An atomically thin graphene monolayer can be considered as a transverse conducting sheet with a complex-valued surface conductivity *σ*(*ω*)=*σ*′+*jσ*″ that can be well modeled by first-principle calculations [61], [62], random phase approximations [17], [22], [63], or semiclassical models [12], [13]; here, we assume an *e*^{jωt} time dependence throughout this work. In the long-wavelength region, the surface conductivity of graphene monolayer depends on its Fermi energy *E*_{F} (or chemical potential), which can be largely tuned either by chemical doping [35], [38], external static electric field [64] (leading to an isotropic scalar surface conductivity), or external static magnetic field via Hall effect [15], [65] (leading to a gyrotropic and tensor surface conductivity). In the absence of static magnetic field, the semiclassical conductivity of graphene, which includes both intraband conductivity *σ*_{intra} and interband conductivity *σ*_{inter}, is given by

$$\begin{array}{c}\sigma (\omega ,{\mu}_{\text{c}},\Gamma ,\text{T})={\sigma}_{\text{intra}}+{\sigma}_{\text{inter}}\\ =-\frac{j{q}^{2}(\omega -j{\tau}^{-1})}{\pi {\hslash}^{2}}[{\displaystyle {\int}_{-\infty}^{+\infty}\frac{\left|\epsilon \right|}{{(\omega -j{\tau}^{-1})}^{2}}\frac{\partial F\left(\epsilon \right)}{\partial \epsilon}}d\epsilon \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-{\displaystyle {\int}_{0}^{+\infty}\frac{F(-\epsilon )-F\left(\epsilon \right)}{{(\omega -j{\tau}^{-1})}^{2}-4{(\epsilon /\hslash )}^{2}}}d\epsilon \text{\hspace{0.17em}}],\end{array}$$(1)

where *F*=1/(1+exp[(*ε*−*E*_{F} )/(*K*_{B} *T*)]) is the Fermi-Dirac distribution, *ε* is the energy, *T* is the temperature, *q* is the electron charge, *ℏ* is the reduced Planck’s constant, *K*_{B} is the Boltzmann’s constant, and *τ* is the impurity-limited relaxation time associated with plasmon loss in graphene. The first and second terms in Eq. (1) account for the intraband and interband contributions, respectively. In the long-wavelength region, *σ*_{intra} dominates over *σ*_{inter}. Therefore, for a doped graphene with |E_{F}|>>*k*_{B} *T* and for photon energy far below the interband transition threshold, *ℏω*<2|*E*_{F} |, Eq. (1) can be conveniently expressed using Drude-type dispersion:

$$\begin{array}{l}{E}_{\text{x}}^{}=\left[{\sigma}_{\text{graphene}}\frac{p}{p-d}-i\frac{{\eta}_{\text{eff}}}{2}{\alpha}_{\text{ABC}}\left(1+\frac{1}{{k}_{\text{eff}}^{2}}\frac{{\partial}^{2}}{\partial {\text{x}}^{2}}\right)\right]\text{\hspace{0.05em}}\left({H}_{\text{y}}^{-}-{H}_{\text{y}}^{+}\right)={Z}_{\text{s}}^{\text{TM}}{J}_{\text{x}}\\ {E}_{\text{y}}^{}=\left[{\sigma}_{\text{graphene}}\frac{p}{p-d}-i\frac{{\eta}_{\text{eff}}}{2}{\alpha}_{\text{ABC}}\right]\text{\hspace{0.05em}}\left({H}_{\text{x}}^{-}-{H}_{\text{x}}^{+}\right)={Z}_{\text{s}}^{\text{TE}}{J}_{\text{y}}\end{array}$$(2)

where the Drude weight *D*=*q*^{2}*E*_{F} /*ℏ*^{2}. The intrinsic relaxation time can be expressed as $\tau =\mu {E}_{F}/q{V}_{F}^{2}$ where *v*_{F} is the Fermi velocity and *μ* is the measured dc mobility. For instance, with a mobility of 10,000 cm^{2}/V s, a conservative value *τ*≈10^{−13} s is measured for *E*_{F} =0.1 eV. In general, the interband conductivity is on the order of *q*^{2}/*ℏ*. For *ℏω*, |*E*_{F} |>>*K*_{B} *T*, *σ*_{inter} can be approximately expressed as [15], [16]

$${\sigma}_{\text{inter}}\approx \frac{{q}^{2}}{4\hslash}\left(\theta (\hslash -2{E}_{F})-j\frac{1}{\pi}\mathrm{ln}\left[\frac{2\left|{E}_{F}\right|-\hslash (\omega -j{\tau}^{-1})}{2\left|{E}_{F}\right|+\hslash (\omega -j{\tau}^{-1})}\right]\right),$$(3)

where *θ*(·) is the step function. In the NIR and visible spectral ranges, where the photon energy *ℏω*>>*E*_{F} , *K*_{B}*T*, the interband contribution dominates and the optical conductivity is constant, given by

$$\sigma =\frac{{q}^{2}}{4\hslash}\mathrm{tanh}\left(\frac{\hslash \omega -2{\epsilon}_{F}}{4{K}_{B}T}\right)\approx \frac{{q}^{2}}{4\hslash}.$$(4)

As a result, graphene is almost transparent (~97.7% transmittance) over a wide range of wavelengths from NIR to visible and cannot sustain surface plasmons.

Typically, the Fermi energy of graphene can be tuned over a wide range by an externally applied bias (electrostatic gating), as the Fermi energy in graphene is related to the carrier concentration *n*_{s} by [6]

$${n}_{s}=\frac{2}{\pi {\left(\hslash {\nu}_{F}\right)}^{2}}{\displaystyle {\int}_{0}^{\infty}\epsilon [F(\epsilon -{E}_{F})-F(\epsilon +{E}_{F})]d\epsilon}.$$(5)

From Eqs. (2) and (5), we note that the Drude weight of graphene $D\propto {E}_{F}\propto \sqrt{{n}_{s}}$ can be tuned by varying the carrier density of graphene. By placing a p^{+}_{/}n^{+}-doped polysilicon or metal gate behind the insulating oxide that supports the graphene monolayer, the carrier concentration in graphene and, therefore, its dynamic conductivity can be tuned over a wide range by applying different gate voltages. Due to the electron-hole symmetry in the electronic band structure of graphene, both negative and positive signs of Fermi energy provide the same value of complex conductivity. For an *N*-layer graphene, one may assume *σ*_{N-layer}=*Nσ*, which is approximately valid for at least up to *N*=10, as validated by the experimental data [33], [58].

The unique features of ballistic transport and ultrahigh electron mobility in a high-quality graphene may provide a relatively large value of |Im[*σ*]/Re[*σ*] [19]. This implies that a graphene monolayer can be seen as a reactive sheet (impedance surface) with low loss in the THz-to-MIR region, effectively playing a role analogous to a frequency selective surface (FSS) without the need for lithographically patterning graphene [20]. Figure 2 reports the complex-valued conductivity of graphene with different Fermi energies. It is seen that the imaginary part of graphene’s conductivity (or surface impedance *Z*_{s} =1/*σ*) is tunable with respect to the Fermi energy. Such fascinating behavior appears naturally suited to manipulate the electromagnetic scattering and near-field enhancement for THz and long-wavelength infrared waves.

Figure 2: Wavelength dependence of the (left) real and imaginary parts and (right) their ratio of dynamic conductivity for a graphene monolayer, whose dual plasmonic or dielectric properties can be tuned by the Fermi energy of graphene.

In the THz and infrared regions, a uniform or patterned graphene (e.g. metasurface) has been demonstrated to achieve the scattering reduction or enhancement for its covered dielectric/conducting objects, which has the iconic term of *graphene cloak* [20]. The physical principle behind the graphene cloak resides in the scattering cancellation effect, which was first validated in microwave experiments by using the copper FSS [66], [67]. For the graphene cloak in Figure 3A, the effective surface current yielded by the plasmon oscillations in graphene can be tailored to radiate “antiphase” scattered fields [66], [67]. Consequently, around the operating wavelength, a cancellation between the scattering fields contributed by the covered object and graphene may restore the incident wavefronts in the near- and far-field. This effect is, in general, independent of the polarization and incidence angle of the impinging wave and the position of the observer, since the surface impedance of graphene is isotropic without an external static magnetic field. Moreover, the tunable surface reactance of graphene related to Im[*σ*] may enable a frequency-reconfigurable cloaking device. In addition to camouflaging and mirage applications, the graphene cloak has been proposed to improve the performance of near-field sensors and photodetectors in the interference/scattering-rich environment [68], [69]. We note that most subdiffractive measurements are usually performed in the very-near-field of the details to be imaged, and therefore, their accuracy is intrinsically limited by the disturbance introduced by the close proximity of the sensing instrument, e.g. a sensing probe that may perturb the near field distributions and influence the measurement [68]. A graphene cloak may greatly suppress the undesired multipath scattering and noise for the highly sensitive sensor and probe.

Figure 3: (A) Normalized total SW of a SiO_{2} cylinder with diameter 2*a*=40 μm and permittivity *ε*_{d} =3.9 *ε*_{0}, without and with graphene cloak (i.e. graphene-wrapped microtube). Here, the Fermi energy of graphene is varied to show its great tunability on the design frequency. The phases of magnetic fields on the E plane (B) with and (C) without the graphene cloak with Fermi energy *E*_{F} =0.25 eV are shown at the operating frequency *f*_{0}=1.45 THz.

Figure 3A considers, as an example, a dielectric cylinder covered by a graphene-microtube cloak [70], whose scattering response can be calculated using the Lorentz-Mie scattering theory with the impedance boundary condition. The induced averaged surface current on the graphene sheet is proportional to the discontinuity of tangential magnetic fields: ${H}_{\mathrm{tan}}{|}_{r={a}^{+}}-{H}_{\mathrm{tan}}{|}_{r={a}^{-}}=\sigma (\widehat{r}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{E}_{\mathrm{tan}}{|}_{r=a}),$ where *a* is the radius of the dielectric cylinder. For an isotropic surface with negligible cross-polarization coupling, such as a graphene monolayer, the total scattering width (SW), as a quantitative measure of the overall visibility of the object at the frequency of interest, can be computed as the sum of Mie scattering coefficients. In the quasi-static limit, the closed-form cloaking condition for a dielectric cylinder under the TM-polarized illumination can be derived as *X*_{diel}=2/[*ωa*(*ε*_{d} −*ε*_{0})], where *ε*_{d} and *ε*_{0} are permittivities of dielectric cylinder and background medium, respectively. Obviously, an inductive surface, such as the doped graphene monolayer, is required for efficiently suppressing the scattering from a moderate-size dielectric object, under the illumination of THz waves. Figure 3A also presents the total SW for an infinite SiO_{2} cylinder with permittivity *ε*_{d} =3.9 *ε*_{0} and diameter 2*a*=40 μm, covered by a graphene-wrapped microtube with different Fermi energies; here, an uncloaked SiO_{2} cylinder is also presented for comparison (gray dash line). From Figure 3A, we see how significant scattering reduction can be achieved by using an ultrathin and conformal graphene cloak. The cloaking frequency may be widely tuned by varying the Fermi energy of graphene, thus realizing a tunable and switchable cloaking device. Around the operating frequency, it may be possible to tailor the total SW by over two orders of magnitude, simply through the variation of Fermi energy. Figure 3B and C show, respectively, the near-field phase contours of the magnetic field on the E plane for a cloaked (*E*_{F} =0.25 eV) and uncloaked SiO_{2} cylinder at the operating frequency *f*_{0}=1.5 THz. In both cases, the plane wave excites the geometry from the left side and contours are plotted on the same color scale. It is found that for all positions around the cylinder, significant scattering reduction and restoration of original phase fronts can be obtained when compared with an uncloaked dielectric cylinder. It is also worth mentioning that the electromagnetic radiation can enter the graphene cloak, thus enabling an “invisible” sensor for applications in the low-noise, cross-talk-free THz sensing, imaging, communication, and spectroscopy systems, as well as wireless networks, such as inter/intra-chip ultrafast links, indoor systems, and internet-of-nanothings highlighted in Refs. [71], [72], [73].

On the other hand, for a conducting cylinder with moderate cross-section covered by a thin spacer, the following cloaking condition is obtained, generalizing the results of Ref. [69]: *X*_{cond}≅−*ωμ*_{0}*a*[(*γ*^{2}−1)/(*γ*(*γ*^{2}+1))], where *γ* is the radius ratio between the graphene cloak and the conducting cylinder and *μ*_{0} is the permeability of background medium. As observed from this formula, an ultrathin cloak with capacitive surface reactance is required to effectively reduce the scattering from a conducting cylinder. A nanostructured graphene monolayer, such as a graphene-nanopatch metasurface [69], can have a surface reactance tuned from inductive to capacitive, as a function of the graphene’s kinetic inductance and the geometric patch capacitance. This dual-reactive property may allow for cloaking both dielectric and conducting objects. The graphene cloak may also be exploited to conceal a 3D dielectric/conducting object, such as a dielectric spherical particle [74], [75] or a more complicated geometry like 3D finite wedges [76]. Moreover, the multiband operation and cloaking of larger objects may also be conceivable by using multiple graphene layers, which offer more degrees-of-freedom and allow suppressing a larger number of scattering orders. A theoretical discussion on the possibility of cloaking a dielectric sphere with multiband operation can be found in Ref. [74].

The research on tailoring light scattering properties using graphene has initiated a substantial number of studies dedicated to graphene nanostructures, such as ribbons [22], [24] and disks [64], [77], [78], of which finite size effects and boundary conditions are important, yielding new electromagnetic phenomena and opening up the way of dynamically modulating the amplitude, phase, and polarization states of THz and infrared radiation over an ultrathin impedance surface. The tunable electromagnetic response is one of the most exciting areas in current plasmonic and metamaterial research [79], [80], since it may add a large degree of control and flexibility to the exotic electromagnetic properties. Here, we note two main features of the graphene-based metamaterial and metasurface: (1) the large kinetic inductance yielded by subwavelength confinement of surface plasmons in the structured graphene may help scaling the size of metamaterial inclusions to deeply subwavelength, thus enhancing the homogeneity and granularity of artificial materials, and (2) the field-effect-tuned surface impedance can be achieved, as the dynamic conductivity of graphene is controlled by its carrier concentration.

Figure 4A presents a metasurface formed by a collection of specific graphene surface inclusions (e.g. a lithographically patterned graphene). Each graphene surface inclusion, when illuminated by external radiation, may generate a localized or mixed-type semilocalized SPPs, leading to the strong resonant scattering. The scattering fields (**E**_{s}, **H**_{s}) are responsible for the localized, field-dependent surface current ${k}_{s}~\sigma {E}_{0}^{\mathrm{tan}}$ inside each graphene inclusion. The polarizability *α* and the equivalent electric dipole moment per unit length **p** (parallel to the surface) are given by $p=(-j/\omega ){\displaystyle {\int}_{{S}_{in}}{k}_{s}ds=\alpha {E}_{0}^{\mathrm{tan}}},$ where *σ* and *S*_{in} are the conductivity and the enclosed area, respectively, of each graphene inclusion [81]. In the metasurface configuration (Figure 4A), the induced dipole moment of an inclusion, considering the interdipole interactions via the dyadic Green function **G** [82], can be expressed as

Figure 4: (A) Schematics of a planar array of doped graphene nanodisks of diameter *D*=60 nm and Fermi energy *E*_{F} =0.4 eV and (B) the corresponding absorption for (left) an asymmetric interface with *ε*_{1}≠*ε*_{2} and (right) a symmetric interface with *ε*_{1}=*ε*_{2}; the lower-index medium has relative permittivity ε_{1}=1. (C) Scanning electron microscopy image for nanofabricated infrared metamaterials made of multilayer circular graphene patches. (D) Measured electromagnetic wave rejection ratio of the graphene metamaterials in the MIR and far-infrared spectrum [41].

$$p={p}_{0}\widehat{x}=\frac{{E}_{0}^{\mathrm{tan}}}{{\alpha}^{-1}-{\displaystyle \sum _{\left({N}_{x}\text{,}{N}_{y}\right)\ne (0,0)}G\left({r}_{{N}_{x}{N}_{y}}\right)\cdot \widehat{x}\cdot \widehat{x}}.}.$$(6)

The equivalent surface impedance of the metasurface, defined as the ratio of local electric field to surface current density ${J}_{s}={J}_{s}\widehat{x}=j\omega p/\left({d}_{x}{d}_{y}\right)$ is given by *Z*_{s} =*E*_{0}/*J*_{s} −*η*_{0}/2, where *η*_{0} is the free space impedance and *d*_{x} , *d*_{y} are the lattice spacing of metasurface. Figure 4B shows the spectral absorption of a graphene metasurface depicted in Figure 4A, considering the dipole-dipole mutual coupling between adjacent graphene inclusions. It is seen that despite its ultrathin profile, the graphene metasurface provides a perfect infrared absorption. Further, the peak frequency may be wideband tunable via the electrostatic gating, thus being of interest for a variety of optical applications, such as tunable filters and absorbers. We also note that in a symmetric environment (e.g. a graphene immersed in the uniform medium), a maximum absorption of 50% can be achieved, which is consistent with what is predicted from the optical theorem [24]. The perfect absorption can be achieved by considering the asymmetric environment (e.g. a dissimilar dielectric interface with *ε*_{1}≠*ε*_{2} or a patterned graphene on top of a metallic ground plane [77], [78], [83], [84]).

Recent progress in the growth and lithographic patterning of large-area epitaxial graphene presents great opportunities for the practice of graphene-based THz and infrared metamaterials. Figure 4C shows a graphene-nanopatch metamaterial recently invented by IBM Corporation [41]. Hence, wafer-scale, large-area graphene metamaterials have been successfully fabricated and characterized. To realize frequency selective optical properties, the plasmonic resonances are introduced in a graphene-insulator stack by lithographically patterning it into microdisks arranged in a triangular lattice. Figure 4D shows the measured rejection ratio (extinction in transmission) spectrum for the graphene metamaterial in Figure 4C with different dimensions; here, the transmission extinction is defined as 1−*T*/*T*_{0}, where *T* and *T*_{0} are the transmission through the quartz substrate with and without the graphene-insulator stack, respectively. A measured peak transmission of ~80% is achieved with a stacked device made of five graphene layers. It is evident that by changing the dimensions of the fabricated graphene patch and lattice constant, the resonant peak can be adjusted in the MIR spectrum. When the number of graphene layers is increased, the peak intensity increases significantly and the resonance frequency upshifts [41].

Another geometry of interest in our investigation is based on lithographic graphene nanoribbon (GNR) arrays shown in Figure 5A [64], where conduction currents are confined within the *disjoint* GNRs with strong reactive coupling between neighboring plasmonic inclusions that have subwavelength width *d* and period *p*. We note that according to the semiempirical results [6], the bandgap of a GNR with width *p*−*d*>500 nm is not open yet. Thus, the previously derived surface conductivity for a graphene monolayer is still valid. One can analytically solve the problem using the averaged boundary condition, for which a discontinuity on the tangential magnetic field on the metasurface is related to the averaged surface current by the equivalent surface impedance *Z*_{s} [82]:

Figure 5: (A) Schematics of THz metaferrite composed of a planar array of GNRs on top of a SiO_{2}/p^{+}-poly-Si slab backed by a metallic backgate, which can be equivalent to a ferrite with relative permeability *μ*=*μ*′−*jμ*″ backed by a conducting plane, as shown in the bottom panel. (B) Frequency dispersion of effective permeability for a graphene metaferrite with different biased Fermi energy (solid line: real, dash line: imaginary). (C) Absorption spectra for the graphene metaferrite of (A). The thickness of =SiO_{2}/p^{+}-poly-Si is 25 nm/2 μm, and the dimensions of nano-patch are period *d*=1 μm and gap *d*/*p*=0.25.

$$\begin{array}{l}{E}_{x}^{-}=-i\frac{{\eta}_{\text{eff}}}{2}{\alpha}_{ABC}\left[{\sigma}_{g}\frac{p}{p-d}+\left(1+\frac{1}{{k}_{eff}^{2}}\frac{{\partial}^{2}}{\partial {x}^{2}}\right)\right]({H}_{y}^{-}-{H}_{y}^{+})={Z}_{s}^{\text{TM}}{J}_{x}\\ {E}_{y}^{-}=-i\frac{{\eta}_{\text{eff}}}{2}{\alpha}_{ABC}({H}_{x}^{-}-{H}_{x}^{+})={Z}_{s}^{\text{TE}}{J}_{y},\end{array}$$(7)

where *α*_{ABC} is called grid parameter and *ε*_{eff}, ${\eta}_{eff}=\sqrt{{\mu}_{0}/{\epsilon}_{eff}{\epsilon}_{0}},$ and ${k}_{eff}=\omega \sqrt{{\mu}_{0}{\epsilon}_{eff}{\epsilon}_{0}}$ are the relative permittivity, wave impedance, and wave number of the effective host medium [82], respectively. The superscripts TE and TM in Eq. (7) represent the transverse-magnetic and transverse-electric incident waves, respectively. The surface reactance of metasurface Im[*Z*_{s} ] may be tuned from inductive to capacitive, as a function of the tunable kinetic inductance of GNRs and the geometric capacitance [66]. It has been shown in Ref. [66] that the effective surface inductance/capacitance can be derived from Eq. (7), and the intriguing dual-reactive property on a single metasurface may provide many new possibilities for tailoring the THz and MIR scattering properties [66]. The dual-reactive property of graphene metasurface also introduces an extra pole and zero in the Lorentzian dispersion of surface impedance, which may enable the enhanced, multiband, or broadband extinction and scattering [78], as well as the possibility of cloaking both dielectric and conducting objects using the same graphene metasurface (but varying *E*_{F} ) [66].

Figure 5A considers a GNR array separated from the metallic ground plane by a thin dielectric layer. Such structure exhibits interesting electromagnetic properties, which are equivalent to a grounded magnetic-metamaterial slab (e.g. arrays of split-ring resonators) with effective permeability being negative [78], [84], [85], [86]. The grounded dielectric slab with a subwavelength thickness offers a magnetic inductance [87] as a function of the height of dielectric slab, which may resonate with the geometric capacitance and the kinetic inductance of the GNR array. Therefore, the frequency dispersion of this graphene metamaterial can be tailored by varying either the period/gap and carrier density of GNRs (which tunes the geometric capacitance and kinetic inductance), as well as the dielectric constant and thickness of the dielectric slab (which tunes the substrate-induced inductance). In addition, the metallic and dielectric layers may enable the electrostatic back-gating, leading to a tunable THz and infrared metamaterial. Figure 5B shows the frequency dependency of effective permeability retrieved for the graphene magnetic metamaterial (or metaferrite) in Figure 5A under normal incidence. The solid and dashed lines represent the real and imaginary parts of effective permeability, respectively. The geometry of GNRs are *g*=0.5 μm and *a*=4 μm, and the dielectric slab is made of SiO_{2} with thickness *t*=1 μm. From Figure 5B, it is seen that the Lorenzian resonance in the effective permeability can be tuned from low or negative to positive. A low reflection corresponding to a high absorption can be accomplished, provided that the equivalent permeability *μ*=*μ*′−*jμ*″ is tailored to the optimum value for a ground-back magnetic-film absorber: *μ*′≈0 and *μ*″≈1/*k*_{0}*t* [78], where *k*_{0} and *t* are the free space wave number and the thickness of graphene metaferrite, respectively. Figure 5C presents the absorption spectra for the graphene metaferrites in Figure 5B, showing large and tunable absorption with respect to the doped Fermi energy of graphene. The GNR-based metamaterial may be of interest for a variety of long-wavelength infrared applications, including the artificial magnetic conductor, ultracompact Salisbury absorbing films [88], polarization filters [84], and selective thermal emitters or absorbers [89]. Recent experimental works have demonstrated that the THz extinction of a planar GNR array can be tailored by varying its geometry or by electrostatically gating GNRs [41].

In addition to their scattering properties, nanostructured graphene also exhibits fascinating properties from the in-plane point of view. Specifically, it is able to implement ultrathin metasurfaces able to support different topologies, from isotropic to hyperbolic, and going through the *σ*-near-zero (canalization) regime [43], [44], [46], [90] (see Figure 6). Hyperbolic metasurfaces have recently been introduced using graphene nanostructures – i.e. ultrathin 2D materials – at THz and far-infrared (FIR) frequencies [43] and patterned silver surfaces at optics [91], and they provide very exciting properties, such a relatively simple fabrication using standard techniques, resilience to volumetric losses, compatibility with integrated circuits, and an easy external access to the propagating surface wave using near-field techniques. Importantly, hyperbolic metasurfaces support the propagation of low-loss and extremely confined surface plasmons able to strongly interact with the surrounding media. In addition, these structures provide large degrees of freedom to manipulate the supported plasmons, enabling advanced functionalities, such as canalization, negative refraction, dramatic enhancement of light-matter interactions, or the routing of plasmons toward specific directions within the layer. Let us assume an infinitesimally thin homogeneous anisotropic metasurface described by conductivity tensor $\overline{\overline{\sigma}}=\text{diag(}{\sigma}_{xx},\text{\hspace{0.17em}}{\sigma}_{yy}\text{)},$ where the different components may be in general complex, and the structure has been assumed to be passive, free of magneto-optical and nonlocal effects, and aligned along our reference system. By investigating the sign of Im[*σ*_{xx} ] and Im[*σ*_{yy} ], the metasurface topology can easily identified and engineered. For example, Figure 6A shows an *isotropic elliptic topology* (Im[*σ*_{xx} ]<0, Im[*σ*_{yy} ]<0), where the excited transverse magnetic SPPs propagate with similar characteristics across the entire layer. A natural example of isotropic metasurface is graphene, an inductive 2D material where Im[*σ*_{xx} ]=Im[*σ*_{yy} ]<0. We should note that unbalancing the conductivity components, but keeping their inductive (metallic) nature, leads to designs able to favor the propagation of SPPs toward one direction over others [43]. Figure 6B shows a metasurface that implements a *hyperbolic topology*, which arises when the structure behaves as a metal (inductive, Im[*σ*]<0) along one direction and as a dielectric (capacitive, Im[*σ*]>0) along the orthogonal one, i.e. sgn(Im[*σ*_{xx} ])≠sgn(Im[*σ*_{yy} ]). The last topology considered here is the *σ-near-zero regime* (see Figure 6C), which appears at the metasurface topological transition [43] and canalizes the propagating energy towards a specific direction. It is important to mention that, even though ideal hyperbolic metasurfaces are expected to provide infinite wave confinement and local density of states, their response is limited in practice by the presence of losses, nonlocality arising due to the periodicity (in case of patterned configurations) and the intrinsic spatial dispersion of the composing materials [45]. An array of densely packed graphene strips [43] (see inset of Figure 6A–C) is an ideal platform to implement any metasurface topology at THz and far infrared frequencies, with the important advantage to provide inherent reconfiguration capability to the resulting structure. The in-plane effective conductivity tensor of such metasurface can be homogenized as ${\sigma}_{xx}^{\text{eff}}=L\sigma {\sigma}_{C}/(W{\sigma}_{C}+G\sigma ),\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}\hspace{0.17em}}{\sigma}_{yy}^{\text{eff}}=\sigma (W/L),\text{\hspace{0.17em}}\text{\hspace{1em}}\text{\hspace{0.17em}}{\sigma}_{xy}^{\text{eff}}={\sigma}_{yx}^{\text{eff}}=0,$ where *σ*_{C} =*jωε*_{0}*ε*_{eff}(*L*/*π*)ln[csc(*πG*/2*L*)] describes the near-field coupling between adjacent strips [69], *W* and *G* are the strips width and separation, and *L* is the unit-cell period. We should stress that the homogeneous condition *L*≪*λ*_{SPP} must be satisfied for such equations to be valid, where *λ*_{SPP} is the plasmon wavelength. An example of this homogeneous model, which takes into account that wave propagation is inductive (capacitive) along (across) the strips, is illustrated in Figure 6D. This figure shows that the proposed structure can indeed implement different topologies versus frequency. Rigorous full-wave simulations shown in Figure 6E confirm the ability of the nanostructured graphene to support hyperbolic plasmons, while dramatically enhancing the spontaneous emission rate of emitters located nearby (Figure 6F). The graphene-based reconfigurable hyperbolic metasurfaces may pave the way toward the development of miniaturized devices able to extremely confine and dynamically guide light over ultrathin surfaces, with direct applications in communications, lensing, and sensors.

Figure 6: Hyperbolic, isotropic, and *σ*-near zero metasurfaces implemented by nanostructured graphene.

(A)–(C) Color maps illustrate the *z*-component of the electric field excited by a *z*-directed emitter (black arrow) located 10 nm above the metasurface [44]. Insets present the isofrequency contour of each metasurface (top) and a possible graphene-based implementation (bottom). (A) Isotropic metasurface. (B) Hyperbolic metasurface. (C) *σ*-near zero metasurface. (D) Example of the in-plane effective conductivity tensor realized by a densely packed array of graphene strips, implementing several types of topologies at different frequencies [43]. (E) Full-wave verification of hyperbolic plasmons excited over nanostructured graphene [44]. (F) Dramatic enhancement of the spontaneous emission rate (in log scale) of a dipole versus its distance over a hyperbolic metasurface [45]. Results are computed considering (nonlocal) and not (local) considering graphene’s inherent spatial dispersion.

As previously pointed out, graphene has been experimentally demonstrated to support the surface wave at the THz and long-wavelength infrared wavelengths, with moderate loss and strong field localization. The extremely confined surface waves sustained by a graphene sheet may be of interest for passively guiding and actively gate-tuning/modulating THz and infrared surface waves. The dynamically tunable dispersion relation and SPP wavelength of propagating surface waves may be expected to realize flatland SPP waveguides and transformation optics [19], [92]. The dispersion relation of SPP modes in graphene can be computed by matching the *two-sided impedance boundary*, which, for TM surface waves illustrated in Figure 7A, can be expressed as

Figure 7: (A) Schematics of a propagating SPP (surface) wave on graphene (left) and the eigenmode dispersion: surface-wave wavenumber *k* and attenuation length *ζ* for a graphene monolayer with Fermi energy of 0.1 eV at room temperature for both TE and TM surface waves (right) [13]. (B) Schematics of a GPPWG and a simplified compact design inspired by the field symmetry of quasi-TEM mode (left). The complex phase constant of a GPPWG with thickness *d*/2=50 nm, supporting a quasi-TEM mode, is shown at the right side. Lines and symbols represent the eigenmodal solutions calculated by Eqs. (12) and (13), respectively.

$$\frac{{\epsilon}_{2}}{{\epsilon}_{1}}\sqrt{{\beta}^{2}-{\text{k}}_{1}^{2}}+\sqrt{{\beta}^{2}-{\text{k}}_{2}^{2}}+\frac{\sigma}{j\omega {\epsilon}_{1}}\sqrt{{\beta}^{2}-{\text{k}}_{1}^{2}}\sqrt{{\beta}^{2}-{\text{k}}_{2}^{2}}=0,$$(8)

whereas for TE, the dispersion equation is

$$\frac{{\mu}_{2}}{{\mu}_{1}}\sqrt{{\beta}^{2}-{\text{k}}_{1}^{2}}+\sqrt{{\beta}^{2}-{\text{k}}_{2}^{2}}+j\omega {\mu}_{2}\sigma =0,$$(9)

where *β* is the complex phase constant of surface wave and *k*_{i} , *ε*_{i} , and *μ*_{i} are the wave number, permittivity, and permeability of the *i*th medium, respectively. The type of surface waves supported on the monolayer depends on both signs and value of imaginary part of graphene’s conductivity. In general, only modes on the proper Riemann sheet may provide meaningful physical wave phenomena, whereas leaky modes on the improper sheet can be used to approximate parts of the spectrum and to explain certain radiation phenomena [13], [14], [15]. Considering a freestanding graphene in vacuum (*ε*_{1}=*ε*_{2}=*ε*_{0} and *μ*_{1}=*μ*_{2}=*μ*_{0}), the eigenmodal solutions for Eq. (8) can be explicitly expressed as [13]

$$\beta =\sqrt{{k}_{0}^{2}-{\left(\frac{\text{2}\omega {\epsilon}_{0}}{{\sigma}^{2}}\right)}^{2}},$$(10)

whereas the similar explicit solution for Eq. (9) is

$$\beta =\sqrt{{k}_{0}^{2}-{\left(\frac{\omega {\mu}_{0}\sigma}{2}\right)}^{2}}.$$(11)

From Eqs. (10) and (11), we notice that if Re[*σ*] is small and Im[*σ*]<0 a TM-mode slow surface-wave exists at THz, FIR, and MIR wavelengths. In this case, the strongly confined TM mode arises from the plasmonic property of graphene with negative Im[*σ*], (see Figure 2). However, if Im[*σ*] is positive, a leaky mode on the improper Riemann sheet is obtained for the TM-mode surface wave. On the other hand, at wavelengths where Im[*σ*]>0, the TE-mode surface wave can exist in the MIR region (see Figure 2). Unlike strongly localized TM-mode surface waves, TE-mode surface waves are only weakly localized at the surface. Besides, the plasmon losses and optical phonon scattering in the infrared spectrum may limit the propagation length of TE-mode surface waves.

Figure 7A shows (left panel) the schematic of a propagating SPP on graphene/insulator interface and (right panel) contours of real and imaginary phase constant (normalized by the free space wave number) for TM surface waves, varying the operating frequency and the Fermi energy of graphene. At very low photon energy, the TM surface wave is relatively fast (*β*≅*k*_{0}) and poorly confined to the graphene surface. In the THz to MIR region, the surface wave is strongly confined and becomes slow (*β*≅*k*_{0}), as energy is concentrated in the near field of graphene surface. We note that around the interband transition threshold (*ℏω*=2|*E*_{F} |), a moderate change in Fermi energy can dramatically change the sign of imaginary conductivity from negative to positive, thereby supporting TE surface waves. In the range of Γ≪*ω*<2|*E*_{F} |/*ℏ*, only a strongly confined TM surface wave propagates. This TM mode is of particular interest since it is dispersive with the Fermi energy in the frequency range of interest. This allows one to tune and modulate the dispersion of surface waves. In contrast, the TE mode is poorly confined to the graphene sheet and it is essentially nondispersive. Since the TE and TM modes do not coexist, this allows one to make a wideband-tunable graphene polarizer, supporting the TE surface wave propagation for frequencies above the interband transition threshold [13], [14], [15].

Vakil and Engheta have theoretically shown that by designing inhomogeneous and nonuniform conductivity patterns across a graphene sheet [19] (e.g. using electrostatic gating), the flatland transformation optics devices can be realized. In Ref. [19], several numerical examples have been presented to demonstrate the infrared modulation and transformation functions. In Ref. [92], a flatland Fourier optics lens has been numerically studied for transforming a broadcasting surface wave from the source into perfect planar wavefronts on a graphene monolayer. So far, several groups have experimentally demonstrated the gate-tuning of SPP waves on a graphene ribbon by using the scatter-type scanning near-field microscopy. These measurements were based on the feedback scheme taken from atomic force microscopy, where the nanomanipulated nanotip is positioned and scanned in the immediate vicinity of graphene surface. The strong local field confinement and gate-tuning standing-wave field patterns due to the interference in a finite-length graphene ribbon have been successfully characterized in Refs. [27] and [28]. We believe that the advancement of nanotechnology and nanofabrication will benefit the rapidly expanding field of graphene plasmonic devices.

Traditionally, a subwavelength parallel-plate plasmonic waveguide, such as the metal-insulator-metal (MIM) heterostructure, can provide better control and confinement of the guided SPP modes [14]. Figure 7B (upper panel) shows a graphene parallel-plate waveguide (GPPWG), constructed with two parallel graphene sheets with widths much larger than the separation distance *d* (with the propagation axis oriented along the *z*-axis). In many senses, the propagation properties of this GPPWG are analogous to those of a MIM plasmonic waveguide, but suitable in the sub-THz to MIR wavelengths. For TM_{z} waves with magnetic field *H*_{x} (*y*) exp[*j*(*ω*_{t} −*β*_{z} )], the complex eigenmodal phase constant *β* can be evaluated by solving the dispersion equation [14], [53]:

$$\begin{array}{l}\text{tanh}[\sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{2}}h/2]\frac{\sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{2}}}{\sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{1}}}\\ \text{\hspace{1em}}=-\frac{{\epsilon}_{2}}{{\epsilon}_{1}-j\sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{1}}\frac{\sigma}{\omega}},\end{array}$$(12)

where *ε*_{2} and *ε*_{1} are the material permittivity inside and outside the waveguide, respectively. When the separation between the graphene monolayers is reduced to a deep-subwavelength scale *h*≪*λ*_{0}, the graphene waveguide supports a quasi-transverse electric-magnetic (TEM) mode [14], [53]. Under the long-wavelength approximation $h\ll \mathrm{min}\left(2\pi \text{/}\right|\omega \sqrt{{\epsilon}_{2}{\mu}_{0}}|,\text{\hspace{0.17em}}2\pi /|\omega \sqrt{{\epsilon}_{1}{\mu}_{0}}\left|\right)$ and $\sigma \sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{2}}/\omega {\epsilon}_{2},$ $\sigma \sqrt{{\beta}^{2}-{\omega}^{2}{\mu}_{0}{\epsilon}_{1}}/\omega {\epsilon}_{1}\gg 1,$ a simple explicit dispersion relation can be obtained as [14]

$$\beta /{k}_{0}\cong \sqrt{\frac{{\epsilon}_{2}}{{\epsilon}_{0}}\left(1-j\frac{2}{{\mu}_{0}\omega \sigma h}\right)}.$$(13)

As expected, for sufficiently large conductivity, the permittivity of the outer cladding layer has a negligible effect on the complex phase constant *β*, since the mode is tightly confined between two graphene layers. For the quasi-TEM mode that, in principle, has no cutoff frequency, the longitudinal field *E*_{z} is nonzero but very small compared to the uniform transverse field *E*_{x} , provided that the waveguide dimension (*h*) is very small. Due to the symmetry of fields for the quasi-TEM mode, the GPPWG can be replaced by a half-size waveguide in Figure 7B (bottom panel), where a single graphene sheet is separated from a conducting ground plane by a half-thickness dielectric layer. Figure 7B (right side) presents the normalized phase constant *β*/*k*_{1} for the graphene waveguide shown in Figure 7B (left side), filled with a 50-nm-thick SiO_{2} (*ε*_{2}=4*ε*_{0} and *d*/2=50 nm) in a background of air (*ε*_{1}=*ε*_{0}). It can be seen that in the low-THz region, the quasi-TEM mode is relatively nondispersive, supporting a slow-wave propagation with strongly confined THz waves inside the waveguide, and a guided wavelength much smaller than the free-space wavelength *λ*_{g} ≪*λ*_{0}, thanks to the large kinetic inductance of graphene. For frequencies above sub-THz, the attenuation constant becomes small and the group velocity is almost constant without dispersion. In this case, a low-loss transmission line with moderate signal attenuation and dispersion is obtained [6]. From Figure 7B, it is seen that the phase constant of graphene waveguides is tunable by shifting the Fermi level of graphene, enabling reconfigurable transmission lines at sub-THz and THz frequencies. It is worth mentioning that at RF and microwave frequencies, some recent experiments have shown that graphene-based transmission lines may make a compact attenuator or terminator [93].

The transmission line model and transfer matrix method can be used to evaluate characteristics of THz/infrared components constituted by single/integrated graphene waveguides. The characteristic impedance of the graphene waveguide shown in Figure 7B can be defined as *Z*_{c} =−*E*_{y} /*H*_{x} =*β*/*ωε*_{2}, By combining the graphene waveguide in Figure 7B with a double gate (or the hybrid graphene/metal [back gate] waveguide configuration), the THz conductivity of graphene can be controlled by the applied gate voltage, thereby adjusting dynamically the propagation constant, phase velocity, and local impedance of a transmission line segment. It can be intuitively understood that the phase constant and characteristic impedance of a graphene transmission line can be tuned from relatively high (e.g. a pristine graphene with a small *E*_{F} ) to low (e.g. a biased graphene with a large *E*_{F} ). This is arguably the most significant advantage of graphene over conventional noble-metal plasmonic waveguides, providing an exciting venue to realize electronically programmable THz transmission lines, similar to their microwave counterparts realized by microstrip-lines loaded with varactors, diodes, or transistors.

Figure 8A illustrates an integrally gated transmission line based on the quasi-TEM GPPWG, backed by a metal gate. As an example to illustrate the potential of such reconfigurable THz transmission line, the device in Figure 8A can be used as a loaded-line phase shifter designed for THz phased arrays or THz modulators, with relatively low insertion loss, low return loss, and small phase error. In practical designs, the return loss is inherently present due to the reflection at the mismatched loaded-line. The design in the inset of Figure 8B consists of a 3-bit phase shifter with eight phase shift states: 0°/45°/90°/135°/225°/270°/315° [53], [94]. When gate voltages are applied, the length of each loaded section must be a multiple of half guided wavelength such that the impedance mismatch and return loss can be minimized. In order to create impedance matching for all binary states, the *i*th transmission line segment must satisfy the conditions *β*_{bias}*l*_{i} =*π*/2 and *β*_{unbias}*l*_{i} −*β*_{bias}*l*_{i} =Δ*ϕ*_{i} , where *l*_{i} and Δ*ϕ*_{i} are defined as the gate length and relative phase shift between the unbiased and biased conditions for the *i*th section, respectively. The total phase shift is therefore obtained as Δ*ϕ*=Δ*ϕ*_{1}+Δ*ϕ*_{2}+Δ*ϕ*_{3}. In order to satisfy these conditions, the required Fermi energy and length for each bits need to be properly designed. For instance, if we want to produce phase shifts of 45° (bit 0), 90° (bit 1) and 180° (bit 2), the propagation constants must satisfy the relationships *β*_{1}=4/5 *β*_{0}, *β*_{2}=2/3 *β*_{0} and *β*_{3}=1/2 *β*_{0}, and the associated length of each line must satisfy the condition *β*_{bias}*l*_{i} =*π*/2. Figure 8B and C show numerical results for the magnitude of *S*_{21} and phase shifts of 45°, 90°, 180°, and 315°, respectively. These phase angles are obtained by applying the proper voltages to gate G1, G2, G3, and all, respectively. It is seen that good input return loss is obtained, with desired relative phase shifts at the design frequency *f*_{0}=1.5 THz. This graphene phase shifter may be applied to sub-THz, THz, and infrared high-gain, steerable phased-array antennas. The gated graphene waveguides, with tunable and strongly confined mode, may enable a number of chip-scale THz nanocircuit components, including switching, matching, coupling, power dividing or combining, and filtering devices [94].

Figure 8: (A) Schematics of a hybrid electronic-plasmonic (plasmoelectronic) nanodevice, comprising a graphene field-effect-transistor (GFET) and a graphene plasmonic waveguide (GPWG). (B) Integrally gated graphene transmission line with digitized, spatially varying phase velocity and characteristic impedance. Such device may realize, for example, a 3-bit graphene-based phase shifter using three integrated gates. The simulated phase shift versus frequency shows accurate angles at the design frequency (5 THz). (C) Transmission (dashed) and reflection (solid) for the device in (B), showing moderate insertion and return losses.

In this context, it is important to emphasize that the response of graphene-based components, such as the aforementioned phase shifters or in-plane low-pass filters [95], can be strongly influenced by the intrinsic nonlocality (or spatial dispersion) of graphene [15], [54], [96], [97], [98]. This phenomenon arises due to the finite Fermi velocity of electrons in graphene, which are unable to follow the expected fast variations of the supported plasmons. In order to model this phenomenon, advanced wavenumber-dependent *σ*(*k*_{ρ
} ) conductivity models of graphene have been derived [18], [96], [99]. Figure 9 illustrates the influence of spatial dispersion on phase shifters and a seventh-order low phase filters [54] implemented using the double gated graphene waveguides described above, and studies their responses as a function of the surrounding media. In this analysis, we first model graphene using the local description and then using a more accurate graphene model that takes nonlocal effects into account [96]. The results reveal that the intrinsic nonlocality of graphene upshifts the operation frequency of plasmonic components, limiting their reconfiguration capabilities, and degrading their overall response. More importantly, it is also found that the influence of spatial dispersion increases with the permittivity of the surrounding media, which is associated to the higher confinement of the supported plasmons in such cases. The study in [97] clearly confirms that graphene nonlocality must be rigorously taken into account in the development of plasmonic THz in-plane components.

Figure 9: Influence of spatial dispersion in the response of graphene-based digital load-line phase shifters (top row) and low-pass filters (bottom row) [54] implemented by double-gated graphene parallel-plate waveguides.

The analysis of the phase shifter includes (A) phase difference between ports, (B) scattering parameters, and (C) phase error due to spatial dispersion versus the permittivity of the surrounding media. The 7th-degree low-pass filter is designed and analyzed in (D) free space and (E) embedded in Si (*ε*_{r}≈11.0). The error in the cutoff frequency and maximum in-band reflection due to graphene nonlocality as a function of surrounding permittivity is shown in panel (F). Solid line: results neglecting spatial dispersion effects; dashed line: results including spatial dispersion effects. Further details of the devices can be found in [54].

The integration of several active graphene-based THz nanocircuit components, including a graphene THz antenna [47], [48], [49], [100], [101], into a single entity will present a fundamental step towards design architectures and protocols for innovative THz communications, biomedicine, sensing, and actuation [70], [71], [72]. In addition, a plasmonically resonant graphene patch is expected to realize a frequency-reconfigurable, electrically small THz antenna with moderate radiation efficiency and directionality [47], [48], [49], [100], [101]. It can be implemented either in planar or in cylindrical configuration, as shown in Figure 10A and B. These configurations [47], [50], [52] realize electrically small THz antennas with tunable properties, providing large tunability of the resonant frequency, while keeping constant a very high antenna input impedance. Very importantly, such large values permit very good matching with photomixers and other embedded optical sources that usually present very high input impedances, thus boosting the overall antenna radiation efficiency, as can be seen in Figure 10C. These resonant configurations are very promising to enhance the response of future THz communication and sensing systems.

Figure 10: Graphene-based reconfigurable resonant antennas excited by a photomixer or a plasmonic optical source at its gap (from [47], [50], [52]).

(A) Planar structure composed of two graphene self-biasing patches. (B) Graphene cylindrical waveguide. (C) H-plane radiation diagram of the planar antenna at 1.8 THz.

In order to further control the direction of the radiated beam, a graphene-based THz leaky-wave antenna for electronic beamscanning was recently proposed [102], as shown in Figure 11A. This THz antenna consists of a graphene monolayer with a set of polysilicon pads located beneath it. Thanks to the gate-tuned conductivity of graphene, the use of multiple cascaded gates may electronically modulate the surface reactance. In the zero bias condition, the graphene monolayer provides a “slow-wave” propagating SPP mode. When the gate electrodes are properly biased, a “digitized” sinusoidal modulation may be applied to the surface reactance of graphene, introducing high-order spatial harmonics that can support a “fast-wave” leaky mode. By varying the modulation periods through different bias setups, the THz beam can be steered over a wide range of scan angles. Another promising structure [51] consists of a width-modulated graphene strip able to couple ultraconfined surface plasmons to free-space propagating waves, while providing customized coupling angle, radiation rate, and exotic beam steering functionalities by simply applying a unique and modest DC bias to the antenna’s ground plane. It is also important to mention that even more advanced designs able to implement nonreciprocal responses can be achieved by exploiting the spatiotemporal modulation of graphene [55], allowing to dramatically modify the radiation pattern of antennas when operating in transmission or reception mode and to, under time-reversal, transmit and receive SPPs oscillating at different frequencies. However, in order to digitize the required sinusoidal surface reactance and reduce the radiation side-lobes, it is necessary to use a large number of gate electrodes underneath the graphene, which increases the fabrication complexity. An alternative solution able to solve such technological challenges is shown in Figure 11B, which employs the exotic acousto-optic effect of graphene.

Figure 11: (A) Graphene-based leaky-wave antenna for electronic beam scanning. By electronically varying the number of bias pads per period (*N*), the THz beam (2 THz) can be scanned in free space. (B) Graphene-based analog leaky-wave antenna for acoustic-based beam scanning, with the angle of radiation as a function of acoustic frequency. Both devices show reconfigurable beamsteering effect at a locked frequency.

It is well known that periodic corrugations on metallic surfaces can generate SPP resonances for various frequencies and angles of arrival of the exciting electromagnetic field. Similar ideas have been applied to excitations of propagating SPPs on a graphene sheet [103], [104]. In order to efficiently couple the incident photon into the propagating SPP mode, metallic gratings with suitable periods are typically required for matching their momentum, ensuring that the incident plane wave from free space can be coupled into the slow surface wave on the graphene sheet. Recently, an acousto-optical approach has been proposed to efficiently couple the incident radiation into propagating SPP mode on the graphene surface. Due to the large Young modulus and extreme thinness of graphene, it is possible to model elastic vibrations of graphene in the transverse direction with the scalar biharmonic equation: $\text{(}{\Delta}^{2}-{\beta}_{b}^{4}+\kappa /D)W=q,$ where *W*, *q*, *β*_{b} , *κ*, and *D* represent the vertical displacement field, source of vibration, flexural wave number, stiffness of the substrate, and the flexural rigidity, respectively [103]. The flexural wave traveling on a graphene surface that satisfies the biharmonic equation can be accurately modeled as the acoustic phonon modes with static grating of period ${\Lambda}_{b}=2\pi /\text{Re}[{\tilde{\beta}}_{b}]$ [103]. The periodic nano-corrugations introduced by elastic vibrations can excite the spatial (Floquet) harmonics, necessary for coupling the infrared incident waves from free space into the propagating SPP modes in graphene. Based on the electromagnetic reciprocity, the acousto-optical effect may realize reconfigurable THz and infrared leaky-wave antennas, which exploit the continuous, sinusoidal diffraction grating on graphene surface to transform the propagating surface wave into a leaky wave, with desired angles of departure, as shown in Figure 11B [105]. Moreover, since the period and depth of nanogratings are controlled by the acoustic frequency, it is possible to steer a directive beam into the far field at a constant infrared frequency, which is very different from conventional leaky-wave antennas that operate based on sweeping the electromagnetic frequency. Since the guided SPP mode excited in graphene is typically a slow wave with complex wave number Re[*β*_{SPP}]>>*k*_{0}, the leaky (radiation) modes require the excitation of negative spatial harmonics, yielding a beam angle as a function of acoustic frequency *ω*_{ac} and electromagnetic waves *ω*, as well as the Fermi energy of graphene *E*_{F} :

$${\theta}_{\text{0}}={\text{sin}}^{\text{\hspace{0.17em}}-\text{\hspace{0.17em}}1}\left(\frac{\mathrm{Re}[{\beta}_{\text{SPP}}(\omega ,\text{\hspace{0.17em}}{E}_{F})]}{{k}_{0}}-N\frac{{\lambda}_{0}}{{\Lambda}_{bh}\left({\omega}_{ac}\right)}\right).$$(14)

For the dominant TM plasmonic mode, *β*_{SPP} can be obtained from the dispersion relation in Eq. (8). It is common to take *N*=−1 spatial harmonic to avoid the undesired grating lobes in the backfire direction. The normalized radiation patterns calculated using a modified array factor method [105] are presented in Figure 11B (right panel), showing that a graphene leaky-wave antenna, with *E*_{F} =0.5 eV, can steer the beam from backfire to endfire at a fixed MIR frequency *ω*/(2*π*)=30 THz, by simply tuning the frequency of flexural wave produced by an acoustic synthesizer (e.g. surface acoustic wave devices), and the results are well predicted by Eq. (14). We note that the switching speed of this acousto-optical device can be quite fast (MHz and GHz modulation speed), which may be of interest for realizing high-speed beamforming and beamsteering antennas fed by guided surface waves.

The interband loss mechanism of graphene becomes dominant at NIR frequencies and surface plasmons cannot be formed and sustained at its surface [18], which is the case for all previously presented examples operating at low THz frequencies. However, it is interesting to note that the properties of graphene can still be electrically controlled [106] even at these high frequencies, due to the strong variation in its carrier density and, as a result, doping level. Therefore, it is expected that its tunable properties will lead to altered scattering or transmitting responses at NIR or even optical frequencies when graphene is ingeniously combined with plasmonic metasurface [107] or nanoantenna [108] structures. However, graphene can only interact with the tangential (in-plane) electric field components of the impinging electromagnetic radiation due to its one-atom thickness. The ultrathin nature of graphene in the out-of-plane direction leads to very weak interaction with the normal (out-of-plane) electric field components of the incident electromagnetic wave.

To this end, an alternative hybrid graphene/all-dielectric metasurface design is presented in Figure 12A to achieve tunable and modulated transmission at NIR frequencies [109]. The presented all-dielectric hybrid metasurface is composed of periodically arranged pairs of asymmetric silicon (Si) nanobars with graphene placed on top of this configuration. The dimensions of the metasurface are highly subwavelength compared with the wavelength NIR radiation. This dielectric metasurface can sustain trapped magnetic resonances with a sharp Fano-type transmission or reflection signature [110]. The strong in-plane electric field distribution at the resonance is computed and shown in the inset of Figure 12B. One-atom-thick CVD graphene can be transferred and placed over this dielectric metasurface using standard transfer techniques [111]. Very strong transmission modulation is obtained at NIR telecom wavelengths when the doping level of graphene is increased, as it can be seen in Figure 12B. The enhanced in-plane fields along the all-dielectric metasurface strongly interact with the tunable properties of graphene. This leads to strong coupling between the incoming electromagnetic radiation and graphene. The transmission amplitude modulation of the proposed structure is presented in a more quantitative way by calculating the difference in transmission between heavily doped (*E*_{F} =0.75 eV) and undoped (*E*_{F} =0 eV) graphene. The absolute value of the transmission difference Δ*T*′=|(*E*_{F} =0.75 eV)−*T*(*E*_{F} =0 eV)| is plotted in Figure 12C as a function of the impinging’s radiation wavelength. Interestingly, the transmission difference Δ*T* (modulation) can reach values higher than 60% at the Fano resonance transmission dip (*λ*=1.59 μm). Note that moderate modulation is also obtained within a narrow wavelength range around the resonance transmission dip. In this hybrid configuration, the in-plane resonant fields interact strongly with the in-plane graphene properties and strong electro-optical modulation is obtained at the transmission spectrum of this device. Several new integrated nanophotonic components are envisioned based on the proposed device, such as efficient ultrathin electro-optical transmission and reflection modulators and switches.

Figure 12: Hybrid graphene/dielectric metasurface configuration.

(A) Unit cell geometry of the proposed hybrid planar modulation device. (B) Strong transmission modulation for different doping levels of graphene. The field enhancement distribution is shown in the inset at the resonance. (C) Transmission difference between undoped and heavily doped graphene. More than 60% transmission modulation can be obtained in a narrow frequency region.

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