# Laser Transverse Modes with Ray-Wave Duality: A Review
## 2 Ideal Spherical Cavity
### Wave Representation for Unifying the Eigenmodes and Geometric Modes  
In this section, we review that the Gaussian wave-packet state can be developed from the coherent state of quantum harmonic oscillator to unify the representation for the HG eigenmodes and geometric modes. From quantum theory, the coherent state for the one-dimensional (1D) harmonic oscillator is given by [58]  
\[\begin{array}{rl}g(x;u)&=\sum\limits_{n=0}^{\infty}\frac{\mu^{n}}{\sqrt{n}}e^ {-|u|^{2}/2}\,\psi_{n}(x)\\ &=\pi^{-1/4}\,e^{-(x^{2}-2\sqrt{2}xu+u^{2}+|u|^{2})/2}\\ &=\pi^{-1/4}\,e^{i\,\text{Im}\left(u\right)\,\left\lfloor\sqrt{2}x-\text{Re} \left(u\right)\right\rfloor}\,e^{-\left[x-\text{Re}\left(\sqrt{2}u\right) \right]^{2}/2}\end{array} \tag{14}\]  
where \(\psi_{n}(x)\) is the HG eigenstates given by  
\[\psi_{n}(x)=(2^{n}n!)^{-1/2}\pi^{-1/4}H_{n}(x)\,e^{-x^{2}/2} \tag{15}\]  
\(H_{n}(\cdot)\) are the Hermite polynomials of order \(n\), and \(u\) is a complex parameter related to the moving path. The maximal point of the Gaussian coherent state \(g(x;u)\) in Equation (14) is just at \(x=\text{Re}(\sqrt{2}u)\). The 2D HG eigenmodes in a spherical cavity with the transverse orders of \(n_{x}\) and \(n_{y}\) are given by [1]  
\[\Phi_{n_{x},n_{y}}^{\pm}(\mathbf{r})=\,e^{-i\,\Theta^{+}(\mathbf{r})}\,\psi_{n _{x}}(\widetilde{x})\,e^{\mp i\,n_{x}\,\theta_{G}(z)}\,\psi_{n_{y}}(\widetilde {y})\,e^{\mp i\,n_{y}\,\theta_{G}(z)} \tag{16}\]  
\[\Theta^{\pm}(\mathbf{r})=\,\frac{z}{z_{R}}\frac{\widetilde{x}^{2}+\widetilde {y}^{2}}{2}\pm\theta_{G}(z) \tag{17}\]where \(\Theta^{\pm}(\mathbf{r})\) comprises the quadratic phase and Gouy phase, \(\mathbf{r}=(\widetilde{x},\widetilde{y},z)\), and the symbol of \(\pm\) represents the propagating directions. From Equation (16), the transverse part of the HG mode can be given by \(\psi^{(HG)}_{\eta_{x},\eta_{y}}(\widetilde{x},\widetilde{y})=\psi_{\eta_{x}}( \widetilde{x})\,\psi_{\eta_{y}}(\widetilde{y})\). Using Equations (12)-(17), the 2D Gaussian wave-packet state with the maximal point at \(\widetilde{x}_{s}(z)=\text{Re}[\sqrt{2}\,u_{s}^{\pm}(z)]\) and \(\widetilde{y}_{s}(z)=\text{Re}[\sqrt{2}\,v_{s}^{\pm}(z)]\) in Equations (12) and (13) can be generalized as  
\[G(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm})=\,e^{-i\Theta^{\pm}(\mathbf{r})}( \widetilde{x};u_{s}^{\pm})\,g(\widetilde{y};v_{s}^{\pm}) \tag{18}\]  
The Gaussian wave-packet state \(G(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm})\) can be seen to be the product of two 1D quantum coherent states except for the phase term \(\exp[-i\,\Theta^{\pm}(\mathbf{r})]\). In terms of \(G(\mathbf{r},u_{s}^{\pm},v_{s}^{\pm})\), the wave representation for the geometric mode with the leftward and rightward parts of a complete tracing can be expressed as [18]  
\[\Psi_{N_{x},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{Q}\sum_{s=0}^{Q-1}\,G(\mathbf{r} ;u_{s}^{\pm},v_{s}^{\pm})\,e^{i(N_{x}+N_{y})\,\theta_{s}} \tag{19}\]  
where the phase term \(\exp[i(N_{x}\,+N_{y})\,\theta_{s}]\) indicates the phase arising from the transverse component. From Equation (19), the intensity of the geometric mode within \(-L_{c}\leq z\leq 0\) can be calculated by \(I(\mathbf{r})=\left|\Psi_{N_{x},N_{y}}^{+}(\mathbf{r})\right|^{2}+\left|\Psi_{ N_{x},N_{y}}^{-}(\mathbf{r})\right|^{2}\), where the intensity emitted from the output coupler only considers the forward part \(\left|\Psi_{N_{x},N_{y}}^{+}(\mathbf{r})\right|^{2}\).