# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 3 Three Kinds of Tunable OAM Modes
A higher-order PS is introduced as the theoretical framework for describing the spatially inhomogeneous SOPs of generalized vortex modes [36; 37], as shown in Figure 3. Similar to the points on the orbital PS, arbitrary ones on the higher-order PS can be obtained by the linear combination of the modes on the two poles. The two poles are orthogonal CP-OAM modes with opposite topological charge. \[\psi_{2}=a_{2}\big{|}\mathrm{N}_{l}^{\mathrm{R}}\big{\rangle}+b_{2}\mathrm{e} ^{i\varphi_{2}}\big{|}\mathrm{S}_{l}^{\mathrm{L}}\big{\rangle} \tag{9}\]  
where  
\[\big{|}\mathrm{N}_{l}^{\mathrm{R}}\big{\rangle}=\mathrm{e}^{-il\theta}( \overrightarrow{\mathrm{x}}+i\overrightarrow{\mathrm{y}})/\sqrt{2} \tag{10}\]  
\[\big{|}\mathrm{S}_{l}^{\mathrm{L}}\big{\rangle}=\mathrm{e}^{+il\theta}( \overrightarrow{\mathrm{x}}-i\overrightarrow{\mathrm{y}})/\sqrt{2} \tag{11}\]  
Equations (10) and (11) represent right and left circularly-polarized OAM modes with topological charge \(-l\) and \(+l\), respectively. The coefficients \(a_{2}\) and \(b_{2}\) are the amplitudes of the Equations (10) and (11), respectively, and the \(\phi_{2}\) is the relative phase between them. The higher-order PS has five salient features: (1) For \(l>1\), the OAM and SAM handedness of each pole can be the same or opposite, therefore two spheres are needed to describe higher-order SOPs of VV modes. (2) All the modes on the PS have annular intensity profiles and a dark hollow center, which possess phase or polarization singularities. (3) The modes can degenerate to the modes on the orbital PS through a linear polarizer, e.g., horizontally orientated as depicted by the double-sided arrows in the Figure 3a,b. (4) When the state of mode changes along the longitude, the average OAM value varies from \(-l\) to \(l\) and the SAM changes from \(-1\) to \(1\) (\(1\) to \(-1\)), correspondingly. Figure 3a,b show intensity profiles, polarization states and phase distributions of six points on the two higher-order PSs with \(l=\pm 1\), respectively. For \(l=+1\), the handedness of OAM and SAM on each pole is opposite, as \(\mathrm{N}_{l}^{\mathrm{R}}\) and \(\mathrm{S}_{l}^{\mathrm{L}}\) show in Figure 3a. This higher-order PS can completely characterize a general cylindrical vector mode [38], such as radial and azimuthal polarization, which are equivalent to TE and TM fiber modes. The H\({}_{l}\) and V\({}_{l}\) points are the TE\({}_{01}\) and TM\({}_{01}\) fiber modes which are obtained by the combination mode of two poles owning same amplitudes with \(\phi_{2}=0\) and \(\pi\), respectively. The deduction processes about H\({}_{l}\) and V\({}_{l}\) points are described by Equation (12). In addition, we choose two specific points (A\({}_{l}\) and B\({}_{l}\)) to illustrate the intensity and polarization distributions on the upper and lower hemisphere with \(-0.5\) and \(0.5\) average OAM values, respectively. \[ \begin{array}{l}|\mathrm{H}_{l}\rangle=\frac{\big{|}\mathrm{N}_{l}^{\mathrm{ R}}\big{\rangle}+\big{|}\mathrm{S}_{l}^{\mathrm{L}}\big{\rangle}}{2}=\cos(l \theta)\overrightarrow{\mathrm{x}}+\sin(l\theta)\overrightarrow{\mathrm{y}} \\ |\mathrm{V}_{l}\rangle=-i\frac{\big{|}\mathrm{N}_{l}^{\mathrm{R}}\big{\rangle}- \big{|}\mathrm{S}_{l}^{\mathrm{L}}\big{\rangle}}{2}=-\sin(l\theta) \overrightarrow{\mathrm{x}}+\cos(l\theta)\overrightarrow{\mathrm{y}}\end{array}  \tag{12}\]  
For \(l=-1\), the handedness of OAM and that of SAM on each pole are the same, shown in the Figure 3b. Similarly, \(\mathrm{N}_{l}^{\mathrm{R}}\) and \(\mathrm{S}_{l}^{\mathrm{L}}\) represent the modes on the north and south poles. This higher-order PS can describe the so-called \(\pi\)-vector modes [39] which are equivalent to the HE\({}_{21}^{\mathrm{e}}\) and HE\({}_{21}^{\mathrm{o}}\) fiber modes. H\({}_{l}\) and V\({}_{l}\) are the two points on the equator, whose average OAM values are both \(0\). Intermediate modes between the pole and equator have the elliptical polarizations and annular intensity profiles.