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{{About|the Fresnel equations describing reflection and refraction of light at uniform planar interfaces|the diffraction of light through an aperture|Fresnel diffraction|the thin lens and mirror technology|Fresnel lens}}

[[File:Partial transmittance.gif|right|thumb|250px|Partial transmission and reflection amplitudes of a wave travelling from a low to high refractive index medium.]]
<!----Since the whole article deals with intensities, it is confusing that the picture deals with amplitudes (there is no negative intensity) I added the word amplitude in the caption of the article---->

The '''Fresnel equations''' (or Fresnel conditions), deduced by [[Augustin-Jean Fresnel]] {{IPAc-en|f|r|ɛ|ˈ|n|ɛ|l}}, describe the behaviour of [[light]] when moving between [[medium (optics)|media]] of differing [[refractive index|refractive indices]]. The [[reflection (physics)|reflection]] of light that the equations predict is known as '''Fresnel reflection'''.

==Overview==
When light moves from a medium of a given [[refractive index]] ''n''<sub>1</sub> into a second medium with refractive index ''n''<sub>2</sub>, both [[reflection (physics)|reflection]] and [[refraction]] of the light may occur. The Fresnel equations describe what fraction of the light is reflected and what fraction is refracted (i.e., transmitted). They also describe the [[phase (waves)|phase shift]] of the reflected light.

The equations assume the interface is flat, planar, and homogeneous, and that the light is a [[plane wave]].

==Definitions and power equations==
[[File:Fresnel1.svg|right|thumb|300px|Variables used in the Fresnel equations.]]
In the diagram on the right, an incident light [[ray (optics)|ray]] '''IO''' strikes the interface between two media of refractive indices ''n''<sub>1</sub> and ''n''<sub>2</sub> at point '''O'''. Part of the ray is reflected as ray '''OR''' and part refracted as ray '''OT'''. The angles that the incident, reflected and refracted rays make to the [[surface normal|normal]] of the interface are given as ''θ''<sub>i</sub>, ''θ''<sub>r</sub> and ''θ''<sub>t</sub>, respectively.

The relationship between these angles is given by the [[law of reflection]]: 

:<math>\theta_\mathrm{i} = \theta_\mathrm{r} </math>

and [[Snell's law]]: 

:<math>\frac{\sin\theta_\mathrm{i}}{\sin\theta_\mathrm{t}} = \frac{n_2}{n_1}</math>

The fraction of the incident [[Power (physics)|power]] that is reflected from the interface is given by the [[reflectance]] ''R'' and the fraction that is refracted is given by the [[transmittance]] ''T''.<ref>Hecht (1987), p. 100.</ref> The media are assumed to be ''non-magnetic''.

The calculations of ''R'' and ''T'' depend on [[polarization (waves)|polarisation]] of the incident ray. Two cases are analyzed:

#The incident light is polarized with its [[electric field]] perpendicular to the plane containing the incident, reflected, and refracted rays, i.e. in the plane of the diagram above. The light is said to be ''s''-polarized, from the German ''senkrecht'' (perpendicular).
#The incident light is polarized with its electric field parallel to the plane described above. Such light is described as ''p''-polarized.

The [[reflectance]] for ''s''-polarized light is
<!----
EDIT WITH CARE! THERE ARE DIFFERENT FORMS OF THE EQUATIONS, AND WHAT IS HERE MAY NOT MATCH *YOUR* BOOK, AND YET MAY BE CORRECT
---->
: <math>R_\mathrm{s} = 
\left|\frac{n_1\cos\theta_{\mathrm{i}}-n_2\cos\theta_{\mathrm{t}}}{n_1\cos\theta_{\mathrm{i}}+n_2\cos\theta_{\mathrm{t}}}\right|^2
=\left|\frac{n_1\cos\theta_{\mathrm{i}}-n_2\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}}{n_1\cos\theta_{\mathrm{i}}+n_2\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}}\right|^2</math>,

while the [[reflectance]] for ''p''-polarized light is 
<!----
EDIT WITH CARE! THERE ARE DIFFERENT FORMS OF THE EQUATIONS, AND WHAT IS HERE MAY NOT MATCH *YOUR* BOOK, AND YET MAY BE CORRECT.
---->
: <math>R_\mathrm{p} =
\left|\frac{n_1\cos\theta_{\mathrm{t}}-n_2\cos\theta_{\mathrm{i}}}{n_1\cos\theta_{\mathrm{t}}+n_2\cos\theta_{\mathrm{i}}}\right|^2
=\left|\frac{n_1\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}-n_2\cos\theta_{\mathrm{i}}}{n_1\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}+n_2\cos\theta_{\mathrm{i}}}\right|^2</math>.

The second form of each equation is derived from the first by eliminating ''θ''<sub>t</sub> using [[Snell's law]] and [[trigonometric identity|trigonometric identities]].

As a consequence of the [[conservation of energy]], the [[transmission coefficient]]s are given by <ref>Hecht (1987), p. 102.</ref> 

:<math>T_s = 1-R_s\,\!</math>

and 

:<math>T_p = 1-R_p\,\!</math>

These relationships hold only for power coefficients, not for amplitude coefficients as defined below.

If the incident light is unpolarised (containing an equal mix of ''s''- and ''p''-polarisations), the reflection coefficient is 

:<math>R = \frac{R_s+R_p}{2}\,\!</math>

For common glass, the reflection coefficient at ''θ''<sub>i</sub>&nbsp;=&nbsp;0 is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2''R''/(1&nbsp;+&nbsp;''R''), when [[Interference (wave propagation)|interference]] can be neglected (see [[#Multiple surfaces|below]]).

The discussion given here assumes that the [[permeability (electromagnetism)|permeability]] ''μ'' is equal to the [[vacuum permeability]] ''μ''<sub>0</sub> in both media. This is approximately true for most [[dielectric]] materials, but not for some other types of material. The completely general Fresnel equations are more complicated.

[[File:fresnel reflection.svg|800 px]]

For low-precision applications where polarization may be ignored, such as [[computer graphics]], [[Schlick's approximation]] may be used.

===Special angles===
{{main|Brewster's angle|Total internal reflection}}
At one particular angle for a given ''n''<sub>1</sub> and ''n''<sub>2</sub>, the value of ''R''<sub>p</sub> goes to zero and a ''p''-polarised incident ray is purely refracted. This angle is known as [[Brewster's angle]], and is around 56° for a glass medium in air or vacuum.  Note that this statement is only true when the refractive indices of both materials are [[real number]]s, as is the case for materials like air and glass.  For materials that absorb light, like [[metal]]s and [[semiconductor]]s, ''n'' is [[complex number|complex]], and ''R''<sub>p</sub> does not generally go to zero.

When moving from a denser medium into a less dense one (i.e., ''n''<sub>1</sub>&nbsp;&gt;&nbsp;''n''<sub>2</sub>), above an incidence angle known as the ''critical angle'', all light is reflected and ''R''<sub>s</sub>&nbsp;=&nbsp;''R''<sub>p</sub>&nbsp;=&nbsp;1. This phenomenon is known as [[total internal reflection]]. The critical angle is approximately 41° for glass in air.

==Amplitude equations==
Equations for coefficients corresponding to ratios of the [[electric field]] complex-valued [[amplitude]]s of the waves (not necessarily real-valued magnitudes) are also called "Fresnel equations". These take several different forms, depending on the choice of formalism and [[sign convention]] used. The amplitude coefficients are usually represented by lower case ''r'' and ''t''.
[[File:Amptitude Ratios air to glass.JPG|thumb|right|Amplitude ratios: air to glass]]
[[File:Amplitude ratios glass to air.JPG|thumb|right|Amplitude ratios: glass to air]]

===Conventions used here===
In this treatment, the coefficient ''r'' is the ratio of the reflected wave's [[complex number|complex]] electric field amplitude to that of the incident wave. The coefficient ''t'' is the ratio of the transmitted wave's electric field amplitude to that of the incident wave. The light is split into ''s'' and ''p'' polarizations as defined above. (In the figures to the right, ''s'' polarization is denoted "<math>\bot</math>" and ''p'' is denoted "<math>\parallel</math>".)

For ''s''-polarization, a positive ''r'' or ''t'' means that the electric fields of the incoming and reflected or transmitted wave are parallel, while negative means anti-parallel. For ''p''-polarization, a positive ''r'' or ''t'' means that the ''magnetic fields'' of the waves are parallel, while negative means anti-parallel.<ref name=Sernelius>Lecture notes by Bo Sernelius, [http://www.ifm.liu.se/courses/TFYY67/ main site], see especially [http://www.ifm.liu.se/courses/TFYY67/Lect12.pdf Lecture 12].</ref> It is also assumed that the magnetic permeability µ of both media is equal to the permeability of free space µ<sub>0</sub>.

===Formulas===
Using the conventions above,<ref name=Sernelius/>
<!----EDIT WITH CARE! ARBITRARY SIGN CONVENTIONS ARE USED IN DERIVING THESE EQUATIONS. DIFFERENT BOOKS USE DIFFERENT CONVENTIONS.---->
:<math>r_s = \frac{n_1 \cos \theta_\text{i} - n_2 \cos \theta_\text{t}}{n_1 \cos \theta_\text{i} + n_2 \cos \theta_\text{t}}</math>

:<math>t_s = \frac{2 n_1 \cos \theta_\text{i}}{n_1 \cos \theta_\text{i} + n_2 \cos \theta_\text{t}}</math>

:<math>r_p = \frac{n_2 \cos \theta_\text{i} - n_1 \cos \theta_\text{t}}{n_1 \cos \theta_\text{t} + n_2 \cos \theta_\text{i}}</math>

:<math>t_p = \frac{2 n_1\cos \theta_\text{i}}{n_1 \cos \theta_\text{t} + n_2 \cos \theta_\text{i}}</math>
<!----EDIT WITH CARE! ARBITRARY SIGN CONVENTIONS ARE USED IN DERIVING THESE EQUATIONS. DIFFERENT BOOKS USE DIFFERENT CONVENTIONS.---->

Notice that <math>t_s = 1 + r_s</math> but <math>t_p \neq 1 + r_p</math>.<ref>Hecht (2003), p. 116, eq.(4.49)-(4.50).</ref>

Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the amplitude reflection coefficient is related to the reflectance ''R'' by <ref>Hecht (2003), p. 120, eq.(4.56).</ref>

:<math>R=\left| r \right|^2</math>.

The transmittance ''T'' is generally not equal to |''t''|<sup>2</sup>, since the light travels with different direction and speed in the two media. The transmittance is related to ''t'' by.<ref>Hecht (2003), p. 120, eq.(4.57).</ref>

:<math>T=\frac{n_2 \cos \theta_\text{t}}{n_1 \cos \theta_\text{i}} \left| t \right|^2</math>.

The factor of ''n''<sub>2</sub>/''n''<sub>1</sub> occurs from the ratio of intensities (closely related to [[irradiance]]). The factor of cos θ<sub>t</sub>/cos θ<sub>i</sub> represents the change in area ''m'' of the [[pencil of rays]], needed since ''T'', the ratio of powers, is equal to the ratio of (intensity × area). In terms of the ratio of refractive indices, 
:<math>\rho = n_2/n_1</math>,
and of the magnification ''m'' of the beam cross section occurring at the interface,
:<math>T = \rho m t^2</math>.

==Multiple surfaces==
When light makes multiple reflections between two or more parallel surfaces, the multiple beams of light generally [[Interference (wave propagation)|interfere]] with one another, resulting in net transmission and reflection amplitudes that depend on the light's wavelength. The interference, however, is seen only when the surfaces are at distances comparable to or smaller than the light's [[coherence length]], which for ordinary white light is few micrometers; it can be much larger for light from a [[laser]].

An example of interference between reflections is the [[iridescence|iridescent]] colours seen in a [[soap bubble]] or in thin oil films on water. Applications include [[Fabry–Pérot interferometer]]s, [[antireflection coating]]s, and [[optical filter]]s. A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

The [[transfer-matrix method (optics)|transfer-matrix method]], or the recursive Rouard method<ref>see, e.g. O.S. Heavens, ''Optical Properties of Thin Films'', Academic Press, 1955, chapt. 4.</ref> can be used to solve multiple-surface problems.

==See also==
*[[Index-matching material]]
*[[Fresnel rhomb]], Fresnel's apparatus to produce circularly polarised light [http://physics.kenyon.edu/EarlyApparatus/Polarized_Light/Fresnels_Rhomb/Fresnels_Rhomb.html]
*[[Specular reflection]]
*[[Schlick's approximation]]
*[[Snell's window]]
*[[X-ray reflectivity]]

==References==
<references/>
*{{cite book | first=Eugene|last=Hecht|year=1987|title=Optics|edition=2nd |publisher=Addison Wesley|isbn=0-201-11609-X}}
*{{cite book | first=Eugene|last=Hecht|year=2002|title=Optics|edition=4th |publisher=Addison Wesley|isbn=0-321-18878-0}}

==Further reading==
* ''The Cambridge Handbook of Physics Formulas'', G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.
* ''Introduction to Electrodynamics (3rd Edition)'', D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3
* ''Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers'', Y.B. Band, John Wiley & Sons, 2010, ISBN 978-0-471-89931-0
* ''The Light Fantastic – Introduction to Classic and Quantum Optics'', I.R. Kenyon, Oxford University Press, 2008, ISBN 978-0-19-856646-5
* ''Encyclopaedia of Physics (2nd Edition)'', R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
* ''McGraw Hill Encyclopaedia of Physics (2nd Edition)'', C.B. Parker, 1994, ISBN 0-07-051400-3

== External links ==
*[http://scienceworld.wolfram.com/physics/FresnelEquations.html Fresnel Equations] – Wolfram
*[http://people.csail.mit.edu/jaffer/FreeSnell FreeSnell] – Free software computes the optical properties of multilayer materials
*[http://thinfilm.hansteen.net/ Thinfilm] – Web interface for calculating optical properties of thin films and multilayer materials. (Reflection & transmission coefficients, ellipsometric parameters Psi & Delta)
*[http://www.calctool.org/CALC/phys/optics/reflec_refrac Simple web interface for calculating single-interface reflection and refraction angles and strengths.]
*[http://wm.eecs.umich.edu/webMathematica/eecs434/f08/ideliz/final.jsp Reflection and transmittance for two dielectrics ] – Mathematica interactive webpage that shows the relations between index of refraction and reflection.
*[http://www.jedsoft.org/physics/notes/multilayer.pdf A self-contained first-principles derivation] of the transmission and reflection probabilities from a multilayer with complex indices of refraction.

[[Category:Geometrical optics]]
[[Category:Physical optics]]
[[Category:Equations]]