## 2 Modeling of Spatially Dependent PQS  
The spatially-dependent lasing photon density and the population inversion density can be expressed as \(\Phi(x,y,z)=\Phi\varphi_{0}(x,y,z)\) and \(n(x,y,z)=Nr_{0}(x,y,z)\) where \(\varphi_{0}(x,y,z)\) and \(r_{0}(x,y,z)\) are the spatially-normalized lasing and pumping photon densities, respectively, \(\Phi\) and \(N\) are the number of the photon in the cavity and the number of the inversion population in the gain medium. For a single fundamental transverse TEM\({}_{00}\) mode, the lasing photon density can be expressed as:  
\[\varphi_{0}(x,y,z)=\frac{2}{\pi w_{c}^{2}l_{c}}e^{-2(x^{2}+y^{2})/w_{c}^{2}} \tag{1}\]  
\[w_{c}=w_{0}\sqrt{1+\left(\frac{\lambda z}{\pi w_{0}^{2}}\right)^{2}}\approx w_ {0}\sqrt{1+\left(\frac{\lambda d_{c}}{\pi w_{0}^{2}}\right)^{2}} \tag{2}\]  
where \(l_{c}\) is the length of the cavity, \(w_{c}\) is the mode size in the gain crystal, \(w_{0}\) is the beam waist of lasing, \(d_{c}\) is the distance from the beam waist to the middle of the length of the gain crystal, and \(\lambda\) is the wavelength of lasing. We assume a constant beam size without diffraction here if the variation of the beam size in the gain medium is small enough and neglectable. For a fiber-coupled laser diode pumping, the population inversion density in the gain medium can be described as a top-hat distribution [16; 17]:  
\[r_{0}(x,y,z)=\frac{\delta e^{-\delta z}}{\pi w_{p}^{2}\left(1-e^{-\delta l_{g} }\right)}\cdot\mathrm{H}\left[w_{p}^{2}-\left(x^{2}+y^{2}\right)\right]\cdot \mathrm{H}\left(l_{g}-z\right) \tag{3}\]  
\[w_{p}=w_{p0}\sqrt{1+\left[\frac{\lambda_{p}M_{p}^{2}}{n_{r}\pi w_{p0}^{2}}(z-z _{0})\right]^{2}} \tag{4}\]  
where \(\delta\) is the absorption coefficient at the wavelength of the pumping source, \(l_{g}\) is the length of the gain crystal, \(w_{p}\) is the pump size in the gain crystal, H is the Heaviside step function, \(w_{p0}\) is the beam waist of the pump, \(\lambda_{p}\) is the wavelength of the pump, \(M_{p}^{2}\) is the beam quality factor, \(n_{r}\) is the refractive index of the gain crystal, and \(z_{0}\) is the location of the focal plane of the pump. We introduce the spatial dependence stated above to the coupled rate equations of the four-level PQS laser. The coupled rate equations can be expressed as:  
\[\frac{d\Phi}{dt}\varphi_{0}=\frac{\Phi}{t_{r}}\varphi_{0}\left[2\sigma Nr_{0} l_{g}-\frac{2\varphi_{gs}N_{gs}l_{s}}{V_{s}}-\frac{2\varphi_{cs}N_{es}l_{s}}{V_{s }}-\ln\left(\frac{1}{R_{\alpha c}}\right)-L_{s}\right] \tag{5}\]  
\[\frac{dN}{dt}r_{0}=-\gamma\frac{2l_{c}\sigma}{t_{r}}\Phi N\varphi_{0}r_{0} \tag{6}\]  
\[\frac{dN_{gs}}{dt}=-\frac{A}{A_{s}}\frac{2l_{c}\sigma_{gs}N_{gs}}{t_{r}}\Phi \varphi_{0} \tag{7}\]where \(\sigma\) is the stimulated emission cross section of the gain medium, \(\sigma_{gs}\) and \(\sigma_{es}\) are the ground-state and excited-state absorption cross section of the absorber, \(N_{gs}\) and \(N_{es}\) are the number of the population in the ground state and in the excited state of the absorber, \(l_{g}\) and \(l_{s}\) are the lengths of the gain medium and the absorber respectively, \(V_{s}\) is the volume of the absorber, \(R_{oc}\) is the output coupler reflectivity, \(L_{s}\) is the dissipative loss in a roundtrip for a photon, \(t_{r}\) is the roundtrip time for a photon in the cavity, \(\gamma\) is the inversion reduction factor, and \(A\) and \(A_{s}\) are the cross areas of lasing in the gain medium and in the absorber, respectively. According to the Lambert-Beer absorption law, the initial transmission of the absorber can be expressed as \(T_{0}=\exp\left(-\sigma_{gs}N_{so}l_{s}/V_{s}\right)\) where \(N_{so}=N_{gs}+N_{es}\).