4
$\begingroup$

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?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.