# In-Band Pumped Thulium-Doped Tellurite Glass Microsphere Laser
## 2 Theoretical Model  
The general scheme of the analyzed system is shown in Figure 1. It consists of a thulium-doped tellurite glass microsphere and a silica fiber taper for both, coupling the CW pump wave and extracting the generated laser wave. We consider in-band CW pump (\({}^{3}\)H\({}_{6}\rightarrow{}^{3}\)F\({}_{4}\)) at a wavelength of 1.55 \(\upmu\)m and lasing in the 2 \(\upmu\)m range (\({}^{3}\)F\({}_{4}\rightarrow{}^{3}\)H\({}_{6}\)) as shown in Figure 2a. The cross-relaxation process, which can play an important role for a pump at \(\sim\)0.79 \(\upmu\)m [32], is neglected here, because levels above \({}^{3}\)F\({}_{4}\) are almost unpopulated under in-band pump. Here, we used the two-level approximation, which is equivalent to the quasi three-level model because the transition from the pump level to the upper laser level (both belong to the \({}^{3}\)F\({}_{4}\) manifold) is very rapid at a room temperature, so the population of the pump level is almost zero (please see Appendix A for details). Let us define the population densities of the \({}^{3}\)H\({}_{6}\) and \({}^{3}\)F  
Figure 1: A simplified scheme of the considered system. Ppump is a pump power and \(\kappa\) is a coupling coefficient. levels as \(n_{1}\) and \(n_{2}\). They are normalized to the concentration of Tm\({}^{3+}\) ions \(N_{Tm}\) and satisfy the expression  
\[n_{1}+n_{2}=1. \tag{1}\]  
An important characteristic is a gain function proportional to (\(\sigma_{em}n_{2}-\sigma_{abs}n_{1}\)), where \(\sigma_{em}\) and \(\sigma_{abs}\) are the emission and absorption cross sections, respectively. The calculated normalized gain function depending on wavelength is given in Figure 2b. The wavelength corresponding to the maximum of the function depends on \(n_{2}\) and determines the generated wavelength. Note that the emission cross-section at the pump wavelength \(\lambda_{p}=1.55\) um is almost zero \(\sigma_{em}(\lambda_{p})\approx 0\) for similar tellurite glasses [33; 34]. The rate equation for the ground state population is [32; 34; 35]:  
\[\frac{dn_{1}}{dt}=-W_{12}n_{1}+\left(W_{21}+\frac{1}{\tau_{21}}\right)n_{2}. \tag{2}\]  
where \(\tau_{21}=3\) ms [32] is the lifetime of the \({}^{3}\)F\({}_{4}\) level and \(W_{12,21}\) are the stimulated absorption and emission rates (assuming \(\sigma_{em}(\lambda_{p})\approx 0\))  
\[W_{12}=\frac{\Gamma\lambda_{p}\sigma_{abs}(\lambda_{p})}{hc\xi_{eff}}|A_{p}|^{ 2}+\frac{\Gamma}{hc\xi_{eff}}\sum_{l}\lambda_{s,l}\sigma_{abs}(\lambda_{s,l}) |A_{s,l}|^{2}, \tag{3}\]  
\[W_{21}=\frac{\Gamma}{hc\xi_{eff}}\sum_{l}\lambda_{s,l}\sigma_{em}(\lambda_{s, l})|A_{s,l}|^{2}. \tag{4}\]  
Here, \(\lambda_{s,l}\) is the laser wavelength with the mode order \(l\) (we consider here only the family of fundamental modes with angular mode order \(l\)), \(h\) is [PERSON]'s constant, \(c\) is the speed of light in vacuum, \(S_{\mathit{eff}}\) is the effective mode field area, \(\Gamma=0.95\) is the overlap factor (defined in [28]), \(A_{p}\) is the intracavity field amplitude at the pump wavelength, and \(A_{s,l}\) is the intracavity field amplitude at laser wavelength. For the fundamental mode family, the overlap factor is almost independent on the laser wavelength in the 1.9-2 um range. We calculated previously that for the fundamental mode family, \(S_{\mathit{eff}}\) is almost a linear function of the tellurite microsphere radius changing from \(S_{\mathit{eff}}\approx 6\) um\({}^{2}\) up to \(S_{\mathit{eff}}\approx 20\) um\({}^{2}\) when the radius increases from 50 um to 200 um [36]. The temporal evolution of the intracavity field amplitudes inside the microsphere can be described by the system of ordinary differential Equations [29; 30; 31; 35]:  
\[\frac{dA_{p}}{dt}=-\frac{1}{2}\Bigg{(}\frac{1}{T_{\mathit{eff}}}-g_{p}-i \Delta\omega\Bigg{)}A_{p}+i\sqrt{\frac{1}{T_{\mathit{ext}}T_{\mathit{RT}}}}E_ {pump} \tag{5}\]  
\[\frac{dA_{s,l}}{dt}=-\frac{1}{2}\Bigg{(}\frac{1}{T_{\mathit{eff}}}-g_{s,l} \Bigg{)}A_{s,l} \tag{6}\]  
Figure 2: (a) A simplified scheme of laser levels for in-band pump.