# Optical Encoding Model Based on Orbital Angular Momentum Powered by Machine Learning
## 2 Concept and Principle of the Optical Encoding Model  
The schematic setup of the conceptual art of this proposed optical encoding model is illustrated in Figure 1. On the transmitter side, an optical system based on a Mach-Zehender interferometer is used to generate a coherent combination of two Laguerre-Gaussian (LG) beams carrying orbital angular momentum (OAM). A laser source provides a coherent fundamental Laguerre-Gaussian (LG00) beam in free space that is launched to a polarization beam splitter (PBS) to control relative power between the reference and the selector arm. Both arms will go through an OAM generator to convert a fundamental LG(\(p=0\), \(\ell=0\)) mode to a higher-order LG(\(p,\ell\)) mode carrying OAM. The reference arm is converted to an LG(\(p,\ell=1\)) mode (via OAM Generator 1 in Figure 1), while selector arm is converted to an LG(\(p,\ell\)) mode carrying OAM (via OAM Generator 2 in Figure 1). Since the topological charge \(\ell\) and the radial index p at the LG(\(p,\ell\)) selector beam can be properly selected to generate the intensity pattern for the optical encoder, this mode index will be the code-key numbers associated with the data symbol. The reference and selector arms are combined through a beam splitter (BS1), and the intensity profile of this superposition will be the pattern corresponding to a unique data symbol associated with (\(p,\ell\)) combination. After encoding, the transmitted output beam is transferred to a communication channel in which different noise sources will be added in order to affect the signal. On the receiver side, the received beam is decoded by a machine learning process using an SVM-based method for image recognition and classification. For the generation stage of the intensity pattern that will be used on the transmitter side, the mathematical formulation of the Laguerre-Gaussian beams [35; 36] has been  
Figure 1: Concept and proposed setup of an optical encoding model. LS: laser source; PBS: polarization beam splitter, M1,2: mirror; BS1,2: beam splitter; PD: photodetector. used, which is characterized by two indices \((p,\ell)\) corresponding to radial and azimuthal distribution, respectively. The optical field of an LG(\(p,\ell\)) mode can be represented by:  
\[LG_{p,\ell}(r,\theta,z) \tag{1}\] \[=\sqrt{\frac{2p!}{\pi(p+|\ell|)!}}\frac{1}{w(z)}\left(\frac{r\sqrt{ 2}}{w(z)}\right)^{|\ell|}L_{p}^{|\ell|}\left(\frac{2r^{2}}{w^{2}(z)}\right)\] \[\exp\left(\frac{-r^{2}}{w^{2}(z)}\right)\exp\left(\frac{ik_{\ell }r^{2}z}{2\left(z^{2}+z_{R}^{2}\right)}\right)\exp(\Phi(z))\exp(i\ell\theta)\]  
where \(w(z)\) is the beam width, \(z_{R}\) is the Rayleigh range and \(\Phi(z)\) is the Gouy phase. \(L_{p}^{|\ell|}\) are the generalized Laguerre polynomials, and \((r,\theta,z)\) represents the cylindrical coordinate. Then, the superposition of two LG modes carrying OAM [37] can be expressed as:  
\[u(r,\theta,z)=LG_{p^{\prime},\ell^{\prime}}(r,\theta,z)+LG_{p,\ell}(r,\theta,z) \tag{2}\]  
The first term in Equation (2) describes the reference field, while the second term represents the optical field that acts as a selector. As mentioned, for the optical encoder presented in this work, the \(LG(p,\ell=1)\) mode will be used as the \(LG_{p^{\prime},\ell^{\prime}}(r,\theta,z)\) reference beam, while the selector beam \(LG_{p,\ell}(r,\theta,z)\) will be a previously selected \(LG_{p,\ell}\) mode. The same radial index \(p\) has been chosen for both reference and selector beam in order to simplify the design of the encoder. An intensity profile of \(u(r,\theta,z)\) is associated with a data-bit sequence according to the \((p,\ell)\) parameters used in the selector beam generation. Since OAM beams have twisted helical phase fronts, often characterized by the azimuthal index \(\ell\) (also named topological charge), while propagating[11], the intensity profile will be most affected in rotations along the propagation axis, without significant changes in the intensity pattern.