## Abstract

The flexible control of surface plasmon polaritons (SPPs) is important and intriguing due to its wide application in novel plasmonic devices. Transformation optics (TO) offers the capability either to confine the SPP propagation on rigid curved/uneven surfaces, or to control the flow of SPPs on planar surfaces. However, TO has not permitted us to confine, manipulate, and control SPP waves on flexible curved surfaces. Here, we propose to confine and freely control flexible SPPs using TO and graphene. We show that SPP waves can be naturally confined and propagate on curved or uneven graphene surfaces with little bending and radiation losses, and the confined SPPs are further manipulated and controlled using TO. Flexible plasmonic devices are presented, including the bending waveguides, wave splitter, and Luneburg lens on curved surfaces. Together with the intrinsic flexibility, graphene can be served as a good platform for flexible transformation plasmonics.

©2013 Optical Society of America

## 1. Introduction

Flexible plasmonics has received great attention by industry and scientific community due to its various and promising applications on nonplanar surfaces [1–5]. However, the current applications of flexible plasmonics only make use of the electromagnetic response of plasmonic structures on flexible substrate (e.g. plasmonic antennas [1,2], absorbers [3,4], and sensors [5]). To the best of our knowledge, no flexible plasmonic devices have been realized by controlling the flow of surface plasmon polaritons (SPPs). This is mainly because the surface topological variation can modify the SPP propagation characteristics [6]. Hence the confinement of flexible SPP waves is still a big challenge.

Recently, transformation optics (TO) has been proposed as a general technique to control the light flow in any desirable ways [7–10], which plays significant role in the design of novel devices with unprecedented properties (e.g. invisibility cloaks [11–13], lenses [14–16], and optical black holes [17,18]). More recently, TO has been used to precisely and efficiently control the propagation of SPP waves [6,19–23], achieving a series of new plasmonic devices such as the invisibility carpet cloaks [19,20], bending waveguides [6,21], cylindrical cloaks [19,20], and Luneburg lenses [6,22]. Although TO has the ability to mould the flow of SPP waves on two-dimensional (2D) planar surfaces or confine the SPP propagation on rigid curved or uneven surfaces, it cannot do both simultaneously. This is because the procedures to confine and precisely control SPPs require two different TO strategies. In addition, TO cannot be used to control the flexible SPP propagation on curved surfaces of traditional noble metals, since the framework of TO used to confine SPPs should be redesigned once the curved surface topology is changed. However, graphene provides a possibility to solve the problem.

Graphene has shown its intriguing characteristics as a good platform for plamonics. The propagation of SPPs on planar graphene has been demonstrated experimentally by two groups almost simultaneously [24,25]. Since 2011, TO has been used to control the propagation of SPPs and design novel plasmonic elements with the aid of graphene [26–28], due to its tunability of the conductivity via electrostatic gating and/or chemical doping [29–31]. Nevertheless, all the transformation plasmonic devices based on graphene are implemented on 2D planar surfaces. Here, we make strong confinements and desirable controls of flexible SPP waves on curved surfaces, taking advantages of both TO and graphene. We show that SPP waves on graphene surfaces have very strong lateral confinements, and SPPs can naturally be restricted on curved graphene surfaces with little bending and radiation losses. Such a priority of graphene can be easily used to design special plasmonic devices such as bending waveguides. We further show that the propagation of SPPs confined on curved graphene surfaces can be freely controlled by using TO. As a result, a Y-shaped beam splitter and a Luneburg lens have been demonstrated on curved graphene surfaces. These findings reveal the remarkable properties of graphene to manipulate SPP waves. Integrated with plasmonic nanofeatures and flexible substrates, graphene is a good alternative platform for transformation flexible plasmonics.

## 2. Strong confinement of SPP waves on graphene

The propagation of SPPs on graphene is strongly dependent on the chemical potential. It has been shown that, for a sufficiently high Fermi level which can be achieved by increasing the chemical potential, the plasmon losses in graphene are very small [32]. In order to investigate the effect of chemical potential on characteristics of SPP waves on graphene, we choose a lower chemical potential ${\mu}_{c}=0.246$ eV, which corresponds to carrier density of ${n}_{s}=\text{5}\text{.10}\times {\text{10}}^{12}$ cm^{−2}, and a higher chemical potential ${\mu}_{c}=0.8$ eV, which corresponds to carrier density of ${n}_{s}=5.23\times {10}^{13}$ cm^{−2}, for comparison [33,34]. The chemical potential of ${\mu}_{c}=0.8$ eV in graphene has been realized experimentally by chemical doping or ion-get gating [29–31]. The phenomenological scattering rate $\Gamma $ is chosen as 3.3 meV, corresponding to the momentum relaxation time of $\tau =0.1$ ps, which coincides with the experiment [35]. According to the Kubo formula [34], the conductivity of graphene can be achieved in these two potentials. Then, combined with $\mathrm{Re}({\epsilon}_{r,eq})=-\frac{{\sigma}_{g,i}}{\omega \Delta {\epsilon}_{0}}$ [26] ($\Delta $ is the thickness of graphene, here we set $\Delta =1$ nm), the real part of the equivalent permittivity, $\mathrm{Re}({\epsilon}_{r,eq})$, of the graphene layer is given in Fig. 1. For ${\mu}_{c}=0.246$ eV, the working frequency of SPP wave ranges from 20 to 80 THz; for ${\mu}_{c}=0.8$ eV, the frequency ranges from 20 to 250 THz. Within such frequencies, the propagation of SPP waves satisfies the condition $\mathrm{Re}({\epsilon}_{r,eq})<-1$ [36]. From Fig. 1 we notice that the SPP frequency range can be dramatically enlarged by increasing the chemical potential in graphene.

The characteristics of SPP waves supported by a flat free-standing graphene layer and a flat thin-silver (with thickness of 30 nm) are compared. For SPPs on graphene, the dispersion relation is [26]

where${\sigma}_{g}$, ${k}_{0}$ and ${\eta}_{0}$ are the conductivity of graphene, the free space wavenumber, and the intrinsic impedance of free space, respectively. The effective mode index is ${n}_{g\_spp}=\sqrt{1-{(\frac{2}{{\sigma}_{g}{\eta}_{0}})}^{2}}$, and the SPP wavelength is calculated by ${\lambda}_{g\_spp}={\lambda}_{0}/{n}_{g\_spp}$, in which ${\lambda}_{0}$ is the free space wavelength. The propagation length is calculated by ${L}_{g}=1/\mathrm{Im}(\beta )$, and the lateral decay length is ${\ell}_{g\_spp}=1/\mathrm{Re}(\sqrt{{\beta}^{2}-{(\omega /c)}^{2}})$ [33]. As comparison, the propagation length ${L}_{Ag\_spp}$and lateral decay length ${\ell}_{Ag\_spp}$ of silver can be calculated according to [36].Figure 2 illustrates the ratios of propagation length to SPP wavelength (Fig. 2(a)) and lateral decay lengths (Fig. 2(b)) for silver and graphene with chemical potentials ${\mu}_{c}=0.246$ eV and ${\mu}_{c}=0.8$ eV. From Fig. 2(b), we observe that the SPP waves on graphene have very strong confinements, with lateral decay length one or two orders better than that on the silver surface. Although the SPP wave on graphene with lower chemical potential (${\mu}_{c}=0.246$ eV) has the best confinement, the propagation length is only several SPP wavelengths, which can hardly be used for real applications. However, the propagation length of SPP wave on graphene with higher chemical potential (${\mu}_{c}=0.8$ eV) can reach dozens of wavelength, which is comparable to that of silver at high frequency. Hence, considering the strong SPP confinement and sufficient propagation length, we predict that graphene with higher chemical potential has great advantages over silver, especially on curved surfaces. The conclusion we obtained here can be a good complement to that achieved in [33], in which the chemical potential was assumed to be 0.246 eV by using back-gate [37].

## 3. The propagation of SPP waves on curved graphene surfaces

Next we investigate the propagation of SPP waves on curved free-standing graphene and thin silver film (i.e., the curved silver and graphene are supposed to be imbedded in air.) The curvature-induced radiative energy loss will affect the SPP propagation efficiencies [38,39], and the attenuation of SPP wavenumber $\mathrm{Re}(\beta )$, a parameter closely related to the confinement of SPPs. Hence, there exists a critical curvature radius ${r}^{\ast}$, when $\mathrm{Re}(\beta )$ is reduced below the photon momentum in free space ${k}_{0}$ [38], or ${n}_{spp}=\frac{\mathrm{Re}(\beta )}{{k}_{0}}<1$, the electromagnetic fields can no longer be confined, and the SPP waves become radiative.

The critical curvature radius can be calculated by

*T*= 300 K, $\Gamma =3.3$ meV,

*f*= 160 THz, and ${\mu}_{c}=0.8$ eV. For a 30-nm-thick silver under the conditions of

*T*= 300 K and

*f*= 500 THz, the SPP wavelength is ${\lambda}_{Ag\_spp}=0.55\text{\hspace{0.17em}}\mu m$. Then we calculate the critical curvature radii (${r}^{\ast}$) for graphene and silver are ${r}_{g}^{\ast}=0.02{\lambda}_{g\_spp}$ and ${r}_{Ag}^{\ast}=3.4{\lambda}_{Ag\_spp}$, respectively. Only when the radius of curvature is larger than ${r}^{\ast}$, the SPP wave could be supported. Hence we conclude that the range of curved graphene surfaces to support SPP waves is much broader than that of silver, which is in accordance with the strong-confinement ability of graphene.

We have simulated the propagations of SPP waves on 30-nm-thick silver films (Fig. 3(a)) and on graphene (Fig. 3(b)) along arc surfaces with different radii but the same electrical length ${l}_{arc}=5{\lambda}_{Ag(g)\_spp}$. Three curvature radii are selected for both silver and graphene surfaces: $3{\lambda}_{Ag(g)\_spp}$, $7.5{\lambda}_{Ag(g)\_spp}$, and $\infty $, from the top to bottom. From Fig. 3, we clearly observe that the SPP waves propagate efficiently on the air-graphene-air interfaces with different radii. However, on the air-silver-air interfaces, when the radius is $7.5{\lambda}_{Ag\_spp}$ (which is larger than ${r}_{Ag}^{\ast}$), the confinement of SPP waves is not as good as on air-graphene-air interfaces, and when the radius of $3{\lambda}_{Ag\_spp}$(which is less than ${r}_{Ag}^{\ast}$), it can no longer support SPP waves.

According to the above analysis, the priority of graphene can be easily used to design special plasmonic devices, including 180° bending waveguides, S-shaped waveguides, spiral waveguides, and curved waveguide. As noted earlier [6], in the bending dielectric-metal surface, almost all energy leaks to free space and metal cannot be directly used for the 180° bending, as shown in Fig. 4(a). However, in Fig. 4(b), one clearly observes that the SPP wave on graphene can propagate through the 180° bend excellently well. In order to estimate the scattering loss or the propagation efficiency, the distribution of tangent magnetic fields H_{y} along the bending area are illustrated in Figs. 4(c) and 4(d). For the 180° graphene bending interface, almost all energies of SPP waves are transported through the bending area; while for silver interface, almost all SPP energies are leaked to free space. We further realize very complicated S-shaped waveguide and spiral waveguide using graphene. The geometry and simulation results are demonstrated in Figs. 4(e) and 4(f), which reveal that the SPP waves can be manipulated along arbitrarily curved trajectories using graphene. The S-shaped waveguide and spiral waveguide could be applied to photonic integrated circuits which can directionally route the SPP energies and reduce the sizes of the devices.

We also realize a curved graphene waveguide across a bump on a flat surface, as shown in Fig. 4(g). This waveguide can be widely used to hide targets on the ground, due to their ability to allow propagation of waves on surfaces even though physical objects exist. We start by considering the propagation of SPPs on air-graphene-air interface, which has a one-third-circular bump with the radius of 30 nm. From Fig. 4(g), we observe that the SPP waves propagate through the surface smoothly even when the bump physically exists.

## 4. Flexible transformation plasmonics using graphene

The propagation of SPPs confined on curved graphene surfaces can be further controlled using TO. Hence we can realize flexible transformation SPP devices based on curved graphene layers. We first consider a Y-shaped waveguide variant, which consists of two distinct regions on graphene under the conditions of *f* = 160 THz, $\Gamma =3.3$ meV, and *T* = 300 K. The Y-shaped region possesses the chemical potential of 0.8eV, which corresponds to the carrier density of ${n}_{s}=5.23\times {10}^{13}$ cm^{−2}, resulting in the equivalent permittivity of ${\epsilon}_{r,eq1}=-8.6+i0.13$ (which can support SPP waves). The remaining region possesses the chemical potential of 0.4 eV, which corresponds to the carrier density of ${n}_{s}=1.32\times {10}^{13}$ cm^{−2}, resulting in the equivalent permittivity of ${\epsilon}_{r,eq2}=0.15+i0.71$ (which cannot support SPP waves). The side view of the Y-shaped waveguide variant is a sinusoidal curve with the function $z(x)=12\mathrm{sin}(\pi x/60)$ nm in the region $0\le x\le 300$nm, where $x$ is the propagation direction and $z$ is normal to the surface. Although the graphene layer has been bent to a curved surface, due to the strong confinement, the SPP waves still propagate well along the graphene and are split into two paths, as shown in Fig. 5(a). Such a Y-shaped waveguide variant can realize the function of power dividers in curved surfaces.

Next we propose a curved-surface Luneburg lens on graphene, in which the graphene layer is bent to a shape of a quarter circular arc with radius *R* = 130 nm. The curved Luneburg lens is designed by seven concentric circular rings with different conductivity values according to $\mathrm{Re}({\epsilon}_{r,eq,n})=\frac{\mathrm{Re}({\epsilon}_{r,eq,back})}{\sqrt{2-{(\frac{{r}_{n}+{r}_{n-1}}{2r})}^{2}}}$ [26], where $\mathrm{Re}({\epsilon}_{r,eq,back})$ is the real part of equivalent permittivity of the background graphene, $\mathrm{Re}({\epsilon}_{r,eq,n})$ and ${r}_{n}$ are the real part of equivalent permittivity and radius of the *n*^{th} ring, and *r* is the radius of Luneburg lens. Under the conditions of $f=160$ THz, $T=300$ K, $\Gamma =3.3$ meV, and ${\mu}_{c}=0.8$ eV, the equivalent permittivity of the background graphene is calculated as ${\epsilon}_{r,eq,back}=-8.6+i0.13$. Figure 5(b) illustrates the simulation results, showing that the spherical waves generated from a point source are transformed to “curved-plane” SPP waves by the curved Luneburg lens on graphene surface. The above two examples suggest that graphene provides a platform for various flexible plasmonic devices.

In summary, compared to noble metal, graphene has much stronger confinement for SPPs on curved surfaces. We have shown that flexible SPP waves can be strongly confined and efficiently manipulated by combining graphene with TO. Novel flexible plasmonic devices with intriguing functions have been realized on curved graphene surfaces, including the bending waveguides, 180°-bending waveguide, S-shaped waveguide, spiral waveguide, curved waveguide, Y-shaped waveguide variant, and curved Luneburg lens. Together with the intrinsic flexibility, graphene is promised as a good platform for transformation flexible plasmonics.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61271057, 61071045, 11104026, and 11174051), Doctoral Fund of Ministry of Education of China (No. 20110092110009). TJC acknowledges supports by the National Science Foundation of China (60990324, 61138001, and 60921063) and the 111 Project (111-2-05).

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