# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 2 Three Types of Fiber Mode
### Orbital Angular Momentum Modes  
OAM modes, are also one of the groups of fiber-based modes. Unlike the conventional plane light wave, OAM modes are characterized by a helical phase front \(\varepsilon^{\pm il\xi}\)[1], where \(\pm\)\(l\) is the TC and \(\xi\) is the azimuthal angle related to the optic axis. In addition, \(l\) can take the integer numbers from zero to \(+\infty\). It should be noted that \(l\) is the same as the azimuthal order of CV modes. For different points on the beam cross-section with the same radius, the polarization states are the same, but with different phases. This indicates the helical phase front of OAM modes. Figure 2e,f show the propagating properties and time average intensity patterns of two typical OAM modes \(\hat{\sigma}^{-}OAM_{+1}\) and \(\hat{x}OAM_{+1}\). Taking \(\hat{\sigma}^{-}OAM_{+1}\) as an example, as indicated in Figure 2e, electric vectors at each point with the same radius on the beam cross section are the right-hand circular polarized (\(\hat{\sigma}^{-}\)). The phase factor of \(OAM_{+l}\) should be \(e^{i(kz-\omega t+l\xi)}\). For \(l\)th order OAM modes, the number of equal phase points on the beam cross section will be \(l\). As shown in Figure 2e, initially, the \(x\)-polarization point is located at \(\xi=0\) (\(kz-\omega t+\xi=0\)). Then, when the field propagates to \(kz-\omega t=\frac{\pi}{4}\), the \(x\)-polarization point (with the same phase) is located at \(\xi=-\frac{\pi}{4}\) (\(kz-\omega t+\xi=0\)). This means that, with \(\xi=-(kz-\omega t)\), the equal phase point appears along the clockwise direction during the propagating, which indicates the factor \(e^{i\xi}\). Thus, Figure 2e indicates \(\hat{\sigma}^{-}OAM_{+1}\) mode. The analysis method is similar to that of \(\hat{x}OAM_{+l}\). The symbol \(\hat{x}\) simply indicates the linear polarization, which can be substituted by another linear polarization symbol when the observation coordinates rotate.  
For a typical combination of CV modes used to generate the OAM mode, \(\hat{\sigma}^{-}OAM_{+1}=TM_{01}-iTE_{01}\), as shown in Figure 2 The term \"\(-iTE_{01}\)\" indicates the figures of the first row (Figure 2a) with a \(-\frac{\pi}{2}\) phase delay. The physical meaning of \(\hat{\sigma}^{-}OAM_{+1}=TM_{01}-iTE_{01}\) is the interference between \(TM_{01}\) and \(TE_{01}\) patterns with a \(-\frac{\pi}{2}\) phase delay of \(TE_{01}\). In Figure 2, when \(TM_{01}\) propagates to \(kz-\omega t=\frac{\pi}{2}\), \(TE_{01}\) reaches \(kz-\omega t=0\). Adding these two electric fields, we obtain the \(\hat{\sigma}^{-}OAM_{+1}\) at \(kz-\omega t=\frac{\pi}{2}\). In addition to \(\hat{\sigma}^{-}OAM_{+1}=TM_{01}-iTE_{01}\), there are a series of transformation relations among CV, LP, and OAM modes. Moreover, we may note that there is no difference among the time average intensity patterns of CV and OAM modes. To further assure the phase information, a fundamental mode is usually used to interfere with a higher order fiber mode. Through the interference patterns we can obtain the phase information to confirm the specific electric vector field of the same doughnut intensity patterns. In the following section, we are going to derive the entire relation for CV modes, LP modes, and OAM modes, and introduce the generation and detection of OAM modes.