Abstract
In this work we study the terahertz light propagation through deeplyscaled graphenebased reconfigurable metasurfaces, i.e. metasurfaces with unitcell dimensions much smaller than the terahertz wavelength. These metasurfaces are analyzed as phase modulators for constructing reconfigurable phase gradients along an optical interface for the purpose of beam shaping. Two types of deeplyscaled metacell geometries are analyzed and compared, which consist of: (i) multi split ring resonators and (ii) multi spiral resonators. Two figures of merit, related to: (a) the loss and (b) the degree of reconfigurability achievable by such metamaterials when applied in beam shaping applications, are introduced and discussed. Simulations of these two types of deepsubwavelength geometries, when changing the metal coveragefraction, show that there is an optimal coveragefraction that gives the best tradeoff in terms of loss versus degree of reconfigurability. For both types of geometries the best tradeoff occurs when the area covered by the metallic region is around 40% of the metacell total area. From this point of view, reconfigurable deeplyscaled metamaterials can indeed provide a superior performance for beam shaping applications when compared to not deeplyscaled ones; however, counterintuitively, employing very highlypacked structures might not be beneficial for such applications.
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Introduction
Terahertz technology is a growing technological field, which in recent years has been finding multiple emerging applications in diverse areas including: medical imaging, biochemical sensing, security, wireless communications and so on^{1}. In this context, future compact lowcost terahertz systems, such as beam steerers for MIMO communications, tunable flat lenses for terahertz cameras, etc., will demand components capable of achieving active beamshaping at some degree. Reconfigurable terahertz metamaterials^{2} were shown capable of modulating the phase of an arbitrary terahertz beam^{3}; these metamaterial phase modulators can be employed to construct arbitrary phase gradients in an optical interface, which is of special interest for terahertz beamshaping applications. In this regard, via independently biasing each metacell, thus spatially controlling the phaseshift, arbitrary phase gradients can be constructed^{3}, which in turn can shape the reflected or transmitted beams in accordance with the recently proposed generalized laws of reflection and refraction (generalized Snell's law)^{4}. Our work: (i) discusses the use of reconfigurable deepsubwavelength metasurfaces for constructing arbitrary phase gradients for beam shaping applications, (ii) introduces two figures of merit that are related to the performance of these metasurfaces in the context of beam shaping applications and (iii) discusses the geometrical tradeoffs is designing such structures and identify the metal coveragefraction as an important parameter in this regard.
When a phase gradient is placed in the interface between two media of refractive index n_{t} and n_{i}, Snell's law of transmission should be rephrased as the generalized law of reflection and refraction^{4}:
where θ_{i} and θ_{t} are the angle of incidence and the transmitted angle, respectively, λ_{0} is the vacuum terahertz wavelength and dϕ/dx represents the phase gradient. Assuming normal incidence and n_{i} = n_{t} = 1, Eqn. (1) can be rewritten as:
Therefore, it can be easily seen that the shape of the transmitted beam can be arbitrarily controlled via designing an adequate phase gradient. For instance, assuming an incident collimated beam, a linear phase gradient can tilt the transmitted beam, whereas a parabolic phase gradient can focus it. It is therefore of interest the use of electricallydriven reconfigurable metamaterial phase modulators for constructing these arbitrary phase gradients. However, in order to enable the design of these arbitrary phase gradients, each metacell in the device should be able to provide: (a) the same transmission amplitude and (b) 360° (2π) control over the transmitted phase. In this context, it can be observed, as discussed by Chen et al^{3}, that the terahertz transmission amplitude and phase through a metacell are not independent of each other, but they are related by KramersKronig(KK) relations. Near frequencies where maximum amplitude modulation is achieved no phase modulation takes place. In contrast, near frequencies where the transmission amplitude is not dependent of the applied voltage (i.e. there is no amplitude modulation), but its slope is, the phase experiences maximum shift. From this point of view at the frequencies where maximum modulation of phase is obtained there is no amplitude modulation and therefore (a) is guaranteed. Since terahertz metamaterial phase modulators proposed to date exhibit phase modulation much smaller than 360° (see Refs. 3, 5), epitaxial stacking of multiple layers^{3} is necessary in order to achieve (b), which in turn increases the loss in the device. Moreover, construction of arbitrary phase gradients is also limited by the geometrical length of each unitcell. This is due to the fact that, when employing metamaterials, a continuous phase gradient is approximated by a discretely spatiallyvarying one. From this point of view, the smaller the unitcell length (when compared to the target terahertz wavelength), the better one can approximate an arbitrary phase gradient, therefore the more functionality and better performance the metamaterial beam shaper might achieve. In this context, a problem of metamaterial structures proposed todate as phase modulators is that the unitcell to wavelength ratio is not small enough to provide good performance. For instance, in order to provide ~90° control over the transmission angle using a 10 element phasegradient discretization, a unitcell length < λ_{0}/10, thus a unitcell to wavelength ratio < 0.1, is required. Terahertz metamaterial phase shifters reported todate have unitcell to wavelength ratios in the order of ~0.15/~0.2 (see Refs. 3, 5). From this point of view, one of the main challenges of terahertz metamaterial phase modulators is: designing a metamaterial with small unitcell to wavelength ratio, which has a large phase modulation and large transmission at the frequencies at which maximum phase modulation takes place. In this work, the terahertz (THz) light propagation through deeplyscaled graphenebased reconfigurable metasurfaces is studied in the context of beamshaping applications. Although graphene is used as an example reconfigurable semiconductor in these devices, the discussion presented here is general enough and the results are also valid if employing other semiconductor materials.
Results
Two types of deepsubwavelength metamaterial geometries are studied and compared. These consist of: (i) multi spiral resonators (MSRs) and (ii) multi split ring resonators (MSRRs), as depicted in Fig. 1(a) and Fig. 1(b), respectively. A sheet of graphene was considered as the tunable element to reconfigure the terahertz transmission properties of the metamaterial^{6,7,8}, which was placed in some strategic regions of the device, as depicted in Fig. 1. For the MSRR structure the graphene sheet is located inside each split, whereas for the MSR structure the graphene sheet is located in the geometriccenter of the structure connecting the four spiral arms. The electromagnetic properties of this graphene layer and therefore the effective properties of the metamaterial, can be adjusted via controlling the Fermi level of graphene therefore its density of states available for intraband transitions and thus its optical conductivity^{6}. Although graphene metamaterials have been widely employed in devices modulating the amplitude of a transmitted terahertz beam^{9,10,11,12}, to the author's knowledge, graphenebased terahertz metasurfaces controlling the phase of a transmitted terahertz beam under normal incidence, have not yet been proposed todate. In terms of reflection, graphene based metamaterials have theoretically been shown capable of modulating phase in reflectarray geometries^{13}; however these structures do not provide constant amplitude of reflection when the phase is reconfigured. Actuation over the graphene terahertz optical conductivity can be achieved electrostatically via either gating graphene with another graphene layer (selfgated structure)^{14}, or via employing iongel as the gating element^{15}.
Shown in Fig. 2 are the characteristic transmission and phase frequency responses as a function of graphene conductivity for one of these metamaterials (a MSR with 30% metal to unitcell area coveragefraction). Maximum phase modulation, 108°, was observed at 500 GHz; at this frequency the transmittance was found to be 20%, independently of the graphene conductivity.
Metacells consisting of MSRRs and MSRs were numerically simulated. In order to extract useful information regarding the design tradeoffs in these structures, simulations were performed by changing the metal coveragefraction in each of both geometries. The width of the metal rings/spirals was set to 2μm, which is a dimension comparable with that of the minimum features achievable in optical lithography; the unitcell edgelength was taken between 52μm and 58μm (depending on the particular metacell). The graphene sheet area was set to 4μm by 4μm in the MSR structure and 2μm by 4μm in the MSRR structure. Therefore, a larger coveragefraction translates into: (a) a larger number of rings and smaller spacing between adjacent rings for the MSRR metacell geometries, or (b) a larger number of turns and smaller spacing inbetween metals for the MSR metacell geometries. Shown in Fig. 3 are the sketches of the eight simulated devices (4 MSRR geometries and 4 MSR geometries, each of them having a different metal coveragefraction); the results of these simulations are shown in Table 1 (where f_{p} stands for the frequency at which maximum phase modulation takes place and PM and T are the phase modulation and the transmittance, respectively, through the metacell at f_{p}).
Discussion
As discussed in the introductory section, for beam shaping applications, an ideal metamaterial geometry should provide: (i) large phase modulation, (ii) large transmittance and (iii) small unitcell to wavelength ratio. Arbitrary phase gradients need to be constructed when reconfiguring the phaseshift inserted by each metacell. Therefore, a full control of the transmitted phase, i.e. between 0 and 360°, is desirable in each unitcell in order to achieve truly arbitrary designs. But the phase modulation achievable by each metacell is finite, e.g. prior metamaterial phasemodulator proposals^{3,5} show phase modulation < 50°; therefore, epitaxial stacking of layers is required in order to obtain a 360° control over phase in each metacell. When many layers are epitaxially stacked, although the phase shifts can be added^{3}, loss increases with number of layers, which is not desirable. From this point of view, the following figure of merit, related to loss, is defined: FoM_{1} = PM × T/([360°] × [100%]). For an ideal metacell geometry FoM_{1} should approach unity (since PM and T are bounded by 360° and 100%, respectively); the larger the FoM_{1} the most suitable a metamaterial geometry is for beam steering, i.e. the less loss the device will provide. But also, a small unitcell to wavelength ratio is required in order to construct sharp phase gradients, which are needed, for instance, in order to achieve large swings in beam steering applications as discussed in the introductory section. From this point of view, a second figure of merit is defined: FoM_{2} = L/λ_{P}, where L is the edgelength of the metacell and λ_{p} is the wavelength associated with the frequency at which maximum phase modulation takes place. For an ideal metacell geometry FoM_{2} should approach zero, the smaller the FoM_{2} the most suitable a metamaterial geometry is for beam steering.
As depicted in Table 1, it was observed that for MSRRs, the resonance always redshifts as the metal coveragefraction is increased. However, for MSRs, when the coveragefraction is increased, the resonance frequency first starts redshifting and then blueshifts. Moreover, if the metal coveragefraction is further increased (as depicted in Fig. 4), the response becomes even less monotonic. The first blueshift is observed when the metal coveragefraction is increased to larger values (i.e. from 50% to 64%). The trends observed in both structures can be qualitatively explained with an equivalent circuit model (the series of an equivalent inductance and an equivalent capacitance), see Refs. 16–17. Although, for the sake of simplicity, a twoarm MSR will be analyzed in the following discussion as an example to illustrate how the resonance frequency evolves when changing the metal coveragefraction, analogous, i.e. nonmonotonic trends, will also hold for fourarm MSRs.
For instance, for the case of MSRRs, the equivalent inductance (L_{0}) is given by the average inductance of the rings. Therefore, the unitcell size will be determinant in L_{0}; since for the simulated structures the unitcell dimensions remain almost constant, the equivalent inductance can be considered as independent of the metal coveragefraction. Shown in Fig. 5(a)–(b) are sketches of the equivalent circuit models for MSR and MSRR geometries, respectively. These equivalent circuit might explain the behavior of these metamaterial with the change in metal coverage ratio. As previously discussed, for simplicity and for illustrative purposes here we discuss the case of a twoarm spiral structure (Fig. 5(a)), however the behavior of a fourarm spiral structure will follow analogous trends. Here C_{0} is the unity capacitance and L_{0} is the unity inductance (capacitance and inductance of two adjacent rings); in the figure it is assumed that this capacitance, which is related to the spacing between rings, remains constant as the spacing changes. The total inductance of these structure will not be largely affected when increasing the numbers of rings^{16,17}, because the unit cell size is nearly constant and it can be approximated by the average ring size in the structure. It is worth mentioning that effects such as the capacitance between nonadjacent rings and the resistances arising from losses in the metal and the dielectric will be neglected. As the metal coveragefraction increases (by adding more turns to the structure) the total effective capacitance of the structure increases as depicted in Fig. 5(b). In practice, because of the smaller spacing when coveragefraction is increased, C_{0} also increases. This increase in effective total capacitance explains the observed redshifting of the resonance frequency in MSRRs as the metal coveragefraction is increased. As the metal coveragefraction is increased, there are two competing effects taking place in this geometry: (i) as the number of turns increases, as depicted in Fig. 4(a), the equivalent capacitance first (at small number of turns) increases and then (at large number of turns) decreases. From this point of view, since indeed the number of turns is increased when the coveragefraction is increased, the resonance is expected to blueshift for very large metal coveragefractions. In the other hand, (ii) as the coveragefraction is increased, C_{0} increases due to a smaller spacing. From this point of view, the resonance is expected to redshift. These two effects, (i) and (ii), are competing and both are important when increasing the coveragefraction. For small coveragefractions the same resonancefrequency evolution trend as in the MSRR is observed. However, for large coveragefractions (i) can become dominant and the overall effect we observe might be a decrease in the equivalent circuit capacitance and therefore a resonance frequency blueshift. However, when the metal coverage fraction is increased even further, (ii) can become dominant again and cause the resonance frequency to again redshift; this leads to a very nonmonotonic characteristic at large coveragefractions.
It can be also observed (Fig. 4) that when the resonance redshifts its strength diminishes, until eventually it becomes so weak that it disappears (blue, red and skyblue data points which correspond to 30%, 42% and 49% metal coveragefractions, respectively). When the metal coveragefraction is further increased then a former higherorder resonance becomes the first resonance (e.g. purple data point, which corresponds to 64% metal coveragefraction), leading to a sawtooth characteristic for resonancefrequency versus metal coveragefraction as observed in Fig. 4. The resonance strengths for metal coveragefractions above 80% become considerably weaker since the gaps between adjacent goldstripes become much smaller than the width of the goldstripes and thus the structure, effectively, is mostly covered with metal.
Shown in Fig. 6 are the plots of FoM_{1} and FoM_{2} versus metal coveragefraction for the two metamaterial geometries that were studied. When analyzing FoM_{1}, it is observed that in MSRs the smallest coveragefractions give the best tradeoffs. However, in MSRRs, decreasing the coveragefraction below 30% significantly decreases transmission and phase modulation due to a very weak interaction between rings. Therefore, it can be noticed that moderate coveragefractions (i.e. around 40%) give the best tradeoff.
When analyzing FoM_{2}, in MSRRs, it can be observed that large coveragefractions give the best tradeoff. This is a result of the monotonically increasing dependence of effective capacitance with coveragefraction in this geometry. However, in contrast, for MSRs a completely different trend is observed when the metal coveragefraction is increased. There is an optimal coveragefraction, which occurs around 40%, that gives the best tradeoff. This is a result of the nonmonotonic dependence of equivalent capacitance with coveragefraction in this geometry.
When (overall) considering all the above described trends, it can be concluded that for both types of geometries the best tradeoff between FoM_{1} and FoM_{2} occurs in the region where coveragefraction is around 40%. From this point of view, deeplyscaled metamaterials can indeed provide a better performance than traditional metamaterials in beamshaping applications. However, counterintuitively, use of very highly packed structures can actually be not beneficial. There is an optimal metal coveragefraction, ~40%, which offers the best tradeoff. Interestingly, this optimal coveragefraction is the same for both types of metamaterial geometries, MSRRs and MSRs.
Methods
Numerical simulations and structural parameters
In the analyzed metamaterials, gold was chosen as the material for the metallic layers, whereas Al_{2}O_{3} was considered as the dielectricinbetween (see Fig. 1). These materials were set in top of a 2μm thick polyimide layer, which has the role of a substrate; the thickness of the gold and Al_{2}O_{3} layers was 1μm. The metamaterials were numerically simulated employing high frequency structural simulator (HFSS). In these simulations graphene was taken as a finitethickness material with a 1nm thickness, as discussed in Refs. 11, 18.
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Acknowledgements
The authors thank David Schurig and Ajay Nahata for useful discussions. The authors acknowledge the support from the NSF MRSEC program at the University of Utah under grant # DMR 1121252 and from the NSF CAREER award #1351389 (monitored by Dimitris Pavlidis).
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S.A. and B.S.R. carried out the numerical simulations, analyzed the data and contributed to the preparation of the manuscript.
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Arezoomandan, S., SensaleRodriguez, B. Geometrical tradeoffs in graphenebased deeplyscaled electrically reconfigurable metasurfaces. Sci Rep 5, 8834 (2015). https://doi.org/10.1038/srep08834
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DOI: https://doi.org/10.1038/srep08834
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