Editing Refractive index (section)

==Complex refractive index==
{{See also|Mathematical descriptions of opacity}}

When light passes through a medium, some part of it will always be [[attenuation|absorbed]]. This can be conveniently taken into account by defining a complex refractive index,
<math display="block">\underline{n} = n - i\kappa.</math>

The real and imaginary part of this refractive index are not independent, and are connected through the [[Kramers–Kronig relations]], i.e. the complex refractive index is a [[linear response function]], ensuring causality. <ref name=lightmatterinteractionbook>{{cite book |last= Stenzel|first=Olaf |date=2022 |title=Light–Matter Interaction |series=UNITEXT for Physics |url=https://link.springer.com/book/10.1007/978-3-030-87144-4 |location= |publisher=Springer, Cham |page=386 |doi=10.1007/978-3-030-87144-4 |isbn=978-3-030-87144-4 }}</ref> Here, the real part {{mvar|n}} is the refractive index and indicates the [[phase velocity]], while the imaginary part {{mvar|κ}} is called the '''extinction coefficient'''<ref name=DresselhausMITCourse>{{cite web
 |url         = http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf
 |title       = Solid State Physics Part II Optical Properties of Solids
 |last        = Dresselhaus
 |first       = Mildred S.
 |author-link = Mildred Dresselhaus
 |date        = 1999
 |website     = Course 6.732 Solid State Physics
 |publisher   = MIT
 |access-date = 2015-01-05
 |url-status   = live
 |archive-url  = https://web.archive.org/web/20150724051216/http://web.mit.edu/course/6/6.732/www/6.732-pt2.pdf
 |archive-date = 2015-07-24
}}</ref>{{rp|36}} indicates the amount of attenuation when the electromagnetic wave propagates through the material.<ref name="Hecht"/>{{rp|p=128}} It is related to the <!-- do not wikilink, the article by that name covers thermal absorption-->'''absorption coefficient''', <math>\alpha_\text{abs}</math>, through:<ref name=DresselhausMITCourse/>{{rp|41}}
<math display="block">\alpha_\text{abs}(\omega) = \frac{2\omega\kappa(\omega)}{c}</math>
These values depend upon the frequency of the light used in the measurement.

That {{mvar|κ}} corresponds to absorption can be seen by inserting this refractive index into the expression for [[electric field]] of a [[plane wave|plane]] electromagnetic wave traveling in the {{mvar|x}}-direction. This can be done by relating the complex [[wave number]] {{mvar|{{uu|k}}}} to the complex refractive index {{mvar|{{uu|n}}}} through {{math|{{uu|''k''}} {{=}} 2π{{uu|''n''}}/''λ''{{sub|0}}}}, with {{math|''λ''{{sub|0}}}} being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the {{mvar|x}}-direction as:
<math display="block">\begin{align}
\mathbf{E}(x, t)
  &= \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(\underline{k}x - \omega t)}\right] \\
  &= \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(2\pi(n + i\kappa)x/\lambda_0 - \omega t)}\right] \\
  &= e^{-2\pi \kappa x/\lambda_0} \operatorname{Re}\! \left[\mathbf{E}_0 e^{i(kx - \omega t)}\right].
\end{align}</math>

Here we see that {{mvar|κ}} gives an exponential decay, as expected from the [[Beer–Lambert law]]. Since intensity is proportional to the square of the electric field, intensity will depend on the depth into the material as

<math display="block">I(x)= I_0 e^{-4\pi \kappa x/\lambda_0} .</math>

and thus the [[Attenuation coefficient#Absorption and scattering coefficients|absorption coefficient]] is {{math|''α'' {{=}} 4π''κ''/''λ''{{sub|0}}}},<ref name="Hecht"/>{{rp|p=128}} and the [[penetration depth]] (the distance after which the intensity is reduced by a factor of {{math|1/''e''}}) is {{math|''δ''{{sub|p}} {{=}} 1/''α'' {{=}} ''λ''{{sub|0}}/4π''κ''}}.

Both {{mvar|n}} and {{mvar|κ}} are dependent on the frequency. In most circumstances {{math|''κ'' > 0}} (light is absorbed) or {{math|''κ'' {{=}} 0}} (light travels forever without loss). In special situations, especially in the [[gain medium]] of [[laser]]s, it is also possible that {{math|''κ'' < 0}}, corresponding to an amplification of the light.

An alternative convention uses {{math|{{uu|''n''}} {{=}} ''n'' + ''iκ''}} instead of {{math|{{uu|''n''}} {{=}} ''n'' − ''iκ''}}, but where {{math|''κ'' > 0}} still corresponds to loss. Therefore, these two conventions are inconsistent and should not be confused. The difference is related to defining sinusoidal time dependence as {{math|Re[exp(−''iωt'')]}} versus {{math|Re[exp(+''iωt'')]}}. See [[Mathematical descriptions of opacity]].

Dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption. However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's [[transparency (optics)|transparency]] to these frequencies.

The real {{mvar|n}}, and imaginary {{mvar|κ}}, parts of the complex refractive index are related through the [[Kramers–Kronig relations]]. In 1986, A.R. Forouhi and I. Bloomer deduced an [[Forouhi–Bloomer model|equation]] describing {{mvar|κ}} as a function of photon energy, {{mvar|E}}, applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for [[Forouhi–Bloomer model|{{mvar|n}} as a function of {{mvar|E}}]]. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988.

The refractive index and extinction coefficient, {{mvar|n}} and {{mvar|κ}}, are typically measured from quantities that depend on them, such as [[Fresnel equations|reflectance, {{mvar|R}}, or transmittance, {{mvar|T}}]], or ellipsometric parameters, [[ellipsometry|{{mvar|ψ}} and {{mvar|δ}}]]. The determination of {{mvar|n}} and {{mvar|κ}} from such measured quantities will involve developing a theoretical expression for {{mvar|R}} or {{mvar|T}}, or {{mvar|ψ}} and {{mvar|δ}} in terms of a valid physical model for {{mvar|n}} and {{mvar|κ}}. By fitting the theoretical model to the measured {{mvar|R}} or {{mvar|T}}, or {{mvar|ψ}} and {{mvar|δ}} using regression analysis, {{mvar|n}} and {{mvar|κ}} can be deduced.

===X-ray and extreme UV===
For [[X-ray]] and [[extreme ultraviolet]] radiation the complex refractive index deviates only slightly from unity and usually has a real part smaller than 1. It is therefore normally written as {{math|{{uu|''n''}} {{=}} 1 − ''δ'' + ''iβ''}} (or {{math|{{uu|''n''}} {{=}} 1 − ''δ'' − ''iβ''}} with the alternative convention mentioned above).<ref name=Attwood/> Far above the atomic resonance frequency delta can be given by
<math display="block"> \delta = \frac{r_0 \lambda^2 n_\mathrm{e}}{2 \pi} </math>
where {{math|''r''{{sub|0}}}} is the [[classical electron radius]], {{mvar|λ}} is the X-ray wavelength, and {{math|''n''{{sub|e}}}} is the electron density. One may assume the electron density is simply the number of electrons per atom {{mvar|Z}} multiplied by the atomic density, but more accurate calculation of the refractive index requires replacing {{mvar|Z}} with the complex [[atomic form factor]] {{nowrap|<math> f = Z + f' + i f'' </math>.}} It follows that
<math display="block">\begin{align}
\delta &= \frac{r_0 \lambda^2}{2 \pi} (Z + f') n_\text{atom} \\
\beta  &= \frac{r_0 \lambda^2}{2 \pi} f'' n_\text{atom}
\end{align}</math>
with {{mvar|δ}} and {{mvar|β}} typically of the order of {{val|e=−5}} and {{val|e=−6}}.