  Electromagnetic waves propagating in a layered superconductor with arbitrary
momentum with respect to the main crystallographic directions display an
unavoidable mixing between longitudinal and transverse degrees of freedom. Here
we show that this basic physical mechanism explains the emergence of a
well-defined absorption peak in the in-plane optical conductivity for light
propagating at small tilting angles with respect to the stacking direction in
layered cuprates. More specifically, we show that this peak, often interpreted
as a spurious leakage of the $c$-axis Josephson plasmon, is instead a signature
of the true longitudinal plasma mode occurring at larger momenta. By combining
a classical approach based on Maxwell's equations with a full quantum
derivation of the plasma modes based on the modelling of the superconducting
phase degrees of freedom, we provide an analytical expression for the
absorption peak as a function of the tilting angle and light polarization. We
suggest that an all-optical measurement in tilted geometry can be used as an
alternative way to access plasma-wave dispersion, usually measured by means of
large-momenta scattering techniques like resonant inelastic X-ray scattering
(RIXS) or electron energy loss spectroscopy (EELS).
