Editing Fresnel equations (section)

== Power (intensity) reflection and transmission coefficients  ==
[[File:Fresnel power air-to-glass.svg|thumb|right|Power coefficients: air to glass]]
[[File:Fresnel power glass-to-air.svg|thumb|right|Power coefficients: glass to air (Total internal reflection starts from 42° making reflection coefficient 1)]]

We call the fraction of the incident [[Power (physics)|power]] that is reflected from the interface the ''[[reflectance]]'' (or '''reflectivity''', or '''power reflection coefficient''') {{math|''R''}}, and the fraction that is refracted into the second medium is called the ''[[transmittance]]'' (or '''transmissivity''', or '''power transmission coefficient''') {{math|''T''}}. Note that these are what would be measured right ''at'' each side of an interface and do not account for attenuation of a wave in an absorbing medium ''following'' transmission or reflection.<ref>Hecht, 1987, p. 100.</ref>

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 display=block>
  R_\mathrm{s} = \left|\frac{Z_2 \cos \theta_\mathrm{i} - Z_1 \cos \theta_\mathrm{t}}{Z_2 \cos \theta_\mathrm{i} + Z_1 \cos \theta_\mathrm{t}}\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 display=block>
  R_\mathrm{p} = \left|\frac{Z_2 \cos \theta_\mathrm{t} - Z_1 \cos \theta_\mathrm{i}}{Z_2 \cos \theta_\mathrm{t} + Z_1 \cos \theta_\mathrm{i}}\right|^2,
</math>
where {{math|''Z''<sub>1</sub>}} and {{math|''Z''<sub>2</sub>}} are the [[wave impedance]]s of media 1 and 2, respectively.

We assume that the media are non-magnetic (i.e., {{math|1=''μ''<sub>1</sub> = ''μ''<sub>2</sub> = [[permeability of free space|''μ''<sub>0</sub>]]}}), which is typically a good approximation at optical frequencies (and for transparent media at other frequencies).<ref>{{Cite book | doi=10.1081/E-EOE|title = Encyclopedia of Optical Engineering|year = 2011|isbn = 978-0-8247-0940-2|last1 = Driggers|first1 = Ronald G.| last2=Hoffman| first2=Craig| last3=Driggers| first3=Ronald}}</ref> Then the wave impedances are determined solely by the refractive indices {{math|''n''<sub>1</sub>}} and {{math|''n''<sub>2</sub>}}:
<math display=block>Z_i = \frac{Z_0}{n_i}\,,</math>
where {{math|''Z''<sub>0</sub>}} is the [[impedance of free space]] and {{math|1=''i'' = 1, 2}}. Making this substitution, we obtain equations using the refractive indices:
<math display=block>
  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>
<math display=block>
  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 {{math|''θ''<sub>t</sub>}} using [[Snell's law]] and [[trigonometric identity|trigonometric identities]].

As a consequence of [[conservation of energy]], one can find the transmitted power (or more correctly, [[irradiance]]: power per unit area) simply as the portion of the incident power that isn't reflected:{{hsp}}<ref>Hecht, 1987, p.{{tsp}}102.</ref>
<math display=block>T_\mathrm{s} = 1 - R_\mathrm{s}</math>
and
<math display=block>T_\mathrm{p} = 1 - R_\mathrm{p}</math>

Note that all such intensities are measured in terms of a wave's irradiance in the direction normal to the interface; this is also what is measured in typical experiments. That number could be obtained from irradiances ''in the direction of an incident or reflected wave'' (given by the magnitude of a wave's [[Poynting vector]]) multiplied by {{math|cos{{nnbsp}}''θ''}} for a wave at an angle {{math|''θ''}} to the normal direction (or equivalently, taking the [[dot product]] of the Poynting vector with the unit vector normal to the interface). This complication can be ignored in the case of the reflection coefficient, since {{math|1=cos{{nnbsp}}''θ''<sub>i</sub> = cos{{nnbsp}}''θ''<sub>r</sub>}}, so that the ratio of reflected to incident irradiance in the wave's direction is the same as in the direction normal to the interface.

Although these relationships describe the basic physics, in many practical applications one is concerned with "natural light" that can be described as unpolarized. That means that there is an equal amount of power in the ''s'' and ''p'' polarizations, so that the ''effective'' reflectivity of the material is just the average of the two reflectivities:
<math display=block>R_\mathrm{eff} = \frac{1}{2}\left(R_\mathrm{s} + R_\mathrm{p}\right).</math>

For low-precision applications involving unpolarized light, such as [[computer graphics]], rather than rigorously computing the effective reflection coefficient for each angle, [[Schlick's approximation]] is often used.

=== Special cases ===
==== Normal incidence ====
For the case of [[normal incidence]], {{math|1=''θ''{{sub|i}} = ''θ''{{sub|t}} = 0}}, and there is no distinction between s and p polarization.  Thus, the reflectance simplifies to
<math display=block>
R_0 = \left|\frac{n_1  - n_2 }{n_1 + n_2 }\right|^2\,.
</math>

For common glass ({{math|''n''<sub>2</sub> ≈ 1.5}}) surrounded by air ({{math|1=''n''<sub>1</sub> = 1}}), the power reflectance at normal incidence can be seen to be about 4%, or 8% accounting for both sides of a glass pane.

==== Brewster's angle ====
{{Main|Brewster's angle}}
At a dielectric interface from  {{math|''n''<sub>1</sub>}} to {{math|''n''<sub>2</sub>}}, there is a particular angle of incidence at which {{math|''R''<sub>p</sub>}} goes to zero and a p-polarised incident wave is purely refracted, thus all reflected light is s-polarised. This angle is known as [[Brewster's angle]], and is around 56° for {{math|1=''n''<sub>1</sub> = 1}} and {{math|1=''n''<sub>2</sub> = 1.5}} (typical glass).

==== Total internal reflection ====
{{Main|Total internal reflection}}
When light travelling in a denser medium strikes the surface of a less dense medium (i.e., {{math|1=''n''<sub>1</sub> &gt; ''n''<sub>2</sub>}}), beyond a particular incidence angle known as the ''critical angle'', all light is reflected and {{math|1=''R''<sub>s</sub> = ''R''<sub>p</sub> = 1}}. This phenomenon, known as [[total internal reflection]], occurs at incidence angles for which Snell's law predicts that the sine of the angle of refraction would exceed unity (whereas in fact {{math|sin{{nnbsp}}''θ'' ≤ 1}} for all real {{math|''θ''}}). For glass with {{math|1=''n'' = 1.5}} surrounded by air, the critical angle is approximately 42°.

==== 45° incidence ====
Reflection at 45° incidence is very commonly used for making 90° turns. For the case of light traversing from a less dense medium into a denser one at 45° incidence ({{math|1=''θ'' = 45°}}), it follows algebraically from the above equations that {{math|''R''<sub>p</sub>}} equals the square of {{math|''R''<sub>s</sub>}}:
<math display=block> R_\text{p} = R_\text{s}^2 </math>

This can be used to either verify the consistency of the measurements of {{math|''R''<sub>s</sub>}} and {{math|''R''<sub>p</sub>}}, or to derive one of them when the other is known. This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required.

Measurements of {{math|''R''<sub>s</sub>}} and {{math|''R''<sub>p</sub>}} at 45° can be used to estimate the reflectivity at normal incidence.{{cn|date=October 2023}} The "average of averages" obtained by calculating first the arithmetic as well as the geometric average of {{math|''R''<sub>s</sub>}} and {{math|''R''<sub>p</sub>}}, and then averaging these two averages again arithmetically, gives a value for {{math|''R''<sub>0</sub>}} with an error of less than about 3% for most common optical materials.{{cn|date=October 2023|reason=source needed to establish accuracy.}} This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other. However, since the dependence of {{math|''R''<sub>s</sub>}} and {{math|''R''<sub>p</sub>}} on the angle of incidence for angles below 10° is very small, a measurement at about 5° will usually be a good approximation for normal incidence, while allowing for a separation of the incoming and reflected beam.