# Laser Transverse Modes with Ray-Wave Duality: A Review
## 2 Ideal Spherical Cavity
### Experimental Results and Theoretical Confirmations  
The experimental setup was a diode-end-pumped Nd:YVO\({}_{4}\) laser with a concave-flat cavity. The gain medium was an \(a\)-cut 2.0 at.% Nd:YVO\({}_{4}\) crystal with a length of 2 mm and an aperture of 10 \(\times\) 10 mm\({}^{2}\). The aperture size of the used Nd:YVO\({}_{4}\) crystal was much larger than the conventional one for generating the transverse modes as high as possible. Both end surfaces of the Nd:YVO\({}_{4}\) crystal were coated as antireflective at 1064 nm (R \(<\) 0.2%). The laser crystal was wrapped with indium foil and mounted into a copper holder with water cooling. A concave mirror with a radius-of-curvature of 20 mm was employed as the input mirror whose entrance face was coated as antireflective at the pump wavelength of 808 nm and the second surface was coated as highly reflective at 1064 nm (R \(>\) 99.8%) and highly transmissive at 808 nm (T \(>\) 95%). A flat mirror with a partial reflection of 97% at 1064 nm was used as the output coupler. The pump source was a 3.0 W 808 nm fiber-coupled laser diode. The core diameter and the numerical aperture of the coupled fiber were 100 \(\upmu\)m and 0.16, respectively. The laser crystal was placed behind the input mirror 1-2 mm. The pump beam from the fiber output was focused into the laser crystal with an average pump radius of approximately \(w_{p}\) = 100 \(\upmu\)m.  
The wave-packet state \(\Psi_{N_{x},N_{y}}^{\pm}(\mathbf{r})\) in Equation (19) can reveal the spatial structure with ray-wave duality. Moreover, the state \(\Psi_{N_{x},N_{y}}^{\pm}(\mathbf{r})\) can unify the representation for HG eigenmodes and geometric modes. Let us set \(N_{x}=0\) and \(N_{y}=N\) to demonstrate the geometric modes associated with the planar rays. From Equation (12), \(N_{x}=0\) indicates \(u_{s}^{\pm}=0\) and \(g(x;0)=\psi_{0}(x)\). As a result, Equation (19) becomes  
\[\Psi_{0,N}^{\pm}(\mathbf{r})=\frac{1}{Q}e^{-i\,\Theta^{\pm}(\mathbf{r})}\psi_ {0}(\widetilde{x})\sum_{s=0}^{Q-1}\,g(\widetilde{y},v_{s}^{\pm})\,e^{i\,N\, \theta_{s}} \tag{20}\]  
Figure 4 depicts the calculated patterns \(I(\mathbf{r})=\left|\Psi_{0,N}^{+}(\mathbf{r})\right|^{2}+\left|\Psi_{0,N}^{- }(\mathbf{r})\right|^{2}\) by using Equation (20) with \(N=100\) to correspond to the trajectories shown in Figure 2. The calculated distributions agree very well with the geometric rays for all cases. On the other hand, if the transverse order \(N\) is not high enough, the distribution cannot exhibit the ray feature. When \(N<Q/2\), the mathematical form of \(\Psi_{0,N}^{\pm}(\mathbf{r})\) can be verified to approach to the HG mode \(\psi_{0,N}^{(HG)}(\widetilde{x},\widetilde{y})\), as discussed later. By using Equation (20), the calculated far-field patterns \(I(\widetilde{x},\widetilde{y})=\left|\Psi_{0,N}^{+}(\widetilde{x},\widetilde{y },\widetilde{z}=20)\right|^{2}\) with \(P/Q=2/7\), \(\phi_{y}=0\), and \(\widetilde{z}=z/z_{R}\) are shown in Figure 5a. The values of \(N\) in Figure 5a cover from 0 to 370 to display the variation from the HG modes to the geometric modes. The experimental results are shown in Figure 5b for comparison. The experimental results were obtained by using a diode-pumped solid-state laser with the off-axis pumping [4; 14; 15; 16]. The experimental results agree very well with the calculated patterns for the variation from the HG modes to the geometric modes. Note that the transverse order \(N\) can be determined from \(\Delta y\) by finding the integer closest to \(\left(\Delta y/\omega_{c}\right)^{2}\), where \(\omega_{c}\) is the fundamental mode size in the gain medium. Figure 4: Calculated patterns \(I(\mathbf{r})=\left|\Psi_{0,N}^{+}(\mathbf{r})\right|^{2}+\left|\Psi_{0,N}^{-} (\mathbf{r})\right|^{2}\) from Equation (20) with \(N=100\), corresponding to the results shown in Figure 2.  
For a nondegenerate cavity, corresponding to \(Q\to\infty\), we can confirm that the state \(\Psi_{0,N}^{\pm}(\mathbf{r})\) is directly related to the HG mode \(\psi_{0,N}^{(HG)}(\widetilde{x},\widetilde{y})\).