Читать книгу Electromagnetic Vortices - Группа авторов - Страница 18

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 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.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 describes a twisted wavefront, as shown in Figure 1.5. Note that the first term is frequency dependent similar to Eq. (1.10). The gradient of the wavefront (i.e. the wavevector direction) is:

(1.12)

Unlike the wavevector of the Airy disk, the wavevector of a Laguerre–Gaussian beam has a radial and azimuthal component. Note that for very large distances compared to the wavelength (k₀r → ∞),

(1.13)

In other words, the wavefront of a Laguerre–Gaussian modes at very large distances compared to the wavelength, i.e. in the very far‐field, resembles a spherical wavefront, as shown in Figure 1.5.

An immediately remarkable trait of the cone‐shaped far‐field pattern is the reduced directivity compared to the conventional beam counterpart. The Airy disk pattern is a highly directive far‐field pattern with 100% aperture efficiency. A Laguerre–Gaussian mode with l = 1, aperture diameter D, and beam waist w_g = 0.415D has an aperture efficiency of 50%. As a result, an antenna with an aperture that is twice as large compared to the conventional counterpart is required to maintain the same directivity level. Figure 1.6 shows the directivity of the Airy disk pattern and the first‐order Laguerre–Gaussian beam for D = 4λ and w_g = 0.415D, normalized to the maximum directivity of the Airy disk pattern. The directivity of the Laguerre–Gaussian beam is 3 dB less than the Airy disk pattern of the same aperture size. Another distinctive characteristic is the beam divergence; the larger the OAM mode number, the larger the cone angle of the beam. That is, higher‐order OAM modes diverge more rapidly with propagating distance.

Figure 1.5 Far‐field wavefront of (a) Airy disk and (b, c) Laguerre–Gaussian modes. The wavefront of the Airy disk is spherical and the wavefront of Laguerre–Gaussian modes is twisted; (c) at very large distances compared to the wavelength (k₀r → ∞), the wavefront of Laguerre–Gaussian approaches a spherical wavefront.

Figure 1.6 Normalized directivities of the Airy disk pattern with aperture diameter D = 4λ and the Laguerre–Gaussian mode with mode number l = 1, same aperture diameter D, and beam waist w_g = 0.415D. The directivity of the Laguerre–Gaussian beam is 3 dB less than the Airy disk pattern for this choice of D and w_g. The curves are normalized to the maximum directivity of the Airy disk pattern.

To provide further insight in the OAM field distribution at various distances far from the antenna, we studied the changes of amplitude pattern shape from the reactive near‐field toward the far‐field. The first case of study is a conventional reflector antenna with a uniform aperture phase and −10 dB edge taper. The aperture field distribution is modeled using the two‐parameter (2P) model [22, eq. (16)] (for more details see Appendix 1.A), and the field at various distances is calculated using the Fresnel–Kirchhoff diffraction integral [24]. The second case of study is a helicoidal reflector [23]. The aperture field is modeled using the [22, eq. (16)] multiplied by the phase term e^−jlϕ, for l = 1. The aperture diameter for both cases is D = 10λ. The changes of amplitude pattern shape from the reactive near‐field toward the far‐field for the two cases is shown in Figure 1.7. The pattern is calculated at r = 4.9λ, 24λ, 8000λ corresponding to the reactive near‐field, radiating near‐field and far‐field regions [25]. It can be observed that the amplitude null at the vortex center is maintained at all distances.