# In-Band Pumped Thulium-Doped Tellurite Glass Microsphere Laser
## 3 Theoretical Results and Discussion  
We consider critical coupling when an intrinsic Q-factor is equal to a coupling Q-factor: \(Q_{0}=Q_{ext}\) (in other words when \(T_{\mathit{eff}}=T_{0}/2=T_{ext}/2\)). We set \(\sigma_{abs}(\lambda_{p})=0.05\cdot 10^{-20}\) cm\({}^{2}\) and the emission/absorption cross sections at laser wavelengths as in the paper [34]. We varied Tm\({}^{3+}\) ion concentrations, microsphere diameters, Q-factors, and pump powers. The calculated coefficients \(\kappa\) as functions of the total Q-factor for different diameters of a tellurite microsphere ranging from 40 \(\upmu\)m to 400 \(\upmu\)m are plotted in Figure 3a as well as expected laser wavelengths (Figure 3b), slope efficiencies (Figure 3c), threshold pump powers (Figure 3d-f), and threshold values of coupling coefficients multiplied by pump powers (Figure 3g-i) as functions of the varied parameters of the system. Figure 3d demonstrates that for a Tm\({}^{3+}\) ion concentration of 1\(\cdot\)10\({}^{19}\) cm\({}^{-3}\), lasing is possible only if the Q-factor exceeds 10\({}^{6}\). For lower Q-factors, losses are too large and the gain cannot compensate them; the laser threshold is not reached. With an increase in active ion concentrations to \(N_{Tm}=5\cdot 10^{19}\) cm\({}^{-3}\) and \(N_{Tm}=10^{20}\)cm\({}^{-3}\), the gain according to Equation (8) grows, and the minimum Q-factors required for generation become equal to 3\(\cdot\)10\({}^{5}\) and 2\(\cdot\)10\({}^{5}\), respectively (Figure 3e,f). Figure 3d-f also demonstrate that for a certain active ion concentration, the larger the microsphere diameter, the higher the threshold pump power is. Figure 3d-f show that with an increase in the Q-factor, the threshold pump powers first decrease, then reach a minimum, and then begin to increase. This has the following explanation. At low Q-factors, a large intracavity pump power is required to overcome losses and reach the lasing threshold. As the Q-factor increases, less losses need to be compensated, so the threshold power decreases. However, at high Q-factors, the coupling coefficient is rather small (see Figure 3a), because \(\kappa\) is inversely proportional to the total Q-factor for the critical coupling. The intracavity pump power becomes lower than for medium Q-factors, which explains the non-monotonic behavior of the curves in Figure 3d-f. For a clearer demonstration, we plot the dependence of the threshold values of coupling coefficients multiplied by pump powers on Q-factor (Figure 3g-i). These curves are monotonic: the higher the Q-factor, the lower intracavity power is required to attain lasing. Let us go back to the expected lasing wavelength. We found that in the used model, it does not depend on the diameter of the microsphere and the pump power, but depends on Q-factor and active ion concentration. At low Q-factors, a higher pump power is required, so the population \(n_{2}\) of the level \({}^{3}\)F\({}_{4}\) is rather high, so the maximum gain is achieved at shorter wavelengths near the maximum of the emission cross section (see Figure 2b). With an improved Q-factor, the laser wavelength grows smoothly. Knowing experimental \(\lambda_{s}\) and \(N_{Tm}\), it is possible to compare the measured value of the Q-factor with the calculated one or even estimate the effective Q-factor (if it is not measured experimentally). The slope efficiencies calculated for pump powers above the threshold are plotted in Figure 3c. They do not depend of microsphere diameters. There is an optimal Q-factor to attain maximum efficiency, which coincides with the general laser theory according to which there is an optimal value of transmission coefficient of a laser resonator (or optical losses) to obtain the maximum output laser power [35]. Our semi-analytical model, which allows fast simulation of a steady-state solution unlike dynamical models used previously, makes it possible to perform a scan over a broad range of laser parameters.