## 3 Generation of High-Purity, Nanosecond HG0\({}_{0n}\) Laser Beams
 \tag{3}\]  
where \(L_{S}=\frac{L}{\alpha\cos(\alpha)}\) is the optical path length in the slab; \(L\) is the length of the Nd:YAG slab; \(n\) is the refractive index of Nd:YAG, which is 1.82; \(\alpha\) is the angle between the laser path and the bottom surface of the slab; \(L_{\rm Q}\) is the length of the AO Q-switch, which is 50 mm; \(n_{\rm q}\) is the refractive index of the AO Q-switch, which is 1.45; and \(R_{2}\) is the curvature radius of OC, which is taken as positive infinity in calculation. With the laser oscillator presented in Figure 1, the beam diameter, under the assumption of the fundamental mode, can be expressed by  
\[d_{0}=2\bigg{(}\frac{\lambda}{\pi}\bigg{)}^{\frac{1}{2}}|B|^{1/2}\bigg{[}1- \bigg{(}\frac{D+A}{2}\bigg{)}^{2}\bigg{]}^{-1/4} \tag{4}\]  
where \(\lambda\) is the wavelength of the oscillating laser. Based on the beam diameter ratio between the HG\({}_{0n}\) mode and the fundamental mode described in Equation (2), for a calculated 1.04 mm diameter fundamental mode at the position of A\({}_{\rm y}\), the corresponding beam diameters of the HG\({}_{01}\) to HG\({}_{09}\) modes are about 1.80 mm, 2.33 mm, 2.76 mm, 3.13 mm, 3.45 mm, 3.76 mm, 4.04 mm, 4.30 mm, and 4.54 mm, respectively. Therefore, considering the 5 mm active aperture of the AO Q-switch, for the width direction of the slab, the width D\({}_{\rm y}\) of aperture A\({}_{\rm y}\) is adjusted to 2.0 mm, 2.4 mm, 3.0 mm, 3.4 mm, 3.8 mm, 4.0 mm, 4.3 mm, 4.5 mm, and 4.8 mm, respectively. A similar calculation is used to select the diameter D\({}_{\rm x}\) of the aperture A\({}_{\rm x}\); in order to obtain the fundamental mode in the thickness direction of the slab, D\({}_{\rm x}\) is selected to be 1.3 mm. A CMOS sensor (Beamage-4M, Gentec Inc., St-Jean-Baptiste, QC, CA) is used to record the intensity distribution of the output laser beams. The laser beams are focused by a lens with a 300 mm focal length. The tilt angle \(\theta\) of OC M2 in the y-o-z plane is adjusted carefully, and then the beam intensity profile changes can be observed. When the HG\({}_{0n}\) mode spatial profile with the strongest contrast is displayed on the CMOS at a given tilt angle \(\theta\), as shown in Table 1, the corresponding intensity distribution is recorded. A beam quality analyzer (Beamage-M2, Gentec Inc., St-Jean-Baptiste, QC, CA) measures the beam quality factors \(M^{2}\) of the output laser beams at different apertures A\({}_{y}\) and M2 tilt angles. Figure 2 shows the typical beam quality factors of the highest order HG\({}_{09}\) mode in the x- and y-directions, which are \(M_{\rm x}\)\({}^{2}\) = 1.15 (blue) and \(M_{\rm y}\)\({}^{2}\) = 19.96 (red), respectively. The inset of Figure 2 displays the corresponding three-dimensional (3D) beam spatial intensity distribution at the far field, indicating nine intensity peaks. Figure 3 shows the measured beam quality factors \(M_{\rm x}\)\({}^{2}\) (blue triangle) and \(M_{\rm y}\)\({}^{2}\) (red circle) of the HG\({}_{01}\) to HG\({}_{09}\) modes in the x- and y-directions, respectively. In Figure 3, it can be seen that the beam quality factors (\(M_{\rm x}\)\({}^{2}\), \(M_{\rm y}\)\({}^{2}\)) of the HG\({}_{01}\) to HG\({}_{09}\) modes are (1.19, 3.44), (1.20, 5.58), (1.14, 7.09), (1.15, 9.98), (1.17, 11.20), (1.26, 13.39), (1.22, 15.79), (1.25, 17.10), and (1.15, 19.96), respectively. Considering the theoretical beam quality factor \(M_{\rm x}\)\({}^{2}\) = 1 for the ideal fundamental mode in the x-direction and the beam quality factor \(M_{\rm y}\)\({}^{2}\) = 2\(n+1\) for the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**Mode** & **HG\({}_{01}\)** & **HG\({}_{02}\)** & **HG\({}_{03}\)** & **HG\({}_{04}\)** & **HG\({}_{05}\)** & **HG\({}_{06}\)** & **HG\({}_{07}\)** & **HG\({}_{08}\)** & **HG\({}_{09}\)** \\ \hline P/mW & 349.8 & 407.0 & 318.7 & 379.5 & 332.2 & 338.8 & 321.2 & 336.6 & 289.3 \\ \(\theta\)/mrad & 0.636 & 0.022 & 0.678 & 0.107 & 0.512 & 0.520 & 0.121 & 0.095 & 0.750 \\ \hline \end{tabular}
\end{table}