Can upconversion pumping initiate co-lasing in \textit{Tm}^\textit{3+} gain media similar to conventional pumping?

Rate-equation formulation was used to compare the thresholds and efficiencies during conventional pumping near 0.8 μm (³H₆→³F_2,3 transition) and upconversion pumping near 1 μm (³H₆→³H₅ and ³F₄→³F_2,3 transitions) to achieve co-lasing near 2 μm and 2.3 μm in Tm³⁺ lasers. Coupled power-gain equations were derived by accounting for concentration-dependent effects of cross relaxation (CR) and energy transfer upconversion (ETU). Analytical results were derived for lasing thresholds and efficiencies in the co-lasing regime. Very good agreement was obtained with experimental co-lasing data obtained in Tm³⁺:YLF lasers under conventional pumping. We show that in both cases of pumping, co-lasing can be achieved and also show that the output powers at both wavelengths vary linearly with pump power in the co-lasing regime. Analytical conditions were further derived that show which transition, 2 μm or 2.3 μm, lases first. Finally, our model predicts a critical Tm³⁺ ion concentration at which the first lasing wavelength switches from 2.3 μm to 2 μm during conventional and upconversion pumping.