# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 4 Advances in Fiber OAM Generation Systems
### Fiber Grating-Based OAM Generation Systems
The first- and second-order CP-OAM modes composed of even and odd \(HE_{21}\), \(HE_{31}\), \(EH_{11}\) and \(TM_{01}\), \(TE_{01}\) modes with a \(\pi/2\) phase shift are experimentally demonstrated in [43] by our group, as shown in Figure 8b,c, using two LPFGs written in a four-mode fiber induced by CO\({}_{2}\) laser. The first LPFG is used to convert fundamental mode to the first order mode. The second LPFG is used to convert the generated the first order mode to the second order mode which has wider bandwidth than the LPFG converting fundamental mode to the second order mode. The OAM mode generated mentioned above is only one pure OAM mode with the same polarization. There are other two types of mixed OAM modes composed of two orthogonal polarized OAM modes with the opposite TCs, according to the polarization state of the two orthogonal modes that can be generated in all-fiber system. One type is LP-OAM, while the other type is CP-OAM. For the first type, the mixed OAM mode, composed of two orthogonal LP-OAM modes with the opposite TCs, can be obtained by the following equation:  
\[E=\left\{ \begin{array}{c}EH_{l-1,1}^{even}\pm iHE_{l+1,1}^{odd}=\hat{\pi} OAM_{\pm l}\pm i\hat{\gamma}OAM_{\mp l}\\ EH_{l-1,1}^{odd}\mp iHE_{l+1,1}^{even}=\hat{\pi}OAM_{\mp l}\pm i\hat{\gamma}OAM_{ \pm l}\end{array} \right. \tag{8}\]  
where \(E\) represents the mixed OAM mode. We can find that the mixed OAM modes are composed of \(x\)- and \(y\)-polarized LP-OAM modes with the opposite TCs, and these two orthogonal LP-OAM modes can be separated by a polarizer. In 2016, [PERSON] et al. experimentally demonstrated a method to generate optical vortices with tunable OAM in optical fiber by using a mechanical LPFG, a mechanical rotator and a metal parallel slab [34]. The tunable OAM mode can be seen as a combination of \(HE_{l+1,m}^{even}\) (\(HE_{l+1,m}^{odd}\)) and \(EH_{l-1,m}^{odd}\) (\(EH_{l-1,m}^{even}\)), with a \(\pi/2\) phase shift. They experimentally achieved the smooth variation of OAM mode from \(l=-1\) to \(l=+1\) by adjusting the polarizer placed at the end of the fiber, as shown in Figure 8d. For the other type, the mixed OAM mode, composed of two orthogonal CP-OAM modes with the opposite TCs, can be obtained by the following equation:  
\[ \begin{array}{l}EH^{even}_{l-1,m}=\frac{1}{2}(\sigma^{-}OAM_{-l}+\sigma^{+}OAM_ {+l})\\ EH^{odd}_{l-1,m}=-\frac{i}{2}(\sigma^{-}OAM_{-l}-\sigma^{+}OAM_{+l})\\ HE^{even}_{l+1,m}=\frac{1}{2}(\sigma^{+}OAM_{-l}+\sigma^{-}OAM_{+l})\\ H^{odd}_{l+1,m}=\frac{i}{2}(\sigma^{+}OAM_{-l}-\sigma^{-}OAM_{+l})\end{array}  \tag{9}\]  
We can see that the arbitrary single CV mode can be seen as a linear combination of a left-handed and a right-handed CP-OAM with the opposite TC. Since one of these two CP-OAM is left-handed, the other one is right-handed, we can use a QWP to make these two CP-OAM have an orthogonal projection. Then we can use a polarizer with optical axis rotating to one projection orientation to filter out the other CP-OAM. The authors of this paper have experimentally demonstrated that using a single CV mode in a two-mode fiber (TMF), the topological charge of generated OAM mode can be switched among \(-1\), \(0\), \(+1\), as shown in Figure 8e [32]. In our work, a CO\({}_{2}\)-laser induced rocking LPFG inscribed in the two-mode fiber is fabricated to efficiently generate the CV mode, including \(TE_{01}\), \(TM_{01}\) and \(TE_{01}\pm TM_{01}\) mode. Then a QWP and a polarizer are used to separate the two CP-OAM modes. Figure 8: (**a**) and (**b**) Operating principle of OAM generation based on LPFGs system; (**c**)-(**e**) mode pattern of corresponding generated OAM modes in (**c**) [43], (**d**) [34], and (**e**) [32], respectively. Reprinted with permission from [35], copyright 2015 The Optical Society; [32], copyright 2018 The Optical society; [34], copyright 2016 The Optical Society; [43], copyright 2018 De Gruyter.