# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 3 Basic Concepts and Theories of OAM Beams Generation and Detection
### Transformation Relation among CV Modes, LP Modes, and OAM Modes
For three groups of fiber mode bases, there exists transformation relation from CV modes to LP modes and OAM modes as follows (the detailed derivation can be found in [54]):  
\[\frac{1}{2}\left( \begin{array}{cccc}1&-i&1&i\\ i&1&-i&1\\ 1&i&1&-i\\ -i&1&i&1\end{array} \right)\left( \begin{array}{c}A\\ B\\ C\\ D\end{array} \right)=\left( \begin{array}{c}x_{-l}\\ y_{-l}\\ x_{+l}\\ y_{+l}\end{array} \right) \tag{2}\]  
for CV modes to OAM modes:  
\[\left( \begin{array}{cccc}1&0&1&0\\ 0&1&0&1\\ 0&-1&0&1\\ 1&0&-1&0\end{array} \right)\left( \begin{array}{c}A\\ B\\ C\\ D\end{array} \right)=\left( \begin{array}{c}x_{e}\\ y_{e}\\ x_{o}\\ y_{o}\end{array} \right) \tag{3}\]for CV modes to LP modes, and:  
\[\frac{1}{2}\left( \begin{array}{cccc}1&0&i&0\\ 0&1&0&i\\ 1&0&-i&0\\ 0&1&0&-i\end{array} \right)\left( \begin{array}{c}x_{e}\\ y_{e}\\ x_{o}\\ y_{o}\end{array} \right)=\left( \begin{array}{c}x_{-}\\ y_{-}\\ x_{+}\\ y_{+}\end{array} \right) \tag{4}\]  
for LP modes to OAM modes. If we define the transformation matrix of CV modes to LP modes (Equation (3)) as M\({}_{1}\), and the transformation matrix of LP modes to OAM modes as M\({}_{2}\) (Equation (4)), then the transformation matrix of CV modes to OAM modes (Equation (2)) is just the matrix product of M\({}_{1}\) and M\({}_{2}\), namely, M\({}_{3}\) = M\({}_{1}\)M\({}_{2}\). Figure 3 shows the intuitive sketch describing the transformation relation among CV modes, LP modes and OAM modes. In Figure 3, the circle indicates all \(l\)th order electric fields, which should be a four-dimensional complex space, however, we simply use a two-dimensional circle to express them. The lines with different colors in the three circles indicate the different divided methods of the four-dimensional complex space. Notice that, the shapes of the lines are irrelevant. They are simply used to clearly denote the different space divided methods. Four degenerated modes operate as the different bases to completely describe the four-dimensional complex space (the entire \(l\)th order modes), where they can be transformed into each other through the transformation matrix derived above. The transformation among the mode bases is equivalent to the base transformation in the four-dimensional complex space, by the aforementioned matrices. The generation of pure OAM modes, is equivalent to adjust the mode field using physical methods to further simplify the expression in OAM mode bases. For example, a pure OAM mode \(\hat{x}OAM_{-l}\) expressed in OAM mode bases is \(\left(1,0,0,0\right)^{T}\), whereas \(0.5\left(EH_{l-1,m}^{ren}+iEH_{l-1,m}^{odd}+HE_{l-1,m}^{ren}-iHE_{l-1,m}^{odd}\right)\) or \(0.5(1,i,1,-i)^{T}\) in CV mode bases and \(\hat{x}LP_{l,m}^{ren}-i\hat{x}LP_{l,m}^{odd}\) or \(\left(1,-i,0,0\right)^{T}\) in LP mode bases. The three expressions denote the same spatial field which is not a pure CV mode or pure LP mode because the expression in the corresponding mode bases contain several components. The generation of pure CV modes or LP modes is similar.