Cyclic concentrator, carpet cloaks and fisheye lens via transformation plasmonics

We first review basic equations of plasmonics in anisotropic media. We recall the origin of Maxwell’s gradient index fisheye lens. We then apply tools of transformation optics to the design of a cyclic concentrator and a variety of plasmonic carpet-cloaks. We further give a brief account of the discovery of spoof plasmon polaritons (SfPPs) by Pendry et al (2004 Science 305 847–8) 150 years after Maxwell invented the fisheye lens. Finally, we experimentally demonstrate a concept of a fisheye lens for SfPPs at microwave frequencies. We stress that perfect metal surfaces perforated with dielectrics offer a playground for moulding surface waves in many areas of physics.


Introduction
Mathematicians and physicists have studied the physics of waves at structured interfaces for more than a century. In 1898 the mathematician Lamb (who gave his name to flexural waves in plates) wrote a paper on reflection and transmission through metallic gratings [1] which could be viewed as an ancestor of the 2004 theoretical proposal of spoof plasmon polaritons (SfPPs) by Pendry, Martin-Moreno and Garcia-Vidal [2]. SfPPs have been at the core of many exciting new developments in the past decade with the emergence of metamaterials in plasmonics [3][4][5] but also metamaterials to mould surface water waves, Lamb waves in structured plates and very recently surface seismic (Rayleigh) waves [6].
Indeed, ideas in so-called transformational plasmonics translate into the world of hydrodynamic and elastodynamic waves. This makes the class of surface electromagnetic waves known as surface plasmon polaritons (SPPs), which can be thought of as collective motions of electrons at structured metal/dielectric interfaces, a unique platform for the investigation of analogous phenomena in large-scale metamaterials, such as ocean wave and seismic metamaterials. Of course, one can argue that transformational plasmonics opens up unprecedented avenues for scaling down optical devices and it makes possible two-dimensional integrated optics capable of supporting both light and electrons with an accurate control of their flow. This in itself fully justifies the keen interest of the metamaterials community in this young topic that appeared in 2010. But while most efforts have so far focused on the extrapolation of concepts borrowed from guided optics to novel plasmonic functionalities, we would like to stress that most of what shall be discussed in the sequel finds some counterpart in the realm of linear (and possibly non-linear e.g. tsunami) water waves and surface seismic waves. The plan of this paper is as follows. We first review basic equations of transformation plasmonics (TP). As a first application of TP (this versatile mathematical toolbox can be seen as a subset of transformation optics), we then propose to design a cyclic SPP concentrator and an omnidirectional SPP carpet cloak, inspired by the famous strategy of Li and Pendry of hiding under a carpet [7]. We then introduce a plasmonic fisheye, counterpart of Maxwell's fisheye in optics [8] (which was revisited by Luneburg [9] with an approach that could be considered as some ancestor of transformation optics) that we experimentally demonstrate by designing a metal grid supporting a spoof plasmon polariton in the microwave regime. We conclude the paper with remarks on potential applications of these concepts in the realm of ocean and seismic waves.

Mathematical tools of transformational optics applied to plasmonics
A few words on the mathematical setup should be in order if one wants to understand the specificity of a transformation plasmonics design in comparison with transformational optics. To simplify the discussion, we consider transverse magnetic (p-polarized) SPPs propagating in the positive xdirection at a flat interface z=0 between metal (z<0) and an isotropic medium (z>0): Note here that R e (k z1 ) and R e (k z2 ) are strictly positive in order to enforce the evanescence of fields above and below the interface z=0.
Such fields are a solution of Maxwell's equations, therefore continuity of their tangential components is required across the interface z=0, which translates into mathematics as k x1 =k x2 =k x and this in turn leads us to the famous dispersion relations for p-polarized SPPs:

( )
This equation shows that one now has additional degrees of freedom to mould the flow of SPPs at interfaces between metal and transformed dielectric media. An even richer dispersion relation can be obtained at an interface between transformed metal and dielectrics [11][12][13][14].
As an alternative to the control of SPPs on flat interfaces between transformed media, one might wish to consider curved interfaces. The groups of Martin-Moreno and Garcia-Vidal [11] and Zhang [12] have explored this fascinating route. To conclude this introduction, we would like to stress that in a similar way to what transformation optics has achieved with antennas, lenses, carpets, cloaks, concentrators, beam-splitters, super-scatterers, optical black holes etc for volume electromagnetic waves in specially designed anisotropic heterogeneous media, transformation plasmonics opens a door to a whole new world of metamaterials for light harvesting in surface science.

Mapping a virtual sphere on plasmonic surfaces
It is well known that the geodesics on a sphere surface are the great circles. As illustrated by the red line in figure 1, this implies that all geodesics passing through a point should cross at the same point on the opposite side. Transformation media have been designed for electromagnetic waves in order to mimic the sphere surface. One famous example of an optical system within which light follows the same trajectories as it would on a sphere is the fisheye lens, which was conceived 160 years ago by James Clerk Maxwell. One might hastily conclude that transformation optics (TO) takes its roots in the 19th century. However, at that time Maxwell did not use TO tools to deduce the spatially varying refractive index of the eponymous fisheye lens: mathematical objects underpinning TO such as tensors would be used by Albert Einstein more than half a century afterwards in the context of general relativity.
Although light rays in the designed transformation medium experience the same focusing phenomenon as on a sphere surface, in the mathematical problem Maxwell proposed (see below), he did not consider any stereographic projection of the sphere surface on the plane where light propagates along circular trajectories (cf problem 3, vol 8 in [8]): A transparent medium is such that the path of a ray of light within it is a given circle, the index of refraction being a function of the distance from a given point in the plane of the circle. Find the form of this function and shew that for light of the same refrangibility: (1) The path of every ray within the medium is a circle.
(2) All the rays proceeding from any point in the medium will meet accurately in another point. (3) If rays diverge from a point without the medium and enter it through a spherical surface having that point for its centre, they will be made to converge accurately to a point within the medium.
Maxwell notes that his solution to this problem (based on Euclidean geometry) came from: The possibility of the existence of a medium of this kind possessing remarkable optical properties, was suggested by the contemplation of the structure of the crystalline lens in fish; and the method of searching for these properties was deduced by analogy from Newton's Principe, lib. I., prop. vii.
The realization that the solution to this problem could be produced via a completely different route [9], nowadays called transformation optics, came 90 years after Maxwell posed his mathematical problem. Depending upon how one projects the points of the sphere on a flat surface (e.g. a cylinder or a plane) one can achieve a cyclic lens or a fisheye. The paper is now devoted to such applications, together with more elaborate, invisibility cloak and carpet, metamaterials.

Plasmonic cyclic concentrator
Let us start with the projection of the surface of a sphere on a cylinder. This amounts to considering a refractive index with a hyperbolic secant profile: where R 0 is the radius of the cylindrical lens, and n 0 and n R are respectively the refractive indices (invariant along the zaxis) at r=0 (along the optical axis) and at r=R 0 (at the outer edge of the lens). Such a design can be deduced from quasi-conformal grids. We would like to propose a structured design of a cyclic concentrator, which mimics the transformed design of such an ideal quasi-conformal grid. We note that the effective permittivity ε e of most two-phase composite media is given by the classical Maxwell where ε 0 and ε are the permittivity of material (in our case PMMA) and background (Air), respectively. We consider that the filling fraction f which is equal to the ratio of cross-section area of material (PMMA) to the elementary cell is that of a hexagonal unit cell (figure 2); thus f=1−(4π/√3) * (r/d) 2 with r the radius of air holes. The concentrator consists of air holes of diameters ranging from 128 nm to 340 nm and the size of each elementary cell is d=267 nm.

Analysis of focusing properties of an elementary cell of the cyclic concentrator
Computing the electromagnetic field for a plane wave incident on the structured concentrator described in figure 2 (using the commercial finite element software COMSOL MULTIPHYSICS), we note in figure 3 that the energy is concentrated at the exit of the lens, and this corresponds to the focusing point in figure 1.

Analysis of focusing properties of the cyclic concentrator
Let us now increase the length of the concentrator in order to achieve multiple focusing points as suggested by figure 1. We note in figure 4 that the energy is concentrated cyclically along the optical axis. In addition, our device works over a large bandwidth (see figure 5) as its structured design is based upon an effective medium theory.

Plasmonic carpet-cloaks
The first example of a cylindrical invisibility cloak for SPPs deduced from Pendry's coordinate transformation [15] was proposed by some of us [9]. Let us recall briefly its design: one need consider an electromagnetic space described by cylindrical coordinates (r′, θ′, z′) where the cloak's inner boundary is chosen to be r′=a and the outer boundary is r′=b. Then, a coordinate transformation is applied to transform (r′, θ′, z′) to (r, θ, z) where the corona a<r′<b is mapped onto a disk 0<r<b. Following [15] the inverse of such a transformation is: Under such an inverse transformation, the region r′<b is compressed to the region a<r<b. It has been established that the material parameters of the invisibility cloak expressed in cylindrical coordinates are given by: One can see that such a design requires strongly anisotropic tensors of permittivity and permeability, so the dispersion relation of the SPP is given by the general formula of section 1.1. A numerical simulation is performed with COMSOL MULTIPHYSICS for such parameters and shown in figure 6.
The implementation of such an invisibility cloak is fairly challenging; we therefore go back to spatially varying, scalarvalued, refractive index metamaterials.

SPP carpet-cloak constructed with quasi-grids
Let us now investigate the application of quasi-grids that preserve right angles in transformed metrics i.e. avoid the use of artificial anisotropy in the metamaterials' designs. In 2010, some of us introduced the design of a broadband plasmonic  carpet [13]. In figure 7, we show a finite element numerical computation performed by Muamer Kadic within the framework of his PhD thesis at Institut Fresnel in 2010 that clearly demonstrates the nearly flat wavefront for a plane SPP wave incident upon a curved mirror at 700 nm. However, such an invisibility effect is achieved thanks to high refractive index pillars deposited at the nodes of a quasi-grid on a metal surface.
In order to meet experimentally the parameters found in Kadic's simulations, Jan Renger (one of the researchers in Quidant's group at ICFO) chose a configuration in which a gold surface is structured with TiO 2 nanostructures. The TiO 2 pillars forming the crescent-moon-like carpet were first fabricated on top of a 60 nm thin Au film by combining electronbeam lithography and reactive ion etching. In a second lithography step, Jan Renger 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. The shape of the obtained TiO 2 particles is conical (h=200 nm, r=100 nm) as a consequence of the etching anisotropy, but the reported experimentally characterized efficiency of the carpet was quite convincing (reduction of scattering up to a factor 3.5 throughout the 700 nm to 900 nm wavelength range) at that time [13]. We would like to revive this successful design of SPP carpet as a uni-directional SPP cloak, as proposed in figure 7(right).
What we have done is to simply consider a combination of two SPP carpets, which we have stuck together, creating a pinched, uni-directional cloak. The parameters used in the COMSOL computation reported in figure 8 are those of figure 8 except that there are twice as many rods, and the SPP should now be incident on the cloak from the side. We place a metallic bump that fills the invisibility region, and we check the scattering by the metallic bump with and without the cloak; see figure 9. From plasmonic carpet to unidirectional cloak: an invisibility carpet deduced from a quasi-conformal grid (left) that serves as a basis for a unidirectional SPP cloak (right). Note that the direction of propagation of SPP has been tilted through an angle of π/4 in the SPP cloak. SPP carpet has been designed and by Muamer Kadic and experimentally tested by Jan Renger [13].

Homogeneous SPP cloak constructed with uniform layered structures
We have successfully designed some carpet cloaks using quasi-grids. There is nonetheless another interesting route towards plasmonic carpets, using anisotropic media. This might prove harder to implement in practice, but for the sake of completeness we show below two possible designs in which the metal surface has defects in the transverse (vertical) direction, a case first studied by the groups of Garcia-Vidal in Madrid [11] and Zhang in Berkeley [12].
Let us first propose a layered carpet cloak with a uniform silicon grating structure. The cloak takes a triangular shape as shown in figure 10. The permittivity tensor of a non-magnetic cloak for transverse-magnetic (TM, magnetic field perpendicular to the cloak device) polarization can be expressed as where H 1 and H 2 represent the heights of the obstacle and the cloak, respectively, and D is half the bottom length of the cloak. The permittivity tensor of the above problem can be deduced by rotating the optical axis through an angle θ: According to the effective medium theory, the anisotropic parameters can be achieved with an alternation of layered materials at sub-wavelength scale.
Therefore, appropriately combining two dielectric layers with permittivities ε 1 and ε 2 would yield the material with desired effective permittivity: e e e = + - where r denotes the filling factor of the constituent materials.
To illustrate how SPPs interact with this structure, we simulate in figure 11 its propagation along a metal bump on a metal surface (upper panel), the same obstacle cloaked with  In fact, as announced before, there is another interesting possibility for a plasmonic carpet. One can consider rods instead of homogeneous layers, following the route of quasiconformal mapping. The point is again to avoid dealing with anisotropy. The quasi-grid is in the transverse (vertical) plane, and placing high refractive index rods at its nodes as shown in figures 12 and 13 allows us to control surface plasmon polaritons (SPPs) and volume electromagnetic waves. In fact, as already pointed out in [14], transformation optics and plasmonics sometimes lead to the same designs in the case of carpets. This does not come as a surprise since surface electromagnetic waves are solutions of Maxwell's equations.

Theory of spoof plasmons and Maxwell's fisheye
In this last section, we would like to experimentally demonstrate some designs of a Maxwell fisheye deduced from a stereographic projection as shown in figure 1. The fabrication of this gradient index lens would be hard to achieve in the range of visible wavelengths, hence we turn our attention to microwaves. It is well known that conditions for the existence of SPPs cannot be met at microwaves. However, thanks to the proposal of surface spoof plasmon polaritons (SfPPs) on structured perfect metal surfaces [2] we know there is a natural path towards the design of transformation-based plasmonic metamaterials in the microwave regimes. We can thus simulate and experimentally characterize a Maxwell fisheye for SfPPs at Institut Fresnel. To do this, we first follow the recipe in the seminal paper by Pendry, Martin-Moreno and Garcia-Vidal [2] in order to generate a spoof plasmon source.

Introduction to spoof plasmon polaritons
Let us recall the main steps of the SfPPs' derivation as proposed by Pendry and his Madrilenian colleagues in [2] where all the details can be found (we note in passing that some elements of the mathematical proof of what we might call the SfPP's existence theorem are analogous to equations in a book chapter on grating anomalies by Daniel Maystre [18]). Assuming that the incident wave wavelength is much larger than the sidelength of square perforations (say, a), and array pitch (say, d), that is a<d=λ 0 , see figure 14, one can use homogenization techniques to derive the effective dispersion relation of SfPPs in the following manner. Notice first that the incident field excites mostly the fundamental mode of each waveguide within the periodically perforated metal (the electromagnetic field vanishes inside the perfect metal parts). In each dielectric waveguide one has: x a x a y a 0,1,0 sin e 0 a n d0 with the z-component of the wave vector such that: Here, k 0 is the wavenumber in a vacuum and ε h and μ h are the permittivity and permeability of the dielectric waveguides (for our experiments, we shall choose sand).
Note that for long wavelengths, due to symmetry, the effective tensors of permittivity and permeability of the periodic set of square waveguides satisfy ε z , ε x =ε y and μ z , μ x =μ y . Moreover, it is well known in the homogenization of perfect metal-dielectric structures [16,17] that e m = = ¥. As noted by Pendry et al in their original derivation, it seems reasonable to assume that the effective field inside the perforated metal can be written as: ) so that identifying incident and reflected wave-fields, and using the tangential continuity of the averaged electromagnetic field at the interface between air and structured metal-dielectric medium, one finds that  In fact, any metal surface structured by a periodic array on a subwavelength scale can be described by such a lowfrequency plasmon resonance and would support a spoof plasmon. In the case of dielectric waveguides of an arbitrary cross section, w p has a slightly more complex expression, but the structure of the effective tensors of permittivity and permeability remains essentially the same (the z-component is infinite) although e x differs from e y (as well as from m x and m y ) so that one essentially adds artificial anisotropy to the effective medium. A more general dispersion relation follows, along the lines of section 1.1.

Maxwell's fisheye in the spotlight
Inspired by John Pendry's revolutionary idea of a perfect lens via negative refraction [24], Ulf Leonhardt proposed, in a mathematically beautiful work, that a closed Maxwell fisheye lens (with a perfect mirror placed at a distance from the centre corresponding to the radius of a virtual sphere used in its design) might also be a good candidate for a lens with high resolution [25]. Some authors questioned this claim [26,27] and [28] failed to observe subwavelength features in the image within the mirrored fisheye lens. Although a microwave experiment suggested some subwavelength nature of the image [29], Roberto Merlin has conclusively shown [30] that a passive drain considered by Leonhardt in [25] (which is nothing but a time-reversed source placed very close to the theoretical position of the image) is necessary in order to beat the Rayleigh diffraction limit in Maxwell's fisheye. Moreover, Merlin pointed out that in [29], the outlets were absorbers identical to the source and impedance matched to the cables, which thus behaved as sources in reverse. In fact evanescent waves, which are essential in Pendry's perfect lens, do not contribute to the image reconstruction in the mirrored Maxwell's fisheye. Actually, the original fisheye lens shares much of the optical properties of a spherical lens, and what Leonhardt added as a further smart twist to its design is reminiscent of time-reversal sources and cavities in Figure 12. Scattering of an incident Gaussian beam by a metallic bump on its own (left) and dressed with a carpet (right). The incident wavelength is 800 nm and its angle made with the normal to the interface has been varied between 0 (normal incidence) to 45°. the work of Mathias Fink's group [31]. Nonetheless, let us stress that Leonhardt's paper has revived the interest of the wave community in Maxwell's lens, notably with some experiments performed in the time domain for elastic waves focusing on a fisheye-like plate [32].

A fisheye for surface plasmon polaritons
Research scientists have shown a keen interest in the Maxwell fisheye in the past few years, and we have mentioned that its design has been translated to the area of elastic waves in plates [32]. This suggests that a similar design might hold in plasmonics. We tested the design of a plasmonic fisheye without boundaries (see figure 16) using COMSOL MUL-TIPHYSICS. It transpires from this figure that, in theory, one can focus SPP with such a lens. However, the required spatially varying refractive index requires some exotic medium that needs to be engineered. Moreover, figure 16 shows some interesting features of field concentration at the antipodal location to the line source at the periphery of the lens that would need to be confirmed with a less challenging numerical model. We therefore decided to opt for a model of plasmonic fisheye within a cylindrical metallic cavity (indeed, in this case, there is no possible issue with perfectly matched layers that are quite subtle to implement for such a case). Figure 17 shows a structured design of fisheye surrounded by a metallic cavity. This design consists of TiO 2 pillar inclusions in air atop a  A perfect metal surface structured with periodically arranged dielectric waveguides behaves in the long wavelength limit like an effective medium with permittivity and permeability tensors displaying some plasma frequency dependence in e , x e y (with infinite e z and μ z ). This makes possible some electromagnetic surface waves with a dispersion reminiscent of plasmon polaritons (its asymptotes are the light line at low frequency and the plasma frequency ω p at large transverse wavenumbers k // in contrast to an isotropic plasma which has this asymptote shifted downwards by a factor of square root of 2. This makes SfPPs a new class of surface waves. metal surface. The source is placed well within the lens (far from the metal cavity) and the image is formed at a location which can be deduced from ray optics considerations on the virtual sphere used to generate the Maxwell fisheye (through stereographic projection as we already discussed). This design will be a source of inspiration for our experimental realization of the plasmonic Maxwell fisheye for SfPPs, which comes next.

A source of spoof plasmon polaritons
For our microwave experiments, we consider a plasmon frequency about 10 GHz (f c =10 GHz), which corresponds to a wavelength λ c =3 cm. Depending upon the medium which we consider within the dielectric inclusions of the metallic array, this will bring specific constraints on the cross-sectional size of inclusions and the array pitch, in order to meet the long wavelength criterion a<d<λ c . In fact, homogenization theory requires that λ c be much larger than d but in practice it is sufficient to have a wave wavelength three times larger than the array pitch.
If we consider air inclusions (n=1), we find that a=15 mm, d=30 mm meets the homogenization criterion. For dry sand inclusions (modelled with n=3.6, as checked with the EpsiMu® tool developed at Institut Fresnel which consists of a coaxial measuring cell and a dedicated software; see [34,35]), we can take a=7.9 mm, d=10 mm. We then used the CST Microwave studio commercial package to generate the dispersion diagram associated with the spoof plasmon polaritons in these cases.
To do this, we consider Floquet-Bloch conditions in the xy-plane: E(x+d, y+d, z)=E(x, y, z,)exp(i(kx+ky)d) (and similarly for the magnetic field) with k=(kx, ky) the Bloch vector that describes the first Brillouin zone ΓXM in the reciprocal space (Γ=(0, 0), X(π/d, 0), M(π/d, π/d)), where d is the pitch of the array in the physical space. We consider a small volume of vacuum above and below the structured metal, and periodic conditions in the z-direction, since CST does not allow for a combination of Floquet-Bloch conditions and perfectly matched layers (this seems to be mainly due to the fact that this would involve search of complex eigenfrequencies associated with so-called quasi-modes [33], which cannot be done in CST). We numerically checked the convergence of eigenfrequencies versus depth of the structured metal (the deeper the metal, the lower the cut-off frequency). For the plasmonic crystal, which consists of a perfect metal with dry sand inclusions, we finally obtain the dispersion of the fundamental mode as shown in figure 18. The SfPP propagates on the structured metal surface below a cut-off frequency of nearly 10 GHz (we numerically find 9.89 GHz) as requested.
In order to confirm the existence of the SfPP below 10 GHz, we now consider a finite plasmonic crystal with dimensions L=l=120 mm and h=40 mm illuminated with an electric source over a large range of microwave frequencies (between 5 GHz and 7 GHz) well below the cut-off frequency of 10 GHz. We show in figure 19 the result of CST simulations for the real part of the electric field component Ez (view from top for a plane located 1 mm above the crystal). The side view is shown in figure 20. The SfPP is clearly visible in figure 19 at the aforementioned frequencies: it propagates with almost circular wavefronts in the xy-plane, which shows that the effective medium is indeed isotropic in this plane. For frequencies higher than 10 GHz, the electromagnetic wave which is evanescent along the dielectric (sand) waveguide axes as clearly seen in figure 20, stops propagating in the xy-plane and propagates through the dielectric waveguides (the field remains of course evanescent in the perfect metal regions), that is, along the z-direction.
For the realization of the plasmonic crystal, we use a metallic grid (11×11 cells) consisting of aluminium tubes with a depth of 40 mm and with a square cross section of smaller and larger sidelengths 8 mm and 10 mm filled with dry sand. We show in figure 20 a comparison between the real part of Ez fields obtained from the simulations (upper panels) and the results of experimental measurements (lower panels) at 5 GHz, 6 GHz and 7 GHz (from left to right) with a network analyzer. One can clearly see the surface wave nature of the electric field, which propagates along the (horizontal) xyplane and is evanescent along the (vertical) z-direction.

A plasmonic fisheye with polystyrene and sand
We are now in a position to propose a practical implementation of a fisheye lens for SfPPs. In figure 21 we show some numerical simulations (CST Microwave Studio) for an electric source generating SfPPs at 5 GHz (upper right), 6 GHz (lower left) and 7 GHz (lower right). The schematic diagram of the fisheye is shown in the upper left panel of figure 21. The symmetry of the field with respect to a line passing through the centre of the fisheye is noted, and is a hallmark of the stereographically projected light rays of the virtual sphere underpinning the design of the lens as shown in figure 1. We numerically checked that fisheye lensing is actually achieved from 4 GHz to 8 GHz so it is a broadband effect (the only constraints being that the frequency is not too small or too large so that the fisheye is not of sub-wavelength overall size, and that its structural details remain smaller than the wave wavelength for effective medium theory to remain valid).
We finally show in figure   plasmonic crystal (the Ez field is zero on the latter since it is perfect metal). This lower panel also shows the plane, located 1 mm above the fisheye, along which we take a slice of the Ez field. In the middle panel, experimental measurements performed with the same setup as in figure 20 clearly demonstrate the lensing effect at 5 GHz. Numerical simulations (see right panel) are in very good agreement with experiments. One limitation of our device is that over the range of working frequencies, the plasmonic crystal is barely large enough to accommodate both the source and the image. However, our experimental results are a conclusive validation of the theoretical proposal of open plasmonic fisheye as shown in figure 16. Other improvements might consist in making the fisheye lying atop the structured metal thicker. One might also wish to place the electric source within the fisheye and look at lensing in a closed fisheye lens, which requires adding a perfect metal cavity around the fisheye (for instance with an aluminium sheet), and in this case one might achieve similar features as in figure 17. We believe that our proof of concept of a plasmonic fisheye can be easily translated into other gradient index lenses, such as Luneburg and Eaton lenses.

Concluding remarks and perspectives in other wave areas
In this paper, we reviewed some properties of transformational plasmonics and proposed some designs of cyclic    concentrators, carpets and cloaks. We investigated the physics of spoof plasmon polaritons introduced by Pendry, Martin-Moreno and Garcia-Vidal in 2004 [2] at a perfect metal surface structured with sand with a fisheye lens lying atop, the latter achieving a spatially varying refractive index with a combination of air and sand. We experimentally demonstrated a prototype of a plasmonic fisheye in the microwave regime between 5 GHz and 7 GHz. To conclude this review article, we would like to point out some connections between these small-scale experiments for electromagnetic waves and largescale experiments conducted in structured soils [6]. Rayleigh waves behave in many ways like SfPPs; at least from a mathematical standpoint they have a similar form (notably the exponential decrease of the displacement field in the plane perpendicular to the air-soil interface). We thus propose that one might test seismic metamaterials at a laboratory scale using analogies between plasmonics and geophysics, for instance in the microwave regime where devices such as cloaks are easier to test [36] than in optics. As a first prediction, we believe that surface polariton waves analogous to SfPPs observed on thin metal stripes almost 15 years ago [37] could be observed in the form of Rayleigh waves in sols structured with trenches. One can also envision the testing of a variety of transformation optics-based plasmonic metamaterials (e.g. cloaks, concentrators, lenses [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55]) and translate the most promising devices in the realm of transformation seismology. This is in essence what our group is currently doing in collaboration with Imperial College London (group of Richard Craster), with the Earth Science Institute ISTerre in Grenoble (group of Philippe Roux) and in partnership with the civil engineering Ménard Company in Lyon (group of Stéphane Brûlé). Moreover, similar studies are being led for the control of surface ocean waves, and here again useful analogies can be drawn with transformation plasmonics in order to design e.g. dykes for the protection of coastlines (based on surface water wave carpets, beam shifters and cloaks [56][57][58][59] whose designs are indeed reminiscent of plasmonic metamaterials). The versatility of transformation optics techniques introduced by John Pendry, David Schurig and David Smith in 2006 therefore represents not only a shift in electromagnetic paradigm, but also in acoustics, hydrodynamics and elastodynamics [59]. Analogues of spoof plasmons could be found in many wave areas where interfaces are structured at a subwavelength scale using effective medium theories [16,60] that unveil exotic plasmonic behaviors such as cloaking [61,62] and extreme control of light notably in metamaterials nanotips [63,64]. The knowledge gained in metamaterial surfaces over the past few years thanks to the advent of plasmonic metamaterials will certainly have deep implications in other disciplines, notably in civil and maritime engineering.