## 3 Concept of Spatial Mode Diversity  
A spatial mode multiplexed (SMM) beam for the proposed system is generated using a spatial phase mask (SPM) and is expressed by  
\[U(\omega_{z},\varphi)=A(\omega_{z})e^{il\phi} \tag{9}\]  
where, \(A(\omega_{z})\)\(\alpha\)\(e^{(-\frac{\omega_{z}}{\omega_{0}})}\) is the electric field strength of the Gaussian beam of waist radius of \(\omega_{z}\) at the transmission distance \(z\) and \(\omega_{0}\) at \(z=0\) respectively. The OAM state phase difference is defined by \(\varphi\). After 16-QAM modulation, the hybrid electric field distribution of \(N\), OAM-OFDM state, spatially multiplexed signal [17] is given by  
\[U_{Mix}(\omega_{z},\phi,t)=\sum_{s=1}^{N}m_{s}(t)A_{s}(\omega_{z})e^{il_{s} \phi_{L}} \tag{10}\]  
In above expression, \(m_{s}(t)\) is the 16-QAM modulated signal and \(l_{s}\) is the topological quantum number which is distinctive by the angle mark of '\(s\)' respectively. The total power in \(k_{th}\) symmetrical OAM mode state Gaussian beam \(g(t)\) is given by \(r_{k}=P_{+m}+P_{-m}\) and the power ratio \(\alpha_{N}\) of \(m_{th}\) specific beam to the total power is expressed by \(\alpha_{N}\)=\(\frac{\sqrt{P_{+m}}}{r_{k}}\). The coherent coupling in the optical light field [18] for the relative phase shift \(\phi_{L}\) between two OAM beams is given by  
\[S_{KNL}(r_{k},\alpha_{N},\phi_{L},t)=U_{1}(r_{k},\alpha_{N},\phi_{L},t)+U_{2}( r_{k},\alpha_{N},\phi_{L},t)=\sqrt{r_{k}g(t)\alpha_{N}^{2}}e^{im\phi}e^{(-\frac{i \phi_{L}}{2}g(t))}+\sqrt{r_{k}g(t)(1-\alpha_{N}^{2})}e^{-im\phi}e^{(\frac{i \phi_{L}}{2}g(t))} \tag{11}\]  
A generalized Laguerre-Gaussian (\(LG_{p,l}\)) beam of twisted photons [19] with wavelength \(\lambda\), angular momentum \(l\), and radial mode number \(p\) is characterized by Equation (12), whereas the received field is given by the Equation (13), respectively:  
\[\omega_{z}=\omega_{0}\sqrt{(2p+l+1)\Bigg{(}1+(\frac{\lambda z}{\pi\omega_{0} ^{2}})\Bigg{)}^{2}} \tag{12}\]  
\[\phi(r,z)=\sum_{i=-\infty}^{+\infty}\sum_{k}\rho_{k}\alpha_{ki}u_{i}(r,z) \tag{13}\]  
After spatial mode de-multiplexing and the photo-detector collecting process, the received power in the \(i_{th}\) channel state of OAM mode [20] is given by  
\[y_{i}=\int\left|\sum_{k\in N}\rho_{k}\alpha_{ki}u_{i}(r,z)\right|^{2}\!dr=\sum _{k\in N}\rho_{k}\alpha_{ki}\Bigg{|}^{2} \tag{14}\]  
As stated in Equation (14), the signal and the inter-channel crosstalk (ICC) are coherently superimposed for the proposed system. Here \(\alpha_{ki}\) refers to the complex coefficient between the \(k_{th}\) transmitted and \(i_{th}\) received mode state of the instantaneous channel state, whereas '\(r\)' refers to the position vector. Hence the received signal vector field can be expressed as  
\[Y=[Y_{1},\ Y_{2},\ldots\ldots\ldots Y_{N}]^{T}=Y=|H_{\rho}|^{2}=\ | \begin{bmatrix} \alpha_{11}&\ldots&\alpha_{N1}\\.&\ldots&.\\ \alpha_{1N}&\ldots&\alpha_{NN}\end{bmatrix} ^{2}| \tag{15}\]where, \(\rho=[\rho_{1},\rho_{2},.........\rho_{N}]^{T}\) is the transmitted signal vector, and \(H\) is the channel matrix consists of \(\alpha_{ij}\) coefficients, respectively. This non-linear transformation between the \(N\) number of transmitted and received spatial mode signals is caused by square law detection, which is used in conventional MIMO digital signal processing techniques [21]. The photon rate \(A_{i}\) of this channel for the photon detector efficiency \(\upmu\) and signal \(\rho_{i}\alpha_{ii}\), calculated from the SMM [20], can be written in the expression below. The following expression can be used to express the photon rate \(A_{i}\) of this channel for the photon detector efficiency \(\upmu\). Here, \(\rho_{i}\alpha_{ii}\), shows the signal from the calculated SMM [20].