# 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  
Free-space generation methods of tunable OAM modes are generally assisted by spatial light modulators (SLMs) [17; 43; 44; 45; 46; 47; 48; 49; 50; 51], deformable mirrors (DMs) [51], q-plate cells [52; 53; 54; 55] and metasurfaces [56], and spiral phase plates (SPPs) [22]. Those optical elements can change the phase distribution of a mode, where the SLM and DM are programmable and can control the phase dynamically. In addition, the DM and SPP are polarization insensitive. SLM is a computer-addressable reflective liquid crystal (LC) display which can impose any desired phase profile onto an incoming collimated beam by controlling the voltage (V) of each SLM pixel [46]. The phase retardation for each SLM pixel can be described as a function of the voltage (V) applied: \(\delta(\text{V})=(2\pi/\lambda)(\text{n}_{\text{e}}(\text{V})-\text{n}_{ \text{o}})\text{d}\), where d is the thickness of the LC layer, \(\text{n}_{\text{e}}\) and \(\text{n}_{\text{o}}\) are the extraordinary and ordinary refractive indices of the LC retarder, respectively. Because of the birefringent nature of LC, when the input polarization state makes a projection on both the fast and slow axes of the SLM, the polarization state can be altered. The polarization property of the SLM can be exploited by the appropriate optical setup to achieve the desired change in the polarization. Moreover, the combination of the SLM and wave plates can be used to control the amplitudes of x and y polarizations due to the birefringent nature. For the SLM, it has the advantage of high flexibility due to the arbitrarily adjustable phase distribution, but also has the disadvantages of maximum power density limitation and large loss. The DM is composed of many units. Each unit has its own independent controller. Under the control of external voltage, it can transform the wavefront phase [51]. In principle, the SLM modulates the wave-front phase by controlling the refractive index, and the DM modulates the wave-front by changing the distance of light propagation. As a phase controller, the DM is energy efficient and highly flexible, while the range of controllable phase is limited. The q-plate cell is essentially birefringent waveplates with a uniform birefringent phase retardation \(\delta\) across the plate thickness (which can be electrically controlled) and a space-variant transverse optical axis distribution exhibiting a topological charge \"2q\" [52]. The charge \"q\" represents the number of rotations of the local optical axis in a path circling once around the center of the plate. When the q-plate cell is illuminated by a circularly-polarized vortex beam, the output beam from the q-plate cell is the combination of two different-order OAM modes with adjustable amplitudes. For the q-plate usually used, it is a q-plate cell with a uniform birefringent phase retardation \(\delta=\pi\). If a Gaussian mode with arbitrary polarization passes through the q-plate, the output mode will perform the following linear transformation, as shown in Equation (20) [54], where \(\sigma^{+}\) and \(\sigma^{-}\) respectively indicate the left and right circular polarization. Thus, a q-plate cell can convert a Gauss beam to a vector beam and generate a vortex phase in one step. However, the \"q\" value is fixed, so the flexibility is poor. \[\text{q}\cdot(\text{A}\sigma^{+}+\text{B}\sigma^{-})=\text{A}\sigma^{-}\text{e }^{i2\text{q}\theta}+\text{B}\sigma^{+}\text{e}^{-i2\text{q}\theta} \tag{20}\]  
The metasurface with tailorable structure geometry, as a two-dimensional electromagnetic nanostructure, possesses unparalleled advantages in optical phase and polarization manipulation, especially in subwavelength scale [57]. The operation principle is the same as the q-plate. But it is the difficult to fabricate and untunable once fabricated. Free-space generation methods can be classified into three types according to the controllable variables.