The 2 μm laser, especially pulsed lasers, has multiple applications in domains like medical treatment, fiber optics, and beam splitter [1,2]. Currently, the output of 2 μm pulsed lasers is primarily achieved through the Q-switching techniques [3,4]. There are two types of Q-switching techniques: active and passive. Using electro-optic crystals or external pulse signal sources, active Q-switching alters the intracavity threshold on a regular basis [5]. Passive Q-switching, on the other hand, uses a saturable absorber put into the cavity to control pulse production using its nonlinear absorption properties [6].

Passive Q-switching lasers differ from active Q-switching in that they have simpler architecture and lower costs. Following their initial establishment by VuyIsteke and Wagner, the passive Q-switching rate equations, refined by Guohua Xiao and J.J. Degnan [[6], [7], [8]], have emerged as crucial theoretical tools for simulating the output characteristics of passive Q-switching pulsed laser. These equations formulate a system of differential equations that describe the temporal evolution of photon numbers in the laser cavity and the population dynamics of various energy states in the gain medium. This formulation accounts for factors such as pump light distribution, decay losses of inverted particle numbers in the gain medium, as well as the influence of the saturable absorber's excited-state absorption. Nevertheless, their theoretical framework is based on the plane wave approximation and assumes uniform pumping, uniform intracavity laser energy, along with homogenous whitening of the saturable absorber. As a result, numerous factors remain unaddressed, prompting ongoing research focused on refining the passive Q-switching rate equations.

In 2006, Guiqiu Li proposed a Gaussian spatial distribution of photon density within the cavity and modified the passive Q-switching rate equations by considering the longitudinal variations of photon density and pump beam spatial distribution. He experimented with Cr4+: YAG as a saturable absorber within a diode-pumped Nd: YVO4 laser, obtaining a relationship between pulse width and the radii of the laser and pump beams [9]. In 2009, M. Lu examined the impact of pump rate and the dynamic changes in the excited states of the absorber on the intracavity photon number, explaining the unexpected decrease in output energy with increasing pump power [10]. In 2013, Y. Guyot et al. studied the effects of upconversion on Nd3+ passive Q-switching lasers [11]. However, despite numerous revisions, the modified passive Q-switching rate equations still generally apply to all passive Q-switching lasers, with experimental validations primarily focused on Nd3+ lasers. Consequently, specific particle dynamics behaviors in other laser types have not been studied, limiting the theoretical guidance for particular lasers.

Single-doped Tm3+ is a vital approach for achieving 2 μm lasers [12]. However, studies on the passive Q-switching output at this wavelength have tended to be more experimental than theoretical, which has led to less-than-ideal outcomes. Although various novel materials can facilitate 2 μm passive Q-switching, they fall short in comparison to active Q-switching outcomes, failing to meet practical application requirements [[13], [14], [15], [16], [17]]. Therefore, theoretical simulations of single-doped Tm3+ passive Q-switching are crucial for guiding experiments. To date, in the 2 μm wavelength range, no studies have specifically established new passive Q-switching rate equations for Tm3+ lasers. We consider the influence of inter-particle energy transfer during the passive Q-switching process of Tm3+ lasers at 2 μm. It presents a theoretical analysis of spontaneous emission, upconversion, cross-relaxation, and reabsorption on the particle dynamics within the resonator during passive Q-switching for the first time. Additionally, it summarizes and organizes the theoretical research on passive Q-switching, establishing a system of rate equations based on Tm3+ for passive Q-switching lasers operating at 2 μm, thus providing significant theoretical support for Tm3+-based passive Q-switching lasers operating at 2 μm.

Moreover, experimental validation of the model is essential. The main materials for direct laser output above 2 μm are Tm: YAG (2013.3 nm) and Tm: LuAG (2020.3 nm). However, Tm: YAG has a small gain cross-section, long upper-state lifetime, and overlaps with water absorption peaks, causing severe thermal effects. This limits its beam quality, output power, and stability. Tm: LuAG, a newer crystal, offers higher thermal conductivity, lower phonon losses, and larger absorption/emission cross-sections. It supports broad tuning, with absorption peaks at 682 nm, 788 nm, 1173 nm, and 1629 nm. Tm: LuAG also features high slope efficiency and damage thresholds, making it superior to Tm: YAG for 2 μm lasers. Its performance advantages make it an excellent material for high-power and stable laser systems [18,19].

Recent advancements in two-dimensional materials, such as black phosphorus, which possesses broad-band saturation absorption, a direct bandgap, and high carrier mobility, have emerged as a focal point of research [20,21]. However, no reports have been on its application in Tm: LuAG lasers. Therefore, utilizing black phosphorus as a saturable absorber for passive Q-switching at 2 μm is crucial for exploring its feasibility in achieving high-performance outputs in Tm3+ lasers. Furthermore, utilizing black phosphorus as a saturable absorber for passive Q-switching at 2 μm is essential for exploring its potential to achieve high-performance outputs in Tm3+ lasers. Integrating experimental results with the established rate equations can improve our understanding of black phosphorus's behavior within the resonator. This promotes its application as a saturable absorber in the domain of 2 μm passive Q-switching lasers.