Electromagnetic Vortices

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 gis the equivalent beam waist that can be related to the antenna aperture diameter D (refer to [5] and Appendix 1.Afor 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 −jlϕvanishes. 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 1.2and 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.Afor the proof):

(1.6)

and Ψ = k 0 w gsin θ ( k 0= 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 −jlϕin 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, the cone angle increases as we increase the mode number l .

In what follows, a comparison between two classes of beams is carried out. The first is the conventional Airy disk, and the second is the OAM‐carrying Laguerre–Gaussian beam. The Airy disk pattern is produced by a circular aperture with uniform amplitude and phase‐field distributions and it is a common and useful model in the design of conventional aperture‐type antennas, such as reflectors. The far‐field electric field of the Airy disk pattern can be written as (see Appendix 1.Afor the proof):

Figure 1.2 Normalized aperture field intensity distributions versus ρ / w gof Laguerre–Gaussian beams with different azimuthal and radial modes l and p . 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.

Figure 1.3 Normalized aperture field intensity line cuts of Laguerre–Gaussian beams with different azimuthal and radial modes l and p .

(1.8)

where is a constant; J 1is the first‐order Bessel function of the first kind; is the free‐space wavenumber; D is the aperture diameter, and a = D /2 is the radius of the aperture.

What are the fundamental differences between conventional and OAM beams? In the near‐field region, the aperture phase of the Airy disk is uniform and the wavefront is planar. The Airy disk can be produced by a uniformly illuminated circular aperture antenna, such as a parabolic reflector antenna [22]. On the other hand, Laguerre–Gaussian modes can be produced by helicoidal reflector antennas [23]. The aperture phase of Laguerre–Gaussian modes twirls around the beam axis and changes 2π l after a full turn ( l is the OAM mode number), resulting in a spiral wavefront. Figure 1.4illustrates the analogies and antitheses between these two types of beams.

Figure 1.4 Comparison between conventional and OAM beams. A uniformly illuminated circular aperture produces a planar wavefront in the near‐field and a highly directive far‐field radiation pattern. An OAM‐carrying Laguerre–Gaussian beam with mode number l produces a spiral wavefront in the near‐field and a cone‐shaped far‐field pattern with an amplitude null at the phase vortex center. The OAM beam divergence increases for larger l .

The far‐field characteristics of the Airy disk and the Laguerre–Gaussian beam are remarkably different. For the Airy disk case, the manifestation of the uniform aperture phase and the planar wavefront is a highly directive far‐field pattern with the maximum gain at the axis of symmetry of the antenna. The locus of the points with constant phase in the far‐field, i.e. the far‐field wavefront S of the Airy disk, can be found from Eq. (1.8):