Analytical conditions for co-lasing in thulium ion-doped lasers at 2 μm and 2.3 μm

By assuming low doping concentration with negligible concentration-dependent energy transfer mechanisms, we employ rate-equation formalism to derive the coupled power-gain equations in steady state and investigate the requirements of co-lasing at 2 μm and 2.3 μm in continuous-wave thulium ion (Tm³⁺)-doped lasers. We obtain an analytical condition which provides a criterion about which optical transition lases first. Analytical expressions are further derived for single-lasing and co-lasing threshold powers. By using the spectroscopic parameters for Tm³⁺:YLiF₄, our model predicts that 2.3 μm lasing can start first in gain media with low Tm³⁺ ion doping if the length of the gain medium is sufficiently long. We then discuss how the results of the low-concentration model developed in this study can be extended to the case of high doping concentration in an approximate way and obtain good agreement between model predictions and experimental co-lasing data previously obtained with Tm³⁺:YLiF₄ lasers.