## 4 Effects of the Parameters of Oceanic Turbulence and Mathieu-Gaussian Beam
### Influence of System Parameters  
Figure 2 reveals that the energy of the emitted vortex modes spreads into the adjacent vortex modes for different values of the vortex topological charge \(2l_{0}\) and the semi-cone angle \(\beta\). Comparatively speaking, the transmitting probability of signal vortex modes of the MG beam decreases and modes crosstalk becomes more significant as \(\beta\) increases. When \(\beta=0.0002\) and \(\beta=0.0003\), the effect of weak-turbulence channel on the transmitting probability of signal vortex mode with different vortex topological charge can be ignored. When \(\beta=0.0001\), the magnitude of the transmitting probability of signal vortex modes decreases with increase of the vortex topological charge \(2l_{0}\). Theoretically, we can improve the information capacity of the communication channel by selecting a suitable semi-cone angle and using the high-order vortex modes of the beam as the information carrier.  
Figure 3 shows the dependence of the transmitting probability of signal vortex modes of the MG beam on initial beam width \(w_{0}\), received diameter \(D\) and propagation distance \(z\). As would be expected, the magnitude of the transmitting probability of signal vortex modes decreases with increase of the propagation distance \(z\). In Figure 3a, the value of the transmitting probability of signal vortex modes decreases with increase of the \(w_{0}\). The reason for this result is that as the increase of the beam diameter, the number of turbulent eddies that have different refractive indexes in the beam also increases, which increases the wavefront aberration of the beam. As a consequence, a larger \(w_{0}\) leads to lower transmitting probability of signal vortex modes. As shown as in Figure 3b, when \(D\) increases, the value of the transmitting probability of signal vortex modes decreases first and then reaches a steady state. The phenomenon could be explained that the wave front aberration increases as increase of the \(D\), which cause larger inter-mode crosstalk of vortex modes [18]. As a consequence,  
the aperture receiver with larger \(D\) leads to the transmitting probability decreases. When the receiver aperture diameter is large enough, the whole beam spot is fully received, the transmitting probability of signal vortex modes is no longer dependent on \(D\).  
Figure 4 illustrates the impact of the wavelength \(\lambda\) on the transmitting probability of signal vortex modes of the MG beam with different values of ellipticity parameter \(q\). We find that the value of the transmitting probability of signal vortex modes decreases as \(q\) increases. It means that the MG beam with small ellipticity parameter is more suitable for underwater wireless optical communication. Like any other type of beam, the transmitting probability of signal vortex modes decreases with decreasing \(\lambda\). This is because the long-wavelength beam has lower scintillation [29].  
Figure 3: The normalized transmitting probability of the signal vortex modes versus (**a**) initial beam width \(w_{0}\) and (**b**) received diameter \(D\) with \(w_{0}=0.05\) m for different values of propagation distance \(z\).  
Figure 2: The normalized energy for each received vortex modes carried by Mathieu-Gaussian beam propagating in oceanic turbulence for different values of vortex topological charge \(2l_{0}\) and semi-cone angle \(\beta\) (\(\Delta l=l-2l_{0}\)).