# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 4 Advances in Fiber OAM Generation Systems
### Fiber Grating-Based OAM Generation Systems
However, this method has broken the fiber structure, and the resonant wavelength and the coupled mode are fixed once the LPFG is fabricated. In general, fiber grating-based methods for pure OAM mode generation, with experimental set up as shown in Figure 8a, can be classified into two types according to the source modes. For one type, the OAM modes can be generated by combining two linearly polarized modes with a \(\pi/2\) phase shift, where the generated OAM modes have no spin angular momentum (SAM). For the other type, the OAM modes can be generated by combining two vector modes (\(HE_{l+1,m}\) or \(EH_{l-1,m}\)) with a \(\pi/2\) phase shift, where the generated OAM modes have the SAM at the same time. There have been many reports about the generation of these two types OAM modes using LPFGs [32; 34; 35; 37; 43]. For the first type, namely the generation of linearly-polarized OAM (LP-OAM) modes, the LP-OAM modes in the \(x\) and \(y\) polarized directions can be obtained by eliminating other component in the OAM mode bases except one, such as \((x_{-l},y_{-l},x_{+l},y_{+l})^{T}=(1,0,0,0)^{T}\), which corresponds to \(\hat{x}OAM_{-l}\), an \(x\)-linear polarized OAM beam with TC = +\(l\). The expression in other mode bases can be obtained by substituting this vector into Equation (2) or Equation (4). \((x_{-l},y_{-l},x_{+l},y_{+l})^{T}=(1,0,0,0)^{T}\) is equivalent as \((A,B,C,D)^{T}=0.5,0.5i,0.5,-0.5i^{T}\) and \((x_{e},y_{e},x_{o},y_{o})^{T}=(1,0,-i,0)^{T}\). Given a physical meaning, that is \(\hat{x}OAM_{-l}=0.5 EH_{l-1,m}^{recur}+0.5 iEH_{l-1,m}^{add}+0.5 EH_{l+1,m}^{recur }-0.5 iEH_{l+1,m}^{odd}=\hat{x}L_{l,m}^{recur}-i\hat{x}L_{l,m}^{odd}\). Thus, the LP-OAM can be obtained by combining the even and odd \(LP_{l,1}\) mode  
Figure 7: Experimental setup of (**a**) mechanical micro-bending and (**b**) CO2 laser irradiation method to fabricate LPFGs; (**c**) transmission spectrum of the LPFGs. Reprinted with permission from [65], copyright 2000 The Optical Society; [60], copyright 2018 The Optical Society; [43], copyright 2018.  
having the same polarized direction with a \(\pm\pi/2\) phase shift. The first order LP-OAM modes, composed of \(LP_{11}^{a}\) and \(LP_{11}^{b}\) mode with a \(\pi/2\) phase shift, using a two-mode fiber (TMF), a mechanical LPFG, a mechanical rotator and metal slabs, as shown in Figure 8a, and the second order LP-OAM modes, composed of \(LP_{21}^{a}\) and \(LP_{21}^{b}\) mode with a \(\pi/2\) phase shift, utilizing an LPFG written in a four-mode fiber induced by CO\({}_{2}\) laser irradiation, have been experimentally demonstrated in [35] and [37], respectively. For the other type, namely the generation of circularly-polarized OAM (CP-OAM) modes, the typical CP-OAM modes with SAM being \(\pm 1\) can be obtained by the following equation [43]:  
\[\left( \begin{array}{c}\hat{\sigma}^{-}OAM_{+1}\\ \hat{\sigma}^{+}OAM_{-1}\\ \hat{\sigma}^{+}OAM_{+1}\\ \hat{\sigma}^{-}OAM_{-1}\end{array} \right)=F_{1,1}(r) \begin{pmatrix}1&i&0&0\\ 1&-i&0&0\\ 0&0&1&i\\ 0&0&1&-i\end{pmatrix} \left( \begin{array}{c}HE_{21}^{even}\\ H_{21}^{Eodd}\\ TM_{01}\\ TE_{01}\end{array} \right),l=1 \tag{7}\] \[\left( \begin{array}{c}\hat{\sigma}^{-}OAM_{+l}\\ \hat{\sigma}^{+}OAM_{-l}\\ \hat{\sigma}^{+}OAM_{+l}\\ \hat{\sigma}^{-}OAM_{-l}\end{array} \right)=F_{l,1}(r) \begin{pmatrix}1&i&0&0\\ 1&-i&0&0\\ 0&0&1&i\\ 0&0&1&-i\end{pmatrix}\begin{pmatrix}HE_{l+1,1}^{even}\\ H_{l+1,1}^{even}\\ EH_{l-1,1}^{even}\\ EH_{l-1,1}^{odd}\end{pmatrix} ,l\geq 2\]  
where \(\hat{\sigma}^{+}\) and \(\hat{\sigma}^{-}\) represent left- and right-handed circular polarization. Taking \(\hat{\sigma}^{+}OAM_{+1}\) as an example, one can just set the expression in the OAM mode bases as \(\left(x_{-l},y_{-l},x_{+l},y_{+l}\right)^{T}=\left(0,0,1,i\right)^{T}\) and calculate the corresponding expression in the CV mode bases or LP mode bases, the results of which are the same as Equation (7).