# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 2 The Classical Orbital Angular Momentum (OAM) Mode  
The light beams carrying orbital angular momentum are spatially structured beams with helical phase fronts. For the points on the mode cross-section with the same radius, the polarization states and amplitudes are the same, but with different phases. The electromagnetic field of an classical OAM beam is identified by a phase term expressed as \(\exp(\pm i\theta)\), where \(\theta\) is the azimuthal angle in the transverse plane of the mode. The \(l\) which called as the topological charge means the number of \(2\pi\) phase shifts along the circle around the beam axis [1; 19]. The sign of \(l\) is relative to the handedness of helical phase front. The positive is for left helical phase front and negative is for right helical phase front (from the point of view of the receiver). In principle, \(l\) can take an arbitrary integer number ranged from \(-\infty\) to \(+\infty\), therefore, the state of OAM-carrying mode is infinite. Figure 1 shows the helical phase fronts of \(l=0\), \(1\), \(-1\) and \(2\). Meanwhile, the OAM is the component of angular momentum of a light beam that is only dependent on the spatial field distribution but not on the polarization. Thus, the OAM mode can be classified into two types according to the state of polarization (SOP) that the beam carries. One type is the linearly-polarized OAM (LP-OAM) whose polarization states of every point on the mode cross-section are the linear polarization. This kind of OAM mode has no SAM. The other type is circularly-polarized OAM (CP-OAM) whose polarization states of every point on the mode cross-section are the circular polarization. In the free-space system, the OAM beams can be generated via numerous methods such as spatial light modulators [20], computer-generated fork holograms [21], spiral phase plates [22], cylindrical lens pairs [23], q-plates [24], etc. Although those spatial generation methods have the advantages of strong scalability and low crosstalk, there are two common defections for them, i.e., high insertion loss and large volume. Compared with free-space generation systems, fiber-based methods have the advantage of the low insertion loss. Moreover, the devices in fiber-based generation methods are smaller, which greatly facilitates miniaturization. In the fiber, the OAM\({}_{\pm l}\) modes can be obtained by combining the two sets of fiber mode bases, LP mode and vector mode bases. The LP mode bases (LP\({}_{lmax}\), LP\({}_{lmay}\), LP\({}_{lmbx}\) and LP\({}_{lmby}\)) and vector mode bases (HE\({}^{\text{e/o}}\), EH\({}^{\text{e/o}}\), TE and TM) are respectively scalar and vector solutions to Maxwell Equation in the fiber [25]. Here, \"e\" and \"o\" refer to the even and odd modes, \"m\" is the radial order, denoting the number of radial nodes of the mode and \(l\) is the azimuthal order. The \(l\) order LP mode bases can be expressed by the Equation (1). \[ \begin{array}{l}\text{LP}_{lmax}=\overrightarrow{\text{x}}\text{F}_{lm}( \text{r})\cos(l\theta)\text{LP}_{lmay}=\overrightarrow{\text{y}}\text{F}_{lm}( \text{r})\cos(l\theta)\\ \text{LP}_{lmbx}=\overrightarrow{\text{x}}\text{F}_{lm}(\text{r})\sin(l\theta )\text{LP}_{lmby}=\overrightarrow{\text{y}}\text{F}_{lm}(\text{r})\sin(l \theta)\end{array}  \tag{1}\]  
where \(\overrightarrow{\text{x}}\) and \(\overrightarrow{\text{y}}\) denote the linear polarization along the x-axis and y-axis, respectively; F\({}_{lm}\)(r) represents the radial field distribution and \(\theta\) is the azimuthal coordinate. The vector mode bases have the following transverse electric field distributions:  
\[\left\{ \begin{array}{l}\text{H}_{l+1,m}^{\text{e}}\\ \text{H}_{l+1,m}^{\text{e}}\\ \text{E}_{l-1,m}^{\text{e}}\\ \text{E}_{l}^{\text{e}}\\ \text{E}_{l-1,m}^{\text{e}}\\ \text{E}_{l-1,m}^{\text{o}}\\ \end{array} \right\}=\text{F}_{l,m}(\text{r})\left\{