Reflection from a dielectric interface can depend on the polarization direction of the incoming light. At one special incident angle, the reflected light contains no electric-field component polarized in the plane of incidence. This is Brewster's angle.

The plane of incidence is the plane containing the incident ray and the surface normal.

• p polarization has its electric field parallel to the plane of incidence.
• s polarization has its electric field perpendicular to that plane.

At Brewster's angle, the reflected p-polarized component vanishes in the ideal dielectric-interface model. Light reflected from initially unpolarized light is therefore strongly s-polarized.

Brewster geometry

At the Brewster condition, the reflected and transmitted rays are perpendicular:

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

The law of reflection makes the reflected angle equal to the incident angle $\theta_B$.

Snell's law gives

$$n_1\sin\theta_B=n_2\sin\theta_t.$$

Since

$$\theta_t=90^\circ-\theta_B,$$

we have

$$\sin\theta_t=\cos\theta_B.$$

Therefore

$$n_1\sin\theta_B=n_2\cos\theta_B,$$

and hence

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

This relation applies when light travels from a medium of refractive index $n_1$ toward one of index $n_2$.

Worked example: air to glass

For light incident from air onto glass, take

$$n_1\approx1.00,\qquad n_2=1.50.$$

Then

$$\tan\theta_B=1.50,$$

so

$$\boxed{\theta_B\approx56.3^\circ}.$$

At this incident angle, the reflected beam is ideally entirely s-polarized if the incoming light contains both polarization components.

The transmitted angle is

$$\theta_t=90^\circ-56.3^\circ=33.7^\circ.$$

A check with Snell's law gives

$$1.00\sin56.3^\circ \approx1.50\sin33.7^\circ.$$

Why reflected glare becomes polarized

Sunlight or room light is often initially unpolarized. Reflection from water, glass, roads and other dielectric surfaces can preferentially remove the p component from the reflected beam, especially near Brewster's angle. The reflected glare is therefore partially polarized.

A linear polarizer oriented to suppress that dominant reflected polarization can reduce glare. This is one reason polarizing filters are useful in sunglasses and photography.

Scope

Brewster's law describes the ideal interface behavior of the p-polarized reflection coefficient. Away from Brewster's angle, both s and p components generally reflect with nonzero amplitudes. Real surfaces, absorbing materials, multilayers and roughness can modify the simple result.

The central idea is that reflection is not only a geometric change of ray direction: electromagnetic boundary behavior can make reflection depend strongly on polarization.