## 2 Kolmogorov Ocean Spectrum and Random Coaxial Double-Position Power Gaussian Vortex
### Random Coaxial Double-Position Power Gaussian Vortex  
In Figure 1, we give a schematic diagram of the rectangular coordinates \((x^{\prime},y^{\prime},z^{\prime})\) and the communication link for signal transmission from the light source plane \((x^{\prime},y^{\prime},z^{\prime})\) to the receiving plane \((x,y,z)\). In rectangular coordinates \((x^{\prime},y^{\prime},z^{\prime})\) and at light source plane, we consider two position power Gaussian vortexes that have the same amplitude at their waist [26]  
\[E\big{(}x^{\prime},y^{\prime}\big{)}=\big{(}x^{\prime}\pm iy^{\prime}\big{)}^{ l_{0}}\exp\left(-\frac{x^{\prime 2}+y^{\prime 2}}{w_{0}^{2}}\right),\ x^{\prime},y^{ \prime}>0 \tag{7}\]  
where \"+\" represents right- spin vortex, \"\(-\)\" represents the left--spin vortex, \(l_{0}\) is the topological charge of vortex, and \(w_{0}\) is the waist of these vortexes. Essentially, \((x^{\prime}\pm iy^{\prime})^{l_{0}}\) is a transmission factor and is a position power vortex. The position power vortex can be produced from \"fork\" hologram [27]. In the cylindrical coordinate system \((r^{\prime},\theta^{\prime},z^{\prime})\) these two position power Gaussian vortexes can be rewritten as follows  
\[E\big{(}r^{\prime},\theta^{\prime},0\big{)}=r^{\prime l_{0}}\exp\left(-\frac{ r^{\prime 2}}{w_{0}^{2}}\pm il_{0}\theta^{\prime}\right)\theta^{\prime}>0 \tag{8}\]  
where \(r^{\prime}=\sqrt{{x^{\prime}}^{2}+{y^{\prime}}^{2}}\), and \(\theta^{\prime}=\arctan(y^{\prime}/x^{\prime})\). Now, according to Equation (8), we form a coaxial double-position power Gaussian vortex with a normalized field and opposite helices in the source plane \(z^{\prime}=0\), as shown in the following equation (see Appendix A)  
\[E\big{(}r^{\prime},\theta^{\prime},0\big{)}=N\left[\frac{1-|\alpha|}{2}r^{\prime l _{0}}\exp\left(-\frac{r^{\prime 2}}{w_{0}^{2}}-il_{0}\theta^{\prime}\right)+\frac{1+| \alpha|}{2}r^{\prime l_{0}}\exp\left(-\frac{r^{\prime 2}}{w_{0}^{2}}+il_{0} \theta^{\prime}\right)\right] \tag{9}\]  
Figure 1: The schematic diagram of a signal transmission from the light source plane \((x^{\prime},y^{\prime},z^{\prime})\) to the receiving plane \((x,y,z)\). where \(N=\left\{\pi\Big{[}(1+|\alpha|)^{2}+(1-|\alpha|)^{2}\Big{]}2^{-l_{0}-3}w_{0}^{2l_{ 0}+2}\gamma\left(l_{0}+1,\frac{D^{2}}{2w_{0}^{2}}\right)\right\}^{-1/2}\) is normalization coefficient, \(D\) is the diameter of the receiver, \(\gamma\left(l_{0}+1,\frac{D^{2}}{2w_{0}^{2}}\right)\) is the incomplete gamma function, and \(\alpha\) denotes a dimensionless structural parameter. When \(|\alpha|=1\), the beam degenerates into left spin or right spin single-position power Gaussian vortex, respectively; when \(|\alpha|\
eq 1\), the beam is a coaxial double-position power Gaussian vortex with reverse helix and different amplitude, and when \(|\alpha|=0\), the beam is a coaxial double-position power Gaussian vortex with inverse spiral and equal amplitude. The field of the coaxial double-position power Gaussian vortex in free space at any receiving lane \(x^{\prime}=x,y^{\prime}=y,z^{\prime}=z\) is given by (see Appendix B)  
\[ \begin{array}{ll}E(r,\theta,z)&=-\frac{ik}{2 Nz}\exp\left[i\left(kz+\frac{kr^ {2}}{2z}\right)\right]\left[(1-|\alpha|)(-i)^{l_{0}}e^{-il_{0}\theta}+(1+| \alpha|)(i)^{l_{0}}e^{il_{0}\theta}\right]\\ &\times\frac{zk^{l_{0}}w_{0}^{2l_{0}(l_{0}+1)}\rho_{0}}{\left(2z-ikw_{0}^{2} \right)^{l_{0}+1}}F_{1}\left(l_{0}+1;l_{0}+1;-\frac{k^{2}r^{2}w_{0}^{2}}{2z \left(2z-ikw_{0}^{2}\right)}\right)\end{array}  \tag{10}\]  
where \(\Gamma\Big{(}\frac{v+\mu+1}{2}\Big{)}\) is gamma function, and \({}_{1}F_{1}\Big{(}\frac{v+\mu+1}{2};v+1;-\frac{b^{2}}{4a}\Big{)}\) is confluent hypergeometric function. \(k\) is the wave number, and \(\lambda\) is the wavelength of light in the vacuum. In weak turbulence and absorbent seawater and by Rytov approximation [25], the field of the RCDPFGV is expressed as (see Appendix C)  
\[