# Laser Transverse Modes with Ray-Wave Duality: A Review
## 3 Spherical Cavity with Astigmatism  
### Eigenfrequency Spectrum and Lissajous Geometric Modes  
Next, we review the eigenfrequency spectrum subject to the influence of a gain medium with birefringence. In the vicinity of \(L_{P/Q}\), let the cavity length be given by \(L_{c}=L_{P/Q}+\delta L_{c}\). The variation of the ratio \(\Delta f_{T}/\Delta f_{L}\) due to an infinitesimal change \(\delta L_{c}\) can be verified as  
\[\delta\bigg{(}\frac{\Delta f_{T}}{\Delta f_{L}}\bigg{)}=\frac{\delta L_{c}}{ \pi R_{c}\sin(2\pi P/Q)} \tag{26}\]  
In terms of the Rayleigh range \(z_{R}=(R_{c}/2)\sin(2\pi P/Q)\), the frequency ratio \(\Delta f_{T}/\Delta f_{L}\) for the cavity length near \(L_{P/Q}\) can be approximated as  
\[\frac{\Delta f_{T}}{\Delta f_{L}}=\frac{P}{Q}+\frac{\delta L_{c}}{2\pi z_{R}} \tag{27}\]  
Figure 8: Calculated results (upper row) for the transverse patterns of \(\Psi_{N,N}^{\pm}(\mathbf{r})\) on the flat mirror with \(P/Q=2/7\) and \(\phi_{x}-\phi_{y}=\pi/2\) for \(N=1,3,14\). Low row: experimental results.  
Figure 7: Calculated results by using Equation (19) with \(N_{x}=N_{y}=50\) for the transverse patterns on the flat mirror, corresponding to the nonplanar geometric modes shown in Figure 3.  
Nd-doped vanadate crystals are frequently utilized as gain materials for implementing diode-pumped solid-state lasers. Due to the large birefringence, the Nd-doped vanadate crystal unavoidably leads to significant astigmatism inside the resonator without compensation. Consequently, there is a considerable difference between the effective cavity lengths \(L_{c,x}\) and \(L_{c,y}\), where \(L_{c,x}\) and \(L_{c,y}\) are the effective cavity lengths for the transverse parts in the directions of \(x/b\) and \(y/c\), respectively. By using the new variables of \(d=L_{c,y}-L_{c,x}\) and \(L_{c}=(L_{c,y}+L_{c,x})/2\), the frequency ratios for \(x\) and \(y\) directions can be individually given by  
\[\frac{\Delta f_{T,x}}{\Delta f_{L}}=\frac{1}{\pi}\sin^{-1}\Biggl{[}\sqrt{\frac {1}{R_{c}}\biggl{(}L_{c}-\frac{d}{2}\biggr{)}}\Biggr{]} \tag{28}\]  
\[\frac{\Delta f_{T,y}}{\Delta f_{L}}=\frac{1}{\pi}\sin^{-1}\Biggl{[}\sqrt{\frac {1}{R_{c}}\biggl{(}L_{c}+\frac{d}{2}\biggr{)}}\Biggr{]} \tag{29}\]  
The difference between \(\Delta f_{T,x}\) and \(\Delta f_{T,y}\) leads the relative eigenfrequency spectrum to be  
\[F(\delta\mathbf{n};\,L_{c},d)=\delta n_{1}\frac{\Delta f_{T,x}}{\Delta f_{L}} +\delta n_{2}\frac{\Delta f_{T,y}}{\Delta f_{L}}+\delta n_{3} \tag{30}\]  
The calculated results for the relative spectrum \(F(\delta\mathbf{n};\,L_{c},d)\) as a function of \(L_{c}/R_{c}\) for \(|\delta n_{i}|\leq 10\) with \(i=1,2,3\) are shown in Figure 9. The values of \(d/R_{c}\) used in Figure 9a,b are \(3.3\times 10^{-3}\) and \(6.6\times 10^{-3}\), respectively. It can be found that the astigmatism causes the degeneracies at \(L_{c}=L_{P/Q}\) to split. Furthermore, the width of the frequency splitting can be seen to increase with increasing the value of \(d\) that measures the degree of astigmatism. Even though the astigmatism destroys the original degeneracies at \(L_{c}=L_{P/Q}\), a variety of new fine degeneracies can be generated in the vicinity, as shown in Figure 9c for the cavity length near \(P/Q=1/4\).  
Equations (27)-(29) can be exploited to obtain the cavity lengths for the new fine degeneracies. From Equations (27)-(29), the frequency spacings \(\Delta f_{T,x}\) and \(\Delta f_{T,y}\) for the cavity length near \(L_{P/Q}\) can be explicitly approximated as  
\[\frac{\Delta f_{T,x}}{\Delta f_{L}