# Laser Transverse Modes with Ray-Wave Duality: A Review
## 4 Damping Effect  
Experimental patterns shown in Figures 11-15 were generated by using an output coupler with a transmission as low as 2% at the lasing wavelength. It has been experimentally observed that the spatial structures of lasing modes can be significantly affected by the transmission of the output coupler. It has been demonstrated [53] that an output coupler with high transmission can be used to deliberately generate the asymmetric lasing modes. In theory, the influence of output transmission on the spatial structures of Lissajous lasing modes can be directly taken into account from Equation (51). Let the effective damping coefficient be considered as a loss factor from the output transmission \(T_{oc}\). The loss factor on the \(s\)th ray in Equation (51) can be given by \(\exp(-\gamma\theta_{s})\) with \(\gamma=T_{oc}/2\pi\). Taking the damping effect into account, the Gaussian wave-packet representation for Lissajous geometric modes can be expressed as [38; 53]  
\[\Psi_{N_{\
u},N_{y}}^{\pm}(\mathbf{r})=\frac{1}{MQ}\sum_{s=0}^{MQ-1}G(\mathbf{ r};u_{s}^{\pm},v_{s}^{\pm})\,e^{i(N_{x}\,\theta_{gs}+N_{y}\,\theta_{gs})}\,e^{- \gamma\,\theta_{s}} \tag{57}\]  
Figures 16a and 17a show experimental results obtained, respectively, with \(T_{oc}=2\%\) and \(15\%\) for the far-field patterns of Lissajous geometric modes with the cavity length near \((P,Q)=(1,\,4)\) for four different \((p,\,q)\). The first and second row in Figures 16a and 17a are the cases for \(\phi_{x}=\phi_{y}=0\) and \(\phi_{x}=\phi_{y}=\pi/4\), respectively. Note that the far-field patterns were measured at the position of \(z=50\)\(L_{c}\). Making a comparison between the patterns of Figures 16a and 17a, a high output transmission can lead the Lissajous transverse pattern to exhibit a feature of broken line. The asymmetric structure arising from a high output transmission can be found to depend on not only \((p,\,q)\) but also \(\left(\phi_{x},\phi_{y}\right)\). By using Equations (51) and (57), the calculated patterns corresponding to Figures 16a and 17a are shown in Figures 16b and 17b with experimental parameters and \(M=200\), \(\left(N_{x},N_{y}\right)=(150,\,100)\), and \(m=1\). All the calculated patterns are in excellent agreement with experimental results to manifest the damping effect on the spatial structures of Lissajous geometric modes. Finally, experimental results for the far-field patterns of Lissajous geometric modes generated by using an output coupler with a higher transmission of \(T_{oc}=25\%\) are shown in Figure 18a. By using Equation (57) with the parameters of \(M=200\), \(\left(N_{x},N_{y}\right)=(75,\,50)\), and \(m=1\), theoretical patterns for \(T_{oc}=25\%\) are calculated and shown in Figure 18b. The superiority of the Gaussian wave-packet representation can be confirmed from the good agreement between experimental and theoretical patterns in manifesting the damping effect arising from cavity losses.  
Figure 16: (**a**) Experimental results for the far-field patterns of Lissajous geometric modes with \((P,Q)=(1,\,4)\) for four different (\(p\), \(q\)), the first row: \(\phi_{x}=\phi_{y}=0\); second row: \(\phi_{x}=\phi_{y}=\pi/4\). (**b**) Numerical patterns calculated by using Equation (51) for one-to-one correspondence with (**a**).  
Figure 17: (**a**) Experimental results for the far-field patterns of Lissajous geometric modes generated by using an output coupler with \(T_{\infty}=15\%\). (**b**) Numerical patterns obtained from Equation (57) with the parameters used in Figure 16b.