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Trapping and guiding surface plasmons in curved graphene landscapes Daria A. Smirnova, S. Hossein Mousavi, Zheng Wang, Yuri S. Kivshar, and Alexander B Khanikaev ACS Photonics, Just Accepted Manuscript • Publication Date (Web): 22 Mar 2016 Downloaded from http://pubs.acs.org on March 22, 2016
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ACS Photonics
Trapping and guiding surface plasmons in curved graphene landscapes Daria Smirnova,† S. Hossein Mousavi,‡ Zheng Wang,‡ Yuri S. Kivshar,† and Alexander B. Khanikaev∗,¶,§ Nonlinear Physics Center, Australian National University, Canberra ACT 2601, Australia, Microelectronics Research Center, Cockrell School of Engineering, University of Texas at Austin, Austin, TX 78758 USA, Department of Physics, Queens College of The City University of New York, Queens, NY 11367, USA, and Department of Physics, The Graduate Center of The City University of New York, NY 10016, USA E-mail:
[email protected] Abstract We demonstrate that graphene placed on top of curved substrates offers a novel approach for trapping and guiding surface plasmons. A monolayer graphene with a spatially varying curvature exhibits an effective trapping potential for graphene plasmons near curved areas such as bumps, humps and wells. We derive the governing equation for describing such localized channel plasmons guided by curved graphene and validate our theory by the first-principle numerical simulations. The proposed confinement mechanism enables plasmon guiding by the regions of maximal curvature, and it offers a versatile platform for manipulating light in ∗ To
whom correspondence should be addressed Physics Center, Australian National University, Canberra ACT 2601, Australia ‡ Microelectronics Research Center, Cockrell School of Engineering, University of Texas at Austin, Austin, TX 78758 USA ¶ Department of Physics, Queens College of The City University of New York, Queens, NY 11367, USA § Department of Physics, The Graduate Center of The City University of New York, NY 10016, USA † Nonlinear
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planar landscapes. In addition, isolated deformations of graphene such as bumps are shown to support localized surface modes and resonances suggesting a new way to engineer graphenebased metasurfaces.
KEYWORDS: Graphene plasmons, Curvature, Trapping potential Unprecedented optical properties make graphene a promising plasmonic material with great potential for practical applications ranging from optical routing 1–5 to nonlinear optics 6 , optomechanics 7 and sensing 8 . In addition to relatively strong and tunable electromagnetic response of this one-atom-thick material, it was shown that at infra-red frequencies graphene supports surface plasmon-polaritons (SPPs) with extreme confinement and propagation characteristics controllable by doping 9 . In particular, thanks to the strong dependence of its optical characteristics from electric bias graphene is on the path to be widely used for electro-optical modulation 1,10 . Possibility of the electrostatic doping enables an efficient control over the electrons density at the Fermi level which defines high frequency optical response of graphene 11–15 . With tremendous success in graphene electronics, these unique capabilities enable integration of plasmonic and electronic devices on the same substrate 16–20 and even a design of tunable cloaking devices 21 . Presently, two of the most common approaches to utilize unique plasmonic response of graphene for guiding applications rely on (i) inhomogeneous gating of graphene 22,23 affecting its local plasmonic response, and (ii) subwavelength-scale patterning of graphene layer giving rise to localized plasmonic resonances 5 . In this paper, we propose an alternative approach to engineer plasmonic guiding properties of graphene by employing its property to confine and scatter light by curved regions such as extended one-dimensional humps and local two-dimensional deformations. Technologically, such curved graphene profiles, shown schematically in Figs. 1(a,b), can be fabricated by either transferring graphene onto curved dielectric substrates or grown directly by laser ablation on structured surface of SiC demonstrated recently 24 . In particular, we demonstrate plasmon guiding by a bump of monolayer graphene which effectively operates as a plasmonic channel waveguide 25–33 . In addition, we show that a local isolated deformation of graphene in the form of a bump supports local 2 ACS Paragon Plus Environment
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(a)
ε+ ε
ξ s
(b)
ε+ ε
Figure 1: Schematic view of convex curved substrates covered by graphene, and related global Cartesian and local curvilinear coordinate systems. (a) and (b) correspond to the cases of one- and two-dimensional curved surfaces, respectively. surface plasmonic resonances. The localized plasmonic modes studied here differ from the modes supported by gratings that can be generated in graphene by acoustic or elastic waves 34,35 . It has been already established that a curved graphene surface can play a role of an effective potential for electrons and phonons, e.g. it may support spatially localized phonons 36 . Similarly, we may expect that graphene plasmons can be modified by effective geometric potentials, as this was shown earlier for curved metal-dielectric interfaces 37 . We notice that the study of the propagation of light in curved dielectric space has attracted some special attention due to novel opportunities to guide and focus light 38–40 . Here, we extend these ideas to the case of plasmons in curved graphene and demonstrate substantially different regimes for engineering plasmons in curved graphene.
RESULTS AND DISCUSSIONS Analytical approach As the first step, we explore the capability of trapping light with the use of an analytical approach assuming graphene surface of small one-dimensional curvature. Specifically, we consider a graphene monolayer placed on a curved interface between two dielectric media with permittivities ε− and ε+ , as shown in Fig. 1(a). Such a graphene-coated surface forms a channel waveguide for 3 ACS Paragon Plus Environment
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surface plasmons, thus enabling a two-dimensional (2D) spatial localization of light, alternative to waveguides created by graphene conductivity modulation 22,23 . We base our analytical consideration on Maxwell’s equations solved in the adiabatic approximation within the small-wavelength limit. We assume that an inhomogeneity in the graphene profile is described by a smooth function x = f (z), and we look for modes propagating along the y axis. For convenience, the curvilinear orthogonal coordinate system {ξ , y, s} bound to the reference curve x = f (z) is used in place of the Cartesian coordinate system {x, y, z}. Accepting harmonic exp(−iωt) time-dependence, Maxwell’s equations can be written as follows ∇ × E = ik0 H , ∇ × H = −ik0 ε(ξ )E + 4π δ (ξ )σ (ω)Eτ , c
(1)
where k0 = ω/c is the wavenumber in free space, σ (ω) ≡ iσ (I) (ω)+σ (R) (ω) is the linear frequencydependent surface conductivity of graphene, the function ε(ξ ) describes the dielectric permittivity distribution, ε(ξ ) =
ε− , ξ < 0,
(2)
ε+ , ξ > 0. The Dirac delta-function δ (ξ ) in Eq. (1) indicates that the graphene layer is placed at ξ = 0, and the subscript τ refers to the field component tangential to the surface. In our case of low-energy excitations close the Dirac points, where transitions occur within the π-π ∗ electronic bands, the normal component of the induced current density in graphene is negligible. The marginal character of the contribution normal to the surface is also evidenced by the earlier experimental studies 41 . Equation (1) leads to
− ∇ × ∇ × H + k02 ε(ξ )H = ik0 ∇ε(ξ ) × Eτ −
4π σ (ω)∇ × δ (ξ )Eτ , c
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(3)
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(a) ×10
−2
8 6
0
g/k2sp
4
−0.001 45
a(z)
2 0
50
� ksp /ksp
−2 −4 −50
−25
0
kspz
25
50
|H|/|H| max
(b) ×10 4
−2
2
a(z) 0
g/k2sp
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ACS Photonics
� ksp /ksp
−2 0
−4
−0.006 45
−6 −8 −50
−25
0
kspz
25
50
50
|H|/|H| max
Figure 2: Top panels in (a,b): Effective potential (black solid curves) and transverse profiles a(z) (a.u.) of the trapped modes (blue dotted curves) calculated for the Gaussian profiles f (z) = ± f0 exp (−z2 /w2 ), f0 = 200 nm, w = 506 nm, and dielectric permittivities ε+ = 1, ε− = 2 at the frequency ω0 /2π = 16 THz, respectively. The red dashed line levels the relative correction ∆ksp /ksp (ω0 ) to the wavenumber ksp (ω0 ) = 22.4k0 corresponding to the fundamental channel mode. Bottom panels in (a,b): Normalized magnetic field distribution |H|/|H|max of the modes shown in top panels and obtained with the use of first-principle FEM solver. i (∇ × H)τ is continuous at ξ = 0, and the vector operations can be calculated k0 ε(ξ ) using the Lame coefficients given as hξ = 1, hy = 1, hs = 1−κ(s)ξ ≡ 1−ξ /R, where κ(s) ≈ f 00 (z)
where Eτ =
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is a signed curvature of the reference cylinder line x = f (z), R = 1/κ(s) is the local radius of curvature. Following the approach elaborated earlier in Refs. 42–45 , we derive the equation for the slowly varying plasmon amplitude, employing the asymptotic description often used in the paraxial optics and physics of optical solitons 46 . To develop the consistent perturbation theory, we introduce a small parameter, ( β 2 = max
) 1 σ (R) , , ksp |R| σ (I)
assuming the losses in graphene and spatial inhomogeneity to be small (β 2 1). Accordingly, the solution of Eq. (3) is sought in the form 2 2 (2) 2 Hs = A(β y, β s)h(ξ ) + β H (β y, β s, ξ ) + . . . eiksp y (2) = A (s, y)h(ξ ) + H (s, y, ξ ) + . . . eiksp y , (1) 2 3 (3) 2 Hy = β H (β y, β s, ξ ) + β H (β y, β s, ξ ) + . . . eiksp y (1) (3) = H (s, y, ξ ) + H (s, y, ξ ) + . . . eiksp y .
(4)
where A (s, y) is slowly varying amplitude of the plasmonic mode propagating along the curved channel and ksp (ω) satisfies the dispersion relation ε+ ε− 4π (I) + = σ (ω) , κ+ κ− ω 2 − k2 ε where κ+,− = ksp 0 +,−
1/2
(5)
.
Substituting Eqs. (4) into Eq. (3) and maintaining zeroth order terms in β lead to the dispersion relation (5) and gives the transverse profile for the plasmon on a locally flat graphene, ε+ e−κ+ ξ , h(ξ ) = −ik0 κ+ − ε− eκ− ξ , κ−
ξ >0, (6) ξ
