporting Examples 6.2 and 7.2 included as ancillary files