6.2.8. Brewster's Angle

Table of Contents

Polarization by Reflection

When light strikes the boundary between two transparent materials, part of it is reflected and part is transmitted. In general, the reflected light contains more of one polarization direction than the other. At one special angle of incidence, called Brewster's angle, the reflected light becomes completely linearly polarized.

This effect is important because it shows that reflection depends on polarization. It is also a practical way to produce polarized light without using a polarizing filter.

The Idea of Polarization Directions

To describe Brewster's angle, we separate the electric field of the incoming light into two polarization components. One component is parallel to the plane of incidence, and the other is perpendicular to that plane. The plane of incidence is the plane containing the incoming ray and the normal to the surface.

At Brewster's angle, the component parallel to the plane of incidence is not reflected. Only the perpendicular component remains in the reflected beam. As a result, the reflected beam is perfectly polarized.

At Brewster's angle, the reflected light is completely linearly polarized because the reflected component parallel to the plane of incidence vanishes.

Geometric Condition

A remarkable geometric property appears at Brewster's angle. The reflected ray and the refracted ray are perpendicular to each other.

If the angle of incidence is $\theta_B$ and the angle of refraction is $\theta_t$, then at Brewster's angle,

$$
\theta_B + \theta_t = 90^\circ
$$

This relation leads directly to the formula for Brewster's angle.

Rendering scientific sketch…

Light at Brewster's angle

Formula for Brewster's Angle

Using Snell's law together with the perpendicular condition, we obtain the standard formula:

$$
\tan \theta_B = \frac{n_2}{n_1}
$$

where $n_1$ is the refractive index of the medium the light comes from, and $n_2$ is the refractive index of the second medium.

So the Brewster angle is

$$
\theta_B = \tan^{-1}\left(\frac{n_2}{n_1}\right)
$$

Brewster's angle formula:
$$
\tan \theta_B = \frac{n_2}{n_1}
$$
This formula applies for light going from medium 1 into medium 2.

Common Example

A common case is light going from air into glass. If we take

$$
n_1 \approx 1.00, \qquad n_2 \approx 1.50
$$

then

$$
\tan \theta_B = \frac{1.50}{1.00} = 1.50
$$

$$
\theta_B = \tan^{-1}(1.50) \approx 56.3^\circ
$$

This means that if unpolarized light hits a glass surface at about $56^\circ$, the reflected light is strongly useful as a source of polarized light, and at the exact Brewster angle it is fully polarized.

Why This Happens

The reflected and transmitted waves are produced by oscillating charges in the material. At the Brewster angle, the geometry is such that the charges cannot reradiate energy in the reflected direction for the polarization parallel to the plane of incidence. Therefore that component disappears from the reflected beam.

For beginners, the most important point is not the microscopic mechanism, but the result: one polarization is removed from the reflected beam at a specific angle.

Relation to Polarizing Sunglasses

Reflected glare from water, roads, and glass is often partially polarized. Since such reflections are often close to horizontal surfaces, the reflected light tends to have a preferred polarization direction. Polarizing sunglasses reduce this glare by blocking much of that polarized reflected light.

Brewster's angle helps explain why reflected glare can be strongly polarized.

Comparison of Incidence Angles

The behavior of reflected light changes with angle of incidence.

Angle of incidence	Reflected light
Small angle	Usually only partially polarized
Brewster's angle	Completely linearly polarized
Larger or smaller than Brewster's angle	Partially polarized again

Important Notes

Brewster's angle exists for reflection at an interface between two media with different refractive indices. Its value depends on both media. If the light travels in the opposite direction, the formula still works, but $n_1$ and $n_2$ must be assigned correctly according to the direction of travel.

Also, Brewster's angle refers specifically to the angle of incidence measured from the normal, not from the surface.

Brewster's angle is measured from the normal to the surface, not from the surface itself.

Summary

Brewster's angle is the angle of incidence at which reflected light becomes completely polarized. At this angle, the reflected and refracted rays are perpendicular, and the angle satisfies

$$
\tan \theta_B = \frac{n_2}{n_1}
$$

This gives a simple and powerful connection between reflection, refraction, and polarization.

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