# In-Band Pumped Thulium-Doped Tellurite Glass Microsphere Laser
## Appendix A  
Here, we show the equivalence of the two-level and quasi three-level models for our system. Let us define in the three-level model: the ground state \({}^{3}\)H\({}_{6}\) as \"1\", the upper laser level as \"2\" and the pump level as \"3\". Note that level 2 and level 3 belong to the same \({}^{3}\)F\({}_{4}\) manifold. For glass matrices, intra-band transitions 3\(\rightarrow\)2 are very rapid at a room temperature (\(\tau_{32}\rightarrow\)0), so the population of level 3 is close to zero (\(n_{3}\to 0\)). Therefore, the  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Thulium Ion Content** & **Diameter of Microsphere, \(\upmu\)m** & **Pump Source; Wavelength, \(\upmu\)m** & **Laser Wavelength, \(\upmu\)m** & **Reference** \\ \hline
5\(\cdot\)10\({}^{19}\) cm\({}^{-3}\) & 320 & Telecommunication laser, 1.55 & \(\sim\)1.9 & Present study \\ \hline
4.2\(\cdot\)10\({}^{20}\) cm\({}^{-3}\) (1 mol\(\%\) Tm\({}_{2}\)O\({}_{3}\)) & 30 & Tunable laser source, 1.504–1.629 & Centered at \(\sim\)1.975 & [20] \\ \hline
0.15\(\%\) Tm\({}_{2}\)O\({}_{3}\) & 104 & Ti: sapphire, 0.8 & \(\sim\)1.5\&1.9 & [39] \\ \hline
0.5 wt\(\%\) Tm\({}_{2}\)O\({}_{3}\) & Not reported & Ti: sapphire, 0.793 & \(\sim\)1.5\&1.9 & [40] \\ \hline
5 wt\(\%\) & 25 & Ti: sapphire, 0.793 & \(\sim\)2.0 & [19] \\ \hline
1 mol\(\%\) Tm\({}_{2}\)O\({}_{3}\) & \(\sim\)60 & Laser diode, 0.793 & \(\sim\)1.92\&2.35; \(\sim\)1.90 & [21] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of Tm\({}^{3*}\)-doped tellurite glass microsphere lasers.  
expression for three-level model \(n_{1}+n_{2}+n_{3}=1\) reduces to \(n_{1}+n_{2}=1\), which coincides with Equation (1).  
Further, the stimulated absorption and emission rates for the three-level model (see Figure A1) are  
\[W*_{12}=\frac{\Gamma}{hcS_{eff}}{\sum\limits_{l}{\lambda_{s,l}\sigma_{abs}( \lambda_{s,l})|A_{s,l}|^{2}}}\] (A1)  
\[W*_{21}=\frac{\Gamma}{hcS_{eff}}{\sum\limits_{l}{\lambda_{s,l}\sigma_{em}( \lambda_{s,l})|A_{s,l}|^{2}}},\] (A2)  
\[W*_{13}=\frac{\Gamma\lambda_{p}\sigma_{abs}(\lambda_{p})}{hcS_{eff}}|A_{p}|^{2}\] (A3)  
\[W*_{31}=\frac{\Gamma\lambda_{p}\sigma_{em}(\lambda_{p})}{hcS_{eff}}|A_{p}|^{2}=0\] (A4)  
The rate equation for the ground state population for the three-level model:  
\[\frac{dn_{1}}{dt}=-W*_{12}n_{1}-W*_{13}n_{1}+\bigg{(}W*_{21}+\frac{1}{\tau_{2 1}}\bigg{)}n_{2}=-(W*_{12}+W*_{13})n_{1}+\bigg{(}W*_{21}+\frac{1}{\tau_{21}} \bigg{)}n_{2}\] (A5)  
Recalling expressions for \(W_{12}\), \(W_{21}\) (Equations (3) and (4)), we obtain:  
\[W_{12}=\frac{\Gamma\lambda_{p}\sigma_{abs}(\lambda_{p})}{hcS_{eff}}|A_{p}|^{2 }+\frac{\Gamma}{hcS_{eff}}{\sum\limits_{l}{\lambda_{s,l}\sigma_{abs}(\lambda_ {s,l})|A_{s,l}|^{2}}}=W*_{13}+W*_{12}\] (A6)  
\[W_{21}=W*_{21}\] (A7)  
and the rate Equation (A5) for the ground state population for the three-level model becomes  
\[\frac{dn_{1}}{dt}=-W_{12}n_{1}+\bigg{(}W_{21}+\frac{1}{\tau_{21}}\bigg{)}n_{2}\] (A8)  
Therefore, Equation (A8) coincides with Equation (2). The expressions for \(g_{\rm P}\) and \(g_{\rm S,l}\) for the three-level model coincide with expressions (7) and (8). Therefore, the two-level and quasi three-level models give the same results for in-band pumped Tm-doped tellurite glass microlasers.