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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 ejlϕ 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]:

is a complex amplitude coefficient, l and p are integers known as azimuthal and radial mode numbers, wg 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)

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