Editing Fresnel equations (section)

== Complex amplitude reflection and transmission coefficients  ==
The above equations relating powers (which could be measured with a [[photometer]] for instance) are derived from the Fresnel equations which solve the physical problem in terms of [[electromagnetic field]] [[complex amplitude]]s, i.e., considering [[phase (waves)|phase]] shifts in addition to their [[Absolute value#Complex numbers|amplitudes]]. Those underlying equations supply generally [[complex-valued]] ratios of those EM fields and may take several different forms, depending on the formalism used. The complex amplitude coefficients for reflection and transmission are usually represented by lower case {{math|''r''}} and {{math|''t''}} (whereas the power coefficients are capitalized). As before, we are assuming the magnetic permeability, {{math|''µ''}} of both media to be equal to the permeability of free space {{math|''µ''<sub>0</sub>}} as is essentially true of all dielectrics at optical frequencies.
[[File:Fresnel amplitudes air-to-glass.svg|thumb|right|Amplitude coefficients: air to glass]]
[[File:Fresnel amplitudes glass-to-air.svg|thumb|right|Amplitude coefficients: glass to air]]

In the following equations and graphs, we adopt the following conventions. For ''s'' polarization, the reflection coefficient {{math|''r''}} is defined as the ratio of the reflected wave's complex electric field amplitude to that of the incident wave, whereas for ''p'' polarization {{math|''r''}} is the ratio of the waves complex ''magnetic'' field amplitudes (or equivalently, the ''negative'' of the ratio of their electric field amplitudes). The transmission coefficient {{math|''t''}} is the ratio of the transmitted wave's complex electric field amplitude to that of the incident wave, for either polarization. The coefficients {{math|''r''}} and {{math|''t''}} are generally different between the ''s'' and ''p'' polarizations, and even at normal incidence (where the designations ''s'' and ''p'' do not even apply!) the sign of {{math|''r''}} is reversed depending on whether the wave is considered to be ''s'' or ''p'' polarized, an artifact of the adopted sign convention (see graph for an air-glass interface at 0° incidence).

The equations consider a plane wave incident on a plane interface at [[Angle of incidence (optics)|angle of incidence]] {{nowrap|1=<math> \theta_\mathrm{i}</math>}}, a wave reflected at angle {{nowrap|1=<math> \theta_\mathrm{r} = \theta_\mathrm{i} </math>}}, and a wave transmitted at angle {{nowrap|1=<math> \theta_\mathrm{t}</math>}}. In the case of an interface into an absorbing material (where {{math|''n''}} is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number. In that case, however, meaningful results can be obtained using formulations of these relationships in which trigonometric functions and geometric angles are avoided; the [[inhomogeneous wave]]s launched into the second medium cannot be described using a single propagation angle.

Using this convention,<ref name=Sernelius>Lecture notes by Bo Sernelius, [http://www.ifm.liu.se/courses/TFYY67/ main site] {{Webarchive|url=https://web.archive.org/web/20120222125421/http://www.ifm.liu.se/courses/TFYY67/ |date=2012-02-22 }}, see especially [http://www.ifm.liu.se/courses/TFYY67/Lect12.pdf Lecture 12] .</ref><ref name="Born 1970">Born & Wolf, 1970, p.{{hsp}}40, eqs.{{tsp}}(20),{{hsp}}(21).</ref>
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<math display="block">\begin{align}
  r_\text{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}}, \\[3pt]
  t_\text{s} &= \frac{2 n_1 \cos \theta_\text{i}}                           {n_1 \cos \theta_\text{i} + n_2 \cos \theta_\text{t}}, \\[3pt]
  r_\text{p} &= \frac{  n_2 \cos \theta_\text{i} - n_1 \cos \theta_\text{t}}{n_2 \cos \theta_\text{i} + n_1 \cos \theta_\text{t}}, \\[3pt]
  t_\text{p} &= \frac{2 n_1 \cos \theta_\text{i}}                           {n_2 \cos \theta_\text{i} + n_1 \cos \theta_\text{t}}.
\end{align}</math>

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For the case where the magnetic permeabilities are non-negligible, the equations change such that every appearance of <math> n_i </math> is replaced by <math> n_i/\mu_i </math> (for both <math> i=1, 2 </math>).{{cn|date=March 2025}}

One can see that {{math|1=''t''<sub>s</sub> = ''r''<sub>s</sub> + 1}}<ref>Hecht, 2002, p.{{hsp}}116, eqs.{{tsp}}(4.49),{{hsp}}(4.50).</ref> and {{math|{{sfrac|''n''<sub>2</sub>|''n''<sub>1</sub>}}''t''<sub>p</sub> {{=}} ''r''<sub>p</sub> + 1}}. One can write very similar equations applying to the ratio of the waves' magnetic fields, but comparison of the electric fields is more conventional.

Because the reflected and incident waves propagate in the same medium and make the same angle with the normal to the surface, the power reflection coefficient {{math|''R''}} is just the squared magnitude of {{math|''r''}}:{{hsp}}<ref>Hecht, 2002, p.{{hsp}}120, eq.{{hsp}}(4.56).</ref>
<math display=block>R = |r|^2.</math>

On the other hand, calculation of the power transmission coefficient {{mvar|T}} is less straightforward, since the light travels in different directions in the two media. What's more, the wave impedances in the two media differ; power ([[irradiance]]) is given by the square of the electric field amplitude ''divided by'' the [[wave impedance|characteristic impedance]] of the medium (or by the square of the magnetic field ''multiplied by'' the characteristic impedance). This results in:<ref>Hecht, 2002, p.{{hsp}}120, eq.{{hsp}}(4.57).</ref>
<math display=block>T = \frac{n_2 \cos \theta_\text{t}}{n_1 \cos \theta_\text{i}} |t|^2</math>
using the above definition of {{math|''t''}}. The introduced factor of {{math|{{sfrac|''n''<sub>2</sub>|''n''<sub>1</sub>}}}} is the reciprocal of the ratio of the media's wave impedances. The {{math|cos(''θ'')}} factors adjust the waves' powers so they are reckoned ''in the direction'' normal to the interface, for both the incident and transmitted waves, so that full power transmission corresponds to {{math|1=''T'' = 1}}.

In the case of [[total internal reflection]] where the power transmission {{mvar|T}} is zero, {{mvar|t}} nevertheless describes the electric field (including its phase) just beyond the interface. This is an [[evanescent field]] which does not propagate as a wave (thus {{math|1=''T'' = 0}}) but has nonzero values very close to the interface. The phase shift of the reflected wave on total internal reflection can similarly be obtained from the [[Argument (complex analysis)|phase angles]] of {{math|''r''<sub>p</sub>}} and {{math|''r''<sub>s</sub>}} (whose magnitudes are unity in this case). These phase shifts are different for  ''s'' and ''p'' waves, which is the well-known principle by which total internal reflection is used to effect [[Fresnel rhomb|polarization transformations]].

=== Alternative forms ===

In the above formula for {{math|''r''<sub>s</sub>}}, if we put <math>n_2=n_1\sin\theta_\text{i}/\sin\theta_\text{t}</math> (Snell's law) and multiply the numerator and denominator by {{math|{{sfrac|1|''n''<sub>1</sub>}}{{tsp}}sin{{tsp}}''θ''<sub>t</sub>}}, we&nbsp;obtain{{hsp}}<ref>Fresnel, 1866, p.{{hsp}}773.</ref><ref>Hecht, 2002, p.{{hsp}}115, eq.{{hsp}}(4.42).</ref>
<math display=block>r_\text{s}=-\frac{\sin(\theta_\text{i}-\theta_\text{t})}{\sin(\theta_\text{i}+\theta_\text{t})}.</math>

If we do likewise with the formula for {{math|''r''<sub>p</sub>}}, the result is easily shown to be equivalent to{{hsp}}<ref>Fresnel, 1866, p.{{hsp}}757.</ref><ref>Hecht, 2002, p.{{hsp}}115, eq.{{hsp}}(4.43).</ref>
<math display=block>r_\text{p}=\frac{\tan(\theta_\text{i}-\theta_\text{t})}{\tan(\theta_\text{i}+\theta_\text{t})}.
</math><!-- DON'T BE THE NEXT PERSON WHO WRONGLY CHANGES THE SIGN OF THIS EXPRESSION JUST BECAUSE SOME TEXTBOOK USES A DIFFERENT CONVENTION FROM THE ONE USED ***CONSISTENTLY*** IN THIS ARTICLE! -->

These formulas{{hsp}}<ref>E. Verdet, in Fresnel, 1866, p.{{hsp}}789n.</ref><ref>Born & Wolf, 1970, p.{{hsp}}40, eqs.{{hsp}}(21a).</ref><ref>Jenkins & White, 1976, p.{{hsp}}524, eqs.{{hsp}}(25a).</ref> are known respectively as ''Fresnel's sine law'' and ''Fresnel's tangent law''.<ref>Whittaker, 1910, p.{{hsp}}134; Darrigol, 2012, p.{{tsp}}213.</ref> Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the [[Limit (mathematics)|limit]] as {{math|''θ''<sub>i</sub> → 0}}.