# In-Band Pumped Thulium-Doped Tellurite Glass Microsphere Laser
## 1 Introduction  
Whispering gallery mode (WGM) resonators with high Q-factors and a small mode volume providing efficient light-matter interaction are very attractive photonic elements for obtaining optical frequency combs for different applications [1; 2; 3; 4; 5; 6; 7; 8; 9], creating optical filters, switches, microlasers, sensors, and other miniature optoelectronic devices [10; 11; 12; 13; 14; 15; 16]. Microlasers in the two-micron range are potentially interesting for extensive application in remote chemical sensing, biomedicine, and atmosphere monitoring. Lasing in the two-micron range is easily achieved in microspheres doped with Tm\({}^{3+}\) and/or Ho\({}^{3+}\) ions (sometimes with the addition of other co-dopants) [17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. Due to high gain of Tm\({}^{3+}\) ions, lasing is easily achieved in various glass matrices such as silica [17], tellurite [18; 19; 20], fluorotellurite [21], fluoride [22], and even chalcogenide [23]. Tellurite glasses are of great interest for the development of microlasers in the two-micron range [18]. Current technologies allow producing low-loss tellurite glasses characterized by substantial solubility of rare-earth ions [27], which is beneficial in fabricating such photonic devices. Additionally, a high linear refractive index \(n\)-2 for tellurite glasses leads to an emission cross section higher than for silica glass with refractive index \(n\)-1.4 [27], since the cross section is proportional to \((n^{2}+2)^{2}/9n\). Despite a large number of experimental studies on laser generation in thulium-doped microresonators made of different glasses, insufficient attention has been paid to theoretical models and numerical investigations. There are only a few theoretical studies of Tm-doped microlasers. A rather complex theoretical model for lasing at \(\sim\)2 \(\upmu\)m in a Tm-doped tellurite microsphere coupled with an undoped microsphere was proposed and studied in the recent work by [PERSON] et al. [28]. Recently, we investigated theoretically the possibility of single-color and multi-color cascade lasing at wavelengths of about 1.9, 1.5, and 2.3 \(\upmu\)m in Tm-doped tellurite microspheres pumped at 792 nm [29]. Diagrams of various regimes were plotted depending on the parameters of the system (such as Q-factors and pumppowers) [29]. After that, [PERSON] et al. demonstrated experimentally dual-wavelength lasing at 1.9 and 2.3 \(\upmu\)m for the first time in a Tm-doped fluorotellurite microsphere [21], thus confirming the performance of our theoretical model [29]. Here, we concentrate on studying laser generation in the 1.9-2 \(\upmu\)m range at the \({}^{3}\)F\({}_{4}\rightarrow{}^{3}\)H\({}_{6}\) transition of Tm\({}^{3+}\) ions with in-band pump in the range of about 1.55 \(\upmu\)m, which is highly attractive for many practical applications. Currently, there are plenty of commercially available laser sources at 1.55 \(\upmu\)m demanded in telecom applications; therefore, their use for pumping thulium microlasers is a fairly straightforward solution. The simple theoretical model presented here can be used to describe Tm-doped microlasers based on various glass matrices, including tellurite (which is also analyzed in detail in this work), silica, fluorotellurite, fluoride, and others. Microlasers based on different active rare-earth ions can be described by a system of ordinary differential equations of the first order in time [30; 31]. The most common way to analyze such microlasers is numerical solution of a dynamical system (commonly by the Runge-Kutta method). A steady state solution that is of the greatest interest can be found by simulating the laser dynamics over a long time sufficient for relaxation oscillations to decay [31]. However, with a large number of WGMs falling into the gain band of an active ion, the dynamical system contains a very large number of coupled equations, which can take a lot of time to solve, especially for many realizations with varied parameters.