# Four-Dimensional Thermal Analysis of 888 nm Pumped Nd:YVO4 Dual-Rod Acousto-Optic Q-Switched Laser
[15] made a thermal analysis in a passively Q-switched diode-pumped Nd:YAG/Cr\({}^{4+}\):YAG laser, but they only discussed the thermal lens effect in a steady state. In 2013, [PERSON] et al. [16] improved the model for passively Q-switched lasers that can make four-dimensional (4D) space-time-dependent thermal analysis. In 2015, [PERSON] et al. [17] used composite Nd:YVO\({}_{4}\) crystals in their passively Q-switched laser to improve laser performance and developed the theoretical model. For actively Q-switched or pulse pumped lasers, in contrast to a passive case, it is a challenge to simulate and discuss the repetition-rate-dependent properties. In 2009, [PERSON] et al. [18] made a detailed 4D thermal analysis for their 946 nm in-band pulse pumped Nd:YAG laser. In 2010, [PERSON] et al. [19] built a thermal model for actively Q-switched lasers, and they solved and discussed the time-dependent transient temperature profiles. In 2015, [PERSON] et al. [20] made a 4D thermal effects analysis in a pulsed pumped composite Tm:YAG laser at a low-repetition-rate. However, all of these previous 4D thermal analysis models roughly expressed the thermal load as a certain periodic function. This assumption make these models fail to describe the dynamic heat generation evolution in the first few unstable pulses region. Actually, thermal load in Q-switched lasers is also coupled with complex 4D physical effects like ground state depletion (GSD) [21], fluorescence branching, energy transfer upconversion (ETU) [22; 23], cross relaxation (CR) and temperature-dependent cross sections [23; 24]. Recently, our group developed the spatial dynamic thermal iteration (SDTI) model [23], which can calculate the 3D thermal load distributions by precisely solving nonradiative transitions using the temperature iteration method. However, the temperature-dependent and repetition-rate-dependent 4D thermal analyses including all of these complex physical processes for actively Q-switched lasers have not been developed. In this paper, we develop powerful 4D dynamics and a thermal model for multi-rod actively Q-switched laser, and we present the experiment of a high-power high-repetition-rate 888 nm in-band pumped Nd:YVO\({}_{4}\) dual-rod acousto-optic Q-switched laser. In our model, we particularly considered the nonradiative transitions rates, fluorescent branching ratios of each energy manifolds and the ETU process contributing to heat generation. The 4D heat generation in crystal is solved by coupled dynamic equations based on the principles of nonradiative transitions and energy levels dynamics. Temperature-dependent thermal conductivity, absorption and emission cross sections, and their effects on output properties are also included for solving the transient temperature, power, and population distributions. Additionally, the anisotropy of gain medium is considered in our heat conduction equation. Thus, our model can precisely simulate the 4D heat generation evolution in not only the steady pulse region but also the unstable region at the beginning. The simulation results are in good agreement with our measured output data and temperature at various repetition-rates and pump powers in our experiment. To the best of our knowledge, it is the first experimental demonstration of 888 nm pumped multi-rod Q-switched laser oscillator.