# Brewster's angle

Brewster's angle (also known as the polarization angle) is an optical phenomenon named after the Scottish physicist, Sir David Brewster (17811868).

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, θB. 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.

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

Using Snell's law,

we can calculate the incident angle θ1B at which no light is reflected:

Rearranging, we get:

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.

## Examples

Polarized sunglasses use the principle of Brewster's angle to eliminate glare from the sun reflecting off of water (or any other reflective surface). 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 mostly s-polarized light will reflect from water. Sunglasses made up of polarizers (e.g. polaroid film) aligned to block this light consequently block reflections from the water. To accomplish this, sunglass makers assume people will be upright while viewing the water and thus align the polarizers to block the polarization which oscillates along the line connecting the sunglass ear-pieces (i.e. horizontal).

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

Photographs taken of mudflats with a camera polarizer filter rotated to two different angles. In the first picture, the polarizer is rotated to maximize reflections, and in the second, it is rotated 90° to minimize reflections - almost all reflected sunlight is eliminated.

## 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).
Optics (ὀπτική appearance or look in Ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter.
Sir David Brewster FRS, (11 December 1781 – 10 February 1868) was a Scottish scientist, inventor and writer.

He was born at Jedburgh, where his father, a teacher of high reputation, was rector of the grammar school.
8th century - 9th century - 10th century
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885 886 887 - 888 - 889 890 891

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Subjects:     Archaeology - Architecture -
18th century - 19th century - 20th century
1830s  1840s  1850s  - 1860s -  1870s  1880s  1890s
1865 1866 1867 - 1868 - 1869 1870 1871

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Light is electromagnetic radiation of a wavelength that is visible to the eye (visible light). In a scientific context, the word "light" is sometimes used to refer to the entire electromagnetic spectrum.
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.
The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.
Reflection is the change in direction of a wave front at an between two dissimilar media so that the wave front returns into the medium from which it originated. Common examples include the reflection of light, sound and water waves.
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 direction of an airfoil with respect to the direction of an airplane.
polarization (Brit., polarisation) is the property of electromagnetic waves, such as light, that describes the direction of the transverse electric field. More generally, the polarization of a transverse wave describes the direction of oscillation in the plane
electric field. This electric field exerts a force on other electrically charged objects. The concept of electric field was introduced by Michael Faraday.

The electric field is a vector field with SI units of newtons per coulomb (N C−1
plane is a two-dimensional manifold or surface that is perfectly flat. Informally it can be thought of as an infinitely vast and infinitesimally thin sheet oriented in some space.
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing.
surface normal, or simply normal, to a flat surface is a vector which 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.
German language (Deutsch, ] ) is a West Germanic language and one of the world's major languages.
dipoles (Hellènic: di(s)- = twi- and pòla = pivot, hinge). An electric dipole is a separation of positive and negative charge. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some, usually small,
Specular reflection is the perfect, mirror-like reflection of light (or sometimes other kinds of wave) from a surface, in which light from a single incoming direction (a ray) is reflected into a single outgoing direction.
Snell's law (also known as Descartes' law or the law of refraction), 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
The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. For example, typical glass has a refractive index of 1.
A polarizer is a device that converts an unpolarized or mixed-polarization beam of electromagnetic waves (e.g., light) into a beam with a single polarization state (usually, a single linear polarization).
Etienne-Louis Malus (July 23, 1775 – February 24, 1812) was a French officer, engineer, physicist, and mathematician.

Malus was born in Paris, France. He participated in Napoleon's expedition into Egypt, 1798 to 1801.