Reflection from uniaxial crystals: symmetries, and formulae for near-normal incidence

The following symmetries and interrelationships are established: the direct reflection amplitudes r ss , r pp are independent of the signs of the direction cosines of the optic axis. For example, they are unchanged by ϕ → π − ϕ or ϕ →− ϕ , where ϕ is the azimuthal angle of the optic axis. The cross-polarization amplitudes r sp and r ps are both odd in ϕ ; they also satisfy the general relations r sp ( ϕ )= r ps ( π + ϕ ) and r sp ( ϕ )+ r ps ( π − ϕ )=0. All of these symmetries apply equally to absorbing media with complex refractive indices, and thus complex reflection amplitudes. Analytic expressions are given for the amplitudes which characterize the reflection from a uniaxial crystal when the incidence is close to normal. The amplitudes for reflection in which the polarization is unchanged ( r ss and r pp ) have corrections which are second order in the angle of incidence. The cross-reflection amplitudes r sp and r ps are equal at normal incidence and have corrections (equal and opposite) which are first order in the angle of incidence. Examples for normal incidence and small-angle (6°) and large-angle (60°) incidence reflection are given for non-absorbing calcite and absorbing selenium.