# Laser Transverse Modes with Ray-Wave Duality: A Review
## 3 Spherical Cavity with Astigmatism
### Experimental and Numerical Results  
We present an overview of the comparison between experimental and theoretical results for Lissajous geometric modes. It has been demonstrated that Lissajous geometric modes can be well generated from an off-axis diode-end-pumped Nd:YVO\({}_{4}\) laser. The detailed description for the experiment can be obtained from previous papers [14; 40; 41]. When a short laser crystal with length of 1-2 mm was used in experiment, the value of \(M\) was found to be greater than 300 by far. Since the transverse orders of \(N_{x}\) and \(N_{y}\) for Lissajous lasing modes can be approximately \(10^{2}\sim 10^{3}\) by varying the off-axis displacement of the pumping, the degree of the overlapping \(D\) in Equation (56) is then significantly larger than 1.0. The first experimental case is shown in Figure 11a for the Lissajous geometric mode corresponding to the parameters of \((P,Q)=(1,\ 3)\), \((p,q)=(4,-1)\), and \(\phi_{x}=\phi_{y}=\pi/3\) for the transverse patterns with propagation dependence. The theoretical results are depicted in Figure 11b by using Equations (14), (46), (47), (50) and (51) with the parameters and \(M=400\), \(\left(N_{x},N_{y}\right)=(400,\ 200)\), and \(m=1\). Theoretical patterns can be seen to agree very well with experimental results. From Equation (56), the value of \(D\) for the case in Figure 11 can be obtained to be approximately 7.07. As expected, the transverse pattern displays a characteristic of continuous curve due to large overlapping between Gaussian wave packets. The condition for the critical overlapping between Gaussian wave packets can be specified as \(D=1\). Figure 12 shows the influence of the degree of the overlapping \(D\) on the transverse far-field patterns with the parameters of \((P,Q)=(1,\ 4)\), \((p,q)=(1,\ 7)\), \(\phi_{x}=\phi_{y}=0\), \(M=100\), and \(m=1\). The values of \(\left(N_{x},N_{y}\right)\) in Figure 12a-d are \((200,\ 200)\), \((400,\ 400)\), \((600,\ 600)\), and \((800,\ 800)\), respectively. Accordingly, the values of \(D\) for Figure 12a-d are \(1.414\), \(1.000\), \(0816\), and \(0.707\), respectively. Figure 12 clearly reveals that the transverse pattern varies from the continuous curves to the discrete spots for the value of \(D\) from greater to less than 1.0. Figure 11: Lissajous geometric mode corresponding to the parameters of \((P,Q)=(1,\,3)\), \((p,q)=(4,-1)\), and \(\phi_{x}=\phi_{y}=\pi/3\) for the transverse patterns at different positions: (**a**) experimental observations; (**b**) theoretical results. Figure 12: Numerical results for Lissajous transverse patterns varying from the continuous curves to the discrete spots with (**a**) \(D=1.414\); (**b**) \(D=1.000\); (**c**) \(D=0.816\); (**d**) \(D=0.707\). Experimentally, the astigmatism needs to be increased to reduce the value of \(M\) for realizing the condition of \(D<1\). One practical way of increasing the astigmatism is to use a gain medium with longer length. By using a length of 10 mm, the value of \(M\) can be reduced to the range between 40 and 120. In addition to reducing the value of \(M\), Equation (56) indicates that the degree of the overlapping \(D\) can be varied by the factor \(C_{d}\) which is the common divisor among \((M-qm)\), \((M+pm)\), and \(MQ\). The influence of the factor \(C_{d}\) is shown in Figure 13 for two cases of \(M=100\) and \(M=101\) for the far-field transverse patterns with the parameters of \((P,Q)=(1,\,4)\), \((p,q)=(1,\,7)\), \(\left(N_{x},N_{y}\right)=(500,\,300)\), \(\phi_{x}=\phi_{y}=0\), and \(m=1\). All far-field patterns were measured at the position of \(z=100\)\(L_{c}\). The results in Figure 13a with \(M=100\) can be found to correspond to \(C_{d}=1\) and \(D=0.898\). On the other hand, the case shown in Figure 13b with \(M=101\) corresponds to \(C_{d}=2\) and \(D=0.454\). It can be found that even though the difference of the astigmatism is very small for \(M=100\) and \(M=101\), the degree of the overlapping \(D\) varies significantly due to the influence of the factor \(C_{d}\).