# A Review of Tunable Orbital Angular Momentum Modes in Fiber: Principle and Generation
## 1 Introduction  
A light beam has two \"rotational\" degrees of freedom: spin angular momentum (SAM) and orbital angular momentum (OAM) [1]. The SAM per photon is \(\sigma\hbar\) (where \(\hbar\) is the Plank's constant h divided by \(2\pi\)), which is related to the state of polarization for left-circular \(\sigma=+1\), for right-circular \(\sigma=-1\), while for linearly polarized light \(\sigma=0\). For elliptically polarized light, the SAM varies from zero to \(\pm\)\(1\hbar\) as the state of polarization varies from linear to circular. The OAM is associated to the phase structure of the complex electric field with a helical phase front defined by the factor of \(\exp(i\hbar\theta)\), which carry a definite amount of OAM per photon equal to \(\hbar\)h. As the OAM beams have a phase singularity, they have a doughnut-shaped spatial profile with zero intensity at the center. Considerable interest in orbital angular momentum arises over its potential applications in multiple fields. For instance, orbital angular momentum can be transferred to trapped suitable material particles causing them to rotate, which enables optical manipulation, trapping and tweezers in fields as diverse as biosciences and micromechanics [2; 3; 4; 5]; the angular momentum of light can be used to encode quantum information that is carried by the corresponding photon states [6]; the exploitation of the orbital angular momentum also opens the door to the generation and manipulation of multi-dimensional quantum entangled states with an arbitrarily large number of entanglement dimensions [7; 8; 9; 10]; the doughnutintensity profile and phase singularity of OAM contribute to contrast enhancement techniques by depleting the fluorescence everywhere except at the dark center of the depletion beam, which enables resolution microscopy beyond the diffraction limit [11; 12; 13]; the OAM modes are also applied in optical communications (fiber and free space) with large capacity and long ranges due to its partial robustness against turbulence [14; 15]; and astrophysical processes may generate photonic OAM, such as light scattering off inhomogeneities in the environments surrounding energetic sources (masers, pulsars, quasars) and light scattering off rotating black holes [16], etc.  
The unique advantage of using optical OAM in these applications relies, to a large extent, on the use of multiple different OAM states. For example, the multiple available OAM states facilitate high-dimensional quantum information processing and large-capacity optical communications. The rotation speed of the trapped micro-particles for optical micro-manipulation is also related to the states of OAM and required to be continuously adjusted [17; 18]. Thus, the control of the orbital angular momentum in a beam is important.  
In this paper, an overview of the basic concepts and generation methods of the tunable OAM modes is given. Firstly, the classical OAM mode is introduced in the Section 2, including the physical concept, mathematical expression and generation methods. Secondly, the basic concepts and theoretical expressions of three kinds of tunable OAM modes are briefly described in Section 3. In order to intuitively understand the OAM evolution, three Poincare spheres (PSs) are used to represent the three kinds of tunable OAM modes, which is similar to the polarization PS. In addition, the relationships among three PSs are concluded. Then, the fiber-based and free-space generation methods are respectively classified into three types according to the controllable variables in the Section 4. Finally, the advantages and disadvantages of each generation method are listed and the key challenges for tunable OAM modes are discussed in Section 5.