A) Graphene conductivity

Graphene is a true 2-D metamaterial [Reference Tamagnone, Gómez-Díaz, Mosig and Perruisseau-Carrier18] whose surface conductivity σ = σ _R  − jσ _J is a complex-valued scalar quantity that, at room temperature and in the low THz range, is sufficiently well described by taking into account only the intraband-contribution terms of the Kubo formula [Reference Hanson13, Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19], which describes σ as a function of the chemical potential μ _c , the relaxation time τ, the temperature T, and the frequency f. Clearly, at standard conditions (T = 300 K) and when f is fixed, σ only depends on τ and μ _c .

In Fig. 2(a), we have reported the values of σ _R and σ _J for f = 1 THz as a function of τ in the range 0–3 ps for different values of the chemical potential μ _c ranging from 0 to 1 eV (similar ranges of variations for both τ and μ _c are widely used in the available literature, e.g. see [Reference Llatser22] and references therein). As can be seen, σ _R decays almost exponentially for values of τ greater than about 0.3 ps for any value of μ _c . In addition, for any given τ, σ _R is a monotonically increasing function of μ _c , whereas |σ _J | is a monotonically increasing function of both τ and μ _c .

Fig. 2.

(a) Graphene Re[σ] and Im[σ] versus τ at f = 1 THz for μ _c ranging from 0 to 1 eV. Re[σ] and Im[σ] curves gradually shade from red to blue and from gray to black, respectively, as μ _c increases from 0 to 1 eV. (b) Ohmic losses of a suspended SPP (in terms of $\hat{\alpha} _{spp} $ ) for values of σ _R and σ _J in the dynamic range shown in (a). Isolines at $\hat{\alpha} _{spp} = 0.01,\;0.05,\;0.1,\;0.5,\;1$ are traced for readability purposes.

However, when σ _R is not much lower than |σ _J |, the dissipation losses of a SPP expressed by α _spp cannot be related only to σ _R , but generally depend on both σ _R and σ _J . In order to clearly emphasize this issue, in the following subsection we derive exact formulas, which express α _spp and β _spp of the SPP supported by a suspended graphene sheet as a function of σ _R and σ _J .

B) Losses in plasmonic propagation

As is known [Reference Maier17], the propagation constant k _spp  = β _spp  − jα _spp of the SPP supported by a conducting sheet is obtained in analytical form only when such a sheet is suspended in a homogeneous medium (e.g. vacuum). It is worth here to remark that this is a good approximation for a graphene sheet in air above a ground plane at a distance greater than half a wavelength (see Fig. 6 in [Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19]). In this case, the normalized plasmonic wavenumber $\hat k_{spp} = \hat{\beta} _{spp} - j \hat{\alpha} _{spp} = k_{spp} /k_0 $ (where k ₀ is the free-space wavenumber) reads:

(1) $$\hat k_{spp} = \sqrt {1 - \displaystyle{4 \over {(\sigma \zeta _0 )^2}}}, $$

where ζ ₀ ≃ 120π Ω is the free-space impedance. After some algebra, it is possible to get exact formulas for the normalized phase constant $\hat{\beta} _{spp} $ and the normalized attenuation constant $\hat{\alpha} _{spp} $ , as functions of σ _R and σ _J :

(2) $$\eqalign{& \hat{\beta} _{spp} = \left[ {\sigma _R \cos \left( {\displaystyle{1 \over 2}\arctan \displaystyle{\Pi \over \Delta}} \right) - \sigma _J \sin \left( {\displaystyle{1 \over 2}\arctan \displaystyle{\Pi \over \Delta}} \right)} \right] \cr & \;\;\;\;\;\;\;\;\displaystyle{{(\Delta ^2 + \Pi ^2 )^{1/4}} \over {\sigma _R^2 + \sigma _J^2}},} $$

(3) $$\eqalign{&\hat{\alpha} _{spp} = - \left[ {\sigma _J \cos \left( {\displaystyle{1 \over 2}\arctan \displaystyle{\Pi \over \Delta}} \right) + \sigma _R \sin \left( {\displaystyle{1 \over 2}\arctan \displaystyle{\Pi \over \Delta}} \right)} \right] \cr &\;\;\;\;\;\;\;\;\displaystyle{{(\Delta ^2 + \Pi ^2 )^{1/4}} \over {\sigma _R^2 + \sigma _J^2}},} $$

where $\Delta = \sigma _R^2 - \sigma _J^2 - 4/\zeta _0^2 $ and Π = −2σ _R σ _J . In Fig. 2(b), equation (3) is implemented for values of σ _R and σ _J in roughly the same range of values of Fig. 2(a). This clearly shows the role played separately by the real and imaginary parts of conductivity in establishing the level of the dissipation losses of a SPP propagating along a conductive sheet.

In order to compare our results with those obtained for a LWA based on SPP propagation along a graphene sheet over a silica (SiO₂) substrate (ε _rs  = 3.8), we have considered values of τ = 1 ps, and μ _c  = 0.436 eV, at f = 1 THz [Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19]. From Fig. 2(a) we get σ _R  ≃ 3 mS and σ _J  ≃ 7 mS, which gives rise through (2) to $\hat{\alpha} _{spp} \;{\rm \simeq}\; 0.08$ (see Fig. 2(b)). According to [Reference Tamagnone, Gómez-Díaz, Mosig and Perruisseau-Carrier18], α _spp linearly depends on f and on a scaling factor S = (ε _rs  + 1)/2. As a result, $\hat{\alpha} _{spp} $ does not depend on f, but it should be scaled by a factor S = 2.4 (since we are considering a half-space of silica), which gives $\hat{\alpha} _{spp} \;{\rm \simeq} \;0.2$ , as is found in [Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19], thus corroborating the validity of our analysis.

The resulting high ohmic losses are the most limiting factor for the antenna efficiency η of this kind of structures, that is restricted to $\eta \;{\rm \simeq}\; 20\% $ [Reference Tamagnone, Gómez-Díaz, Mosig and Perruisseau-Carrier18, Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19].

C) Quality measure of surface plasmons

In the previous subsection, we have considered the quantity $\hat{\alpha} _{spp} = \alpha _{spp} \lambda _0 /2\pi $ , where λ ₀ is the free-space wavelength, as a figure of merit (FoM) for the dissipation losses of the SPP. This choice is motivated by our interest in antenna applications, where the relevant dimensions are typically related to λ ₀. However, a measure of the quality of surface plasmons in waveguiding structures (e.g. nano-interconnects, nano-resonators, Bragg gratings, etc.) is given by different FoM. In this context, three FoMs have been proposed as benefit-to-cost ratios (where the benefit is represented by confinement and the cost is the attenuation) [Reference Berini23]. Specifically, we focus on the M ₂ and the M ₃ FoM, which are defined as

(4) $$M_2 = (\hat{\beta} _{spp} - 1)/\hat{\alpha} _{spp}, $$

(5) $$M_3 = \hat{\beta} _{spp} /(2\pi \hat{\alpha} _{spp} ).$$

M ₂ gives a direct measure of the confinement of an SPP for 1-D and 2-D structures, whereas M ₃ is strictly related to the quality factor Q [Reference Berini23]. In Figs 3(a) and 3(b), M ₂ and M ₃ are reported, respectively, as functions of σ _R and σ _J for a SPP propagating along a conductive sheet. As is seen, both M ₂ and M ₃ considerably improve for rather low values of σ _R and sufficiently high values of σ _J . This clearly emphasizes that the use of SPP along graphene sheets is of potential interest for future applications only for sufficiently high values of τ and μ _c , for which graphene shows a relatively high σ _J and a moderate σ _R (see Fig. 2(a)). As a final remark, it is worth here to note that equations (2) and (3), allow for deriving closed-form analytical expressions for the evaluation of both M ₂ and M ₃. Hence, equations (2) and (3), as well as Figs 2 and 3, may provide a useful tool for the design of plasmonic-based devices either for antenna applications or for guided-wave structures.

Fig. 3.

SPP figures of merit: (a) $M_2 = (\hat{\beta} _{spp} - 1)/\hat{\alpha} _{spp} $ and (b) $M_3 = \hat{\beta} _{spp} /(2\pi \hat{\alpha} _{spp} )$ versus σ _R and σ _J in the dynamic range shown in Fig. 2(a). The former (a) gives a measure of the confinement of a suspended SPP for 1-D and 2-D waveguide structures. The latter (b) is strictly connected to the quality factor Q [Reference Berini23].

A) Graphene-based planar single-slab antenna

A graphene-based planar single-slab antenna [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14] is a reconfigurable LWA [Reference Galli, Baccarelli and Burghignoli24] based on a graphene metasurface as the one depicted in Fig. 1(b). The structure consists of a grounded dielectric layer of relative permittivity ε _r1, covered with a graphene sheet, possibly biased through the application of a DC voltage between the graphene and an extremely-thin polysilicon layer placed just underneath. In these results, the thickness of the dielectric layer h ₁ is chosen equal to half the wavelength in the medium at the operating frequency (parameters f = 1 THz and ε _r1 = 3.8, as well as τ = 3 ps, are chosen as in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14]), so that the structure acts as a 2-D FPCA [Reference Jackson and Oliner25].

Based on a transverse-resonance technique [Reference Sorrentino and Itoh26, Reference Valerio, Jackson and Galli27], a thorough modal analysis of this structure in the low THz range [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14] revealed us that for values of μ _c in the range 0.25–1 eV the fundamental TE and TM leaky modes exhibit sufficiently low leakage rates $\hat{\alpha} _z = \alpha _z /k_0 $ , thus opening very interesting scenarios in the context of reconfigurable THz LWAs based on graphene metasurfaces (we skip here the details about the actual design of the graphene-based planar single-slab antenna that can be found in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14]). More interestingly, using the equivalent circuit model of the graphene-based planar single-slab antenna (see [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14]), and with the aid of the reciprocity theorem [Reference Lovat, Burghignoli and Celozzi28], the far-field expression of the magnetic field radiated on the H-plane considering a horizontal magnetic dipole (HMD) source directed along the y-axis (see Fig. 1), has been found:

(6) $$F(\theta ) = H_0 \;\cos \theta \displaystyle{{2Y_{cc}^{TE} \;\cos (k_{x1} h_s )} \over {(\sigma + Y_0^{TE} + Y_{cc}^{TE} )\;\cos (k_{x1} h_1 )}}$$

with

(7) $$\eqalign{Y_{cc}^{TE} = & - jY_1^{TE} \cot (k_{x1} h_1 ),\quad Y_1^{TE} = \displaystyle{{k_{x1}} \over {\omega \mu _0}}, \quad Y_0^{TE} = \displaystyle{{k_{x0}} \over {\omega \mu _0}}, \cr k_{x1} = & \sqrt {k_0^2 \varepsilon _{r1} - k_z^2}, \quad k_{x0} = \sqrt {k_0^2 - k_z^2}, \quad H_0 = - j\displaystyle{{k_0 \zeta _0} \over {4\pi}},} $$

where h _s is the distance of the source with respect to the ground plane. The expression for the E-plane pattern can be obtained by suppressing in (6) the factor cosθ and replacing the TE admittances with the TM ones:

(8) $$F(\theta ) = H_0 \displaystyle{{2Y_{cc}^{TM} \cos (k_{x1} h_s )} \over {(\sigma + Y_0^{TM} + Y_{cc}^{TM} )\cos (k_{x1} h_1 )}}$$

with

(9) $$\eqalign{Y_{cc}^{TM} = & - jY_1^{TM} \cot (k_{x1} h_1 ),\quad Y_1^{TM} = \displaystyle{{\omega \varepsilon _0 \varepsilon _{r1}} \over {k_{x1}}}, \quad Y_0^{TM} = \displaystyle{{\omega \varepsilon _0} \over {k_{x0}}}, \cr k_{x1} = & \sqrt {k_0^2 \varepsilon _{r1} - k_z^2}, \quad k_{x0} = \sqrt {k_0^2 - k_z^2}, \quad H_0 = - j\displaystyle{{k_0 \zeta _0} \over {4\pi}}} $$

We note that the intensity of the radiation pattern over both planes is maximized when the source is placed on the ground plane h _s  = 0. This location is quite convenient, since a HMD source can be used to model a slot etched in the ground plane and back-illuminated by a coeherent THz source as a photomixer.

These analytical results have been further validated through full-wave simulations on CST Microwave Studio [29]. In fact, according to reciprocity theorem [Reference Lovat, Burghignoli and Celozzi28, Reference Lovat, Burghignoli and Jackson30], the magnetic far field at the observation point can be calculated by evaluating the field at the source dipole location due to the incident wave from a testing dipole, for any angle of observation. This scenario has been modeled in CST, by exciting a unit cell of period l = λ ₀/4 with a Floquet port at a proper distance. Full-wave simulations and analytical results have shown a remarkable agreement, as can be inferred looking at the graphene-based planar single-slab antenna radiation patterns reported in Fig. 4, where the cyan circles (CST) and the black lines (equation (6)) are actually overlapped. It is worth here to stress that in the CST model graphene is modeled with the full integral version of the Kubo formula, which takes into account also the interband contributions. Hence, the remarkable agreement between our results, based solely on the intraband contributions of Kubo formula, and full-wave simulations, further corroborates the validity of our initial hypotheses.

Fig. 4.

Radiation patterns versus elevation angle θ on the H-plane for a graphene-based planar single-slab antenna. Results are obtained through equation (6) (solid black lines) and through CST full-wave simulations (cyan circles). The beam-scanning behavior at a fixed frequency is shown for pointing angles θ _p  = 0°, 15°, 30°, 45° in connection with the different values of μ _c . Similar results are obtained on the E-plane. In all cases, a remarkable agreement is observed among analytical and numerical results.

In conclusion, both analytical and numerical results have shown that the tunability of graphene conductivity allows for achieving beam scanning at fixed frequency, an uncommon feature for conventional LWAs [Reference Galli, Baccarelli and Burghignoli24, Reference Lovat, Burghignoli and Celozzi28, Reference Baccarelli, Di Nallo, Frezza, Galli and Lampariello31]. However, the directivity of the graphene-based planar single-slab antenna is rather low due to the relatively high values of the leakage rate. Hence, in the next subsection, novel multilayered structures based on graphene metasurfaces are considered in order to improve the radiative performance.

B) Graphene-based SS antenna

A graphene-based SS antenna [Reference Fuscaldo, Burghignoli, Baccarelli and Galli15] is a reconfigurable leaky-wave SS antenna including a graphene metasurface as the one depicted in Fig. 1(c). The structure is a conventional SS antenna [Reference Jackson and Oliner25], except for the introduction of a graphene sheet at a suitable position within the substrate. In brief, starting from the design of the graphene-based planar single-slab antenna described in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14], a cover dielectric layer (or “superstrate”) of higher relative permittivity ε _r2 (e.g. Hafnium oxide (HfO₂) for which ε _r2 = 25), and thickness h ₂ equal to a quarter of the wavelength in the superstrate [Reference Jackson and Oliner25], is mounted on top of the graphene layer to increase directivity at broadside (the interested reader can find more details about the actual design of the graphene-based SS antenna in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli15]).

Then we have considered different positions for the graphene sheet within the substrate in order to find an optimum location where it simultaneously allows for good beam reconfigurability at fixed frequency and an improved directivity. The former feature is given by positions where the graphene sheet strongly interacts with the tangential electric field, the latter is a result of the “resonance gain” phenomenon in SS configurations [Reference Jackson and Oliner25]. Thus, an optimum position is expected to be found between the middle plane of the substrate and the interface between the dielectric layers.

Henceforth, a parametric analysis has been carried out showing that an optimum position exists for which the $\hat{\alpha} _z $ of the dominant leaky mode is minimum at the cut-off frequency f _c (i.e. the frequency for which $\hat{\beta} _z (f_c ) = \hat{\alpha} _z (f_c )$ [Reference Lovat, Burghignoli and Jackson30]) (see Fig. 5), and in turn the directivity is maximum at broadside [Reference Jackson and Oliner25, Reference Lovat, Burghignoli and Jackson30].

Fig. 5.

Cut-off frequency f _c (blue to cyan solid lines) and relevant value of $\hat{\beta} _z (f_c ) = \hat{\alpha} _z (f_c )$ (red to yellow solid lines) as a function of the distance of the graphene sheet from the ground plane x ₀ normalized to the substrate thickness h ₁, for the fundamental TM leaky mode in the graphene-based SS antenna. Similar results are found for the fundamental TE leaky mode. As the dielectric contrast spans the following values R = 2, 5, 10, 20, 50, the curves shade from blue to cyan and from red to yellow for values of f _c and of $\hat{\alpha} _z (f_c )$ , respectively.

Following the same approach outlined in the previous subsection, the far-field expression of the magnetic field radiated on the H-plane for a graphene-based SS antenna, considering a HMD source directed along the y-axis (see Fig. 1), is found:

(10) $$F(\theta ) = H_0 \cos \theta \displaystyle{{2Y_{in3}^{TE}} \over {Y_{in3}^{TE} + Y_0^{TE}}} \displaystyle{{Y_{cc}^{TE}} \over {Y_{cc}^{TE} + \sigma}} \displaystyle{{\cos (k_{x1} h_s )} \over {\cos (k_{x1} h_1 )}}M$$

(11) $$\eqalign{Y_{cc}^{TE} &= - jY_1^{TE} \cot (k_{x1} \xi h_1 ), Y_2^{TE} = \displaystyle{{k_{x2}} \over {\omega \mu _0}}, Y_1^{TE} = \displaystyle{{k_{x1}} \over {\omega \mu _0}}, \cr Y_0^{TE} &= \displaystyle{{k_{x0}} \over {\omega \mu _0}}, \xi = x_0 /h_1 \cr k_{x2} &= \sqrt {k_0^2 \varepsilon _{r2} - k_z^2}, k_{x1} = \sqrt {k_0^2 \varepsilon _{r1} - k_z^2}, k_{x0} = \sqrt {k_0^2 - k_z^2}, \cr Y_{in3}^{TE} &= Y_2^{TE} \displaystyle{{Y_{in2}^{TE} \cos (k_{x2} h_2 ) + jY_2^{TE} \sin (k_{x2} h_2 )} \over {Y_2^{TE} \cos (k_{x2} h_2 ) + jY_{in2}^{TE} \sin (k_{x2} h_2 )}},} $$

(12) $$\eqalign{Y_{in2}^{TE} &= Y_1^{TE} \displaystyle{{Y_{in1}^{TE} \cos (k_{x1} (1 - \xi )h_1 ) + jY_1^{TE} \sin (k_{x1} (1 - \xi )h_1 )} \over {Y_1^{TE} \cos (k_{x1} (1 - \xi )h_1 ) + jY_{in1}^{TE} \sin (k_{x1} (1 - \xi )h_1 )}}, \cr Y_{in1}^{TE} &=\sigma + Y_{cc}^{TE}, \cr M &= {\cos(k_{x1}(1-\xi)h_1) \over \sec(k_{x2}h_2)} - {Y_{1}^{\rm TE} \over Y_2^{\rm TE}} {\sin(k_{x1}(1-\xi)h_1) \over \csc(k_{x2}h_2)} \cr &\quad - {1 \over Y_{\rm in3}^{\rm TE}} \left[jY_2^{\rm TE} {\cos(k_{x1}(1-\xi)h_1) \over \csc(k_{x2}h_2)}+jY_1^{\rm TE} {\sin(k_{x1}(1-\xi)h_1) \over \sec(k_{x2}h_2)} \right]} $$

It is straightforward to verify that for ξ = 1 (i.e. when the graphene sheet is placed at the SS interface) and h ₂ = 0 (i.e. without the superstrate), equation (10) coincides with equation (6). (The expression for the E-plane can be obtained as described in the previous Section III.A).

Both full-wave simulations (CST) and analytical results for the radiation patterns (see Fig. 6) confirm that the optimized graphene-based SS antenna shows substantially improved directivities with respect to graphene-based planar single-slab antenna solutions (see Fig. 4) for all the considered pointing angles (θ _p  = 0°, 15°, 30°, 45°). Note that the obtained directivities of both the graphene-based planar single-slab and the graphene-based SS antennas are comparable with those obtained for the graphene-based SPP antennas proposed in [Reference Tamagnone, Gómez-Díaz, Mosig and Perruisseau-Carrier18, Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19].

Fig. 6.

Radiation patterns  versus elevation angle θ on the H-plane for a graphene-based SS antenna. Results are obtained through equation (10) (solid black lines) and through CST full-wave simulations (cyan circles). The beam-scanning behavior at a fixed frequency is shown for pointing angles θ _p  = 0°, 15°, 30°, 45° in connection with the different values of μ _c . Similar results are obtained on the E-plane. In all cases, a remarkable agreement is observed among analytical and numerical results.

It is worth noting that these results have been obtained for a given dielectric contrast of the layers R = ε _r2/ε _r1 ≃ 6. However, in Fig. 5 it is shown that, when different dielectric contrasts are considered, this minimum condition still exists but is less pronounced and achieved for graphene positions shifted toward the interface between the layers, as R increases. This is exactly what we expect from theory, since, for a higher R, “cover” effects [Reference Jackson and Oliner25] dominate over graphene effects. Consequently, it is convenient to place graphene in a position where it weakly interacts with the tangential electric fields (we recall here that in a standard SS antenna the tangential electric field has a minimum at the interface between the layers).

C) Efficiencies of graphene-based planar single-slab and graphene-based SS antennas

In order to evaluate the efficiencies of both the graphene-based planar single-slab and the graphene-based SS antennas we have calculated the ratio η = P _rad /(P _rad  + P _g ), where P _rad is the power radiated in space and P _g is the power dissipated along the graphene sheet [Reference Di Nallo, Frezza, Galli and Lampariello32]. As is typical, these structures are assumed to be electrically large in the transverse plane so that the power dissipated at the end of the line is negligible. Note also that more sophisticated approaches, which take into account ohmic and dielectric losses as well as dyadic nonlocal models for σ, have already been investigated in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14], showing very similar results to those considered here.

The quantities P _rad and P _g have been evaluated using the transmission-line method and the ABCD-matrix approach [Reference Pozar33] based on the transverse equivalent networks of the structures [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14]. The calculation revealed us that the graphene-based planar single-slab antenna achieves an efficiency of about 70%, whereas the graphene-based SS antenna achieves efficiencies in the range 40–90%, depending on the graphene position. As an example, considering a graphene-based SS antenna with R = 5 (see Fig. 5), the optimum position of graphene for achieving maximum directivity is reached approximately at x ₀/h ₁ ≃ 0.8 for f _c  ≃ 1.12 THz, resulting in $\hat{\alpha} _z \;{\rm \simeq}\; 0.16$ and $\eta \;{\rm \simeq}\; 60\% $ . Now, if we move the graphene position toward the interface, e.g. x ₀/h ₁ ≃ 0.9, this would lead to a slight increase of $\hat{\alpha} _z (f_c )\;{\rm \simeq}\; 0.18$ at f _c  ≃ 1.05 THz, but a quite increased efficiency $\eta \;{\rm \simeq}\; 80\% $ . It is worth here to stress that the graphene position, which leads to the maximum directivity at broadside does not generally produce maximum efficiency. Hence, a trade-off exists between directivity and efficiency, which gives designers a further degree of freedom for the synthesis of a graphene-based SS antenna with respect to the graphene-based planar single-slab antenna and above all to the previously analyzed SPP structures [Reference Tamagnone, Gómez-Díaz, Mosig and Perruisseau-Carrier18, Reference Esquius-Morote, Gómez-Díaz and Perruisseau-Carrier19].

D) Graphene-strip grating antenna

The graphene-strip grating antenna proposed finally here is a reconfigurable LWA based on a patterned graphene metasurface as the one depicted in Fig. 1(d). The structure is equal to the graphene-based planar single-slab antenna [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14], except for the patterning of the graphene sheet. Specifically, we have considered a capacitive grid of infinitely-long graphene strips aligned along the y-axis. The proposed grating is characterized by a subwavelength period p = λ/5 and a very small gap between the strips w = p/10 = λ/50 at the design frequency f = 1 THz (which corresponds to λ ≃ 300 μm). Under these conditions, the patterned graphene sheet can be accurately described by a single homogenized surface impedance [Reference Luukkonen34–Reference Tretyakov36]. Note that, under these hypotheses, the proposed graphene-strip grating antenna is considerably different from those proposed in [Reference Shapoval, Gómez-Díaz, Perruisseau-Carrier, Mosig and Nosich21], where the period of the grating is comparable with the wavelength, and hence a single impedance would not suffice for the description of the properties of the surface [Reference Luukkonen34].

For an imperfect conductor, or more generally for an arbitrary 2-D material characterized by a complex surface conductivity as graphene, the following expressions for the homogenized TE and TM impedances hold [Reference Yakovlev, Padooru, Hanson, Mafi and Karbasi35, Reference Tretyakov36]:

(13) $$Z_s^{TM} = \displaystyle{p \over {\sigma (p - w)}} - j\displaystyle{{\eta _{eff}} \over {2\alpha}}, $$

(14) $$Z_s^{TE} = \displaystyle{p \over {\sigma (p - w)}} - j\displaystyle{{\eta _{eff}} \over {2\alpha (1 - (k_z /k_{eff} )^2 /2)}}$$

with

$$\alpha = ( {\rm k_{eff}} p/\pi )\ln \;\csc \left( {\displaystyle{{\pi w} \over {2p}}} \right),$$

where $k_{eff} = k_0 \sqrt {\varepsilon _{eff}} $ , $\eta _{eff} = \zeta _0 /\sqrt {\varepsilon _{eff}} $ , and ε _eff  = (ε _r1 + 1)/2.

Following the same procedure outlined in [Reference Fuscaldo, Burghignoli, Baccarelli and Galli14, Reference Yakovlev, Zouhdi, Sihvola and Vinogradov37], the dispersion curves of the graphene-strip grating antenna are readily obtained using equations (13) and (14) to describe the sheet impedance. Results are shown in Fig. 7 where the dispersion curves of the fundamental TE (see Figs 7(a) and 7(c)) and TM (see Figs 7(b) and 7(d)) leaky modes of the graphene-strip grating antenna (see Figs 7(c) and 7(d)) have been reported and compared with those of the graphene-based planar single-slab antenna (see Figs 7(a) and 7(b)) for four different values of the chemical potential μ _c . The values of μ _c have been selected in order to get the same theoretical pointing angles [Reference Lovat, Burghignoli and Jackson30] analyzed for the graphene-based planar single-slab and the graphene-based SS antennas, i.e. θ _p  = 0°, 15°, 30°, 45°, at the fixed frequencies of f ≃ 0.922 THz for the first one and of f ≃ 1.12 THz for the second one. As is shown, except for a frequency shift, the behaviors are similar, and hence comparable radiative performances are expected from the contribution of the relevant leaky modes.

Fig. 7.

Dispersion diagrams of $\hat{\beta} _z $ and $\hat{\alpha}_z $ versus f for (a)–(c) TE and (b)–(d) TM fundamental leaky modes of a (a, b) graphene-based planar single-slab antenna (solid lines) and a (c, d) graphene-strip grating antenna (dashed lines). The values of the chemical potentials μ _c are reported in the legends.

To verify this, the radiation patterns (see Fig. 8) have been calculated analytically by taking into account only the contribution of the leaky mode, as described in [Reference Lovat, Burghignoli and Jackson30]. As expected, the radiation patterns at broadside (black lines) of a graphene-strip grating antenna show a slightly larger beamwidth due to the small increase of the leakage rate, as can be inferred from the inspection of Figs 7 and 8. However, as the bias is decreased from its maximum value μ _c  = 1 eV (broadside condition), graphene ohmic losses increase as extensively commented in Section II.A. As a consequence, in the graphene-based planar single-slab antenna (see Figs 7(a) and 7(b)), the leakage rates at the cut-off frequency f ≃ 0.922 THz tend to remain almost the same (note that, in 2-D LWAs, as the phase constant increases beyond the cut-off frequency, the leakage rate usually decreases [Reference Lovat, Burghignoli and Jackson30]), and in turn the corresponding half-power beamwidths are rather large over the considered scanning range (see solid lines in Figs 8(a) and 8(b)). Conversely, in the graphene-strip grating antenna (see Figs 7(c) and 7(d)) the leakage rates at the cut-off frequency f ≃ 1.12 THz decrease as μ _c decreases, and in turn the corresponding half-power beamwidths (see dashed lines in Figs 8(a) and 8(b)) are narrower than those of the graphene-based planar single-slab antenna (see solid lines in Figs 8(a) and 8(b)) as the beam is scanned for a wider angle. This behavior has a simple physical explanation. As can be inferred from equations (13) and (14), the dependence of σ on the homogenized impedance of the graphene-strip grating, is “weighted” by the geometrical properties of the grating. As a consequence, any change in the graphene conductivity σ is reflected in a weaker effect whose intensity depends on the “filling-factor” w/p. The results of this section are thus a direct consequence of the choice w/p = 0.1.

Fig. 8.

Normalized radiation patterns P(θ)/P _max versus θ for (a) TE (H-plane) and (b) TM (E-plane) fundamental leaky mode of a graphene-based planar single-slab antenna (solid lines) and a graphene-strip grating antenna (dashed lines).

As a final remark, the different equalization of the TE–TM modes in the case of a graphene-strip grating antenna is worth to be commented. As is seen, the values of the chemical potentials needed to obtain the same pointing angle are different over the H- and the E-plane, whereas they are almost the same for the graphene-based planar single-slab antenna. This is a consequence of the different expression (see equations (13) and (14)) of the equivalent impedance shown by the graphene-strip grating for the TE and TM cases. However, the graphene-strip grating antenna, is still quite attractive thanks to the additional degrees of freedom provided to antenna designers. In fact, the possibility to use a patterned graphene sheet, instead of a uniform unpatterned graphene sheet, allows for independently biasing each strip, thus permitting the realization of tunable LWAs with a tapered aperture distribution.

We finally note that the radiating features of the graphene-strip grating antenna have been preliminarily analyzed assuming a homogenized impedance for the graphene-strip grating. In this context, full-wave simulations are even more required in order to assess the validity of the homogenization formulas in equations (13) and (14) for conductive strips characterized by a finite complex conductivity as graphene.