Geometric Mechanisms of Radiation for Periodic Elliptic Operators

Far-field radiation in periodic media is governed by the geometry of the Fermi surface, yet the geometric mechanisms underlying this relationship remain poorly understood. Existing Floquet-Bloch representations separate propagating and regular spectral contributions, but this analytic decomposition does not distinguish the geometric mechanisms underlying radiation.
We show that outgoing limiting absorption solutions naturally decompose into three geometrically distinct components: evanescent, non-grazing, and grazing. The non-grazing and grazing components are associated with different geometric structures on the Fermi surface and exhibit different asymptotic regimes, under natural non-degeneracy assumptions on the underlying Fermi geometry.
To reveal this structure, we develop a sliced spectral analysis (SSA), in which the observation direction is incorporated into the spectral representation, reducing the Floquet-Bloch integral to a family of one-dimensional analytic problems. This reduction restores the analytic structure required for contour deformation and leads naturally to the geometric decomposition of radiation.
We establish the far-field asymptotics of each component and show that the grazing contribution forms an independent radiation mechanism, capable of becoming the leading-order far-field term. Consequently, the classical propagating--regular decomposition mixes geometrically distinct radiation mechanisms, whereas the present decomposition separates them according to their underlying spectral geometry.