McCumber relation

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. It is named after Dean McCumber, who proposed the relationship in 1964.
Definition.
Let formula_1 be the effective absorption cross-section formula_2 be effective emission cross-sections at frequency formula_3, and let formula_4 be the effective temperature of the medium. The McCumber relation is
where formula_6 
is thermal steady-state ratio of populations; frequency formula_7 is called "zero-line" frequency;
formula_8 is the Planck constant and 
formula_9 is the Boltzmann constant. Note that the right-hand side of Equation (1) does not depend on formula_10.
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 
formula_1 
and the emission cross-section
formula_2 
at frequency formula_10 can be related to the lasers gain in such a way, that the gain at this frequency can be determined as follows:
D.E.McCumber had postulated these properties and found that the emission and absorption cross-sections are not independent; they are related with Equation (1).
Idealized atoms.
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 )
Deduction of the McCumber relation.
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 formula_15 and formula_16 do not depend on the populations formula_17 and formula_18.
Therefore, we can deduce the relation, assuming the thermal state.
Let formula_19 be group velocity of light in the medium, the product formula_20 is spectral rate of 
stimulated emission, and formula_21 is that of absorption; formula_22 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:
Which can be rewritten as
The thermal distribution of density of photons follows from blackbody radiation 
Both (4) and (5) hold for all frequencies formula_10. For the case of idealized two-level active centers, formula_27, and formula_28, which leads to the relation between the spectral rate of spontaneous emission formula_29 and the emission cross-section formula_30. (We keep the term probability of emission for the quantity formula_31, which is probability of emission of a photon within small spectral interval formula_32 during a short time interval formula_33, assuming that at time formula_34 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 formula_35, for each sublevel number formula_36, the partial spectral emission probability formula_37 can be expressed from consideration of idealized two-level atoms:
Neglecting the cooperative coherent effects, the emission is additive: for any concentration formula_39 of sites and for any partial population formula_40 of sublevels, the same proportionality between formula_41 and formula_42 holds for the effective cross-sections:
Then, the comparison of (D1) and (D2) gives the relation
This relation is equivalent of the McCumber relation (mc), if we define the zero-line frequency formula_45 as solution of equation
the subscript formula_4 indicates that the ratio of populations in evaluated in the thermal state. The zero-line frequency can be expressed as 
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.
In particular, relation (1) makes it possible to approximate two functions of frequency, emission and absorption cross sections, with single fit
Violation of the McCumber relation and perpetual motion.
In 2006 the strong violation of McCumber relation was observed for Yb:Gd2SiO5 and reported in 3 independent journals. Typical behavior of the cross-sections reported is shown in Fig.2 with thick curves. The emission cross-section is practically zero at wavelength
975 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 
 were refuted in 2007. With the McCumber theory, the correction was suggested for the effective emission cross section (black thin curve).
Then this correction was confirmed experimentally.