Theoretical background

Fabry–Perot cavity antennas have been studied with different approaches, such as the ray-optics model [Reference von Trentini23], the application of the reciprocity theorem on their equivalent transmission line [Reference Zhao, Jackson, Williams and Oliner21], and the leaky-wave perspective [Reference Burghignoli, Fuscaldo and Galli2]. Regardless the considered method, the most important design rule in FPCAs is related to the proper choice of the cavity height h as a function of X_s and the working frequency f with the aim of maximizing the radiated power on a certain direction θ ₀. For an air-filled cavity pointing at broadside, i.e., with $\theta_0=0$, it results [Reference von Trentini23]:

(1)\begin{equation} h(f) = \frac{\lambda_0}{4\pi}\left[\phi_{\rm PRS}(f)-\pi(2n-1)\right], \; n = 0,\pm 1, \dots, \end{equation}

where λ ₀ is the vacuum wavelength at the frequency f and ϕ _PRS is the phase of the reflection coefficient seen from the ground plane, i.e., $\Gamma_{\rm PRS} = |\Gamma_{\rm PRS}|e^{j\phi_{\rm PRS}}$ (see Fig. 1). The evaluation of the $\Gamma_{\rm PRS}$ value can be addressed in any commercial solver by considering the simulation of the unit cell in a periodic environment (see e.g., [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli6, Reference Almutawa, Hosseini, Jackson and Capolino29] for further details).

In this paper, a metal strip grating with periodicity $p=\lambda_0/6$ and strip width $w=0.125p$ is considered (see Fig. 1), obtaining $X_s\simeq100\;\Omega$ (in good agreement with homogenization formulas in [Reference Luukkonen, Simovski, Granet, Goussetis, Lioubtchenko, Raisanen and Tretyakov19]). The reflection coefficient $\Gamma_{\rm PRS}$ at the substrate-air interface has been retrieved for three different examples in the 45–75 GHz frequency range through a frequency-domain simulation on CST Microwave Studio [31]: the case of the simple thin PRS, and a pair of thick-PRS configurations realized through the same metal strip grating with a lossless superstrate on top whose relative permittivity is $\varepsilon_r=6.15$ and whose heights are $h_{\rm sup}=0.5$ mm or $h_{\rm sup}=1.3$ mm. This dielectric choice is representative of a realistic Rogers TC600 laminate which presents the same relative permittivity ɛ_r with a negligible, loss-tangent value of $\tan\delta=0.002$.

Figure 1. (a) Schematic representation of a Fabry–Perot cavity antenna, its thin-PRS unit cell, and its broadside pencil-beam radiation pattern. While the gray solid describes the air-filled cavity, the black color shows the perfect electric conductor which is used to implement the thin-PRS metallic pattern. The semi-transparent green solid box describes the dielectric superstrate which is present only for Fabry–Perot cavity antennas based on thick PRSs. The red rectangular area shows the slot on the ground plane fed by the rectangular waveguide. On the right, the TEN of the device is reported. The PRS contribution, both in the thin and in the thick cases, is described by the reflection coefficient $\Gamma_{\rm PRS}$ of the PRS looking upwards from the cavity. (b) Rectangular-waveguide feeder and its matching network constituted by two capacitive irises with their relevant design parameters. The red area on top represents the section at which the source is connected to the Fabry–Perot cavity antenna.

As one can infer from Fig. 2, the curve of the optimal substrate height h in the case of the thin-PRS case and the case of a thick PRS with $h_{\rm sup}=0.5$ mm intersects the design substrate height $h_{\rm sub}=2.3$ mm only at a single frequency. This condition results from the Foster-like behavior of the phase of the reflection coefficient ϕ _PRS, viz. $\partial\phi_{\rm PRS}/\partial f \lt 0$. In order to enhance the device working bandwidth, the condition (1) has to be satisfied over a wide frequency range. For this reason, a thicker superstrate (viz., $h_{\rm sup}=1.3$ mm) has been placed above the thin PRS, obtaining a thick PRS with a non-Foster behavior. Interestingly, this phase envelope is present only near the condition $h_{\rm sup}\simeq\lambda_d/2$ (with λ_d being the operative wavelength in the dielectric superstrate) and not for a thinner or thicker superstrate. As shown in Fig. 2, equation (1) is now satisfied over a wide frequency range for a substrate height ${h=h_{\rm sub}=2.3}$ mm (the relevant curve of h has a flat behavior vs. f around 60 GHz), leading to the desired wide bandwidth.

Figure 2. Cavity height h which satisfies the resonance condition (1) vs. frequency f in the presence of a thin PRS (black solid line) or of a thick PRS with $h_{\rm sup}=0.5$ mm (green solid line) or $h_{\rm sup}=1.3$ mm (blue solid line). The thin, black, dashed horizontal line represents the substrate height $h_{\rm sub}=2.3$ mm chosen in this work. The insets pictorially show the device for each configuration.

At this point, it is worth noting that in FPCAs based on thin [Reference Burghignoli, Fuscaldo and Galli2] or thick [Reference Almutawa, Hosseini, Jackson and Capolino29] PRS, the radiating features – such as the radiation pattern, the directivity, and the bandwidth – can be predicted through the evaluation of the leaky radial wavenumber $k_\rho = \beta-j\alpha$ (with β and α being the leaky phase and attenuation constants, respectively). This task can be addressed by numerically solving the dispersion equation [Reference Galdi and Pinto32] obtained by applying the transverse resonance technique [Reference Sorrentino and Mongiardo30] to the TEN of the device. For the considered structures the dispersion equation reads (see Fig. 1):

(2)\begin{equation} Y_{\rm up}+Y_{\rm down}=Y_{\rm up}-jY_0\cot(k_zh_{\rm sub})=0, \end{equation}

where Y ₀ is the wave admittance of the substrate transmission line and $k_z=\sqrt{k_0^2-k_\rho^2}$ its propagation constant (being k ₀ the vacuum wavenumber). The value Y _up represents the admittance of the PRS looking upwards from the cavity, thus, it can easily be retrieved through the full-wave evaluation of the reflection coefficient $\Gamma_{\rm PRS}$ for a normal incidence (which represents with a good approximation the PRS behavior for low-angle oblique incidences):

(3)\begin{equation} Y_{\rm up} = Y_0\frac{1-\Gamma_{\rm PRS}}{1+\Gamma_{\rm PRS}}. \end{equation}

The results of the proposed analysis are shown in Fig. 3, where the leaky phase and attenuation constants are reported normalized with respect to the vacuum wavenumber with $\hat{\beta}=\beta/k_0$ and $\hat{\alpha}=\alpha/k_0$, respectively. It is worthwhile to point out that, as theoretically expected from the ray-optics approximation in (1) and shown in Fig. 2, the condition for which occurs the maximum radiated power at broadside, i.e., $\beta\simeq\alpha$ [Reference Lovat, Burghignoli and Jackson33], is verified just at a single frequency for the case of a thin PRS and the case of a PRS which is not thick enough ( $h_{\rm sup}=0.5$ mm). On the other hand, for a properly designed thick PRS with $h_{\rm sup}=1.3$ mm, the leaky phase β and attenuation α constants are very similar to each other over a wider frequency range, thus obtaining a theoretically wider −3 dB gain bandwidth.

Figure 3. Dispersion curves vs. frequency f of the leaky-wave phase (solid lines) and attenuation (dashed lines) constants normalized with respect to the vacuum wavenumber k ₀, i.e., $\hat{\beta}=\beta/k_0$, and $\hat{\alpha}=\alpha/k_0$, respectively. The cases of a thin PRS, a thick PRS with $h_{\rm sup}=0.5$ mm, and a thick PRS with $h_{\rm sup}=1.3$ mm are reported in black, green, and blue curves, respectively.

Field matching technique

In this section, the leaky modes in an FPCA transitioning from a thin- to a thick-PRS case are investigated through an FMT approach. Without loss of generality, we assume the FPCA excited by an HMD source aligned with the x-axis, and we obtain the tangential electromagnetic fields over the E-plane, $\phi=90^\circ$. We recall that this source excites a purely $\mathrm{TM}_z$ ( $\mathrm{TE}_z$) electromagnetic field over the E-plane (H-plane) [Reference Ip and Jackson34]. Thanks to the geometry of the problem, as shown in [Reference Negri, Ballarini, Fuscaldo, Burghignoli and Galli1], the electric-field modal solution for an FPCA based on a thin PRS can be written as:

(4)\begin{equation}\left\{\begin{array}{lcl} E_\rho(z)&=A_2e^{-jk_z(z-h_{\mathrm s\mathrm u\mathrm b})},& z \gt h_{\mathrm s\mathrm u\mathrm b}\\ E_\rho(z)&=A_1\sin{(k_zz)},& z \leq h_{\mathrm s\mathrm u\mathrm b}\end{array}\right. \end{equation}

where A ₁ and A ₂ are complex amplitude coefficients. This is due to the propagative nature of the electric field above the PRS (for $z\ge h_{\rm sub}$) and its resonant behavior in the FPCA (for $z\le h_{\rm sub}$) with a null on the ground plane at z = 0. At this point, from Maxwell’s equations, the magnetic field reads:

(5)\begin{equation}\left\{\begin{array}{lcl}H_\phi(z) & = Y_0A_2e^{-jk_z(z-h_{\rm sub})},& z \gt h_{\rm sub}\\H_\phi(z) & = jY_0A_1\cos{(k_zz)},& z \le h_{\rm sub}\end{array}\right.\end{equation}

By applying the boundary conditions at $ z=h_{\rm sub}$ related to the homogenized metasurface representing the PRS (viz., $E_\rho(z=h_{\rm sub}^-)=E_\rho(z=h_{\rm sub}^+)$ and $H_\phi(z=h_{\rm sub}^+)-H_\phi(z=h_{\rm sub}^-)=Y_sE_\rho(z=h_{\rm sub})$), the problem can be recast in the following matrix form:

(6)\begin{equation} \begin{bmatrix} \sin{(k_zh)} & -1\\ -jY_0\cos{(k_zh)}+Y_s\sin{(k_zh)} & Y_0 \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \end{bmatrix}=0, \end{equation}

with $Y_s=1/Z_s$ being the equivalent admittance sheet of the PRS. A nontrivial solution for (6) is obtained by enforcing the determinant to zero, which yields the determinantal equation (note that the analogous formula, i.e. (7) of [Reference Negri, Ballarini, Fuscaldo, Burghignoli and Galli1], has an incorrect sign of the second addend):

(7)\begin{equation} Y_0+Y_s-jY_0\cot{(k_zh)}=0, \end{equation}

which is equivalent to (2), since $Y_{\rm up}=Y_0+Y_s$, but a full-wave simulation is not needed for the evaluation of $\Gamma_{\rm PRS}$. It is worthwhile to note that, assuming an Y_s dependent on θ, this approach is valid for any angle of incidence. Moreover, by considering PRS unit-cell configurations that are almost spatially nondispersive [Reference Fuscaldo, Tofani, Zografopoulos, Baccarelli, Burghignoli, Beccherelli and Galli6], a constant Y_s value for different incident angles can be assumed, still providing a valid antenna model. Equation (7) can then be numerically solved [Reference Galdi and Pinto32], obtaining the same leaky wavenumbers of “Theoretical background” section. In addition, the leaky-wave fields can be determined by solving for the unknown coefficients in (6). (The two equations are linearly dependent, so one of the two coefficients is arbitrarily set to unity and the reduced system solved for the other coefficient.)

A similar FMT approach can be repeated to obtain the leaky-wave fields for the thick-PRS case. By following the same procedure of the thin-PRS structure but assuming the presence of an additional layer, one has the following magnetic-field distribution:

(8)\begin{equation} \begin{cases} H_\phi(z) &= A_1e^{-jk_{z0}(z-h_{\rm tot})}, \;\; z\ge h_{\rm tot}\\ H_\phi(z) &= A_2\cos{\left[k_{zd}(z-h_{\rm sub})\right]}+\\ &A_3\sin{\left[k_{zd}(z-h_{\rm sub})\right]}, \;\; h_{\rm sub} \lt z \lt h_{\rm tot}\\ H_\phi(z) &= A_4\cos{\left(k_{z0}z\right)}, \;\; 0\le z\le h_{\rm sub}, \end{cases} \end{equation}

where $h_{\rm tot}=h_{\rm sub}+h_{\rm sup}$ is the total device height, ${k_{z0}=\sqrt{k_0^2-k_\rho^2}}$, and $k_{zd}=\sqrt{\varepsilon_rk_0^2-k_\rho^2}$ are the propagation constants in the vacuum and dielectric, respectively.

By applying Maxwell’s equation for this TM-polarized field, one can retrieve the electric-field distribution as:

(9)\begin{equation} \begin{cases} E_\rho(z) &= Z_0A_1e^{-jk_{z0}(z-h_{\rm tot})}, \;\; z\ge h_{\rm tot}\\ E_\rho(z) &= jZ_d\{A_3\cos{\left[k_{zd}(z-h_{\rm sub})\right]}-\\ & A_2\sin{\left[k_{zd}(z-h_{\rm sub})\right]}\}, \;\; h_{\rm sub} \lt z \lt h_{\rm tot}\\ E_\rho(z) &= -jZ_0A_4\sin{\left(k_{z0}z\right)}, \;\; 0\le z\le h_{\rm sub}, \end{cases} \end{equation}

where $Z_0=k_{z0}/(\omega\varepsilon_0)=1/Y_0$ and $Z_d=k_{zd}/(\omega\varepsilon_r\varepsilon_0)$.

At this point the boundary conditions have to be enforced. In particular, while the continuity of the tangential components of the electric and magnetic field has to be considered in $z=h_{\rm tot}$, only the tangential electric field is invariant at $z=h_{\rm sub}$. As for the ‘thin’ case, this aspect is due to the presence of the PRS which enforces the discontinuity of the tangential magnetic field components caused by an induced current as $\mathbf{J}_s=Y_s\mathbf{E}_\rho=\mathbf{z}\times(\mathbf{H}^+-\mathbf{H}^-)$. In this manner, a system with four equations and four unknowns is obtained and reported in its matrix form:

(10)\begin{equation} \begin{aligned} \left[\begin{matrix} 1 & -\cos{\left(k_{zd}h_{\rm sup}\right)} &-\sin{\left(k_{zd}h_{\rm sup}\right)} \\ Z_0 & jZ_d\sin{\left(k_{zd}h_{\rm sup}\right)} & -jZ_d\cos{\left(k_{zd}h_{\rm sup}\right)}\\ 0 & 0 & jZ_d\\ 0 & 1 & jZ_dY_s\\ \end{matrix}\right. \quad\dotsm \\ \dotsm\quad\left.\begin{matrix} 0\\ 0\\ jZ_0\sin{(k_{z0}h_{\rm sub})}\\ -\cos{(k_{z0}h_{\rm sub})} \end{matrix}\right] \begin{bmatrix} A_1 \\ A_2 \\ A_3 \\ A_4 \end{bmatrix}=0 \end{aligned} \end{equation}

This system has a nontrivial solution only if the determinant is set to zero. As for the thin PRS case, it is thus possible to retrieve the dispersion equation of the structure without full-wave simulations. After few mathematical steps and by indicating $Y_d=1/Z_d$, the determinantal equation reads:

(11)\begin{equation} Y_s+Y_d\frac{Y_0+jY_d\tan{(k_{zd}h_{\rm sup})}}{Y_d+jY_0\tan{(k_{zd}h_{\rm sup})}}-jY_0\cot{(k_{z0}h_{\rm sub})}=0. \end{equation}

It is worth noting that equation (11) is crucial for further studies and optimization procedures in FPCAs since it provides for the first time the dispersion analysis of these structures without requiring a full-wave simulation of the thick-PRS behavior.

When (11) is satisfied, all the equations in (10) are not independent. It is thus possible, for instance, to remove the first row and assume $A_1=1$. In this way, one has:

(12)\begin{equation} \begin{aligned} \left[\begin{matrix} jZ_d\sin{\left(k_{zd}h_{\rm sup}\right)} & -jZ_d\cos{\left(k_{zd}h_{\rm sup}\right)}\\ 0 & jZ_d\\ 1 & jZ_dY_s \end{matrix}\right. \quad\dotsm \\ \dotsm\quad\left.\begin{matrix} 0\\ jZ_0\sin{(k_{z0}h_{\rm sub})}\\ -\cos{(k_{z0}h_{\rm sub})}\\ \end{matrix}\right] \begin{bmatrix} A_2 \\ A_3 \\ A_4 \end{bmatrix}= \begin{bmatrix} -Z_0 \\ 0 \\ 0 \end{bmatrix} \end{aligned} \end{equation}

By inverting the matrix in (12), the coefficient are determined as shown in (13), thus achieving the complete field-distribution evaluation.

(13)\begin{equation} \begin{cases} A_4 &= \{-j\sin{\left(k_{z0}h_{\rm sub}\right)}\cos{\left(k_{zd}h_{\rm sup}\right)} + \\ &+\frac{Z_d}{Z_0} \sin{(k_{zd}h_{\rm sup})}[-j\cos{\left(k_{z0}h_{\rm sub}\right)}+\\ &+Z_dY_s\sin{\left(k_{z0}h_{\rm sub}\right)}]\}^{-1}\\ A_3 &= -A_4\frac{Z_0}{Z_d}\sin{\left(k_{z0}h_{\rm sub}\right)}\\ A_2 & = A_4\left[\cos{\left(k_{z0}h_{\rm sub}\right)}+jZ_0Y_s\sin{\left(k_{z0}h_{\rm sub}\right)}\right] \end{cases} \end{equation}

Some representative results of the FMT analyses are reported in Fig. 4. The tangential electric-field components, normalized with respect to their maxima, are plotted vs. z, normalized with respect to h _tot ( $h_{\rm sup}=0$ for the FPCA based on a thin PRS). As expected, for a thin-PRS case (see Fig. 4(a)) the tangential electric field follows a sine-like envelope inside the cavity and a typical leaky pattern outside. We note that the value assumed by $|E_\rho|$ at the air-substrate interface strongly depends on the reactance value: a lower X_s generates a lower $|E_{\rho}(z=z_{\rm sub})|$ (for the limit case $X_s\to0$, the tangential electric field tends to zero as in the asymptotic case of a parallel-plate waveguide). It is even more interesting to observe the thick-PRS cases. As preliminarily discussed, a superstrate thickness $h_{\rm sup}=0.5$ mm is not enough to achieve a PRS non-Foster behavior (see Fig. 2 and Fig. 3) and, in turn, a bandwidth enhancement. This condition occurs, instead, for $h_{\rm sup}=1.3$ mm. This difference is present also for the modal-field distribution obtained through the FMT. As can be inferred by Fig. 4(b) and Fig. 4(c), the bandwidth enhancement in thick PRS is appreciable only for superstrate modes that properly resonate above the cavity. It is worthwhile to point out that this effect occurs only for superstrate thicknesses in the order of $h_{\rm sup}=\lambda_d/2$.

Figure 4. Results of the FMT analysis when (a) a thin PRS, (b) a thick PRS with $h_{\rm sup}=0.5$ mm, or with (c) $h_{\rm sup}=1.3$ mm are considered. While the grey and dark-grey areas represent the substrate and superstrate, respectively, the black dotted line is the thin-PRS discontinuity represented by X_s. The absolute value of the electric-field leaky-wave mode is reported normalized with respect to its maximum $|E_\rho|_{\rm max}$ through a solid line vs. z normalized with respect to h _tot.

Full-wave analysis

In order to corroborate the proposed analysis and observe the simulated radiating features of the device, the entire three-dimensional (3-D) model of the FPCA based on the thick PRS with $h_{\rm sup}=1.3$ mm has been implemented on CST Microwave Studio [31] assuming a lateral dimension of the device $L=10\lambda_0$. The latter provides a radiation efficiency of about 99% through the typical formulas of leaky-wave antennas $\eta_r=1-e^{-2\pi\hat\alpha L/\lambda_0}$ [Reference Burghignoli, Fuscaldo and Galli2], with L being the lateral dimension of the Fabry–Perot cavity (see Fig. 1). (It is worth noting that the considered expression is an underestimation of the actual radiation efficiency due to the effect of the cylindrical radial spreading in 2-D leaky-wave antennas, which is not present in their 1-D counterpart).

Figure 5(a) shows how the normalized far-field radiation pattern at broadside, evaluated through both a theoretical and a simulative approach, evolves for different working frequencies f. By exploiting the leaky-wave theory in [Reference Almutawa, Hosseini, Jackson and Capolino29], it is indeed possible to find the far-field radiation intensity at broadside as a function of the leaky phase and attenuation constants as follows:

(14)\begin{equation} U(\omega,\theta=0)=\frac{4\sqrt{\varepsilon_r}U_0}{k_0h_{\rm sub}}\left[\frac{\hat{\alpha}\hat{\beta}}{\left(\hat{\alpha}^2+\hat{\beta}^2\right)^2}\right], \end{equation}

Figure 5. (a) Radiation pattern of a Fabry–Perot cavity antenna based on a thick PRS. Simulated results (black solid line) and theoretical results (red dashed line) are in a good agreement. The vertical solid lines represent the −3 dB power limits. (b) Reflection coefficient of a realistic truncated Fabry–Perot cavity antenna based on a thick PRS and fed through a rectangular waveguide by means a quasi-resonant slot on the ground plane. By considering a waveguide of length $L_{\rm WG}=\lambda_0$ (see Fig. 1(b)), the simulated magnitude of the S ₁₁ parameter is reported when a matching network is not considered (blue solid line) or two optimized capacitive irises are introduced (black solid line). The red dashed line represents the S ₁₁ parameter predicted by the feeding-network optimization procedure proposed in [Reference Negri, Fuscaldo, Tofani, Burghignoli and Galli17].

where U ₀ is a multiplicative excitation constant related to the amplitude of the leaky-wave mode. It is worthwhile to point out that the slight disagreement between theoretical and full-wave results in terms of radiation pattern in Fig. 5(a) are due to the edge effects which are present when a realistic, finite-size antenna is considered. In this regard, it is worthwhile to point out that it is also possible to further enlarge the radiation-pattern bandwidth by properly truncating the device (see, e.g., [Reference Baba, Hashmi and Esselle35, Reference Baba, Hashmi, Esselle and Weily36]).

As concerns the source, in order to realize a realistic counterpart of an HMD on the ground plane [Reference Negri, Fuscaldo, Burghignoli and Galli14], a quasi-resonant slot fed by a rectangular waveguide – with transverse dimensions given by the width a and height b reported in Table 1 – has been considered (see Fig. 1). However, by simply connecting a rectangular-waveguide feeding scheme to the antenna, the device is mismatched in the theoretical bandwidth (see the blue curve in Fig. 5(b)). For this reason, a matching network constituted by two capacitive irises has been designed and implemented on the 3-D model of the device. The choice of the geometrical parameters – i.e., the iris gaps g ₁ and g ₂ and the distances d ₁ and d ₂ (see Fig. 1(b)) – has been carried out through an efficient semi-analytical matching technique based on a transmission-line equivalent model. The latter, as shown in [Reference Negri, Fuscaldo, Tofani, Burghignoli and Galli17], only requires a single full-wave simulation of the device by de-embedding the waveguide port of a distance $L_{\rm RW}=\lambda_0$ (without the guide discontinuities). In particular, once the reflection coefficient and, thus, the device input impedance at the ground-plane level are known, they can be exploited in a transmission-line model to find the best configuration of the considered capacitive irises in a fast numerical manner (since such waveguide discontinuities are represented through well-known closed formulas [Reference Collin37, Reference Marcuvitz38]). By defining the capacitive-irises design parameters as in Fig. 1, the matching network has been implemented on CST Microwave Studio [31] with the feeder dimensions in Table 1. Simulated results obtain a very good reflection-coefficient profile which is in an impressive agreement with the S ₁₁ theoretically predicted with the approach shown in [Reference Negri, Fuscaldo, Tofani, Burghignoli and Galli17] (see Fig. 5(b)). In this manner, the impedance bandwidth, viz., the working frequencies for which the reflection coefficient is lower than −10 dB, covers almost the overall theoretical bandwidth.

Table 1. Design parameters of the feeding network used for matching the thick-PRS-based Fabry–Perot cavity antenna.