## 2 Theoretical Analysis  
According to the mode theory of the optical resonator, the self-reproducing field pattern of a square aperture flat-concave resonator can be expressed in rectangular coordinates as [19]  
\[ \begin{array}{c}u_{mn}(x,y,z)=\frac{C_{mn}}{w(z)}H_{m}\Big{(}\frac{\sqrt{2}} {w(z)}x\Big{)}H_{n}\Big{(}\frac{\sqrt{2}}{w(z)}y\Big{)}\exp\biggl{(}-\frac{x^ {2}+y^{2}}{w(z)^{2}}\biggr{)}\\ \times\exp\biggl{(}-ik\frac{x^{2}+y^{2}}{2R(z)}\biggr{)}\exp\bigl{[}-i(m+n+1) \psi(z)\bigr{]},\end{array}  \tag{1}\]  
where \(w(z)=w_{0}\sqrt{1+z^{2}/z_{R}^{2}}\) is the beam radius on the axial position \(z\); \(R(z)=\left(z_{R}^{2}+z^{2}\right)/z\) is the radius of curvature of the wavefront that intersects the axis at \(z\); \(C_{mn}\) is the normalization constant; \(w_{0}\) is the radius of the fundamental mode Gaussian beam at \(z=0\), which can be calculated by the propagation matrix; \(z_{R}=\pi{w_{0}}^{2}/\lambda\) is the Rayleigh range of the mode; and \(H_{m}\) and \(H_{n}\) are the \(m\)th- and \(n\)th-order Hermite polynomial, respectively. In the transverse field distribution, the number of nodes with zero intensity in the x- and y-directions are \(m\) and \(n\), respectively, so that for the one-dimensional HG\({}_{mn}\) beams mentioned in this letter, \(m=0\). The relationship between the radius of HG\({}_{0n}\) mode and HG\({}_{00}\) mode can be obtained from Equation (1) [20]:  
\[w_{0n}=\sqrt{2n+1}w_{00}. \tag{2}\]  
Combined with the ABCD propagation matrix, the beam radius of HG\({}_{0n}\) mode at any position in the resonator can be calculated. Considering the ratio of the diameter of the one-dimensional HG0\({}_{0n}\) beam in the x- and y-directions, the slab is selected as the gain medium due to the inherent advantage of its large aspect ratio. To generate high-purity, one-dimensional HG0\({}_{0n}\) laser beams, the fundamental mode in the slab's thickness direction, i.e., the x-direction, needs to be determined first. A small aperture in the thickness direction of the slab provides an advantage for the fundamental mode output. On the other hand, an aperture that matches the diameter of the fundamental mode in the thickness direction of the slab can be inserted into the resonator to limit the transverse mode field in this direction; thus, only one-dimensional HG0\({}_{0n}\) laser beams exist in the cavity. Then, the high-order mode in the slab's width (y-) direction must be ensured. Because the diameter of high-order modes is larger than that of the low-order mode, high-order modes can be suppressed using an intracavity aperture with proper width [20]. The width of the aperture used for mode-limiting is adjusted according to the theoretical mode diameter of its position to ensure the highest order of the HG0\({}_{0n}\) mode that can exist in the cavity is \(N\). Due to the limitation of the aperture, only the HG0\({}_{00}\) to HG0\({}_{N}\) modes exist in the cavity. The width ratio of the two apertures in the width and thickness direction is just suitable due to the large aspect ratio of the slab. Furthermore, to extract the high-purity HG0\({}_{N}\) mode from the HG0\({}_{00}\) to HG0\({}_{0N}\) modes, it is necessary to perform differential regulation on each order mode's gain G0\({}_{0n}\) and loss L0\({}_{0n}\). It can be seen from Equation (2) that the higher-order mode has a larger mode volume and a greater overlap with the gain region so that for slab configuration, the higher-order mode can occupy more population inversion; that is, the gain G0\({}_{0n}\) is larger. On the other hand, when the Fresnel number of the resonator is determined, the diffraction loss of the higher-order mode is higher than that of the lower-order mode [21], so the threshold of the higher-order mode is generally higher than that of the low-order mode. But this can be changed by introducing OC tilt angles into the resonator. Previous studies have shown that for a given resonator, the misalignment of two cavity mirrors has less effect on higher-order modes [22].