## 2 Modeling of Spatially Dependent PQS
By integrating coupled rate equations with respect to the cavity volume, we have  
\[\frac{d\Phi}{dt}=\frac{\Phi}{t_{r}}\left[\frac{2\sigma Nl_{g}}{V_{eff}}-\frac{ 2\sigma_{gs}N_{gs}l_{s}}{V_{s}}-\frac{2\sigma_{es}N_{es}l_{s}}{V_{s}}-\ln\left( \frac{1}{R_{oc}}\right)-L_{s}\right] \tag{8}\]  
\[\frac{dN}{dt}=-\gamma\frac{2l_{c}\sigma N\Phi}{t_{r}V_{eff}S} \tag{9}\]  
\[\frac{dN_{gs}}{dt}=-\frac{A}{A_{s}}\frac{2l_{c}\sigma_{gs}N_{gs}\Phi}{t_{r}V_{ eff}S} \tag{10}\]  
where the effective mode volume \(V_{eff}\) and the overlap efficiency \(S\)  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]0  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]1  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]2  
From Equation (8), with the condition that \(d\Phi/dt=0\) and \(N_{es}=0\), the initial population in the gain medium for generating a pulse is given by  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]3  
Dividing the Equation (9) by the Equation (10) and integrating the result can give \(N_{gs}\) as a function of \(N\), that is \(N_{gs}=N_{so}(N/N_{i})^{\alpha}\) where \(\alpha=A\sigma_{gs}/(\gamma A_{s}\sigma)\). Dividing the rate Equation (8) by the rate Equation (9) and using the Equation (14) give  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]4  
where  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]5  
and \(\beta=\sigma_{es}/\sigma_{gs}\). The second derivative of \(\Phi\) with respect to \(N\) can be expressed as  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]6The condition \(d^{2}\Phi/dN^{2}>0\) at \(N=N_{i}\) is referred to as the second threshold for generating a stable pulse train in PQS lasers and can be given by \(\xi>\alpha/(\alpha-1)\) where \(\xi=N_{i}/N_{t}\) is the ratio of the initial inversion to the critical inversion. Integrating the equation of \(d\Phi/dN\) yields  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]7  
The final inversion population \(N_{f}\) can be determined by the condition \(\Phi\Big{(}N_{f}\Big{)}=0\):  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]8  
Defining \(\eta_{E}=\Big{(}N_{i}-N_{f}\Big{)}/N_{i}\) as the energy-utilization factor, the Equation (19) can be expressed as a transcendental equation:  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]9  
In the condition of no gain saturation, the larger value of the \(\eta_{E}\) means the more efficient population depleted in the saturable absorber, since \(\eta_{E}\) cannot be directly solved as a function of \(\xi\) and \(\alpha\). A powerful expression derived from numerical fitting [18] for the energy utilization efficiency \(\eta_{E}\) is given by  
\[r_{0}(x,y,z)=\frac{\delta e^{-\delta z}}{\pi w_{p}^{2}\left(1-e^{-\delta l_{g} }\right)}\cdot\mathrm{H}\left[w_{p}^{2}-\left(x^{2}+y^{2}\right)\right]\cdot \mathrm{H}\left(l_{g}-z\right) \tag{3}\]0  
with \(\xi>\alpha/(\alpha-1)\). The transcendental equation quantitatively determines the characteristics of PQS.