# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 4 Methods for Generation of Tunable OAM
### Free Space Method for Generation of Tunable OAM
Figures 6-8 simply show schematic diagrams used to generate the tunable OAM mode  
field, e.g., the phase, amplitude and retardation between the x and y components [47; 48]. The section of amplitude is achieved by the combination of two quarter-wave plates (QWPs) and a SLM [46] or the combination of a half-wave plate (HWP) and two diffraction gratings [47]. In ref. [46], the fast axes of the QWPs are along 45\({}^{\circ}\) and 135\({}^{\circ}\) with respect to the horizontal axis, respectively. By loading appropriate phase information on the three sections, this system can generate arbitrary modes on the three PSs. The modes on the three PSs can be obtained when the mathematical expression of the output mode equals to the one of Equations (17)-(19). When a Gaussian beam with circularly polarization is launched to two q-plate cells and a SLM, two different-order classical OAM modes with orthogonally-circular polarization are generated. The amplitude and phase of two OAM modes can be adjusted by changing the retardations of the q-plate cells and the SLM [52], which can go through all the points on the hybrid PS, as shown in the third row of Figure 6.  
For these kinds of generation method, the greatest advantage is high flexibility due to the phase distributions and retardations can be arbitrarily electrically controlled. However, the common shortcoming is limited response speed. For the second type, the tunable OAM modes can be generated by interfering two Laguerre-Gaussian modes with same or different topological charges, as shown in Figure 7. The modulation of amplitude is realized by rotating the optical elements [49; 50], where \"\(a\)\" and \"\(b\)\" are the amplitudes of x and y polarizations. The modulation of phase between two Laguerre-Gaussian modes can be realized by rotating the optical elements [48] or changing the phase distribution of SLM/DM [49]. Because the phase difference between the two split and recombined beams determines the properties of the generated vector beams, this kind of method employing interferometry may be vulnerable to environmental noise like vibrations or air circulation. Another weakness is the slow rotating speed for the optical elements. The third type of generating the beam with tunable OAM in the free-space system is achieved by continuously changing the state of input polarization [51; 55; 56; 57; 58; 59], as shown in Figure 8. A Gaussian beam with arbitrary polarization state can be generated by using a polarizer followed by an arbitrarily oriented QWP. The arbitrary polarization state can be expressed as the \(\overrightarrow{Ax}+\overrightarrow{By}\) in the basis of x and y polarizations or \(\mathrm{A}\sigma^{+}+\mathrm{B}\sigma^{-}\) in the basis of right- and left-hand circular polarizations, where the symbols \"A\" and \"B\" are complex amplitudes. When a Gaussian mode with arbitrary polarization is injected to the combination of a SLM, a SPP/DM and a QWP, the SLM is used to generate the helical phase distributions of x and y polarizations. The SPP/DM is for compensating superfluous phase factor, and the QWP can convert the orthogonal linear polarizations to orthogonal circular polarizations. As shown in the top branch of Figure 8, when \(\varphi=l\theta\), the modes on the higher-order PS can be generated by adjusting the input polarization [50]. When \(\varphi=m\theta\), where \(m\
eq l\), the system can generate the modes on the hybrid-order PS. If the input beam is launched to a q-plate, the output mode will be located on higher-order PS according to Equation (20), as shown in the bottom branch of Figure 8 [53; 54; 55; 56; 57]. In addition, by adding a phase factor \(\exp(\mathrm{i}(l+n)\theta/2)\) into the bottom branch of Figure 8, the tunable OAM on the hybrid-order PS can also be generated by adjusting the input polarization [52; 55]. The extra phase can be achieved by numerous methods, such as spiral phase plates, SLM, diffractive elements and fork gratings. This kind of generation methods involve fewer components.