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 a n d 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 a n d r _pp) have corrections which are second order in the angle of incidence. The cross-reflection amplitudes r _sp a n d 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.