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{{Expert-subject|Physics|date=March 2009}}
The '''McCumber relation''' (or McCumber theory) refers to the effective cross-sections of absorption and emission of light in the physics of [[solid-state laser]]s
.<ref name="mc">D.E.McCumber. Einstein relations connecting broadband emission and absorption spectra. [[w:Physical Review B|PRB]] ''' 136''' (4A), 954&ndash;957 (1964)
</ref><ref name="B">P.C.Becker, N.A.Olson, J.R.Simpson. ''Erbium-doped fiber amplifiers: fundamentals and theory'' (Academic, 1999).
</ref>

==Definition==
Let <math>\sigma_{\rm a}(\omega)</math> be the effective absorption cross-section <math>\sigma_{\rm e}(\omega)</math> be effective emission cross-sections at frequency <math>\omega</math>, and let <math>~T~</math> be the effective temperature of the medium. The McCumber relation is

:(1) <math>\frac{\sigma_{\rm e}(\omega)}{\sigma_{\rm a}(\omega)}\exp\!\left( \frac{\hbar \omega}{k_{\rm B} T}\right)
=\left(\frac{N_1}{N_2}\right)_T
=\exp\!\left( \frac{\hbar \omega_{\rm z}}{k_{\rm B} T}\right)</math>
where <math> \left(\frac{N_1}{N_2}\right)_T</math> 
is thermal steady-state ratio of populations; frequency <math>\omega_{\rm z}</math> is called "zero-line" frequency;<ref name="comment">
{{cite journal
|author=D. Kouznetsov
|title=Comment on Efficient diode-pumped Yb:Gd<sub>2</sub>SiO<sub>5</sub> laser (Appl.Phys.Lett.88,221117(2006))
|journal=[[Applied Physics Letters|APL]]
|volume=90
|pages=066101 
|year=2007
|url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000090000006066101000001&idtype=cvips&gifs=yes
}}
</ref><ref name="chi">{{cite journal
|author= D.Kouznetsov 
|title=Broadband laser materials and the McCumber relation 
|journal=[[Chinese Optics Letters]]
|volume= 5 |pages=S240–S242 |year=2007
|url=http://col.org.cn/getArticle.aspx?id=OJ070715000250D0GcIf 
|format= &ndash; <sup>[http://scholar.google.co.uk/scholar?hl=en&lr=&q=author%3A+intitle%3ABroadband+laser+materials+and+the+McCumber+relation&as_publication=%5B%5BChinese+Optics+Letters%5D%5D&as_ylo=2007&as_yhi=2007&btnG=Search Scholar search]</sup>
}} {{Dead link|date=June 2008}}</ref>
<math>\hbar </math> is the [[Planck constant]] and 
<math>k_{\rm B} </math> is the [[Boltzmann constant]]. Note that the right-hand side of Equation (1) does not depend on <math>~\omega~</math>.

==Gain==
It is typical that the lasing properties of a medium are determined by the temperature and the population at the excited laser level, and are not sensitive to the method of excitation used to achieve it. In this case, the absorption cross-section 
<math>\sigma_{\rm a}(\omega)</math> 
and the emission cross-section
<math>\sigma_{\rm e}(\omega)</math> 
at frequency <math>~\omega~</math> can be related to the lasers [[gain(lasers)|gain]] in such a way, that the [[gain(lasers)|gain]] at this frequency can be determined as follows:
:(2) <math>~~~~~~~~~~~~~~~G(\omega)=N_2 \sigma_{\rm e}(\omega)-N_1 \sigma_{\rm a}(\omega)</math>

D.E.McCumber had postulated these properties and found that the emisison and absorption cross-sections are not independent
;<ref name="mc" /><ref name="B"/> they are related with Equation (1).

==Idealized atoms==
In the case of an idealized [[two-level atom]] the [[detailed balance]] for the [[Emission (electromagnetic radiation)|emission]] and [[absorption (electromagnetic radiation)|absorption]] which preserves the [[Planck|Max Planck]] formula for the 
[[black body radiation]] leads to equality of cross-section of absorption and emission. In the solid-state lasers the splitting of each of laser levels leads to the broadening which greatly exceeds the [[natural spectral linewidth]]. In the case of an ideal two-level atom, the product of the linewidth and the lifetime is of order of unity, which obeys the [[Heisenberg uncertainty principle]]. In solid-state laser materials, the linewidth  is several orders of magnitude larger so the spectra of emission and absorption are determined by distribution of excitation among sublevels rather than by the shape of the spectral line of each individual transition between sublevels. This distribution is determined by the effective temperature within each of laser levels. The McCumber hypothesis is that the distribution of excitation among sublevels is thermal. The effective temperature determines the spectra of emission and absorption ( The ''effective temperature'' is called a ''temperature'' by scientists even if the excited medium as whole is pretty far from the thermal state )

==Deduction of the McCumber relation==
[[Image:McCumberRelationActiveCemters.png|right|500px|thumb|Fig.1. Sketch of sublevels]]
Consider the set of active centers (fig.1.). Assume fast transition between sublevels within each level, and slow transition between levels.
According to the McCumber hypothesis, the cross-sections <math>\sigma_{\rm a}</math> and <math>\sigma_{\rm e}</math> do not depend on the populations <math>N_1</math> and <math>N_2</math>.
Therefore, we can deduce the relation, assuming the thermal state.

Let <math>~v(\omega)~</math> be group velocity of light in the medium, the product <math>~n_2\sigma_{\rm e}(\omega) v(\omega)D(\omega)~</math> is spectral rate of 
[[stimulated emission]], and <math>~n_1\sigma_{\rm a}(\omega) v(\omega)D(\omega)~</math> is that of absorption; <math>a(\omega)n_2</math> is spectral rate of [[spontaneous emission]]. (Note that in this approximation, there is no such thing as a spontaneous absorption)
The balance of photons gives:
:(3)  <math>~~~
n_2\sigma_{\rm e}(\omega) v(\omega)D(\omega)+n_2 a(\omega)=
n_1\sigma_{\rm a}(\omega) v(\omega)D(\omega)
~~~~~~~~~~~~~~~{\rm (balance)}
 </math>

Which can be rewritten as

:(4) <math>~~~
D(\omega)=
\frac{\frac{a(\omega)}{\sigma_{\rm e}(\omega) v(\omega)}}
{\frac{n_1}{n_2} \frac{\sigma_{\rm a}(\omega)}{\sigma_{\rm e}(\omega)}-1}
~~~~~~~~~~~~~~{\rm (D1)} </math>

The thermal distribution of density of photons follows from 
blackbody radiation <ref name="e2">e2</ref>

:(5) <math>~~~
D(\omega)~=~
\frac{\frac{1}{\pi^2} \frac{\omega^2}{c^3}}
{\exp\!\left(\frac{\hbar\omega}{k_{\rm B}T}\right)-1}
~~~~~
{\rm (D2)}
</math>

Both (4) and  (5) hold for all frequencies <math>~\omega~</math>.
For the case of idealized two-level active centers, 
<math>~\sigma_{\rm a}(\omega)=\sigma_{\rm e}(\omega)~</math>, and <math>~n_1/n_2=\exp\!\left( \frac{\hbar\omega}{k_{\rm B}T} \right)</math>,
which leads to the relation between the spectral rate of spontaneous emission 
<math>a(\omega)</math> and the emission cross-section <math>~\sigma_{\rm e}(\omega)~</math> 
.<ref name="e2" /> 
(We keep the term [[probability of emission]] for the quantity <math>~a(\omega){\rm d}\omega{\rm d}t~</math>,
which is probability of emission of a photon within small spectral interval
<math>~(\omega,\omega+{\rm d}\omega)~</math> 
during a short time interval <math>~(t,t+{\rm d}t)~</math>, assuming that at time <math>~t~</math> the atom is excited.)
The relation (D2) is a fundamental property of spontaneous and stimulated emission, and perhaps the only way to prohibit a spontaneous break of the thermal equilibrium in the thermal state of excitations and photons.

For each site number <math>~s~</math>, for each sublevel number <math>j</math>, the partial spectral emission
probability <math>~a_{s,j}(\omega)~</math> can be expressed from consideration of idealized two-level atoms
:<ref name="e2" />
:(6) <math>~~~
a_{s,j}(\omega)=\sigma_{s,j}(\omega)
\frac{\omega^2 v(\omega)}{\pi^2c^3}~~.
~~~~~~~~~~~~~~~~{\rm comparison1}
~~{\rm partial}
</math>

Neglecting the cooperative coherent effects, the emission is additive:
for any concentration <math>~q_{s}~</math> of sites and for any partial population <math>~n_{s,j}~</math>
of sublevels, the same proportionality between <math>~a~</math> and <math>~\sigma_{\rm e}~</math>
holds for the effective cross-sections:
:(7) <math>
\frac{a(\omega)}{\sigma_{\rm e}(\omega)}=
\frac{\omega^2 v(\omega)}{\pi^2c^3}
~~~~~~~~~~~~~~~~~~(\rm comparison)(av)
 </math>

Then, the comparison of (D1) and  (D2) gives the relation

:(8) <math>
\frac{n_1}{n_2}
\frac{\sigma_{\rm a}(\omega)}
{\sigma_{\rm e}(\omega)}=
\exp\!\left( \frac{\hbar\omega}{k_{\rm B}T}\right)~~.
~~~~~~~~{\rm (n1n2) (mc1)}
 </math>

This relation is equivalent of the McCumber relation (mc), if we define the
zero-line frequency <math>\omega_{Z}</math> as solution of equation

:(9) <math>~\left(\frac{n_2}{n_1}\right)_{\!T}=
\exp\!\left(\frac{\hbar \omega_{\rm Z}}{k_{\rm B}T}\right)~~~~,~~~
</math>

the subscript <math>~T~</math> indicates that the ratio of populations in evaluated in the
thermal state. The zero-line frequency can be expressed as 
:(10) <math> \omega_{\rm Z}=\frac{k_{\rm B}T}{\hbar}
\ln \left(\frac{n_1}{n_2}\right)_{T} ~~~~~~~~~~~~~~~~.~~{(\rm oz)} </math>
Then (n1n2) becomes equivalent of the McCumber
relation (mc).

No specific property of sublevels of active medium is required to keep the McCumber relation. It follows from the 
assumption about quick transfer of energy among excited laser levels and among lower laser levels. 
The McCumber relation (mc) has the same range of validity as the concept of the emission cross-section 
itself.

===Confirmation of the McCumber relation===
The McCumber relation is confirmed for various media.<ref name="validity2">{{cite journal
|author=R.S.Quimby
|title=Range of validity of McCumber theory in relating absorption and emission cross sections
|journal=[[Journal of Applied Physics|J. Appl. Phys.]]
|volume=92
|pages=180–187
|doi=10.1063/1.1485112
|url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JAPIAU000092000001000180000001 
|year=2002|bibcode = 2002JAP....92..180Q }}
</ref><ref name="validity1">{{cite journal
|author=R.M.Martin
|coauthors=R.S.Quimby
|title=Experimental evidence of the validity of the McCumber theory relating emission and absorption for rare-earth glasses
|journal=[[Journal of the Optical Society of America B|JOSAB]]
|volume=23
|issue=9
|pages= 1770–1775
|url=http://josab.osa.org/abstract.cfm?id=97826
|year=2006
|doi=10.1364/JOSAB.23.001770|bibcode = 2006JOSAB..23.1770M }}
</ref>
In particular relation (1) makes it possible to approximate two functions of frequency, emission and absorption cross sections, with single fit
.<ref name="uns">{{cite journal
| author=D.Kouznetsov
| coauthors=J.-F.Bisson, K.Takaichi, K.Ueda
| title=Single-mode solid-state laser with short wide unstable cavity
|url=http://josab.osa.org/abstract.cfm?id=84730
|journal=[[JOSAB]]|volume=22| issue=8| pages=1605–1619
| year=2005
| doi=10.1364/JOSAB.22.001605
| bibcode=2005JOSAB..22.1605K
}}</ref>

==Violation of the McCumber relation and perpetual motion==
[[image:McCumberViolation.png|right|400px|thumb|Fig.2. Cross-sections for Yb:Gd<sub>2</sub>SiO<sub>5</sub>
versus <math> \lambda=\frac{2\pi c}{\omega}</math>]]

In 2006 the strong violation of McCumber relation was "observed" for  Yb:Gd<sub>2</sub>SiO<sub>5</sub> and reported in 3 independent journals.<ref name="apl">
{{cite journal
|author=W. Li
|coauthors=H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, J. Xu.
|title=Efficient diode-pumped  Yb:Gd<sub>2</sub>SiO<sub>5</sub> laser
|journal=[[Applied Physics Letters|APL]] 
|volume= 88
|pages=221117
|url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000088000022221117000001
|year=2006}}
</ref><ref name="oe">
{{cite journal
|author=W.Li
|coauthors=H.Pan, L.Ding, H.Zeng, G.Zhao, C.Yan, L.Su, J.Xu
|title=Diode-pumped continuous-wave and passively mode-locked Yb:Gd<sub>2</sub>SiO<sub>5</sub>laser
|journal=[[Optics Express]]
|volume=14
|url=http://www.opticsexpress.org/search.cfm year=2006
|pages=686–695
|doi=10.1364/OPEX.14.000686
|year=2006|bibcode = 2006OExpr..14..686L
|issue=2 }}
</ref><ref name="ss">{{cite journal
|author=C.Yan
|coauthors=G.Zhao, L.Zhang, J.Xu, X.Liang, D.Juan, W.Li, H.Pan, L.Ding, H.Zeng.
|title=A new Yb-doped oxyorthosilicate laser crystal: Yb:Gd<sub>2</sub>SiO<sub>5</sub>
|journal=[[Solid State Communications]]
|volume=137
|pages=451–455
|doi=10.1016/j.ssc.2005.12.023 
|url=http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVW-4HYTYFW-3&_user=10&_coverDate=02%2F28%2F2006&_alid=615430684&_rdoc=1&_fmt=summary&_orig=search&_cdi=5545&_sort=d&_docanchor=&view=c&_ct=1&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=b6673cac47762a0184b56a206d31ccd6
|year=2006|bibcode = 2006SSCom.137..451Y
|issue=8 }}
</ref> Typical behaivor of the cross-sections reported is shown in FIg.2 with thick curves. The emission cross-section is practically zero at wavelength
975&nbsp;nm; this property makes  Yb:Gd<sub>2</sub>SiO<sub>5</sub> an excellent material for efficient [[solid-state laser]]s.

However, the property reported (thick curves) is not compatible with the  [[Second Law of thermodynamics]].  With such a material, the [[Perpetual motion]] device would be possible. It would be sufficient to fill a box with reflecting walls with  Yb:Gd<sub>2</sub>SiO<sub>5</sub> and allow it to exchange radiation with a [[blackbody]] through a spectrally-selective window which is transparent in vicinity of 975&nbsp;nm and reflective at other wavelengths. Due to the lack of emissivity at 975&nbsp;nm the medium should warm, breaking the thermal equilibrium.

On the base of the [[Second Law of thermodynamics]], the experimental results 
<ref name="apl"/><ref name="oe"/><ref name="ss" />  were refuted in 2007. WIth the McCumner theory, the correction was suggested for the effective emission cross section (black thin cirve).<ref name="comment">
D. Kouznetsov. Comment on Efficient diode-pumped 
Yb:Gd<sub>2</sub>SiO<sub>5</sub> laser. [[Applied Physics Letters|APL]] '''90''', 066101 (2007),
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000090000006066101000001&idtype=cvips&gifs=yes
</ref> Then this correction was confirmed experimentally.<ref name="respo">{{cite journal
|author=G.Zhao
|coauthors=L.Su, J.Xua, and H.Zeng
|title=Response to ''Comment on Efficient diode-pumped Yb:Gd<sub>2</sub>SiO<sub>5</sub> laser'' (Appl. Phys. Lett. '''90''', 066101 2007)
|journal=[[Applied Physics Letters|APL]]
|volume=90
|pages=066103
|url=http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000090000006066103000001&idtype=cvips&gifs=yes
|year=2007}}
</ref>

==References==
{{reflist}}

[[Category:Spectroscopy]]
[[Category:Solid-state lasers]]