text: 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  and  do not depend on the populations  and .
  Therefore, we can deduce the relation, assuming the thermal state.
  Let  be group velocity of light in the medium, the product  is spectral rate of 
  stimulated emission, and  is that of absorption;  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 . For the case of idealized two-level active centers, , and , which leads to the relation between the spectral rate of spontaneous emission  and the emission cross-section . (We keep the term probability of emission for the quantity , which is probability of emission of a photon within small spectral interval  during a short time interval , assuming that at time  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 , for each sublevel number , the partial spectral emission probability  can be expressed from consideration of idealized two-level atoms:
  Neglecting the cooperative coherent effects, the emission is additive: for any concentration  of sites and for any partial population  of sublevels, the same proportionality between  and  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  as solution of equation
  the subscript  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.