## 2 Light Propagation in Water
### Turbulence  
Turbulence in the atmosphere as well as in water is caused by refractive index fluctuations. In the atmosphere, the refractive index is primarily driven by temperature, whereas in an underwater medium, temperature and salinity change the refractive index [15]. [PERSON] and [PERSON] introduced the power spectrum of refractive index fluctuations \(\Phi(\kappa)\) in isotropic oceanic turbulence based on temperature and salinity as follows:  
\[\Phi(\kappa)=0.388\times 10^{-8}e^{-\frac{1}{3}}\kappa^{-\frac{11}{3}}\left(1+2.35 (\kappa\eta)^{\frac{2}{3}}\right)\times\left[\frac{\chi_{T}}{w^{2}}\right] \left(w^{2}e^{-A_{T}\delta}+e^{-A_{S}\delta}-2 we^{-A_{TS}\delta}\right), \tag{2}\]  
where \(\kappa=\sqrt{\kappa_{x}^{2}+\kappa_{y}^{2}}\) is the spatial frequency of oceanic turbulence and \(\epsilon\) denotes the turbulent dissipation of kinetic energy rate per fluid mass unit. It has a high range of variations extending from \(10^{-2}\,\mathrm{m}^{2}/\mathrm{s}^{3}\) to \(10^{-8}\,\mathrm{m}^{2}/\mathrm{s}^{3}\), and \(\eta\) is the Kolmogorov microscale. The symbol \(\chi_{T}\) indicates the dissipation rate of mean-square temperature, which ranges from \(10^{-4}\,\mathrm{K}^{2}/\mathrm{s}\) to \(10^{-10}\,\mathrm{K}^{2}/\mathrm{s}\). Furthermore, the coefficients are defined as follows:  
\[ \begin{split} A_{T}=1.863\times 10^{-2},& A_{S}=1.9 \times 10^{-4},& A_{TS}=9.41\times 10^{-3},\\ &\delta=8.284(\kappa\eta)^{\frac{4}{3}}+12.978(\kappa\eta)^{2}. \end{split}  \tag{3}\]  
The dimensionless measure \(w\) quantifies the relative impact of temperature and salinity fluctuations on underwater optical turbulence, ranging from \(-5\) (temperature dominance) to \(0\) (salinity dominance) [16; 17]. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Attenuation coefficient in clean ocean & \(c=\ 0.151\ 1/\mathrm{m}\) \\ LC-SLM aperture diameter & \(D=4\,\mathrm{mm}\) \\ LC-SLM fitting error constant & \(\alpha=0.349\) \\ Rate of dissipation of turbulent kinetic energy & \(\epsilon=10^{-2}\,\mathrm{m}^{2}/\mathrm{s}^{3}\) \\ Kolmogorov microscale & \(\eta=10^{-2}\,\mathrm{m}\) \\ Rate of dissipation of mean-square temperature & \(\chi=10^{-4}\,\mathrm{K}^{2}/\mathrm{s}\) \\ Carrier frequency & \(f_{c}=600\,\mathrm{THz}\) \\ Laser launch power & \(P_{t}=10\,\mathrm{mW}\,(10\,\mathrm{d}\mathrm{B}\mathrm{m})\) \\ Insertion loss of MZM & \(\eta_{\mathrm{MZM}}=31.6\%\,(-5\,\mathrm{d}\mathrm{B})\) \\ Insertion loss of OPA & \(\eta_{\mathrm{Tx}/\mathrm{Rx}}=50\%\,(-3\,\mathrm{d}\mathrm{B})\) \\ Number of OPA antenna elements & \((N\times M)=(16\times 16)\) \\ Transmitter and receiver antenna & \(D_{t/r}=28.9\,\mathrm{d}\mathrm{B}\mathrm{m}\) \\ directivity & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used for numerical simulations. The effect of different water parameters on the turbulence strength has been extensively studied by researchers [18; 19; 20]. In this study, due to the significant influence of \(w\) on turbulence strength, we fix the water parameters as listed in Table 1 for our numerical simulations and choose variations of \(w\) covering weak, moderate, and strong turbulence. In Figure 1, the impact of the ratio of temperature to salinity fluctuations \(w\) on the scintillation index, which is a measure of turbulence strength, is illustrated. As distance increases, the scintillation index also rises, indicating greater turbulence strength. The comparison shows that salinity-dominated fluctuations have a stronger influence on turbulence strength than temperature-dominated fluctuations. It can also be seen that the curves for \(w=-0.2\) and \(w=-0.5\) rise beyond unity and proceed into saturation as distance increases. The reason behind this saturation is the loss of the spatial coherence of the optical wave while propagating through the turbulent cells. Turbulent cells that are larger than the spatial coherence radius do not affect the wave.