## 3 OAM Beams and Laguerre-Gaussian (LG) Modal Analysis  
Light waves can carry OAM beams, which are characterized by the helical phase front of a light beam that is represented by \(e^{im\varnothing}\), where m and \(\varnothing\), respectively, represent the number of \(2\pi\) phase shifts in the phase profile of beam and the azimuthal angle [30].  
Figure 2 shows the donut shape of the OAM beam (wave front, intensity, and phase) with different values of \(m\).  
OAM beams can have an infinite number of states, where \(m\) can be any integral value. OAM beams are orthogonal while propagating coaxially with each other. As an example, consider two different OAM beams with \(m_{1}\) and \(m_{2}\), respectively [31]:  
\[u_{1}(\rho,\varnothing,z)=A_{1}(\rho,z)e^{im_{1}\varnothing} \tag{4}\]  
\[u_{2}(\rho,\varnothing,z)=A_{1}(\rho,z)e^{im_{2}\varnothing} \tag{5}\]  
where \(\rho\) refers to radial and \(z\) is the direction of propagation and \((\rho,\varnothing,z)\) is the cylindrical coordinate. The orthogonality between two different OAM beams can be given as [32]:  
\[\int\limits_{0}^{2\pi}u_{1}(\rho,\varnothing,z){u_{2}}^{*}(\rho,\varnothing,z )d\varnothing=\left\{\begin{array}{cc}0&if\ m_{1}\
eq m_{2}\\ A_{1}{A_{2}}^{*}&if\ m_{1}=m_{2}\end{array}\right. \tag{6}\]  
As an OAM beam refers to any helically phased light beam, regardless of its radial distribution, the Laguerre-Gaussian (LG) modes are used for a complete two-dimensional (2D) modal analysis. The LG modes have both m for azimuthal distribution and p for radial distribution [33; 34].  
The mathematical description of the electric field of the LG, \(\psi(\rho,\phi,z;m,p)\), in cylindrical coordinate is given by [30; 35]:  
\[\begin{array}{ll}\psi(\rho,\phi,z;m,p)=&\sqrt{\frac{2p!}{\pi(p+|m|)!}}\frac{1 }{\omega(z)}\big{[}\frac{\
u\sqrt{2}}{\omega(z)}\big{]}^{|m|}L_{p,m}\Big{[} \frac{2\
u^{2}}{\omega^{2}(z)}\Big{]}\\ &\times exp\left[\frac{-r^{2}}{\omega^{2}(z)}\right]exp\Big{[}\frac{-icr^{2}z }{2(z^{2}+z^{2}k^{2})}\Big{]}\\ &\times exp\left[i(2p+|m|+1)tan^{-1}\frac{z}{z_{k}}\right]\exp(-im\varnothing) \end{array} \tag{7}\]  
Figure 2: Donut shape of OAM beam with m = 0, 1, 2, 3 [29].  
where \(\rho\), \(\phi\), \(z\), \(\omega(z)\), and \(z_{R}\) represent radial coordinate, angular coordinate, direction of propagation along z-axis, waist size of beam at \(z\)-distance, and Rayleigh range, respectively. \(L_{p,m}\) is \(LG\) polynomial (\(p\) and m are modal indices in \(\rho\) direction and \(\phi\) direction, respectively) and \(c\) is the optical wave number that is equal to \(2\pi/\lambda\).  
For a given \(\omega(z)\) and \(z\), the orthogonality between LG beams that have different values of m and/or p can be represented as [33; 34; 36]:  
\[\int\limits_{0}^{\infty}\int\limits_{0}^{2\pi}\psi(\rho,\phi,z;m,p)\psi^{*}( \rho,\phi,z;m,p)\rho d\rho d\rho=\left\{\begin{array}{ll}0&if\ m_{1}\
eq m_{2} \ or\ p_{1}\
eq p_{2}\\ 1&if\ m_{1}=m_{2}\ and\ p_{1}=p_{2}\end{array}\right. \tag{8}\]  
OAM can be utilized in FSO communication systems and enhance the capacity of the system through mode division multiplexing (MDM), which is a subset of SDM [37; 38].  
Each OAM beam can carry the same p value and different values of m in LG modes or the same m value and different p values. In this work, we considered the same m value that is 0 and different p values that are 0 and 10. Theoretically infinite modes are possible, but due to the limitations of OAM modes multiplexing and de-multiplexing techniques and higher inter-modal crosstalk due to a large number of modes, we used only two modes in this study.