I understand that ordinary ray (O-ray) and extra-ordinary ray (e-ray) have different refractive indices.

This should mean O-ray and e-ray move with different velocities in a substance and they should refract at different angle according to Snell's law. If the angle of incidence is zero, the angle of refraction should also be zero, i.e a ray would move undeviated if the angle of incidence is zero. However, I have seen that e-ray bends even if the angle of incidence is zero (which means e-rays don't obey Snell's law).

Why this is the case?


The distinction between "ordinary" and "extraordinary" rays arises when you have a birefringent material, in which light with different polarizations has different speed.

Snell's Law, like many "laws" in physics, is not an absolute description of all behavior but a mathematical shortcut based on a more fundamental description which applies in many common circumstances. You can derive Snell's Law, for light traveling from a medium with permittivity and permeability $\epsilon,\mu = \epsilon_1,\mu_1$ to a medium with $\epsilon_2,\mu_2$, by demanding that the following boundary conditions are satisfied: \begin{align} \epsilon_1 \vec E{}_1^\perp &= \epsilon_2 \vec E{}_2^\perp & \vec B{}_1^\perp &= \vec B{}_2^\perp & \text{normal to surface} \\ \vec E{}_1^\parallel &= \vec E{}_2^\parallel & \frac 1{\mu_1} \vec B{}_1^\parallel &= \frac 1{\mu_2} \vec B{}_2^\parallel & \text{parallel to surface} \end{align} Here the superscripts $^\perp$ and $^\parallel$ indicate orientation with respect to the surface, not to any polarization axis. The external wave will have the form $$ \vec E_1 = \vec E_A \exp i\left(\vec k_1\cdot \vec x - \omega t \right) + \vec E_B \exp i\left(\vec k_1\cdot \vec x - \omega t \right) $$ where $\vec E_A$ and $\vec E_B$, representing the two plane polarization components of the incident wave, are orthogonal to each other and to the wavevector $\vec k_1$. The magnetic field $\vec B_1$ is orthogonal to $\vec E_1$ and to $\vec k_1$. The details of the derivation can be found in many E&M and optics textbooks.

In a birefringent material you have the complication that the permittivity, $\epsilon$, may be given by a tensor rather than a scalar. Now the orientation of the surface is not the only property of the material to define a direction, and the E&M boundary-value problem must be solved again; this time it doesn't lead to Snell's Law, especially if the relative orientation of the optical axis (or axes) of the material and its surface is a wonky angle.

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