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November 2013

EPL, 104 (2013) 37001 www.epljournal.orgdoi: 10.1209/0295-5075/104/37001

Mid-infrared, plasmonic switches and directional couplers inducedby graphene sheets coupling system

Hongju Li, Lingling Wang(a), Zhenrong Huang, Bin Sun, Xiang Zhai and Xiaofei Li

School of Physics and Microelectronic and Key Laboratory for Micro-Nano Physics and Technology of Hunan Province,Hunan University - Changsha 410082, China

received 29 September 2013; accepted in final form 30 October 2013published online 25 November 2013

PACS 78.67.Wj – Optical properties of graphenePACS 73.20.Mf – Collective excitations (including excitons, polarons, plasmons and other

charge-density excitations)PACS 52.75.Kq – Plasma switches (e.g., spark gaps)

Abstract – In this letter, the original switch effect induced by a double-layer graphene sheetscoupling system is investigated comprehensively by using the finite-difference time-domain method(FDTD). Simulation results reveal that the transmission varies periodically with the increase ofthe graphene sheets’ overlapping length. The switch effect can be modulated by an ingenious andsimple way of displacing one monolayer graphene horizontally in the graphene coupling system.At the same time, tuning the space between the graphene sheets and its chemical potential canalso give rise to the perfect switch effect. The FDTD results are consistent with the theoreticalcalculations. As an application, a three-layer graphene coupling system is constructed. By thesame principle, it works not only as an effective optical spatial switch but also as a perfectdirectional coupler. The proposed structures are operational in the mid-infrared region and willplay a significant role in nano-integrated circuits for optical processing and switching.

Copyright c© EPLA, 2013

Introduction. – Graphene [1–3], a two-dimensionalmaterial with only one atom thickness, exhibits a greatdiversity of electronic and optical features such as extremeconfinement, advantageous tunability, and low losses [4].So it has been considered as a promising candidate forplasmonic material for constructing nano-optical devicesand systems for a wide wavelength ranging from near-infrared to THz [5], and a growing number of researchfoci are pointing to studying plasmonic effects [6] anddesigning novel optical devices based on graphene. Forexample, the monolayer graphene sheet [7–10] has beenused to fabricate a surface plasmon resonance sensor [11].It has been also reported that the S-shaped waveguide,spiral waveguide, and curved waveguide [12] based onbending graphene sheet, are proposed, in order to makefull use of its properties of tight confinement and lowlosses. Surface Plasmon Polariton (SPP) modes excitedon graphene nano-ribbons [13–15] by guided-mode res-onances have been discussed in detail. More recently,more and more attention has been paid to optical couplingof surface plasmons between multilayer graphene sheets.

(a)E-mail: llwang@hnu.edu.cn

The characteristics of symmetric and antisymmetric SPPmodes [16–18] supported by a parallel graphene pair sep-arated by small gaps, have been investigated comprehen-sively. At the same time, strong coupling SPPs in mono-layer graphene sheet arrays [19] have also been analyzedtheoretically and numerically. As applications, zero in-sertion loss optical splitters [20] and spatial switches [21]based on a three-layer graphene coupling system havebeen covered not long ago, and then the ultra-compactMach-Zehnder interferometer [22] is demonstrated. Thoseunique structures based on graphene sheets take full ad-vantage of the dynamical tunability of the surface con-ductivity of graphene, by means of gate voltage [23–26]or chemical doping [27], which affect directly the chemicalpotential of graphene.Through previous works about graphene, we were im-

pressed to find out that plasmonic devices based ongraphene present indeed more wonderful properties thanthat consisting of conventional noble metals [7,12,21].They include the advantageous propagation propertiesof SPPs and of engineering easily the SPPs. Hencegraphene yields a new promising platform for new-generation nano-photonics in the mid-infrared region.

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Hongju Li et al.

However, the changes of the graphene sheets’ overlappinglength, interlayer space and chemical potential that canlead to a lot of unusual optical effects have not been re-ported systematically so far. In the present letter, weconcentrate on investigating numerically the novel switcheffect induced by the double-layer graphene coupling sys-tem by using the FDTD method with perfectly matchedlayer absorbing boundary conditions. Simulation resultsshow that the switch effect can be realized easily by theingenious way of displacing one monolayer graphene hor-izontally in the double-layer graphene coupling system.The transmission varies periodically with the overlappinglength increase. At the same time, tuning the space be-tween the graphene sheets and its chemical potential canalso give rise to the perfect switch effect. The FDTDresults are consistent with the theoretical calculations.As an application, one three-layer graphene coupling sys-tem is fabricated. By the same principle, displacing thegraphene sheet of different outputs horizontally can makethe SPPs propagate on different outputs. Thus, the three-layer graphene coupling system acts not only as a perfectdirectional coupler but also as an effective optical switch.Undoubtedly, the ways to realize switch effect are moresimple and meaningful. The proposed structures are op-erational in the mid-infrared region and our studies willbenefit the fabrication of the ultra-compact plasmonic de-vices for optical switching and processing.

Theory and simulations. – To start with, the ba-sic double-layer graphene coupling system is shown infig. 1(a), where the parallel interlaced graphene pair isembedded in air. The overlapping length and the spacebetween the graphene sheets are denoted by L, d, respec-tively. During calculations, the graphene sheet is consid-ered as an ultra-thin film with a thickness t (= 1 nm). Thesurface conductivity (σg) of graphene needed for this sim-ulation is governed by the Kubo formula [9,23,28], whichdepends on the chemical potential (Fermi energy) μc, in-cident wavelength λ (frequency ω), temperature T , andmomentum relaxation time τ . In the implementation, μc,λ, T , and τ are assumed to be 0.15 eV, 10μm, 300K, and0.5 ps, respectively. The equivalent permittivity (εeq) ofgraphene is attained by the following equation [21]:

εeq = 1 + iσgη0/(k0t), (1)

where k0 = 2π/λ and η0 (≈ 377Ω) is the impedance of air.Thus, εeq = −45.05 + 0.72i can be calculated easily. It isobvious that the real part of the equivalent permittivity isless than 0, so the graphene sheet works as a thin metalfilm and supports a TM mode SPP wave. With regardto the proposed structure of the double-layer graphenecoupling system shown in fig. 1(a), one phenomenon canbe predicted easily that the SPP wave propagating onthe left input graphene sheet will couple into the outputsheet, according to the general regulations of the tradi-tional dielectric optical directional couplers. One unusualprocess should appear repeatedly that the SPP wave firstly

Fig. 1: (Color online) Schematic diagram of the proposeddouble-layer graphene coupling system (a). Magnetic-field(Hz) distributions of the SPPs at the overlapping length L =220 nm (b), 420 nm (c).

transforms into the output sheet from the input graphenelayer and then couples to the input sheet anew, as longas the overlapping length is sufficient. In other words, thetransmission of the right output will vary periodically withthe increase of the graphene sheets’ overlapping length.For the sake of verifying our analysis, a series of numer-

ical simulations are carried out by using the 2D FDTDmethod. In this case, the minimum mesh size (Δ) insidethe graphene layer equals 0.1 nm and increases graduallyoutside it, for saving storage space and computing time.The temporal step is Δ/2c, where c is the velocity of lightin vacuum, for the good convergence of the numerical cal-culations. A single dipole source is placed 2 nm over theinput graphene sheet, so as to excite the TM mode SPPwave. Simulation results are tidied clearly in figs. 1(b)and (c). In fig. 1(b) where the L and d are chosen tobe 220 nm, 50 nm, respectively, the incident SPP poweris transferred almost completely to the output sheet fromthe left input graphene layer. Moreover, the overlappinglength is about λ/45, illustrating that the structure couldbe ultra-compact and works deeply under the diffractionlimit. When the overlapping length increases to 420 nmby displacing horizontally the output graphene layer, theSPPs propagating on the output channel couple into theinput sheet anew, as shown in fig. 1(c), resulting in a lowtransmission at the output sheet. Consequently, these sim-ulation results confirm the above predictions.Next, we display further the transmission spectrum

of the double-layer graphene coupling system withoverlapping length L, shown in fig. 2(a). Obviously,the transmission varies periodically with the overlap-ping length increase, because of the repetitive processby which the incident power is transferred to the out-put sheet and then is transferred back again gradually.It is consistent with previous analysis. The couplinglength, over which the power is completely transferredfrom the input graphene to the output, is governed byLc = π/

√2|β+ − β−| [29], where β± stand, respectively,

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Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system

Fig. 2: (Color online) (a) The transmission spectrum of theoutput channel with overlapping length L. Magnetic-field (Hz)distributions of SPPs at the overlapping length L = 80nm (b),160 nm (c), respectively.

for the wave vectors of the symmetric and antisymmetricSPP modes supported by the double-layer graphene. β±can be obtained by the following equation [21]:

β± = kspp +2ik0/ (η0σg)− kp (1∓ up)(1∓ up) kspp/kp ± upksppd, (2)

where kspp = k0

(1− (2/η0σg)2

) 12

, kp =(k2spp − k20

) 12 ,

and up = exp(−kpd). In this case, β+ = 47.8+0.60i μm−1and β− = 37.71+0.79i μm−1 [21]. So Lc = 220.1 nm is ver-ified by the simulation results shown in fig. 2(a). Besides,the power loss will decrease with the increase of the over-lapping length and the corresponding maximal transmis-sion increases at L = 720 nm, as seen in fig. 2(a). Onthe other hand, two abnormal points, L = 80 nm andL = 160 nm, can be found easily in the first quarter-periodof the transmission spectrum. They break away from theintrinsic regulations of the transmission spectrum, andthe corresponding magnetic-field (Hz) distributions of theSPPs are shown in figs. 2(b) and 3(c), respectively.Clearly, the reason why the two remarkable points ap-

pear in the transmission spectrum is that the reflectedwave from the end of the input graphene layer has an in-fluence on the incident wave. In the first quarter-period,only the SPP waves transfer from the input graphene tothe output. When the L = mλ−/2 (m = 1, 2), where theλ− = 2π/Re(β−) is the wavelength of the antisymmetric

Fig. 3: (Color online) The magnetic-field (Hz) distributionsof the SPPs at L = 100 nm (a), 220 nm (b) with d = 30 nm;L = 220 nm (c) with d = 50 nm.

modes (166 nm), the antisymmetric coupling resonanceforms in the overlapping section resulting from the effectof the reflected wave. Therefore, most power is confinedto the overlapping section shown in figs. 2(b) and (c),where the corresponding overlapping lengths are 80 nmand 160 nm, leading to the sharp decrease of the trans-mission in fig. 2(a). However, when the overlapping lengthis larger than 220.1 nm, the process that the SPP wavesfirstly transform into the output sheet from the inputgraphene layer and then couple to the input sheet anewwill appear in the overlapping section. There will be a va-cant region without SPP waves in the overlapping section,such as the M area shown in fig. 1(c). So the antisym-metric coupling resonance modes form hardly in the wholeoverlapping section and there is no obvious abnormal pointappearing in the other periods of the transmission spec-trum. Certainly, in most instances, the transmission variesregularly with the overlapping length increase and a per-fect switch effect can be brought about in the double-layer graphene coupling system easily by the simple wayof displacing the output monolayer graphene horizontallyto change the overlapping length.

Based on eq. (2), it is well know that β± are also relatedto the space d between the graphene sheets in addition toL, and β± affect the coupling length Lc directly. Hence,the change of d will have an effect on the Lc and thetransmission spectrum with the overlapping length. Thesimulation result is presented in fig. 3(a) where d = 30 nmand Lc is changed to 100 nm, which is consistent with thetheoretical value 99 nm. At the same time, the transmis-sion spectrum can also be modified by the change of d. Sothe switch effect is also induced by displacing the outputmonolayer graphene perpendicularly to tune the space d.This is indeed confirmed by the FDTD results, shown infigs. 3(b) and (c) where the values of the d are respec-tively 30 nm and 50 nm, with the same overlapping lengthL = 220 nm. On the other hand, β± can be influenced bythe chemical potential μc of the graphene. According tothe above analysis, the role of μc will be the same as thatof d and L. Thus, we can tune the chemical potential of

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Hongju Li et al.

Fig. 4: (Color online) The magnetic-field (Hz) distributionsof the SPPs at µc = 0.15 eV (a), 0.08 eV (b) with the samestructural parameters d = 50nm and L = 420 nm.

the graphene sheets to achieve the switch effect by dopingor gate voltage. As shown in fig. 4 where L = 420 nm andd = 50 nm, only the chemical potential is changed and itexhibits a perfect switch effect. As a result, the switcheffect can be easily realized in the double-layer graphenecoupling system not only by displacing the output mono-layer graphene horizontally or perpendicularly but also bytuning the chemical potential of the graphene.

As an application, one three-layer graphene couplingsystem, consisting of one input graphene sheet and twosymmetric output sheets, is constructed. Similarly, thetransmission of the three-layer graphene coupling sys-tem will vary periodically with the overlapping length in-crease, and an abnormal point only appears in the firstquarter-period, according to the above analysis about thestructure displayed in fig. 1(a). The related transmis-sion spectrum is displayed in fig. 5(a). The overlappinglength is changed by displacing horizontally the two out-put graphene sheets simultaneously. When the trans-mission is a peak value, the three-layer graphene systembehaves as a superior optical splitter, described in thefig. 5(b), where the overlapping length is 160 nm and theincident power is transferred equably to the two outputs.The theoretical value of the Lc = π/2|β+ − β−| is 156 nmand is verified by the simulation result shown in fig. 5(a).On the contrary, little SPP power passes the outputs whenthe transmission is a valley value, as shown in fig. 5(c)where the overlapping length is 280 nm and d is kept as50 nm. As a result, the two-channel switching effect canbe generated on the three-layer graphene coupling systemeasily, by means of displacing horizontally the two sym-metric graphene output sheets simultaneously.

In addition, the three-layer graphene coupling systemprovides a more flexible tunability for practical appli-cation. We can displace horizontally the two outputgraphene sheets asynchronously, resulting in differentoverlapping lengths at the upper output and lower out-put. For instance, the overlapping length of the upperoutput is altered to be 300 nm and the other is changedto be 200 nm, and the corresponding magnetic-field (Hz)distributions are shown in fig. 6(a). Clearly, the inci-dent SPP power is only coupled to the upper output sheet

Fig. 5: (Color online) The transmission spectrum of the three-layer graphene coupling system with respect to the overlappinglength (a). The magnetic-field (Hz) distributions of the SPPsat L = 160 nm (b), 280 nm (c).

Fig. 6: (Color online) Magnetic-field (Hz) distributions of theSPPs only coupled to the upper output (a), or the lower out-put (b).

directionally and little power passes to the lower output,and the corresponding transmissions are 0.182, 0.006, re-spectively. Furthermore, the power output is switchedinversely as demonstrated in fig. 6(b), where the upperand lower overlapping lengths are changed to be 200 nm,300 nm, respectively. In short, the three-layer graphenecoupling system acts not only as an effective digital op-tical switch but also as a perfect directional coupler, bydisplacing horizontally the two output graphene sheets tochange the overlapping lengths. This way it is simpler andmore convenient to operate.

Conclusions. – In a word, the novel switch effectcan be realized based on a double-layer graphene sheets

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Mid-infrared, plasmonic switches and directional couplers induced by graphene sheets coupling system

coupling system, which is predicted theoretically andinvestigated numerically by using the 2D FDTD method.Simulation results reveal that the transmission varies pe-riodically with the overlapping length increase, exhibitingperfect switch effect. It is changed by the simple wayof displacing the output graphene layer horizontally. Inaddition to this, displacing perpendicularly the outputgraphene to tune the space between the sheets or changingthe chemical potential of the graphene sheets by means ofthe gate voltage can also give rise to the perfect switcheffect. The FDTD results are verified by the theoreticalcalculations. As an application, one three-layer graphenecoupling system is fabricated. By the same principle, thethree-layer graphene coupling system behaves as not onlya perfect optical spatial switch but also an effective di-rectional coupler. Undoubtedly, the ways to induce theswitch effect are ingenious and meaningful, and the pro-posed structures are operational in the mid-infrared re-gion. Our studies will benefit the fabrication of the highlyintegrated plasmonic devices for optical processing andswitching.

∗ ∗ ∗

This work was supported by the National NaturalScience Foundation of China (Grant Nos. 11074069,61176116, 11264021), the Specialized Research Fund forthe Doctoral Program of Higher Education of China(Grant No. 20120161130003), and Aid program for Sci-ence and Technology Innovative Research Team in HigherEducational Institutions of Hunan Province.

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