# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 3 Basic Concepts and Theories of OAM Beams Generation and Detection
### Detection of OAM Beams  
Beams from the fiber will be converted into a divergent wave once they leave the fiber. For conciseness, we use a spherical wave to discuss the same physical process in this review. Indeed, they could be other types of divergent waves besides spherical waves, as long as their wavevectors are different from the other interference beam, such as a Gaussian light, whose curvatures of these lights are not the same on the beam cross-section when propagating while a spherical wave is a light wave with an invariant curvature. Without loss of generality, the physical meaning of the discussion below will not change under the assumption of a spherical wave. The spherical wave is characterized by the factor \(e^{ikR}\), where \(k\) is the wavevector and \(R=\left(x^{2}+y^{2}+z^{2}\right)^{\frac{1}{2}}\) is the radius relative to the light source. A spherical wave carrying OAM TC = +  
Figure 4: Sketch of fiber OAM modes generation system. Reprinted with permission from [54], copyright 2018 De Gruyter. \(l\) can be expressed as \(E_{s}(r,\xi,z)=F_{l,m}(r)e^{i(l\xi+k_{s}(r^{2}+z^{2})^{\frac{1}{2}})}\), where \(k_{s}\) denotes the wavevector of OAM mode, \(r=\left(x^{2}+y^{2}\right)^{\frac{1}{2}}\) and \(\xi\) denote the transverse radius and the spatial angle on the beam cross section respectively and \(z\) denotes the propagating distance along with the optics axis. In addition, \(F_{l,m}(r)\) is the radial field distribution, as mentioned above. A fundamental mode with a different divergence is usually used to interfere with OAM beam to detect the phase information of OAM beams. A spherical fundamental mode can be expressed as \(E_{f}(r,\xi)=F_{0,1}(r)e^{ik_{f}(r^{2}+z^{2})^{\frac{1}{2}}}\), where \(k_{f}\) denotes the wavevector of fundamental mode. In general, the radial field distribution only affects the spot size but do not contribute to the final interference shapes. Herein, we omit the discussion about the radial field. Assume that the polarizations of the signal OAM beams and the interference beams are the same, the interference of the electric field should be:  
\[\left|E\right|^{2}=\left(E_{s}+E_{f}\right)^{*}\left(E_{s}+E_{f}\right)=\left| E_{s}\right|^{2}+\left|E_{f}\right|^{2}+E_{s}^{*}E_{f}+E_{f}^{*}E_{s} \tag{5}\]  
where the left term \(\left|E_{s}\right|^{2}+\left|E_{f}\right|^{2}\) is the direct current (DC) component invariant with the spatial angle \(\xi\), which acts as an intensity base in the interference patterns. Because the DC component doesn't affect the shapes of the patterns, we emphatically show the interference term \(E_{s}^{*}E_{f}+E_{f}^{*}E_{s}\) through Equation (6):  
\[E_{s}^{*}E_{f}+E_{f}^{*}E_{s}\propto F_{l,m}(r)F_{0,1}(r)\cos\left(l\xi+\left( k_{s}-k_{f}\right)\left(r^{2}+z^{2}\right)^{\frac{1}{2}}+\phi_{s}-\phi_{f}\right) \tag{6}\]  
where the term \(F_{l,m}(r)F_{0,1}(r)\) indicates the area of interference related to the spot sizes of OAM beam and fundamental beam. Here, \(\phi_{s}-\phi_{f}\) indicates the initial phase difference of OAM beam and the fundamental beam. Based on Equation (6), Figure 4 shows the interference patterns of \(OAM_{+1}\), \(OAM_{+2}\) and \(OAM_{+3}\) with the different wavevector difference \(k_{s}-k_{f}\) and initial phase difference \(\phi_{s}-\phi_{f}\). As can be seen, when \(k_{s}\
eq k_{r}\), the patterns exhibit the vortex shapes, while the vortex number indicates \(\left|l\right|\), the absolute value of TC. Meanwhile, the vortex rotation direction is closely related to the difference between wave vectors of two beams. When \(k_{s}-k_{f}<0\), which indicates that the fundamental beam is more divergent, a counter-clockwise vortex indicates \(l\) as the positive value while a clockwise vortex indicates the negative value. Contrary patterns exist when \(k_{s}-k_{f}>0\). When \(k_{s}=k_{f}\), the interference patterns do not exhibit the vortex shapes and we are unable to judge the specific TC for this type of interference.