# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 3 Three Kinds of Tunable OAM Modes  
The OAM Varies from \(-l\) to \(l\) with Homogeneous State of Polarization (SOP) along the Longitude of Orbital Poincare Sphere (PS)  
The mode with OAM from \(-l\) to \(l\) and homogeneous SOP can be produced by overlapping two collinear classical OAM modes with variable relative amplitudes and equal but opposite phase chirality [16]. This kind of tunable OAM mode could maintain a constant geometry and total intensity during tuning, which is valuable for some applications such as the vortex tweezers and optical manipulation [2; 3; 4; 5; 17; 18]. It is worth noting that the polarizations of two collinear classical OAM modes are the same. The superimposed mode is described by the equation:  
\[\psi_{1}=a_{1}|\mathrm{N}_{l}\rangle+b_{1}\mathrm{e}^{i\varphi_{1}}|\mathrm{S}_ {l}\rangle \tag{5}\]  
with positive, real amplitudes \(a_{1}\), \(b_{1}\) and relative phase \(\phi_{1}\). In addition, the squares of amplitudes add up to unity. \(\mathrm{N}_{l}\) and \(\mathrm{S}_{l}\) are the two OAM modes owning topological charge \(-l\) and \(+l\) with the same polarization, respectively. \[ \begin{split}|\mathrm{N}_{l}\rangle&=\mathrm{e}^{- il\theta}(\mathrm{E}_{\mathrm{x}}\overset{\rightarrow}{\propto}+\mathrm{E}_{ \mathrm{y}}\overset{\rightarrow}{\rightarrow})/\sqrt{(|\mathrm{E}_{\mathrm{x }}|_{2}+|\mathrm{E}_{\mathrm{y}}|_{2})}\\ |\mathrm{S}_{l}\rangle&=\mathrm{e}^{+il\theta}( \mathrm{E}_{\mathrm{x}}\overset{\rightarrow}{\rightarrow}+\mathrm{E}_{ \mathrm{y}}\overset{\rightarrow}{\rightarrow})/\sqrt{(|\mathrm{E}_{\mathrm{x }}|_{2}+|\mathrm{E}_{\mathrm{y}}|_{2})}\end{split}  \tag{6}\]  
where \(\mathrm{E}_{\mathrm{x}}\) and \(\mathrm{E}_{\mathrm{y}}\) are the complex amplitudes of \(\mathrm{x}\) and \(\mathrm{y}\) polarizations, respectively. In agreement with intuitive arguments, the amplitudes \(a_{1}\) and \(b_{1}\) govern the relative contribution of OAM\({}_{+\ell}\) and OAM\({}_{-\ell}\) to the local and total orbital angular momentum. In general, the average OAM value that mode carries is calculated from the power in each OAM mode, as shown in Equation (7). The \(\mathrm{P}_{l}\) represents the power in each OAM mode [27]. \[\mathrm{L}_{\mathrm{ave}}=\frac{\sum lp_{l}}{\sum P_{l}}=\frac{l\times|a|^{2}+( -l)\times|b|^{2}}{|a|^{2}+|b|^{2}} \tag{7}\]  
In 1999, [PERSON] and [PERSON] proposed an orbital PS to represent this kind of tunable OAM mode intuitively [28]. The north and south poles of the orbital PS represent the OAM modes with equal \(\ell\) value but opposite helicity, respectively. Similar to the polarization PS [29], all the points on the orbital PS can be described as the superposition of the two poles. Therefore, according to Equation (5), the orbital PS can be used to describe completely all the states for this kind of tunable OAM. Figure 2a,b show the mode patterns, phase distributions and polarization states on the two orbital PSs when \(l=1\) and \(l=4\), respectively. Unlike the classical OAM mode, the amplitude and phase of the superimposed mode vary with azimuthal coordinate, which can be changed by modulating the relative amplitudes of OAM\({}_{+l}\) and OAM\({}_{-\ell}\) modes. When \(\mathrm{a}=0\) or \(\mathrm{b}=0\), the superimposed mode has the phase of a classical OAM\({}_{+l}\) or OAM\({}_{-\ell}\) mode which lies on the south or north pole, as the \(\mathrm{S}_{l}\) and \(\mathrm{N}_{l}\) points shown in Figure 2a,b; When \(\mathrm{a}=\mathrm{b}\), the phase of the mode is binary with \(2l\) alternating phase segments of \(0\) and \(\pi\), which is equivalent to the phase of the optical cogwheel. This mode carries no orbital angular momentum and possesses \(2l\) intensity peaks about the azimuthal coordinate, which is the \(\mathrm{LP}_{\ell\mathrm{m}}\) mode of fiber and lies on the equator. The intensity fringes will occur to rotate about the center of the resultant beam by adjusting the relative phase (\(\phi_{1}\)) of the two overlapping modes.