Vortex circular airy beams through leaky-wave antennas

A novel method to design leaky-wave antennas radiating vortex cylindrical Airy beams at microwave frequencies is here presented. Two different approaches are adopted to produce waves with a nonzero orbital angular momentum (OAM): one based on a bull’s eye design excited by a uniform circular array of vertical coaxial probes with proper azimuthal phase delay, and one based on a single coaxial feeder exciting a multi-spiral radiator. Both of them take advantage of backward radial propagation of cylindrical leaky waves promoting circular Airy beams with vortex patterns. The OAM state can be changed by either varying the probe phasing or the number of spiral units. A reference profile is designed under transverse-electric and transverse-magnetic excitation independently. Numerical full-wave analysis are performed using different angular states to validate the antenna design, as well to highlight the different advantages of the two alternative design approaches.


Introduction
Vortex electromagnetic waves with helical wavefronts, capable of carrying orbital angular momentum (OAM), have been the subject of a large and ever growing number of investigations in the last decades [1,2].The seminal paper by Allen et al [3] first considered optical vortex paraxial beams and triggered a whole literature on the OAM of light (see, e.g.[4][5][6] and references therein).However, applications of OAM beams have been found in all the electromagnetic spectral regions, above the optical frequencies, e.g. in the x-ray [7,8] and Gamma-ray regions [9][10][11], as well as below, in the radio frequency (RF) ranges (see, e.g.[12] and references therein).
A variety of designs have been proposed for the generation of RF OAM beams including spiral phase plates, deformed Cassegrain reflectors, circular traveling-wave antennas, and photonic crystals [13][14][15][16].Interesting realizations have also been based on metasurfaces, either based on discrete geometric-phase-based scatterers or quasi-continuous PEC/PMC structures [17][18][19][20][21][22][23].However, designs based on circular phased arrays have proven to be the most effective in terms of reconfigurability of the OAM state [24][25][26].In particular, the use of concentric uniform circular arrays (CUCAs) allows for reconfiguring the OAM state while maintaining the elevation angle of the conical far-field pattern [27,28].
Recently, the use of single uniform circular arrays (SUCAs) capable of achieving the same goal was proposed to feed planar two-dimensional leaky-wave antennas (LWAs), either uniform [29] or radially periodic [30].The latter structures, the so called bull's eye (BE) LWAs, are constituted by an arrangement of concentric annular metal strips printed on a grounded dielectric slab (GDS) or, equivalently, of concentric slots etched in the upper plate of a radial parallel plate waveguide (PPW).For a schematic side view of a GDS and a PPW, see figure 1(b).BE-LWAs are particularly versatile microwave (MW) emitters, which can be designed to radiate both conical far-field patterns angularly scannable by frequency [31] and near-field configurations with limited-diffraction features, such as Bessel and Bessel-Gauss beams [32,33].
In this paper we consider another interesting near-field configuration, the so-called circular Airy beam (CAB), which exhibits abruptly focusing and self-healing properties [34].Adding vorticity to CABs may be useful in near-field wireless communication links, whose capacity may be enhanced by exploiting different OAM states, as well as for nanomanipulation purposes [35,36].
In particular, two distinct designs will be presented for operation at MW frequencies.The first one is based on a BE-LWA, already considered in [37] with a single centered coaxial probe for the synthesis of azimuthally invariant CABs with zero OAM, now fed by a circular phased array of sources (see figure 1, up-left scheme).The second one is based on a multispiral (MS) LWA fed by a single coaxial source (see figure 1 top-right scheme), already considered in [38] for the emission of Bessel beams with tunable nondiffracting range.
Array-fed BE-LWAs and MS-LWAs with a single feeder have distinct advantages and disadvantages.As mentioned above, the former allow for reconfiguring the OAM order by rephasing the array elements.On the other hand, the array feeder suffers from potential cross-talk between emitters; this could be reduced by increasing the element spacing, but at the expense of the increase of the structure dimensions and of the purity of the excited field [39].Furthermore, such arrays are typically narrow-band feeders.In contrast, MS-LWAs can excite a desired OAM order in a wide range of frequencies thanks to the natural twisting capability of spiral structures; the only disadvantage of this choice is the impossibility to dynamically reconfigure the OAM order.
We investigate here the accuracy in reproducing an ideal vortex CAB profile by means of the two proposed designs, and we also combine them in order to evaluate their interchangeability.Both TM and TE polarizations are separately considered.Since the CAB emission relies on the excitation of an aperture field with a fast radially oscillating profile, a special adaptation has been involved to smoothen such oscillations and thus make the CAB design affordable with LWAs.Moreover, according to the paraxial character of CAB beams, the Airy profile has been assigned to the dominant aperture-field component parallel to the aperture plane (namely to H ϕ and E ϕ for transverse-magnetic (TM) and transverse-electric (TE) polarizations, respectively, i.e. the azimuthal field components in the cylindrical coordinate system).The paper is organized as follows.In section 2 , the main analytical foundations of the radiator designs are presented.In section 3, the design principle adopted here is reviewed, based on the excitation of cylindrical leaky waves with a radial modulation of their radial propagation wavenumber.In section 4, BE-LWA and MS-LWA are tested for the excitation of TM H ϕ -CABs.In section 5, the same analysis is carried out for TE E ϕ -CABs.Finally, in section 6 are drawn.

Theoretical analysis
Before entering the main discussion, we first review some important concepts at the root of the design and of the test processes.
In the analysis of optical 2D Airy beams under the scalar approximation, the optical disturbance U 2D (x, z) has the classical expression [40]: where In the previous formulas, k 0 = ω √ µ 0 ε 0 is the vacuum wavenumber (where ω is the angular frequency and µ 0 ,ε 0 the magnetic permeability and the electric permittivity of vacuum, respectively), x 0 is the horizontal offset, a is the additional decay factor required to make the infinite Airy tail exponentially small at infinity and thus enable the physical realization of the beam, and b is the primary acceleration factor.Assuming b < 0, the beam accelerates towards x < 0. The analysis of 0 th -order scalar CABs is usually based on the assumption that the aperture distribution of the relevant azimuthally symmetric optical disturbance U 3D (ρ, z) is given by the corresponding 2D distribution: U 3D (ρ, z = 0) = U 2D (ρ, z = 0) [41,42].
The analysis of vortex RF CABs presented in this work is based on a vector treatment in which a specific CAB profile is assigned to the main on-plane field component.In particular, assuming TM and TE excitation separately, we will define the leading field components respectively as: where n is the azimuthal (OAM) order, z = z E is the emission plane, and (ρ, ϕ) are the usual polar coordinates in the xyplane.The remaining field components are calculated according to the theory of vector cylindrical harmonics [43][44][45], as shown in the appendix.
It is important here to highlight that a strict and exact CAB emission at MW frequencies is complicated due to the extremely fast oscillations of the Airy-beam tail, which hinder the leaky-wave synthesis procedure with annular slots (the interested reader can find more details in [37]).
In order to study the entire excitation and emission process, we chose a reference CAB function to be radiated by a cylindrical antenna with maximum radius ρ max based on a platform similar to the one used in [33,46].The relevant parameters are: By setting n = {0, 1, 2, 3} in equation ( 3), and calculating in Matlab the remaining field components as described in equations (A.6) and (A.7), we obtained field profiles as shown in figure 2 (xz elevation plane, absolute values) and figure 3 (cross plane, real parts), where we reported only the E field for the sake of simplicity.In figure 3, the field is calculated at z = z F,0 = 22.8 λ 0 , with z F,0 being the z coordinate where the maximum of |E z | occurs for the non-vortex (i.e.n = 0) case.
The corresponding spectrum b n (k ρ ) is reported in absolute value in figure 4, where we also added the analytical profile of b 0 , which has been calculated by means of some approximations (see the appendix for details):

Leaky-wave design
The design principle adopted here is based on i) the excitation of a dominant TM or TE surface wave supported by the background GDS; ii) the conversion of such wave into a radiative (i.e.leaky) wave by means of a radially periodic structure; iii) the radial tapering of the geometrical features of such structure, aimed at synthesizing the desired illumination function, i.e. the aperture field on the plane z = 0 [37].
To this purpose, the radially periodic structure is linearized, thus obtaining a 1-D periodic metal strip grating printed on a grounded slab.This is a canonical periodic structure whose modal spectrum can be determined through full-wave simulations of the unit cell, thanks to the Floquet theorem.The relevant leaky Bloch modes are characterized by a complex propagation constant k ρ = β − jα, whose dispersion features are determined as a function of the two parameters that characterize the unit-cell geometry, namely the (radial) period Λ and the metal strip width W (or equivalently the filling factor FF = W/Λ).In order to get the finest design resolution of metal strips/spirals, we need both a wide and smooth distribution of {Λ, FF} values.A suitable thickness h of the GDS was selected to enhance the effective index of the outward traveling wave (i.e.controlling the dispersion); the latter allows for smaller Λ to be usable for the beam irradiation.This is achieved in TM polarization by setting h to 3.14 mm, a value already used in [33] and corresponding to the commercially available Rogers Duroid 5880 laminate, having a relative permittivity ε r = 2.2 and tan δ < 10 −3 .The latter is ignored in the simulations to reduce the computational burden for its negligible effects on the radiation performance.Differently, for TE polarization a minimum h of 3.86 mm is required to have propagating waves in a GDS and 5.64 mm in a PPW, as it can be inferred from figure 5(a).
When h exceeds 5.64 mm under TM polarization and 11.22 mm under TE polarization the platform turns into a multimodal one.As a consequence, this would prevent the unique determination of the leaky-wave attenuation constant α and the phase constant β.
As we shall see, both solutions adopted here to achieve vortex CAB emissions do not promote a perfectly pure TM or TE irradiation when the azimuthal order n is different from 0, because the wavefronts of the purely TM/TE polarized waves excited either by the phased probe array in the BE-LWA, or by the single central probe in the MS-LWA, are no longer parallel to the metal structure.Therefore, we shall observe small spurious field contributions associated to the opposite polarization, increasing in magnitude with the order n.Assuming 'small' values of n will ensure a reasonable polarization purity of the radiated fields, as a spectrum analysis stage will subsequently demonstrate.

Vortex CAB beams under TM polarization
By selecting a BE-LWA, a phased probe array (here modeled through idealized vertical electric-dipole sources) is required to promote OAM by setting a constant phase delay between probes; due to system symmetry, the azimuthal order of excitation field will be directly transposed on the final outward radiated fields, inducing a twisting character to the promoted CAB.In MS-LWAs, this character is directly promoted by the spiral geometry itself by interacting with the perfectly isotropic wave radiated by the central probe; the field scattered by homologous portions of spirals will necessarily share the same amplitude and phase, featuring a repeated pattern within an angular period of 2π/n.Thanks to the smooth radial evolution of spirals, the azimuthal momentum will be homogeneously spread along the entire extent of each angular sector.Therefore, the spiral order of MS-LWA will be automatically converted into the azimuthal order of radiated fields.
We used the dataset values of figure 6(a) to design a BE-LWA radiating a TM-polarized CAB as described in the previous section and featuring a null OAM (i.e. with n = 0), with the resulting discrete parametric set of figure 6(b).More precisely, after calculating the required attenuation (α) and phase (β) profiles, we extracted discrete values (black asterisks) chosen by selecting the best match within the datasets in (a) and by using the corresponding Λ value as spacing from the two  closest points.The width and radial positions of metal slots are shown in table 1.
The MS scheme involves a more accurate design procedure than that used for the BE type (see figure 7 and the relevant caption).The two resulting structures have been displayed in figure 8.
We used the two LWA variants to radiate the target CAB mode with N S = 0 and N S = 3.In all the presented numerical results, here and in the following section, CST Microwave Studio [47] and Ansys Lumerical [48] have been used to calculate the aperture field over the plane of metallizations (i.e. the antenna aperture plane); subsequently, the near-field distribution has been calculated in Matlab by evaluating the Stratton-Chu radiation integrals.
We first combined the BE-LWA with a single TM-polarized ideal excitation placed at the antenna center; the results are shown in the upper rows of both figures 9 and 10 to radiate the target CAB with N S = 0. Then (2nd rows) we used the same ideal emitter with the MS-LWA to rise N S to 3; azimuthal order N S = 3 has been achieved also by feeding the BE-LWA with a continuous phase probe array of ideal emitters (located at the radial distance of ρ = 1λ 0 ), as shown in the 3rd rows.Finally (4th rows), we inverted the phasing in the probe array  (i.e.setting the array azimuthal order to n = −3) and used it with the MS-LWA to rise the target CAB with N S = 0 again, in order to demonstrate the remarkable equivalence of the two approaches.
As synthesized by the field plots in figure 10 and the spectra in figure 11, we observe (in 2nd rows) that the MS-LWA induces a cleaner emission in term of polarization with respect to the BE-LWA fed with a phased probe array (3rd rows).Differently, in terms of field profile reconstruction accuracy, the field radiated by the array-fed BE-LWA appears more consistent with the ideal pattern.This aspect can be seen both from graphical comparison with the 4th row of figure 2, and by observing the more precise alignment of peaks with the ideal spectrum in 3rd row of figure 11.Both approaches appear convincing for the promotion of higher order CABs, and their combination by setting opposite azimuthal order to promote a non-vortex 0th-order CAB is effective, as displayed in the 4th row of figure 11, where we find a spectral behavior closely resembling the first one shown in the 1st row.

Vortex CAB beams under TE polarization
For TE polarization, we followed the same procedure adopted for TM irradiation, now considering vertical magnetic dipoles as idealized primary sources.The design of this second LWA is fully described by the maps incorporated in figures 12 and 13.The basic emitting configuration resides in the BE-LWA whose geometrical description is explicited in table 2.
As done in the previous sections, we converted the BE-LWA irradiation pattern, graphically presented in the left side  As before, full-wave simulations specifically for TE CAB emission have returned the vector field profiles described by the plots of figures 14-16.Note that in this instance we chose to plot the vector components to be complementary to those selected for TM polarization.
After an accurate investigation for this specific polarization, we drew the same conclusions of section 4. Indeed, as  shown in the 2nd row of figure 16, we see a high purity in polarization of the TE-polarized MS-CAB emission of azimuthal order 3, with almost null TM-polarized components.At the same time, the 3rd row reports a more accurately reconstructed spectrum by the TE-polarized phased arrayfed BE-LWA, and correspondingly we get a better visual resemblance of the ideal plots in the 4th row of figure 2 with those in the 3rd row of figure 14.These differences are contained and, as in the TM case, the two approaches result both convincing for the irradiation of high order TE CABs, and they can also be combined by setting opposite azimuthal orders to induce the emission of a non-vortex 0th-order CAB.
One difference of the TE design with the TM one is that the former always presents a weak spectral power in the region of lower wavenumbers; this is not necessarily a defect, since the spectral cylindrical components corresponding to k values close to 0 are consistent with vertical irradiation, whilst CAB emission requires a rather oblique one to induce an effective field acceleration.[TE,an] on the left and [TM,bn] on the right, for 4 distinct cases: (a) a BE-LWA fed by a single TE-polarized probe, (b) a MS-LWA of azimuthal order N S = 3 fed by the same probe, (c) a BE-LWA fed with a TE-polarized probe array with azimuthal order N S = 3, and finally (d) MS-LWA of azimuthal order N S = 3 fed by a probe array of opposite azimuthal order.All plots have been normalized to the total upward radiated power P z,Tot .The uppermost and lowermost left insets display also the respective function related to ideal CAB generation.

Conclusions
In this work we originally presented and numerically demonstrated the effectiveness of two equivalent methods for generating CAB with vortex wavefronts at MW frequencies.
Both methods share a common architecture, a circular GDS which differs in the two cases for the feeding scheme and the radiating aperture.In the first approach, the promotion of higher azimuthal order resides on the proper phasing of an array feeder, whereas the radiating aperture only serves to allow for effectively radiating backward leaky waves.In the second approach, a single feeder is used to launch an isotropic cylindrical wave in the GDS, whose phase wavefront is suitably manipulated on the radiating aperture thanks to the use of an arrangement of metallic spirals.Design criteria are proposed to design both structures, and numerical and full-wave simulations demonstrate the consistency of both methods.
Compared with other existing solutions, the approach proposed in our manuscript, based on the excitation of radially propagating (i.e.cylindrical) leaky waves, has attractive features in the simplicity of the realization (based on planar metallizations, readily realizable with standard printed-circuit technology) and in the computational efficiency of the design, which is based on the dispersive analysis of a two-dimensional structure supporting one-dimensional propagation (namely, a grating of parallel microstrips).
After some algebra, it is easy to demonstrate that the vertical emission power P z , consisting in the integration of the effective directional energy flux Π z,n along x and y axes, is given by: For a detailed derivation and analysis of the vector cylindrical eigenfunctions M n and N n , we redirect the reader to [44].
Given the shape of the vector functions M n and N n , the vector field H (E) possesses just two non-null vector components under TM (TE) polarization, while E (H) has all the three components active.
We consider H ϕ (E ϕ ) as the primary field under TM (TE) polarization.For TM polarization, the five field components will be given by: where we deliberately neglected to show the b n dependence on k ρ , and we defined Our goal is to make the primary component H ϕ assume the desired CAB pattern of order n.Then, we proceed by extracting the spectral descriptor b n as a function of k ρ , and consequently this will allow for the derivation of the remaining field components.From the 2nd of (A.6), we see that H n,ϕ transformation kernel consists in the derivative of J n rather than in this same function, thus we require to adopt a further step to finally get b n .So, we first use the following definitions: and subsequently we use them to calculate: By inserting the b n spectrum given by (A.10) into expressions (A.6) and (A.7), all the remaining field components can be calculated for any z exceeding z E .We remark that in order to get a precise evaluation of b n , the g function in (A.9) must be calculated along a large ρ range, so inducing a close approach of g to 0. Similarly, for the TE polarization we shall have the field components: a n e jkzz k 2 ρ J n (k ρ ρ) dk ρ .(A.12) As for the previous case, after modeling the E ϕ component with the selected CAB function of order n, we shall first state the following definitions: and the use of the a n spectrum given by (A.15) into (A.11) and (A.12) will return the remaining four field components.
In the case of non-rotating CAB, we can extract the spectrum expression by using simple approximations; we start by defining the scalar field: √ ρ e −aρ (A.16) whose spectrum must be calculated from: (A.17) The last expression in (A.17 Thus, after some algebra, we can recast the spectrum (A.17

Figure 1 .
Figure 1.(a) Generation of a vortex beam (represented with a violet transparent flow) with non-zero OAM order by feeding a BE-LWA with a phased coaxial array (whose primary excited field has a vortex front, represented by curled black arrows) and MS-LWA fed by a single probe (whose primary excited field is an isotropic cylindrical wave, represented by radial black arrows).(b) Schematic cut view of a grounded dielectric slab (GDS) and a parallel plate waveguide (PPW), which represent the opposite reference configurations of the radial waveguide used to calculate the effective index of guided modes.

Figure 2 .
Figure 2. Absolute value of the field components Eρ, E ϕ and Ez in the xz-plane, for a CAB with parameters x 0 = 7λ 0 , b = −80 m −1 , ρmax = 15λ 0 , a = 0.026k 0 , calculated for N S = 0,1,2,3 and shown in the 1st, 2nd, 3rd and 4th lines, respectively.The components Hρ and H ϕ are not reported since they are highly similar to E ϕ and Eρ, respectively, while Hz is zero.This CAB is TM-polarized; anyway, TE-polarized CABs features the same diagrams with exchanged roles between H and E.

Figure 3 .
Figure 3. Real part of the same fields shown in figure 2, evaluated at the z = Z F = 22.8λ 0 , corresponding to the non-vortex (i.e.n = 0) Ez-norm hot spot level.

Figure 5 .
Figure 5. (a) Effective index profiles for the GDS and (b) the PPW under TM (thin blue line) and TE (thick red line) polarization, as a function of the platform width h.A schematic depiction of a GDS and a PPW is presented in the lower section of figure 1.In (a), a magenta dashed line indicates the dielectric medium refractive index.

Figure 6 .
Figure 6.(a) Dataset used to design the 0th-order BE-LWA for TM polarization, represented as a double set of 2D maps of attenuation constant α and phase constant β values as a function of (Λ; FF).(b) Continuous α and β profiles (blue straight lines) as a function of ρ for the 0th-order BE-LWA radiating the TM-polarized CAB mode of figure 2, and extracted discrete values (black asterisks) chosen by selecting the best match within the datasets (a), and by using the corresponding Λ value as a spacing from the two closest points.(c) Functional representation of the CAB-radiating MS-LWA associated to N S = 3, as a function of ρ C , i.e. the radial coordinate of the spiral wire center; ϕ is the angular coordinate, and W the width function.

Figure 7 .
Figure 7.After setting a minimum radius and number of spiral units N S , we set the first point (indicated with 1 in the picture) and clone it with an azimuthal period of 2π/N S .Then again we calculate its projected point (indicated with 2) by means of recursive adjustments.Then the first spiral segment (indicated with 3) is completed by linearly increasing the radial coordinate.The subsequent spiral segments will be calculated by progressively calculating the normal and tangent directions across all the points of the closest preceding segment (as done with point 4), and then applying a rotation of 2π/N S radiants.A final interpolation will suffice to smooth the overall design result.

Figure 9 .
Figure 9. Front view of the three E field components (norm values) respectively promoted by a BE-LWA fed by a single emitter (1st row), an MS-LWA with N S = 3 (2nd row), a BE-LWA fed by a phased array of coaxial cables with azimuthal order 3 (3rd row), and the same MS-LWA of the 2nd row fed by a phased array with azimuthal order -3.All values have been normalized to the maximum of their respective ∥Ez∥ value across the focusing region.

Figure 10 .
Figure 10.Transverse profiles (real values with arbitrary phase) corresponding to the field distributions of figure 9, calculated at the 0 th -order Ez-focus level z = 22.8λ 0 ; all values have been normalized to the maximum value of their respective ∥Ez∥ across the focusing region.

Figure 11 .
Figure 11.Spectral profiles showing f X = π ωϵ0 with [X,Y] being [TE,an] on the left and [TM,bn] on the right, for 4 distinct cases: (a) a BE-LWA fed by a single TM-polarized probe, (b) a MS-LWA of azimuthal order N S = 3 fed by the same (c) a BE-LWA fed with a TM-polarized probe array with azimuthal order N S = 3, and finally (d) MS-LWA of azimuthal order N S = 3 fed by a probe array of opposite azimuthal order.All plots have been normalized to the total upward radiated P z,Tot .The uppermost and lowermost right insets display also the respective function related to ideal CAB generation. of figure 13), into the MS-LWA to N S = 3, shown in the right side.

Figure 12 .
Figure 12.(a) Dataset used for the design of the TE samples (platform width h = 10 mm).(b) Continuous straight lines refer to α and β profiles associated to the CAB emission of section 2 and by setting the emission at z E = 1λ 0 , while asterisks refer to discrete set used to design the BE-LWA of table 2. (c) Design scheme for the corresponding MS-LWA with azimuthal order N S = 3.

Figure 13 .
Figure 13.(Left) Bull-Eye and (right) Multi-Spiral (N S = 3) LWA meant for TE polarization studied in this chapter.These antennas feature a diameter of 50 cm.

Figure 14 .
Figure 14.Vertical cut (in norm) of the H LWA meant for TE polarization studied in this chapter 9, normalized to the maximum value of their ∥Hz∥ across the focusing region.

Figure 15 .
Figure 15.Side view (real part) of the H field components following the same scheme of figure 10, normalized to the maximum value of their ∥Hz∥ across the focusing region.

Table 2 .
Width W k and radial position ρ c,k (k = 1, . . ., 17) expressed in (mm) for each metal strip of the designed BE-LWA radiating a TE-polarized 0th-order CAB.