## 2 Experimental Section  
### Theoretical Analysis  
The Q-switched pulse duration is so short that both spontaneous emission and optical pumping can be neglected in the rate equations. The two coupled differential equations for photon density \(\phi\) and population inversion density \(n\) are as follows [16]:  
\[\frac{\partial\phi}{\partial t}=\phi(c\sigma n\frac{l}{l^{\prime}}-\frac{ \varepsilon}{t_{R}}) \tag{1}\]  
\[\frac{\partial n}{\partial t}=-\gamma n\phi\sigma c \tag{2}\]  
where \(\sigma\) is the stimulated emission cross section, \(l\) is the length of the  
where \(\tau_{c}\) is the photon lifetime.  
The output energy of the Q-switched laser can be expressed as follows:  
\[E=\frac{h\
u A}{2\sigma\
u}\ln\frac{1}{R}\cdot\ln\frac{n_{i}}{n_{f}} \tag{4}\]  
where \(h\
u\) is the laser photon energy, and \(A\) is the effective beam cross-sectional area. The initial and final population inversion densities, \(n_{i}\) and \(n_{f}\), are obtained through the transcendental equation:  
\[n_{i}-n_{f}=n_{t}ln\frac{n_{i}}{n_{f}} \tag{5}\]  
where \(n_{t}\) is the population inversion density at threshold, that is,  
\[n_{t}=\frac{1}{2\sigma l}\ln(\frac{1}{R}+L) \tag{6}\]  
The pulse duration of the Q-switch pulse can also be expressed as a function of the inversion level, \(n_{i},n_{f}\), and \(n_{t}\):  
\[\Delta t_{p}=\tau_{c}\frac{n_{i}-n_{f}}{n_{i}-n_{t}(1+\ln(n_{i}/n_{t}))} \tag{7}\]  
The equations for pulse energy and pulse duration (and, therefore, peak power) are expressed in terms of the initial and finial population inversion densities, which are obtained via a transcendental equation. A successful analytical solution was reported in a study by [PERSON] [17], which revealed that key parameters, such as optimum reflectivity, output energy, extraction efficiency, pulse duration, and peak power, can all be expressed as a single dimensionless variable:  
\[\mathrm{z}=\frac{2g_{0}l}{L} \tag{8}\]  
where \(2g_{0}l\) is the logarithmic small-signal gain, and \(L\) is the round-trip loss.  
The following expression for optimum reflectivity was derived:  
\[R_{opt}=e^{-L\frac{(\mathrm{z}-1-\ln\mathrm{z})}{\ln\mathrm{z}}} \tag{9}\]  
The energy output for an optimized system is:  
\[E_{out}=E_{sc}(\mathrm{z}-1-\ln\mathrm{z}) \tag{10}\]  
where \(E_{sc}\) is a scale factor with the dimension of energy that contains a number of constants:  
\[E_{sc}=\frac{Ah\
u L}{2\sigma\
u} \tag{11}\]  
The FWHM pulse duration _versus_\(z\) is obtained as follows:  
\[t_{p}=\frac{t_{R}}{L}\cdot\frac{\ln\mathrm{z}}{z}\frac{1}{1-(\frac{\mathrm{z} -1}{\ln\mathrm{z}})\cdot(1+\ln\frac{\mathrm{z}\ln\mathrm{z}}{\mathrm{z}-1})} \tag{12}\]  
The energy extraction efficiency can then be obtained:  
\[\eta_{Ex}=1-\frac{1+\ln\mathrm{z}}{\mathrm{z}} \tag{13}\]The laser crystal we used had a diameter of 7 mm, with a laser resonator of 280 mm in length. Assuming a 5% round-trip cavity loss ( \(L=0.05\)), and with \(h\
u=1.86\times 10^{-19}\) J and \(\sigma=2.8\times 10^{-19}\)cm\({}^{2}\), we calculated \(E_{SC}=2.13\) mJ. Based on Equation (10), the single dimensionless variable was \(z=170\). Finally, we obtained the optimum reflectivity of 20.2%, according to Equation (9).