# 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  
As the three groups of fiber mode bases, there is a transformation relation among CV modes, LP modes and OAM modes. Among these three modes, CV modes are the eigenmodes in fiber, which are able to propagate stably in fiber. We start our introduction of the transformation relation with CV modes. Figure 2: The propagation properties during half a period and the integer-period time average patterns of two typical CV modes, (**a**) \(TE_{01}\) and (**b**) \(TM_{01}\); and two typical LP modes, (**c**) \(\hat{\kappa}L_{11}^{even}\) and (**d**) \(\hat{\kappa}L_{11}^{odd}\); and two typical OAM modes, (**e**) \(\hat{\sigma}^{-}OAM_{+1}\) and (**f**) \(\hat{\kappa}OAM_{+1}\). In an axisymmetric index profile fiber, the intrinsic electric field is under the restriction of a cylindrical waveguide. By solving the Helmholtz equation in cylindrical coordinates, the eigenmodes in fiber can be derived, that is the CV modes. The solutions are given in Equation (1):  
\[\left( \begin{array}{c}E_{x}(r,\xi,z)\\ E_{y}(r,\xi,z)\end{array} \right)=\left\{ \begin{array}{ll}F_{l,m}(r)\left( \begin{array}{c}\cos(l\xi)\\ \sin(l\xi)\end{array}\right)e^{i\beta_{1}z};&EH_{l-1,m}^{even}/TM_{0,m}\\ F_{l,m}(r)\left(\begin{array}{c}-\sin(l\xi)\\ \cos(l\xi)\end{array}\right)e^{i\beta_{2}z};&EH_{l-1,m}^{odd}/TE_{0,m}\\ F_{l,m}(r)\left(\begin{array}{c}\cos(l\xi)\\ -\sin(l\xi)\end{array}\right)e^{i\beta_{3}z};&HE_{l+1,m}^{even}\\ F_{l,m}(r)\left(\begin{array}{c}\sin(l\xi)\\ \cos(l\xi)\end{array}\right)e^{i\beta_{4}z};&HE_{l+1,m}^{odd}\end{array} \right. \tag{1}\]  
where \(\xi\) is the spatial angle, \(F_{l,m}(r)\) is radial field distribution, \(l\) is the azimuthal order of CV modes, \(m\) is the radial order and \(\beta_{1-4}\) are the propagation constants. Usually, \(F_{l,m}(r)\) is the Bessel function in a step index fiber. For \(l=1\), \(EH_{l-1,m}^{odd}\) should be substituted by \(TM_{0,m}\) and \(EH_{l-1,m}^{even}\) should be substituted by \(TE_{0,m}\). In this section, however, we simply use \(EH_{l-1,m}^{even}\) and \(EH_{l-1,m}^{odd}\) to express the corresponding CV modes for conciseness, even for \(l=1\). Any \(l\)th-order electric field in fiber can be decomposed into the superposition of the four degenerated eigenmodes, that is, \(E=AEH_{l-1,m}^{even}+BEH_{l-1,m}^{odd}+CHE_{l+1,m}^{even}+DHE_{l+1,m}^{odd}\), where \((A\,,B\,,C\,,D)^{T}\) is an arbitrary complex vector. The amplitudes and the phases of \((A\,,B\,,C\,,D)^{T}\) represent the amplitudes and the relative phases of \(EH_{l-1,m}^{even}\), \(EH_{l-1,m}^{odd}\), and \(HE_{l+1,m}^{even}\), \(HE_{l+1,m}^{odd}\), respectively. In OAM mode bases, the four mode bases are \(\hat{x}OAM_{-l},\hat{y}OAM_{-l},\hat{x}OAM_{+l}\), and \(\hat{y}OAM_{+l}\), where \(\hat{x}(\hat{y})\) represents the \(x(y)\) linearly polarized direction and \(OAM_{\pm l}\) represents the OAM modes with TCs \(\pm l\). With the aid of Jones calculus, any \(l\)th-order electric field can be expressed as \(E=OAM_{-l}\left( \begin{array}{c}x_{-l}\\ y_{-l}\end{array} \right)+OAM_{+l}\left( \begin{array}{c}x_{+l}\\ y_{+l}\end{array} \right)\). Likely, as an arbitrary complex vector, \((x_{-l},y_{-l},x_{+l},y_{+l})^{T}\) completely describes the entire \(l\)th-order electric field. In LP mode bases, the four mode bases are \(\hat{x}LP_{l,m}^{even},\hat{y}LP_{l,m}^{even},\hat{x}L_{l,m}^{odd}\), and \(\hat{y}LP_{l,m}^{odd}\), where \(LP_{l,m}^{even}\) and \(LP_{l,m}^{odd}\) represent the two LP modes with complementary intensity patterns. Any \(l\)th-order electric field can be expressed in LP mode bases as \(E=LP_{l,m}^{even}\left( \begin{array}{c}x_{e}\\ y_{e}\end{array} \right)+LP_{l,m}^{odd}\left( \begin{array}{c}x_{o}\\ y_{o}\end{array} \right)\). The corresponding complex vector is \((x_{e},y_{e},x_{o},y_{o})^{T}\).