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Brewster's angle

Brewster's angle (also known as the polarization angle) is an angle of incidence at which light with a particular polarization is perfectly transmitted through a surface, with no reflection. The angle at which this occurs is named after the Scottish physicist, Sir David Brewster (1781–1868).

Explanation

When light moves between two media of differing refractive index, generally some of it is reflected at the boundary. At one particular angle of incidence, however, light with one particular polarization cannot be reflected. This angle of incidence is Brewster's angle, θ_B. The polarization that cannot be reflected at this angle is the polarization for which the electric field of the light waves lies in the same plane as the incident ray and the surface normal (i.e. the plane of incidence). Light with this polarization is said to be p-polarized, because it is parallel to the plane. Light with the perpendicular polarization is said to be s-polarized, from the German senkrecht—perpendicular. When unpolarized light strikes a surface at Brewster's angle, the reflected light is always s-polarized. Although 's' and 'p' polarization states were not named for this convention, it is convenient to remember that 's' polarized light will "skip" off a Brewster boundary and 'p' polarized light will "plunge" through.

The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction along which they oscillate. Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles will not create any reflected light. Since, by definition, the s-polarization is parallel to the interface, the corresponding oscillating dipoles will always be able to radiate in the specular-reflection direction. This is why there is no Brewster's angle for s-polarized light.

With simple trigonometry this condition can be expressed as:

theta_1 + theta_2 = 90^circ,

where θ₁ is the angle of incidence and θ₂ is the angle of refraction.

Using Snell's law,

n_1 sin left(theta_1 right) =n_2 sin left(theta_2 right),

we can calculate the incident angle θ₁=θ_B at which no light is reflected:

n_1 sin left(theta_B right) =n_2 sin left(90 - theta_B right)=n_2 cos left(theta_B right).

Rearranging, we get:

theta_B = arctan left(frac{n_2}{n_1} right),

where n₁ and n₂ are the refractive indices of the two media. This equation is known as Brewster's law.

Note that, since all p-polarized light is refracted (i.e transmitted), any light reflected from the interface at this angle must be s-polarized. A glass plate or a stack of plates placed at Brewster's angle in a light beam can thus be used as a polarizer.

For a glass medium (n₂≈1.5) in air (n₁≈1), Brewster's angle for visible light is approximately 56° to the normal while for an air-water interface (n₂≈1.33), it's approximately 53°. Since the refractive index for a given medium changes depending on the wavelength of light, Brewster's angle will also vary with wavelength.

The phenomenon of light being polarized by reflection from a surface at a particular angle was first observed by Etienne-Louis Malus in 1808. He attempted to relate the polarizing angle to the refractive index of the material, but was frustrated by the inconsistent quality of glasses available at that time. In 1815, Brewster experimented with higher-quality materials and showed that this angle was a function of the refractive index, defining Brewster's law.

Although Brewster's angle is generally presented as a zero-reflection angle in textbooks from the late 1950s onwards, it truly is a polarizing angle. The concept of a polarizing angle can be extended to the concept of a Brewster wavenumber to cover planar interfaces between two linear bianisotropic materials.

Applications

Polarized sunglasses use the principle of Brewster's angle to reduce glare from the sun reflecting off of horizontal surfaces such as water or road. In a large range of angles around Brewster's angle the reflection of p-polarized light is lower than s-polarized light. Thus, if the sun is low in the sky reflected light is mostly s-polarized. Polarizing sunglasses use a polarizing material such as polaroid film to block horizontally-polarized light, preferentially blocking reflections from horizontal surfaces. The effect is strongest with smooth surfaces such as water, but reflections from road and the ground are also reduced.

Photographers use the same principle to remove reflections from water so that they can photograph objects beneath the surface. In this case, the polarizing filter camera attachment can be rotated to be at the correct angle (see figure).

Brewster windows

Gas lasers typically use a window tilted at Brewster's angle to allow the beam to leave the laser tube. Since the window reflects some s-polarized light but no p-polarized light, the gain for the s polarization is reduced but that for the p polarization is not affected. This causes the laser's output to be p polarized, and allows lasing with no loss due to the window.

References

A. Lakhtakia, "Would Brewster recognize today's Brewster angle?" OSA Optics News, Vol. 15, No. 6, pp. 14–18 (1989).
A. Lakhtakia, "General schema for the Brewster conditions," Optik, Vol. 90, pp. 184–186 (1992).