Gate-Programmable Electro-Optical Addressing Array of Graphene-Coated Nanowires with Sub-10 nm ResolutionHuawei Liang,†,‡ Lei Zhang,‡ Shuang Zhang,§ Tun Cao,∥ Andrea Alu,⊥ Shuangchen Ruan,*,†

and Cheng-Wei Qiu*,‡

†Shenzhen Key Laboratory of Laser Engineering, College of Optoelectronic Engineering, Shenzhen University, Shenzhen, 518060,People’s Republic of China‡Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore§School of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, U.K.∥Department of Biomedical Engineering, Dalian University of Technology, Dalian, 116024, People’s Republic of China⊥Department of Electrical and Computer Engineering, University of Texas at Austin, Austin, Texas 78712, United States

*S Supporting Information

ABSTRACT: The rapid development of highly integrated photoniccircuits has been driving electro-optic (EO) devices to increasinglycompact sizes, with the perspective of being able to control light atthe nanoscale. However, tunability with spatial resolution below 10nm scale with conventional approaches, such as metallic nanowires,remains a challenge. Here, we show a graphene-coated nanowiresystem aiming at beam spatial modulation at a deeply subwavelengthscale. By analytically and numerically investigating the eigenmodalproperties of this system, we found that beam power can propagatealong either a swinging or a helical path in the hybrid nanowire. Inparticular, the period of the swing beam and the chirality and periodof the helix beam can be flexibly controlled by tuning the chemicalpotential of graphene via the gate voltage. Significantly, due to itsgood modal confinement, such a beam can be independentlymanipulated even in the presence of another nanowire at a separation of 40 nm, which opens a realistic path toward gate-programmable EO addressing or data storage with ultrahigh density (64 terabyte/μm). At the same time, by fulfilling the phasematching condition between the two supported guided modes operating at different wavelengths, either a full band or band-tunable terahertz wave at the nanoscale may be achieved by nonlinear difference frequency generation. Our proposed hybridnanowire system opens interesting potentials to accomplish gate-programmable EO devices at sub-10 nm scale.

KEYWORDS: electro-optic addressing, sub-10 nm scale, graphene, phase matching, terahertz-wave generation

Optoelectronic devices based on high-speed electro-optic(EO) modulation, such as directional coupler

switches1−3 and optical modulators,4−6 have been keycomponents in optical communications and signal processing.The sustained interest in highly integrated and miniaturizedphotonic circuits requires that the size of each component isbrought down to the nanoscale.7,8 However, this requirement isdifficult to fulfill by conventional optoelectronic devices,because of the diffraction limit.9 Although surface plasmonpolaritons (SPPs) supported by metallic structures offer apromising solution to circumvent the diffraction issue,10−13

there is lack of tunability, due to their fixed material properties.It has been demonstrated that, by combining SPPs with EOmaterials, signals at deeply subwavelength scale may betemporally modulated via an applied voltage.14−18 Theemerging graphene platform may serve as an alternative EOmaterial to realize similar control,19−23 since its complexconductivity can be tuned by changing its chemical potential via

the gate voltage, electric field, temperature, chemical doping,and so on.24,25 Noteworthy, in all previous demonstrations,only the intensity of the signal was modulated, while itsposition remained the same. So far, beam spatial modulationwithin a sub-10 nm scale has yet to be attained, which couldshrink the size of nanophotonic devices to a large extent.In this work, we present a scheme to attain gate-

programmable EO addressing, including both spatial andtemporal modulation, using achievable graphene-coated nano-wires.21,26 The proposed waveguide supports two types oftightly confined SPP modes: hybrid plasmonic modesdominated by a longitudinal electric field (EH) and a transversemagnetic (TM) SPP.27,28 Different from optical modes innormal fibers, the polarization state of EH SPP modes may beeither quasi-linearly polarized (QLP) or left-/right-handed

Received: May 28, 2016Published: September 6, 2016

Article

pubs.acs.org/journal/apchd5

© 2016 American Chemical Society 1847 DOI: 10.1021/acsphotonics.6b00365ACS Photonics 2016, 3, 1847−1853

quasi-elliptically polarized (QEP). Interestingly, the modalwidths can be reduced to less than 10 nm. It is found that, bysatisfying the mode phase matching (MPM) condition of theEH and TM SPP modes, their coherent superposition can yielda stable crescent beam, just as a nanoscale crescent propagatesalong the graphene-coated nanowire. Once deviating from theMPM condition, the beam can propagate along either aswinging or a helical path depending on whether QLP or QEPEH SPPs are coherently superimposed to the TM mode. Thispeculiar effect is attributed to modal interference along thebeam propagation direction. Furthermore, both the period ofthe swing beam and the chirality and period of the helix can beconveniently modulated by tuning graphene’s chemicalpotential via the gate voltage. As a result, the beam at a fixedoutput plane can be temporally and spatially varied, enablinggate-programmable EO modulation at sub-10 nm scale.Importantly, even if the pitch in a graphene-coated nanowirearray is as small as 40 nm, each crescent beam can still bemanipulated independently. Furthermore, each nanowire canprovide at least four addressing positions, so an array of 1 μmwidth can be used for gate-programmable EO addressing in 425

space positions, which corresponds to 64 terabyte storagecapacity. Besides, the MPM condition can also be applied tononlinear generation for EH and TM SPPs operating atdifferent wavelengths, which can be used in gate-programmablenonlinear nanophotonic devices. Our theoretical analysis isnumerically verified in the following using COMSOL. Theproposed graphene-coated nanowire system may lead tounprecedented beam control in nanostorage, nanocommunica-tions, and chemical characterization.

■ RESULTS AND DISCUSSIONFigure 1a schematically shows the proposed gate-program-mable EO addressing scheme based on a graphene-coated

silicon (Si) nanowire. Ion gel is used not only to apply the gatevoltage on the graphene but also to achieve index-matchingwith the insulating layer SiO2 preserving the cylindricalsymmetry of the system. The distribution of time-averagedPoynting vectors ⟨S⟩ of the supported SPP modes in thiswaveguide is shown in Figure 1b. In a real experiment, the

guided modes in a nanowire of radius a = 5 nm can be excitedwith various mechanisms, such as hybrid plasmonic wave-guides29 and tapered graphene.30 It is found that the real partsof the propagation constant β1 of EH and TM SPPs are highlydependent on the chemical potential of graphene, allowingelectrical modulation of the supported modes. A swing crescentbeam can be excited via the coherent superposition of QLP EHand TM SPPs. The distribution of the time-averaged Poyntingvector ⟨S⟩ at the output plane can be modulated via the gatevoltage, as shown in the upper panel of Figure 1c. When thereal part of the propagation constants of the two modes areequal, i.e., Δβ1 = 0, the crescent beam will retain the sameshape along the propagation direction, as the pattern labeled bythe address 00 in the upper panel of Figure 1c. Because of thedependence of propagation constants on the chemicalpotential, the position of the crescent beam at the outputplane can be controlled in real time by tuning the appliedvoltage. As a demonstration, when Δβ1L increases from 0 to π,where L is the length of the nanowire, the power of the outputbeam shifts from left to right, as shown by the patternscorresponding to addresses 00, NA, and 01 in the upper panelof Figure 1c. If Δβ1L further increases from π to 2π, the opticalpower will shift back from right to left. When Δβ1L continuesto increase, the evolution will repeat. Movie 1 shows thevariation (in a period 0 ≤ Δβ1L ≤ 2π) of the swing beam at theoutput plane. The period of the swing behavior is determinedby LT = 2π/Δβ1, which is also tunable due to the dependenceof β1 on the chemical potential. Therefore, this waveguide canwork as a dynamically reconfigurable directional coupleroperating with 10 nm scale resolution.Similarly, a helix crescent beam can be excited via the

coherent superposition of EH+1 (QEP EH) and TM SPPs. Alsoin this case, the distribution of ⟨S⟩ can be modulated by theapplied gate voltage. Three representative profiles are labeledwith addresses 10, 00, and 11 in the lower panel of Figure 1c. Itis found that the crescent beam rotates anticlockwise as Δβ1L isincreased with the gate voltage. The evolution in a period 0 ≤Δβ1L ≤ 2π of the output helix beam is shown in Movie 2.While Δβ1L decreases, the crescent beam will rotate clockwise.When the helix crescent beam is superposed by EH−1 (QEPEH) and TM SPPs, the rotation behavior will be reversed. Theperiod of the helix beam is determined in the same manner asthat of the swing one. When the wavelength is far away fromthe MPM points, it is hard to gain a stable crescent shape, butthe beam can still propagate along a swinging or a helical path.It is worth emphasizing that both the period of the swing beamand the chirality and period of the helix beam can bemanipulated by tuning graphene’s chemical potential via thegate voltage, so the position of the crescent beam at the outputplane is electrically controllable, which enables a gate-programmable EO addressing at sub-10 nm scale, just asshown in Figure 1c. Furthermore, the graphene can operate at500 GHz, determined by its intrinsic carrier relaxation time,31,32

so the speed of the graphene-based EO addressing is mainlylimited by the RC time constant of the modulation system. It isreported in ref 33 that the resistance (R) was directlyproportional to the thickness (2.2−13.4 μm) and the reciprocalarea (0.01−0.06 cm2), whereas the specific capacitance (C′)was insensitive to the film geometry. The gel polarization timeconstants RC (where C = C′ × area) could be as small as 2.8 μsfor 2.2 μm thick ion gel films. In our case, the thickness of theion gel is 20 nm and the area is 40 × 200 nm2. That may inprinciple lead to an even faster modulation rate, although this is

Figure 1. Gate-programmable EO addressing at 10 nm scale. (a)Schematic architecture. (b) Time-averaged Poynting vector distribu-tions of guided modes in the graphene-coated nanowire. (c) Spatialmodulation of time-averaged Poynting vector distributions of crescentbeams at the output plane via gate voltages. Each distribution can beassigned to an address for programming purposes. The wavelength ofincident light is λ = 2960 nm. Parts b and c share the same colorlegend.

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quite subjective to the satisfactory material candidate in theexperiment, which is still quite challenging now.As shown in Figure 1b and c, the optical power is mainly

confined inside the nanowire, so it is possible to manipulateeach crescent beam independently in a graphene-coatednanowire array even if the pitch of the nanowire array isultrasmall. In the following simulations using COMSOL, twocrescent beams propagating along two nanowires with a spaceof 40 nm are shown to be manipulated independently. Asindicated in Figure 1c, each nanowire can provide at least fourspatial addresses. Therefore, the proposed nanowire system canbe used for highly integrated gate-programmable EO storagewith an address density 425/μm, corresponding to a 64 terabytestorage capacity per micrometer.Theoretical Model. A dielectric nanowire with a cross-

sectional radius a and a permittivity ε1 is coated by monolayergraphene, which is embedded in a homogeneous medium ofpermittivity ε2, as shown in the inset of Figure 2a. Graphene is

modeled as an infinitely thin surface characterized by a surfaceconductivity σg,

34 which can be calculated using the Kuboformula.24 Our study focuses on the case in which both theexternal magnetic field and excitonic gap are zero, so the surfaceconductivity of graphene is isotropic. Both its calculationequation and the dependence of the chemical potential on thegate voltage are given in Supporting Information A.35−37 In thecylindrical coordinates, the longitudinal electric and magneticfields can be expressed as Ejz = AjZm

j (hjr)e−i(mφ+ωt−βz) and Hjz =

BjZmj (hjr)e

−i(mφ+ωt−βz), respectively, where j = 1 or 2corresponds to the region inside the nanowire or outside thegraphene layer, and the azimuthal quantum number m = 0, ±1,±2, etc. Aj and Bj are mode coefficients; ω is the angularfrequency; β = β1 + iβ2 is the complex propagation constant; hj

2

= β2 − μjεjk02, where k0 = 2π /λ is the wavenumber in a vacuum

and μ1 = μ2 = 1 for nonmagnetic dielectrics, and Zmj (hjr) is the

modified Bessel function Im(h1r) for j = 1 or Km(h2r) for j = 2.According to Ejz, Hjz, and the relationship between thetransverse and longitudinal components of electromagneticfields,38 we attain the transverse field components Ejr, Ejφ, Hjr,and Hjφ. Then by using the boundary conditions E1z = E2z, E1φ= E2φ, H2z − H1z = −σgE1φ and H2φ − H1φ = σgE1z at r = a, wederive the dispersion equation of guide modes as follows

β

− +

+

×+ − +

−=

ωεσ

ωεσ

ωε ε εσ

β σ

ωεε ε σ

ωεσ

ωεε ε

′ ′ ′ ′

′ −

′ ′

′ −

⎛⎝⎜

⎞⎠⎟

mh h a

i i

i

i i

i

h hI h aI h a

K h aK h a h k

I h aI h a k h

K h aK h a

hK h aK h a h

m

h h a hI h aI h a h

K h aK h a

hI h aI h a h

1 ( )( )

( )( )

( )( )

( )( )

1 ( )( )

( )

( )( )

( )( )

( )( )

1 2

2

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

m

1 2

1

1

2

2

0

g 1 02

1

1

0

g 02

2

2

2

2

2

2

0 2 1

g 22

2 2g

0 12

22 2

1

1

1

1

2

2

2

2

g

0

g

0 1

1

1

2 1

12

(1)

When m = 0, eq 1 can be reduced to

ωεε ε

σ+ =⎡⎣⎢

⎤⎦⎥h

I h aI h a h

K h aK h a

i( )( )

( )( )0

1

1

1 1

0 1

2

2

1 2

0 2g

(2)

ωεσ− + =

⎡⎣⎢

⎤⎦⎥k

hI h aI h a

hK h aK h a

i( )( )

( )( )

0

02 1

0 1

1 12

0 2

1 2g

(3)

Equation 2 corresponds to TM SPPs, which is the same asthe equation obtained before.39 The TE modes determined byeq 3 are weakly confined, so we will not discussed them here. m≠ 0 corresponds to the hybrid modes. According to eq 1, thelowest order EH modes (including EH+1 and EH−1 modes) aredegenerate, and their dispersion relation can be rewritten as

β

+ +

−

×+ + +

−=

ωεσ

ωεσ

ωε ε εσ

σ β

ωε

σ

ωεε ε

σ

ωεε ε

+ + + +

+ −

+ +

+ −

⎛⎝⎜

⎞⎠⎟h h a

i i

i

i i

i

h hI h a I h a

I h aK h a K h a

K h a h kI h a I h a

I h a k hK h a K h a

K h a

hK h a K h a

K h a h

h h a hI h a I h a

I h a hK h a K h a

K h a

hI h a I h a

I h a h

14

( ) ( )( )

( ) ( )( ) 2

( ) ( )( ) 2

( ) ( )( )

12

( ) ( )( )

( )

2( ) ( )

( ) 2( ) ( )

( )

2( ) ( )

( )( )

1 2

2

1 2

0 1 2 1

1 1

0 2 2 2

1 2

0

g 1 02

0 1 2 1

1 1

0

g 02

2

0 2 2 2

1 2

2

0 2 2 2

1 2

0 2 1

g 22

g2

0 12

22 2

g

0

1

1

0 1 2 1

1 1

2

2

0 2 2 2

1 2

g

0 1

0 1 2 1

1 1

2 1

12 (4)

The nanowire is assumed to be made of Si with radius a = 5nm and permittivity ε1 = 11.7, and the outer medium is SiO2 ofpermittivity ε2 = 2.25. The parameters for graphene are the

following: the charged particle scattering rate Γ = 0.1 meV, thetemperature T = 300 K, and the chemical potential μc = 0.8 eVexcept where otherwise stated. According to eqs 4 and 2, we

Figure 2. Comparison of propagation characteristics of EH±1 and TMSPPs. (a and b) Real and imaginary (loss coefficients) parts ofpropagation constants of EH±1 (solid line) and TM (dashed line)modes, respectively. The inset of part a shows the graphene-coatednanowire and the coordinate system adopted. (c) Differences of thereal (red line) and imaginary (blue line) parts of propagation constantsbetween the two modes.

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numerically calculate the complex propagation constants β = β1+ iβ2 of EH±1 and TM SPPs, respectively. The dependences ofβ1 (as shown in Figure 2a) and β2 (as shown in Figure 2b) onthe wavelength λ can thus be obtained. We also calculate thedependences of β1 and β2 on the chemical potential (as shownin Figure B1 in Supporting Information B) and the nanowireradius (as shown in Figure B2 in Supporting Information B). Itcan be concluded that both SPPs have high modal confinement,because β1 is much larger than the wave vector in SiO2 ( ε2 k0,1.9−7.3 × 106 m−1 in the given wavelength range). Figure 2bindicates that both EH±1 and TM modes can propagate over amicroscale distance. Δβ1 and Δβ2 are the values for β1 and β2 ofthe EH modes minus the counterparts of the TM mode,respectively, shown in Figure 2c. The difference of the two losscoefficients Δβ2 is very small over a large wavelength range,which is useful for modal interference. It is particularly notedthat, as shown in Figure 2c, there are two MPM wavelengths(2960 and 4430 nm) where Δβ1 = 0; that is, the EH±1 and TMmodes have the same phase velocity. Because β1 is dependenton graphene’s surface conductivity, the MPM wavelengths canbe tuned by varying the chemical potential μc, as shown inFigure B3a in Supporting Information B. The MPM wave-lengths can also be varied by changing the waveguide structure,e.g., the nanowire radius, as shown by Figure B3b in SupportingInformation B, due to the dependence of β1 on the waveguideparameters.As shown in Figure 2b, the loss of the EH mode exhibits a

sudden increase when the wavelength exceeds 4000 nm. Thephenomenon is mainly introduced by the enlarged fieldoverlapping in the Si rod, causing additional attenuationalong the mode propagation, as shown in Figure B4 inSupporting Information B. According to the azimuthal phasedistribution of the EH mode (e−imφ), the phases of those twodecaying fields (from graphene radial positions toward the Sicenter) carry opposite signs, which is dominant for λ > 4000nm and then responsible for the increasing loss. The mode fielddistributions of the TM mode are shown in Figure B5 inSupporting Information B for comparison, namely, in theabsence of field overlapping.We further calculate the dependences of β1 and β2 on the

wavelength with considering the permittivity dispersions of Siand SiO2, as shown in Figure B6 in Supporting Information B.The results show that the mode characteristics are wellmaintained, so the mode interference and the EO addressingcan still work as mentioned above.Mode Interference for Gate-Programmable Crescent

Beams. The analytical expressions of all electromagnetic fieldcomponents for both EH and TM SPPs are given in SupportingInformation C. On the basis of the propagation constants at theMPM wavelength λ = 2960 nm and the field componentexpressions, we calculate the time-averaged Poynting vectors ofQEP EH, QLP EH, and TM SPPs using ⟨S⟩ = Re[ErHφ* −EφHr*]/2,

40 as shown in Figure 1b. ⟨S⟩ of QEP (including EH+1and EH−1) modes have the same profile, which are similar tothat of the TM mode. However, at a given cross section, thephase of the QEP EH modes varies azimuthally following e−imφ,where m = +1 or −1, while it is spatially uniform for the TMmode. All the modal widths (determined by the 1/e2 decaypoints of ⟨S⟩)13 are ∼10 nm (diameter of the nanowire, ∼λ/300). In contrast, the optical power of the hybrid SPPs on ametal nanowire is largely outside the nanowire. In particular,the transverse size of a metal nanowire for guiding hybrid modeSPPs must be comparable to the excitation wavelength,41 but

the diameter of the dielectric nanowire in our waveguide can beas small as several nanometers, which is much smaller than thewavelength. Therefore, the modal width of hybrid SPPs on thegraphene-coated nanowire may be several orders of magnitudesmaller than that for the metal nanowire.42

The polarization properties of EH modes are very differentfrom those of the TM one. The transverse electric field of theTM SPPs is radially polarized, but both EH+1 and EH−1 SPPsare quasi-elliptically polarized. At a given position, |Er| ≠ |Eφ| forboth EH modes and their electric fields rotate elliptically.Obviously, the polarization characteristics are also differentfrom those of circularly polarized modes in common fibers,40

where |Er| = |Eφ|. In particular, the polarization directions varywith positions at the cross section, so they are dubbed QEPSPPs. Moreover, the coherent superposition of EH+1 and EH−1SPPs with the same phase and magnitude yields a QLP mode,in which the polarization directions at different positions arealso different. The polarization characteristics of all EH+1 QEP,EH−1 QEP, and QLP SPPs are discussed in detail in SupportingInformation D, and their polarization changing with time t overa period T = 2π/ω is shown in Movies 3, 4, and 5, respectively.The power flow ⟨S⟩ of the QLP mode is shown in Figure 1b. Inthe designed waveguide, the modal energy is mainly confinedinside the nanowire, but it can also turn to be outside thenanowire by some other designs such as adopting a nanowirewith a lower permittivity.According to Figure 1b and Supporting Information D, EH

and TM SPPs on the graphene-coated nanowire have nearly thesame modal widths, but their polarization properties aresignificantly different; hence their coherent superposition couldyield to either constructive or destructive interference atdifferent positions. Significantly, the mode moving paths arequite robust to amplitude differences between two modesparticipating in the interference (see detailed discussion inSupporting Information E). It is verified numerically that thesuperposition of EH and TM modes at MPM points can yield astable crescent beam due to stable mode interference; namely,the ⟨S⟩ shape of the beam is unchanged along the propagationdirection. When deviating from the MPM points, the modalinterference becomes unstable and therefore the superpositionof QLP or QEP EH and TM SPPs can yield a swing or helixcrescent beam, respectively. The chirality of the helix crescentbeam is dependent on the sign of Δβ1. The helix beamsupported by the EH+1 and TM SPPs rotates counterclockwiseor clockwise when the wavelength is between (Δβ1 > 0) or outof (Δβ1 < 0) the two MPM ones shown in Figure 2. Onanother hand, the rotation behavior will be reversed for thehelix beam supported by EH−1 and TM SPPs. In fact, the stablecrescent beam may be seen as a special case of the swing orhelix one. More interestingly, the controllability of both theperiod of the swing beam and the chirality and period of thehelix one via the applied gate voltage on graphene makes theposition of the crescent beam at the output plane electricallycontrollable, which enables gate-programmable EO spatial aswell as temporal modulations at sub-10 nm scale.

Numerical Simulations. In order to better understand thecrescent beam propagation properties, the evolution of thetime-averaged Poynting vector ⟨S⟩ along the propagationdirection is simulated numerically in graphene-coated nano-wires with different shapes, as shown in Figure 3. Thesimulations are performed applying the finite element method,with details provided in Supporting Information F. In Figure 3aand b, the nanowire radius, the operating wavelength, and the

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temperature are assumed to be the same as in the previousfigures, while the chemical potential μc is 0.9 eV in order tobetter highlight the evolution of the swing and helix crescentbeams in a waveguide of length L = 200 nm, half of the period.The white dashed lines in the plots correspond to the locationof the graphene layer. When QLP EH and TM SPPs propagatealong the waveguide, the power shifts gradually from the leftcrescent beam to the right one due to the modal interference, asshown in Figure 3a. If the waveguide is long enough, the opticalpower will swing periodically. When QEP EH and TM SPPspropagate along the waveguide together, as shown in Figure 3b,the crescent beam will rotate around the waveguide’s centralaxis, which is just like a helix beam from an overall perspective.Since the periods of both the swing and helix crescent beamscan be tuned by changing the chemical potential μc, theposition of the crescent beam at the output plane can bemanipulated electrically. The simulation results agree well withthe theoretical analyses and numerical calculations. Thecrescent beam can also propagate in a curved waveguide, asshown in Figure 3c. The waveguide consists of a straightsection with a 100 nm length and a bend with a 300 nm arcradius and 0.1π radians. By comparing Figure 3c with Figure 3a,it is found that the small bending has negligible influence on theswing characteristics of the crescent beam.

As shown in Figure 3a, b and c, the optical power is mainlyconfined inside the nanowires. It is thus expected that the modecoupling between two nanowires is weak, even though thedistance between them is very small. In order to verify thenegligible coupling at a deeply subwavelength scale, twographene-coated nanowires are put close to each other with acenter-to-center interval of 40 nm, as shown in Figure 3d. Theswing characteristics of the crescent beams are studied here. Bycarefully choosing the chemical potentials μc of the left andright graphene layers to be 0.95 and 0.85 eV, respectively, theoptical power in the left nanowire shifts from the right crescentbeam to the left one completely, while half of the optical powerin the right nanowire shifts from the lower crescent beam to theupper one in a waveguide length L = 150 nm. It is found thattheir swing has negligible difference when compared to theisolated case. Therefore, by packing an array of graphene-coated nanowires, an ultrahigh density data storage device maybe envisioned with parallel gate-programmable EO addressing.

Difference Frequency Generation (DFG) of TerahertzWaves. The MPM condition can also be satisfied for EH±1 andTM SPPs operating at different wavelengths, which provides aninteresting approach to realize dynamically configurablenonlinear nanophotonic systems.43−46 For example, when thewaveguide is used for DFG, the required phase matchingcondition is nearly the same as the MPM, since the β1 of thegenerated electromagnetic wave is several orders of magnitudesmaller than that of light in free space. Herein, SiC is used asthe nonlinear material instead of Si, which has no second-ordernonlinear response. The permittivity of SiC is obtained from ref47. The other parameters of the waveguide are the same as inthe previous examples. When the chemical potential is μc = 0.8eV, the wavelength relationship between EH and TM SPPs atthe MPM points is shown by the red line in Figure 4a, while thegenerated THz frequency is shown by the blue line. When asupercontinuum light spanning a wavelength range from 3.9 to5 μm is incident,48 the generated frequency can range over 10THz, covering the whole THz spectrum. If an incident beam

Figure 3. Evolutions of the swing and helix crescent beams ingraphene-coated nanowires. The evolutions of the swing (a) and helix(b) crescent beams in a straight waveguide. (c) Evolution of the swingbeam in a curved waveguide. (d) Evolutions of two swing beams in adouble-nanowire waveguide. The four parts share the same colorlegend.

Figure 4. Graphene-coated nanowire for the electrically tunablegeneration of a THz wave. (a) Wavelength relationship between EH±1and TM SPPs at the MPM points (red line) and the frequency of aTHz wave generated by DFG (blue line). (b) Dependence of thefrequency range (red region) of the THz wave on the chemicalpotential when the incident wavelength ranges from 3.6 to 3.7 μm.

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with a narrower bandwidth is employed, the frequency andbandwidth of THz generation can be modulated at the sametime by tuning the applied gate voltage on graphene. Forexample, Figure 4b shows the dependences of the highest(green) and lowest (blue) generated frequencies on thechemical potential, when the incident wavelength ranges from3.6 to 3.7 μm. The red region between the two lines shows thegenerated THz spectrum. Therefore, a nanoscale THz wavemay be excited in the waveguide, and its frequency andbandwidth can be tuned by the gate voltage due to thedependence of β1 on the chemical potential. At the same time,the two MPM SPPs operating at different wavelengths havenearly the same modal widths, which is important for efficientDFG.

■ CONCLUSIONSTo conclude, in this paper we derived the eigenmode equationsof both tightly confined EH and TM SPPs supported bygraphene-coated nanowires and analytically and numericallystudied their modal characteristics. In contrast to guided modesin normal fibers, the polarization state of EH SPPs can be eitherQLP or QEP and their mode widths could be reduced to lessthan 10 nm. The existence of MPM points for EH and TMSPPs enables the generation of a stable crescent beam by thecoherent superposition of the two modes. While deviating fromthe MPM points, their coherent superposition can yield a swingor a helix crescent beam. Significantly, both the period of theswing beam and the chirality and period of the helix can bemodulated by tuning the applied gate voltage on graphene. As aresult, the crescent beam can be varied temporally and spatiallyat a fixed output plane, which offers a way for gate-programmable EO addressing at sub-10 nm scale. In agraphene-coated nanowire array, the crescent beam in eachnanowire can be manipulated independently, even if theseparation between two nanowires is only 40 nm, so a gate-programmable EO addressing, including both spatial andtemporal modulations, with ultrahigh density (64 terabyte/μm) might be attained. Moreover, the MPM can also beachieved for EH and TM SPPs at different wavelengths, whichis useful in gate-programmable nonlinear nanophotonic devices.We expect that the dynamically configurable crescent beamswill motivate a new class of studies and applications innanostorage, nanocommunications, and chemical character-ization.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acsphoto-nics.6b00365.

Variations of the swing and helix beams at the outputplane; polarization equations and evolutions of EH+1

QEP, EH−1 QEP, and QLP SPPs; equations forcalculating the surface conductivity of graphene; modecharacteristics and dependences of MPM wavelengths onthe graphene’s chemical potential and the nanowireradius; electromagnetic field components of both EH andTM SPPs; mode interference characteristics; steps forbuilding the models by using COMSOL (PDF)Movie 1 (AVI)Movie 2 (AVI)Movie 3 (AVI)

Movie 4 (AVI)Movie 5 (AVI)

■ AUTHOR INFORMATIONCorresponding Authors*E-mail (S. Ruan): scruan@szu.edu.cn.*E-mail (C.-W. Qiu): chengwei.qiu@nus.edu.sg.NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSH.L. acknowledges the National Natural Science Foundation ofChina under Grant 61405124, Natural Science Foundation ofGuangdong Province, China, under Grant 2014A030313560,Specialized Research Fund for the Doctoral Program of HigherEducation of China under Grant 20134408120002, and theShenzhen Science and Technology Project of Shenzhen, China,under Grants JCYJ20160308092830132 and JCYJ201404-18091413577. L.Z. and C.-W.Q. acknowledge the financialsupport from A*STAR Pharos Programme (Grant No. 152 7000014, with Project No. R-263-000-B91-305). This work waspartially supported by National Natural Science Foundation ofChina (Grant Nos. 61571186 and 51302026) and InternationalScience & Technology Cooperation Program of China (GrantNo. 2015DFG12630).

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