## 3 Generation of High-Purity, Nanosecond HG0\({}_{0n}\) Laser Beams  
The experimental setup of the oscillator for generating high-purity, nanosecond HG0\({}_{0n}\) laser beams is shown in Figure 1. The laser oscillator includes a laser module, a pair of apertures A\({}_{\rm x}\) and A\({}_{\rm y}\), a rear cavity mirror M1, an OC M2, and an acousto-optic (AO) Q-switch. The laser module contains a 0.6 at. % doped trapezoidal Nd:YAG slab crystal with dimensions of 139 mm \(\times\) 30 mm \(\times\) 4 mm (length \(\times\) width \(\times\) thick), having a large aspect ratio of 7.5. The cut angle of the two end facets of the slab is 56\({}^{\circ}\), and both are coated with anti-reflective (AR) film at 1064 nm. The top surface of the slab is coated with an AR film for 808 nm, and the bottom surface of the slab is highly reflective (HR) at 808 nm for double-pass pumping. The top surface of the slab is edge-sealed with an O-ring, while the bottom surface is soldered on the oxygen-free copper heat sink with microchannels in order to maintain the temperature of the laser module at 20 \({}^{\circ}\)C (\(\pm\)0.2 \({}^{\circ}\)C) through a water cooler. The slab is face-pumped by six 808 nm quasi-continue-wave (QCW) microchannel-cooled laser diode arrays (LDA), with each array containing six bars. The temperature of cooling water for the LDA is tuned to 20 \({}^{\circ}\)C (\(\pm\)0.2 \({}^{\circ}\)C) to match the absorption spectrum of the crystal medium [23]. A pump beam with a 300 \(\upmu\)s pump pulse duration and 500 Hz pulse repetition rate (PRR) is normally incident on the top face of the slab via a quartz duct. The size of the pump region is 66 mm \(\times\) 20 mm, so the effective aspect ratio in the pump region is 20 mm/4 mm = 5. The laser beam propagates along a zig-zag optical path in the slab, and the internal reflection angle between the optical path and the normal of the bottom face of the slab is 56.0\({}^{\circ}\). The rear cavity mirror M1 is a concave mirror with curvature radius \(R_{1}\) of 1600 mm coated with HR film at 1064 nm. The output coupler M2 with a transmittance (T) of 10% is placed on a mount with an adjustable tilt angle, while the tilt angle \(\theta\) in the y-o-z plane is the key to obtaining HG\({}_{0n}\) laser beams. The AO Q-switch (SGQ27-1064-5 QD, CETC) with a 5 mm \(\times\) 5 mm active aperture is located in the front of the slab crystal. The arm lengths from the end facets of the slab to M\({}_{1}\) and A\({}_{\rm y}\) are \(L_{1}\) = 110 mm and \(L_{2}\) = 575 mm, respectively. Two apertures, A\({}_{\rm x}\) and A\({}_{\rm y}\), with adjustable sizes in the thickness and width directions of the slab, are placed at \(L_{\rm x}\) = 20 mm and \(L_{\rm y}\) = 40 mm from M2, respectively. The total optical length of the resonator is 800 mm. The sizes of apertures A\({}_{\rm x}\) and A\({}_{\rm y}\) in the thickness and width direction of the slab denote D\({}_{\rm x}\) and D\({}_{\rm y}\), respectively. As mentioned above, D\({}_{\rm x}\) and D\({}_{\rm y}\) should be selected based on the resonator shown in Figure 1. Since the zig-zag optical path averages the thermal  
Figure 1: Schematic of the slab laser oscillator from top and side views. M1: rear cavity mirror; M2: output coupler; LDA: laser diode array; AO: acousto-optic Q-switch; A\({}_{\rm x}\): aperture in the x-direction; A\({}_{\rm y}\): aperture in the y-direction. gradient, the first-order thermal lens effect is almost ignored. The diameter of the beam at the position of A\({}_{\rm y}\) can be calculated by the ABCD propagation matrix, which is given by  
\[ \begin{array}{l}T=\begin{pmatrix}1&L_{S}+L_{1}+L_{2}-L_{\rm Q}+\frac{L_{Q}}{n _{\rm q}}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -\frac{2}{R_{1}}&1\end{pmatrix}\begin{pmatrix}1&L_{S}+L_{1}\\ 0&1\end{pmatrix}\\ \times\begin{pmatrix}1&L_{2}-L_{\rm Q}+\frac{L_{Q}}{n_{\rm q}}+L_{y}\\ 0&1\end{pmatrix}\begin{pmatrix}1&0\\ -\frac{2}{R_{2}}&1\end{pmatrix}\begin{pmatrix}1&L_{y}\\ 0&1\end{pmatrix}=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\end{array}