Tutorial : brewster angle

The Snell descartes Law describes the reflexion and transmission of a light ray at the interface between two homogeneous media. This law applies also for waves. The incident wave can be split in two waves : one wave with the electic field perpendicular to the incidence plane (defined as TE polarization - see "glossary") and the second one with the electric field in the incident plane (defined as TM polarization - see "glossary").

In this section, the magnetic field H is used instead of the magnetic induction B. They have the same direction and their amplitudes are linked by the relation :

.

In the following formulas :

- index I is related to the parameters of the incident wave,
- index R is related to the parameters of the reflected wave,
- index T is related to the parameters of the refracted ( transmitted ) wave.

Wirh the same logic, n_I and n_T are the refraction indexes respectively of the first and second media.

For TE polarization, the boundary conditions at the interface are :

.

This brings to :

.

Therefore, the reflexion and transmission ratios in amplitude at the interface are :

.

For TM polarization, the boundary conditions at the interface are :

.

This brings to :

.

Whatever the polarization, if r ant t are respectively the reflexion and transmission ratio in amplitude, the reflexion and transmission ratios ( respectively R and T ) are given by :

.

The reflexion coefficient is never null whatever the incidence angle for TE polarization. It can be zero for TM polarization for an incidence angle said Brewster angle θ_B satisfying :

.

Incidence at Brewter angle is currently used to control the polarization. Indeed, the reflected light from an incident wave at Brewter angle is linearly polarized with an electric field perpendicular to the incidence plane (TE polarization). It is also commonly used in gas lasers. Indeed, the gas tube is closed by windows at Brewter angle. Therefore, the wave travels in the laser cavity without Fresnel reflexions at the windows interface for the TM polarization that can then oscillate more efficiently than TE polarization.