Brewster's angle

Polarization by reflection

When unpolarized light reflects from a dielectric surface, the reflected light is often partially polarized. At one special angle, called Brewster's angle, the reflected light is completely linearly polarized perpendicular to the plane of incidence.

This effect is important in glare reduction, photography, lasers, and optical coatings.

Plane of incidence

The plane of incidence is the plane containing the incident ray, reflected ray, refracted ray, and surface normal. Polarization perpendicular to this plane is called s-polarization. Polarization parallel to this plane is called p-polarization.

At Brewster's angle, the reflected p-polarized component goes to zero, leaving reflected light purely s-polarized.

Brewster condition

At Brewster's angle, the reflected and refracted rays are perpendicular:

$heta_B+ heta_t=90^circ$

Snell's law gives

$n_1sin heta_B=n_2sin heta_t$

Since $heta_t=90^circ- heta_B$,

$sin heta_t=cos heta_B$

Substitute into Snell's law:

$n_1sin heta_B=n_2cos heta_B$

Therefore

an heta_B=rac{n_2}{n_1}

This is Brewster's law.

Air to glass example

For light traveling from air into glass, take $n_1approx1.00$ and $n_2approx1.50$. Then

$an heta_B=1.50$

so

$heta_Bapprox56.3^circ$

At this incidence angle, reflected light is strongly polarized parallel to the surface and perpendicular to the plane of incidence.

Why p-polarized reflection vanishes

A deeper explanation uses electromagnetic boundary conditions and oscillating dipoles in the material. The refracted wave drives charges to oscillate, and oscillating dipoles do not radiate along their axis. At Brewster's angle, the reflected direction aligns with the direction where p-polarized radiation vanishes.

This gives a physical wave explanation behind the geometric condition.

Polarized sunglasses

Glare from horizontal surfaces such as water or roads is often strongly horizontally polarized because reflection occurs near Brewster-like angles. Polarized sunglasses have vertical transmission axes, reducing horizontally polarized glare while still allowing much useful light through.

This improves contrast and comfort outdoors.

Brewster windows

Lasers sometimes use Brewster-angle windows to minimize reflection losses for p-polarized light. If the window is set at Brewster's angle, p-polarized light transmits with minimal reflection.

This can also help select a preferred polarization in laser cavities.

Limitations

Brewster's law in the simple form applies to ideal dielectric interfaces and nonmagnetic media. Metals behave differently because they absorb strongly and have complex refractive indices. Real surfaces, coatings, roughness, and wavelength dependence can modify the effect.

The big idea

Brewster's angle is the incidence angle at which reflected p-polarized light vanishes, leaving reflected light linearly polarized. Brewster's law, $an heta_B=n_2/n_1$, follows from Snell's law plus the perpendicular reflected-refracted geometry. The effect is central to glare reduction and polarization optics.