Theoretical framework

The reference antenna structure is reported in Fig. 1 and consists of a grounded dielectric slab with a PRS on top. When excited with a vertical electric dipole or a horizontal magnetic dipole on the ground plane (VED or HMD), it produces a null or a pencil beam at broadside, respectively. (Similar results apply for a vertical magnetic dipole (VMD) and a horizontal electric dipole (HED) in the middle of the cavity.) As shown in [Reference Fuscaldo, Galli and Jackson13], for an infinite structure, the normalized leaky-wave far-field power pattern of 2D LWA fed with an HMD source has the following expression [Reference Ip and Jackson4] (as a consequence of [Reference Ip and Jackson4, Eq. (33)] under the condition [Reference Ip and Jackson4, Eq. (35)]):

(1)\begin{equation} P_{\mathrm{HMD}}(\theta)=\frac{1}{|\hat{k}_\rho^2-\sin^2\theta|^2}, \end{equation}

Figure 1.

Side views of a typical 2D LWA are reported on the left, fed with an HMD (top left) and a VED (bottom left) on the ground plane. On the right side are corresponding representative patterns. The HMD produces a pencil beam at broadside, while the VED produces a narrow null pattern at broadside.

where θ is the elevation angle (measured from the vertical z-axis) and $\hat{k}_\rho=\hat{\beta}-j\hat{\alpha}$ is the normalized (with respect to the free-space wavenumber $k_0=2\pi/\lambda_0$, λ ₀ being the free-space wavelength) leaky wavenumber, with $\hat{\beta}$ and $\hat{\alpha}$ being the normalized leaky phase and attenuation constants. When the excitation is a VED source, the normalized leaky-wave far-field power pattern reads [Reference Fuscaldo, Galli and Jackson13]:

(2)\begin{equation} P_{\mathrm{VED}}(\theta)=\sin^2\theta P_{\mathrm{HMD}}(\theta). \end{equation}

(The radiated far field from the VED will be polarized in the E_θ direction.) When the antenna is of finite radius R (assumed to be truncated by ideal absorber at this radius), these formulas might not be accurate, especially for low values of the radiation efficiency e_r. For 2D LWAs, an exact formula for the radiation efficiency is not available, but it is typically evaluated with the same formula used for 1D bidirectional LWAs [Reference Fuscaldo, Jackson and Galli15], viz., $e_r\simeq 1-\exp(-2\alpha R)$. Formulas for the radiation patterns in the finite case are provided in [Reference Ip and Jackson4] and are not repeated here for brevity.

Numerical results

A comprehensive analysis of the optimum conditions should consider the simultaneous variation of the relative permittivity ɛ_r, the PRS susceptance B_s, the antenna radius R, the cavity height h, and the operating frequency f. In this regard, we should stress that, as far as we assume a nondispersive behavior for B_s (a reasonable assumption for the small fractional bandwidths examined in this work), the optimization with respect to h or f are two sides of the same coin: results will look the same in terms of a normalized cavity height $h/h_0$ or frequency $f/f_0$ where h ₀ and f ₀ represent the optimum cavity height and frequency, respectively for obtaining maximum radiated power at broadside in an infinite 2D LWA [Reference Lovat, Burghignoli and Jackson19]. This condition also corresponds to the leaky cutoff of a Fabry–Perot cavity (FPC) LWA, i.e., when ${r\equiv\hat{\beta}/\hat{\alpha}=1}$.

Equations for obtaining h ₀ as a function of the other parameters have already been provided in [Reference Fuscaldo, Galli and Jackson20] and can be expressed as:

(3)\begin{equation} \frac{h_0}{h_{\mathrm{PPW}}}=\mathrm{Re}\left[1+\frac{1}{\pi}\cot^{-1}\left(\frac{B_s \eta_0-j}{\sqrt{\varepsilon_r}}\right)\right], \end{equation}

with

(4)\begin{equation} h_{\mathrm{PPW}}=\frac{c}{2f_0\sqrt{\varepsilon_r}}=\frac{\lambda_0|_{f=f_0}}{2\sqrt{\varepsilon_r}}\,, \end{equation}

c being the speed of light in vacuum. (Although derived in [Reference Fuscaldo, Galli and Jackson20] for a 1D PRS LWA, the same result holds for a 2D PRS LWA.) Equation (4) corresponds to the cutoff condition of the equivalent parallel-plate waveguide (PPW) structure obtained for $B_{\mathrm{s}}\to\infty$. With these two equations (viz. (3) and (4)) at hand, the antenna designer has multiple choices to optimize the structure. If the cavity thickness h ₀ is provided as a requirement, $h_{\mathrm{ppw}}$ can be obtained from (3), and used to obtain the optimum frequency from (4). Vice versa, if the frequency f ₀ is provided as a requirement, $h_{\mathrm{ppw}}$ can be obtained from (4) and used to obtain the optimum cavity thickness from (3).

In the following, we consider the air-filled case $\varepsilon_{r}=1$ and fix $\bar{B}_s\equiv B_s\eta_0=20$ where $\eta_0\simeq377\,\Omega$ is the free-space impedance, and $f_0=60$ GHz. For this choice of parameters, we set $h_0/\lambda_{0}=0.5079$ according to (3) and explore the optimum normalized frequency $f/f_0$ to get either the minimum HPBW or the minimum HPNW. The HPNW is defined as twice the angle between the null angle (broadside) and the angle where the power pattern is down by 3 dB from that at the peak of the conical beam. The minimum HPBW condition is known to occur in infinite 2D LWAs for $r\simeq0.5176$ [Reference Fuscaldo, Galli and Jackson13, Reference Lovat, Burghignoli and Jackson19]. Conditions for minimum HPBW in finite-size 2D LWAs and for minimum HPNW in both infinite and finite-size 2D LWAs are, instead, new. From the analysis of this case, similar conclusions can be drawn for other choices of B_s, ɛ_r, and f ₀.

Examining the complex wavenumbers helps provide insight into the radiation performance. Taking advantage of the azimuthal symmetry of the considered 2D LWAs, we obtained the dispersion equations in the TE and the TM cases by considering a 1D transverse section of the structure. The expressions are obtained in compact form in [Reference Fuscaldo, Galli and Jackson20] and reported here for the reader’s convenience (assuming an air substrate):

(5)\begin{align} &\mathrm{TE:}(j\bar{B}_s+\hat{k}_{z})\frac{\sin(k_0 h\hat{k}_{z})}{\hat{k}_{z}}-j\cos(k_0 h\hat{k}_{z})=0 \end{align}

(6)\begin{align} &\mathrm{TM:}\bigl(j\bar{B}_s+\hat{k}_{z}^{-1}\bigr)\sin(k_0 h\hat{k}_{z}) -\frac{j\cos(k_0 h\hat{k}_{z})}{\hat{k}_{z}}=0, \end{align}

where $\hat{k}_z = \sqrt{1-\hat{k}_\rho^2}$ is the vertical wavenumber in free space. (The radical sign (principal square root) ensures the improper choice of wavenumber in the air.) We numerically solved the dispersion equations above for both TE and TM leaky modes for various values of f in the range $0.99\leq f/\ f_0\leq1.01$, which correspond to values of r in the range ${0.04\leq r\leq25}$. The dispersion curves for the fundamental TE-TM leaky mode pair (dashed light blue and solid black line, respectively) are shown in Fig. 2. In the case of a VED excitation, only the fundamental TM and an extra quasi transverse electromagnetic (TEM) leaky mode are excited, whereas for an HMD excitation, all modes are excited. In addition, for the optimum frequency f ₀, we decided to inspect the behavior of the TE and TM dispersion equations as functions of $\hat{\beta}$ and α in the range $0 \lt \hat{\beta} \lt 1.5$, $|\hat{\alpha}| \lt 0.05$ to verify the presence of other zeros (or, equivalently, poles of the relevant dyadic Green’s functions of the structure) over the proper ( $\mathrm{Im}\{\hat{k}_z\} \lt 0$) and improper ( $\mathrm{Im}\{\hat{k}_z\} \gt 0$) Riemann sheets (see Fig. 3(a)–(d)). We recall here that leaky modes possess a quadrantal symmetry for a lossless planar stratified structure [Reference Tamir and Oliner21, Reference Tamir and Oliner22]: given a leaky mode with complex wavenumber $k_{\mathrm{LW}}=\beta_{\mathrm{LW}}-j\alpha_{\mathrm{LW}}$, the solutions $\pm k_{\mathrm{LW}}$ and $\pm k^*_{\mathrm{LW}}$ are also mathematically found. In our problem, we only consider solutions with $\beta_{\mathrm{LW}} \gt 0$ and refer to the solution with $\alpha_{\mathrm{LW}} \gt 0$ ( $\alpha_{\mathrm{LW}} \lt 0$) as a physical (nonphysical) leaky mode [Reference Fuscaldo, Burghignoli and Galli23]. (Here, with nonphysical we refer to leaky-wave solutions that exponentially grow along the radial direction.) In the representation of Fig. 3(a)–(d), for each leaky mode, we thus see a complex conjugate leaky-mode pair (the remaining two mathematical solutions are not displayed because we only consider solutions with β > 0).

Figure 2.

Dispersion curves $\hat{\beta}$ and $\hat{\alpha}$ vs. $f/f_0$ of the fundamental TM-TE leaky mode pair (TM in solid black line, TE in dashed light blue line) and the quasi-TEM leaky mode (in solid orange line in the inset).

Figure 3.

Normalized surface plots of the magnitude of the dispersion equation (the left-hand side of (5) or (6)) as a function of $\hat{\beta}$ and $\hat{\alpha}$ reported in dB scale (i.e., $10\log(\cdot)$) for the following cases: (a) TM, improper sheet, (b) TM, proper sheet, (c) TE, improper sheet, (d) TE, proper sheet. Pole singularities of the relevant 1D Green’s functions correspond to zeroes of the dispersion equation and look as sharp minima in this representation. Leaky-mode solutions appear as a complex conjugate pair in the TM and TE cases for the improper sheet. The weak dip appearing in the TM case for the proper sheet is due to the presence of the branch-point singularity at k ₀.

This analysis proved to be very useful to find another TM leaky mode which appears quite close to the branch point, i.e., $\hat{k}_\rho=1$ (see the two narrow dark dips in Fig. 3(a)). This is the quasi-TEM leaky mode that exists in an air-filled Fabry–Perot cavity and which may contribute to endfire radiation, as will be shown next in Section 4. The frequency dispersion of this quasi-TEM leaky mode is then obtained by solving again the TM dispersion equation, providing the wavenumber of the unperturbed TEM parallel-plate waveguide mode as initial guess, and is reproduced with solid orange lines in the inset of Fig. 2.

The fundamental TE-TM leaky-mode pair and the quasi-TEM leaky mode are the only three modes supported by this structure, as can be inferred from Fig. 3(a)–(d). In this regard, we should note that the weak dip appearing in the proper sheet for the TM case is not due to a pole, but is simply part of the natural variation of the function. In the TM case the function actually tends to infinity at the branch point, but this is not seen due to the sample density in the plot.

Since we are interested in 2D LWAs radiating at broadside we will now examine the contribution of the fundamental TE-TM leaky mode pair only. Specifically, for each value of f, we evaluated the radiation patterns (a few relevant examples are shown in Fig. 4(a)–(b)) considering either a VED or an HMD source for the infinite case (through (1) and (2), respectively), $R\to\infty$ and for ${R=5\lambda_0,\,10\lambda_0,\,15\lambda_0,\,20\,\lambda_0}$ (through [Reference Ip and Jackson4, Eqs. (10)–(12)] and [Reference Ip and Jackson4, Eqs. (24)–(34)] for the VED and HMD cases, respectively). For each broadside pattern (HMD case) we evaluated the HPBW, whereas for each null pattern (VED case), we evaluated the HPNW. Both the HPBW and HPNW are then normalized to the HPBW obtained from a 2D LWA when r = 1, which is equal to $2\sqrt{2}\hat{\alpha}$. These normalized beamwidths $\overline{\mathrm{HPBW}}$ and $\overline{\mathrm{HPNW}}$ are then reported in Fig. 4(c)–(d) as functions of r for the different choices of R. The different values of r were obtained by changing the substrate thickness while keeping the frequency fixed at 60 GHz. Colored circles denote the values of r that give the minimum beamwidths or nullwidths. (Circles are not shown for the two smallest apertures (the black and red curves), since either the minima are off the plotted scale, or the curves are so flat for the larger r values that the minimum is inconsequential.)

Figure 4.

(a)–(b) Normalized leaky radiation patterns for r = 1 (solid lines) and r = 0.5176 (dashed lines) for the infinite case, i.e., $R\to\infty$ (in gray) and for $R=\,20\lambda_0$ (in blue). (a) HMD case and (b) VED case. (c)–(d) Normalized half-power beamwidth and normalized half-power nullwidth vs. r for different values of the aperture radius R. A colored circle highlights the minimum beamwidth or nullwidth for each curve.

In this regard, it is worthwhile to stress that the radiation patterns of infinite 2D LWAs are similar to those of 1D infinite bidirectional LWAs, and thus the beamsplitting and dual-beam conditions are expected to occur for r > 1 and $ r \gt 1+\sqrt{2}\simeq2.414$, respectively. As a result, for values of r > 2.414, it is not always possible to have a well-defined HPBW at broadside. However, as we consider finite-size 2D LWAs, these boundaries are expected to change, although for 2D LWAs a rigorous examination as those provided in [Reference Sutinjo, Okoniewski and Johnston14, Reference Fuscaldo, Jackson and Galli15] is still lacking and will be subject of future studies. Therefore, results for $r\gg1$ have to be taken with care, and this is the reason why we reported results in the limited range ${0 \lt r \lt 2.5}$.

The results for the normalized HPBW in the infinite case agree with the theoretical expectations: the minimum is found approximately at r = 0.5176 and at r = 1 the normalized HPBW approaches unity, as per definition (see Fig. 4(c)). In this regard, we should recall that 0.5176 is actually an approximation of the exact minimum HPBW condition for the infinite-aperture case, which reads ${r=\sqrt{2-\sqrt{3}}}$ as shown in [Reference Lovat, Burghignoli and Jackson19]. Remarkably, this quantity, viz., $\sqrt{2-\sqrt{3}}$, is also connected to another condition, as it represents the ratio between the HPNW and the HPBW when r = 1, i.e., $\overline{\mathrm{HPNW}}(r=1)\simeq 0.5176$ (see the gray curve in Fig. 4(d)) as already discussed in [Reference Fuscaldo, Galli and Jackson13]. The behavior of the HPNW/HPBW ratio as a function of r can be inferred from Fig. 4(c) and (d): the HPNW/HPBW ratio in the infinite case increases for r → 0 reaching values as large as 0.64.

The results in the finite case are rather interesting. As R decreases, the minimum region becomes less pronounced (the curve becomes flatter) and the minimum condition shifts to higher values of r. Even for a rather electrically large antenna of radius equal to $15\lambda_0$, the minimum HPBW occurs at values of r close to 1 rather than 0.5176. It is seen that for small apertures, e.g., $R=5\lambda_0$, the normalized HPBW is only slightly affected by increasing r beyond 1. The beam narrowing for the sought conditions can also be appreciated from Fig. 4(a) where the leaky radiation patterns for the infinite case (in gray) and the finite case with $R=20\lambda_0$ (in blue) are shown for r = 1 and the value of r that leads to minimum beamwidth for the infinite and finite cases.

Similar considerations can be drawn for the normalized HPNW (see Fig. 4(b) and (d)). Again, as R decreases, the minimum region becomes less pronounced and the minimum condition shifts to higher values of r (see Fig. 4(d)). However, the HPNW is much less affected by variations of r, the region of the curves near the minimum being very flat. Interestingly, for the infinite case, the condition for the minimum HPNW differs from that for the minimum HPBW and it is found to be about r = 0.6258 (see the gray curve in Fig. 4(d)). In conclusion, the range $0.5\leq r\leq 1$ seems appropriate if the goal is the minimization of either the HPBW or the HPNW, whatever the aperture size is. The weaker sensitivity of the nullwidth with respect to the beamwidth to variations of r and of the lateral truncation are also manifest in 4(b) where it is seen that the condition for minimum nullwidth only slightly narrows the null region with respect to the condition that leads to maximum radiated power at broadside, viz., r = 1. As for the HPNW/HPBW ratio, it is noted that as R decreases the HPNW/HPBW ratio (for r = 1) shows a rather constant behavior, and the value slightly increases with respect to the value of 0.5176 found in the infinite case. To give some numbers: for $R=5\lambda_0$, we have $\mathrm{HPNW/HPBW}=0.73$, for $R=10\lambda_0$ we have $\mathrm{ HPNW/HPBW}=0.68$.

Full-wave validation

Results from previous sections discussed results from either an infinite structure or one terminated with ideal absorber. This section aims at bringing the previous analysis to a more practical level. First, we demonstrate with full-wave simulations that the leaky-wave pattern provides a remarkably accurate representation of the total pattern near the peak. Second, we investigate the effect of the lateral truncation when different boundary conditions are considered.

For this purpose, we designed in CST the 2D LWA described in the previous section with $R=6.6\lambda_0$ which approximately leads to a radiation efficiency of $90\%$ at f ₀; a typical criterion for the design of practical 2D LWA [Reference Jackson, Caloz and Itoh1, Reference Jackson, Burghignoli, Lovat, Capolino, Chen, Wilton and Oliner3]. The PRS is implemented as a surface impedance boundary condition (SIBC) with $B_{\mathrm{s}}=20\eta_0$ at f ₀. This assumption is well satisfied by typical PRS in the homogenized regime, such as patch arrays [Reference Luukkonen, Simovski, Granet, Goussetis, Lioubtchenko, Raisanen and Tretyakov24]. Therefore, the use of the SIBC in place of the patterned PRS allows for saving significant computational resources without affecting the accuracy of the results, as already shown in [Reference Fuscaldo, Galli and Jackson20] where this approach has been used to simulate a 1D LWA based on a metal strip-grating PRS.

At the lateral edge, two different boundary conditions are considered: (i) a perfect electric conductor (hereafter referred as “PEC”), and (ii) an SIBC (hereafter referred as “ $Z_{\mathrm{TM}}$”) using the wave impedance of the radially propagating TM leaky mode in the leaky radial waveguide (see [Reference Harrington25, Eq. (5.38)] for n = 1). (These boundary conditions extend from the ground plane to the PRS at the lateral edge.) We should stress that the latter condition acts as a nearly perfect absorbing surface for VED excitation, but not for HMD excitation. In this latter case, both TE and TM leaky modes are excited and the wave impedance of a TE mode is different than of a TM mode [Reference Harrington25, cf., Eqs. (5.38)–(5.40)]).

The structure fed with either a simple VED or a simple HMD is finally simulated considering the two different lateral boundary conditions, and the resulting total far-field patterns at f ₀ are compared against the theoretical leaky patterns in Fig. 5(a)–(b). (The leaky-wave patterns are based on the dominant leaky waves, and ignore the quasi-TEM leaky wave that is a TM wave.) More precisely, a 1D cut at $\phi=0^\circ$ of the power pattern normalized to its maximum is reported in Fig. 5(a)–(b). (Results look very similar for $\phi=90^\circ$ in the case of HMD excitation, and for any ϕ in the case of VED excitation thanks to the azimuthal symmetry of the aperture field.) In the case of HMD excitation (Fig. 5(a)), there are no appreciable differences among the two different lateral boundary conditions, and the leaky-wave pattern agrees very well with the CST pattern. In the VED case, when either the PEC or the $Z_{\mathrm{TM}}$ boundary condition is applied, the agreement between the leaky pattern and the CST pattern is satisfactory only near the peak; a secondary peak near endfire is always present. As we discussed in Section 3, this secondary beam is most likely due to the excitation of the quasi-TEM leaky mode that we clearly identified in Fig. 3(a).

Figure 5.

(a)–(b) Normalized leaky radiation patterns (in dashed black line) for (a) the HMD case and (b) the VED case, compared against normalized total (CST) radiation patterns when a PEC boundary condition (in solid green line) and an SIBC (in solid blue line) matching the wave impedance of the TM leaky mode are applied to the radial edge.

This secondary unwanted beam severely hinders the use of 2D LWAs for producing null beams in practical scenarios. Fortunately, there exist several methods for mitigating this issue, each one with its pros and cons. One possibility is to introduce a dielectric filling which converts the quasi-TEM leaky mode into a TM surface wave that no longer radiates, but still substracts power thus reducing the radiation efficiency. In this regard, the use of a waveguide-fed double iris slot originally proposed in [Reference Hickey, Qiu and Eleftheriades26] or that of reduced surface-wave microstrip circular patches in [Reference Jackson, Williams, Bhattacharyya, Smith, Buchheit and Long27] allow the feed to avoid exciting the TM surface wave. However, both feeders are alternatives to horizontal dipoles, not vertical dipoles, and thus they are not effective for obtaining a null beam at broadside with no endfire radiation. Taking cues from [Reference Jackson, Williams, Bhattacharyya, Smith, Buchheit and Long27], one may consider to feed the cavity with a circular array of vertical dipoles (which still excites a purely TM leaky mode [Reference Burghignoli, Fuscaldo, Comite, Baccarelli and Galli28]) and place the array at a critical radius $\rho_{\mathrm{c},\mathrm{TM}}$ to avoid exciting the TM surface wave, given by $\rho_{\mathrm{c},\mathrm{TM}}=j_{01}/\beta_{\mathrm{TM}_0}$, where $j_{01} = 2.4048$ is the first root of the $J_0(\cdot)$ Bessel function and $\beta_{\mathrm{TM}_0}$ is the phase constant of the TM₀ surface-wave mode. Another possibility is to excite the structure with a VMD: in this case, there will be no quasi-TEM mode excited (there is no TEM mode coming from TE modes in a PPW) and no TE surface wave is thus excited, and only the fundamental TE leaky mode seen in Fig. 3(c) is excited. However, the practical realization of either a circular array of VEDs or a VMD is not as simple as that of a single VED, the latter being well represented by a coaxial feeder. (A VMD will also radiate an E_ϕ polarized far field instead of an E_θ polarized far field). Possible implementations of VMD sources have recently been proposed and consist of loop antennas [Reference Lu, Voyer, Bréard, Huillery, Allard, Lin-Shi and Yang29, Reference Lu, Bréard, Huillery, Yang and Voyer30] and a circular array of radial slots [Reference Negri, Benassi, Fuscaldo, Masotti, Burghignoli, Costanzo and Galli31] (possibly fed with microstrip lines as in [Reference Benassi, Negri, Fuscaldo, Paolini, Maita, Burghignoli, Masotti, Galli and Costanzo32]) for designs at microwave frequencies, or a circular waveguide operating in its higher-order TE₀₁ mode [Reference Negri, Fuscaldo, González-Ovejero, Burghignoli and Galli33] (a Marié transducer is used to correctly excite this mode while preventing the excitation of other unwanted modes) for operation at (sub-)millimeter-wave frequencies. Although these feeders have all been used for generating TE-polarized Bessel beams, they can suitably be used in Fabry–Perot structures as Bessel-beam launchers and 2D LWAs share a very similar architecture, viz., a grounded dielectric slab with a PRS on top.