1. Introduction

In 2006, Pendry et al. [1] and Leonhardt [2] independently showed the possibility of designing a cloak that renders any object inside it invisible to electromagnetic radiation. This coating consists of a meta-material whose physical properties (permittivity and permeability) are deduced from a coordinate transformation in the Maxwell system. The anisotropy and the heterogeneity of the parameters of the coat work as a deformation of the optical space around the object. The first experimental validation [3] of these theoretical considerations was given, a few months later for a copper cylinder invisible to an incident plane wave at 8.5 GHz as predicted by the numerical simulations. This markedly enhances the capabilities to manipulate light, even in the extreme near field limit [4]. However, such cloaks suffer from an inherent narrow bandwidth as their transformation optics design leads to singular tensors on the frontier of the invisibility region (obtained by blowing up a point [5]). To remove the cloak’s singularity, Jiang et al. proposed to consider the blow-up of a segment instead of a point [6], but this cloak only works for certain directions. On the other hand, Leonhardt and Tyc considered a stereographic projection of a virtual hyper-sphere in a four dimensional space [7], which bears some resemblance with the construction of a Maxwell fisheye.

As an alternative to non-singular cloaking Li and Pendry proposed a one-to-one geometric transform from a flat to a curved ground: their invisibility carpet [8] is by essence non singular and thus broadband [9]. This proposal led to a rapid experimental progress in the construction of carpets getting close to optical frequencies [10–13].

Another way to make cloaks broadband is to approximate their parameters using a homogenization approach, which leads to nearly ideal cloaking [14–16], as it does not rely upon locally resonant elements. In 2008, some of us [17] demonstrated broadband cloaking of surface liquid waves using a micro-structured metallic cloak which was experimentally validated around 10 Hz. Ultra-broadband cloaking can even be achieved in this way for flexural surface waves in thin-plates [18]. This naturally prompts the question of whether, at optical frequencies, an object lying onto a metal film could be cloaked from propagating Surface Plasmon Polaritons (SPPs).

The extraordinary physics of the transmission of light through holes small compared with the wavelength is by now well known [19], but some heralding earlier work is less well known (for example combining both theory and experiment [20]). Pendry, Martin-Moreno and Garcia-Vidal showed in 2004 that one can manipulate surface plasmon ad libitum via homogenization of structured surfaces [21]. In the same vein, pioneering approaches to invisibility relying upon plasmonic metamaterials have already led to fascinating results [22–25]. 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 be hidden, and external cloaking, whereby a plasmonic resonance cancels the external field at the location of a set of electric dipoles. In 2008, Smolyaninov, Hung and Davis achieved a noticeable reduction in the scattering of SPPs incident upon a cloak consisting of polymethylmethacrylate at a wavelength of 532 nm [24]. More recently, Baumeier, Leskova and Maradudin have demonstrated theoretically and experimentally that it is possible to reduce significantly the scattering of an object by an SPP at a wavelength of 632.8 nm when it is surrounded by two concentric rings of point scatterers [25]. However these two experiments rely upon the resonant features of the plasmonic cloak.

In the present paper, we extend the design of invisibility carpets to SPPs on a finite frequency range. The cornerstone of our approach is the identification of the dispersion relation of SPPs at the interface between a metal and a heterogeneous anisotropic carpet. On this basis, we design a dielectric carpet and use full wave computations to study its properties. Finally, we measure the SPP intensity when interacting with the carpet to validate the concept.

2. Transformational plasmonics

2.1. Surface plasmon polaritons on a metal-dielectric interface

We consider a transverse magnetic (p-polarized) SPP propagating in the positive x direction at the interface $z=0$ between metal ($z<0$) and air ($z>0$):

For this field to be solution of Maxwell’s equations, continuity of its tangential components is required across the interface z = 0 and this brings $k_{x1}=k_{x2}=k_x$ together with the dispersion relations:

2.2. Geometric transform for a plasmonic carpet

So far, such a mathematical setting is fairly standard. However, we now wish to analyze the interaction of SPPs with an anisotropic heterogeneous structure, in the present case an invisibility carpet, deduced from the following geometric transformation:

The permittivity and permeability tensors in the transformed coordinates are now given by:

The symmetric tensors $\underset{¯}{\underset{¯}{{\epsilon }^{\prime }}}$ and $\underset{¯}{\underset{¯}{{\mu }^{\prime }}}$ are fully described by five non vanishing entries in a Cartesian basis:

2.3. Surface plasmon polaritons on a transformed metal-dielectric interface

In a diagonal basis associated with a quasi-conformal grid the fully anisotropic tensors reduce to: $\underset{¯}{\underset{¯}{{\epsilon }^{\prime }}}=\text{diag}({\epsilon }_{xx2},{\epsilon }_{yy2},{\epsilon }_{zz2})$ and $\underset{¯}{\underset{¯}{{\mu }^{\prime }}}=\text{diag}({\mu }_{xx2},{\mu }_{yy2},{\mu }_{zz2})$. Further assuming some low spatial variation of both tensors within the carpet, we define local transverse wave numbers associated with SPPs propagating at the interface between the metal and the carpet that should satisfy (2) for $i=1$.

Let us derive the dispersion relation for a surface plasmon at the interface between a metal and the above anisotropic medium described by diagonal tensors of relative permittivity and permeability $\underset{¯}{\underset{¯}{{\epsilon }^{\prime }}}=\text{diag}({\epsilon }_{xx2},{\epsilon }_{yy2},{\epsilon }_{zz2})$ and $\underset{¯}{\underset{¯}{{\mu }^{\prime }}}=\text{diag}({\mu }_{xx2},{\mu }_{yy2},{\mu }_{zz2})$. From the first Maxwell equation, we know that

2.3.1. Fundamental properties of SPPs in transformed media

The penetration length of the SPPs in both media depends on the dielectric constant and can be easily expressed as:

For the sake of clarity we define:

2.4. Reduced form of the permittivity and permeability tensors for a structured plasmonic carpet above the metallic surface

We can compute the eigenvalues of $T^{-1}$ as these are the relevant quantities for the tensor components along the main optical axes:

The expression for the relative permittivity and permeability tensors becomes $\underset{¯}{\underset{¯}{{\epsilon }^{\prime }}}=\text{diag}({\Lambda }_1,{\Lambda }_2,{\Lambda }_3)$ and $\underset{¯}{\underset{¯}{{\mu }^{\prime }}}=\text{diag}({\Lambda }_1,{\Lambda }_2,{\Lambda }_3)$. If we now multiply these expressions by α, we obtain the reduced tensor components along the main optical axes, and importantly ${\Lambda }_3=1$. This means we now have only two varying tensor components for the permittivity, with the permeability tensor reducing to the identity. We further looked at a design for which${\Lambda }_1$~${\Lambda }_2$ .

The dispersion relation (13) reduces to:

We point out that there exists a family of coordinate transformations that would map a flat mirror onto a curved one. However, if we now want to reduce the anisotropy of the carpet (e.g. to avoid dealing with magnetism), we need to work with quasi-conformal coordinates, as proposed by Li and Pendry [8]. We deduced those from the minimization of the modified Liao functional [28], and the resulting grid is shown in Fig. 1(a) .

 

Fig. 1 (a) Quasi-conformal grid associated with transformed plasmonic space in a crescent carpet; (b) SPP incident from the top on the heterogeneous carpet deduced from the quasi-conformal mapping (c) SEM micrograph of the structure realized by single-step electron-beam-lithography. The cloak is made of TiO₂ cones as shown in the zoom (upper right). The TiO₂ particles have a conical shape numerically approximated by a cylindrical one (h=200 nm, r=100 nm).

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In Fig. 1(b) we have simulated a SPP incident on a heterogeneous carpet whose refractive index distribution is related to the quasi-conformal map via $n^2=1/\sqrt{\text{det}(J^{\text{T}}J)}$. We are in a position to mimic the properties of such a spatially varying isotropic metamaterial simply by placing some identical dielectric pillars at the nodes of the quasi-conformal grid, as shown in Fig. 1(a). The diameter of these pillars has been found via a simple optimization algorithm with an objective function taking as argument the refractive index of pillars and their diameter.

Moreover, it is easily seen that the anisotropy of the crescent carpet reduces when we flatten its boundaries, and this allows us to find a good compromise between control of the SPPs and complexity of the arrangement of pillars. We discovered that dielectric cylindrical pillars with a refractive index$n=\sqrt{5.76}$, of height 200 nm and identical diameter 200 nm lead to nearly perfect cloaking for SPPs at λ = 800 nm when they are located at the quasi-conformal grid nodes.

3. Numerical modelling of surface plasmon polaritons with finite elements

In order to compute the total electromagnetic field for a SPP normally incident upon a three-dimensional carpet, we implemented the weak form of the scattering problem in the finite element package COMSOL using second order finite edge elements which behave nicely under geometric changes. Perfectly Matched Layers (PMLs), which can be seen as a stretch of coordinates, further enable us to model the unbounded three-dimensional domain. We choose the magnetic field $H=(H_1,H_2,H_3)(x,y,z)$ as the unknown and therefore look for solutions of

Also, $H=H_i+H_d$, where $H_i$ is the SPP incident field given by (1). The expression for $H_i$ has been enforced on a flat surface on the leftmost part of the computational domain and $H_d$ is the diffracted field which decreases inside the PMLs. We note that PMLs involve both an anisotropic heterogeneous medium with absorption above the interface z = 0, which is fairly standard, but also an anisotropic heterogeneous metal with gain below the interface. The typical number of degrees of freedom used in our computations is around ${5.10}^6$ and computational time is approximately 15 hours on a supercomputer with 256 Gb of RAM.

3.1. Mimicking the scattering of a SPP by a flat Bragg-mirror with a curved Bragg-mirror dressed with a structured carpet

Figures 2(a) and (d) display the computed intensity pattern for a SPP at 800 nm incident on a curved metallic reflector dissimulated behind a cloak of dielectric pillars. We also show for comparison the cases of a flat (Fig. 2(b)) and curved reflector without cloak (Fig. 2(c)). One can clearly see how the designed dielectric cloak transforms the back-reflected light pattern to make it mimic that of the flat reflector. Importantly, we further numerically checked that the SPP cloaking is robust with respect to the shape of pillars, and for instance conical pillars produce nearly the same overall scattering as cylindrical pillars of the same height (not shown for sake of brevity). We attribute this property to the fact that the dielectric pillars are non-resonant elements whose scattering cross-section is only slightly modified by moderate changes in shape.

 

Fig. 2 Numerical diffraction of a SPP incident from the top for (b), (c) and (d). Magnitude of the magnetic field is represented: (a) 3D structure used in the simulations. (b) The incident SPP (yellow arrow) hits the straight reflector (red line). (c) The incident SPP (yellow arrow) hits the curved reflector (red line). (d) Placing the cloak in front of the curved reflector (red line) nearly compensates for the curved reflector leading to a straight beating pattern.

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3.2. Broadband plasmonic cloaking in the visible spectrum

One of the main features of the SPP carpet we designed is that it consists of fairly low contrast structural dielectric elements, thereby ensuring a smooth dependence upon the working wavelengths: This was made possible thanks to the removal of the magnetic component of the transformed medium using some reduced form of the permittivity and permeability tensors. Indeed, artificial magnetism is one of the key challenges in metamaterials as it involves strongly resonant structural elements. The invisibility carpets proposed by the groups of Pendry and Shalaev [3,14] are inherently narrowband as they rely upon resonant elements (the former proposal makes use of split ring resonators while the latter is based upon thin metallic wires). The touchstone of the 2008 proposal of an invisibility carpet by Li and Pendry is indeed to broaden the range of wavelengths over which cloaking occurs [8]. The first experimental realizations of structured invisibility carpets are in essence broadband [10–13], but importantly not in the visible spectrum. The numerical achievements within this paper are twofold: we design a broadband carpet for surface plasmon polaritons, which extends the seminal work of Pendry, Schurig and Smith [1] to control of plasmonic fields; besides from that, our carpet works at visible wavelengths (650nm-900nm), see Fig. 4 . We note that while Smolyaninov et al. [24] already proposed a plasmonic cloak working at 532nm, the underlying mechanism was much different (no geometric transform involved).

 

Fig. 4 Numerical simulation of SPPs interacting with a dielectric carpet for different wavelengths. Magnitude of the magnetic field is represented for wavelengths (a) 650 nm, (b) 700 nm, (c) 800 nm, and (d) 900 nm, respectively.

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4. Experimental results

In order to meet experimentally the parameters found in our simulations, we chose a configuration in which a gold surface is structured with TiO₂ nanostructures. The TiO₂ pillars forming the crescent-moon-like carpet were first fabricated on top of a 60-nm-thin Au film by combining electron-beam lithography and reactive ion etching. In a second lithography step, we added a curved Bragg-type reflector (formed by 15 gold lines (section =150 nm×150 nm) periodically separated by half of the SPP wavelength), acting as the object to be hidden behind the carpet (see Fig. 1(c)). The shape of the obtained TiO₂ particles is conical (h=200 nm, r=100 nm) as a consequence of the etching anisotropy.

The SPP was launched at a ripple-like, 200-nm-wide TiO₂-line [29] placed 44 µm away from the reflector as shown in Fig. 5 . SPPs propagating on thin metal films deposited on dielectric substrates have radiative losses into the substrates. This leakage radiation was collected using a high-numerical aperture objective to map the SPP fields [30]. Additionally for the sake of clarity, we employed spatial filtering in the conjugated (Fourier-) plane to suppress the direct transmitted light from the excitation spot and scattered light in order to isolate the carpet properties. Original attempts at reflecting SPPs with flat and curved homogeneous metallic step-like mirror turned out being inefficient because the SPPs tend to transmit across the step and also radiate. We therefore decided to consider instead flat and curved Bragg mirrors, formed by periodically arranged metal ridges, which show a much higher reflectivity [31].

 

Fig. 5 Experimental image of the leakage radiation of the SPPs interacting with a curved Bragg mirror. The SPPs are excited by slightly focussing the light at the lithographically structured defect line (marked by the yellow rectangular) and propagate under 90° at the surface away in both directions as indicated by the yellow arrows. The doted yellow rectangular indicates the area zoomed in Fig. 6.

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The leakage radiation microscopy (LRM) images map the distribution of the SPPs propagating at the gold/air interface and interacting with the different structures fabricated at the gold surface. In the case of a bare curved Bragg-reflector, the reflected SPPs are propagating into different directions depending on their relative angle to the normal to the mirror lines (see green arrows in Fig. 6 (c) ), thus leading to a curved wave front. Conversely, adding the crescent-moon-like TiO₂ carpet re-establishes a fringe pattern with a nearly straight wave front (see Fig. 6 (b)) similar to the case of a flat Bragg-mirror. The remaining small lateral modulations are attributed to imperfections in the manufacturing. Further, data analysis has been used to quantify the modification in the wave front curvature induced by the presence of the crescent-moon-like TiO₂ carpet. Comparing the areas under the numerically averaged curves b (curved mirror with carpet) and c (curved mirror without carpet) leads to reduction by a factor 3.7 as shown in Fig. 6(d).

 

Fig. 6 Experimental image of the leakage radiation of the SPP at λ=800 nm. (a) The incident SPP (yellow arrow) hits the straight Bragg mirror (red). The interference with the back-reflected SPPs (green) results in a straight beating pattern. (c) The incident SPPs (yellow arrow) hit a curved Bragg mirror (red). The backreflected SPPs (green) have different directions due to the curved shape of the reflector resulting in the curved intensity pattern dominated by the beating of the counter-propagating SPPs. (b) Cloak is placed in front of the curved Bragg mirror. The beating pattern in reflection is clearly visible and similar to a straight Bragg reflector. (d) Averaged relative position of the interference fringes

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5. Conclusion

In conclusion, we have studied analytically and numerically the extension of cloaking to near infrared SPP waves propagating at a metal/dielectric interface. These waves obey the Maxwell equations at a flat interface and are evanescent in the transverse direction, so that, the problem we have treated is somewhat isomorphic to the case of linear surface water waves which satisfy a similar dispersion relation.

Nevertheless, our analytical derivation of the dispersion relation for SPPs propagating at the interface between metal and a medium conceived by transformational optics greatly enhances our capabilities to design metamaterials for plasmonics such as invisibility carpets. Moreover, 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 a plasmonic carpet consisting either of an anisotropic heterogeneous medium or dielectric pillars regularly spaced in air. These two media are in any case described by permittivities of opposite sign in order to leave enough room for the existence of SPPs.

One of the main achievements of the present work is to bring cloaking a step closer to visible wavelengths as we consider a SPP at wavelength λ = 800 nm. We also emphasize that the manufactured crescent carpet should be broadband. Indeed, numerical results for a range of wavelengths λ=650 to 900 nm demonstrate the principle underlying our approach to plasmonic cloaking should also work in the visible spectrum. However, SPPs attenuate faster at such wavelengths what poses future experimental challenges. Plasmonic meta-surfaces provide a natural experimental platform to demonstrate principles of transformational optics in the context of two-dimensional optics [32]. Using transformation based methods, it is also possible to control SPPs propagating on curved metallic surfaces by manipulating the refractive index of the medium above the surface [33,34]. Recent proposals of metamaterials emulating anisotropic permittivity and permeability [35] could be implemented to engineer a three-dimensional structured carpet in a way similar to what was chiefly achieved at infrared wavelengths [13]. However, in the present paper, we have focused on control of SPPs on flat structured meta-surfaces, so that the prerequisite metamaterials are in essence planar. The ideas developed can be also applied for the design of structured meta-surfaces for cloaks, concentrators or rotators, foreseen in [36] (whereby only geometric transformations were considered).