Establishing the new 2 μm passive Q-switching rate equation: Incorporating energy transfer based on dual energy level system

New 2 μm passive Q-switching rate equations based on a dual energy level system were established in our research. These equations comprehensively account for the energy transfer mechanisms between Tm³⁺ ions and the saturable absorber, including spontaneous emission, cross relaxation, up-conversion, and reabsorption effects. Building on this foundation, we experimentally employ black phosphorus, a new two-dimensional material with a bandgap ranging from 0.3 eV (bulk) to 1.5 eV (monolayer), into the Tm: LuAG laser as a saturable absorber for the first time. Consequently, this provides a new technical pathway for achieving efficient and stable passive Q-switching operation. In the experiment, with a pump power of 16.22 W, the laser produced an average output of 1.42 W, a repetition rate of 47.23 kHz, and a pulse width of 1.02 μs Compared with conventional models, the new model improved predictive accuracy for pulse width, repetition rate, and peak power by 1.54 %–5 %. These findings demonstrate the new model's significant advantages in describing the output characteristics of 2 μm passively Q-switched lasers and establish a solid theoretical foundation for further investigations into passive Q-switching rate equations for specific laser systems.