# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 3 Three Kinds of Tunable OAM Modes
The A\({}_{l}\) and B\({}_{l}\) are, respectively, the points with \(-0.5\) and \(0.5\) average OAM values. The OAM Varies from \(l\) to \(n\) with Inhomogeneous SOP Along the Longitude of Hybrid-Order PS  
For the second kind of tunable OAM mode, the polarization states and OAMs of modes on the higher-order PS are still confined to some special cases. For example, the modes on the equator have the azimuthally and radially linear polarization, but they are only the single vector beams possessing spatially inhomogeneous SOP and carrying zero-order OAM. The OAM modes on the poles of higher-order PS are only the single vortex beams with spiral wavefronts. Compared to a single vector mode and a single vortex mode, a vector vortex mode provides more degrees of freedom in optical manipulation [40; 41]. Hence, in 2015, the hybrid-order PS is proposed to describe the evolution of the OAM and SOP, which extends the orbital PS and higher-order PS to a more general form [42]. The representation of the modes on the hybrid-order PS is the same as those on the higher-order PS except the orbital states on the poles. The orbital states of the two poles on orbital and higher-order PSs have the same value but opposite signs. Unlike the previous PSs, the orbital states of the poles on the hybrid-order PS are not confined to the same order topological charge and can be chosen arbitrarily. Any one mode on the hybrid-order PS can be expressed as the superposition of two poles:  
\[\psi_{3}=a_{3}\big{|}\mathrm{N}_{l}^{\mathrm{R}}\big{)}+b_{3}\mathrm{e}^{i \varphi_{3}}\big{|}\mathrm{S}_{n}^{\mathrm{L}}\big{\rangle} \tag{13}\]  
\[\big{|}\mathrm{N}_{l}^{\mathrm{R}}\big{\rangle}=\mathrm{e}^{i\theta}(\overset{ \rightarrow}{\propto}+i\overset{\rightarrow}{\gamma})/\sqrt{2} \tag{14}\]  
\[\big{|}\mathrm{S}_{n}^{\mathrm{L}}\big{\rangle}=\mathrm{e}^{i\pi\theta}( \overset{\rightarrow}{\propto}-i\overset{\rightarrow}{\gamma})/\sqrt{2} \tag{15}\]  
Equations (14) and (15) represent right and left circularly-polarized modes with different topological charges \(l\) and \(n\), respectively. Any mode on the hybrid-order PS can be achieved by changing the coefficients \(a_{3}\) and \(b_{3}\mathrm{e}^{i\varphi_{3}}\). Generally, the equatorial points on the hybrid-order PS represent the superposition of the two poles with equal intensities. The H\({}_{l,n}\) and V\({}_{l,n}\) of Figure 4 are the two equatorial points when \(\phi_{3}=0\) and \(\pi\), respectively. \[ \begin{array}{c}\big{|}\mathrm{H}_{l,n}\big{\rangle}=\frac{\big{|}\mathrm{N }_{l}^{\mathrm{L}}\big{\rangle}+\big{|}\mathrm{S}_{n}^{\mathrm{L}}\big{\rangle} }{2}=\exp\frac{i(l+n)\theta}{2}[\cos\frac{(l-n)\theta}{2}\overset{ \rightarrow}{\propto}+\sin\frac{(l-n)\theta}{2}\overset{\rightarrow}{ \rightarrow}]\\ \big{|}\mathrm{V}_{l,n}\big{\rangle}=-i\frac{\big{|}\mathrm{N}_{l}^{\mathrm{L }}\big{\rangle}-\big{|}\mathrm{S}_{n}^{\mathrm{L}}\big{\rangle}}{2}=\exp\frac {i(l+n)\theta}{2}[\cos(\frac{l-n}{2}\theta+\frac{\pi}{2})\overset{ \rightarrow}{\propto}+\sin(\frac{l-n}{2}\theta+\frac{\pi}{2})\overset{ \rightarrow}{\gamma}]\end{array}  \tag{16}\]  
From Equation (16), it should be noted that the equatorial points represent modes carrying \((l+n)/2\) per photon. The relative phase of the superposition determines the orientation of the longitude. Figure 3: (a) Intensity profiles, polarization states and phase distributions of superimposed modes when \(l=1\) and (b) \(l=-1\). The north pole \(\mathrm{N}_{l}^{\mathrm{R}}\) and south pole \(\mathrm{S}_{l}^{\mathrm{L}}\) represent orthogonal circularly-polarized modes with topological charges of \(-l\) and \(+l\); The points H\({}_{l}\) and V\({}_{l}\) represent the two points on the equator; The A\({}_{l}\) and B\({}_{l}\) points represent the two modes with \(-0.5\) and \(0.5\) average OAM values, respectively. Figure 4 depicts a hybrid-order PS at the situation of the north pole with state \(\sigma=+1\) and \(l=0\), while the south pole with \(\sigma=-1\) and \(n=+2\).