Electromagnetic Vortices. Группа авторов

two OAM beams with modes l₁, l₂, and radial distributions A₁(ρ, z), A₂(ρ, z), the following orthogonality relation is satisfied [11]:

      (1.2)

      where the asterisk (^*) denotes the complex conjugate. It follows that: (i) there is an infinite number of OAM modes, with each mode identified by the mode number l, and (ii) the infinite set of OAM states forms an orthogonal basis.

      The second special feature of OAM beams is the beam divergence. The far‐field signature of the helical wavefront is an amplitude null at the phase vortex center. Accordingly, the null size can be described in terms of a divergence angle, which represents the angle from the null to the maximum gain [12]. As the OAM beam travels through space, the radius of the ‘dark zone’ around the amplitude null in the center of the beam increases.

      1.2.1 Laguerre–Gaussian Modes

      In general, an OAM‐carrying beam could refer to any beam that carries the e^jlϕ term, regardless of the radial distribution A(ρ, z) in Eq. (1.1). The Laguerre–Gaussian modes are a special subset among all OAM‐carrying beams that are cylindrically symmetric solutions to the paraxial wave equation in the cylindrical coordinate system [3]. The Laguerre–Gaussian modes are chosen to be presented because they are one of the most popular examples of OAM‐carrying beams (see, for example [13–18]), and a general OAM‐carrying beam can be expanded in a complete basis of Laguerre–Gaussian modes [11, 19, 20]. The electric field of a linearly polarized Laguerre-Gaussian beam at z = 0 can be written as [3, 5]:

(1.3)

where ρ and ϕ are the radial and azimuthal coordinates in the cylindrical coordinate system;

is a complex amplitude coefficient, l and p are integers known as azimuthal and radial mode numbers, w_g is the equivalent beam waist that can be related to the antenna aperture diameter D (refer to [5] and Appendix 1.A for more details) and is equal to the half‐width of the normalized aperture field amplitude at 1/e controlling the transverse extent of the beam,

is the associated Laguerre polynomial [21]:

      (1.4)

      where the binomial coefficient is [21]:

      (1.5)

when k ≤ n and is zero when k > n. For l = 0, the Laguerre–Gaussian beam carries no OAM since the phase term e^−jlϕ vanishes. For any other l, the field carries the phase term e^−jlϕ, which gives rise to an OAM state of −l‐order. The normalized electric field intensity distributions of Laguerre–Gaussian beams with different azimuthal and radial modes l and p are shown in Figures 1.2 and 1.3. It can be observed that the number of side lobe intensity rings is equal to the integer p. For the same p, the null size (i.e. the divergence angle) increases as the azimuthal mode number l increases.

The far‐field features of Laguerre–Gaussian beams were studied in [5]. The far‐field expression can be found from Eq. (1.3) using the aperture field method [5] (see Appendix 1.A for the proof):

(1.6)

and Ψ = k₀w_g sin θ (k₀ = 2π/λ is the free‐space wavenumber). Equation (1.6) is a cone‐shaped pattern with azimuthal symmetry. Note that the electric field maintains the phase term e^−jlϕ in the far‐field. This is a general characteristic of OAM fields (for example, the same feature is observed for the case of Bessel–Gaussian beams [5]) and a proof can be found in Appendix 1.A. The far‐field