I applied the snell's law to find the angle of refraction of the Ordinary and Extraordinary ray. And I got the correct answer 3.51. But I know my approach to the question is wrong because I applied the snell's law to the extraordinary rays that do not follow the snell's law. But I still don't understand how I got the correct answer. Any guidance on analysing the extraordinary rays in this context would be greatly appreciated.
For a double-refractive birefringent materials like calcite,- Snell's law does apply for the rays of different polarization, like Wikipedia states :
According to Snell's law of refraction, the two angles of refraction are governed by the effective refractive index of each of these two polarizations. This is clearly seen, for instance, in the Wollaston prism which separates incoming light into two linear polarizations using prisms composed of a birefringent material such as calcite.
That's why we can calculate the answer to the problem as being:
$$ \arcsin\left(\frac{n_{air}}{n_e}~\sin(\theta_1)\right) − \arcsin\left(\frac {n_{air}}{n_o}~\sin(\theta_1)\right) $$
Which when evaluated will give about $3.5^\circ$ angle separation between $\text{o/e}$ rays.
