# Laser Transverse Modes with Ray-Wave Duality: A Review
## 3 Spherical Cavity with Astigmatism
### Wave Representation for Lissajous Geometric Modes  
Next, we review that the parametric equations in Equations (44) and (45) for geometric rays can be implanted into the coherent state in Equation (14) to derive the Gaussian wave-packet representation for Lissajous geometric modes. By using the dimensionless variables \(\widetilde{x}=\sqrt{2}x/w_{x}(z)\) and \(\widetilde{y}=\sqrt{2}y/w_{y}(z)\), the parametric equations in Equations (44) and (45) can be written as \(\widetilde{x}_{s}(z)=\mathrm{Re}[\sqrt{2}\,u_{s}^{\pm}(z)]\) and \(\widetilde{y}_{s}(z)=\mathrm{Re}[\sqrt{2}\,v_{s}^{\pm}(z)]\) with  
\[u_{s}^{\pm}(z)=\sqrt{N_{x}}\,\exp\bigl{\{}-i\,\bigl{[}\theta_{q,s}+\phi_{x}\pm \theta_{G,x}(z)\bigr{]}\bigr{\}} \tag{46}\]  
\[v_{s}^{\pm}(z)=\sqrt{N_{y}}\,\exp\bigl{\{}-i\,\bigl{[}\theta_{p,s}+\phi_{y}\pm \theta_{G,y}(z)\bigr{]}\bigr{\}} \tag{47}\]  
Considering the effect of astigmatism in Equations (16)-(19), these equations for the HG eigenmode and geometric modes are revised as  
\[\Phi_{n_{x},n_{y}}^{\pm}(\mathbf{r})=\,e^{-i\,\Theta^{(k)}}\,\psi_{n_{x}}( \widetilde{x})\,e^{\mp i\,n_{x}\,\theta_{G,x}(z)}\,\psi_{n_{y}}(\widetilde{y} )\,e^{\mp i\,n_{y}\,\theta_{G,y}(z)} \tag{48}\]  
\[\Theta^{\pm}(\mathbf{r})=\,\frac{1}{2}\biggl{\{}\biggl{[}\widetilde{x}^{2} \biggl{(}\frac{z}{z_{R,x}}\biggr{)}+\widetilde{y}^{2}\biggl{(}\frac{z}{z_{R,y }}\biggr{)}\biggr{]}\pm\bigl{[}\theta_{G,x}(z)+\theta_{G,y}(z)\bigr{]}\biggr{\}} \tag{49}\]  
\[G(\mathbf{r};u_{s}^{\pm},v_{s}^{\pm})=\,e^{-i\,\Theta^{\pm}(\mathbf{r})}g( \widetilde{x};u_{s}^{\pm})\,g(\widetilde{y};v_{s}^{\pm}) \tag{50}\]  
\[\Psi_{N_{x},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{MQ}\sum_{s=0}^{MQ-1}\,G( \mathbf{r};u_{s}^{\pm},v_{s}^{\pm})\,e^{i\,(N_{x}\,\theta_{qs}+N_{y}\,\theta _{ps})} \tag{51}\]  
Equations (49)-(51) stand for the Gaussian wave-packet representation for the Lissajous geometric modes.  
The spatial characteristics of Lissajous geometric modes can be deeply comprehended from the parametric equations in Equations (44) and (45) and the wave representation in Equations (49)-(51). We can use Equations (44) and (45) to verify that the total number of output rays is \(N=MQ/C_{d}\), where \(C_{d}\) is the common divisor among \((M-qm)\), \((M+pm)\), and \(MQ\). The total path length on the transverse plane for Lissajous geometric modes can be expressed as  
\[L_{p,q}(N_{x},N_{y};z)=\int_{0}^{2\pi}\sqrt{\left[\frac{dx(\theta;z)}{d\theta }\right]^{2}+\left[\frac{dy(\theta;z)}{d\theta}\right]^{2}}d\theta \tag{52}\]  
where  
\[x(\theta;z)=\sqrt{N_{x}}\,w_{x}(z)\cos[q(\theta+\phi_{x})] \tag{53}\]  
\[y(\theta;z)=\sqrt{N_{y}}\,w_{y}(z)\cos[p(\theta+\phi_{y})] \tag{54}\]Numerical calculations revealed that the path length \(L_{p,q}(N_{x},N_{y};z)\) relies primarily on the parameters \((p,q)\) and \(\left(N_{x},N_{y}\right)\) and is nearly independent of the phase factors \((\phi_{x},\phi_{y})\). Moreover, the path length \(L_{p,q}(N_{x},N_{y};z)\) can be accurately approximated with  
\[L_{p,q}(N_{x},N_{y};z)=4\sqrt{q^{2}N_{x}w_{x}^{2}(z)+p^{2}N_{y}w_{y}^{2}(z)} \tag{55}\]  
Since both values of \(w_{x}(z)\) and \(w_{y}(z)\) are rather close, we can use \(\sqrt{2}w_{x}(z)\approx\sqrt{2}w_{y}(z)\) as the effective width of the Gaussian wave packets. Based on the number of Gaussian wave packets \(N=MQ/C_{d}\) and the overall length \(L_{p,q}(N_{x},N_{y};z)\) in Equation (55), the degree of the average overlapping between two contiguous Gaussian spots can be estimated as  
\[D=\frac{\sqrt{2}\ n\ MQ/C_{d}}{4\sqrt{q^{2}N_{x}+p^{2}N_{y}}} \tag{56}\]  
where \(n\) is the greatest common divisor of \(p\) and \(q\). The value of \(n\) means the number of Lissajous curves appearing in the transverse plane of the corresponding geometric mode. When \(D\gg 1\), the Gaussian wave packets in Equation (51) substantially overlap with each other to display a feature of continuous curves, not the original discrete spots. In contrast, when the value of \(D\) is considerably less than 1.0, the transverse pattern can be seen to reveal a feature of separate spots corresponding to geometric rays.