Gate-controlled mid-infrared light bending with aperiodic graphene nanoribbons array
Eduardo Carrasco, Michele Tamagnone, Juan R. Mosig, Tony Low, Julien Perruisseau-Carrier
GGate-controlled mid-infrared light bending with aperiodic graphene nanoribbons array
Eduardo Carrasco † , Michele Tamagnone † ,
1, 2
Juan R. Mosig, Tony Low ∗ ,
3, 4 and Julien Perruisseau-Carrier Adaptive MicroNano Wave Systems, Ecole Polytechnique Federale de Lausanne (EPFL), 1015 Lausanne, Switzerland Laboratory of Electromagnetics and Acoustics (LEMA),Ecole Polytechnique Federale de Lausanne (EPFL), 1015 Lausanne, Switzerland Department of Physics & Electrical Engineering,Columbia University, New York, NY 10027, USA Department of Electrical & Computer Engineering,University of Minnesota, Minneapolis, MN 55455, USA (Dated: August 13, 2018)
Graphene plasmonic nanostructures enable subwavelength confinement of elec-tromagnetic energy from the mid-infrared down to the terahertz frequencies. Byexploiting the spectrally varying light scattering phase at vicinity of the resonantfrequency of the plasmonic nanostructure, it is possible to control the angle of reflec-tion of an incoming light beam. We demonstrate, through full-wave electromagneticsimulations based on Maxwell equations, the electrical control of the angle of reflectionof a mid-infrared light beam by using an aperiodic array of graphene nanoribbons,whose widths are engineered to produce a spatially varying reflection phase profilethat allows for the construction of a far-field collimated beam towards a predefineddirection. † These authors contributed equally ∗ Corresponding email: [email protected]
PACS numbers:
Introduction—
Graphene plasmonics has emerged as a very promising candidate for terahertz to mid-infraredapplications [1], the frequency range where it supports plasmonic propagation. [1, 3–10]. Today the terahertz to mid-infrared spectrum, ranging from 10 cm − to 4000 cm − , is finding a wide variety of applications in information andcommunication, homeland security, military, medical sciences, chemical and biological sensing, spectroscopy, amongmany others [11–13]. The two-dimensionality of graphene and its semi-metallic nature allow for electrical tunability notpossible with conventional metals by simply biasing electrostatically the graphene device. In addition, graphene is alsoan excellent conductor of electricity, with highest attained carrier mobility reaching 1 , ,
000 cm /Vs in suspendedsamples [14] and 100 ,
000 cm /Vs for ultra-flat graphene on boron nitride [15]. These attributes, in addition to itsstability and compatibility with standard silicon processing technologies, have also raised interest in a myriad ofgraphene-based passive and active photonic [2, 22, 23] and terahertz devices [3, 7, 17–21].Electromagnetic surfaces able to dynamically control the reflection angle of an incident beam have been studiedfor decades in the microwave community [16, 24], with applications in satellite communications, terrestrial anddeep-space communication links. Non-conventional reflecting surfaces for optical frequencies have also been proposedrecently, using metal elements having certain fixed configurations and operating in the plasmonic regime [25, 26].However, the high carrier concentration in metals prohibits dynamic control of the reflected beams via these surfaceelements. In this letter, we show how the reflection angle of a mid-infrared incident beam can be electricallycontrolled with an aperiodic array of graphene nanoribbons, by utilizing its intrinsic plasmonic resonance behavior.We demonstrate that the proposed effect is experimentally realizable by performing numerical Maxwell simulationsof the device with experimentally feasible parameters. Basic Concepts—
Consider an aperiodic array of graphene nanoribbons of varying widths lying at the interface oftwo half spaces as illustrated in Fig. 1a, with p -polarized light at incidence angle θ i , which is reflected and transmittedat angle θ r and θ t respectively. Each nanoribbon constitutes a plasmonic resonator, which can effectively produce ascattering phase φ between 0 and − π depending on the frequency of free-space light ω with respect to the plasmonresonance frequency ω . In plasmonic metasurfaces consisting of nanoribbons of varying widths, the scattering phasein general can also vary across the interface, namely as φ ( x j ), where x j = j ∆ x and ∆ x being the distance betweenthe centers of adjacent ribbons, which is considered constant in this contribution. In the limit where ray optics is a r X i v : . [ phy s i c s . op ti c s ] D ec applicable, i.e. λ (cid:29) ∆ x , where λ is the free-space wavelength, the generalized Snell’s law dictates that [25],sin( θ t ) √ (cid:15) − sin( θ i ) √ (cid:15) = λ π dφdx sin( θ r ) − sin( θ i ) = λ π √ (cid:15) dφdx (1)Eq. 1 implies that the reflected or transmitted beam can be effectively bent such that θ r (cid:54) = θ i,t if a constant spatialgradient in the scattering phase is imposed (i.e. dφ/dx = constant). Simple estimates from Eq. 1 suggest that it ispossible to bend normal incident ( θ i = 0) mid-infrared light far from broadside (i.e. θ r (cid:54) = 0). Assuming λ = 10 µ m(e.g from a CO laser), ∆ x = 100 nm (typical ribbon width where plasmon resonance resides in the mid-infrared), oneobtains ∆ φ (cid:28) π , suggesting that it is indeed possible to induce a gradual spatial variation in φ across the interface.The scattering phase due to a graphene plasmonic resonator, and its design space, can be examined more quan-titatively as follows. We consider graphene on a SiO substrate and with an electrolyte superstrate, which can alsoserve as a top gate for inducing high doping in graphene [28]. We solve the Maxwell equations for the reflectioncoefficient of a p -polarized light, r p ( q, ω ), where graphene is modeled by its dynamic local conductivity σ ( ω ) obtainedfrom the random phase approximation [29, 30]. The scattering phase φ then follows from φ ( q, ω ) = arg[ r p ( q, ω )].The nanoribbon resonator frequency ω can be estimated from the scattering coefficients for a continuous monolayer,where the in-plane wave-vector q is related to the width W of the nanoribbon via q = 3 π/ W , after accounting forthe anomalous reflection phase off the edges [27]: ω = (cid:115) e µ W (cid:126) (cid:15) env (2)where (cid:15) env = ( (cid:15) + (cid:15) ). The Lorentz oscillator model provides then a simple expression for the scattering phase: φ ( W, ω ) ≈ tan − (cid:32) ωτ (cid:18) ω − e µ W (cid:126) (cid:15) env (cid:19) − (cid:33) (3)In this simple model calculation, we simply take (cid:15) = 3 . (cid:15) = 6. Assuming λ = 10 µ m, Fig. 1b-c depicts φ ( W, ω )for varying W , with graphene at different chemical potentials µ , and electronic lifetimes τ . The phase φ changesmost rapidly with W when the ribbon is at resonance. For a given ribbon plasmon resonance frequency, increasingthe doping would allow for the same resonance frequency at a larger W as depicted in Fig. 1b. On the other hand,decreasing τ dampens the plasmon resonance, causing a smoother variation in φ with W as shown in Fig. 1c. Eq. 3therefore provides a simple intuitive understanding of the scattering phase of a graphene plasmonic resonator. Device Simulation—
In principle, the tunable scattering phase of a graphene plasmonic resonator allows the designof surface elements which controls the angle of reflection or transmission of an incoming beam. In this work, weconsider the former i.e. a reflectarray [16]. Fig. 2a-b shows a detailed schematic of the proposed device, designedfor a working frequency of 27 THz (900 cm − ). The graphene nanoribbons array is between an electrolyte gatingsuperstrate of 200 nm and a 1 . µ m SiO dielectric substrate with a metal layer underneath, which serves as a reflector,reflecting most of the incoming light. The full dielectric function of SiO is used in the simulation [31]. For theelectrolyte superstrate, we assumed a dielectric constant of 6. Hence, the effective dielectric constant of the graphene’senvironment, which ultimately determines its plasmonic response, is approximately 5, similar to Ref. [3]. The aperiodicarray of nanoribbons has a designed inter-ribbon separation p taken to be 140 nm, and the width of each nanoribbonsis to be chosen so that the spatial variation in φ r satisfies dφ r /dx = const., as discussed previously. As explainedlater, the final reflection phase φ r of the reflecting cell is not given simply by the response φ of the ribbon, since thecontribution of the ground plane must also be taken into account. We consider a Gaussian beam illuminating thearray at an incident angle 45 o with respect to the normal of the xz plane as illustrated. Fig. 2c depicts the top view ofthe nanoribbons array, including the elliptical projection of the impinging beam on the xz plane, where b x = 31 . µ mand b y = 22 . µ m.We use CST Microwave Studio to numerically compute the reflection coefficient and φ r for a nanoribbon of particularwidth. Using the Floquets theory for periodic arrays, mutual coupling between neighboring nanoribbons is accountedfor. Graphene is modeled by its dynamic local conductivity σ ( ω ), assuming typical electronic lifetime of τ = 0 . φ r of the reflection coefficient, as a function of the chemical potential µ and width W of the nanoribbon. For a chemical potential µ = 1 . φ r varies between 0 o and − o by adjusting the widthof the ribbons between 40 nm and 140 nm. As µ decreases, the range of φ r decreases, which approaches a constant φ r = − o at µ = 0 . o ) and a reflected far-field beam in the specular direction (45 o ). The spatial phase profilerequired to achieve these reflection angles can be determined based on Eq. 1, and are shown in Fig. 3b for these twocases. For specular beam, a zero phase difference between the nanoribbons would suffice, regardless of the absolutevalue of that phase. Otherwise, the sequence of ribbons’ widths has to be chosen such that it provides the neccessaryspatial phase profile to produce a reflected far-field beam at µ = 1 . φ r can be made to vary between 0 o and − o . The missing phase values (from − o to − o )were found not to have noticeable impact in the far-field generation. The widths of the nanoribbons along the arrayare displayed in Fig. 3c. In order to produce a reflected far-field beam in the specular direction, a constant φ r isrequired along all the elements of the array. This can be achieved by decreasing µ to 0 . µ .In Fig. 4a and b, we compute the far-field produced by our reflectarray device, for broadside ( µ = 1 . µ = 0 . Discussion—
A constant phase gradient is needed for producing a collimated beam. At intermediate dopings, wherethe ribbon array phases are not designed with constant phase gradient, the produced far field beam can be highlydistorted. The design of smooth beam steering is possible, but would require a more complicated gating schemethat addresses the doping of individual ribbons separately as typically done in microwave reflectarrays [33] and morerecently also for terahertz graphene-based reflectarrays [32]. The simplicity in the reflectarray array scheme proposedin this work has the obvious appeal of providing a design much easier to be implemented experimentally.A simple circuit model for each nanoribbon cell, as shown in Fig. 2d, is useful for better understanding of the pro-posed device. The substrate and superstrate can be modelled with two transmission line segments having propagationconstants and characteristic impedance equal to the two media which they model. The equivalent of the ground planeis simply a short circuit, while the graphene ribbons can be represented by an equivalent Z g impedance in parallel,that models the surface currents induced in the ribbons by the tangential electric field. An approximate closed formexpression for Z g can be found by modeling resonant ribbons as Lorentz oscillators as explained previously. Thiscan be represented by a simple RLC series circuit, where R and L are simply found from the real and imaginaryparts of the surface conductivity of graphene σ , while C represents the quasi-static electric fields associated with theplasmons, and can be obtained by enforcing the resonant frequency of the RLC circuit (1 / √ LC ) to be equal to theribbons resonant frequency ω : L = π (cid:126) e µ R = τ − L C = 1 ω L (4)where ω ≈ (cid:115) e µ W (cid:126) (cid:15) env (5)The final equivalent impedance of graphene nanoribbons is then, Z g = R + jωL + 1 jωC = π (cid:126) e µ (cid:18) ω + jωτ − − ω jω (cid:19) (6)This circuit model provides for an estimated value for the reflection phase φ r as function of W and µ . Fig. 3d showsthe calculated reflection phase φ r which agrees qualitatively with the numerical simulations presented in Fig. 3a. Thefull wave simulations account for higher order phenomena not captured by this simple model. Importantly, while inthe Lorentz oscillator the phase range is always less than π for the nanoribbons alone (i.e. Eq. 3), the presence of thesubstrate and of the ground plane allows here a much wider phase range, which is a key point for the device presentedin our work.Our device has an average element loss of 1 .
32 dB for the µ = 0 . .
77 dB for µ = 1 eV.Hence, most of the incoming light is reflected. The loss of the device is directly related to the assumed electroniclifetime τ , as evident from Eq. 4, and can be improved with better graphene quality or by using a better substratesuch as boron nitride [34]. Conclusion—
In this work, we show how the reflection angle of a mid-infrared beam can be dynamically controlledby employing tunable graphene plasmons in an aperiodic array of graphene nanoribbons according to the generalizedlaws of reflection. The attractive feature of our device proposal, the use of a simple gating scheme, renders it moreexperimentally feasible and practical. Numerical Maxwell device simulations convincingly demonstrate the dynamicbending of mid-infrared beams, assuming experimentally accessible physical parameters.
Acknowledgements—
This work was supported by the European Union (Marie Curie IEF 300934 RASTREOproject), the Hasler Foundation (Project 11149) and the Swiss National Science Foundation (Project 133583), andthe University of Minnesota. E. Carrasco, M. Tamagnone, T. Low and J. R. Mosig would like to dedicate this paperto the memory of J. Perruisseau-Carrier, who passed away during the preparation of this work.
Figure CaptionsFigure 1: (a) Schematic illustrating the dynamic control of the angle of reflected and transmitted light byintroducing gradient of scattering phases at the interface using graphene nanoribbon resonators with different widths.(b-c) Calculated scattering phase φ for varying ribbons’ width for p -polarized light with free-space wavelength λ = 10 µ m, e.g from a CO laser. Calculations are done for different chemical potentials µ and electronic lifetimes τ as shown. Symbols represent calculation from Maxwell equation, while solid lines are from simple model in Eq. 3. Figure 2: (a) Schematic of the gate-controlled reflectarray device based on aperiodic array of graphene nanorib-bons. (b) Top view of the graphene nanoribbons array, including the elliptical projection of the impinging beam.(c) Lateral view of the device, highlighting the superstrate, substrate, ribbons, and metal reflector. (d) Equivalentcircuit for one element of the proposed array.
Figure 3: (a) Phase φ of the reflection coefficient as a function of chemical potential µ for different widths W ofa nanoribbon, ranging from 40 nm to 140 nm. Mutual coupling between neighboring ribbons is taken into account.(b) Spatial phase profile required at each nanoribbon along the x -axis in order to produce a far-field beam in thebroadside direction (0 o ) and a reflected far-field beam in the specular direction (45 o ). (c) Width of the nanoribbonsacross the array ( x -axis) of the reflectarray device. (d) Phase φ of the reflection coefficient as a function of chemicalpotential µ for different widths W of a nanoribbon using the equivalent circuit model described in Fig. 2d. Figure 4:
Far-field radiation pattern produced by the reflective array for the two bias condition; (a) Beam towardsbroadside (0 o ) at µ = 1 . o ) at µ = 0 . (a) xz ∆ x W W W W W θ i θ r θ t t =0.1ps 150 m =1.0eV (b) (c) -180 -120 -60 0306090120150 W i d t h s ( n m ) Phase f (degrees) t =0.1ps0.8eV m =1.0eV0.6eV -180 -120 -60 0306090120150 W i d t h s ( n m ) Phase f (degrees) t =0.01ps0.1ps m =1.0eV1ps Figure 1 b y Incident beam Reflected beamGraphene nanoribbons z (b) pW b x h G SuperstrateSubstrateSuperstrate nanoribbonsMetal reflector (a)
WSubstrate xz (c) (d) p Z o sup , b sup h Z o sub , b sub Z g (R L C) Figure 2 P ha s e f ( deg r ee s ) Chemical potential m (eV) W=140nmW=40nm 0 50 100 150 200-360-300-240-180-120-600 P ha s e f ( deg r ee s ) Nanoribbons Number m =0.3eV m =1.0eV (a) (b) W i d t h s W ( n m ) Nanoribbons Number (c) P ha s e f ( deg r ee s ) Chemical potential m (eV) W=140nmW=40nm (d)
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