= Birefringent walk-off
{title2=$\rho=\phi-\theta$}
The <extraordinary wave>'s <Poynting vector> can differ in direction from its <wavevector>. The latter specifies <wave phase> propagation, while the former specifies <energy flux>. This angular separation is walk-off.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[§2.8.3, printed pp. 157–158, equations 2.414–2.419: Poynting direction and walk-off]{ref}
In the lossless, nondispersive model above, take the <gradient> of $\omega=c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the <group velocity>, parallel to the <Poynting vector>. If its <angle> to the <optic axis> is $\phi$, this gives $\tan\phi=(n_o^2/n_e^2)\tan\theta$. Consequently the <extraordinary wave> generally carries <energy> away from its <wavevector> direction. An internal <wavevector> normal to a slab face therefore does not guarantee a normal energy path. <Birefringent walk-off> vanishes at $\theta=0$ and $\theta=\pi/2$; the latter direction still has unequal <refractive indices>. This is a useful <waveplate> geometry: overlapping components can acquire <phase retardation> without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one <linear polarization> can also hide a double image, for different reasons.
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