# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 4 Advances in Fiber OAM Generation Systems
### Mode Selective Coupler (MSC)-Based OAM Generation Systems  
Optical fiber couplers have been widely used in many research areas, such as optical communication, optical sensing, and fiber lasers. In addition, the fiber couplers can be fabricated from various types of fibers, such as single mode fiber, few mode fiber, polarization-maintaining fiber, and ring core fiber. The operation principle of the MSC is shown in Figure 10.  
Composed of SMF and FMF, MSC is based on the phase-matching condition between the fundamental mode in the SMF and higher order modes in the FMF. This can be achieved by satisfying the ERI of the fundamental mode in the SMF equaling to that of the higher order mode in the FMF. When the phase-matching condition is satisfied, the fundamental mode in the SMF will be converted to the particular higher order mode in the FMF. In order to determine the coupling efficiency between the fundamental mode and high-order modes on the fiber tapering diameter, the following coupled equations are solved [73]:  
\[ \begin{array}{l}\frac{dA_{1}(z)}{dz}=i(\beta_{1}+C_{11})A_{1}+iC_{12}A_{2}\\ \frac{dA_{2}(z)}{dz}=i(\beta_{2}+C_{22})A_{2}+iC_{21}A_{1}\end{array}  \tag{10}\]  
where \(z\) is the distance along the coupling region of the MSC, \(A_{1}\) and \(A_{2}\) are the slowly-varying field amplitudes in the SMF and FMF of the MSC, and \(\beta_{1}\) and \(\beta_{2}\) are the propagation constant of fundamental mode in the SMF and higher order mode to be coupled in the FMF, respectively. Due to the phase matching condition that ERI of fundamental mode in the SMF equaling to that of the first or second order modes in the FMF, \(\beta_{1}\) should be equal to \(\beta_{2}\). \(C_{11}\) and \(C_{22}\), \(C_{12}\), and \(C_{21}\) are the self-coupling and mutual coupling coefficients, respectively. Self-coupling coefficients are small relative to mutual coupling coefficients, and can be ignored. Moreover, the mutual coupling coefficients \(C_{12}\approx C_{21}\approx C\), where \(C\) is a coefficient depending on the width and length of the coupling region. Thus, the power distribution in coupler can be given as follows [74]:  
\[ \begin{array}{l}P_{1}(z)=\left|A_{1}(z)\right|^{2}=1-F^{2}\sin^{2}(\frac{C}{F }z)\\ P_{2}(z)=F^{2}\sin^{2}(\frac{C}{F}z)\end{array}  \tag{11}\]  
where \(F=\left(1+\frac{\beta_{1}-\beta_{2}}{4C^{2}}\right)^{-\frac{1}{2}}\), \(F^{2}\) is the maximum coupling power between two fibers. According to Equation (11), it can be found that the power in coupling region exchanges periodically. The MSC can be divided into two types according the fiber type used in its fabrication. One type is the MSC composed of the SMF and the FMF or multimode fiber (MMF). The FMF and MMF are  
Figure 10: Schematics of the MSCs, composed of the SMF and the FMF. SMF: single-mode fiber; FMF: four-mode fiber. Figure 9: The mode pattern of generated OAM mode based on the (**a**) LCFG and (**b**) RCFG respectively; RCP: right circularly-polarized; LCP: left circularly-polarized. Reprinted with permission from [44], copyright 2019 The Optical Society. similar to the SMF, which is composed of a core, cladding, and coating. The RI difference between the fiber core and cladding or the radius of the fiber core of FMF/MMF is larger than that of SMF. The other type is the MSC composed of the SMF and the ring core fiber (RCF), also called vortex fiber. The RCF is usually composed of three parts, namely, the fiber core, cladding, and ring. In these three parts, the RI of the ring is the highest. Thus, the light is able to be restricted and propagate in the ring. While it's invariant for the RI of the fiber core and cladding. The RCF can limit the radial order \(m\) to 1 and thereby fix the number of degenerated modes in each high azimuthal order mode group to be 4, which will decrease the multi-input-multi-output complexity. In addition, the RCF is more suitable for transmitting OAM modes than the FMF or MMF. However, the fabrication of RCFs is more difficult than the FMF/MMF.