and Jl(⋅) is the lth 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]:
Note the far‐field Eq. (1.A.9) maintains the e−jlϕ phase 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
(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)]:
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 1.2 and 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:
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 1.2 and 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
