# Research Progress on Router Devices for the OAM Optical Communication
## 1 Introduction  
In recent years, with the continuous development of the modern information industry and the rapid growth of large traffic consumable businesses, such as the Internet of Things, cloud computing, e-commerce, and online gaming, there has been an increasing demand for high-speed and large-capacity communication among people. This trend is expected to pose a significant challenge to existing communication technologies [1; 2; 3; 4; 5]. In the field of free-space optical communication, multi-dimensional multiplexing technologies, such as wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and time-division multiplexing (TDM), are commonly employed to enhance system capacity, along with advanced signal modulation techniques, such as quadrature amplitude modulation (QAM), pulse position modulation (PPM), and multi-phase shift keying [6; 7; 8; 9; 10]. However, in recent years, the development of these resource dimensions has nearly reached its limits. Network bandwidth has gradually become a bottleneck, and capacity crisis remains a significant challenge for optical communication. Therefore, expanding the communication system's capacity has become an urgent problem to be addressed in the field of communication [11]. At present, a novel research concept involves utilizing spatialdimensions for multiplexing. Among these dimensions, orbital angular momentum (OAM) stands out as a distinct degree of freedom from polarization, amplitude, phase, frequency, and time due to its exceptional advantages in terms of transmission speed, capacity, and stability. As such, OAM has become the primary focus of current research [12; 13; 14; 15; 16; 17]. In 1992, [PERSON] et al. [18] experimentally verified that a beam with a helical phase wavefront carries the OAM. The most common type of OAM beam in optical communication systems, the Laguerre-Gaussian (LG) beam's field distribution, can be mathematically expressed as follows [19]:  
\[ \begin{split} u_{LG(l,p)}(r,\varphi,z)&=\sqrt{ \frac{2p!}{(\pi(p+|l|)!)}}\frac{1}{w(z)}\Big{(}\frac{r\sqrt{z}}{w(z)}\Big{)}^{ |l|}L_{p}^{|l|}\Big{(}\frac{2r^{2}}{w^{2}(z)}\Big{)}\\ &\times\exp\Big{(}\frac{-r^{2}}{w^{2}(z)}-\frac{ik^{2}z}{2R(z)} \Big{)}\\ &\times\exp\left[i(2p+|l|+1)\tan^{-1}\frac{z}{z_{R}}\right]\exp( il\varphi)\end{split}  \tag{1}\]  
where \(l\) is referred to as the topological charge or OAM mode value, indicating the number of wavefront rotations within one wavelength propagation distance; \(p\) is the radial indices; \(\lambda\) is the wavelength; \(k=2\pi/\lambda\) is the wavenumber; \(w(z)=w_{0}\sqrt{1+(z/z_{R})^{2}}\) represents the Gaussian beam radius, in which \(w_{0}\) is the beam waist and \(Z_{R}=\frac{\pi w_{0}^{2}}{\lambda}\) is the Rayleigh range; \(L_{p}^{|l|}(\cdot)\) represents the generalized Laguerre polynomial; and \((2p+|l|+1)\tan^{-1}(z/z_{R})\) is the Gouy phase. Due to the existence of a phase singularity in the center of the beam, the distribution of the vortex annular field intensity is 0. This type of beam is commonly referred to as an OAM beam or vortex beam [19; 20; 21; 22; 23]. The helical phase structure characteristics of a vortex beam are determined by the phase factor \(\exp(il\varphi)\), [24; 25; 26]. \(\varphi\) represents azimuths, with each photon in a vortex beam carrying an OAM of \(l\hbar\) (where \(\hbar\) is Planck's constant divided by \(2\pi\)) [27; 28; 29]. Theoretically, any value of \(l\) can be selected, and OAM beams carrying different topological charges are mutually orthogonal [30; 31; 32]. These exceptional qualities of OAM have guaranteed promising results in addressing the problem of low information capacity in optical communications. With the advancement of OAM optical communication technology, OAM router devices (OAM-RDs) have played a key role in significantly improving the flexibility and practicability of communication systems.