Editing Fresnel equations (section)

=== 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|)}}