## 4 Results  
The experimental values of NDSG gain are shown in Figure 4a. With a pumping energy of 7.79 J, the maximum gain is 1.16, and the corresponding absorption coefficient is 0.053 cm\({}^{-1}\). However, for the same pumping conditions, the Nd:glass square rod achieved a gain of 2.44. The high uniformity of pump intensity distribution in the horizontal and the vertical directions is evident in Figure 4b. Because the NDSG laser is amplified in a four-level system, the ideal gain can be calculated using the equations  
\[g_{0}=\frac{E_{st}}{E_{s}} \tag{2}\]  
\[E_{s}=\frac{\textit{ln}}{\sigma}. \tag{3}\]  
Here, \(g_{0}\) is the small-signal gain coefficient; \(E_{st}\) is the energy storage density; \(E_{s}\) is the saturation energy density; and for the NDSG stimulated emission cross section, \(\sigma=1.4\) pm\({}^{2}\). The saturation energy density of NDSG, obtained by calculation, was 13.47 J/cm\({}^{2}\). The calculated energy storage density was 1.74 J/cm\({}^{2}\) (assuming a fluorescence efficiency of 53%) when the pump energy was 7.79 J, the absorption coefficient was 1.6 cm\({}^{-1}\) and the pump spot size was 8 mm \(\times\) 8 mm\({}^{2}\). Under ideal conditions, the calculated absorption coefficient is 0.32 cm\({}^{-1}\), and, using Equation (2), the corresponding gain at 7.79 J is 1.43, as seen in Figure 4a. Comparing the theoretical and experimental results, we find that the measured gain is much lower than the calculated value, and far less than the gain of the neodymium-glass square rod. A large number of excitation particles were consumed by amplified spontaneous emission (ASE), tending to reduce the gain. However, the uniformity of the material affects the gain amplification result. Most photons fail to pass through NDSG due to scattering losses. The wavefront distortion, when the NDSG rod was clamped using O-rings, is shown in Figure 5. The observed distribution of wavefront distortion at the pumping frequency of 1 Hz and a pulse energy of 7.79 J is characterized by central axis symmetry (Figure 5a). The peak-to-valley (P-V) wavefront distortion reached a maximum value of 6.96 \(\lambda\) (\(\lambda\) = 1053 nm) at the corners of the square. When the pump frequency was increased, the distributions of wavefront distortion were similar to those shown of Figure 5a, but the maximum values became larger, and showed a linear relationship with frequency. As shown in Figure 5b, when the frequency was increased by 1 Hz, the increase in wavefront distortion was approximately 7.4 \(\lambda\), reaching 37.14 \(\lambda\) at 5 Hz. The wavefront distortion distribution, as presented in Figure 5a, was decomposed using Legendre polynomials, to yield the coefficients given in Figure 6. We found that the main distortions were due to the 1 st, 4 th and 6 th terms. Of these, the 4 th and 6 th coefficients correspond to positive spherical aberration. The large stress caused by the O-rings caused a certain degree of deformation of the NDSG rod. Further, the end surfaces were not cooled; thus, heat could accumulate, causing thermal lens effects. Finally, the poor doping of NDSG would also affect the measured wavefront distortion. Figure 4: (**a**) Graph of theoretical and measured gain parameters of Nd-doped silica glass versus pump energy and (**b**) pump light intensity distribution. In the second test, the O-rings were removed, and the resulting wavefront distortion distribution at a pump frequency of 1 Hz is shown in Figure 7a. Compared with Figure 5a, the change in wavefront distortion is obvious. Here, the maximum wavefront distortion at 1 Hz is 2.67 \(\lambda\), much smaller than the 6.96 \(\lambda\) observed with the O-rings. A comparison of the results of these two tests reveals that stress had a great influence on the wavefront distortion for NDSG, which has a larger stress constant than that of quartz glass. The dependence of maximum wavefront distortion versus pump frequency, as shown in Figure 7b, displays a linear relationship similar to that in Figure 5b.