We are indebted to all the coauthors of each of the 14 chapters for their excellent contributions to this book, which provides the readers with an invaluable resource documenting the cutting‐edge research outcomes in the field. The cooperation of the contributing authors during the course of this book preparation as well as their dedication to delivering a high‐quality product, especially under such globally difficult circumstances arising from the pandemic, is sincerely and deeply appreciated. Finally, we also gratefully acknowledge the editorial assistance provided by the highly professional staff of Wiley/IEEE Press. It has truly been a pleasure working with them on this book project.
Zhi Hao Jiang
Nanjing, P.R. China
Douglas H. Werner
University Park, PA, USA
1 Fundamentals of Orbital Angular Momentum Beams: Concepts, Antenna Analogies, and Applications
Anastasios Papathanasopoulos and Yahya Rahmat‐Samii
Department of Electrical and Computer Engineering, University of California, Los Angeles, CA, USA
1.1 Electromagnetic Fields Carry Orbital Angular Momentum
Angular momentum is the rotational equivalent of linear momentum. An object moving in a straight line carries linear momentum. Along with its linear motion, an object can also rotate in two distinct ways. A spinning object carries spin angular momentum (SAM) and an orbiting object carries orbital angular momentum (OAM), with Earth being an interesting example for consideration. The analogy between the angular momentum of a moving object and the electromagnetic wave’s angular momentum is shown in Figure 1.1. Earth’s spinning is associated with SAM; Earth’s orbital motion around the sun is linked to OAM. Electromagnetic beams may also carry these two types of momenta. If the electric field vector rotates along the beam axis, the beam has SAM; if the electromagnetic field’s wavefront is twisting, with the wave vector forming a helix, the beam has OAM.
The history of angular momentum dates back to the early twentieth century. The SAM of light was theoretically studied for the first time by Poynting in 1909 [1] and experimentally studied by Beth in 1936 [2]. SAM is intrinsic, since it does not depend on the choice of an axis, and it is only polarization dependent. If s is the SAM mode number, then s = ±1 corresponds to right‐ and left‐hand polarized waves, and s = 0 corresponds to linearly polarized waves. Although the experimental demonstration of the exchange of angular momentum between circularly polarized beams and matter was performed more than 80 years ago, the work associated with light’s angular momentum has been almost exclusively concerned with SAM.
It was not until 1992 that Allen et al. [3] showed that helically phased beams with a phase term ejlϕ (where
Figure 1.1 Analogy between moving object’s and electromagnetic wave’s angular momentum: (a) Earth spins and carries SAM; (b) Earth orbits around the sun and carries OAM; (c) a circularly polarized wave carries SAM; the electric field vector rotates along a straight beam axis; (d) a circularly polarized OAM‐carrying wave carries both SAM and OAM; the electric field vector rotates along a spiral beam axis; (e) beam with planar wavefront; the wavevector is straight; (f) OAM‐carrying beams with spiral wavefront; the wavevector is twisted.
1.2 OAM Beams; Properties and Analogies with Conventional Beams
Helically phased beams exhibit two unique properties: (i) the orthogonality of different OAM modes and (ii) the OAM beam divergence. As we will discuss in Section 1.3, the infinite number of orthogonal OAM modes provides an additional set of data carriers, thus OAM has the potential to increase the capacity and spectral efficiency of wireless communication links [4]. The beam divergence can be advantageous for applications that require cone‐shaped patterns, such as satellite‐based navigation and guidance systems that serve moving vehicles [5] but may also pose a challenge for long‐distance communication links [6]. In the subsequent paragraphs, these two peculiar features are explicated, and a comparative study between conventional and OAM beams is performed to underline their differences.
The first characteristic property of OAM beams is the orthogonality of distinct OAM modes. In general, the electric field of an OAM beam can be written as:
where ρ is the radial distance, ϕ is the azimuthal angle, and z is the propagation distance in the cylindrical coordinate system. There exist various subsets of OAM‐carrying beams, for example Laguerre–Gaussian [3], Bessel–Gaussian [7], hypergeometric–Gaussian [8], vector vortex [9], and other beams [10]. The radial distribution A(ρ, z) in Eq. (1.1) defines the special subset among all OAM‐carrying beams. All the previous beams carry the ejlϕ term, which gives rise to an OAM mode of l‐order. If we consider the electric fields E1(ρ, ϕ,