Electromagnetic Vortices

where ρ ′ and ϕ ′ are the radial and azimuthal coordinates in the cylindrical coordinate system; a is the transverse extend of the aperture field of the beam; l is the OAM order. A schematic of the generation of the OAM aperture field is shown in Figure 1.A.1. Typically, the input beam from the feed is a Gaussian‐type beam with a tapered amplitude distribution. The beam waist w g, i.e. the half‐width of the normalized aperture field amplitude at 1/ e , is directly related to the OAM antenna aperture diameter D , as shown in Figure 1.A.1. For example, the equivalent beam waist in Ref. [5] was w g= 0.415 D for a −12 dB taper illumination. Higher taper illumination would lead to a smaller equivalent beam waist. The role of the OAM antenna (helicoidal reflector in Figure 1.A.1) is to create the desired exit‐aperture amplitude and phase distribution at the infinite exit‐aperture plane. A common model for the aperture field of an OAM antenna is the Laguerre–Gaussian distribution Eq. (1.3).

The equivalent magnetic current density is calculated from [122, eq. 6‐129b]:

(1.A.2)

The radiation integrals can be written as [122, eqs. 6‐125c, 6‐125d]:

(1.A.3)

(1.A.4)

Using the integral identity [5, eq. (5)]:

(1.A.5)

we find the expression of the far‐field integrals:

(1.A.6)

(1.A.7)

where

(1.A.8)

and J l(⋅) is the l th order Bessel function of the first kind [21]. The expression of the far‐field electric field can be written as [122, eqs. 6‐122b, 6‐122c]:

(1.A.9)

Note the far‐field Eq. (1.A.9)maintains the e −jlϕphase term.

Special Case 1: Airy Disk.The aperture field of a uniform amplitude and phase distribution is the special case of Eq. (1.A.1), where l = 0 (no OAM) and . Using the integral identity [21, eq. (6.561–5)]:

(1.A.10)

we find I from Eq. (1.A.8)and the far‐field expression from Eq. (1.A.9):

(1.A.11)

Special Case 2: Tapered‐aperture Distribution.A physically meaningful and mathematically simple model for aperture‐like antennas with uniform phase distribution is the two‐parameter (2P) model [22, eq. (16)]:

(1.A.12)

where P and C are parameters that control the shape and amplitude distribution of the circular aperture. In particular, C can be related to the edge taper by ET = 20 log C . A generalized three‐parameter aperture distribution model for elliptical aperture has also been developed in [123]. The far‐field can be evaluated in the closed form [22, eq. (18–20)]:

(1.A.13)

where

(1.A.14)

in which Γ(⋅) is the gamma function [21]. The far‐field of the tapered‐aperture distribution was studied in Section 1.2and is shown in Figure 1.7a. The OAM tapered‐aperture distribution counterpart was modeled based on Eq. (1.A.12)multiplied by the phase term e −jlϕ:

(1.A.15)

The changes of amplitude pattern shape based on the aperture field of Eq. (1.A.15)from the reactive near‐field toward the far‐field were studied in Section 1.2and are shown in Figure 1.7b.

Special Case 3: Laguerre–Gaussian beam.The aperture field of the Laguerre–Gaussian beams is given by Eq. (1.3). Using the integral identity [21, eq. (7.421‐4)]

(1.A.16)

we find I from Eq. (1.A.8)and the far‐field expression from Eq. (1.A.9):

(1.A.17)

where Ψ = k 0 w gsin θ . For the definition of w grefer to the first paragraph of the appendix and Figure 1.A.1. The previous discussion refers to the far‐field where the radiation integral can be found in closed form. The near‐field calculation using the Fresnel–Kirchhoff diffraction integral [24] was carried out numerically in Section 1.2and the results are shown in Figure 1.7.

1 1 Poynting, J.H. (1909). The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light. Proceedings of the Royal Society of London 82 (557): 560–567.

2 2 Beth, R.A. (1936). Mechanical detection and measurement of the angular momentum of light. Physical Review 50: 115–125.

3 3 Allen, L., Beijersbergen, M.W., Spreeuw, R., and Woerdman, J. (1992). Orbital angular momentum of light and the transformation of Laguerre‐Gaussian laser modes. Physical Review A 45 (11): 8185.

4 4 Drysdale, T.D., Allen, B., Stevens, C. et al. (2018). How orbital angular momentum modes are boosting the performance of radio links. IET Microwaves, Antennas & Propagation 12 (10): 1625–1632.