An approach to couple free-space waves and non-resonant plasmons propagating along graphene strips is proposed based on the periodic modulation of the graphene strip width. The solution is technologically very simple, scalable in frequency, and provides customized coupling angle and intensity. Moreover, the coupling properties can be dynamically controlled at a fixed frequency via the graphene electrical field effect, enabling advanced and flexible plasmon excitation-detection strategies. We combine a previously derived scaling law for graphene strips with leaky-wave theory borrowed from microwaves to achieve rigorous and efficient modeling and design of the structure. In particular we analytically derive its dispersion, predict its coupling efficiency and radiated field structure, and design strip configurations able to fulfill specific coupling requirements. The proposed approach and developed methods are essential to the recent and fundamental problem of the excitation-detection of non-resonant plasmons propagating along a continuous graphene strip, and could pave the way to smart all-graphene sensors and transceivers.
© 2013 OSA
The important advantages of surface plasmon polaritons  (SPPs) supported by graphene structures over plasmons in conventional materials (e.g. silver and gold), which include relative low-losses , strong light-matter interactions , electronic tunability , and large field confinement, have triggered a plethora of potential applications of graphene plasmonics [5–9] at THz and infrared frequencies. One of the main bottlenecks for the further development of this field is the difficult excitation of SPPs in graphene due to the large momentum mismatch between incoming free-space electromagnetic waves and plasmons. Recently, resonant plasmons in arrays of graphene ribbons  and disks [11, 12] have been experimentally demonstrated. Resonant plasmons have also been proposed to create a patch antenna in the THz band . However, these plasmons arise from resonant electronic oscillations bounded to ribbons or disks, and thus such setups are of little practice to excite SPPs propagating along continuous graphene sheets [14, 15] or strips [2, 9, 16].
There are several mechanisms to enable the coupling of incoming free-space light to guided, non-resonant, plasmons . One possibility consists of using the Otto configuration , composed of a bulk prism of high dielectric constant and two dielectrics cladding a graphene sheet, to excite SPPs by evanescent waves . Another option is to modulate the conductivity of an infinite graphene sheet , which can then be seen as a 1D polaritonic crystal [19, 20]. The required position-dependent conductivity profile in the sheet can be obtained through different techniques, such as inducing an inhomogeneous strain on graphene, using a set of patterned gates to locally modify the conductivity via the electric field effect , or by producing an inhomogeneous distribution of absorbed molecules or atoms in the sheet . Interestingly, the underlying coupling principle of modulating the conductivity of a graphene sheet is analogous to the sinusoidally modulated leaky-wave antennas investigated in the 50s , which have a renewed interest today in the microwave antenna community [22,23]. This concept was also recently applied to propose a graphene-based leaky-wave antenna in 2D surfaces . Other possibilities of exciting SPPs include covering diffraction gratings with a graphene sheet [25, 26] or using a periodic corrugation of the graphene surface . In addition, a novel excitation method based on enhancing the in-plane momentum of the incoming light by using the apex of an illuminated nanotip was recently demonstrated [27, 28].
However, the coupling mechanisms discussed above still present some important practical limitations. First, their integration with current graphene-based devices is challenging. These structures are based either on bulky configurations or on infinite 2D planar sheets, and have to be interconnected with other plasmonic components. This interconnection is difficult because plasmonic devices and all-graphene integrated circuits  require the use of planar and finite-width strips able to confine the propagating energy in a reduced area, thus allowing the use of circuit elements such as transistors. Second, some structures require complex fabrication processes or the use of large biasing networks to control graphene properties, as in the case of conductivity-modulated sheets . Third, the SPPs-light coupling occurs in a unique specific angle with respect to the graphene surface [18, 26]. The possible control of this angle would result in novel functionalities, such as detecting the position in space of a given molecule  or transmitting energy towards a specific spatial region. Fourth, the coupling efficiency of the different mechanisms is low and has not been considered in detail . Finally, the analysis and design of such structures is generally not available in closed-form and the use of time-consuming computational tools is required.
Here, we propose to modulate the width of a graphene strip to couple free-space propagating waves and plasmons on realistic finite-width strips. First, we demonstrate that the guiding properties of plasmons can be largely tuned by modifying the strip width [2, 16]. The relation between plasmon momentum and the width of graphene strips is then exploited to adequately control the guiding properties of the ribbon, thus compensating the momentum mismatch between plasmons and the incoming electromagnetic waves. We supplement the reported coupling phenomenon by providing an analytical formulation of the propagating plasmon based on the scaling law of graphene strips  combined with leaky-wave theory [31, 32], thereby allowing a rigorous understanding, analysis, and design of the structure. In addition, we show that the proposed mechanism permits electrically tuning the plasmon-light coupling angle at a single operating frequency by simply adjusting the Fermi level in graphene. This property provides an additional degree of freedom to selectively excite plasmons at different frequencies by using electromagnetic waves coming from the same spatial region. Moreover, the structure can behave as a smart plasmonic antenna able to radiate energy towards different positions in space at a fixed operating frequency, thereby providing beamscanning capabilities.
Compared to previously discussed plasmon-light coupling mechanisms, the proposed structure is fully planar, technologically very simple, based on a finite-width strip (therefore easy to integrate with current graphene-based plasmonic devices [6, 7, 9, 29]), provides customized coupling patterns in terms of angle and intensity, and is electrically reconfigurable. Possible applications of width-modulated graphene strips include sensors, able to detect incoming radiation from specific chemical or biological processes and determine their location in space, reconfigurable antenna components in all-integrated plasmonic devices [6, 33], high resolution image scanners, intrasatellite communications systems  or THz transceivers .
2. Plasmons on finite-width graphene strips
The electronic properties of monolayer graphene strips have been extensively investigated in the literature [2,16], and it has been demonstrated that they support highly confined transverse-magnetic plasmonic edge and waveguide-like modes.
In this work, we focus on the fundamental plasmonic mode propagating along a graphene strip of width W transferred onto a dielectric with relative permittivity εr (see Fig. 1). This is a dominant mode with symmetrical field distribution  and a cut-off frequency located in the low microwave band and determined by the imaginary part of graphene conductivity . The first step consists of computing the dispersion relation of the plasmon mode as a function of the strip width. Unfortunately, this dispersion relation cannot be derived analytically and one has to resort to purely numerical techniques . An accurate approximation of this relation can efficiently be computed by the universal scaling law of graphene strips . This law, which is based on an electrostatic approach assuming a strip width (W) much smaller than the wavelength, establishes that the allowed plasmon frequencies versus momentum are solely determined by graphene features and W. Neglecting losses, the scaling parameter η is given by
Instead of using time-consuming purely numerical techniques, we apply this scaling law to efficiently compute the propagation properties of plasmons guided along individual strips as a function of their width W. For this purpose, η is computed once for the fundamental plasmon mode  propagating along a given strip using the FEM software HFSS (see Appendix A) and Eq. (1). Then, the plasmon momentum related to strips with different characteristics is rapidly retrieved [2, 36] using η and Eq. (1). Importantly, dissipation losses are accurately taken into account by the graphene scaling law . Specifically, losses are incorporated by considering the propagation distance of a plasmon, i.e. the distance in which the power decays by an amount of 1/e, which is defined as Lp = vgτ = 1/e where vg is the plasmon’s group velocity vg = ∂ω/∂βy,spp. This allows determining the dissipation losses as αy,spp = 1/(2Lp) = 1/(2vgτ). In this last expression, τ is graphene relaxation which is related to the average time between two consecutive collisions of an electron propagating in graphene. Finally, the propagation constant of the surface plasmon along the strip is denoted as ky,spp = βy,spp + jαy,spp.
The dispersion characteristics of plasmons propagating along a graphene strip are illustrated in Fig. 2. The figure shows the normalized phase constant and a specific figure of merit [33, 37] (FoM) of plasmons versus different graphene strip widths and operating frequencies for various chemical potential values. Here, we define the plasmon figure of merit [33, 37] as FoM = Re[ky,spp]/Im[ky,spp]. This FoM can be understood as the double of the propagation length Lspp (1/e decay distance of the power) normalized by the plasmon wavelength. Note that the plasmon confinement is high for very narrow strips, but it reduces when the strip width increases . Moreover, the FoM increases with frequency and thinner strips. These results demonstrate that the adequate manipulation of the strip width provides an additional degree of freedom to control the guiding characteristics of plasmons, which is the property that will be employed in this work to propose a novel light-plasmon coupling mechanism.
We introduce here for convenience an equivalent modal surface impedance ZES = RES + jXES of a graphene strip with width W, defined as the surface impedance of an infinitesimally thin 2D sheet that supports an SPP with exactly the same propagation characteristics as the fundamental plasmon mode of the graphene strip. The modal surface impedance of this sheet can be obtained as1, 14, 15, 21], while rigorously taking into account the effect of the finite width of the strip on which the proposed coupling mechanism relies. Fig. 3 illustrates the equivalent surface impedance ZES of a graphene strip versus its width W and operating frequency for various chemical potentials, showing very similar trends to the dispersion features of the plasmons shown in Fig. 2.
3. Coupling plasmons and free-space waves in width-modulated graphene strips
As explained earlier, here we propose to adequately modulate the width of a graphene strip to compensate the momentum mismatch between guided plasmons and free-space waves. The proposed structure is shown in Fig. 4. It consists of a width-modulated graphene strip located at the interface between free-space and a dielectric with a permittivity εr. A polysilicon layer is placed beneath the strip to modify the graphene conductivity σ as a function of an applied DC bias voltage VDC.
The periodic nature of the width-modulated strip creates a guided plasmon mode composed of an infinite number of Floquet space harmonics [31, 32]. Properly designed, the structure periodicity forces one or more spatial harmonics to lie within the fast-wave region, i.e. the frequency region where electromagnetic coupling between guiding waves and free-space waves occurs, thus creating a leaky-wave mode. Note that most of the spatial harmonics are slow waves and do not effectively contribute to the coupling phenomenon. A leaky-wave mode radiates energy meanwhile propagates along the strip, and can be described using a complex propagation constant ky = βy + j(αy,rad +αy,spp) where βy is the phase constant, αy,spp is the dissipation constant associated with the unmodulated strip, and αy,rad is the coupling rate of the structure. The phase constant βy controls the coupling angle between the plasmons and surface waves, i.e. the angle of the radiated beam. In turn, the coupling rate αy,rad controls the amount of energy coupled per unit length, determining the beamwidth.
We take advantage of the equivalent modal surface impedance ZES = RES + jXES of a finite-width graphene strip to apply the well-developed theory of sinusoidally-modulated reactance surfaces . This type of structure was proposed in the 50s to create leaky-wave antennas, and its principle is currently being applied in the microwave community to develop 1D antennas with high-directive beams , and 2D spiral  and holographic  antennas with enhanced radiation features. Here, we employ this concept to analytically determine the dispersion characteristics ky of a given equivalent modulated 2D surface and, vice versa, to obtain the equivalent surface modulation required by a desired dispersion relation (see Appendix B). The sinusoidal modulation of an equivalent surface reactance may be expressed asAppendix B), which in turn determines the coupling characteristics of the surface . This approach also allows analytically computing the total radiation efficiency of the structure as  31] of the structure for a certain amount of dissipated and coupled power.
Once the properties of an equivalent sinusoidally-modulated surface impedance able to achieve a desired dispersion relation are determined, the physical profile of the width-modulated strip [see Fig. 4(b)] that supports the same plasmonic leaky-mode is obtained using the methodology previously explained. Note that there is a unique correspondence between any equivalent surface impedance and the width W of a graphene strip. This relation is not linear, hence a sinusoidal modulation of an equivalent surface impedance does not lead to a sinusoidally-modulated strip width. Importantly, the approach described here allows determination of the physical dimensions of a width-modulated strip able to fulfill a desired plasmon-light coupling without requiring any full-wave numerical computations.
One of the main advantages of the proposed structure is that its plasmon-light coupling angle θ0 (measured from the direction perpendicular to the strip) and rate αy,rad can be controlled almost independently. Specifically, the period p and the phase constant βy,spp of the unmodulated strip (which has a width WM, as shown in Fig. 4(a), corresponding to an equivalent surface reactance XES) mainly determine the coupling angle. This angle may be approximated as
Importantly, graphene’s field effect can be exploited to tune in real time the characteristics of the propagating plasmons, thus providing reconfigurable capabilities to the proposed structure. For this purpose, graphene’s Fermi level is modified by applying a voltage bias VDC between the strip and the polysilicon layer (see Fig. 4 and Appendix A). This allows tuning the characteristics of the leaky-wave mode supported by a width-modulated strip, fully controlling the coupling mechanism between plasmons and free-space waves. Specifically, this property permits excitation of plasmons at different operating frequencies by simply illuminating the structure with light coming from the same spatial position. In addition, the proposed structure can operate as a smart reconfigurable antenna, able to radiate energy towards different directions in free-space at a single operating frequency.
4. Design examples
Here we design and analyze a width-modulated graphene strip able to couple plasmons to free-space at a desired design frequency. In addition, we study the variations of the coupling angle versus frequency and show how tuning the Fermi level of graphene allows electrically steering this angle. Furthermore, we briefly illustrate the plasmon coupling between two width-modulated strips separated in free-space. These cases are clear examples of enhanced plasmon excitation not achievable with current coupling mechanisms , and constitute the basic building blocks of a reconfigurable and directional point-to-point plasmonic communication link, where less radiated power is wasted thanks to the use of directional radiation patterns. In addition, these examples may represent different realistic scenarios, such as the sensing of explosive and biological agents or intra/inter chip wireless communications. The analysis and design of the proposed structures are performed using the analytical developments proposed in this work, and are validated using results from the commercial package HFSS  (see Appendix A), which implements the finite element method (FEM).
In the first example, we design a width-modulated graphene strip to couple propagating plasmons to free-space in the direction normal to the strip (broadside, θ0 = 0°) at f0 = 1.5 THz. This frequency was chosen to overcome the usual difficulties of exciting plasmons and generating radiation in the THz band [17, 41]. However, note that the structure proposed here is scalable to other frequencies, provided that transverse-magnetic plasmons are supported there by graphene strips [2, 16]. In the design, we consider an SiO2 dielectric  with thickness d = 24 μm and permittivity ε = 3.9, and a s = 100 nm-thick polysilicon layer  with εr ≈ 3 located at t = 50 nm beneath graphene [see Fig. 4(c)]. In addition, note that since the polysil-icon layer is extremely thin and has a relative permittivity very similar to the one of SiO2, it barely affects the propagation of surface plasmons. This allows us to safely neglect this layer in the analytical approach, simplifying its numerical implementation. However, this layer is rigorously taken into account in the full-wave simulations. The first step in the design procedure consists of determining the average strip width WM [see Fig. 4(a)] and the chemical potential μc of the graphene that will be transferred into the dielectric. In order to obtain these values, the normalized phase constant, specific figure of merit, and equivalent modal surface impedance of the plasmons versus the strip width are shown at the operating frequency in Fig. 2(a), Fig. 2(b), and Fig. 3(a), respectively. An average strip width WM = 7.8 μm and a chemical potential μc = 0.7 eV are chosen for the following reasons: i) there is a large range of possible variations for the plasmon phase constant (or, similarly, for the equivalent modal surface impedance) achievable by increasing or decreasing WM, and ii) a high figure of merit is obtained, leading to relatively low dissipation losses. These values lead to an average equivalent surface reactance XES = 1265 Ω/□.
The period p of the strip width modulation is then determined using Eq. (5) with θ0 = 0°, yielding p = 57.14 μm. Note that this design considers radiation from the first radiating spatial harmonic, i.e. n = −1, with the aim of reducing the physical dimensions of the final structure. The modulation index M is set to 0.28, which can be obtained using realistic values of the strip width. The normalized phase constant and equivalent surface reactance versus the strip axis are shown in Fig. 5(a). Two different regions of the structure are shown there, namely i) “Guided region”, which is related to an unmodulated graphene strip employed to connect the device to other plasmonic components, and ii) “SPPs-light coupling region”, which is the section where the electromagnetic coupling between plasmons and free-space waves really occurs. The physical dimensions of the width-modulated strip are plotted in Fig. 5(b) versus the distance along the strip axis. Importantly, these dimensions are obtained analytically, without requiring the use of full-wave numerical techniques or optimization methods.
The coupling properties of the designed structure are shown in Fig. 6. As expected, the propagating plasmons are radiated towards the direction normal to the strip at the operating frequency f0 = 1.5 THz [see Fig. 6(a) with μc = 0.7 eV]. This figure also demonstrates the variation of the coupling angle between plasmons and free-space waves versus frequency, which scans the whole space (θ0 ∈ [−90° + 90°]) in the frequency range f0 ∈ [1.2 − 1.9] THz. The normalized coupling rate of the structure is depicted in Fig. 6(b), showing the typical behavior of sinusoidally-modulated surfaces. The coupling efficiency of the structure, assuming that it is long enough to radiate all the input power, i.e. Le → ∞ in Eq. (4), is shown in Fig. 7. The efficiency is around 5% and 7.5% for negative and positive coupling angles, respectively, with a peak of 20% close to the direction normal to the strip (f0 = 1.5 THz). This result is remarkable, and comparable to the radiation efficiency achieved by usual antennas in this frequency band . These favorable values have been obtained because in contrast to the usual strong mismatch that occurs between the actual radiator and the feeding section in THz antennas , here the propagating plasmons are naturally matched to the coupling section of the structure. Moreover, larger efficiencies may be obtained by increasing the modulation index M.
Simulated radiation patterns in the ZY plane are shown in Fig. 8(a) for different operating frequencies, confirming the beam scanning behavior of the structure. A very good agreement between the proposed analytical technique and numerical FEM simulations is obtained, thus validating the reported plasmon - free space waves coupling and the analysis and design approach detailed here. Note that such level of agreement is only possible if the complex propagation constant of the propagating SPPs is accurately computed by the two different techniques. Importantly, since dissipation losses also contribute to define the antenna radiation pattern , these results confirm that they are adequately taken into account in the proposed analytical procedure. To further illustrate the radiation phenomenon, Fig. 9 shows the magnitude of the electric field in the ZY plane of the proposed structure at the operating frequency f0 = 1.7 THz. In the figure, a plasmon propagates along an unmodulated graphene strip (“Guided region”) and is subsequently radiated towards free-space (θ0 ≈ 30°) due to the periodic modulation of the strip width.
One unique property of the proposed structure is that its radiation characteristics can be controlled in real-time by modifying the graphene’s Fermi level via an applied DC bias. Fig. 6(a) demonstrates that increasing the Fermi level up-shifts the fast-wave frequency region of the structure, i.e. the frequencies where the plasmon-light coupling occurs. Moreover, Fig. 6(b) shows that the coupling rate remains relatively constant within the fast-wave region for different chemical potential values. Interestingly, the coupling efficiency increases for larger chemical potentials, as indicated in Fig. 7. This behavior is related to the lower dissipation losses (or high figure of merit) associated with plasmons propagating in graphene strips with large chemical potentials [see Fig. 2(d)]. These results confirm that the proposed structure can be employed for electronic beam scanning at a single operating frequency. This is further demonstrated in Fig. 8(b), which shows the structure radiation patterns at the operating frequency f0 = 1.5 THz for different chemical potential values.
Finally, to show the versatility of the design and a possible application, a wireless plasmon coupling between two width-modulated graphene strips is investigated in Fig. 10. The two structures, which have the same physical dimensions as that previously designed, are located in free-space and separated by a distance of 375 μm in the z axis. To clearly illustrate the coupling phenomenon, the first structure (lower) is DC biased to radiate towards a positive angle (μc = 0.67 eV → θ0 = +7°) and the second strip (upper) to couple these incoming waves to plasmons (μc = 0.78 eV → θ0 = −7°) at the operating frequency f0 = 1.5 THz. Propagating plasmons are then fed into the first strip. These plasmons propagate along the structure, and are subsequently radiated towards free-space and coupled into the receiving width-modulated strip. The coupling phenomenon is clearly illustrated in Fig. 10, which also shows the specific field distribution of the plasmons excited in the upper structure and their subsequent propagation towards the output port. These results further suggest the direct use of the proposed structure for sensing and wireless applications in the THz band.
We have reported a plasmons to free-space waves coupling mechanism based on the periodic modulation of a graphene strip. Our main findings can be summarized as (1) guiding properties of plasmons are shown to be easily controlled by modifying the width of a graphene strip; (2) the periodic modulation of a graphene strip allows compensation of the momentum mismatch between plasmons and free-space propagating waves, which can be rigorously explained using leaky-wave theory ; (3) the physical dimensions of the width-modulated strip can be engineered to modify the features of allowed leaky-wave plasmons, thus controlling the characteristics (angle and efficiency) of the coupling phenomenon with free-space waves at any desired operating frequency; and (4) we have combined the universal scaling law of plasmons in graphene strips  with standard leaky-wave theory [21, 31, 32] borrowed from microwaves to analytically i) derive the dispersion relation of leaky-wave plasmons on a given structure and predict their coupling features, and ii) engineer a particular width-modulated strip able to achieve a required coupling pattern. In brief, width-modulated graphene strips provide a convenient way to couple plasmons to free-space with highly customizable radiation properties. The results shown here are based on the classical description of graphene, where it is described as frequency-dependent conductive layer. A careful study of non-local effects [44–46] must be performed before extrapolating these results to strip widths within the nanometer scale.
The electrostatic tunability of graphene has been exploited to achieve beam scanning capabilities at a fixed operating frequency. Indeed, the variation of the Fermi level allows a smooth control of plasmon properties, dynamically modifying the spatial direction of the coupling angle. It has also been shown that higher Fermi levels lead to larger coupling efficiencies. In this respect, it should be noted that efficiency is closely related to graphene relaxation time τ, which determines the plasmon propagation distance. Specifically, larger plasmon propagation distances allow increasing the coupling to free-space and achieving highly directive beams . Consequently, graphene relaxation time is a critical parameter on which this and a wide range of plasmonic applications are dependent . Even though graphene τ values are larger than in standard plasmonic metals, some phenomena such as phonon emission , many-body interactions  and impurities may influence and deteriorate it. However, it is expected that the improvement of fabrication techniques and the use of composite/active materials will extend the lifetime of propagating plasmons in the near future [48,49]. In addition, no power handling issues are foreseen in the proposed structure since monolayer graphene configurations have very good performance in terms of thermal conductivity .
The proposed structure is easily scalable to other frequency bands, provided that plasmons are supported by graphene strips there [2, 16]. Compared to current plasmon-light coupling mechanisms , the main advantages of width-modulated strips are i) fully planar geometry, ii) relatively simple fabrication process, iii) easy integration with actual graphene-based plasmonic devices, iv) fully customized coupling between plasmon and free-space waves, including coupling angle and intensity, and v) reconfiguration capabilities, thus allowing the excitation of plasmons at different operating frequencies using incoming light from the same spatial region or providing beam scanning functionalities at a fix operating frequency.
From the application point of view, width-modulated graphene strips can easily be engineered to sense incoming radiation and detect the presence of specific gases, or chemical and biological process . Furthermore, its use as a sensor would provide valuable information about the spatial position of the radiation source, thanks to its unique beam scanning capabilities. Note that once the incoming signal has been detected, it can easily be guided, processed, and transmitted through the unmodulated graphene strip. Moreover, the structure can also be arranged in an array configuration to increase its sensitivity. In addition, the proposed configuration can directly be employed as an antenna component in all-integrated plasmonic devices [6,33], extending their current functionalities and leading to applications such as reconfigurable intrasatellite  and wireless intra/inter chip communications, THz transceivers, and high resolution image scanning.
A. Appendix A: Simulation of the proposed structure and graphene description
The proposed structures are simulated with the frequency-domain finite element method (FEM) using the 3D full-wave electromagnetic solver Ansys HFSS v14. In the simulations, graphene is modeled by applying a surface impedance boundary condition (ZS = 1/σ) to a 2D layer, avoiding a numerically inefficient bulk description of the material. The conductivity σ is modeled using the Kubo formalism  as52], and takes into account intraband and interband contributions of graphene conductivity as well as finite temperature. In addition, we do not consider in this work ultra-narrow graphene strips (i.e., W ≫ 50 nm), thus allowing to safely neglect edge and nanoscale effects .
We consider graphene at T = 300° K and with a relaxation time τ = 1.0 ps. The value of τ was estimated from the measured impurity-limited DC graphene mobility on boron nitride  of μ ≈ 60000 cm2/(Vs), which leads to ps for a Fermi level of EF = 0.2 eV. Note that even higher mobilities have been observed in high quality suspended graphene . The estimated value of τ was reduced to 1 ps because in our analysis we ignore the additional losses that may arise from electron-phonon coupling , the deviation from a perfect Dirac-cone structure , and many-body interactions .
Graphene’s Fermi level can be modified by applying a transversal electric field via a DC bias gating structure [see Fig. 4(c)], thus controlling its conductivity. Thanks to the electron-hole symmetry , both positive and negative Fermi level lead to the same complex conductivity. The applied electric field modifies graphene carrier ns as Eq. (8). This relation enables controlling graphene conductivity and consequently the guiding properties of propagating surface plasmons in real time. Note that we have neglected here the possible influence of the quantum capacitance , which may be important in those situations where an extremely thin dielectric of high permittivity is employed within the gate structure.
B. Appendix B: Sinusoidally-modulated 2D surfaces
Electromagnetic propagation and radiation from sinusoidally-modulated 2D surfaces have been intensively investigated in the microwave regime and applied to develop leaky-wave antennas [21, 22, 38, 60]. These structures are based on the sinusoidal modulation of their modal surface impedance. This impedance is defined as the ratio between tangential electric and magnetic fields of the waves guided by the 2D surface , and it can be expressed14, 16, 35].
The periodic nature of the modulated impedance allows expanding the electromagnetic fields above the sheet using spatial harmonics . The nth Bloch wave number tangential to the surface is denoted by kyn = ky0 + 2πn/p, which allows expressing the normal wavenumbers kzn as
The dispersion relation of a sinusoidally-modulated surface impedance can be obtained in a continued fraction form as 21].
Once the fundamental propagation constant ky0 is known, all other spatial harmonics can easily be determined. At this point, dissipation losses are included in the final propagation constant of the sinusoidally-modulated surface as31, 32].
This work was supported by the Swiss National Science Foundation (SNSF) under grant 133583 and by the EU FP7 Marie-Curie IEF grant “Marconi”, with ref. 300966.
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