## 1 Introduction  
The laser diode (LD) side-pumped is a standard pumping scheme for laser generation and amplification, particularly for lasers with moderate to high energy output. Side-pumping Q-switched lasers have been widespread in diverse applications, including laser-induced fluorescence and plasma, particle image velocimetry, and material research and processing [1; 2; 3; 4; 5; 6; 7; 8]. Using PIV (particle image velocimetry) as an example, this technology collects real-time information about the flow field via laser [9; 10; 11]. Hence, the laser must have a high repetition and pulse energy to satisfy the detection requirements. The utilization of a Q-switch can enable the laser to consider the output requisites. Compared with passively Q-switched (PQS) ones, actively Q-switched (AQS) lasers have the advantages of faster switching, better hold-off ability, more considerable output energy, and controllable repetition rates [7; 12; 13]. Among various active Q-switching methods, electro-optical (EO) Q-switching is a competitive and practical option. However, EO Q-switched lasers typically require a high-voltage electronic or radio frequency transducer and have a thermal effect in the gain medium under conditions of high pumping power. To solve these issues, numerous research teams have conducted extensive research. One recent study has confirmed that some electro-optical crystals, such as Relaxor-lead titanate (PbTiO\({}_{3}\)), are tiny, and the driving voltage needed for the crystals can be decreased to a few hundred volts [14]. Hence, the volume of AQS lasers can be further compressed. In addition, periodic pulse pumping with short pumping durations and high pumping outputs can considerably mitigate the thermal effect in side-pumped lasers [9; 10; 11; 12; 13]. Mode matching in a laser refers to matching the resonator's oscillated and pumping laser modes. If the two laser modes match well, a substantial power gain will be generated, and the output power of the laser will be increased [15; 16; 17; 18; 19]. The fundamental mode (TEM\({}_{00}\)), which approximates a Gaussian distribution, is the most robust mode of the resonator's laser field distribution and has high beam quality. Excellent mode matching between TEM\({}_{00}\) and pump beam in the resonator enables lasers with superior beam quality and excellent output power. Therefore, researching relationships of mode matches in laser designs is necessary. In addition, the rate equations represent a dynamic equation for laser output derived from altering population inversion at the atomic level within the gain medium, which can be utilized to derive the characteristic output curve of the laser. Specific research teams have successfully solved the rate equations to a greater extent to simulate and control the pulse waveform of the laser output [20; 21; 22]. For similar purposes, by incorporating the spatial overlap efficiency into the rate equations, the laser's output performance can be more accurately simulated and calculated to reflect the resonator's actual oscillation. Nevertheless, most contemporary models concerning coupled rate equations containing the overlap effect concentrate on end-pumped solid-state lasers [17; 18; 19; 23]. The primary reason is that the direction of the pumping beam propagation is paralleled to the direction of the oscillating laser propagation in an end-pumped laser. Moreover, the fundamental transverse laser mode (TEM\({}_{00}\)) oscillates easily in an end-pump laser, making the spatial overlap effect easier to calculate. In [PERSON]'s publication [24], although he computed the spatially linked rate equations for a side-pumped laser without Q-switch operation, he barely considered several parameters in the computation, such as the complex and variable intensity distribution of pumping power over the section of the gain medium. In addition, he assumed that only a fundamental transverse mode (TEM\({}_{00}\)) oscillated in the resonator, resulting in some discrepancies between the estimated and actual results.