Question #43193

A monochromatic light beam propagating in vacuum has a wave vector $\vec{k}_i$ (units: $\text{rad}\cdot\text{m}^{-1}$) and is incident on the flat surface of a uniaxial birefringent crystal. The unit normal vector to the surface, pointing into the crystal, is $\hat{n}$. The quantity $\vec{b} = \vec{k}_i \times \hat{n}$ acts as a conserved optical impact parameter across the interface. Inside the crystal, the light splits into an ordinary wave and an extraordinary wave. For the extraordinary wave, let $\vec{k}_e$ be its wave vector and $\vec{S}_e$ be its Poynting vector (units: $\text{W}\cdot\text{m}^{-2}$). The crystal's optic axis $\hat{c}$ is oriented such that it is neither parallel nor perpendicular to the plane of incidence. Which of the following vector relationships is strictly valid for the extraordinary wave at the interface?