The McCumber relation (or McCumber theory) is a relationship between the effective cross-sections of absorption and emission of light in the physics of solid-state lasers.[1] [2] It is named after Dean McCumber, who proposed the relationship in 1964.
Let
\sigma\rm(\omega)
\sigma\rm(\omega)
\omega
~T~
(1)
| \sigma\rm(\omega) | |
| \sigma\rm(\omega) |
\exp\left(
| \hbar\omega | \right) =\left( | |
| k\rmT |
| N1 | |
| N2 |
\right)T =\exp\left(
| \hbar\omega\rm | |
| k\rmT |
\right)
| \left( | N1 |
| N2 |
\right)T
\omega\rm
\hbar
k\rm
~\omega~
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
\sigma\rm(\omega)
\sigma\rm(\omega)
~\omega~
(2)
~~~~~~~~~~~~~~~G(\omega)=N2\sigma\rm(\omega)-N1\sigma\rm(\omega)
D.E.McCumber had postulated these properties and found that the emission and absorption cross-sections are not independent;[2] they are related with Equation (1).
In the case of an idealized two-level atom the detailed balance for the emission and absorption which preserves the 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)
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
\sigma\rm
\sigma\rm
N1
N2
Let
~v(\omega)~
~n2\sigma\rm(\omega)v(\omega)D(\omega)~
~n1\sigma\rm(\omega)v(\omega)D(\omega)~
a(\omega)n2
(3)
~~~ n2\sigma\rm(\omega)v(\omega)D(\omega)+n2a(\omega)= n1\sigma\rm(\omega)v(\omega)D(\omega) ~~~~~~~~~~~~~~~{\rm(balance)}
Which can be rewritten as
(4)
| ~~~ D(\omega)= |
| ||||||
|
~~~~~~~~~~~~~~{\rm(D1)}
The thermal distribution of density of photons follows from blackbody radiation [5]
(5)
| ~~~ D(\omega)~=~ |
| |||||||
|
~~~~~ {\rm(D2)}
Both (4) and (5) hold for all frequencies
~\omega~
~\sigma\rm(\omega)=\sigma\rm(\omega)~
~n1/n2=\exp\left(
| \hbar\omega | |
| k\rmT |
\right)
a(\omega)
~\sigma\rm(\omega)~
~a(\omega){\rmd}\omega{\rmd}t~
~(\omega,\omega+{\rmd}\omega)~
~(t,t+{\rmd}t)~
~t~
For each site number
~s~
j
~as,j(\omega)~
(6)
~~~ as,j(\omega)=\sigmas,j(\omega)
| \omega2v(\omega) | |
| \pi2c3 |
~~. ~~~~~~~~~~~~~~~~{\rmcomparison1} ~~{\rmpartial}
Neglecting the cooperative coherent effects, the emission is additive: for any concentration
~qs~
~ns,j~
~a~
~\sigma\rm~
(7)
| a(\omega) | = | |
| \sigma\rm(\omega) |
| \omega2v(\omega) | |
| \pi2c3 |
~~~~~~~~~~~~~~~~~~(\rmcomparison)(av)
Then, the comparison of (D1) and (D2) gives the relation
(8)
| n1 | |
| n2 |
| \sigma\rm(\omega) | |
| \sigma\rm(\omega) |
= \exp\left(
| \hbar\omega | |
| k\rmT |
\right)~~. ~~~~~~~~{\rm(n1n2)(mc1)}
This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency
\omegaZ
(9)
| ~\left( | n1 |
| n2 |
\right)T= \exp\left(
| \hbar\omega\rm | |
| k\rmT |
\right)~~~~,~~~
the subscript
~T~
(10)
\omega\rm=
| k\rmT | |
| \hbar |
ln\left(
| n1 | |
| n2 |
\right)T~~~~~~~~~~~~~~~~.~~{(\rmoz)}
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.
The McCumber relation is confirmed for various media.[6] [7] In particular, relation (1) makes it possible to approximate two functions of frequency, emission and absorption cross sections, with single fit.[8]
right|400px|thumb|Fig.2. Cross-sections for Yb:Gd2SiO5versus
| λ= | 2\pic |
| \omega |
In 2006 the strong violation of McCumber relation was observed for Yb:Gd2SiO5 and reported in 3 independent journals.[9] [10] [11] Typical behavior of the cross-sections reported is shown in Fig.2 with thick curves. The emission cross-section is practically zero at wavelength975 nm; this property makes Yb:Gd2SiO5 an excellent material for efficient solid-state lasers.
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:Gd2SiO5 and allow it to exchange radiation with a black body through a spectrally-selective window which is transparent in vicinity of 975 nm and reflective at other wavelengths. Due to the lack of emissivity at 975 nm the medium should warm, breaking the thermal equilibrium.
On the base of the second Law of thermodynamics, the experimental results [9] [10] were refuted in 2007. With the McCumber theory, the correction was suggested for the effective emission cross section (black thin curve).[3] Then this correction was confirmed experimentally.[12]