# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 4 Advances in Fiber OAM Generation Systems
### Micro-Structured Optical Fiber-Based OAM Generation System  
The generation of OAM mode by the studies mentioned above are based on the all-fiber system concluded in Figure 4. The mode coupling module can only couple the fundamental mode to the mixed mode composed of four degenerated modes in the lth-order CV mode group with random phase and amplitude. A field control module is used to redistribute the phase and amplitude of four degenerated modes to generate the OAM mode. It is usually difficult to obtain one particular OAM mode by adjusting the PC in the all-fiber system. It is more convenient if we can use a special fiber device to directly couple the fundamental mode to the OAM mode we want. There have been some reported works generating OAM modes directly through a special micro-structured optical fiber design [49; 50; 51]. There are two main operating principles of designing fiber for converting the fundamental mode to the high order OAM modes. One is based on the mode coupling theory. The designed fibers based on this principle are usually composed of one or several single-mode cores used to support the fundamental mode and one ring core used to support OAM mode. The electric fields of the fundamental core mode and the high order mode in the ring can be expressed as:  
\[ \begin{array}{l}E_{A}(r)=A(z)E_{A}(x,y)\exp(i\beta_{A}z)\\ E_{B}(r)=B(z)E_{B}(x,y)\exp(i\beta_{B}z)\end{array}  \tag{12}\]  
where the coefficients \(A(z)\) and \(B(z)\) vary with \(z\). And according to coupling mode theory, the two coefficients satisfy:  
\[ \begin{array}{l}\frac{dA(z)}{dz}=i\kappa_{AA}A+i\kappa_{AB}Be^{i(\beta_{B}- \beta_{A})z}\\ \frac{dE(z)}{dz}=i\kappa_{BB}B+i\kappa_{BA}Ae^{i(\beta_{A}-\beta_{B})z}\end{array}  \tag{13}\]The mode coupling coefficient \(\kappa\) is given by:  
\[\kappa_{\varphi t}=\frac{\omega}{4}\int\limits_{-\infty}^{\infty}\int\limits_{- \infty}^{\infty}E_{v}^{*}(x,y)\cdot\Delta\varepsilon\cdot E_{\mu}(x,y)dxdy \tag{14}\]  
where \(\omega\) is the optical frequency, and \(\triangle\varepsilon\) is the perturbation to the permittivity. The light power is supposed to only be launched into the fundamental mode core, so the coupling mode equation can be solved as:  
\[ \begin{array}{c}A(z)=(\cos\gamma z-\frac{i\delta}{\gamma}\sin\gamma z)e^{i \delta z}\\ B(z)=\Big{(}\frac{i\kappa_{BA}}{\gamma}\sin\gamma z\Big{)}e^{-i\delta z}\end{array}  \tag{15}\]  
where \(\gamma=(\kappa_{AB}\times\kappa_{BA}+\delta^{2})^{1/2}\) and \(\delta\) is the phase mismatching coefficient, where, \(\delta=(\beta_{B}+\kappa_{BB}-\kappa_{AA}-\beta_{A})/2\). The mode coupling efficiency can be expressed as:  
\[\eta=\frac{|\kappa_{BA}|^{2}}{\gamma^{2}}\sin^{2}\gamma z \tag{16}\]  
The authors of this paper have proposed and investigated a tunable microstructure optical fiber for different OAM mode generation by simulation based on the mode coupling theory. The microstructure optical fiber is composed of a high RI ring and a hollow core surrounded by four small air holes as shown in Figure 12[49]. The hollow core and the surrounded four air holes are infiltrated by optical functional material whose RI can be modulated by physical parameters, leading to conversion between circularly-polarized fundamental mode and different OAM mode in the high RI ring with tunable operating wavelengths. The OAM modes are composed by \(\hat{\sigma}^{\pm}OAM_{\pm l,1}=HE_{l+1,1}^{even}\pm iHE_{l+1,1}^{odd}\) and \(\hat{\sigma}^{\mp}OAM_{\pm l,1}=EH_{l+1,1}^{even}\pm iEH_{l+1,1}^{odd}\), where \(l\) ranges from 2 to 8. The other is to mimic the refractive spiral phase element [50; 51]. This method typically uses the multi-core fiber (MCF) [50] or photonic crystal fiber (PCF) [51]. For the MCF, the phase change is a function of the core RI. Therefore, by arranging the RI distribution among the multi cores of the MCF The phase difference between the adjacent core exactly equal \(2\pi l/N\) where \(l\) is the TC of the desired OAM mode and \(N\) is the number of cores of MCF.