Analytical Model of Thulium-Doped Fiber Laser Pumped by Two-for-One Process

Analytical models of rare-earth-doped fiber lasers were proposed in the past as very efficient yet reliable alternative to comprehensive numerical models. Cross-relaxation in thulium-doped fiber lasers has a great impact on their efficiency through a two-for-one process but it has not been addressed in analytical models so far. Here, the analytical model of thulium-doped fiber laser is proposed that respects the cross-relaxation mechanism of a type: $^{3}$H$_{4}\rightarrow ^{3}$F$_{4}$, $^{3}$H$_{6}\rightarrow ^{3}$F$_{4}$. The model is based on a two-level model and enables to express the threshold power and slope efficiency in a closed form. The effect of intrinsic losses is modeled by a physically motivated semi-empirical formula. The model is compared with the numerical one and with experimental data. Its application limits are discussed. An approximate method for checking validity conditions of the model is proposed.