## 3 Results and Discussion
### Energy Characteristics
For a beam with a flat top intensity distribution, we can represent this integral in a more evident form:  
\[1/\text{A}_{\text{eff}}=\text{const}\times\int\text{f}_{559}\left(\text{x}, \text{y}\right)\text{dx}\text{dy}\]  
where f\({}_{559}\) (x, y) is the normalized 559 nm intensity distribution:  
\[\text{I}_{559}\left(\text{x},\text{y}\right)=\text{I}_{559}\left(\text{max} \right)\times\text{f}_{559}\left(\text{x},\text{y}\right)\text{; here f}_{559}\left(\text{x},\text{y}\right)\leq 1.\]  
We can observe that for a flat top distribution of the pumping radiation, the maximum value of 1/A\({}_{\text{eff}}\) is obtained when the distribution of the Stokes radiation is also flat top, which can be realized in a flat cavity. Any spatial variation in the Stokes radiation intensity would reduce 1/A\({}_{\text{eff}}\). In stable spherical cavities with a large Fresnel number, the generation of high-order Gaussian (Laguerre-Hermite) modes predominates [10; 12], resulting in less overlap with the pump radiation and a lower conversion efficiency. At the same time, the spatial distribution of 589 nm radiation should be very similar to that of 559 nm due to the common laser cavity. Bearing in mind the narrow spectral width (<0.1 nm) of 532 nm radiation and the relatively small difference in wavelengths, we can expect that the spatial structure of 559 nm radiation should be \"imprinted\" [11] into 589 nm radiation. A good overlap should result in a high-efficiency 559 nm\(\rightarrow\)589 nm conversion. The worsening of the overlap between 532 nm and 559 nm fields in stable spherical cavities also can lead to an increase in the 559 nm generation threshold. The threshold pump intensity I\({}_{\text{TH}}\) at 532 nm depends on cavity losses and the effective gain [11] at 559 nm. We measured I\({}_{\text{TH}}\) in different Raman laser cavities with the same output mirror (98% transmission). Diffraction losses in flat and spherical cavities were small in comparison with transmission losses, so the relationships between different I\({}_{\text{TH}}\) values represent the relationships between the effective gain coefficients obtained in these cavities. The measurement results for the 4 mm pumping spot are shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Pump Mirror** & **I\({}_{\text{532}}\) = 120 MW/cm\({}^{2}\)** & **I\({}_{\text{532}}\) = 220 MW/cm\({}^{2}\)** & **I\({}_{\text{532}}\) = 350 MW/cm\({}^{2}\)** \\ \hline Concave, R = 1000 mm & 5 & 17 & 34 \\ \hline Concave, R = 500 mm & 4 & 14 & 20 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._ \begin{table}
\begin{tabular}{c c c c c} \hline
**Pump Mirror** & **Flat** & **Convex,** & **Convex,** & **Concave,** \\  & & **R2000 mm** & **R750 mm** & **R1000 mm** & **R500 mm** \\ \hline I\({}_{\text{TH}}\), MW/cm\({}^{2}\) & 45 & 50 & 50 & 72 & 110 \\ \hline \end{tabular}
\end{table} 
Table 3: Threshold pump intensity of first-Stokes-order generation. One can see that \(\rm I_{TH}\) is larger in spherical cavities than in flat cavities, and grows when the radius of the spherical mirror's curvature is reduced. These results are consistent with the results in Table 2 and support the above explanation of the behavior of \(\rm\eta_{1559}\) and \(\rm\eta_{1589}\) in Raman lasers. The diffraction losses in unstable cavities are larger than in flat cavities. The magnification coefficients, calculated using formulas [12] for cavities with convex mirrors (R2000 and R750), are 1.56 and 2.05, so there are additional light losses of 59% and 76%, respectively. We assume that the slightly larger \(\rm I_{TH}\) for lasers with unstable cavities than for those with a flat cavity is caused by these losses, whereas the effective gain in unstable cavities can be almost at the same level as in flat cavities. The exact answer can be obtained from the experiments, similar to that observed in [13].