Introduction
An earlier version of this paper was presented at the 52nd European Microwave Conference and was published in its Proceedings [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1]. In paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], the authors have shown the possibility to exploit a purely transverse electric (TE-) Bessel beam in a wireless power transfer (WPT) scenario. In this work, a thorough analysis of the devices discussed in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1] is presented both from a theoretical and a practical viewpoint.
Bessel beams are monochromatic solutions of Helmholtz equation in a cylindrical reference frame [Reference Durnin2] with focusing, limited-diffraction, and self-healing properties (i.e., the capability to reconstruct themselves after an obstacle placed along the beam axis) [Reference Hernández-Figueroa, Zamboni-Rached and Recami3–Reference Pimenta, Soriano, Paschaloudis, Ettorre, Zerrad and Amra6]; such features are particularly attractive for WPT applications in the microwave and/or millimeter-wave range (see, e.g., [Reference Negri, Fuscaldo, Burghignoli and Galli7–Reference Negri, Del Biondo, Fuscaldo, Burghignoli and Galli13]).
Many kinds of Bessel-beam launchers (BBLs), i.e., devices able to generate a Bessel beam, have been proposed in the microwave and millimeter-wave range [Reference Ettorre, Pavone, Casaletti, Albani, Mazzinghi, Freni, Boriskin and Sauleau14]. Two families of launchers can be distinguished in terms of their frequency behavior: wideband BBLs [Reference Comite, Fuscaldo, Podilchak, Hílario-Re, Buendía V, Burghignoli, Baccarelli and Galli15–Reference Pavone, Ettorre and Albani18] feature a large bandwidth at the expense of an electrically large aperture size, whereas resonant BBLs [Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19–Reference Negri, Fuscaldo, Ettorre, Burghignoli and Galli21] are compact in size but typically feature a narrow frequency band.
In a WPT scenario, the bandwidth is of minor concern, whereas a compact size might play a crucial role as, e.g., in WPT for wearable and implantable devices [Reference Costanzo, Apollonio, Baccarelli, Barbiroli, Benassi, Bozzi, Burghignoli, Campi, Cruciani, di Meo, Feliziani, Fuscaldo, Galli, Liberti, Maradei, Marracino, Masotti, Paolini, Pasian, Perregrini, Schettini and Silvestri22]. As a result, resonant BBLs are preferable due to their limited transverse size. Resonant BBLs are typically constituted by a metallic cavity whose upper plate is replaced by a partially reflecting sheet (PRS), which is often a homogenized metasurface. Dipole-like sources are often considered for exciting such cavities [Reference Negri, Fuscaldo, Burghignoli and Galli23].
As already discussed in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], in most of the reported experiments on BB launchers (see, e.g., [Reference Comite, Fuscaldo, Podilchak, Hílario-Re, Buendía V, Burghignoli, Baccarelli and Galli15, Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19, Reference Ettorre, Rudolph and Grbic24]), these are fed with coaxial probes, which are ideally equivalent to vertical electrical dipoles (VEDs), and thus excite transverse magnetic (TM) polarized Bessel beams. From a theoretical viewpoint, a TE-polarized Bessel beam has to be excited by means of a vertical magnetic dipole (VMD). The latter, however, does not have a simple physical implementation. As shown in papers [Reference Lu, Bréard, Huillery, Yang and Voyer25, Reference Lu, Voyer, Bréard, Huillery, Allard, Lin-Shi and Yang26], the VMD could be implemented by loop antennas or coils but the feeding point breaks the azimuthal symmetry, thus generating undesired TM-field components [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8]. The possibility to generate TE-polarized focused beams, however, is important not only from a theoretical viewpoint but also from a practical one to mitigate dielectric losses [Reference Lu, Bréard, Huillery, Yang and Voyer25]. Indeed, in wearable and implantable WPT applications [Reference Costanzo, Apollonio, Baccarelli, Barbiroli, Benassi, Bozzi, Burghignoli, Campi, Cruciani, di Meo, Feliziani, Fuscaldo, Galli, Liberti, Maradei, Marracino, Masotti, Paolini, Pasian, Perregrini, Schettini and Silvestri22], it is important to have focused fields with a negligible on-axis electric-field component so as to minimize the coupling with the very large dielectric losses exhibited by human tissues in the microwave and millimeter-wave ranges.
In this context, the authors investigated the possibility to realize a WPT link between TE-polarized BBLs in papers [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1, Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8]. While in paper [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8] a hybrid-TE (HTE-) polarized Bessel beam has been obtained by means of a loop antenna excitation and an annular-strip grating metasurface, in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1] a pure TE-polarized Bessel beam has been theoretically excited through a radial slot array on the ground plane. In this paper, an effective and original theoretical analysis for the annular strip grating used in the HTE-polarized BBL is presented and corroborated through full-wave simulations on CST Microwave Studio [27]. Afterwards, the WPT performance of TE- and HTE-polarized BBLs is evaluated in terms of link budget, obtaining improved results with a purely TE-polarized BBL.
The paper is organized as follows: section “Design of TE-polarized BBLs” shows the theoretical leaky-wave approach needed to design a TE-polarized BBL. In section “Physical implementation,” full-wave results of HTE- and TE-polarized BBLs are shown along with their physical implementation and the theoretical description of their metasurface and feeder. In section “Wireless power transfer,” the WPT performance of the different resonant BBLs analyzed in this work are shown and compared. In particular, the transmission efficiency has been computed with an effective hybrid numerical and full-wave approach. Conclusions are finally drawn in section “Conclusion.”
Design of TE-polarized BBLs
As mentioned in section “Introduction,” resonant BBLs consist of a circular grounded dielectric slab enclosed by a metallic rim and with a PRS on top [Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19, Reference Negri, Fuscaldo, Ettorre, Burghignoli and Galli21] (see Figure 1). Purely TM(TE)-polarized BBLs can only be obtained by using ideal VED(VMD) sources, which generate a zeroth-order Bessel beam over the vertical electric(magnetic)-field component. Such beams maintain their transverse profile (thanks to the limited-diffraction property) up to the so-called nondiffractive range z ndr, which is given by the following ray-optics approximation [Reference Durnin2]:
Figure 1. Pictorial representation of a TE-polarized resonant BBL and of its Hz field distribution through a three-dimensional colormap. The BBL consists of a metallic cavity of height h and radius ρ ap with a PRS (reported through an aqua green color) on top. TE-polarized resonant BBLs can be excited by means of either loop-antenna feeder (on the bottom-right corner) or a radial slot array on the ground plane (on the bottom-left corner). Two different geometries are considered for the PRS: a fishnet-like metasurface (on the top-left corner) and a dichroic annular strip-grating metasurface (on the top-right corner).
\begin{equation}
z_{\rm ndr}=\rho_{\rm ap}\cot{\theta_0},
\end{equation}where ρ ap is the aperture radius, and θ 0 is the so-called axicon angle (measured with respect to the vertical z-axis). Resonant BBLs belong to the family of leaky-wave BBLs which possesses slightly different properties with respect to classical BBLs due to the intrinsic amplitude exponential decay on the aperture field distribution [Reference Fuscaldo, Comite, Boesso, Baccarelli, Burghignoli and Galli28, Reference Fuscaldo, Benedetti, Comite, Baccarelli, Burghignoli and Galli29].
In leaky-wave BBLs, the axicon angle θ 0 is related to the real part β of the complex, leaky, radial wavenumber 
$k_\rho = \beta-j\alpha$ [Reference Fuscaldo, Comite, Boesso, Baccarelli, Burghignoli and Galli28]. While the attenuation constant α describes the power decay due to the radiation during the propagation inside the cavity, β, i.e., the leaky phase constant, is related to the axicon angle through the relation 
$\beta=k_0\sin\theta_0$ (being k 0 the vacuum wavenumber).
In each polarization type, resonant BBLs generate the desired limited-diffractive beam through the interference of the outward cylindrical leaky wave coming from the source and the inward one given by the reflection on the circular metallic rim [Reference Ettorre and Grbic20]. In particular, in order to enforce the correct constructive interference among such cylindrical waves, the outward and the inward contributions should add in phase and with almost the same amplitude. While the latter constraint is fulfilled by requiring a “small” value of the leakage constant α, the former has to be achieved by enforcing a correct radial resonance. From the boundary conditions, it follows that a resonance is achieved when the tangential electric field at the rim location is zero. In this regard, it is worthwhile noticing that depending on the polarization type, the tangential electric field will show a different aperture distribution, thus different design conditions have to be enforced for the TE and TM case. In particular, a null for the first or the zeroth-order of the first-kind Bessel function has to be placed on the circular metallic rim depending on the specific polarization [Reference Negri, Fuscaldo, Burghignoli and Galli23]. As concerns the TE case, the following relation has to be verified [Reference Lu, Voyer, Bréard, Huillery, Allard, Lin-Shi and Yang26]:
\begin{equation}
\beta\rho_{\rm ap} = j_{1q},
\end{equation} where 
$j_{1q}$ is the qth null of the first-order Bessel function of the first kind.
As in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], the choice of the working frequency, the aperture radius, and the number of nulls is fixed as follows: 
$f_0=30$ GHz, 
$\rho_{\rm ap}=15$ mm, and q = 2, respectively. At this point, the normalized leaky phase constant is fixed at 
${\hat{\beta}=\beta/k_0\simeq0.7436}$ through equation (2). As concerns the normalized attenuation constant 
$\hat{\alpha}=\alpha/k_0$, it should be as low as possible in order to maintain almost the same amplitude between outward and inward cylindrical leaky waves [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli30]. A good design rule is to consider the inward wave power at the antenna rim equal to the 95% at least of the outward one at the antenna feeder which corresponds here to a normalized attenuation constant equal to 
$\hat{\alpha}=0.0027$ as in papers [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1, Reference Benassi, Fuscaldo, Masotti, Galli and Costanzo9].
It is worth pointing out that a low value of α does not mean that the radiation efficiency of the proposed launcher is low. It is true that, in typical leaky-wave antennas (see, e.g., [Reference Galli, Baccarelli, Burghignoli and Webster31] and reference therein), the radiation efficiency ηr is often evaluated through the well-known expression 
$\eta_{r}=1-\exp(-2\alpha\rho_{\rm ap})$. However, this formula ignores the cylindrical spreading factor that characterize cylindrical leaky waves, and assumes the structure to be terminated with an ideal absorber, whereas the leaky-wave launcher studied here is terminated over a circular metallic rim, which would make the radiation efficiency tend to one. This efficiency evaluation, however, does not take into account material losses, mismatching, and potential coupling of the source with other modes different from the dominant leaky wave. All these quantities could affect the total efficiency of the launcher. Since an air-filled cavity, a lossless dielectric and ideal metal are considered in the following, dielectric and ohmic losses are negligible. Therefore, the efficiency is not affected by material losses in our work (this assumption is totally reasonable in this frequency range [Reference Fuscaldo32]). The same holds for mismatching losses which are negligible (and not shown for brevity) since ideal sources have been assumed. The interesting point is related to the potential coupling with other existing modes (e.g., surface waves) that may subtract power to the leaky mode responsible for radiation. In this context, we properly designed the structure to avoid resonances with those modes (more details can be found in papers [Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19, Reference Negri, Fuscaldo, Ettorre, Burghignoli and Galli21]) and to limit the excitation of spurious modes that arise in HTE-polarized configurations, as shown next.
At this point, once the desired leaky wavenumber has been computed, it is necessary to suitably design the cavity to excite the targeted kρ value at f 0. To this aim, the design parameters available in a resonant BBL are the cavity height h and the equivalent PRS reactance X s that can be computed through the methods described in paper [Reference Fuscaldo, Galli and Jackson33]. The PRS is indeed typically realized through an isotropic metallic metasurface [Reference Burghignoli, Fuscaldo and Galli34] which is represented by an imaginary scalar impedance 
$Z_{\rm s}=jX_{\rm s}$ since losses are negligible. This assumption is in general fully satisfactory for resonant BBLs [Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19–Reference Negri, Fuscaldo, Ettorre, Burghignoli and Galli21]. Therefore, as shown in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1] and by following the analysis in papers [Reference Negri, Fuscaldo, Burghignoli and Galli23, Reference Fuscaldo, Galli and Jackson33], the design parameters for the TE-polarized BBL analyzed in this work are fixed at: h = 7.194 mm and 
$X_{\mathrm s}=67.235\;\Omega$.
In order to verify the validity of the design, the transverse resonant technique [Reference Sorrentino and Mongiardo35] is applied on the transverse equivalent network (TEN) of the device, thus obtaining the relevant dispersion equation of the structure, as in papers [Reference Fuscaldo, Valerio, Galli, Sauleau, Grbic and Ettorre19, Reference Ettorre and Grbic20, Reference Negri, Fuscaldo, Burghignoli and Galli23]. In particular, the dispersion curve of the leaky mode can be found by solving the dispersion equation for the complex improper roots, using the Padé algorithm [Reference Galdi and Pinto36].
The results of this complex-mode analysis are reported in Figure 2. As shown, the dispersion curve of the leaky phase constant intersects the dispersive radial resonance at the working frequency f 0. Moreover, as concerns the dispersion curve of the leakage constant, very low values for 
$\hat{\alpha}$ are achieved around the working frequency, as desired in order to achieve the Bessel-beam distribution.
Once the theoretical design parameters are verified, it is necessary to implement from a practical viewpoint the excitation of the device and the PRS. These aspects are discussed in the following section “Physical implementation.”
Figure 2. Dispersion curves of the normalized leaky-wave phase 
$\hat{\beta}$ (blue solid line) and attenuation 
$\hat{\alpha}$ (green solid line) constants vs. frequency f. The black dashed lines represent the dispersion curves of different radial resonances for TE-polarized BBLs.
Physical implementation
In this section, we analyze the possibility to excite TE-polarized and HTE-polarized BBLs through two different feeders. In particular, an effective theoretical analysis for the metasurface used here to enhance the TE components of a non-ideal TE excitation is presented.
Feeder
As discussed in section “Introduction,” a purely TE-polarized BBL can only be obtained with an ideal VMD source. As shown in papers [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8, Reference Lu, Bréard, Huillery, Yang and Voyer25, Reference Lu, Voyer, Bréard, Huillery, Allard, Lin-Shi and Yang26], loop antennas and feeding coils are not ideal VMD sources. Therefore, even if a well-designed TE-polarized resonant BBL is considered (as discussed in section “Design of TE-polarized BBLs”), the excitation through a simple loop feeder generates a hybrid-polarized field with a generally nonnegligible TM-field contribution (see Figure 4). This effect is due to the feeding point (with height 
$h_{\rm lf}\simeq\lambda_0/10$, being λ 0 the vacuum wavenumber at the working frequency f 0) of the loop antenna (with radius 
$R_{\rm l}\simeq\lambda_0/(2\pi)$) which breaks the azimuthal symmetry of the structure (see Figure 1). Therefore, a simple loop antenna does not excite a purely TE-polarized Bessel beam but rather an HTE one. However, as shown in paper [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8], it is possible to reduce the undesired TM contributions by using an original homogenized metasurface consisting of an annular strip grating (see its pictorial representation on the top-right corner of Figure 1 and in Figure 3) whose innovative theoretical description and its dichroic behavior are presented in the following subsection “Metasurface.”
Figure 3. Pictorial representation of the proposed PRS composed by an annular strip grating with periodicity p s and width w s. The nonvanishing electromagnetic tangential field components in the case of an azimuthally symmetric device with a TE (light blue color) and TM (red color) are represented.
As recently demonstrated in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], an option to achieve the excitation of a purely TE-polarized Bessel beam is given by a radial slot array on the ground plane. In this work and in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], the radial slots on the ground plane are ideally excited by dipole-like sources but, from a practical viewpoint, an ad hoc feeding scheme of the slots should be considered. The design and the optimization of the physical excitation of the slots will be addressed in future works.
Once the radial slot array on the ground plane is correctly excited, it represents the discrete counterpart of a continuous loop of radially directed magnetic surface current. The ideal, continuous source couples only with irrotational magnetic fields and, hence, it excites a pure, azimuthal symmetric, TE-polarized field. In this work and in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], an array of N = 8 rectangular radial slots of width w = 1 mm and length l = 2 mm are considered with a uniform azimuthal distribution and a fixed radial distance from the vertical z-axis given by R = 2.5 mm (see Figure 1). It is worth pointing out that the slots have to be electrically small (please note that, in this work, 
$w=\lambda_0/10$ and 
$l=\lambda_0/5$), simultaneously excited with the same amplitude, and their number should be sufficiently high to well represent their continuous counterpart. The purity of the polarization is corroborated by the field envelope further reported in the analysis (see Figure 6): the Ez field component is indeed negligible with respect to the 
$||\bf{E}_t||$ contribution. Therefore, since a realistic pure TE excitation has recently been achieved in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], it can be exploited to demonstrate and quantify the dichroic behavior of the annular strip grating proposed in paper [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8].
Metasurface
The main difference between HTE-polarized and TE-polarized BBLs is given by their excitation scheme. In the former case, the hybrid character of the source call for radiating apertures able to enhance the TE field components with respect to the undesired TM ones.
As discussed in section “Design of TE-polarized BBLs,” an inductive-like metasurface with 
$X_{\mathrm s}\simeq67.235\;\Omega$ is needed in order to excite the desired leaky wavenumber and achieve the correct radial resonance at the working frequency 
$f_0=30$ GHz. A simple structure that allows for synthesizing a wide range of inductance values is the fishnet-like metasurface [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli30]. In particular, the desired X s value is obtained by setting the distance among patches to g = 0.86 mm and the width of the metallic bridge among them to w = 0.3 mm, when the period is set to p = 2 mm (which is fully within the homogenization limit, viz. 
$p \lt \lambda_0/4$). Such a metasurface shows almost the same behavior for both TE and TM polarization [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli30] and thus is not well suitable for HTE-polarized BBLs that instead call for dichroic metasurfaces, such as the annular strip grating (see Figure 3).
An original approach is used here to theoretically describe the latter. As mentioned earlier, this kind of metasurface allows for enhancing the TE field components with respect to the TM ones. This effect clearly emerges from a direct comparison between the field envelopes in Figure 4, where a non-ideal TE source (the loop antenna) is used with a fishnet-like metasurface, and that in Figure 5, where the same source is used with an annular strip grating. In the latter case, the Ez component is considerably reduced.
Figure 4. Full-wave results for the designed TE-polarized BBL with a typical fishnet-like metasurface as a PRS and a loop antenna near the ground plane as a feeder. All the components are normalized with respect to the maximum of their respective field on the 
$z=z_{\rm ndr}/2$ plane, where their absolute values are reported in dB.
Figure 5. Full-wave results for the HTE-polarized BBL [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8] obtained with a loop-antenna feeder and annular-strip-grating metasurface. All the components are normalized with respect to the maximum of their respective field on the 
$z=z_{\rm ndr}/2$ plane, where their absolute values are reported in dB.
However, there is still a nonnegligible vertical component of the electric field that should be zero [Reference Negri, Fuscaldo, Burghignoli and Galli23], even though its maximum value is much lower than the maximum value of 
$\bf{E}_{\rm t}$. For this reason, we refer to both these cases as a HTE polarization and not as a purely TE-polarized Bessel beam as the one achieved with the radial-slot-array feeder and the typical fishnet-like metasurface (whose field distribution is reported in Figure 6).
Figure 6. Full-wave results for the designed TE-polarized BBL proposed in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1]. In this case, a typical fishnet-like metasurface has been considered as PRS and a radial slot array on the ground plane as a feeder. All the components are normalized with respect to the maximum of their respective field on the 
$z=z_{\rm ndr}/2$ plane, where their absolute values are reported in dB.
The effect of the annular-strip-grating metasurface is strictly related to its dichroic nature. In order to understand this behavior, it is important to recall that the only nonvanishing contributions for the TE and the TM polarizations are Hz, Hρ, Eϕ, and Ez, Eρ, Hϕ, respectively [Reference Negri, Fuscaldo, Burghignoli and Galli23]. As shown in Figure 3, the only tangential components on the PRS are 
$H_\rho(E_\rho)$ and 
$E_\phi(H_\phi)$ for the TE(TM) polarization. Therefore, the tangential electric-field components in the TE(TM) case are locally parallel(perpendicular) to the metallic strips, thus revealing the dichroic nature of the metasurface which shows an inductive (capacitive) behavior in the TE(TM) case. Moreover, by virtue of the azimuthal symmetry of the structure, we can consider a section at constant ϕ and consider a local linear approximation of annular strip grating to treat it as a conventional metal strip grating [Reference Luukkonen, Simovski, Granet, Goussetis, Lioubtchenko, Raisanen and Tretyakov37] (see Figure 3). Since the period of the unit cell is much smaller than the wavelength (viz., 
$p\ll\lambda_0$), we can also exploit the homogenization principle and characterize the whole metasurface by studying the scattering properties of the zeroth-order Floquet harmonic under TE and TM incidence. As for more conventional homogenized metasurfaces, the annular strip grating admits a simple surface-impedance representation: an inductive one for the TE case and a capacitive one for the TM case. These impedance values can in principle be found using either approximate analytical formulas [Reference Luukkonen, Simovski, Granet, Goussetis, Lioubtchenko, Raisanen and Tretyakov37] or conventional full-wave approaches [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli30].
However, it is worthwhile to note that such PRS has to be printed on some thin dielectric substrates placed on the BBL top with negligible losses, height h d and relative dielectric constant 
$\varepsilon_\mathrm{r}$. Therefore, a more precise TEN has to be considered in order to correctly enforce the radial resonance at the desired working frequency f 0. As shown in Figure 7, the TEN is given by two cascaded transmission lines: one of length h with the characteristic admittance of the TE mode in the air Y 0 and one of length h d with the characteristic admittance of the TE mode in the dielectric Y d. The metasurface and the air medium are represented by their equivalent admittances 
$Y_{\rm s}^{\rm TE}=-j/X_{\rm s}^{\rm TE}$ and Y 0, respectively.
Figure 7. The TEN for the TE-polarized contribution in the resonant BBL with the annular strip grating as PRS is represented along with its dispersive diagram.
After numerical optimization, it is found that the desired resonance at f 0 is obtained for an annular strip grating with parameters 
$p_{\rm s}=\lambda/10$ and width 
$w_{\rm s}=0.26$ mm printed on a lossless dielectric with 
$\varepsilon_{\rm r}=3$ and thickness 
$h_{\rm d}=0.127$ mm, as shown in Figure 7. It is worth mentioning that this choice of parameters leads to 
$X_s^{\mathrm T\mathrm E}=35.18\;\Omega$, which considerably differs from the value reported at the beginning of subsection “Metasurface”; this is a consequence of the phase shift introduced by the thin dielectric layer considered here.
However, if loop-antenna feeders are considered, non-negligible TM components are excited and need to be taken into account. For this purpose, we analyzed the behavior of the annular strip grating under TM polarization and found 
$X_s^{\mathrm T\mathrm M}=-5530.66\;\Omega$. As opposed to the TE case, the surface impedance in the TM case is large in absolute value and negative. As a result, while the PRS is rather reflective for the TE case, it is almost transparent for the TM case, thus strongly promoting the efficient radiation of TE leaky waves with respect to TM leaky waves. This statement is corroborated by the dispersive diagram reported in Figure 8 for the TM case along with the TEN of the TM wave (the meaning of each admittance is the same of the TE case but for the TM polarization). As shown, the TM leakage constant 
$\hat{\alpha}$ is very high (due to the very low reflectivity of the PRS for this polarization), thus a TM-polarized Bessel beam can no longer be effectively generated by this leaky mode (because the reflected inward cylindrical leaky wave would be significantly damped, thus a stationary aperture field would not be efficiently created).
Figure 8. The TEN for the TM-polarized contribution in the resonant BBL with the annular strip grating as PRS is represented along with its dispersive diagram.
The surface impedance models of the annular strip metasurface, its dichroic nature, and the related dispersive properties of the leaky modes supported in the BBLs can be verified by considering a purely TE (as the radial slot array proposed in paper [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1]) and a purely TM (a coaxial cable [Reference Negri, Fuscaldo, Burghignoli and Galli23]) excitation scheme.
In the former case, a comparison between the results achieved through the realistic metasurface and its surface impedance model 
$X_s = X_s^{\rm TE}$ has been considered in Figure 9(a), where the Hz field radial profile (normalized to its maximum) at 
$z=z_{\rm ndr}/2$ is reported. (All TE field components have been evaluated, but not shown for brevity; similar agreement is obtained in all cases). The analysis of the dual case is shown in Figure 9(b). The impressive agreement between the realistic metasurface and the surface impedance model with an ideal source corroborate the theoretical analysis of the proposed metasurface: the same field distribution has indeed been achieved with the theoretical X s value and the annular strip grating for both polarizations.
Figure 9. Full-wave validation of the theoretical description of the PRS when a purely (a) TE- or (b) TM-polarized source are considered inside the resonant BBL designed in section “Design of TE-polarized BBLs.” The blue dashed lines represent the normalized (a) Hz and the (b) Ez field components with respect to their maximum at 
$z=z_{\rm ndr}/2$ when the annular strip grating is considered as PRS. The red solid lines report the same field components when the PRS is implemented by a surface impedance boundary condition with (a) 
$Z_{\rm s}=jX_{\rm s}^{\rm TE}$ or (b) 
$Z_{\rm s}=jX_{\rm s}^{\rm TM}$.
Wireless power transfer
After conducting a thorough characterization of the HTE-polarized BBL, it is interesting to evaluate its performance in terms of WPT. The intrinsic high-focusing capabilities, together with the system compactness, make BBLs valuable candidates to be exploited within these types of applications. As shown in papers [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8–Reference Heebl, Ettorre and Grbic10], by placing two identical launchers one in front of the other, it is possible to create limited-diffractive WPT links. Here, we focus specifically on the launchers that were created using the specifications provided in Section II.
When using two BBLs within a wireless link, the calculation of power budget becomes a crucial aspect, taking advantage of their unique propagation properties within the nondiffractive range. While in previous works the evaluation of the WPT efficiency has been achieved through an accurate, analytical, and numerical approach based on the equivalence theorem in the spatial [Reference Borgiotti38] and spectral [Reference Pavone and Albani39] domains, a fast, straightforward, yet effective method has been considered in this work. In particular, the general-purpose link budget model given in paper [Reference Rizzoli, Masotti, Arbizzani and Costanzo40] is used to produce an estimation of the power budget of the presented WPT scenario when two BBLs are used as the transmitting (TX) and receiving (RX) antennas [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1]. In order to facilitate the comparison of obtained performance, the two BBLs are chosen of the same polarization. The Norton equivalent circuit of a rectenna can be exploited to rigorously estimate the received power [Reference Rizzoli, Masotti, Arbizzani and Costanzo40]:
\begin{equation}
P_{\rm r}=(1/8)|I_{\rm eq}|^2/\mathrm{Re}\left[Y_{\rm a}(\omega)\right],
\end{equation} where 
$Y_{\rm a}(\omega)$ is the internal complex admittance of the considered antenna, i.e., the HTE-polarized BBLs. As can be inferred from equation (3), the correct computation of I eq allows for safely estimating the overall link power budget. According to the numerical algorithm [Reference Rizzoli, Masotti, Arbizzani and Costanzo40], the electromagnetic field components of the TX and RX launchers are replaced by the equivalent electric and magnetic surface currents computed on a plane positioned between the two launchers, and the current I eq is then precisely determined by solving the surface integral of the following expression [Reference Rizzoli, Masotti, Arbizzani and Costanzo40]:
\begin{multline}
\boldsymbol{\hat{n}} \cdot[ \mathbf{E}_{\rm i}(P_{\rm S})\times\mathbf{H}_{\rm R}(P_{\rm S}) - \mathbf{E}_{\rm R}(P_{\rm S}) \times \mathbf{H}_{\rm i}(P_{\rm S}) ]
\end{multline} where P S are the points belonging to the interposed surface S between the TX and the RX, with unit vector 
$\hat{\rm n} $, whereas 
$\mathbf{E}_{\rm i}$, 
$\mathbf{H}_{\rm i}$, 
$\mathbf{E}_{\rm R}$, and 
$\mathbf{H}_{\rm R}$ are the electric and magnetic fields of the TX and RX launchers, respectively.
The fields 
$\mathbf{E}_{\rm i}$ and 
$\mathbf{H}_{\rm i}$ are computed and extracted by the full-wave characterization of a single launcher radiating in free space, acting as the TX, whereas 
$\mathbf{E}_{\rm R}$ and 
$\mathbf{H}_{\rm R}$ refer to the receiver and are numerically evaluated, reducing remarkably the global computational time required for a full-wave simulation of the studied WPT link.
Figure 10(a) shows the computed equivalent current density for a link made of two pure TE-polarized BBLs (fed with radial slot arrays) [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1], whereas Figure 10(b) and (c) refer to the equivalent current density evaluated for the HTE-polarized BBLs (fed with loop antenna) designed with an annular strip grating and a fishnet-like metasurface, respectively. Both cases are computed and plotted for a reference distance of z = 10 mm. As can be noticed, they both present four main lobes in which the overall equivalent current density is mainly concentrated. However, the pure TE-polarized case exhibits higher peaks that reflects in a higher receiver power, as is further addressed in the following results.
Figure 10. Computed equivalent current density for the (a) TE-polarized BBL [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli1] and two HTE-polarized BBLs achieved through a loop antenna feeder, and (b) an annular strip grating [Reference Benassi, Fuscaldo, Negri, Paolini, Augello, Masotti, Burghignoli, Galli and Costanzo8], or (c) a fishnet-like metasurface. All the plots are evaluated on a xy-plane at 
$z=10 \, \mathrm{mm}$ and reported in a dB scale after the normalization with respect to the maximum among them.
The receiver power levels for the analyzed WPT links are computed for three different TX-RX operating distances, namely 20, 30, and 40 mm. The link performances are listed in Table 1 and the comparison is carried out for the three different polarization types. The obtained values are computed considering an input power at the TX side equal to 21 dBm.
As can be inferred from Table 1, when a pair of pure TE-polarized BBLs is considered, the received power levels are higher than both cases with an HTE polarization due to the higher polarization purity and the correct excitation of the desired dominant leaky mode. It is worth pointing out that the HTE-polarized WPT link performance is greater when a dichroic metasurface is considered rather than a typical isotropic one (such as the fishnet-like metasurface). This effect is related to the promising ability of the innovative strip-grating metasurface to enhance the TE field components with respect to the TM ones when the device is excited with a non-ideal VDM source. Therefore, since the BBL parameters are chosen in order to excite a TE-polarized Bessel beam, the WPT efficiency is higher as the TE polarization purity increases.
Table 1. Received power for BB launchers with different feeders, metasurfaces, and polarizations
Conclusion
In this work, it is shown how a TE-polarized Bessel-beam is excited by a leaky-wave resonant cavity with a radial slot array on the ground plane. As expected from a theoretical viewpoint, a dominant zeroth-order Bessel beam over the vertical magnetic field component Hz and a negligible vertical electric field Ez have been achieved. This innovative radial slot array and a more conventional coaxial cable are then respectively used as ideal TE and TM sources to corroborate the theoretical analysis of an innovative dichroic metasurface under TE and TM polarization, respectively. This kind of metasurface is very useful when a non-ideal source, such as a loop antenna, is considered and the device is designed to resonate with a TE polarization. Finally, the link power budget is evaluated for different cases obtaining interesting results with respect to the state-of-the-art and showing a higher power transmission efficiency as the polarization purity increases.
Supplementary material
The supplementary material for this article can be found at https://doi.org/10.1017/S1759078723001538.
Acknowledgements
This work was funded by the Italian Ministry of Education, University and Research (MIUR) within the framework of the PRIN 2017-WPT4WID (“Wireless Power Transfer for Wearable and Implantable Devices”) ongoing project.
Competing interests
The authors declare none.
Edoardo Negri was born in Orvieto, Italy, in 1997. He received the B.Sc. and M.Sc. degrees (cum laude) in electronic engineering, with honorable mention for the academic curriculum, from Sapienza University of Rome, Rome, Italy, in July 2019 and 2021, respectively, where he is currently pursuing the Ph.D. degree in Information and Communications Technologies (applied electromagnetics curriculum). From March to May 2023, Mr. Negri has been a visiting PhD student in the Institut d’Électronique et de Télécommunications de Rennes (IETR), Université de Rennes 1, Rennes, France. His main research interests are leaky waves, metasurfaces, focusing devices, and wireless power transfer at high frequencies (ranging from microwaves to terahertz). Mr. Negri received the “Antonio Ventura” Award as one of the best engineering students at the Sapienza University of Rome in 2022, the IEEE Antennas and Propagation Society (APS) Fellowship Award in 2023, and the European Microwave Association (EuMA) Internship Award in 2023.
Francesca Benassi received the Ph.D. degree (cum laude) in Electronics, Telecommunications and Information Technologies Engineering from the University of Bologna, Bologna, Italy, in 2022. She is currently working as junior assistant professor at the Department of Electrical, Electronic and Information Engineering (DEI) “Guglielmo Marconi.” Her main research interests involve the design of wearable rectennas for Wireless Power Transfer applications both at microwave and at millimeter waves. Dr. Benassi has been awarded with the MTT-S Graduate Fellowship Award and the EuMA Internship Award in 2021.
Walter Fuscaldo received the B.Sc. and the M.Sc. (cum laude) degrees in Telecommunications Engineering from Sapienza University of Rome, Rome, Italy, in 2010 and 2013, respectively. In 2017, he received the Ph.D. degree (cum laude and with the Doctor Europaeus label) in Information and Communication Technology (applied electromagnetics curriculum) from both the Department of Information Engineering, Electronics and Telecommunications (DIET) and the Institut d’Électronique et de Télécommunications de Rennes (IETR), Université de Rennes 1, Rennes, France, under a cotutelle agreement between the institutions.
In 2014, 2017, and 2018, he was a Visiting Researcher and in 2023 a Visiting Scientist with the NATO-STO Center for Maritime Research and Experimentation, La Spezia, Italy. In 2016, he was a Visiting Researcher with the University of Houston, Houston, TX, USA. Since July 2017, he was a Postdoc Researcher at Sapienza University of Rome, and since July 2020, he joined the Institute for Microelectronics and Microsystems (IMM), Rome, Italy, as a Researcher of the National Research Council of Italy. His current research interests include propagation of leaky waves, surface waves and plasmonic waves, analysis and design of leaky-wave antennas, generation of localized electromagnetic waves, graphene electromagnetics, metasurfaces, and THz spectroscopy and antennas.
Dr. Fuscaldo was awarded several prizes, among which are the prestigious Young Engineer Prize for the Best Paper presented at the 46th European Microwave Conference in 2016 and the Best Paper in Electromagnetics and Antenna Theory at the 12th European Conference on Antennas and Propagation in 2018. He is currently an Associate Editor of the IET Microwaves, Antennas and Propagation journal and IET Electronic Letters.
Diego Masotti (M’00, SM’16) received the Ph.D. degree in electric engineering from the University of Bologna, Italy, in 1997. In 1998, he joined the University of Bologna where he now serves as an Associate Professor of electromagnetic fields. From 2021, he has the role of coordinator of the Telecommunications Engineering Master Degree course. His research interests are in the areas of nonlinear microwave circuit simulation and design, with emphasis on nonlinear/electromagnetic co-design of integrated radiating subsystems/systems for wireless power transfer and energy harvesting applications. He authored more than 70 scientific publications on peer-reviewed international journals and more than 140 scientific publications on proceedings of international conferences. Dr. Masotti serves in the Editorial Board of Electronic Letters, of the Hindawi journal of Wireless Power Transfer, of IEEE Access, and is a member of the Paper Review Board of the main Journals of the microwave sector.
Paolo Burghignoli was born in Rome, Italy, in 1973. He received the Laurea degree (cum laude) in Electronic Engineering and the Ph.D. degree in Applied Electromagnetics from Sapienza University of Rome, Rome, Italy, in 1997 and 2001, respectively. In 1997, he joined the Department of Electronic Engineering, Sapienza University of Rome, where he is currently an Associate Professor with the Department of Information Engineering, Electronics and Telecommunications. He has authored about 250 articles in international journals, books, and conference proceedings. He is a co-author of the book Electromagnetic Shielding: Theory and Applications (Wiley-IEEE Press, 2023, 2nd ed.). His research interests include the analysis and design of planar antennas and arrays, leakage phenomena in uniform and periodic structures, numerical methods for integral equations and periodic structures, propagation and radiation in metamaterials, electromagnetic shielding, transient electromagnetics, and graphene electromagnetics. He is currently serving as an Associate Editor of the IET Electronics Letters and the International Journal of Antennas and Propagation (Hindawi).
Alessandra Costanzo is Full Professor of Electromagnetic Fields at the University of Bologna, Campus of Cesena, since September 2018. She carries out her research in the field of the CAD of RF and microwave circuits and systems such as: Multiple-input-Multiple Output (MIMO), UWB systems, radio frequency identification systems (RFID), and “rectenne” (multi-band rectifying antennas) for the collection of RF energy, specialized for wearable applications. This activity is carried out through the development of innovative project procedures based on the combination of electromagnetic theory, numerical simulation, and nonlinear analysis based on the harmonic balancing method. She studies and develops systems for wireless power transmission (WPT) adopting both near-field and far-field techniques for different operating frequencies and different power levels. She has developed innovative solutions in the field of non-invasive structural monitoring, exploiting electromagnetic interference, and reverse-modeling of antenna arrays.
Alessandro Galli received the Laurea degree in Electronic Engineering and the PhD degree in Applied Electromagnetics from Sapienza University of Rome, Italy. Since 1990, he has been with the Department of Information Engineering, Electronics and Telecommunications of the same university. In 2000, he became an Assistant Professor and in 2002 an Associate Professor in the sector of Electromagnetic Fields; in 2012, he passed the National Scientific Qualification and then definitively achieved the role of Full Professor in the same sector and university. He authored about 300 papers in indexed journals, books, and conference proceedings. His research interests include theoretical and applied electromagnetics, mainly focused on modeling, numerical analysis, and design of antennas and passive devices from microwaves to terahertz. His research activities also concern the areas of geoelectromagnetics, bioelectromagnetics, and plasma heating. Dr. Galli was the Italian representative of the Board of Directors of the European Microwave Association (EuMA) from 2010 to 2015. He was the recipient of various grants and prizes for his research activity; in 2017, he was elected as the “Best Teacher” of the European School of Antennas (ESoA).
Open access
