# Laser Transverse Modes with Ray-Wave Duality: A Review
## 3 Spherical Cavity with Astigmatism
### Parametric Equations for Lissajous Geometric Rays  
From Equations (31), (32), and (35), the frequency spacings \(\Delta f_{T,x}\) and \(\Delta f_{T,y}\) for the new degeneracy \((p,q)\) can be rewritten as  
\[\Delta f_{T,x}/\Delta f_{L} =(1-\beta q)(P/Q) \tag{36}\] \[\Delta f_{T,y}/\Delta f_{L} =(1+\beta p)(P/Q) \tag{37}\]  
where \(\beta=d/(2\pi z_{R}KP)\). The transverse patterns of the high-order modes at the degeneracy \((p,q)\) have been observed to be localized on the Lissajous figures. Consequently, these observed modes are called the Lissajous geometric modes. The Rayleigh ranges for \(x\) and \(y\) directions can be given by  
\[z_{R,x} =\frac{R_{c}}{2}\sin[2\pi(\Delta f_{T,x}/\Delta f_{L})] \tag{38}\] \[z_{R,y} =\frac{R_{c}}{2}\sin\left[2\pi(\Delta f_{T,y}/\Delta f_{L})\right] \tag{39}\]with \(z_{R,x}\) and \(z_{R,y}\), the beam waists are \(w_{o,x}=\sqrt{\lambda z_{R,x}/\pi}\) and \(w_{o,y}=\sqrt{\lambda z_{R,y}/\pi}\), where \(\lambda\) is the laser wavelength. In terms of \(\Delta f_{T,x}\), \(\Delta f_{T,y}\), \(z_{R,x}\) and \(z_{R,y}\), the parametric equations for an astigmatic cavity can be derived as [18, 41]  
\[x_{s}(z)=\sqrt{N_{x}}\,w_{x}(z)\cos[(2\pi s)(\Delta f_{T,x}/\Delta f_{L})+\phi _{x}\pm\theta_{G,x}(z)] \tag{40}\]  
\[y_{s}(z)=\sqrt{N_{y}}\,w_{y}(z)\cos[(2\pi s)\big{(}\Delta f_{T,y}/\Delta f_{L} \big{)}+\phi_{y}\pm\theta_{G,y}(z)] \tag{41}\]  
where \(\mathrm{s}=0,\ 1,\ 2,\ \cdots\) is the running index for different paths, \(w_{x}(z)=w_{o,x}\sqrt{1+(z/z_{R,x})^{2}}\), \(w_{y}(z)=w_{o,y}\sqrt{1+(z/z_{R,y})^{2}}\), \(\theta_{G,x}(z)\)\(=\tan^{-1}(z/z_{R,x})\), and \(\theta_{G,y}(z)\)\(=\tan^{-1}(z/z_{R,y})\), \(\phi_{x}\) and \(\phi_{y}\) are associated with the initial condition, \(N_{x}\) and \(N_{y}\) are the transverse orders in the \(x\)- and \(y\)-directions, and \(-\) and \(+\) in the symbol of \(\pm\) correspond to the backward and forward rays, respectively.  
The parameter \(\beta\) for the geometric mode needs to be a rational number \(m/M\) for forming a complete orbit with finite rays, where \(m\) and \(M\) are coprime positive integers. By using \(\beta=m/M\), the mode spacings \(\Delta f_{T,x}\) and \(\Delta f_{T,y}\) for the degeneracy \((p,q)\) are given by  
\[\Delta f_{T,x}/\Delta f_{L}=(PM_{x})/(QM) \tag{42}\]  
\[\Delta f_{T,y}/\Delta f_{L}=(PM_{y})/(QM) \tag{43}\]where \(M_{x}=M-mq\) and \(M_{y}=M+mp\). Substituting Equations (42) and (43) into Equations (40) and (41), the parametric equations for the periodic rays of the Lissajous geometric mode related to the degeneracy \((p,q)\) in the vicinity of \(L_{P/Q}\) could be derived as  
\[x_{s}(z)=\sqrt{N_{x}}\,w_{x}(z)\cos\bigl{[}\theta_{q,s}+\phi_{x}\pm\theta_{G,x}( z)\bigr{]} \tag{44}\]  
\[y_{s}(z)=\sqrt{N_{y}}\,w_{y}(z)\cos\bigl{[}\theta_{p,s}+\phi_{y}\pm\theta_{G,y} (z)\bigr{]} \tag{45}\]  
where \(s=0,1,2,\cdots,MQ-1\) is the running index, \(\theta_{s}=2\pi s(P/Q)\), \(\theta_{q,s}=\theta_{s}(M-mq)/M\), \(\theta_{p,s}=\theta_{s}(M+mp)/M\), \(w_{x}(z)=w_{0,x}\sqrt{1+\left(z/z_{R,x}\right)^{2}}\), \(w_{y}(z)=w_{o,y}\sqrt{1+\left(z/z_{R,y}\right)^{2}}\), \(\theta_{G,x}(z)=\tan^{-1}(z/z_{R,x})\), and \(\theta_{G,y}(z)=\tan^{-1}(z/z_{R,y})\). The total ray number for forming a geometric period is \(MQ\). Typically, the value of \(M\) ranges from \(10^{2}\) to \(10^{4}\).