## 4 Experimental Results and Verification of Spatially Coupled Rate Equations  
To evaluate the validity of the developed model, an electro-optic (EO) Q-switched laser was constructed employing a five-dimensional LD side-pumped Nd:YAG module with the same structural parameters as the side-pumped module used in the simulation experiments. A pulse pumping scheme with a duty ratio of 12.5% was implemented to reduce the thermal effect in the Nd:YAG rod [30]. Under an EO Q-switched operation with a frequency of 200 Hz, the output laser characteristics of our side-pumped laser with various Nd:YAG rod diameters were determined.  
Figure 9 depicts the relationship between the single pulse energy, pulse width, peak power, and average pump power for various Nd:YAG rod diameters. When the side-pumped module was assembled with 3 mm Nd:YAG crystal rods, a maximum pulse width of 18 ns and a maximum peak power of 103 kW were obtained. The maximum single-pulse energy of 2.3 mJ was obtained using a 5 mm Nd:YAG crystal rod. Experimental results demonstrated that when the side-pumped module was assembled with a crystal rod of a small diameter, the spatial overlap efficiency was higher, and the pulse width was narrower. However, utilizing a crystal rod with a large diameter significantly decreased the peak power and expanded the pulse width. The observed phenomenon was ascribed to the non-uniform distribution of pumping power intensity over the cross-sectional area of the Nd:YAG rod with a large diameter. As a result, the oscillation of high-order modes was  
Figure 8: The diagram illustrates a side-pumped Nd:YAG laser.  
aggravated in the resonator, which diminished the overlap efficiency and affected the output pulse width [31].  
In general, the experimental values obtained by employing 3 mm and 4 mm Nd:YAG rods corresponded well with the fitting function curve. Under the same conditions, the theoretical values are within the experimental values' 5% error bar range. There were some deviations between the experimental values and the simulated curve upon employing a crystal rod of 5 mm. Due to using a crystal rod with a large diameter, higher-order modes oscillated in the resonator than expected, resulting in deviations between the experimental and theoretical values.  
A high-speed silicon photo-detector was utilized to detect the output laser pulse, which had a maximum peak power of 103 kW. The detected pulse was subsequently displayed on a digitizing oscillograph. The uniform pulse width of a 200 Hz pulse train was demonstrated in Figure 10, with a pulse width of 18 ns. The M\({}^{2}\) factors were also measured using a traveling 90/10 knife-edge method when the laser was at maximum peak power. Figure 11 shows that the output laser beam possessed a uniform cross-sectional intensity distribution and excellent beam quality.  
Figure 9: Comparison between experimental and theoretical results for output laser characteristics at different Nd:YAG rod diameters. (**a**) Single pulse energy versus average pump power; (**b**) pulse width versus average pump power; (**c**) peak power versus average pump power.  
In conclusion, the spatially coupled rate equations were proven to be valid by our Q-switched side-pumped Nd:YAG laser experiment. The laser developed in this paper will be used as the main oscillation stage and cooperate with the subsequent amplification stage to form a complete laser system. The laser system will then undergo ambient temperature and vibration testing to fully meet the reliability requirements of laser applications such as PIV.  
Figure 11: The beam cross-sectional intensity distribution profile of the output laser and beam quality factors (M\({}^{2}\)) at the maximum peak power of 103 kW. (**a**) Intensity distribution profile. (**b**) Beam quality factors.  
Figure 10: Pulse train and pulse width of the Q-switched Nd: YAG side-pumped laser. (**a**) 200 Hz Laser pulse train; (**b**) single pulse profile.