# Four-Dimensional Thermal Analysis of 888 nm Pumped Nd:YVO4 Dual-Rod Acousto-Optic Q-Switched Laser
## 4 Results and Discussion
### Q-Switched Condition  
For the Q-switched condition, the power distributions in crystals should obey the 4D coupled equations, Equations (1)-(6) and Equations (12) and (13). In addition, because the intracavity round-trip time of light is much larger than the time of pulse producing, the \(z\)-derivative terms have slowly varying functions in comparison with the \(t\)-derivative terms. Thus, the time-dependent principle barely changes with the effective average of spatial intracavity value being taken into account, as Equation (16) described. Similarly, the space-dependent principle also rarely changes with the effective average of temporal intracavity value considered. Figure 4 shows the numerical results of the temporal evolution of population inversion (a,b), heat generation (c,d) and temperature distribution (e,f) along the \(z\)-axis at the 100 kHz Q-switched condition. For clearly observing the pulse sequence, the maximum time domain span is set as 0.15 ms, before which the laser pulse can reach the steady state but the temperature pulsing growth cannot. The numerical results of output laser power pulse evolution are shown in Figure 5 by solving the coupled rate Equations (1)-(6) and Equation (16). The simulated stable peak power is 17.5 kW and pulse width is 35.4 ns, which are corresponding with the measured value 16.2 kW and 37.5 ns within the scope of tolerance. Additionally, the weak parasitic noisy pulses are also simulated as shown in Figure 5, which actually exist and are observed in our experiment.  
In our simulation model, the coupled dynamic equations without any certain periodic function hypotheses are directly solved, calculating thermal power via nonradiative transition rates and energy gaps. Thus, the first few unstable 4D heat generation pluses within two crystals can be solved, as shown in Figure 4c,d, corresponding to the physical reality. To the best of our knowledge,  
Figure 4: The temporal evolution of **(a)** population inversion in Xtal 1; **(b)** population inversion in Xtal 2; **(c)** heat generation in Xtal 1; **(d)** heat generation in Xtal 2; **(e)** temperature distribution in Xtal 1; and **(f)** temperature distribution in Xtal 2 along the \(z\)-axis at 100 kHz Q-switched condition.  
almost all of previous 4D thermal analysis models [19; 20; 21] roughly expressed the thermal load as certain periodic functions using some classical formulas, thus they cannot work out the unstable region, i.e., the first few unstable heat generation periods. The comparison shows an improved accuracy of our new model for simulating 4D heat generation in Q-switched laser crystals. Just like the laser pulse sequence, the heat generation pulses in unstable regions are a mixed train of high-intensity and low-intensity pulses. Figure 4c,d show the temporal evolution of temperature growth in two crystals. After 1.5 ms, the increment of maximum temperature in each crystal is about 0.12 \({}^{\circ}\)C, while the temperature distributions do not reach the steady state.  
The parasitic sub-pulses, as shown in Figure 5, are usual phenomenon in active Q-switched lasers, the detailed explanation of which can be Referred to Ref. [33]. This unwanted phenomenon can be eliminated by adjusting duty cycle, but the parasitic pulses here are so weak that cannot impact on output properties. Additionally, the existing parasitic pulses can reflect the accuracy of our numerical model. Thus, we did not address it in this work. The experimental pulse duration is 37.5 ns, which is in accordance with the simulation result that 35.4 ns in a tolerable error.