## 4 Power Scaling in MOPA System Using Cascaded Slab Module  
The feasibility of power scaling is the inherited advantage of selecting a large aspect ratio slab module. A cascaded slab gain module is used for laser amplification. This amplification module is the same as the laser module mentioned earlier in the article, which includes a 0.6 at. % Nd\({}^{3+}\) doped 139 mm \(\times\) 45 mm \(\times\) 4 mm (length \(\times\) width \(\times\) thick) trapezoidal Nd:YAG slab, a 6 \(\times\) 6 LDA with 808 nm pump wavelength, 300 \(\upmu\)s pump pulse duration and 500 Hz PRR, and a quartz duct. The temperature of the amplification module is also maintained at 20 \({}^{\circ}\)C (\(\pm\)0.2 \({}^{\circ}\)C). The experimental configuration scheme of the amplifier is shown in Figure 4a, and the picture illustrating the real system is also shown in Figure 4b. The seed pulse output from the oscillator is injected into the slab amplification module. To achieve higher energy extraction, a double-pass amplification geometry is adopted. The internal reflection angles between the two optical paths and the normal of the bottom face of the plate are 56.0\({}^{\circ}\) and 68.9\({}^{\circ}\), respectively. The change in the optical path is due to the use of flat mirrors M3 and M4 coated with 35\({}^{\circ}\)\(\sim\)45\({}^{\circ}\) HR film at 1064 nm. After passing through the amplifier, the beam is divided into two parts by a beam splitter. The main part enters the power meter for the power measurement, and the leaking part is incident on CMOS to record the beam intensity distribution. Figure 5 shows the amplified two-dimensional (2D) intensity distribution of HG\({}_{01}\) to HG\({}_{09}\) beam modes. Meanwhile, the normalized intensity curves (red) in the y-direction are shown in Figure 5, and the corresponding theoretical normalized intensity curves (blue) are also shown in Figure 5. In general, the experimental results are in reasonable agreement with the numerical simulation. Compared to the theoretical model, the slight intensity deviation of the beam in the experiment may be caused by the residual lower-order HG\({}_{0n}\) modes. The purity of the mode is described by using the Pearson correlation coefficient \(\rho\):  
\[\rho=\frac{\sum_{i=1}^{N}\left(li-\overline{I}\right)\left(li^{*}-\overline{I^ {*}}\right)}{\sqrt{\sum_{i=1}^{N}\left(li-\overline{I}\right)^{2}}\sqrt{\sum_{ i=1}^{N}\left(li^{*}-\overline{I^{*}}\right)^{2}}} \tag{5}\]  
where \(I_{i}\) is the measured normalized intensity of each pixel, \(\overline{I}\) is the average value of normalized intensities of all pixels, \(I_{i}\)* is the normalized intensity obtained by the theoretical calculation, and \(\overline{I^{*}}\) is the corresponding average value. The correlation coefficient of the highest order HG\({}_{09}\) mode is calculated as \(\rho_{09}=0.944\), while the correlation coefficient of other modes \(\rho_{0n}>0.93\). It can be observed that the measured laser beams all have longer tails in the edge region of the x-direction. The diffraction effect of the intracavity aperture and the beam diffusion caused by the off-axis aberration of the cavity mirror are the reasons for the insufficient sharpness of the edge reduction. Figure 4: The double-pass slab amplifier. (**a**) The experimental configuration scheme of the amplifier. (**b**) The picture of the real system. Figure 6a shows the amplified output power of typical mode HG\({}_{09}\) (blue) versus the LD pump power. From Figure 6a, it can be seen the output power shows a monotonic increase with the incident pump power, amplifying from 289 mW to 4.73 W. The increase of output power is faster with the increase of the incident pump power and does not show any saturation effect up to around 850 W pump power.