# Bit Error Rate of Multi-Gaussian Correlated Asymmetric Bessel Beam Through Turbulent Ocean
## 3 Numerical Simulations and Discussion  
In this section, we investigate the reception probability and BER of OAM modes for the MGCAB beam propagating in oceanic turbulence. The effects of turbulent environment are numerically analyzed and recommendations on beam parameters are given to optimize beam transmission under QAM. Except for specific instruction, the parameters of beam and turbulence in the following simulations are set as follows: \(l_{0}=1\), \(c=0.1\), \(SNR_{0}=25\) dB, \(N=5\), \(\theta_{0}=0.05\) mrad, \(\omega_{0}=0.01\) m, \(\sigma=0.05\) m, \(\lambda=532\) nm, \(z=100\) m, \(R=0.02\) m, \(M=8\), \(\chi_{T}\)= \(10^{-7}\) K\({}^{2}\)/s, \(\varepsilon=10^{-3}\) m\({}^{2}\)/s\({}^{3}\), \(\langle T\rangle=15\)\({}^{\circ}\)C, \(\langle S\rangle=15\) ppt, and \(H=-10\)\({}^{\circ}\)C/ppt. Figure 2 illustrates the received mode probability distribution of the MGCAB beam and the effects of average temperature \(\langle T\rangle\) on the reception probability of OAM signal mode under different transmission distances \(z\). When \(\Delta l=0\) (i.e., \(l=l_{0}\) in Equation (19)), the reception probability refers to that of signal mode. Otherwise, \(\Delta l\
eq 0\) refers to the crosstalk case. As shown in Figure 2a, the signal reception probability gradually decreases as the average temperature rises, while the mode crosstalk continuously enhances. This phenomenon can be attributed to the fact that the higher temperature accelerates the thermal motion and expansion of water molecules. Then, the instability of stratified turbulence structure is enhanced, leading to stronger turbulence effects on beam propagation. As a result, the relatively severe wavefront distortion elevates crosstalk within OAM modes. Moreover, the received modes probability distribution exhibits asymmetry with respect to the signal mode (i.e., the crosstalk probability \(P_{\Delta l=1}>P_{\Delta l=-1}\)), and this can be ascribed to the high-order mode components in the amplitude of the MGCAB beam. Figure 2b shows that the transmission distance has a large influence on the signal reception probability of the MGCAB beam. When the transmission distance increases, the signal reception probability reduces due to the cumulative turbulence effects on beam intensity and phase patterns, and this phenomenon is more pronounced in high-temperature oceanic conditions. The received mode probability distribution of the MGCAB beam versus average salinity concentration \(\langle S\rangle\) is reported in Figure 3a. Similar to the case presented in Figure 2a, the probability distribution of received modes also exhibits asymmetry relative to the signal mode. The signal reception probability of the MGCAB beam slightly decreases with the increment in salinity concentration. On one hand, the rise of salinity leads to smaller seawater density fluctuation, which causes less refractive effect on beam propagation, thereby reducing the wavefront distortion of OAM mode. On the other hand, the higher salinity causes larger concentration fluctuation and thus more scattering effect in seawater, resulting in the attenuation of signal mode [36]. Since the scattering effect of seawater is greater than refraction, the signal reception probability reduces with the rising salinity. When the transmission distance \(z\) increases, there is a relatively obvious decrease in the signal reception probability of the MGCAB beam, as shown in Figure 3b. This is mainly due to the cumulative effect of salinity concentration in turbulent ocean. Figure 2: Effects of average temperature \(\langle T\rangle\) on (**a**) the received mode probability distribution of the MGCAB beam and (**b**) the signal reception probability under different transmission distances \(z\). Based on the analysis of reception probability, Figure 4 shows the effects of average temperature \(\left\langle T\right\rangle\) and average salinity concentration \(\left\langle S\right\rangle\) on the BER of the MGCAB beam at different transmission distances \(z\).