Editing Brewster's angle (section)

==Explanation==
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 depends on the incoming light's polarization and angle of incidence.

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]] at the point of incidence) will not be reflected if the angle of incidence is 
:<math>\theta_\mathrm{B} = \arctan\!\left(\frac{n_2}{n_1}\right)\!,</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.

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 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 [[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)

With simple geometry this condition can be expressed as
:<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]],
:<math>n_1 \sin \theta_1 = n_2 \sin \theta_2,</math>
one can calculate the incident angle {{nowrap|1=''θ''<sub>1</sub> = ''θ''<sub>B</sub>}} at which no light is reflected:
:<math>n_1 \sin \theta_\mathrm{B} = n_2 \sin(90^\circ - \theta_\mathrm{B}) = n_2 \cos \theta_\mathrm{B}.</math>

Solving for ''θ''<sub>B</sub> gives
:<math>\theta_\mathrm{B} = \arctan\!\left(\frac{n_2}{n_1}\right)\!.</math>


The physical explanation of why the transmitted ray should be at <math>90^\circ</math> to the reflected ray can be difficult to grasp, but the Brewster angle result also follows simply from the [[Fresnel equations]] for reflectivity, which state that for p-polarized light
:<math>
  R_\mathrm{p} = \left|\frac{n_1 \cos \theta_2 - n_2 \cos \theta_1}{n_1 \cos \theta_2 + n_2 \cos \theta_1}\right|^2,
</math>
The reflection goes to zero when 
:<math>
 n_2 \cos \theta_1 = n_1 \cos \theta_2 
</math>
We can now use Snell's Law to eliminate <math> \theta_2 </math> as follows: we multiply Snell by <math> n_1 </math> and square both sides; multiply the zero-reflection condition just obtained by <math> n_2 </math> and square both sides; and add the equations. This produces
:<math>
 n_1^4 \sin^2 \theta_1 + n_2^4\cos^2\theta_1 = n_1^2n_2^2 \sin ^2\theta_2 + n_1^2n_2^2 \cos^2 \theta_2 = n_1^2n_2^2 = n_1^2n_2^2 \sin^2 \theta_1 + n_1^2n_2^2 \cos^2 \theta_1
</math>
We finally divide both sides by <math> n_1^4\cos^2\theta_1 </math>, collect terms and rearrange to produce <math> \tan^2\theta_1 = n_2^2/n_1^2</math>, from which the desired result follows (which then allows reverse proof that <math> \theta_1 + \theta_2 = 90^\circ </math>).

For a glass medium ({{nowrap|1=''n''<sub>2</sub> ≈ 1.5}}) in air ({{nowrap|1=''n''<sub>1</sub> ≈ 1}}), Brewster's angle for visible light is approximately 56°, while for an air-water interface ({{nowrap|1=''n''<sub>2</sub> ≈ 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 [[Étienne-Louis Malus]] in 1808.<ref>See:
*  Malus (1809) [https://books.google.com/books?id=hnJKAAAAYAAJ&pg=PA143 "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.
*  Malus, E.L. (1809) [https://www.biodiversitylibrary.org/item/24789#page/278/mode/1up "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 cristallisées'' [Theory of the double refraction of light in crystallized substances] (Paris, France:  Garnery, 1810), ''Chapitre troisième.  Des nouvelles propriétés physiques que la lumière acquiert par l'influence des corps qui la réfractent ou la réfléchissent.'' (Chapter 3.  On new physical properties that light acquires by the influence of bodies that refract it or reflect it.), [https://books.google.com/books?id=1LNIv6MFH2cC&pg=PA413 pp. 413–449.]</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 [[plane of incidence]] ("''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.

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.<ref>{{Cite journal |last1=Giles |first1=C. L. |last2=Wild |first2=W. J. |year=1985 |title=Brewster angles for magnetic media |url=http://clgiles.ist.psu.edu/pubs/brewster-magnetic.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://clgiles.ist.psu.edu/pubs/brewster-magnetic.pdf |archive-date=2022-10-09 |url-status=live |journal=International Journal of Infrared and Millimeter Waves |volume=6 |issue=3 |pages=187–197 |bibcode=1985IJIMW...6..187G |doi=10.1007/BF01010357|s2cid=122287937 }}</ref>  This has implications for the existence of generalized Brewster angles for dielectric metasurfaces.<ref>{{Cite journal |last1=Paniagua-Domínguez |first1=Ramón |last2=Feng Yu |first2=Ye |last3=Miroshnichenko |first3=Andrey E. |last4=Krivitsky |first4=Leonid A. |last5=Fu |first5=Yuan Hsing |last6=Valuckas |first6=Vytautas |last7=Gonzaga |first7=Leonard |display-authors=etal |year=2016 |title=Generalized Brewster effect in dielectric metasurfaces |journal=Nature Communications |volume=7 |pages=10362 |arxiv=1506.08267 |bibcode=2016NatCo...710362P |doi=10.1038/ncomms10362 |pmc=4735648 |pmid=26783075}}</ref>