  In a layered and strongly anisotropic superconductor the hybrid modes
provided by the propagation of electromagnetic waves in the matter identify two
well separate energy scales connected to the large in-plane plasma frequency
and to the soft out-of-plane Josephson plasmon. Despite the wide interest in
their detection and manipulation by means of different experimental protocols,
a unified description of plasma waves valid at arbitrary energy and momentum is
still lacking. Here we provide a complete description of generalized plasma
waves in a layered superconductors by taking advantage of their connection to
the gauge-invariant superconducting phase. We show that the anisotropy of the
superfluid response leads to two intertwined hybrid light-matter modes with
mixed longitudinal and transverse character, while a purely longitudinal
plasmon is only recovered for wavevectors larger than the crossover scale set
in by the plasma-frequencies anisotropy. Interestingly, below such scale both
modes appears with equal weight in the physical density response. Our results
open a promising perspective for plasmonic applications made possible by the
next-generation spectroscopic techniques able to combine sub-micron momentum
resolution with THz energy resolution.
