Optical resonances in reflectivity due to crystal modes with spatial dispersion

Abstract

We study the effect of spatial dispersion of crystal modes on optical properties such as the reflectivity R. As an example for isotropic media, we investigate the simplest model for phonons in ionic crystals and compare with previous results for highly anisotropic plasmons, which are now understood from a more general point of view. As a consequence of the wave vector dependence of the dielectric function small changes in the lineshape are predicted. Beyond that, if the frequency of minimal R is near a pole of the dispersionless dielectric function, the relative amplitude of dips in R with normal and anomalous dispersion differ significantly, if dissipation and disorder are low.Usually, in crystal optics the dependence of the dielectric function on the wave vector k is neglected because the wave length λ is much larger than interatomic length scales s and effects of spatial dispersion are expected to be of the order of s/λ ≪ 1, see Refs. 1,2 and references therein. Recently, it was shown for the reflectivity near the Josephson plasma resonance (JPR) in highly anisotropic superconductors 3,4 , which is an interlayer charge oscillation of Cooper pairs, that spatial dispersion provides a novel way to stop light and can have effects of order unity on the relative amplitude of plasma modes with different dispersion. This system is exceptional with respect to the symmetry (high anisotropy), the fact that light is screened away from the JPR, and the type of the excitation (plasmon). Here we discuss the possibility that such effects appear under more general conditions. As a generic example of a spatially dispersive resonance we choose the simplest model of one optical phonon band in an ionic crystal 5 (also describing certain excitons 1 ) and study the optical properties of the isotropic phonon-polariton, which is the lattice vibration coupled to light. Thereby, we determine under what circumstances similar effects can be expected in a large class of systems and clarify the origin of previous results for the highly anisotropic plasmon obtained in Ref. 3. We show that the effect of spatial dispersion is of order unity, if the resonance in reflectivity is in a frequency region of size ∆ω/ω 0 = s/λ near the pole ω 0 of the dispersionless dielectric function ǫ(ω); the amplitudes in reflectivity of phonon bands with normal and anomalous dispersion then differ significantly.Weak disorder and small dissipation turn out to be essential for the observation of significant effects due to spatial dispersion, which is the case in the plasmon system of the JPR below the superconducting gap. In insulators, e.g. in ionic crystals with perfect lattice symmetry and weak disorder, such favorable conditions might be realized as well, while in metals similar effects would be strongly overdamped.In the following, we mainly consider an electromagnetic wave of frequency ω in normal incidence, with a wave vector k = (0, 0, k z ) at the surface z = 0 of a semiinfinite dielectric medium. The macroscopic Maxwell equations describing the electroma...

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