Brewster's angle

# Brewster's angle

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Encyclopedia

Brewster's angle is an angle of incidence
Angle of incidence
Angle of incidence is a measure of deviation of something from "straight on", for example:* in the approach of a ray to a surface, or* the angle at which the wing or horizontal tail of an airplane is installed on the fuselage, measured relative to the axis of the fuselage.-Optics:In geometric...

at which light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...

with a particular polarization is perfectly transmitted through a transparent dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

surface, with no reflection
Reflection (physics)
Reflection is the change in direction of a wavefront at an interface between two differentmedia so that the wavefront returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves...

. When unpolarized light is incident at this angle, the light that is reflected from the surface is therefore perfectly polarized. This special angle of incidence is named after the Scottish physicist Sir David Brewster
David Brewster
Sir David Brewster KH PRSE FRS FSA FSSA MICE was a Scottish physicist, mathematician, astronomer, inventor, writer and university principal.-Early life:...

(1781–1868).

## Explanation

When light encounters a boundary between two media
Medium (optics)
An optical medium is material through which electromagnetic waves propagate. It is a form of transmission medium. The permittivity and permeability of the medium define how electromagnetic waves propagate in it...

with different refractive indices
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

, some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the Fresnel equations
Fresnel equations
The Fresnel equations , deduced by Augustin-Jean Fresnel , describe the behaviour of light when moving between media of differing refractive indices...

, and is dependent upon the incoming light's polarization and angle of incidence.

The Fresnel equations predict that light with the p polarization (electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

polarized in the same plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

as the incident ray and the surface normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

) will not be reflected if the angle of incidence is

where n1 and n2 are the refractive indices
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

of the two media. This equation is known as Brewster's law, and the angle defined by it is Brewster's angle.

The physical mechanism for this can be qualitatively understood from the manner in which electric dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

s 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 of the dipole moment
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...

. Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected
Specular reflection
Specular reflection is the mirror-like reflection of light from a surface, in which light from a single incoming direction is reflected into a single outgoing direction...

, the dipoles cannot create any reflected light.

With simple geometry this condition can be expressed as:
where θ1 is the angle of incidence and θ2 is the angle of refraction.

Using Snell's law
Snell's law
In optics and physics, Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water and glass...

,

one can calculate the incident angle θ1 = θB at which no light is reflected:

Solving for θB gives:

For a glass medium (n2 ≈ 1.5) in air (n1 ≈ 1), Brewster's angle for visible light is approximately 56°, while for an air-water interface (n2 ≈ 1.33), it is 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
Étienne-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.

Brewster's angle is often referred to as the "polarizing angle", because light that reflects from a surface at this angle is entirely polarized perpendicular to the incident plane ("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
Polarizer
A polarizer is an optical filter that passes light of a specific polarization and blocks waves of other polarizations. It can convert a beam of light of undefined or mixed polarization into a beam with well-defined polarization. The common types of polarizers are linear polarizers and circular...

. 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
Sunglasses
Sunglasses or sun glasses are a form of protective eyewear designed primarily to prevent bright sunlight and high-energy visible light from damaging or discomforting the eyes. They can sometimes also function as a visual aid, as variously termed spectacles or glasses exist, featuring lenses that...

use the principle of Brewster's angle to reduce glare from the sun reflecting off 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
Photographic filter
In photography and videography, a filter is a camera accessory consisting of an optical filter that can be inserted in the optical path. The filter can be a square or oblong shape mounted in a holder accessory, or, more commonly, a glass or plastic disk with a metal or plastic ring frame, which...

camera attachment can be rotated to be at the correct angle (see figure).

### Brewster windows

Gas laser
Gas laser
A gas laser is a laser in which an electric current is discharged through a gas to produce coherent light. The gas laser was the first continuous-light laser and the first laser to operate "on the principle of converting electrical energy to a laser light output...

s 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.