Editing Fresnel equations (section)

== Derivation ==
Here we systematically derive the above relations from electromagnetic premises.

=== Material parameters ===
In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) [[linearity|linear]] and [[homogeneity (physics)|homogeneous]]. If the medium is also [[isotropy|isotropic]], the four field vectors {{math|'''E''',{{nnbsp}}'''B''',{{nnbsp}}'''D''',{{nnbsp}}'''H'''}}{{tsp}} are [[Constitutive equation#Electromagnetism|related]] by
<math display=block>\begin{align}
\mathbf{D} &= \epsilon \mathbf{E} \\
\mathbf{B} &= \mu \mathbf{H}\,,
\end{align}
</math>
where {{math|''ϵ''}} and {{math|''μ''}} are scalars, known respectively as the (electric) ''[[permittivity]]'' and the (magnetic) ''[[permeability (electromagnetism)|permeability]]'' of the medium. For vacuum, these have the values {{math|''ϵ''<sub>0</sub>}} and {{math|''μ''<sub>0</sub>}}, respectively. Hence we define the ''relative'' permittivity (or [[dielectric constant]]) {{math|''ϵ''<sub>rel</sub> {{=}} ''ϵ''/''ϵ''<sub>0</sub>}}, and the ''relative'' permeability {{math|''μ''<sub>rel</sub> {{=}} ''μ''/''μ''<sub>0</sub>}}.

In optics it is common to assume that the medium is non-magnetic, so that {{math|1=''μ''<sub>rel</sub> = 1}}. For [[ferromagnetic]] materials at radio/microwave frequencies, larger values of {{math|''μ''<sub>rel</sub>}} must be taken into account. But, for optically transparent media, and for all other materials at optical frequencies (except possible [[metamaterial]]s), {{math|''μ''<sub>rel</sub>}} is indeed very close to 1; that is, {{math|''μ'' ≈ ''μ''<sub>0</sub>}}.

In optics, one usually knows the [[refractive index]] {{math|''n''}} of the medium, which is the ratio of the speed of light in vacuum ({{mvar|c}}) to the speed of light in the medium. In the analysis of partial reflection and transmission, one is also interested in the electromagnetic [[wave impedance]] {{mvar|Z}}, which is the ratio of the amplitude of {{math|'''E'''}} to the amplitude of {{math|'''H'''}}. It is therefore desirable to express {{math|''n''}} and {{mvar|Z}} in terms of {{math|''ϵ''}} and {{math|''μ''}}, and thence to relate {{mvar|Z}} to {{math|''n''}}. The last-mentioned relation, however, will make it convenient to derive the reflection coefficients in terms of the wave ''admittance'' {{mvar|Y}}, which is the reciprocal of the wave impedance {{mvar|Z}}.

In the case of ''uniform [[plane wave|plane]] [[sine wave|sinusoidal]]'' waves, the wave impedance or admittance is known as the ''intrinsic'' impedance or admittance of the medium. This case is the one for which the Fresnel coefficients are to be derived.

=== Electromagnetic plane waves ===
In a uniform plane sinusoidal [[electromagnetic radiation|electromagnetic wave]], the [[electric field]] {{math|'''E'''}} has the form
{{NumBlk|:|<math>\mathbf{E_k}e^{i(\mathbf{k\cdot r}-\omega t)},</math>|{{EquationRef|1}}}}
where {{math|'''E<sub>k</sub>'''}} is the (constant) complex amplitude vector, {{math|''i''}} is the [[imaginary unit]], {{math|'''k'''}} is the [[wave vector]] (whose magnitude {{mvar|k}} is the angular [[wavenumber]]), {{math|'''r'''}} is the [[position (vector)|position vector]], {{math|''ω''}} is the [[angular frequency]], {{math|''t''}} is time, and it is understood that the ''real part'' of the expression is the physical field.<ref group=Note>The above form ({{EquationNote|1}}) is typically used by physicists. [[electrical engineering|Electrical engineers]] typically prefer the form {{math|'''E<sub>k</sub>'''{{hsp}}''e''<sup>''j''(''ωt''−'''k⋅r''')</sup>;}} that is, they not only use {{math|''j''}} instead of {{math|''i''}} for the imaginary unit, but also change the sign of the exponent, with the result that the whole expression is replaced by its [[complex conjugate]], leaving the real part unchanged {{bracket|Cf.&nbsp;(e.g.) Collin, 1966, p.{{hsp}}41, eq.{{tsp}}(2.81)}}. The electrical engineers' form and the formulas derived therefrom may be converted to the physicists' convention by substituting {{math|''−i''}} for {{math|''j''}}.</ref>&nbsp; The value of the expression is unchanged if the position {{math|'''r'''}} varies in a direction normal to {{math|'''k'''}}; hence {{math|'''k'''}} ''is normal to the wavefronts''.

To advance the [[phase (waves)|phase]] by the angle ''ϕ'', we replace {{math|''ωt''}} by {{math|''ωt'' + ''ϕ''}} (that is, we replace {{math|−''ωt''}} by {{math|−''ωt'' − ''ϕ''}}), with the result that the (complex) field is multiplied by {{math|''e<sup>−iϕ</sup>''}}. So a phase ''advance'' is equivalent to multiplication by a complex constant with a ''negative'' [[argument (complex analysis)|argument]]. This becomes more obvious when the field ({{EquationNote|1}}) is factored as {{math|'''E<sub>k</sub>'''{{hsp}}''e''<sup>''i'''''k'''⋅'''r'''</sup>''e''<sup>''−iωt''</sup>}}, where the last factor contains the time-dependence. That factor also implies that differentiation w.r.t.&nbsp;time corresponds to multiplication by {{math|''−iω''}}.{{hsp}}<ref group=Note>In the electrical engineering convention, the time-dependent factor is {{math|''e''<sup>''jωt''</sup>}}, so that a phase advance corresponds to multiplication by a complex constant with a ''positive'' argument, and differentiation w.r.t.&nbsp;time corresponds to multiplication by {{math|+''jω''}}. This article, however, uses the physics convention, whose time-dependent factor is {{math|''e''<sup>−''iωt''</sup>}}. Although the imaginary unit does not appear explicitly in the results given here, the time-dependent factor affects the interpretation of any results that turn out to be complex.</ref>

If ''ℓ'' is the component of {{math|'''r'''}} in the direction of {{math|'''k'''}}, the field ({{EquationNote|1}}) can be written {{math|'''E<sub>k</sub>'''{{hsp}}''e''<sup>''i''(''kℓ''−''ωt'')</sup>}}.&nbsp; If the argument of {{math|''e''<sup>''i''(⋯)</sup>}} is to be constant,&nbsp; ''ℓ''&nbsp;must increase at the velocity <math>\omega/k\,,\,</math> known as the ''[[phase velocity]]'' {{math|(''v''<sub>p</sub>)}}. This in turn is equal to {{nowrap|1=<math>c/n</math>.}} Solving for {{mvar|k}} gives
{{NumBlk|:|<math>k=n\omega/c\,.</math>|{{EquationRef|2}}}}

As usual, we drop the time-dependent factor {{math|''e''<sup>−''iωt''</sup>}}, which is understood to multiply every complex field quantity. The electric field for a uniform plane sine wave will then be represented by the location-dependent ''[[phasor]]''
{{NumBlk|:|<math>\mathbf{E_k}e^{i\mathbf{k\cdot r}}.</math>|{{EquationRef|3}}}}
For fields of that form, [[Faraday's law of induction|Faraday's law]] and the [[Ampère's circuital law|Maxwell-Ampère law]] respectively reduce to{{hsp}}<ref name=berry-jeffrey-2007>Compare M.V. Berry and M.R. Jeffrey, "Conical diffraction: Hamilton's diabolical point at the heart of crystal optics", in E.&nbsp;Wolf&nbsp;(ed.), ''Progress in Optics'', vol.{{tsp}}50, Amsterdam: Elsevier, 2007, pp.{{tsp}}13–50, {{doi|10.1016/S0079-6638(07)50002-8}}, at p.{{hsp}}18, eq.{{tsp}}(2.2).</ref>
<math display=block>\begin{align}
  \omega\mathbf{B} &=  \mathbf{k}\times\mathbf{E}\\
  \omega\mathbf{D} &= -\mathbf{k}\times\mathbf{H}\,.
\end{align}</math>

Putting {{math|'''B''' {{=}} ''μ'''''H'''}} and {{math|'''D''' {{=}} ''ϵ'''''E'''}}, as above, we can eliminate {{math|'''B'''}} and {{math|'''D'''}} to obtain equations in only {{math|'''E'''}} and {{math|'''H'''}}:
<math display=block>\begin{align}
  \omega\mu\mathbf{H} &=  \mathbf{k}\times\mathbf{E}\\
  \omega\epsilon\mathbf{E} &= -\mathbf{k}\times\mathbf{H}\,.
\end{align}</math>
If the material parameters {{math|''ϵ''}} and {{math|''μ''}} are real (as in a lossless dielectric), these equations show that {{math|'''k''', '''E''', '''H'''}}  form a ''right-handed orthogonal triad'', so that the same equations apply to the magnitudes of the respective vectors. Taking the magnitude equations and substituting from ({{EquationNote|2}}), we obtain
<math display=block>\begin{align}
  \mu cH &= nE\\
  \epsilon cE &= nH\,,
\end{align}</math>
where {{mvar|H}} and {{mvar|E}} are the magnitudes of {{math|'''H'''}} and {{math|'''E'''}}. Multiplying the last two equations gives
{{NumBlk|:|<math>n = c\,\sqrt{\mu\epsilon}\,.</math>|{{EquationRef|4}}}}
Dividing (or cross-multiplying) the same two equations gives {{math|''H'' {{=}} ''YE''}}, where
{{NumBlk|:|<math>Y = \sqrt{\epsilon/\mu}\,.</math>|{{EquationRef|5}}}}
This is the ''intrinsic admittance''.

From ({{EquationNote|4}}) we obtain the phase velocity {{nowrap|1=<math>c/n=1\big/\!\sqrt{\mu\epsilon\,}</math>.}} For vacuum this reduces to {{nowrap|1=<math>c=1\big/\!\sqrt{\mu_0\epsilon_0}</math>.}} Dividing the second result by the first gives
<math display=block>n=\sqrt{\mu_{\text{rel}}\epsilon_{\text{rel}}}\,.</math>
For a ''non-magnetic'' medium (the usual case), this becomes {{tmath|1= n=\sqrt{\epsilon_{\text{rel} } } }}.
{{larger|(}}Taking the reciprocal of ({{EquationNote|5}}), we find that the intrinsic ''impedance'' is {{nowrap|1=<math display="inline">Z=\sqrt{\mu/\epsilon}</math>.}} In vacuum this takes the value <math display="inline">Z_0=\sqrt{\mu_0/\epsilon_0}\,\approx 377\,\Omega\,,</math> known as the [[impedance of free space]]. By division, {{nowrap|1=<math display="inline">Z/Z_0=\sqrt{\mu_{\text{rel}}/\epsilon_{\text{rel}}}</math>.}} For a ''non-magnetic'' medium, this becomes <math>Z=Z_0\big/\!\sqrt{\epsilon_{\text{rel}}}=Z_0/n.</math>{{larger|)}}

=== Wave vectors ===
[[File:Wave vectors n1 to n2.svg|thumb|305px|Incident, reflected, and transmitted wave vectors ({{math|'''k'''<sub>i</sub>, '''k'''<sub>r</sub>}}, and {{math|'''k'''<sub>t</sub>}}), for incidence from a medium with refractive index {{math|''n''<sub>1</sub>}} to a medium with refractive index {{math|''n''<sub>2</sub>}}. The red arrows are perpendicular to the wave vectors.]]

In Cartesian coordinates {{math|(''x'', ''y'', ''z'')}}, let the region {{math|''y'' < 0}} have refractive index {{math|''n''<sub>1</sub>}}, intrinsic admittance {{math|''Y''<sub>1</sub>}}, etc., and let the region {{math|''y'' > 0}} have refractive index {{math|''n''<sub>2</sub>}}, intrinsic admittance {{math|''Y''<sub>2</sub>}}, etc. Then the {{math|''xz''}} plane is the interface, and the {{math|''y''}} axis is normal to the interface (see diagram). Let {{math|'''i'''}} and {{math|'''j'''}} (in bold [[roman type]]) be the unit vectors in the {{math|''x''}} and {{math|''y''}} directions, respectively. Let the plane of incidence be the {{math|''xy''}} plane (the plane of the page), with the angle of incidence {{math|''θ''<sub>i</sub>}} measured from {{math|'''j'''}} towards {{math|'''i'''}}. Let the angle of refraction, measured in the same sense, be {{math|''θ''<sub>t</sub>}}, where the subscript {{math|''t''}} stands for ''transmitted'' (reserving  {{math|''r''}} for ''reflected'').

In the absence of [[Doppler effect|Doppler shifts]], ''ω'' does not change on reflection or refraction. Hence, by ({{EquationNote|2}}), the magnitude of the wave vector is proportional to the refractive index.

So, for a given {{math|''ω''}}, if we ''redefine'' {{mvar|k}} as the magnitude of the wave vector in the ''reference'' medium (for which {{math|''n'' {{=}} 1}}), then the wave vector has magnitude {{math|''n''<sub>1</sub>''k''}} in the first medium (region {{math|''y'' < 0}} in the diagram) and magnitude {{math|''n''<sub>2</sub>''k''}} in the second medium. From the magnitudes and the geometry, we find that the wave vectors are
<math display=block>\begin{align}
  \mathbf{k}_\text{i} &= n_1 k(\mathbf{i}\sin\theta_\text{i} + \mathbf{j}\cos\theta_\text{i})\\[.5ex]
  \mathbf{k}_\text{r} &= n_1 k(\mathbf{i}\sin\theta_\text{i} - \mathbf{j}\cos\theta_\text{i})\\[.5ex]
  \mathbf{k}_\text{t} &= n_2 k(\mathbf{i}\sin\theta_\text{t} + \mathbf{j}\cos\theta_\text{t})\\
          &= k(\mathbf{i}\,n_1\sin\theta_\text{i} + \mathbf{j}\,n_2\cos\theta_\text{t})\,,
\end{align}</math>
where the last step uses Snell's law. The corresponding [[dot product]]s in the phasor form ({{EquationNote|3}}) are
{{NumBlk|:|<math>\begin{align}
  \mathbf{k}_\text{i}\mathbf{\cdot r} &= n_1 k(x\sin\theta_\text{i} + y\cos\theta_\text{i})\\
  \mathbf{k}_\text{r}\mathbf{\cdot r} &= n_1 k(x\sin\theta_\text{i} - y\cos\theta_\text{i})\\
  \mathbf{k}_\text{t}\mathbf{\cdot r} &= k(n_1 x\sin\theta_\text{i} + n_2 y\cos\theta_\text{t})\,.
\end{align}</math>|{{EquationRef|6}}}}
Hence:
{{NumBlk|:|At&nbsp; <math>y=0\,,~~~\mathbf{k}_\text{i}\mathbf{\cdot r}=\mathbf{k}_\text{r}\mathbf{\cdot r}=\mathbf{k}_\text{t}\mathbf{\cdot r}=n_1 kx\sin\theta_\text{i}\,.</math>|{{EquationRef|7}}}}

=== ''s'' components ===
For the ''s'' polarization, the {{math|'''E'''}} field is parallel to the {{math|''z''}} axis and may therefore be described by its component in the {{math|''z''}}&nbsp;direction. Let the reflection and transmission coefficients be {{math|''r''<sub>s</sub>}} and {{math|''t''<sub>s</sub>}}, respectively. Then, if the incident {{math|'''E'''}} field is taken to have unit amplitude, the phasor form ({{EquationNote|3}}) of its {{math|''z''}}-component is
{{NumBlk|:|<math>E_\text{i}=e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}},</math>|{{EquationRef|8}}}}
and the reflected and transmitted fields, in the same form, are
{{NumBlk|:|<math>\begin{align}
  E_\text{r} &= r_{s\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  E_\text{t} &= t_{s\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|9}}}}

Under the sign convention used in this article, a positive reflection or transmission coefficient is one that preserves the direction of the ''transverse'' field, meaning (in this context) the field normal to the plane of incidence. For the ''s'' polarization, that means the {{math|'''E'''}} field. If the incident, reflected, and transmitted {{math|'''E'''}} fields (in the above equations) are in the {{math|''z''}}-direction ("out of the page"), then the respective {{math|'''H'''}} fields are in the directions of the red arrows, since {{math|'''k''', '''E''', '''H'''}} form a right-handed orthogonal triad. The {{math|'''H'''}} fields may therefore be described by their components in the directions of those arrows, denoted by {{math|''H''<sub>i</sub>, ''H''<sub>r</sub>, ''H''<sub>t</sub>}}. Then, since {{math|''H'' {{=}} ''YE''}},
{{NumBlk|:|<math>\begin{align}
  H_\text{i} &=\, Y_1 e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}}\\
  H_\text{r} &=\, Y_1 r_{s\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  H_\text{t} &=\, Y_2 t_{s\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|10}}}}

At the interface, by the usual [[interface conditions for electromagnetic fields]], the tangential components of the {{math|'''E'''}} and {{math|'''H'''}} fields must be continuous; that is,
{{NumBlk|:|<math>\left.\begin{align}
  E_\text{i} + E_\text{r} &= E_\text{t}\\
  H_\text{i}\cos\theta_\text{i} - H_\text{r}\cos\theta_\text{i} &= H_\text{t}\cos\theta_\text{t}
\end{align}~~\right\}~~~\text{at}~~ y=0\,.</math>|{{EquationRef|11}}}}
When we substitute from equations ({{EquationNote|8}}) to ({{EquationNote|10}}) and then from ({{EquationNote|7}}), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations
{{NumBlk|:|<math>\begin{align}
  1 + r_\text{s} &=\, t_\text{s}\\
  Y_1\cos\theta_\text{i} - Y_1 r_\text{s}\cos\theta_\text{i} &=\, Y_2 t_\text{s}\cos\theta_\text{t} \,,
\end{align}</math>|{{EquationRef|12}}}}
which are easily solved for {{math|''r''<sub>s</sub>}} and {{math|''t''<sub>s</sub>}}, yielding
{{NumBlk|:|<math>r_\text{s}=\frac{Y_1\cos\theta_\text{i}-Y_2\cos\theta_\text{t}}{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}</math>|{{EquationRef|13}}}}
and
{{NumBlk|:|<math>t_\text{s}=\frac{2Y_1\cos\theta_\text{i}}{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\,.</math>|{{EquationRef|14}}}}
At ''normal incidence'' {{math|(''θ''<sub>i</sub> {{=}} ''θ''<sub>t</sub> {{=}} 0)}}, indicated by an additional subscript 0, these results become
{{NumBlk|:|<math>r_\text{s0}=\frac{Y_1-Y_2}{Y_1+Y_2}</math>|{{EquationRef|15}}}}
and
{{NumBlk|:|<math>t_\text{s0}=\frac{2Y_1}{Y_1+Y_2}\,.</math>|{{EquationRef|16}}}}
At ''grazing incidence'' {{math|(''θ''<sub>i</sub> → 90°)}}, we have {{math|cos{{tsp}}''θ''<sub>i</sub> → 0}}, hence {{math|''r''<sub>s</sub> → −1}} and {{math|''t''<sub>s</sub> → 0}}.

=== ''p'' components ===
For the ''p'' polarization, the incident, reflected, and transmitted {{math|'''E'''}} fields are parallel to the red arrows and may therefore be described by their components in the directions of those arrows. Let those components be {{math|''E''<sub>i</sub>, ''E''<sub>r</sub>, ''E''<sub>t</sub>{{hsp}}}} (redefining the symbols for the new context). Let the reflection and transmission coefficients be {{math|''r''<sub>p</sub>}} and {{math|''t''<sub>p</sub>}}. Then, if the incident {{math|'''E'''}} field is taken to have unit amplitude, we have
{{NumBlk|:|<math>\begin{align}
  E_\text{i} &=         e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}}\\
  E_\text{r} &= r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  E_\text{t} &= t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|17}}}}
If the {{math|'''E'''}} fields are in the directions of the red arrows, then, in order for {{math|'''k''', '''E''', '''H'''}} to form a right-handed orthogonal triad, the respective {{math|'''H'''}} fields must be in the {{math|−''z''}}-direction ("into the page") and may therefore be described by their components in that direction. This is consistent with the adopted sign convention, namely that a positive reflection or transmission coefficient is one that preserves the direction of the transverse field {{larger|(}}the {{math|'''H'''}} field in the case of the ''p'' polarization{{larger|)}}. The agreement of the ''other'' field with the red arrows reveals an alternative definition of the sign convention: that a positive reflection or transmission coefficient is one for which the field vector in the plane of incidence points towards the same medium before and after reflection or transmission.<ref>This agrees with Born & Wolf, 1970, p.{{hsp}}38, Fig.{{tsp}}1.10.</ref>

So, for the incident, reflected, and transmitted {{math|'''H'''}} fields, let the respective components in the {{math|−''z''}}-direction be {{math|''H''<sub>i</sub>, ''H''<sub>r</sub>, ''H''<sub>t</sub>}}. Then, since {{math|''H'' {{=}} ''YE''}},
{{NumBlk|:|<math>\begin{align}
  H_\text{i} &=\, Y_1 e^{i\mathbf{k}_\text{i}\mathbf{\cdot r}}\\
  H_\text{r} &=\, Y_1 r_{p\,} e^{i\mathbf{k}_\text{r}\mathbf{\cdot r}}\\
  H_\text{t} &=\, Y_2 t_{p\,} e^{i\mathbf{k}_\text{t}\mathbf{\cdot r}}.
\end{align}</math>|{{EquationRef|18}}}}

At the interface, the tangential components of the {{math|'''E'''}} and {{math|'''H'''}} fields must be continuous; that is,
{{NumBlk|:|<math>\left.\begin{align}
  E_\text{i}\cos\theta_\text{i} - E_\text{r}\cos\theta_\text{i} &= E_\text{t}\cos\theta_\text{t}\\
  H_\text{i} + H_\text{r} &= H_\text{t}
\end{align}~~\right\}~~~\text{at}~~ y=0\,.</math>|{{EquationRef|19}}}}
When we substitute from equations ({{EquationNote|17}}) and ({{EquationNote|18}}) and then from ({{EquationNote|7}}), the exponential factors again cancel out, so that the interface conditions reduce to
{{NumBlk|:|<math>\begin{align}
  \cos\theta_\text{i} - r_\text{p}\cos\theta_\text{i} &=\, t_\text{p}\cos\theta_\text{t}\\
  Y_1 + Y_1 r_\text{p} &=\, Y_2 t_\text{p} \,.
\end{align}</math>|{{EquationRef|20}}}}
Solving for {{math|''r''<sub>p</sub>}} and {{math|''t''<sub>p</sub>}}, we find
{{NumBlk|:|<math>r_\text{p}=\frac{Y_2\cos\theta_\text{i}-Y_1\cos\theta_\text{t}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}</math>|{{EquationRef|21}}}}
and
{{NumBlk|:|<math>t_\text{p}=\frac{2Y_1\cos\theta_\text{i}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\,.</math>|{{EquationRef|22}}}}
At ''normal incidence'' {{math|(''θ''<sub>i</sub> {{=}} ''θ''<sub>t</sub> {{=}} 0)}} indicated by an additional subscript 0, these results become
{{NumBlk|:|<math>r_\text{p0}=\frac{Y_2-Y_1}{Y_2+Y_1}</math>|{{EquationRef|23}}}}
and
{{NumBlk|:|<math>t_\text{p0}=\frac{2Y_1}{Y_2+Y_1}\,.</math>|{{EquationRef|24}}}}
At {{itco|''grazing incidence''}} {{math|(''θ''<sub>i</sub> → 90°)}}, we again have {{math|cos{{tsp}}''θ''<sub>i</sub> → 0}}, hence {{math|''r''<sub>p</sub> → −1}} and {{math|''t''<sub>p</sub> → 0}}.

Comparing ({{EquationNote|23}}) and ({{EquationNote|24}}) with ({{EquationNote|15}}) and ({{EquationNote|16}}), we see that at ''normal'' incidence, under the adopted sign convention, the transmission coefficients for the two polarizations are equal, whereas the reflection coefficients have equal magnitudes but opposite signs. While this clash of signs is a disadvantage of the convention, the attendant advantage is that the signs agree at ''grazing'' incidence.

=== Power ratios (reflectivity and transmissivity) ===
The ''[[Poynting vector]]'' for a wave is a vector whose component in any direction is the ''[[irradiance]]'' (power per unit area) of that wave on a surface perpendicular to that direction. For a plane sinusoidal wave the Poynting vector is {{math|{{sfrac|2}}{{px2}}Re{{mset|'''E''' × '''H'''<sup>∗</sup>}}}}, where {{math|'''E'''}} and {{math|'''H'''}} are due ''only'' to the wave in question, and the asterisk denotes complex conjugation. Inside a lossless dielectric (the usual case), {{math|'''E'''}} and {{math|'''H'''}} are in phase, and at right angles to each other and to the wave vector {{math|'''k'''}}; so, for s polarization, using the {{mvar|z}} and {{mvar|xy}} components of {{math|'''E'''}} and {{math|'''H'''}}  respectively (or for p polarization, using the {{mvar|xy}} and {{math|−''z''}} components of {{math|'''E'''}} and {{math|'''H'''}}), the [[irradiance]] in the direction of {{math|'''k'''}} is given simply by {{math|''EH''/2}}, which is {{math|''E''<sup>2</sup>/2''Z''}} in a medium of intrinsic impedance {{math|''Z''{{nnbsp}}{{=}}{{nnbsp}}1/''Y''}}. To compute the irradiance in the direction normal to the interface, as we shall require in the definition of the power transmission coefficient, we could use only the {{mvar|x}} component (rather than the full {{mvar|xy}} component) of {{math|'''H'''}} or {{math|'''E'''}} or, equivalently, simply multiply {{math|''EH''/2}} by the proper geometric factor, obtaining {{math|{{big|(}}''E''<sup>2</sup>/2''Z''{{big|)}}cos{{tsp}}''θ''}}.

From equations ({{EquationNote|13}}) and ({{EquationNote|21}}), taking squared magnitudes, we find that the ''[[reflectivity]]'' (ratio of reflected power to incident power) is
{{NumBlk|:|<math>R_\text{s}=\left|\frac{Y_1\cos\theta_\text{i}-Y_2\cos\theta_\text{t}}{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\right|^2</math>|{{EquationRef|25}}}}
for the s polarization, and
{{NumBlk|:|<math>R_\text{p}=\left|\frac{Y_2\cos\theta_\text{i}-Y_1\cos\theta_\text{t}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\right|^2</math>|{{EquationRef|26}}}}
for the p polarization. Note that when comparing the powers of two such waves in the same medium and with the same cos{{tsp}}''θ'', the impedance and geometric factors mentioned above are identical and cancel out. But in computing the power ''transmission'' (below), these factors must be taken into account.

The simplest way to obtain the power transmission coefficient (''[[transmittance|transmissivity]]'', the ratio of transmitted power to incident power ''in the direction normal to the interface'', i.e. the {{mvar|y}} direction) is to use {{math|''R'' + ''T'' {{=}} 1}} (conservation of energy). In this way we find
{{NumBlk|:|<math>T_\text{s}
=1-R_\text{s}
=\,\frac{4\,\text{Re}\{ Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}\}}{\left|Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}\right|^2}</math>|{{EquationRef|25T}}}}
for the s polarization, and
{{NumBlk|:|<math>T_\text{p}
=1-R_\text{p}
=\,\frac{4\,\text{Re}\{Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}\}}{\left|Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}\right|^2}</math>|{{EquationRef|26T}}}}
for the p polarization.

In the case of an interface between two lossless media (for which ϵ and μ are ''real'' and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations ({{EquationNote|14}}) and ({{EquationNote|22}}). But, for given amplitude (as noted above), the component of the Poynting vector in the {{mvar|y}} direction is proportional to the geometric factor {{math|cos{{nnbsp}}''θ''}} and inversely proportional to the wave impedance {{mvar|Z}}. Applying these corrections to each wave, we obtain two ratios multiplying the square of the amplitude transmission coefficient:
{{NumBlk|:|<math>T_\text{s}
=\left(\frac{2Y_1\cos\theta_\text{i}}{Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}}\right)^2\frac{\,Y_2\,}{Y_1}\,\frac{\cos\theta_\text{t}}{\cos\theta_\text{i}}
=\frac{4Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}}{\left(Y_1\cos\theta_\text{i}+Y_2\cos\theta_\text{t}\right)^2}</math>|{{EquationRef|27}}}}
for the s polarization, and
{{NumBlk|:|<math>T_\text{p}
=\left(\frac{2Y_1\cos\theta_\text{i}}{Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}}\right)^2\frac{\,Y_2\,}{Y_1}\,\frac{\cos\theta_\text{t}}{\cos\theta_\text{i}}
=\frac{4Y_1 Y_2\cos\theta_\text{i}\cos\theta_\text{t}}{\left(Y_2\cos\theta_\text{i}+Y_1\cos\theta_\text{t}\right)^2}</math>|{{EquationRef|28}}}}
for the p polarization. The last two equations apply only to lossless dielectrics, and only at incidence angles smaller than the critical angle (beyond which, of course, {{math|''T'' {{=}} 0}}).

For unpolarized light:
<math display="block">T={1 \over 2}(T_s+T_p)</math>
<math display="block">R={1 \over 2}(R_s+R_p)</math>
where <math>R+T=1</math>.

=== Equal refractive indices ===
From equations ({{EquationNote|4}}) and ({{EquationNote|5}}), we see that two dissimilar media will have the same refractive index, but different admittances, if the ratio of their permeabilities is the inverse of the ratio of their permittivities. In that unusual situation we have {{math|''θ''<sub>t</sub> {{=}} ''θ''<sub>i</sub>}} (that is, the transmitted ray is undeviated), so that the cosines in equations ({{EquationNote|13}}), ({{EquationNote|14}}), ({{EquationNote|21}}), ({{EquationNote|22}}), and ({{EquationNote|25}}) to ({{EquationNote|28}}) cancel out, and all the reflection and transmission ratios become independent of the angle of incidence; in other words, the ratios for normal incidence become applicable to all angles of incidence.<ref>{{cite journal |first1=C.L. |last1=Giles |first2=W.J. |last2=Wild |title=Fresnel Reflection and Transmission at a Planar Boundary from Media of Equal Refractive Indices |journal=Applied Physics Letters |volume=40 |pages=210–212 |year=1982|issue=3 |doi=10.1063/1.93043 |bibcode=1982ApPhL..40..210G |s2cid=118838757 }}</ref> When extended to spherical reflection or scattering, this results in the Kerker effect for [[Mie scattering]].

=== Non-magnetic media ===
Since the Fresnel equations were developed for optics, they are usually given for non-magnetic materials. Dividing ({{EquationNote|4}}) by ({{EquationNote|5}})) yields
<math display=block>Y=\frac{n}{\,c\mu\,}\,.</math>
For non-magnetic media we can substitute the [[vacuum permeability]] {{math|''μ''<sub>0</sub>}} for {{math|''μ''}}, so that
<math display=block>Y_1=\frac{n_1}{\,c\mu_0} ~~;~~~ Y_2=\frac{n_2}{\,c\mu_0}\,;</math>
that is, the admittances are simply proportional to the corresponding refractive indices. When we make these substitutions in equations ({{EquationNote|13}}) to ({{EquationNote|16}}) and equations ({{EquationNote|21}}) to ({{EquationNote|26}}), the factor ''cμ''<sub>0</sub> cancels out. For the amplitude coefficients we obtain:<ref name=Sernelius /><ref name="Born 1970" />

{{NumBlk|:|<math>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}}</math>|{{EquationRef|29}}}}
{{NumBlk|:|<math>t_\text{s}=\frac{2n_1\cos\theta_\text{i}}{n_1\cos\theta_\text{i}+n_2\cos\theta_\text{t}}\,</math>|{{EquationRef|30}}}}

{{NumBlk|:|<math>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}}</math>|{{EquationRef|31}}}}<!-- PLEASE DON'T CHANGE THE SIGN OF THIS EXPRESSION JUST BECAUSE YOUR FAVORITE TEXTBOOK USES A DIFFERENT SIGN CONVENTION FROM THE ONE DEFINED IN THIS ARTICLE AND ITS REFERENCES. -->
{{NumBlk|:|<math>t_\text{p}=\frac{2n_1\cos\theta_\text{i}}{n_2\cos\theta_\text{i}+n_1\cos\theta_\text{t}}\,.</math>|{{EquationRef|32}}}}

For the case of normal incidence these reduce to:

{{NumBlk|:|<math>r_\text{s0}=\frac{n_1-n_2}{n_1+n_2}</math>|{{EquationRef|33}}}}
{{NumBlk|:|<math>t_\text{s0}=\frac{2n_1}{n_1+n_2}</math>|{{EquationRef|34}}}}

{{NumBlk|:|<math>r_\text{p0}=\frac{n_2-n_1}{n_2+n_1}</math>|{{EquationRef|35}}}}
{{NumBlk|:|<math>t_\text{p0}=\frac{2n_1}{n_2+n_1}\,.</math>|{{EquationRef|36}}}}

The power reflection coefficients become:
{{NumBlk|:|<math>R_\text{s}=\left|\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}}\right|^2</math>|{{EquationRef|37}}}}
{{NumBlk|:|<math>R_\text{p}=\left|\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}}\right|^2\,.</math>|{{EquationRef|38}}}}
The power transmissions can then be found from {{math|''T''{{nnbsp}}{{=}}{{nnbsp}}1{{nnbsp}}−{{nnbsp}}''R''}}.

=== Brewster's angle ===
For equal permeabilities (e.g., non-magnetic media), if {{math|''θ''<sub>i</sub>}} and {{math|''θ''<sub>t</sub>}} are ''[[complementary angles|complementary]]'', we can substitute {{math|sin{{tsp}}''θ''<sub>t</sub>}} for {{math|cos{{tsp}}''θ''<sub>i</sub>}}, and {{math|sin{{tsp}}''θ''<sub>i</sub>}} for {{math|cos{{tsp}}''θ''<sub>t</sub>}}, so that the numerator in equation ({{EquationNote|31}}) becomes {{math|''n''<sub>2</sub>{{px2}}sin{{tsp}}''θ''<sub>t</sub> − ''n''<sub>1</sub>{{px2}}sin{{tsp}}''θ''<sub>i</sub>}}, which is zero (by Snell's law). Hence {{math|''r''<sub>p</sub> {{=}} 0}}{{tsp}} and only the s-polarized component is reflected. This is what happens at the [[Brewster angle]]. Substituting {{math|cos{{tsp}}''θ''<sub>i</sub>}} for {{math|sin{{tsp}}''θ''<sub>t</sub>}} in Snell's law, we readily obtain
{{NumBlk|:|<math>\theta_\text{i}=\arctan(n_2/n_1)</math>|{{EquationRef|39}}}}
for Brewster's angle.

=== Equal permittivities ===
Although it is not encountered in practice, the equations can also apply to the case of two media with a common permittivity but different refractive indices due to different permeabilities. From equations ({{EquationNote|4}}) and ({{EquationNote|5}}), if {{math|''ϵ''}} is fixed instead of {{math|''μ''}}, then {{mvar|Y}} becomes ''inversely'' proportional to {{mvar|n}}, with the result that the subscripts 1 and 2 in equations ({{EquationNote|29}}) to ({{EquationNote|38}}) are interchanged (due to the additional step of multiplying the numerator and denominator by {{math|''n''<sub>1</sub>''n''<sub>2</sub>}}). Hence, in ({{EquationNote|29}}) and ({{EquationNote|31}}), the expressions for {{math|''r''<sub>s</sub>}} and {{math|''r''<sub>p</sub>}} in terms of refractive indices will be interchanged, so that Brewster's angle ({{EquationNote|39}}) will give {{math|''r''<sub>s</sub> {{=}} 0}} instead of {{math|''r''<sub>p</sub> {{=}} 0}}, and any beam reflected at that angle will be p-polarized instead of s-polarized.<ref>More general Brewster angles, for which the angles of incidence and refraction are not necessarily complementary, are discussed in C.L.&nbsp;Giles and W.J.&nbsp;Wild, [http://clgiles.ist.psu.edu/pubs/brewster-magnetic.pdf "Brewster angles for magnetic media"], ''International Journal of Infrared and Millimeter Waves'', vol.{{tsp}}6, no.{{tsp}}3 (March 1985), pp.{{tsp}}187–97.</ref> Similarly, Fresnel's sine law will apply to the p polarization instead of the s polarization, and his tangent law to the s polarization instead of the p polarization.

This switch of polarizations has an analog in the old mechanical theory of light waves (see ''[[#History|§{{nnbsp}}History]]'', above). One could predict reflection coefficients that agreed with observation by supposing (like Fresnel) that different refractive indices were due to different ''densities'' and that the vibrations were ''normal'' to what was then called the [[plane of polarization]], or by supposing (like [[James MacCullagh|MacCullagh]] and [[Franz Ernst Neumann|Neumann]]) that different refractive indices were due to different ''elasticities'' and that the vibrations were ''parallel'' to that plane.<ref>Whittaker, 1910, pp. 133, 148–149; Darrigol, 2012, pp. 212, 229–231.</ref> Thus the condition of equal permittivities and unequal permeabilities, although not realistic, is of some historical interest.