# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 3 Three Kinds of Tunable OAM Modes
We show the mode patterns of two points (\(\mathrm{H}_{l}\) and \(\mathrm{V}_{l}\)) which are located at the start and end points of the diameter on the equator. The \(\mathrm{H}_{l}\) and \(\mathrm{V}_{l}\) points are the combination mode of two poles owning same amplitude with \(\phi_{1}=0\) and \(\pi\), respectively. \[ \begin{split}\left|\text{H}_{l}\right\rangle=\frac{\left|\text{N}_{l} \right\rangle+\left|\text{S}_{l}\right\rangle}{\sqrt{2\left\langle\left| \text{E}_{\text{x}}\right|\text{2+}\left|\text{E}_{\text{y}}\right|\text{2} \right\rangle}}&=\frac{\sqrt{2}[\cos(l\theta)((\text{E}_{\text{x}} \overset{\rightarrow}{\propto}+\text{E}_{\text{y}}\overset{\rightarrow}{ \rightarrow})]}{\sqrt{\left\langle\left|\text{E}_{\text{x}}\right|\text{2+} \left|\text{E}_{\text{y}}\right|\text{2}\right\rangle}}\\ \left|\text{V}_{l}\right\rangle=-i\frac{\left|\text{N}_{l} \right\rangle-\left|\text{S}_{l}\right\rangle}{\sqrt{2\left\langle\left|\text{ E}_{\text{x}}\right|\text{2+}\left|\text{E}_{\text{y}}\right|\text{2} \right\rangle}}&=\frac{-\sqrt{2}[\sin(l\theta)((\text{E}_{\text{x}} \overset{\rightarrow}{\propto}+\text{E}_{\text{y}}\overset{\rightarrow}{ \rightarrow})]}{\sqrt{\left\langle\left|\text{E}_{\text{x}}\right|\text{2+} \left|\text{E}_{\text{y}}\right|\text{2}\right\rangle}}\\ \end{split}  \tag{8}\]  
When \(0<\left|a_{1}-b_{1}\right|<1\), the local curvature of the helical wavefront is no longer constant nor linear. However, the phase singularity remains, qualifying the beam as a kind of optical vortex. Meanwhile, the intensity distribution also becomes an intermediate state between the LP mode and OAM mode. For \(0<a_{1}-b_{1}<1\), the superimposed mode has the negative average OAM value and hence lies on the upper hemisphere. For \(-1<a_{1}-b_{1}<0\), the superimposed mode has the positive average OAM value and lies on the lower hemisphere. The A\({}_{l}\) and B\({}_{l}\) show two specific points on the upper and lower hemisphere, respectively. Therefore, when the relative amplitude (\(a_{1}-b_{1}\)) varies from \(-1\) to \(1\), the superimposed mode will change along the longitude and the average OAM value will also vary from \(l\) to \(-l\). The OAM Varies from \(-l\) to \(l\) with Inhomogeneous SOP Along the Longitude of Higher-Order PS  
The above tunable OAM modes have conventional homogeneous polarizations. In other words, the SOP of each point in the mode is same and invariant along the azimuthal coordinate. Recently there has been increasing interest in the modes with inhomogeneous SOPs. For those vector vortex (VV) modes, each point of the electrical field is the same polarization (linear, elliptical and circular polarization), but the polarized direction of each point is related to the azimuthal coordinate, such as the radial and azimuthal polarized cylindrical vector (CV) beams [30]. The VV modes extend the properties of conventional homogeneous polarization, such as the ability to produce strong longitudinal field  
Figure 2: **(a)** Intensity profiles, polarization states and the phase distributions of superimposed mode when \(l=1\) and (b) \(l=4\). The north pole N\({}_{l}\) and south pole S\({}_{l}\) 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 of (**a**) represent the two modes with average orbital angular momentum (OAM) values of \(-0.5\) and \(0.5\) for \(l=1\). The A\({}_{l}\) and B\({}_{l}\) points of (**b**) represent the two modes with average OAM values of \(-2\) and \(2\) for \(l=4\). components and smaller waist sizes upon focusing by high numerical aperture objectives, which may have important applications in nanoscale optical imaging and manipulation [31; 32; 33; 34; 35].