www.sciencemag.org/cgi/content/full/332/6035/1291/DC1

Supporting Online Material for

Transformation Optics Using Graphene

Ashkan Vakil, Nader Engheta*

*To whom correspondence should be addressed. E-mail: engheta@ee.upenn.edu

Published 10 June 2011, Science 332, 1291 (2010)

DOI: 10.1126/science.1202691

This PDF file includes:

Materials and Methods

SOM Text

Figs. S1 to S9

References

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Materials and Methods 

We have used commercially available full-wave electromagnetic simulator software, CST Studio Suite (S1) in order to obtain the 3D numerical simulations presented in Figs 2, 3 and 4. For the purpose of our 3D simulation, the thickness of graphene is assumed to be 1 nm, although other extremely small values for this thickness lead to similar results. We have assumed 1 nm thickness for graphene and assigned the corresponding permittivity in our simulations according to the derivation shown below in this supporting online material (SOM). Note that as long as the thickness chosen is extremely small compared to the wavelength, this particular choice is not essential—we could assume, for instance, thickness of 0.5 nm and find the corresponding permittivity value. Due to the large contrast in the dimensions of the graphene layer, i.e., contrast between thickness and width and between thickness and length, and due to the special form of the conductivity function of graphene, we have chosen frequency-domain Finite Element Method (FEM) solver in the CST Studio Suite. This solver solves the problem for a single frequency at a time. For each frequency sample, the system of linear equation is solved by an iterative solver. Adaptive tetrahedral meshing with a minimum feature resolution of 0.5 nm has been used in the simulations. A point source (equivalent of an infinitesimal dipole antenna) has been utilized for the excitation of surface-Plasmon polariton (SPP) wave in the structures. All the simulations reached proper convergence; a residual energy of 10-5 of its peak value has been obtained in the computation region. In order to absorb all the energy at the end of the computational domain and to have approximately zero-reflection boundary on the receiving side, in all the simulations a technique similar to the well-known Salisbury Shield method has been implemented (with proper modifications for a TM SPP mode). Depending on the nature of the problem, perfect magnetic conducting (PMC), perfect electric conducting (PEC) and open boundary conditions have been applied to different boundaries, to mimic the two-dimensionality of the geometry.

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SOM Text

Discussion on the Effective Permittivity of Graphene and the Sign of Imaginary Part of Graphene Conductivity

To address the issue of σg,i attaining positive or negative values in the context of metamaterials, we momentarily assume that graphene has a very small thickness Δ (later we shall let Δ→0). Defining a volume conductivity for this Δ-thick graphene layer as g,v g /σ σ≡ Δ , we can write its volume current density as g,vσ=J E . Assuming

exp( )i tω− time harmonic variations, we can rearrange the Maxwell equation

0iωε∇× = −H J E for the Δ-thick graphene layer as g,v 0( )iσ ωε∇× = −H E . Denoting

the equivalent complex permittivity of the Δ-thick graphene layer by εg,eq, we obtain g,eq g,i 0 g,r/ /iε σ ω ε σ ω≡ − Δ + + Δ . For a one-atom-thick layer, bulk permittivity

cannot be defined. However, here we have temporarily assumed that the thickness of layer is Δ , associating an equivalent permittivity with this single layer. This approach allows us to treat the graphene sheet as a thin layer of material with εg,eq. At the end we let Δ→0, and recover the one-atom-thick layer geometry. Specifically, we note that

g,eq g,i 0 g,iRe( ) / /ε σ ω ε σ ω≡ − Δ + ≈ − Δ for a very small Δ, and g,eq g,rIm( ) /ε σ ω≡ Δ . This

shows that the real part of equivalent permittivity for this Δ-thick graphene layer can be positive or negative depending on the sign of the imaginary part of the graphene conductivity. Therefore, when σg,i > 0 , i.e., Re(εg,eq) < 0, a single free-standing layer of graphene effectively behaves as a very thin “metal” layer, capable of supporting a transverse-magnetic (TM) SPP surface wave and the possibility of existence of SPP along the graphene (20-23).

A slab of a material with complex permittivity εm with negative real part (e.g., Ag or Au) and thickness Δ, surrounded by free space can support an odd transverse-magnetic (TM) electromagnetic guided mode with wave number β expressed as (S2)

( )2 2

0 02 2 m0 m 2 2

0 0 m

coth / 2 .β ω μ εεβ ω μ ε

ε β ω μ ε

−− Δ = −

− (S1)

By substituting mε with the equivalent permittivity of the Δ-thick graphene layer derived

above, and letting Δ→0, we get ( )22 20 0 g1- 2 /kβ η σ⎡ ⎤= ⎢ ⎥⎣ ⎦

, which is the dispersion relation

of the TM SPP optical surface wave along a graphene layer (20-23).

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Now consider a slab of a material with complex permittivity εd with positive real part and thickness Δ, surrounded by free space. This slab can support an odd transverse-electric (TE) electromagnetic guided mode with wave number β expressed as (S3)

( )2 2 2 2 2 20 d 0 d 0 0tan / 2 .ω μ ε β ω μ ε β β ω μ ε− − Δ = − (S2)

We show the mathematical steps for this case. By substituting εd with the equivalent permittivity of a Δ-thick graphene layer—whose equivalent permittivity εeq, using Maxwell’s equation, can be written as -σg/iωΔ+ε0—the Eq. (S2) can be recast in the following form

( )( )2 2 2 2 2 2 20 g 0 0 g 0 0tan / 2 .i k i k kωμ σ β ωμ σ β βΔ + − Δ + − Δ = − (S3)

Now let Δ go to zero. In that limit Eq. (S3) is written as

0 g 2 20 .

2i

kωμ σ

β= − (S4)

Solving for β from Eq. (S4), one can simply derive that the following dispersion relation for weakly guided TE SPP surface wave, which has been already reported by other groups [e.g., reference (20) of the report]

2g 0

0 1 .2

kσ η

β⎛ ⎞

= − ⎜ ⎟⎝ ⎠

(S5)

This clearly shows that our approach to re-derive the dispersion relation is simple yet powerful, providing intuition on the physics of the problem.

Comparison between Graphene and Silver as Platform for Surface Plasmon Polaritons

For a free-standing graphene at T = 3 K, with Γ = 0.43 meV, and chemical potential μc = 0.15 eV at frequency 30 THz, the ratio σg,r/|σg,i|, which is known as the loss tangent and equivalent of Im(εg,eq) /Re(εg,eq), is about 1.22×10-2. On the other hand, the ratio Im(εg,eq) /Re(εg,eq) for silver at frequency 30 THz and at room temperature is about 5.19×10-1 (S4). One might argue that the comparison between graphene at 3 K and silver at room temperature is not fair. In the fields of plasmonic optics and surface plasmon photonics it is known that the silver at room temperature is lossy in the visible domain, where the SPP characteristics are desirable. Numerous efforts are under way to explore methods to reduce the loss of SPP in the visible frequencies, and extend the length of propagation of SPP along the silver-dielectric interface. If it were possible to decrease the loss of silver noticeably by simply lowering the temperature, many groups would have already demonstrated such possibility. The very fact that the existing literature is sparse on the

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topic of reduction of loss in silver—or other noble metals—at visible frequencies by lowering temperature indicates that there is little potential to reduce the loss by using this technique. The reason is that most of the silver loss in the visible region of interest to the SPP wave at the silver-air interface is due to the interband electronic transitions. So the silver loss may not be significantly reduced by merely lowering the temperature. Of course for longer wavelengths, it is possible to reduce loss in silver or gold by lowering the temperature (S5), since the loss at the longer wavelengths is mostly due to the electron scattering rate in the Drude model. However, the loss of graphene can be relaxed by temperature reduction in the infrared region, where surface polaritons modes exhibit promising characteristics such as tight confinement to the surface. The comparison between graphene and silver at different frequencies once again might seem unfair at the first glance. But at low frequencies (such as 30 THz), where the fields do not penetrate significantly into the silver, due to large values of imaginary and negative real parts of permittivity, this material supports a SPP mode that is poorly confined to the silver-dielectric interface. Such SPP, clearly, will have little utility, since it is weekly confined. This comparison shows that at mid infrared region graphene definitely provides a much better platform to support highly confined and low loss SPP than silver does at the same wavelengths. We note that at the wavelength of about 6 μm or shorter, the optical phonons may be excited in graphene and consequently the loss will increase. But for longer wavelengths graphene is an excellent choice, as far as low loss and mode confinement is desired. Additionally, once again we would like to emphasize other advantages that graphene has over silver, i.e., tunability and excellent SPP confinement characteristics.

Supporting Examples

Fig. S1, A, B, C, and D, depicts the quantities Re(β), Im(β), figure-of-merit (FOM) known as Re(β)/Im(β), and propagation length defined as 1/Im(β) for TM SPP surface waves on a free-standing isolated graphene in vacuum as a function of frequency and chemical potential (T = 3 K and Γ = 0.43 meV).

Fig. S2 displays the numerical simulation of the SPP mode at 30 THz guided by a uniformly biased graphene layer. As mentioned in our report, the guided wavelength λSPP on this free-standing graphene in air is λ0 /69.34 = 144.22 nm. This highly compressed mode yields an effective SPP index (nSPP ≡ βSPP/k0) of 69.34 and has a relatively large propagation length of 15.6λSPP = 0.225λ0 = 2.25 μm. The color variation shows the y component of the electric field, Ey (snap shot in time).

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We mentioned in our report that in the scenario shown in Fig. 2 of the report the reflection of SPP at the line resembles the Fresnel reflection of a plane wave from a planar interface between two media. We noted that we might analogously think of this phenomenon as the 2D version of the Fresnel reflection from a planar interface between a medium that supports a propagating wave (e.g., a medium with a real refractive index such as a dielectric) and another medium that does not support a propagating wave (e.g., a medium with no real index, such as a noble metal). Based on the established analogy, we can have a guided IR edge wave along the boundary line between these two sections. Fig. S3 represents the numerical simulations from which the presence of such guided IR edge wave could be predicted. This special guided wave propagates along a “one-atom-radius” boundary line. By post processing the simulation results, we estimate the wavelength of the guided edge wave to be around λSPP = 61.5 nm.

Fig. S4 shows a bent waveguide structure with an arbitrarily chosen narrow width of 30 nm, showing that coupled edge wave exists in this ribbon-like “one-atom-thick” waveguide. This scenario may, for example, be realized by using an inhomogeneous distribution of permittivity of dielectric spacer (not shown here) under the graphene. Fig. S4 shows that by spatially varying the conductivity of graphene, we can bend the guided wave, and still maintain the bounded SPP guided through the bend.

As another example, a flat version of the “superlensing” effect (S6, S7) is presented on the graphene (Fig. S5). In this simulation, a point source on a graphene biased at voltage Vb1 (equivalent chemical potential of μc1), is situated near a “strip” region of graphene biased at Vb2 (or μc2). With proper choice of biases to create required conductivity values for the strip and the region outside the strip, and proper adjustment of width of the strip and separation between the source and the strip, we can implement an approximate superlensing effect. The 2-dimensional (2D) superlens could be considered as the basic building block for a 2D hyperlens (S8, S9), providing applications in transforming the deeply subwavelength “input” field distribution to a desired “output” field distribution. This can be useful in optical signal processing and subwavelength imaging on graphene.

Besides the waveguiding mechanisms described in Fig. 3 of the report, we can also have “one-atom-thick” scenarios analogous to 3D guided wave propagation in optical fibers with two different dielectric media of different refractive indices as the core and cladding. In other words, the two sections of graphene could be biased to support two TM SPP modes with two distinct effective indexes nSPP,1 and nSPP,2. We can then realize waveguiding effects based on this notion. So as a final supplementary example, in Fig. S6, full wave simulation of 2D variant of optical fiber on graphene is displayed. It can be

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seen that the mode stays bounded in the “one-atom-thick” core and no energy penetrates to the “cladding” region of graphene.

More on Ideas to Achieve Desired Conductivity Distribution across Graphene

One of our ideas for construction of the desired nonuniform conductivity patterns on graphene is to produce an uneven ground plane—which can be realized using the lithography of the surface of highly doped Silicon—i.e., to design the specific profile of the ground plane underneath the dielectric spacer holding the graphene in order to have nonuniform static biasing electric field under the graphene while the voltage applied between the entire sheet of graphene and ground plane is kept fixed (Figs. S7 and S8). Since in this scenario the separation between the graphene and the ground plane is not uniform, the static electric field due to the single bias voltage between the graphene and ground plane is nonuniform. Therefore the distribution of local carrier densities—hence the spatial distribution of chemical potential—on the sheet of graphene will be nonuniform, resulting in different conductivity values at different segments; the nonuniform profile of the ground plane can be achieved by the standard techniques in nanolithography, e.g., optical lithography and/or electron-beam lithography, etc. This can be an exciting possibility to make the conductivity of a single sheet of graphene inhomogeneous with the specific pattern of its ground plane. This structure is experimentally feasible and within the realm of possibility and current technology.

Another idea to create an inhomogeneous conductivity across a single sheet of graphene is to etch a narrow strip on top of the spacer dielectric that holds the entire graphene and to fill this strip with a different material (Fig. S9). When a DC voltage is applied between the graphene and the ground plane, the static electric field in this narrow strip will be different from the static field elsewhere in the spacer, and thus the section of the graphene on top of this strip will have a different conductivity. Since the width of this strip on top of the spacer can be small (but not too small to become comparable with the nanostructured dimensions within a graphene, in which case the quantum nature of the structure should be taken into account), the width of the graphene with the different conductivity can be as small. It is worth mentioning that the dimensions we have chosen for our examples are larger than the nanostructured dimensions within graphene, so the conductivity model still holds true. Also, in our numerical simulations, for the sake of simplicity and to keep the concepts easily graspable without making them unnecessarily too complicated, we have assumed the “sharp” inhomogeneity in conductivity distributions of the two neighboring sections of a single flake of graphene. However, our

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ideas of transformation optics will still be valid and applicable even when the inhomogeneity between the two sections is smooth.

Derivation of Approximate Expression for Conductivity Distribution of 2D Luneburg Lens on Graphene

In this section we show derivation of the approximate formula we used to find conductivity gradient of the proposed 2D Luneburg lens. It is well known for a 3D Luneburg lens the refractive index (n) of the lens must have the following spatial form (S10):

2

( ) 2 ,rn rR

⎛ ⎞= − ⎜ ⎟⎝ ⎠

(S6)

where R and r denote, respectively, the radius of the lens and radius at which the index is n(r). By mimicking the same spatial variation for effective SPP index on graphene, one can expect to see the same functionality for the 2D variant of Luneburg lens. We mentioned earlier that for a TM SPP surface wave on graphene dispersion relation is

expressed as ( )2

0 0 g1- 2 / .kβ η σ= Since the real part of graphene conductivity is much

smaller than its imaginary part when it can support TM SPP waves (i.e., when the imaginary part is positive), using the dispersion relation, one can write SPP 0 g,i2 / .n η σ≈

Then according to Eq. (S6) we must have

g,i

1/22out

g,i ( ) 2 ,rrR

σ σ−

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ (S7)

where g,i

outσ is the conductivity of the background graphene on which the Luneburg lens is

designed. Moreover in order to perform the numerical simulation we would like to discretize Eq. (S7), which results in the form presented in the report—i.e.,

( )( )1/22

i, i,out 1 2 / .n n nr r Dσ σ−

−⎡ ⎤= − +⎢ ⎥⎣ ⎦

Features of Luneburg Lens

In general, discontinuities in the geometries and/or conductivity distributions in graphene-based IR elements described in the manuscript such as waveguides, beamsplitters and reflectors, is one way to manipulate the 2D surface plasmon waves. Transformation optics, however, provides an alternative approach to control the propagation of light by spatially varying the optical properties of a material [please see

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e.g., (5-7)]. Because the optical properties of this type of lens are changed gradually rather than abruptly, losses due to scattering can be reduced in comparison with previously reported plasmonic elements. In addition to their imaging functionalities, lenses of any kind are basic elements for optical signal processing (e.g., Fourier transforming). Thus, Luneburg lens on graphene could be used to design optical signal processing elements such as spatial filters, correlators, and convolvers in two dimensions. The possibility to create one-atom-thick 2D lens may witness rebirth of the field of Fourier optics on a monolayer of carbon atoms. The Luneburg lens is an aberration-free lens that focuses light from all directions equally well. Other advantages of Luneburg lens are high efficiency and accuracy in control of the focal length, which are essential requirements for signal processing purposes. Moreover, the flat Luneburg lens on graphene may become an integrated photonic component—created just by changing conductivity profile on graphene layer—which due to its 2-dimensionality is easier to realize compared to the original 3-dimensional (3D) variant.

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Fig. S1. (A) Real part and (B) Imaginary part of the normalized wave number (β/k0) of the TM SPP supported by the single sheet of graphene free-standing in air, as a function of chemical potential μc and frequency f, according to Kubo formula (17) for T = 3 K, Γ = 0.43 meV. (C) Figure of merit (FOM) for the SPP mode as a function of μc and f. (D) The propagation length of the SPP mode.

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Fig. S2. Simulation results showing the y-component of the electric field, Ey, (snap shot in time) for a TM SPP mode at 30 THz guided by a uniformly biased graphene layer with dimensions L = 350 nm, w = 235 nm (T = 3 K, Γ = 0.43 meV and μc = 0.15 eV across the entire graphene). This chemical potential can be achieved, for example, by a bias voltage of 22.84 V across a 300-nm SiO2 spacer between the graphene and the Si substrate (but Si substrate and SiO2 spacer are not present here in our simulations). The SPP wavelength along the graphene, λSPP is much smaller than free-space wavelength λ0, i.e., λSPP = 0.0144λ0.

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Fig. S3. Distribution of Ey (snap shot in time) of a guided IR edge wave at f = 30 THz, supported along the boundary line between the two sections of the same sheet of graphene, which has two different conductivity regions (L = 250 nm, w = 80 nm).

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Fig. S4. Simulation results of Ey (snap shot in time) for an IR guided wave at f = 30 THz along a bent ribbon-like section of graphene with the chemical potential μc1, which may be achieved by raising the ground plane underneath this region. This ribbon-like path is surrounded by the two other sections of the same sheet of graphene but with a different chemical potential μc2, which may be obtained by lowering the ground plane beneath these regions. The IR signal is clearly guided along this “one-atom-thick ribbon”. The computational region has the length L = 370 nm and total width w1+w2+w3 = 120 nm + 30 nm + 60 nm while for the bent region w4+w2+w5 = 30 nm + 30 nm + 120 nm.

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Fig. S5. Flatland “superlens”: Simulation results for Ey of SPP at f = 30 THz on the graphene with a subwavelength strip region with conductivity σg2, while the rest of graphene has the conductivityσg1. The object—a point source—and image planes are assumed to be, respectively, 10 nm away from the left and right edges of the strip (w = 2d = 20 nm). The normalized intensity of Ey at the image plane is shown for two cases with and without the strip (Normalization is with respect to their respective peak values). The subwavelength “focusing” is observable due to presence of the strip with conductivityσg2—Im(σg2)< 0.

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Fig. S6. Simulation results of Ey (snap shot in time) for an IR guided wave at f = 30 THz along a ribbon-like section of graphene with the chemical potential μc2 = 0.15 eV. This ribbon-like path is surrounded by the two other sections of the same sheet of graphene, but with a different chemical potential μc1 = 0.3 eV. Both of these chemical potentials result in positive imaginary part of conductivity, but different values for effective SPP index μc2 = 0.15 eV results in higher index for the “core” region (similar to a 3D fiber for which core has higher index for light). In this simulation, the IR signal is clearly seen to be guided along this “one-atom-thick ribbon”. The computational region has the length L = 1 μm and total width w2+w1+w2 = 150 nm + 200 nm + 150 nm.

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Fig. S7. Schematic of the proposed idea of uneven ground plane underneath the graphene layer in order to construct inhomogeneous conductivity pattern across a single flake of graphene: By biasing the graphene with a single static voltage, the static electric field is distributed according to the height of the spacer between the graphene and the uneven ground plane, leading to the unequal static electric field. This results in unequal carrier densities and chemical potentials μc1 and μc2 on the surface of the single graphene and thus different conductivity distributions across the graphene. Here in this schematic and in our numerical simulations, for the sake of simplicity in the simulation and to keep the concepts easily and intuitively understandable, we have assumed the “sharp” inhomogeneity in conductivity distributions of the two neighboring sections of a single flake of graphene. However, since the static biasing electric field in the space underneath of graphene is expected to be gradually varied in going from one region to another, the chemical potential and the conductivity distribution will have a smooth transition. It is important to emphasize that our ideas of transformation optics on graphene will still be valid and applicable even when the conductivity inhomogeneity between the two sections is reasonably smooth in the form of a transition region.

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Fig. S8. Another schematic for the uneven ground plane idea for constructing inhomogeneous conductivity across a single sheet of graphene.

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Fig. S9. Sketch of another proposed idea to create inhomogeneous conductivity across a single sheet of graphene. Here several dielectric spacers with unequal permittivity functions are assumed to be used underneath of the graphene. This can lead to unequal static electric field distributions, resulting in inhomogeneous carrier densities and conductivity patterns across a single sheet of graphene.

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Technical Digest (CD) (Optical Society of America, October 24, 2010), paper FThR4.

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(2006). S9. A. Salandrino and N. Engheta, Phys. Rev. B 74, 075103 (2006). S10. B. E. A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley and

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