## 1 Introduction  
The simultaneous dual-wavelength laser (SDWL) [1] has been regarded for its applications in differential lidar [2; 3; 4], terahertz generation [5; 6; 7] and medical diagnosis [8; 9; 10]. The SDWL in a wavelength range around 2 \(\upmu\)m holds great potential because of its human-eye safety and good transparency features [11; 12; 13]. Tm and Ho co-doped laser materials have been identified as prospective candidates for realizing a 2 \(\upmu\)m SDWL because they possess many sharp fluorescent lines with the transitions of the splitting energy level in \({}^{51}\!_{7}\) and \({}^{51}\!_{8}\). Tm and Ho co-doped dual-wavelength fiber lasers operating at room temperature have been successfully demonstrated through the use of a cascaded fiber Bragg grating array [14] and a spatial filter [13]. In typical all-fiber dual-wavelength laser systems, the wavelength spacing is usually less than 10 nm. Recently, a solid-state laser utilizing Tm,Ho:GdYTaO\({}_{4}\) crystal with a wavelength separation of 120 nm has been developed. Different lasing wavelengths and maximum output powers were realized by adjusting the concentration ratio of Tm and Ho ions [15]. Another work has demonstrated the feasibility and potential of using a Tm,Ho:YAG ceramic in diode-pumped dual-wavelength lasers, which can lead to high beam quality and high power density [16]. Moreover, recent research has reported a cryogenic SDWL operation of Tm:YLF using a modular setup around 2 \(\upmu\)m. As the pump power increases, the output wavelength combination can transition from 1876 nm and 1901 nm to 1901 nm and 1912 nm [17]. However, there is a lack of sophisticated technological methodologies for the generation of 2 \(\upmu\)m dual-wavelength solid-state lasers, which forces researchers to rely on specific laser crystals with particular doping concentrations or pumping powers to achieve simultaneous dual-wavelength output. Moreover, balancing the power ratio between the dual-wavelength components of the laser output remains a challenge for stable and consistent performance. Currently, the growth techniques of Tm,Ho:YAP have become highly mature, and their structural anisotropy allows them to produce polarized emission spectra with multiple emission peaks, which provides strong support for the generation of multi-wavelength laseroscillation. [PERSON] et al. founded that laser output with wavelengths of 2000 nm and 2119 nm can be achieved by Tm,Ho:YAP under CW and Q-switched modes, respectively [18]. However, achieving the simultaneous output of these two wavelengths has been a long-standing challenge. For solid-state laser, the achievement of dual wavelength relies on controlling the gain and loss of different transition lines in the resonator to maintain a balance of net gain, thereby meeting the laser's threshold conditions. The threshold pump power is influenced by several factors, including the reflectivity of the output mirror, intracavity round-trip loss, the stimulated emission cross-section, and the overlap of pump and oscillating light. Thus, SDWL generation can be attained by regulating the aforementioned parameters. Adjustment of the output mirror reflectivity can be achieved by applying a specific coating or utilizing an F-P filter as the output mirror, as demonstrated in previous studies [19; 20]. For strongly gain-enhanced transition lines, the reflectivity of the output mirror at the two wavelengths needs to be determined based on the threshold conditions for dual-wavelength pumping, ensuring similar or equal threshold pump powers. For weakly gain-enhanced transition lines, it is essential to suppress the adjacent, strongly gain-enhanced transition lines. This ensures that the threshold pump power for the weakly gain-enhanced transition line is lower than that of the adjacent, strongly gain-enhanced transition line, ultimately facilitating dual-wavelength laser output. However, when the wavelengths of the dual-wavelength laser are close, controlling the threshold via coating becomes a major challenge.