Brewster's angle (also known as the polarization angle) is an
angle of incidence at which
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
with a particular
polarization is perfectly transmitted through a transparent
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
surface, with ''no
reflection Reflection or reflexion may refer to:
Science and technology
* Reflection (physics), a common wave phenomenon
** Specular reflection, reflection from a smooth surface
*** Mirror image, a reflection in a mirror or in water
** Signal reflection, in ...
''. 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
Sir David Brewster KH PRSE FRS FSA Scot FSSA MICE (11 December 178110 February 1868) was a British scientist, inventor, author, and academic administrator. In science he is principally remembered for his experimental work in physical optics, ...
(1781–1868).
Explanation
When light encounters a boundary between two
media
Media may refer to:
Communication
* Media (communication), tools used to deliver information or data
** Advertising media, various media, content, buying and placement for advertising
** Broadcast media, communications delivered over mass el ...
with different
refractive indices
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
, some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the
Fresnel equations
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
, and depends on the incoming light's polarization and angle of incidence.
The Fresnel equations predict that light with the ''p'' polarization (
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
polarized in the same
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
as the
incident ray
In optics a ray is an idealized geometrical model of light, obtained by choosing a curve that is perpendicular to the ''wavefronts'' of the actual light, and that points in the direction of energy flow. Rays are used to model the propagation o ...
and the
surface normal
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
at the point of incidence) will not be reflected if the angle of incidence is
:
where ''n''
1 is the
refractive index
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium.
The refractive index determines how much the path of light is bent, or ...
of the initial medium through which the light propagates (the "incident medium"), and ''n''
2 is the index of the other medium. 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
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this system i ...
s in the media respond to ''p''-polarized light. One can imagine that light incident on the surface is absorbed, and then re-radiated 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. If the refracted light is ''p''-polarized and propagates exactly perpendicular to the direction in which the light is predicted to be
specularly reflected
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surfa ...
, the dipoles point along the specular reflection direction and therefore no light can be reflected. (See diagram, above)
With simple geometry this condition can be expressed as
:
where ''θ''
1 is the angle of reflection (or incidence) and ''θ''
2 is the angle of refraction.
Using
Snell's law
Snell's law (also known as Snell–Descartes law and ibn-Sahl law and 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 ...
,
:
one can calculate the incident angle at which no light is reflected:
:
Solving for ''θ''
B gives
:
For a glass medium () in air (), Brewster's angle for visible light is approximately 56°, while for an air-water interface (), 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
Étienne-Louis Malus
Étienne-Louis Malus (; ; 23 July 1775 – 23 February 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) and was a member o ...
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
plane of incidence
In describing reflection and refraction in optics, the plane of incidence (also called the incidence plane or the meridional plane) is the plane which contains the surface normal and the propagation vector of the incoming radiation. (In wave optic ...
("''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
A polarizer or polariser is an optical filter that lets light waves of a specific polarization pass through while blocking light waves of other polarizations. It can filter a beam of light of undefined or mixed polarization into a beam of well ...
. The concept of a polarizing angle can be extended to the concept of a Brewster wavenumber to cover planar interfaces between two linear
bianisotropic material In physics, engineering and materials science, bi-isotropic materials have the special optical property that they can rotate the polarization of light in either refraction or transmission. This does not mean all materials with twist effect fall i ...
s. In the case of reflection at Brewster's angle, the reflected and refracted rays are mutually perpendicular.
For magnetic materials, Brewster's angle can exist for only one of the incident wave polarizations, as determined by the relative strengths of the dielectric permittivity and magnetic permeability. This has implications for the existence of generalized Brewster angles for dielectric metasurfaces.
Applications
While at the Brewster angle there is ''no'' reflection of the ''p'' polarization, at yet greater angles the
reflection coefficient
In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium. It is equal to the ratio of the amplitude of the reflected wa ...
of the ''p'' polarization is always less than that of the ''s'' polarization, almost up to 90° incidence where the reflectivity of each rises towards unity. Thus reflected light from horizontal surfaces (such as the surface of a road) at a distance much greater than ones height (so that the incidence angle of specularly reflected light is near, or usually well beyond the Brewster angle) is strongly ''s''-polarized.
Polarized sunglasses
Polarization ( also polarisation) is a property applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave, the direction of the oscillation is perpendicular to the direction of motion of th ...
use a sheet of
polarizing material to block horizontally-polarized light and thus reduce
glare
Glare (derived from GLAss REinforced laminate ) is a fiber metal laminate (FML) composed of several very thin layers of metal (usually aluminum) interspersed with layers of S-2 glass-fiber ''pre-preg'', bonded together with a matrix such as epo ...
in such situations. These are most effective with smooth surfaces where
specular reflection
Specular reflection, or regular reflection, is the mirror-like reflection of waves, such as light, from a surface.
The law of reflection states that a reflected ray of light emerges from the reflecting surface at the same angle to the surf ...
(thus from light whose
angle of incidence is the same as the angle of reflection defined by the angle observed from) is dominant, but even
diffuse reflection
Diffuse reflection is the reflection (physics), reflection of light or other radiation, waves or particles from a surface such that a ray (optics), ray incident on the surface is scattering, scattered at many angles rather than at just one angle ...
s from roads for instance, are also significantly reduced.
Photographers also use polarizing filters to remove reflections from water so that they can photograph objects beneath the surface. Using a
polarizing camera attachment which can be rotated, such a filter can be adjusted to reduce reflections from objects other than horizontal surfaces, such as seen in the accompanying photograph (right) where the ''s'' polarization (approximately vertical) has been eliminated using such a filter.
When recording a classical
hologram
Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other Holography#Applications, applic ...
, the bright reference beam is typically arranged to strike the film in the ''p'' polarization at Brewster's angle. By thus eliminating reflection of the reference beam at the transparent back surface of the holographic film, unwanted interference effects in the resulting hologram are avoided.
Entrance windows or prisms with their surfaces at the Brewster angle are commonly used in optics and laser physics in particular. The polarized laser light enters the prism at Brewster's angle without any reflective losses.
In surface science,
Brewster angle microscope
A Brewster angle microscope (BAM) is a microscope for studying thin films on liquid surfaces, most typically Langmuir films. In a Brewster angle microscope, both the microscope and a polarized light source are aimed towards a liquid surface at ...
s are used to image layers of particles or molecules at air-liquid interfaces. Using illumination by a laser at Brewster's angle to the interface and observation at the angle of reflection, the uniform liquid does not reflect, appearing black in the image. However any molecular layers or artifacts at the surface, whose refractive index or physical structure contrasts with the liquid, allows for some reflection against that black background which is captured by a camera.
Brewster windows
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 lase ...
s using an external
cavity
Cavity may refer to:
Biology and healthcare
*Body cavity, a fluid-filled space in many animals where organs typically develop
**Gastrovascular cavity, the primary organ of digestion and circulation in cnidarians and flatworms
*Dental cavity or too ...
(reflection by one or both mirrors ''outside'' the
gain medium
The active laser medium (also called gain medium or lasing medium) is the source of optical gain within a laser. The gain results from the stimulated emission of photons through electronic or molecular transitions to a lower energy state from a ...
) generally seal the tube using windows tilted at Brewster's angle. This prevents light in the intended polarization from being lost through reflection (and reducing the round-trip gain of the laser) which is critical in lasers having a low round-trip gain. On the other hand it ''does'' remove ''s'' polarized light, increasing the round trip loss for that polarization, and insuring the laser only oscillates in one linear polarization, as is usually desired. And many sealed-tube lasers (which don't even need windows) have a glass plate inserted within the tube at the Brewster angle, simply for the purpose of allowing lasing in only one polarization.
[''Optics'', 3rd edition, Hecht, ]
Pseudo-Brewster's angle
When the reflecting surface is absorbing, reflectivity at parallel polarization (''p'') goes through a non-zero minimum at the so-called pseudo-Brewster's angle.
See also
*
Brewster angle microscope
A Brewster angle microscope (BAM) is a microscope for studying thin films on liquid surfaces, most typically Langmuir films. In a Brewster angle microscope, both the microscope and a polarized light source are aimed towards a liquid surface at ...
References
Further reading
*
External links
Brewster's Angle Extractionfrom Wolfram Research
TE, TM Reflection Coefficients– interactive phase and magnitude plots showing Brewster's angle
{{DEFAULTSORT:Brewster's Angle
Geometrical optics
Physical optics
Angle
Polarization (waves)