The stationary solution of the system

The stationary solution of the system of equations (1) has the form: where
Since the intermode interval Δ0 = 0.05, for calculation convenience, similarly to the supposition made in [6], let us assume that the axial mode of the resonator corresponding to the maximum gain at the 1064.15 nm wavelength is number 70. Then the modes corresponding to the maximum gains at the 1052.10, 1061.50, 1064.40, 1068.20, 1073.70, and 1077.90 nm wavelengths are number 746, 222, 56, –156, –464 and –699. To account for the influence of the homogeneous broadening for a 30 mm-long resonator, all subsequent calculations will assume n equals 2 in Eq. (2) when generation occurs at four axial modes. To account for the inhomogeneous broadening of the laser generation spectrum with an increase in the YAG crystal temperature, the Lorentzian profiles (2) need to be widened using the s(T)-function of inhomogeneous gain line broadening as follows:
The calculations showed that the following changes occur with an increase in inhomogeneous gain line broadening associated with a temperature increase:
The shift to shorter wavelengths that occurs in the generation spectrum with a temperature increase is due to the proximity of the 1061.5 nm gain line. As the Tang–Statz–DeMars system of equations (1) implies that the lower level is not populated, the calculations were performed under the condition that the lower level (the 4I11/2 multiplet) is cleared instantly. It follows from calculations in the system of equations (3) factoring in Eq. (4) that the shift to longer wavelengths with gain line broadening is possible only when the transition cross-sections at the 1061.5 and 1064.4 nm wavelengths tend to zero. As seen in Table 1, the laser transitions with the 1061.50 and 1064.40 nm wavelengths occur between the 4I11/2 multiplet sublevels with the energies of 2002 and 2028 cm−1, respectively. The laser transitions with the 1064.15 and 1068.20 nm wavelengths are to higher sublevels of the 4I11/2 multiplet with the energies of 2110 and 2146 cm−1, respectively. Accepting that the dna ligase of the 4I11/2 multiplet sublevels follows the Boltzmann distribution, the possible difference between the populations of the 4I11/2 multiplet sublevels does not exceed 25%. Therefore, the calculations do not confirm the Boltzmann distribution of the 4I11/2 multiplet populations. Balance equations were composed to determine the population distribution of the 4I11/2 multiplet sublevels. The following assumptions were made to simplify the calculations: where n1, n2 and n3 are the populations of the sublevels of the 4I11/2 multiplet with the energies of 2146, 2110 and 2028 cm−1, respectively; () are the rates of relaxation transitions between the i and j sublevels caused by the phonon–electron influence of the crystal lattice on neodymium ions. The probability of a multi-phonon transition is described [7] by the following dependence on the crystal temperature T and the energy gap ΔЕ: where р is the number of phonons resulting from the phonon–electron interaction; C and α are constants characterizing the basis, in this case, the YAG crystal. It is apparent from (5) that the probability of a relaxation transition is exponentially dependent on the energy gap. Therefore, the probability of a phonon–electron transition between sublevels 1 (2146 cm−1) and 2 (2110 cm−1) is more than the probability of a transition between sublevels 1 and 3 (2028 cm−1), and also 2 and 3. We assumed for the calculations that Since the lifetime of the 4I11/2 multiplet is approximately 10–8s and the time of the phonon–electron transition ≈10–9–10–10 s, the dependence we used was The calculation results showed that the relative population inversion of the multiplet sublevels has the following values:
It was found that this distribution is strongly dependent on the /w and /w20 ratios. The population of the 4I11/2 multiplet sublevels with the energy of 2110 cm−1 depends on the rate of thermal relaxation transitions providing the Boltzmann distribution (Fig. 1a). The calculations showed that in order to adequately model the changes of the emission spectrum from the broadening of gain lines, it is necessary that the relative population of sublevel 2 was not less than 0.3 under heating. We may thus conclude that the phonon–electron relaxation rate in an activated crystal must be higher by at least an order of magnitude than the thermal equilibrium rate in accordance with the Boltzmann distribution.