## Abstract

We adapt tools of transformation optics to surface plasmon polaritons (SPPs) propagating at the interface between two anisotropic media of opposite permittivity sign. We identify the role played by entries of anisotropic heterogeneous tensors of permittivity and permeability-deduced from a coordinate transformation- in the dispersion relation governing propagation of SPPs. We apply this concept to an invisibility cloak, a concentrator and a rotator for SPPs.

©2010 Optical Society of America

## 1. Introduction

Invisibility is a very old subject mixing fascinating and elusive features. It is by now well known that one can reverse the flow of light with negative refractive index materials, within which light takes the wrong turn in accordance with inverted Snell-Descartes laws of refraction [1]. There has been a growing interest in a better control of light through transformational optics, following the recent proposals by Pendry et al. [2] and Leonhardt [3]. The former seminal paper demonstrates the possibility of designing a cloak that renders any object inside it invisible to electromagnetic radiation (using the covariant structure of Maxwell’s equations), while the latter concentrates on the ray optics limit (using conformal mappings in the complex plane). In both cases, the cloak consists of a meta-material whose physical properties (permittivity and permeability) are spatially varying and matrix valued. These theoretical considerations might have remained an academic curiosity, but an experimental validation [4] chiefly achieved in the Gigahertz regime came a few months later for a copper cylinder invisible to an incident plane wave. This markedly enhances our capabilities to manipulate light, even in the extreme near field limit, when a sources lies in the close neighbourhood of the cloak [5]. However, all of these cloaks suffer from an inherent narrow bandwidth as their transformational optics design leads to singular tensors on the frontier of the invisibility region, as first analysed in the context of inverse problems by [6]. The anisotropy and the heterogeneity of these optical parameters work as a deformation of the optical space around the object, in a way similar to what a heavy mass does for gravitational waves in Einstein’s theory of general relativity.

Back in 1998, Ebbesen et al. established that resonant excitation of surface plasmons enhance electric fields at a surface that force light through its tiny holes, giving unusually high transmission coefficients in the sub-wavelength regime [7]. Pendry, Martin-Moreno and Garcia-Vidal further showed in 2004 that one can manipulate surface plasmon ad libitum via homogenization of structured surfaces [8]. In the same vein, pioneering approaches to invisibility relying upon plasmonic metamaterials have already led to fascinating results [9–12]. These include plasmonic shells with a suitable out-of-phase polarizability in order to compensate the scattering from the knowledge of the electromagnetic parameters of the object to hide, and external cloaking, whereby a plasmonic resonance cancels the external field at the location of a set of electric dipoles. Recently, Baumeier et al. have demonstrated theoretically and experimentally that it is possible to reduce significantly the scattering of an object by a surface plasmon polariton, when it is surrounded by two concentric rings of point scatterers [12].

In the present paper, we extend the design of transformation based metamaterials to the area of surface plasmon polaritons (SPPs). The field of plasmonics has matured to the point where an introduction to the optical properties of metals and sometimes even surface plasmons is included in many standard texts in solid-state physics and optics, with a state of the art on meta-surfaces available in [1].

Our main contribution here is that we explain the physics underlying cloaking mechanism by deriving the dispersion relation of SPPs at the interface between two heterogeneous anisotropic media. We then design a cloak, a concentrator and a rotator and use full wave computations to back up our claims of an unprecedented control of SPPs through transformational plasmonics.

## 2. Transformational plasmonics

Let us consider two semi-infinite regions separated by a plane interface at *z* = 0. The upper region is filled with air i.e. with relative permittivity *ε*
_{1} = 1 (resp. relative permeability *μ*
_{1} = 1) for *z* > 0, while the lower region is filled with a Drude metal i.e. with relative permittivity ${\epsilon}_{2}=1-\frac{{\omega}_{p}^{2}}{{\omega}^{2}+1\mathrm{\gamma \omega}}$ (resp. relative permeability *μ*
_{2} = 1) for *z* < 0 : here, some gold with the plasma frequency (*ω _{p}* = 2175 THz) and characteristic collision frequency (

*γ*= 4.35 THz).

We would like to map these two isotropic homogeneous media on two metamaterials described by anisotropic heterogeneous matrices of permittivity and permeability given by [5]

where **T** = **J ^{T}J**/det(

**J**) is the transformation matrix constructed using the Jacobian associated with the change of coordinates. Let us emphasize here that in our numerical implementation, we make use of finite edge elements which are nothing but discrete Whitney differential forms which behave nicely under pull-back transforms. Thus, in what follows, we always map the destination domain onto the original one (so we consider the inverse transforms).

Let us now derive the dispersion relation for a surface plasmon at the interface between two anisotropic media described by diagonal tensors of relative permittivity and permeability *ε _{i}*̳ = diag(

*ε*) and

_{xxi},ε_{yyi},ε_{zzi}*μ*̳ = diag(

_{i}*μ*) with

_{xxi},μ_{yyi},μ_{zzi}*i*= 1 when

*z*> 0 and

*i*= 2 when

*z*< 0, see Fig. 1(a).

From the first Maxwell equation, we know that

where **H**
_{i} is defined by [see Fig.1(b) for the axis system]:

with ℜ(*k*
_{z1}) > 0 and ℜ(*k*
_{z2}) < 0 in order to maintain evanescent fields above and below the interface *z* = 0. This leads to

with **E**
_{j} = (*E _{xj}*, 0,

*E*) and

_{zj}*c*= (

*ε*

_{0}

*μ*

_{0})

^{-1/2}. The transverse wavenumbers are found by invoking the other Maxwell equation

which leads to

The boundary condition at the interface *z* = 0 requires continuity of the tangential components of the electromagnetic field, which brings

The existence of a surface plasmon thus requires *ε*
_{xx1} and *ε*
_{xx2} to be of opposite sign.

Substituting Eq. (6) into Eq. (7), we obtain the dispersion relation for a surface plasmon at the interface between two anisotropic media

Assuming that *ε _{xxi}* =

*ε*=

_{zzi}*ε*and

_{i}*μ*= 1 for

_{yyi}*i*= 1, 2, we retrieve the well-known dispersion relation for two homogeneous media:

## 3. Numerical illustrations

We use the full wave finite element package COMSOL MULTIPHYSICS to model a p-polarized SPP i.e. satisfying Eqs. (3)–(4). The transformed coordinates are computed using a similar approach to the design of cylindrical optical cloaks [5]. Note that due to the localization of the plasmon in the vicinity of the surface the extent of the cloak along the z-direction could be finite without changing the behaviour of the device.

In the present paper, all three geometrical transformations which we consider amount to mapping the electromagnetic field from a circular region to an annular region.

More precisely, we consider four functions *f _{i}*(

*r*),

*g*(

_{i}*r*),

*i*= 1,2 such that:

In the sequel, we launch a SPP at wavelength *λ* = 800 nm on a coated cylinder with an external boundary at *R*
_{3} = 2000 nm.

#### 3.1. Cloaking

We first want to hide a semi-infinite metallic cylinder (*z* > 0) of radius *R*
_{2} = 800 nanometers. For this, we need to coat it with a heterogeneous metamaterial with anisotropic permittivity and permeability. The underpinning geometrical transformation consists in sending the electromagnetic field from the disc 0 < *r* < *R*
_{2} onto the annulus *R*
_{2} < *r*
^{′} < *R*
_{3}. For this, we implement Eq. (10) with linear functions *f*
_{1}(*r*) = *g*
_{1}(*r*) = *αr* + *β* and *f*
_{2}(*r*, *θ*) = *θ*, *g*
_{2}(*r, θ*) = *θ*, where *α* = *R*
_{3} - *R*
_{2}
*R*
_{3} and *β* = *R*
_{2}.

We obtain the following transformation matrix [2, 5]:

where **R**(*θ*) is the rotation matrix of angle *θ* about the *z*-axis. We report in Fig. 2 the diffraction of a SPP by a metallic cylinder [Fig. 2(a)] on its own, and when it is surrounded by the heterogeneous anisotropic cloak [Fig. 2(b)]. In the latter case, the phase and amplitude of the SPP are clearly recovered in the forward scattering, while the backward scattering vanishes.

#### 3.2. Concentrator

Regarding Fig. 3(a), we want to concentrate the electromagnetic field within the central region *r*
^{′} < *R*
_{1} with *R*
_{1} = 800 nm (thereby enhancing light and matter interaction) at the expense of an expansion of space within the ring *R*
_{1} < *r*
^{′} < *R*
_{3}. For this, we consider a geometrical transformation which maps a disc of radius *R*
_{2} = 1400 nm on a disc of radius *R*
_{1} (compression of space) which in the same time maps a ring (*R*
_{2} < *r* < *R*
_{3}) onto a larger ring (*R*
_{1} < *r*
^{′} < *R*
_{3}) (expansion of space). Importantly, the transformation is continuous to free space at *r*
^{′} = *R*
_{3}, and consequently the resulting metamaterial will be impedance matched to air on its outer boundary.

In terms of the system Eq. (10), this means we consider some linear functions [13]

with *f*
_{2}(*r, θ*) = *g*
_{2}(*r, θ*) = *θ*,${\alpha}_{1}=\frac{{R}_{1}}{{R}_{2}}$, *β*
_{1} = 0, and ${\alpha}_{2}=\frac{{R}_{3}-{R}_{1}}{{R}_{3}-{R}_{2}}$, ${\beta}_{2}=\frac{{R}_{1}-{R}_{2}}{{R}_{3}-{R}_{2}}{R}_{3}$.

We obtain the same structure for the transformation matrix as in Eq. (11) with however *α* = *α*
_{1} in the disc *r*
^{′} < *R*
_{1} and *α* = *α*
_{2} in the ring *R*
_{1} < *r*
^{′} < *R*
_{3}.

#### 3.3. Rotator

We finally want to rotate the electromagnetic field inside the inner cylinder of radius *r* ≤ *R*
_{2}, with *R*
_{2} = 800 nm by an angle *θ*
_{0} = *π*/4. We consider the system Eq. (10) with [14]

where (*α*
_{1} = 0 , *β*
_{1} = *θ*
_{0}, ${\alpha}_{2}=\frac{{\theta}_{0}}{{R}_{2}-{R}_{3}}$ and ${\beta}_{2}=\frac{{\theta}_{0}}{{R}_{3}-{R}_{2}}{R}_{3}$.

This leads to the following transformation matrix in the region *R*
_{2} < *r* < *R*
_{3}:

where the eigenvalues ${\lambda}_{1}=1+1/2{\alpha}_{2}^{2}{r}^{2}+1/2\sqrt{4{\alpha}_{2}^{2}{r}^{2}+{\alpha}_{2}^{4}{r}^{4}},$, ${\lambda}_{2}=1+1/2{\alpha}_{2}^{2}{r}^{2}-1/2\sqrt{4{\alpha}_{2}^{2}{r}^{2}+{\alpha}_{2}^{4}{r}^{4}}$ and *λ*
_{3} = 1 are associated with eigenvectors $\mathbf{u}=(\frac{{\alpha}_{2}r}{1/2{\alpha}_{2}^{2}{r}^{2}+1/2\sqrt{4{\alpha}_{2}^{2}{r}^{2}+{\alpha}_{2}^{4}{r}^{4}}},\mathrm{1,0}),\mathbf{v}=(\frac{{\alpha}_{2}r}{1/2{\alpha}_{2}^{2}{r}^{2}-1/2\sqrt{4{\alpha}_{2}^{2}{r}^{2}+{\alpha}_{2}^{4}{r}^{4}}},\mathrm{1,0})$, and **w** = (0,0,1).

As a result of this geometric transform, the polarization of the SPP is clearly rotated within the inner disc *r* ≤ *R*
_{1} in Fig. 3(b), whereas the field outside the device goes unperturbed.

## 4. Conclusion

In conclusion, we have studied analytically and numerically the extension of transformational optics to the domain of surface plasmon waves propagating at the interface between two anisotropic heterogeneous media resulting from a geometric transformation in a lower-half space filled with a Drude metal and and an upper half-space filled with dielectric/air. We have derived the dispersion relation Eq. (8) of p-polarized SPPs at such an interface.

Our numerical computations based on the finite element method take into account the three dimensional features of the problem, such as plasmon polarization and jump of permittivity at the interface between metal and dielectric/air which are described here by tensors of permittivity with one diagonal entry of opposite sign, see Eq. (7), with obvious changes in case of a s-polarized SPP.

For illustrative purpose, we have opted for the designs of an invisibility cloak, a concentrator, and a rotator for surface plasmon polaritons based on the geometric transforms introduced by Pendry and co-workers in optics. In the same way, one could control SPPs on curved surfaces with other transformation based metamaterials, [15, 16]. These theoretical concepts have been validated experimentally with a broadband plasmonic carpet [17].

## Acknowledgements

The authors are grateful for insightful comments by Mr G. Dupont and Prof. R. Quidant.

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