In a layered and strongly anisotropic superconductor, the hybrid modes provided by the propagation of electromagnetic waves in the matter identify two well-separated energy scales connected to the large in-plane plasma frequency and to the soft out-of-plane Josephson plasmon.Despite wide interest in their detection and manipulation by means of different experimental protocols, the physical ingredients underlying a unified description of plasma waves valid at arbitrary energy and momentum are still poorly understood.Here we provide a complete description of generalized plasma waves in layered superconductors in terms of the Work Boots gauge-invariant superconducting phase by including both the Coulomb interaction and the relativistic effects.We show that the anisotropy of the superfluid response SEA VEGETABLES PURE GOLD leads to two intertwined hybrid light-matter modes with mixed longitudinal and transverse character, while a purely longitudinal plasmon is only recovered for wave vectors larger than the crossover scale set in by the plasma-frequency anisotropy.
Interestingly, below such scale both modes appear with equal weight in the physical density response.Our results open a promising perspective for plasmonic applications made possible by next-generation spectroscopic techniques able to combine submicron momentum resolution with THz energy resolution.