# Generation of Orbital Angular Momentum Modes Using Fiber Systems
## 3 Basic Concepts and Theories of OAM Beams Generation and Detection
### Detection of OAM Beams
Moreover, Figure 5a,b show the effects of the initial phase difference \(\phi_{s}-\phi_{f}\) between the two interference beams on the interference patterns. When the initial phase difference changes, the patterns simply rotate an angle but do not change the number and direction of the vortex. This means that the initial phase difference does not disturb OAM beam detection under the interference condition. If the physical process is angle-independent, the distinction between \(OAM_{+l}\) and \(OAM_{-l}\) is not important. We can simply define one direction of the rotated vortex as \(OAM_{+l}\) and the other as \(OAM_{-l}\). The counter-clockwise rotated vortex is usually defined as \(OAM_{+l}\). However, if the physical process is related to the meta-surface, chiral devices, and some other angle-dependent devices, the distinction between \(OAM_{+l}\) and \(OAM_{-l}\) is crucial, and a meticulous method must be used, as discussed above. That is, the divergence of the signal OAM beam and the interference fundamental beam must be given. Figure 6a shows another influencing factor in addition to the difference in wavevector, that is, the tilted degree, which indicates the difference in the propagating direction of the reference beam from the signal OAM beam. The larger the tilted degree is, the larger the angle of the propagating direction between the OAM beam and the reference beam. The effects of the TC and wavevector have been discussed above (Figure 5). Here, we highlight the effect of the tilted degrees. In Figure 6a, we set the propagating direction of the signal OAM beam as the reference axis (red arrow). On the top of Figure 6a, the red arrow indicates that the propagation direction of the signal OAM beam and the reference beam are the same. In addition, black arrows indicate the tilted degree between these two beams. As can be seen, the interference patterns shift into a fork-like shape from the vortex shape when the tilted degree increases. In addition, the open direction of the \"fork\" is closely related to the tilted direction. As shown in the second row in Figure 6a, for \(OAM_{-1}\), the fork is upward open when the fundamental mode is left-tilted in relation to signal OAM beam. While it is downward open direction for a right-tilted fundamental mode. For \(OAM_{+1}\) (the first row of Figure 6a), the results are opposite, compared to the first and the second rows in Figure 6a. Thus, interference patterns are closely related to the left-tilted or right-tilted between the OAM mode and the fundamental mode. As for the effect of the difference of wavevectors, compared with the second and the fourth row of Figure 6a, the divergence (wavevector) difference \(k_{s}-k_{f}\) does not disturb the judgement for nonzero tilted degree. Unlike a case of normal incidence (the middle red arrow), the vortex shape is different when the divergence difference \(k_{s}-k_{f}\) changes, as shown in the middle column in Figure 6a, which will disturb the judgment about the TC of signal OAM beam. In addition, when disregarding TC = \(+l\) or TC = \(-l\) of the OAM mode, upward-open-fork shape patterns are usually defined as \(+l\) while the downward-open-fork shape patterns are defined as \(-l\). Otherwise, it's necessary to introduce the reference beam is left-tilted or right-tilted relative to the propagation direction of signal OAM beam. In addition to using a fundamental beam with a different divergence to interfere with OAM beams, there are still some other methods to obtain the TC information of an OAM beam. The reference light \(E_{f}\) can be other types, such as a tilted plane light wave and an OAM beam itself with a spatial translation. Without proof, we provide the typical interference patterns of these two types. Figure 6b shows the interference between a non-divergent OAM beam (\(OAM_{-1}\), \(OAM_{-2}\) and \(OAM_{-3}\)) and right-tilted plane light wave (\(k_{s}=k_{f}=0\)). This is the degenerated case when \(k_{s}-k_{f}=0\) and titled degree is non-zero in Figure 6a.