## 2 Laguerre-Gaussian Beams Carrying OAM  
In theory, any field in free space can be described by a continuous spectrum of cylindrical waves which is characterized by Laguerre polynomials [26]. Different expressions strictly satisfy the orthogonal relation. Beams propagating in the -direction in free space can be expressed as:  
\[E(r,\varphi,z)=u(r,\varphi,z)e^{jkz} \tag{1}\]  
where \(u(r,\varphi,z)\) is the amplitude of beams in cylindrical coordinates, \(k\) is the wave number in free space. The space EM waves also satisfy the Helmholtz equation:  
\[\big{(}\
abla^{2}+k^{2}\big{)}E(r,\varphi,z)=0 \tag{2}\]  
The substitution of Equation (1) into the scalar wave Equation (2) gives:  
\[\frac{1}{r}\frac{\partial}{\partial r}\bigg{(}r\frac{\partial u}{\partial r} \bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}u}{\partial\varphi^{2}}+\frac{ \partial^{2}u}{\partial z^{2}}+2 jk\frac{\partial u}{\partial z}=0 \tag{3}\]  
The electric field amplitude in the -direction varies much slowly compared to the transverse field in the beam. Therefore, the paraxial wave equation is derived when the second derivative in the -direction is ignored. Then, the cylindrically symmetric solutions Laguerre-Gaussian beams are given by [1]:  
\[\begin{array}{ll}LG_{pl}=&\sqrt{\frac{2p!}{\pi(p+|l|)!}}\frac{1}{w(z)}\Big{[} \frac{r\sqrt{z}}{w(z)}\Big{]}^{|l|}\exp\Big{[}\frac{-r^{2}}{w^{2}(z)}\Big{]} \times L_{p}^{|l|}\Big{[}\frac{2r^{2}}{w^{2}(z)}\Big{]}\exp(jl\varphi)\\ &\qquad\times\exp\Big{[}\frac{j_{0}r^{2}}{2\big{(}2^{2}+z_{R}^{2}\big{)}} \Big{]}\times\exp\Big{[}j(2p+|l|+1)\tan^{-1}\Big{(}\frac{z}{z_{R}}\Big{)} \Big{]}\end{array} \tag{4}\]  
where \(w(z)\) is the radius of the beams, \(z_{R}\) is the Rayleigh range. \(l\) and \(p\) are angular index and radial index, respectively. \(L_{p}^{|l|}(x)\) is the associated Laguerre polynomial, which is derived from:  
\[L_{p}^{|l|}(x)=(-1)^{|l|}\frac{d^{|l|}}{dx^{|l|}}L_{p+|l|}(x) \tag{5}\]  
Different LG beams are determined by different number of \(l\) and \(p\). All LG beams constitute a set of complete orthogonal basis vectors that can represent any spatial beams. The amplitude and phase distributions with different \(l\) and \(p\) values are shown in Figure 1. The radius of the ring-shaped beam will diffract when \(l\) increases, while the parameter \(p\) will lead to a change in the number of rings.  
In 1992, [PERSON] et al. pointed out that the LG beams have a well-defined orbital angular momentum where the spatial phase factor \(e^{-\beta\varphi}\) corresponds to the orbital angular momentum. The angular momentum density can be expressed as:  
\[M=\varepsilon_{0}r\times(E\times B) \tag{6}\]  
where \(\varepsilon_{0}\) is the permittivity, \(E\) and \(B\) represent the electric field and magnetic field, respectively. In classical mechanics and atomic physics, angular momentum can be decomposed into SAM and OAM. Therefore, the total angular momentum is:  
\[J=L+S=\varepsilon_{0}\int r\times(E\times B)dr \tag{7}\]  
where \(L\) represents OAM and \(S\) represents SAM. For a linear polarized wave, the time averaged angular momentum density is obtained when applied to LG beams expression:  
\[M=-\frac{l}{w(z)}\frac{z}{r}|u|^{2}r+\frac{r}{\sqrt{\frac{2p^{l}}{\pi(p+||)!}} }\Bigg{[}\frac{z^{2}}{\left(z^{2}+z_{R}^{2}\right)}-1\Bigg{]}|u|^{2}\varphi+ \frac{l}{w(z)}|u|^{2}z \tag{8}\]  
The first two terms of the expression are symmetrical about the axis which are related to SAM. Therefore, the integration the angular momentum \(J\) will only remain the third term which is related to OAM. Similarly, for circularly polarized wave, the angular momentum power of \(z\) component is:  
\[M_{z}=\frac{l}{w(z)}|u|^{2}+\frac{\sigma_{z}r}{2w(z)}\frac{\partial|u|^{2}}{ \partial r} \tag{9}\]  
where \(\sigma_{\text{z}}=\pm 1\) for left handed and right handed circularly polarized (LHCP and RHCP) beams. As such, orbital angular momentum is theoretically demonstrated in the LG beams.