The McCumber relation can be deduced without assuming that all active centers have the same structure of sublevels. The range of validity of the McCumber relation is the same as that of the effective emission cross-section.
Introduction
The concept of effective cross-sections allow to treat the laser medium as a two-level system. Such a concept is widely used in physics of solid-state lasers; often, these effective cross-sections are called simply cross-sections. The McCumber relation expresses the emission cross-seccion in terms of the absorption cross-section :
where is temperature, is Boltzmann constant and is zero-line frequency, at which the emission and absorption cross-sections are equal. The relation (mc) is validated for various media .
The original deduction of the McCumber relation , as well as the adaptation in the textbook assume, that all active centers are equal. It cannot be applied as is to the broadband laser materials with different sites of the active centers.
This allowed the interpretation of results for Yb:GdSiO by
cites as an indication, that the effective cross-sections of broad-band composite materials have no need to satisfy the McCumber relation: the peak of at wavelength 950 nm corresponds to the gap of (See Fig.0.)
However, a medium with such effective cross-sections
would be good not only for an efficient laser, but also for a Perpetual Motion of Second Kind; the correction of the emission cross-section was suggested (thin black curve in Fig.0) and confirmed. In order to avoid such confusions, the deduction of the McCumber relation should be generalized.
After the presentation , I was asked for the general deduction of the McCumber relation as a substitute of the speculation about the w:gedanken experiment with perpetual motion. Below, such a deduction is suggested.
In this paper, the generalization of the deduction of the McCumber relation is suggested. I show, that the McCumber relation follows from the fundamental properties of the Einstein coefficients , and applies to any material with fast transitions within each of two sets of levels and relatively slow transitions between these two sets.
Active centers
The sketch
of sublevels of active centers
is shown in fig.1. Consider two subsets of quantum states: level 1 and level 2. Assume slow optical transitions from level 1 to level 2.
(This property makes the medium suitable for a laser action.)
Assume quick transfer of energy between neighbors, which leads to
the fast thermalization within each of laser levels.
Then, the refractive index and gain are determined by the populations and of the the laser levels. In this case, and only in this case, the effective cross-sections and of absorption and emission have sense.
Thermalization
Use of effective cross-sections assumes the thermalization
of quantum states within each of laser levels. However, the
population of the laser levels can be far from a thermal state, allowing the lasing. The gain can be expressed as
where and are population of lower and upper laser levels.
Keeping the consideration phenomenological, the spontaneous emission can be characterized with the Einstein coefficients ; the rate of emission of spontaneous photons at frequency can be expressed as
where is probability of spontaneous emission by a random active
center per time per frequency, assuming that it is excited. is equivalent of the Einstein coefficient . Notation is used here to avoid confusion with the Einstein coefficient , which has no established expression (see notes at Table 7.7 of ); not only value, but even dimensions of the Einstein coefficients depend on scale we use: frequencies or wavelengths.
Decay
The decay rate of the excited level
can be expressed in terms of the coefficient :
The cross-section and and the coefficient
do not depend on the populations and of the active medium
and the density of photons of frequency . In this approximation, the properties of the medium are determined by 3 functions , and , and we have no need to consider noninear processes \cite{ desu} which produced a given population; as gain, as refraction index as function of frequency are determined by the populations and . The functions , and are equivalent of the Einstein coefficients, but have an advantage: their values do not depend on system of notations. In the following, the consideration of relations between Einstein coefficients is rewritten, taking into account sublevels (Fig.1).