## 3 Experimental Setup  
The experimental setup for the measurements presented in this paper is shown in Figure 2. The injected laser light was a pulsed circular Gaussian beam with a center wavelength of 1053 nm and a repetition frequency of 1 Hz. Through the Galilean telescope with a magnification of 15x, the diameter of the beam was enlarged to 22.5 mm. A 7-mm soft-edge diaphragm was used to spatially reshape the beam, changing its light-field distribution characteristics from circular Gaussian to a flat-top square beam. Hence, when the incident beam entered the pumping area of the NDSG element, the light intensity was evenly distributed. The beam energy before entering the amplifier head was measured as 2.24 mJ. The amplifier head, the structure of which is illustrated in Figure 3a, was made up of a NDSG rod with circulating deionized water as a cooling liquid. The NDSG rod was fixed by the O-rings at both ends of the surface, which meets the sealing conditions and achieves side cooling. As shown in Figure 3b, the dimensions of the NDSG cylindrical rod were \(\O\)28 mm \(\times\) 28 mm, and its absorption coefficient at wavelength 802 nm was 1.6 cm\({}^{-1}\). The anti-reflection coatings were fabricated onto both end surfaces of the NDSG rod, which have a transmittance greater than 99.5% for the 802-nm and 1053-nm wavelengths. The one-way transmission rate of the NDSG rod at wavelength 1053 nm was measured as 98.21%. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **NDSG** & **LHG-8** & **Nd:YAG** \\ \hline Fluorescence lifetime (\(\upmu\)s) & 376 & 315 & 230 \\ Linewidth (nm) & 51.7 & 26.5 & - \\ Emission cross section @ 1 \(\upmu\)m (pm\({}^{2}\)) & 1.4 & 4.2 & 65 \\ Thermal conductivity (W/(m-K) at 25 \({}^{\circ}\)C) & 1.27 & 0.58 & 7.94 \\ Thermal expansion coefficient (10\({}^{-7}\)/\({}^{\circ}\)C) & 6.5 & 115 & 82 \\ Nonlinear refractive index (10\({}^{-20}\) m\({}^{2}\)/W) & 2.89 & 3.27 & 11.79 \\ Thermal shock parameter (W/cm) & 14.5 & 0.4 & 7.9 \\ Saturation energy density (J/cm\({}^{2}\)) & 13.47 & 4.7 & 0.66 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of three gain materials. Figure 2: Layout of the LD-pumped NDSG rod laser amplification experiment. M1–M2: 45\({}^{\circ}\) mirrors; M3–M4: dichroic mirrors; f1–f2: 4F imaging system; L1–L6, L’1–L’6: lenses. The gain module was longitudinally pumped by two symmetrical LD arrays (LD1 and LD2 in Figure 1) operating with a pulse duration of 550 \(\upmu\)s. The emission area of each array was 11 \(\times\) 1.5 cm\({}^{2}\), and the circulating deionized water was used to maintain the temperature at 25 \({}^{\circ}\)C. A cylindrical micro-lens was used to collimate the fast axis. For each array, the LD light was shaped into a square outline through a cylindrical lens system, and concentrated on a homogenizer of area 10 \(\times\) 10 mm\({}^{2}\). Finally, two spherical lenses were used to image the output square pump beam, via the dichroic mirror, onto the end surface of the NDSG. The size of the pump spot was 8 \(\times\) 8 mm\({}^{2}\), as shown in Figure 3b, and it was centrally positioned on the face of the NDSG rod. A Nd:glass square rod with an absorption coefficient of 1 cm\({}^{-1}\) was used for gain comparison. The following quantities were measured: the light energy, \(E_{0}\), of the 1053 nm pulse signal without amplification; the laser output energy, \(E_{a}\), at different pump energies; and the energy, \(E_{p}\), measured through the gain medium when only the pump source was turned on. The energy of the pump light was calculated and measured using a sampling mirror and an energy meter. The gain was calculated using the formula  
\[g=\frac{\left(E_{a}-E_{p}\right)}{E_{0}} \tag{1}\]  
A wavefront sensor was used to measure the wavefront distortion in the NDSG rod at different pump frequencies. In order to measure the wavefront distortion of the entire pumping area, the injected light was replaced with a continuous point light source, with a center wavelength of a 1053-nm fiber laser.