Electromagnetic Vortices. Группа авторов

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.A for the proof):

Figure 1.2 Normalized aperture field intensity distributions versus ρ/w_g of 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; J₁ is 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 2πl after a full turn (l is the OAM mode number), resulting in a spiral wavefront. Figure 1.4 illustrates 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):

      (1.9)

      which describes a spherical wavefront. The gradient of the wavefront gives the direction of the wavevector (i.e. the geometrical optics ray direction):

(1.10)

      For the Laguerre–Gaussian beam, the far‐field signature of the vortex phase is a cone‐shaped pattern with an amplitude null at the center. The locus of the points with constant phase in the far‐field can be found from Eq. (1.6) and is described by the following equation:

      (1.11)

which