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[[Image:Brewsters-angle.svg|thumb|250px|An illustration of the polarization of light that is incident on an interface at Brewster's angle.]]
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'''Brewster's angle''' (also known as the '''polarization angle''') is an [[angle of incidence]] at which [[light]] with a particular [[Polarization (waves)|polarization]] is perfectly transmitted through a transparent [[dielectric]] surface, with no [[Reflection (physics)|reflection]]. 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 [[David Brewster|Sir David Brewster]] (1781–1868).<ref>David Brewster (1815) [http://books.google.com/books?id=U-U_AAAAYAAJ&pg=PA125#v=onepage&q&f=false "On the laws which regulate the polarisation of light by reflection from transparent bodies,"] ''Philosophical Transactions of the Royal Society of London'', '''105''': 125-159.</ref>


==Explanation==
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When light encounters a boundary between two [[medium (optics)|media]] with different [[refractive index|refractive indices]], some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the [[Fresnel equations]], and is dependent upon the incoming light's polarization and angle of incidence.  
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The Fresnel equations predict that light with the ''p'' polarization ([[electric field]] polarized in the same [[Plane (mathematics)|plane]] as the [[incident ray]] and the [[surface normal]]) will not be reflected if the angle of incidence is
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:<math>\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right), </math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


where ''n''<sub>1</sub> is the [[refractive index]] of the initial medium through which the light propagates (the "incident medium"), and ''n''<sub>2</sub> 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.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


The physical mechanism for this can be qualitatively understood from the manner in which electric [[dipole]]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 [[Electric dipole moment|dipole moment]]. If the refracted light is p-polarized and propagates exactly perpendicular to the direction in which the light is predicted to be [[specular reflection|specularly reflected]], the dipoles point along the specular reflection direction and therefore no light can be reflected. (See diagram, above)
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


With simple geometry this condition can be expressed as
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:<math> \theta_1 + \theta_2 = 90^\circ,</math>
where θ<sub>1</sub> is the angle of reflection (or incidence) and θ<sub>2</sub> is the angle of refraction.


Using [[Snell's law]],
==Demos==


:<math>n_1 \sin \left( \theta_1 \right) =n_2 \sin \left( \theta_2 \right),</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


one can calculate the incident angle θ<sub>1</sub>&nbsp;=&nbsp;θ<sub>B</sub> at which no light is reflected:


:<math>n_1 \sin \left( \theta_\mathrm B \right) =n_2 \sin \left( 90^\circ - \theta_\mathrm B \right)=n_2 \cos \left( \theta_\mathrm B \right).</math>
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Solving for θ<sub>B</sub> gives
==Test pages ==


:<math>\theta_\mathrm B = \arctan \left( \frac{n_2}{n_1} \right) .</math>
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
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For a glass medium (''n''<sub>2</sub>&nbsp;≈&nbsp;1.5) in air (''n''<sub>1</sub>&nbsp;≈&nbsp;1), Brewster's angle for visible light is approximately 56°, while for an air-water interface (''n''<sub>2</sub>&nbsp;≈&nbsp;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.
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The phenomenon of light being polarized by reflection from a surface at a particular angle was first observed by [[Étienne-Louis Malus]] in 1808.<ref>See:
==Bug reporting==
*  Malus (1809) [http://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA143#v=onepage&q&f=false "Sur une propriété de la lumière réfléchie"] (On a property of reflected light), ''Mémoires de physique et de chimie de la Société d'Arcueil'', '''2''' :  143-158.
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* Malus, E.L. (1809) "Sur une propriété de la lumière réfléchie par les corps diaphanes" (On a property of light reflected by translucent substances), ''Nouveau Bulletin des Sciences'' [par la Societé Philomatique de Paris], '''1''' : 266-270.
*  Etienne Louis Malus, ''Théorie de la double réfraction de la lumière dans les substances christallisées'' [Theory of the double refraction of light in crystallized substances] (Paris, France:  Garnery, 1810).</ref> 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]]. 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]]s. In the case of reflection at Brewster's angle, the reflected and refracted rays are mutually perpendicular.
 
== Applications ==
 
Polarized [[sunglasses]] 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 (polarizer)|Polaroid]] sheets to block horizontally-polarized light, preferentially blocking reflections from horizontal surfaces. The effect is strongest with smooth surfaces such as water, but reflections from roads 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 (photography)|polarizing filter]] camera attachment can be rotated to be at the correct angle (see figure).
 
[[Image:Poloriser-demo.jpg|center|frame|Photograph taken of a window with a camera polarizer filter rotated to two different angles. In the picture at left, the polarizer is aligned with the polarization angle of the window reflection. In the picture at right, the polarizer has been rotated 90° eliminating the heavily polarized reflected sunlight.]]
 
===Brewster windows=== <!--Brewster window redirects here.-->
[[Image:Brewster window.svg|thumb|right|A Brewster window]]
[[Gas laser]]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 round trip loss for the ''s'' polarization is higher than that of the ''p'' polarization. This causes the laser's output to be ''p'' polarized due to competition between the two modes.<ref name=Hecht>''Optics'', 3rd edition, Hecht, ISBN 0-201-30425-2</ref>
 
== See also ==
* [[Brewster angle microscope]]
 
==Notes==
<references/>
 
==References==
*A. Lakhtakia, "Would Brewster recognize today's Brewster angle?" ''OSA Optics News'', Vol. 15, No. 6,  pp.&nbsp;14–18 (1989).
*A. Lakhtakia, "General schema for the Brewster conditions," ''Optik'', Vol. 90, pp.&nbsp;184–186 (1992).
 
==External links==
*[http://scienceworld.wolfram.com/physics/BrewstersAngle.html Brewster's Angle Extraction] from Wolfram Research
*[http://www.rp-photonics.com/brewster_windows.html Brewster window at RP-photonics.com]
 
{{DEFAULTSORT:Brewster's Angle}}
[[Category:Geometrical optics]]
[[Category:Physical optics]]
[[Category:Angle]]
[[Category:Polarization (waves)]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

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