Electromagnetic Vortices. Группа авторов

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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 expression for the Laguerre–Gaussian mode with p = 0 can be simplified as:

For the dominant radial mode p = 0, the far‐field expression Eq. (1.7) peaks at the elevation angle of:

(1.7)

      Equation (1.7) shows that the cone angle depends both on the azimuthal mode number l and the beam waist (i.e. aperture diameter, as was shown in [5]). For constant l, the cone angle decreases as we increase the beam waist w_g, i.e. the aperture diameter. For constant w_g,

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