IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013
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Reflectarray Antenna at Terahertz Using Graphene Eduardo Carrasco, Member, IEEE, and Julien Perruisseau-Carrier, Member, IEEE
Abstract—The use of graphene for fixed-beam reflectarray antennas at Terahertz (THz) is proposed. Graphene’s unique electronic band structure leads to a complex surface conductivity at THz frequencies, which allows the propagation of very slow plasmonic modes. This leads to a drastic reduction of the electrical size of the array unit cell and thereby good array performance. The proposed reflectarray has been designed at 1.3 THz and comprises more than 25 000 elements of size about . The array reflective unit cell is analyzed using a full vectorial approach, taking into account the variation of the angle of incidence and assuming local periodicity. Good performance is obtained in terms of bandwidth, cross-polar, and grating lobes suppression, proving the feasibility of graphene-based reflectarrays and other similar spatially fed structures at Terahertz frequencies. This result is also a first important step toward reconfigurable THz reflectarrays using graphene electric field effect. Index Terms—Complex conductivity, graphene, plasmonic, reconfigurable antenna, reflectarrays, Terahertz (THz).
I. INTRODUCTION
T
HE POTENTIAL applications exploiting Terahertz (THz) frequencies, notably in imaging and radioastronomy, together with the recent advances in THz sources and detectors, have recently attracted the attention of the research community on THz antenna systems [1], [2]. Simultaneously, it was understood that graphene’s high electron mobility and field effect, among others, are extremely interesting for developing high-frequency nanoelectronics [3]. Concerning passive devices, the interest in graphene mainly originates from its complex surface conductivity, which allows the propagation of plasmonic slow-wave modes [3]–[5]. Planar reflectarrays are a very attractive antenna technology that combines the main advantages of parabolic reflectors and phased arrays [6], resulting in low loss, compact profile, low cross polarization, easy manufacturing, and high efficiency. They consist of an array of reflective cells that introduce a phase shift upon reflection of an incident wave; see Fig. 1(a). As the elements are spatially illuminated by a feed source, the bulky and lossy beamforming networks present in conventional arrays are not necessary. Depending on the phase distribution on the array surface, the far-field radiation pattern can be collimated or shaped. This kind of antenna has been extensively studied in Manuscript received November 23, 2012; revised January 18, 2013; accepted February 12, 2013. Date of publication February 15, 2013; date of current version March 14, 2013. This work was supported by the European Union under Grant FP7-300934 (IEF Marie-Curie Project RASTREO), the Swiss National Science Foundation (SNSF) under Grant 133583, and the Hasler Foundation under Project 11149. The authors are with the Adaptive MicroNano Wave Systems Group, LEMA/ Nanolab, Ecole Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland (e-mail:
[email protected]; julien.perruisseau-carrier@epfl. ch). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2013.2247557
Fig. 1. (a) Reflectarray architecture. (b) Graphene-based patch.
microwave and millimeter-wave bands [6], and more recently at THz [7]. In all cases, standard metallic conductors have been used as the electrical conductor in the reflectarray elements. This letter evaluates for the first time the feasibility of a fixedbeam reflectarray antenna at THz based on graphene and compares its performance to a similar implementation using gold. The designed graphene antenna operates at 1.3 THz and comprises 25 448 electrically small unit cells. Compared to a goldpatches implementation, it exhibits better bandwidth, slightly lower cross polarization, and easier design thanks to the low sensitivity of the elements to the angle of incidence, but at the cost of increased loss. Since graphene complex conductivity can be efficiently controlled via electric-field biasing [5], [8], reflectarrays based on this material could potentially be dynamically reconfigured. Though corresponding analyses are beyond the scope of this initial demonstration, the use of graphene for efficient dynamic reconfiguration of reflectarrays at THz is another major longer-term motivation for this work. II. GRAPHENE-BASED REFLECTARRAY ELEMENT The proposed reflectarray element is shown in Fig. 1(b) and consists of a square patch of graphene transferred over and a grounded quartz substrate ( at the frequencies of interest [9]) of thickness m and cell size m. Thanks to its mono-atomic thickness, graphene can be accurately modeled as an infinitely thin surface of complex conductivity . This conductivity is expressed by the Kubo formula and depends on the frequency , the absolute temperature , the transport relaxation time , and the chemical potential [10]. Values of K and ps (measured in [11]) are used in this work. is set to 0.19 eV, which can be achieved via chemical doping or permanent electrostatic bias [12], [13]. Fig. 2 shows the resulting complex conductivity in the band from 1.1 to 1.5 THz. It is worth mentioning that this conductivity quite accurately corresponds to the series connection of a real and inductive impedance components, namely . At the central frequency of 1.3 THz,
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Fig. 2. Complex conductivity of graphene in the band of interest. Fig. 4. Cross-polar component of the reflection coefficient, for the proposed element at 1.3 THz (only cases above 50 dB are shown).
Fig. 3. Reflection coefficient of the proposed reflectarray element, as a function of the patch size, including the variation with the incidence angle, at the central frequency of 1.3 THz. (a) Phase. (b) Amplitude.
, revealing the highly inductive nature of graphene conductivity at such frequencies. The reflection coefficient on the surface of the proposed element was full-wave computed using the Floquet’s boundary conditions from CST Microwave Studio, taking into account the interelement coupling and the actual plane-wave incidence angle. Fig. 3 shows the phase and amplitude of the reflection coefficient under normal incidence, as a function of the patch size. The patch resonance occurs when its size is around (into a unit cell) and , with and as the wavelength in free space and the dielectric media , respectively. This phenomenon is due to the well-known slow-wave propagation associated with graphene plasmonic modes [3]–[5]. Indeed,
while the graphene patches resonate when their length is about half of a guided wavelength, namely similar to metallic patches, the aforementioned highly inductive nature of graphene conductivity results in very small guided plasmon wavelengths [5]. It is worth mentioning that the use of subwavelength elements had been proposed to increase the bandwidth in moderate-size reflectarrays, demonstrating that closely spaced elements have a smaller phase variation and reduced frequency phase error [14]. A phase range of 290 is obtained in the whole band from 1.1 to 1.5 THz, which is sufficient to efficiently collimate a beam as shown later. If needed, the range could be increased while preserving the bandwidth and improving the loss by using multiresonant elements in single- [15], [16] or multilayer [17] configuration. The limited surface conductivity of graphene is the dominant contribution to the loss, which varies between 0.5 and 8 dB within the band, as shown in Fig. 3(b), resulting in a gain reduction of the complete reflectarray antenna of only 2.3 dB (see Section III). This is higher than using metal since the loss obtained at 1 THz with a resonating gold patch is around 1.2 dB [7]. This effect is compensated by the benefits of smaller element size demonstrated next. On the other hand, as aforementioned, graphene is extremely interesting for its potential for field-effect dynamic reconfiguration and is expected to favorably compete with other technologies in terms of loss as well at such frequencies. Fig. 3 also includes the results, at the central frequency, for 17 different combinations of oblique angles of incidence , in the range , . Logically, the largest phase variation with regard to normal incidence occurs for the most severe oblique incidence considered ( , ). However, even in this case, the maximum deviation is 35 for patches around 9 m, while in most cases the phase variation is less than 20 . This small dependence to the angle of incidence also results from the extremely small size of the reflective cell. It is finally noticeable that though a small dependence to the incidence angle is generally preferred, the design method used here and detailed in Section III computes the adequate element size at any position considering a more realistic incidence angle on each cell. Fig. 4 shows the cross-polarization components such as defined in [6] for the same angle of incidence variations. It can be seen that the cross polarization is below 20 dB in almost all cases. The thick solid line shows the highest cross polarization, which again occurs at , . However, such
CARRASCO AND PERRUISSEAU-CARRIER: REFLECTARRAY ANTENNA AT THz USING GRAPHENE
Fig. 5. Phase of the reflection coefficient for a gold patch in a 14- m lattice (low-frequency regime) and in a 100.8- m lattice (resonance).
TABLE I MAIN FEATURES OF THE GRAPHENE REFLECTARRAY ANTENNA
angle of incidence only concerns a few elements of the proposed reflectarray designed in Section III, which will be shown to exhibit very low overall cross polarization. It is noticeable in Fig. 5 that, for the same cell size, a patch made of gold and characterized by a Drude model (with S/m, Hz [7]) achieves less than 200 of phase range and with very abrupt change since gold still mostly behaves as standard conductor at low THz. This case is depicted in the left-hand side of Fig. 5. However, a phase shift similar to that of the graphene patch (290 ) can be achieved by adjusting the dielectric thickness to 16 m and the reflective cell size to around ( m), as depicted in the right-hand side of Fig. 5. Here, it is important to remark that at the element level, for a bandwidth of 15.4%, an average phase error smaller than 17 is achieved in the case of the graphene cell, while in the case of the resonant gold cell (100.8 m), this error is about 30 . This will lead to a larger array bandwidth as shown in Section III. III. REFLECTARRAY DESIGN In this section, a complete reflectarray based on the graphenebased cell presented previously is designed and analyzed. The main characteristics of the proposed circular reflectarray with offset configuration are summarized in Table I. The radiating surface is centered at the origin of the coordinates system. Note that in the case of a conventional reflectarray using gold, this aperture and geometry result in 489 elements with a cell size of 100.8 m (25 elements in the main axis), while more than 25 000 elements are used in the case of the graphene-based reflectarray. This is, of course, not an issue since such a structure would be fabricated monolithically, and thus the number of elements does not impact on the cost or manufacturing complexity. The field incident at each reflectarray element depends on the element position and the feeding source radiation pattern.
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Fig. 6. Required phase distribution on the reflectarray limited to the available 290 element phase range.
function has For the concept introduced in this letter, a been used to model the source. The illumination level at the edges of the antenna is 12 dB below the maximum, which is a typical value for achieving good efficiency [6]. Obviously, any other feeding pattern could be considered as well thanks to the generality of the design method. Fig. 6 shows the phase shift, which must be introduced at each reflectarray element in order to radiate a pencil beam toward , . As the proposed element only allows shifting the phase in a range of 290 , that phase distribution has been discretized, meaning a phase error of up to 35 for just a few elements in the array. , 20 , 30 , The discretized angles of incidence are 40 , and , , , 180 . The 17 combinations that were previously evaluated in the reflection coefficient computation (see Fig. 3) correspond to these angles. Though the results of Section II showed the phase and amplitude variations with the angle of incidence could possibly be neglected, here, for a more accurate design, the discretized oblique angles of incidence are considered. As the phase excursion between adjacent angles of incidence is very small, this discretization approach provides accurate results while reducing computation efforts. The far-field radiation patterns have been computed for the following four cases: 1) Ideal elements: phases available in the whole 360 range and no loss; 2) Ideal elements in a restricted range of 290 : The losses are neglected, and the phase required to focus the beam has been restricted to the 290 provided by the proposed element; 3) Designed patches assuming normal incidence in the 290 range: The size of the patches has been computed from the normal incidence curves, considering the loss of the element; 4) Designed patches assuming oblique incidence in the 290 range: This approach is the most realistic one, limiting the phase range to 290 and taking into account the discretized angles of incidence. A rigorous assessment of the cross-polarization can only be carried out using this accurate description, and this is made here using the full vectorial 2 2 reflection coefficient description of the cells [6]. Fig. 7 shows the radiation patterns in elevation and azimuth for all four cases described above at 1.3 THz. The gain obtained in each case is, respectively: 29.30, 29.15, 26.90, and 26.80 dBi.
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 12, 2013
that obtained with a reflectarray of the same aperture and phase range using the resonant gold patches of Fig. 5. Finally, Fig. 9 compares the gain curves for both cases. A 1-dB-gain bandwidth of 15% and 11% is achieved for graphene and gold reflectarrays, respectively. The graph also reveals the lower maximum gain of the array made with graphene, which is due to the higher ohmic loss in graphene than gold at such frequencies. IV. CONCLUSION A reflectarray antenna based on square graphene patches has been proposed for the first time and rigorously modeled. The plasmonic propagation supported by the element enables drastically reduced interelement spacing and good array performance in terms of bandwidth, cross polarization, and design simplification at the cost of slightly higher loss. This letter demonstrates the feasibility of using graphene in—potentially reconfigurable—reflectarrays for future low-cost and efficient directive THz antennas. Fig. 7. Radiation patterns in gain for the different approaches. (a) Elevation. (b) Azimuth.
Fig. 8. Cross-polarization level computed from the case with oblique inci, . dence,
Fig. 9. Theoretical gain curve for the proposed graphene-based reflectarray and the gain provided by a gold-based reflectarray.
It is observed that the limited 290 phase range has noticeable impact only on the secondary lobe levels, which nevertheless remain more than 25 dB below the main beam. The loss in the graphene patches produces less than 2.3 dB of gain reduction, which, as expected, roughly corresponds to the average element loss in Fig. 3(b). Because of the small dependence of the element response to the incident angle, neglecting this effect only leads to a small 0.1-dB gain difference, without considerable changes in the shape of the beam even far away from the main beam. The cross-polarization is more than 35 dB below the maximum of the main beam, as can be seen in Fig. 8 in the conventional - coordinates. This value is 4 dB better than
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