Refraction

Light is refracted as it passes from one material to another. In the illustration below, the material on the left has an index of refraction $n_1$ (white background) and a material on the right has an index of refraction $n_2$ (gray background). A light ray (the red line) is moving to the right at an angle $\theta_1$ from the normal to the interface. At the interface, part of the wave is reflected (the light blue line) and part is refracted into the material on the right. The refracted ray makes an angle $\theta_2$ with the normal that is given by Snell's law,

$\large n_1\sin\theta_1 = n_2\sin\theta_2$.

If $n_1>n_2$ then there is no solution of Snell's law for $\theta_2$ when $n_1\sin\theta_1/n_2 >1$. In these cases, there is no refracted ray and all of the light is reflected. This is called total internal reflection.

Refraction and total reflection is due to the wave interference. Below are two videos of waves striking an interface at $y=125\,\mu\text{m}$ between two materials. The frequency $f$ of the oscillations are the same everywhere but wavelength is longer in the material at the bottom of the video so the speed of waves $c=\lambda f$ is higher in that material. The upper video shows conditions where refraction takes place and the lower video show conditions where total internal reflection takes place.

Both videos were created with this MATLAB script.