## 2 Normalized Amplitude Perfect Optical Vortex in Weak Turbulent and Absorptive Seawater  
Figure 1 depicts a schematic representation of an underwater wireless optical communication system model. In this model, the transmitter (Tx) is composed of M-QAM modulator, optical driver, modulated light source and projection lens. The receiver (Rx) is made of an optical band-pass filter, receiver lens, photodetector, front-end amplifier and demodulator. The light POV is studied in weakly scintillated turbulent seawater, \(\sigma_{1\epsilon\chi T}^{2}=1.23C_{\epsilon\chi T}^{2}K^{2}/6z^{11/6}<1\)[30]. Paraxial transmission and Rytov approximations are adopted [31]. In cylindrical coordinates \((\rho,\theta,z)\) and turbulent absorbent seawater, the normalized complex amplitude of POV  
For stable stratification and isotropic turbulence, and taking into account the integral [33]  
\[\int_{0}^{2\pi}\exp[\tau\cos(\theta-\phi)-\mathrm{i}m\theta]d\theta=2\pi\mathrm{I }_{m}(\tau)\exp(-\mathrm{i}m\phi), \tag{5}\]  
and applying Equation (1) in (4), we have the conditional probability distribution of the OAM mode carried by the POV [23]  
\[p(m/m_{0}) =\frac{2\exp\left(\frac{\sigma_{0}^{2}}{\omega^{2}}-2\mathrm{I}_{ 0}n_{12}\left(1+\frac{\sigma_{0}^{2}}{2\kappa(z)}\right)\right)}{\pi w(z)^{2} \left|\left[\frac{\sigma_{0}^{2}}{\omega^{2}}\right]\right|^{2}\mathrm{I}_{m-m _{0}}\left[\frac{2\rho^{2}}{\rho^{2}}\right]} \tag{6}\] \[\times\exp\left(-\frac{\rho^{2}}{\
u_{eff}^{2}}\right)\left| \mathrm{J}_{m_{0}}\left[\frac{2\rho r_{0}}{2\overline{w}w(z)}\right]\right|^{2 }\mathrm{I}_{m-m_{0}}\left[\frac{2\rho^{2}}{\rho^{2}}\right],\]  
where \(\mathrm{I}_{m_{0}}(x)\) is the modified Bessel function of the first kind and \(m_{0}\)-th orders. In Equation (6) \(\rho_{0}\) is the transverse spatial coherence radius of the plane wave and is given by  
\[\rho_{0} =\Bigg{\{}\frac{9.444C_{\xi\chi T}^{2}(n_{r}^{2}+n_{1}^{2})z}{ \lambda^{2}(1-\overline{\omega})^{2}}\Bigg{[}\overline{\omega}^{2}\kappa_{0}^ {1/3}\mathrm{U}(2;\frac{7}{6};\frac{\kappa_{0}^{2}\eta^{2}}{R_{T}^{2}})+ \overline{\omega}^{2}4.6\eta^{2/3}\kappa_{0}\Gamma(\frac{7}{3})\mathrm{U}( \frac{7}{3};\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{T}^{2}}) \tag{7}\] \[\times\kappa_{0}^{1/3}\mathrm{U}(2;\frac{7}{6};\frac{\kappa_{0}^ {2}\eta^{2}}{R_{S}^{2}})+4.6\eta^{2/3}\kappa_{0}\Gamma(\frac{7}{3})\mathrm{U}( \frac{7}{3};\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{S}^{2}})-2\overline{ \omega}\kappa_{0}^{1/3}\mathrm{U}(2;\frac{7}{6};\frac{\kappa_{0}^{2}\eta^{2} }{R_{TS}^{2}})\] \[-9.2\eta^{2/3}\kappa_{0}\Gamma(\frac{7}{3})\mathrm{U}(\frac{7}{3} ;\frac{3}{2};\frac{\kappa_{0}^{2}\eta^{2}}{R_{TS}^{2}})\Bigg{]}\Bigg{\}},\]  
where \(C_{\xi\chi T}^{2}=0.809\times 10^{-7}\varepsilon^{-1/3}\chi_{T}\overline{ \omega}^{-2}(1-\overline{\omega})^{2}\Big{(}\mathrm{\ K}^{2}/\mathrm{m}^{2/3} \Big{)}\) is the structural constant of the salinity-temperature fluctuation of oceanic turbulence that is a function of the rate of dissipation of kinetic energy per unit mass of fluid \(\epsilon\) in the range \(\left[10^{-10},10^{-1}\right]\mathrm{m}^{2}/\mathrm{s}^{3}\) the dissipation rate of the mean-squared temperature \(\chi_{T}\) in the range \(\left[10^{-10},10^{-2}\right]\mathrm{K}^{2}/\mathrm{s}\) and \(\overline{\omega}\) is the ratio of temperature and salinity contributions to the refractive index fluctuations, which can vary in the interval \((-5,0)\). Here \(\overline{\omega}=-5\) temperature fluctuation is dominant in the induction of optical turbulence, while \(\overline{\omega}=0\) salinity fluctuation is dominant in the induction of optical turbulence.