“…The branching ratios, in which we consider fast multiphonon-relaxation processes 4 F 9/2 → 4 I 9/2 → 4 I 11/2 in the intermediate levels, are calculated from the electric-dipole radiative-transition rate constants, radiative lifetimes, and branching ratios given in [29] cm −3 ( = 6.3at.%), a maximum gain of ∼15 dB/cm is calculated, slightly depending on the assumed value of the propagation loss. The simulation overestimates the experimentally achieved gain of ∼12 dB/cm [47] by approximately 25%, which may be due to (i) a non-negligible influence of the ETU 2 and CR processes, (ii) errors in the approximation of the relevant parameter values, and most likely (iii) heating of the waveguide when pumping in the experiment, which leads to a temperature increase and, consequently, a reduction of transitions cross sections. This reduction, which leads to line broadening, is a fundamental process in rare-earth ions.…”