1. Introduction

Eight years ago, it was suggested that an object surrounded by a coating consisting of a heterogeneous anisotropic [1] or even isotropic heterogeneous [2] metamaterial might become invisible for electromagnetic waves. The theoretical idea based on geometric transformations in the Maxwell’s equations has been since then confirmed by some experiments. Plasmonic routes to invisibility were also proposed [3, 4], which rely on a specific knowledge of the shape and material properties of the object being concealed, and possibly include negatively refracting metamaterials [5]. These studies sparked an interest in transformational optics, giving rise to miscellenaous optical devices such as concentrators [6], rotators [7] and superscatterers [8].

However until now transformational optics has been used to design a metamaterial responsible for a single optical function (e.g. cloak, concentrator, rotator, superscatterer) expected not to be dependent on the illumination conditions (e.g. wavelength, incidence and polarization). In this paper we adress the opportunity to modify this statement, that is, to design a unique cloak able to deliver a specific and predetermined effect for each polarization of the incident light.

2. Merging two transformation matrices in one metamaterial

Let us map a co-ordinate system {u, v, w} to the co-ordinate system {x, y, z}, where x(u, v, w), y(u, v, w) and z(u, v, w). The Jacobian of the transformation is $\mathbf{J}=\frac{\partial (x,y,z)}{\partial (u,v,w)}$ and the transformation matrix $\mathbf{T}=\frac{{\mathbf{J}}^T\mathbf{J}}{\text{det}(\mathbf{J})}$ is a representation of the metric tensor. This tells us that in the transformed coordinates the materials are inhomogeneous (their characteristics are no longer piecewise constant but merely depend on u, v, w co-ordinates) and anisotropic (tensorial nature) [9]:

We would like in this paper to study the electromagnetic scattering of a metamaterial from an incident plane wave. Importantly, we restrict our analysis to media which are invariant along the z-direction. Thanks to this cylindrical geometry, the problem splits into p and s polarizations:

It is illuminating to recast these two equations with the longitudinal fields as unknowns. For this we recall a result useful in the context of duality correspondences in checkerboards[10]:

Property: Let M be a spatially varying real symmetric matrix defined as follows

Then for R(θ) denoting a rotation matrix through an angle θ, one has:

Let us now assume we perform two different kinds of geometric transforms for the s and p polarizations. They are characterized by transformation matrices T^★ and T^♯ in (1). Using the property with $\mathbf{M}={\mu }_0^{-1}{\mu }_r^{-1}{\mathbf{T}}^{★}$ in Eq. (2) and $\mathbf{M}={\epsilon }_0^{-1}{\epsilon }_r^{-1}{\mathbf{T}}^{♯}$ in Eq. (3) we obtain

By inspection of (6) and (7), we note that a metamaterial defined by tensors

3. Metamaterial concentrating p-polarized field and rotating s-polarized field

The above construction is fairly general. To illustrate this fact, we shall design a bicephal meta-material concentrating the field in one polarization and rotating the field in the other polarization, which has inner and outer boundaries described as follows:

More precisely, we consider the following geometric transform for the concentrator:

The transformation matrix T^★ of the resulting concentrator is defined through its inverse:

We now consider the geometric transform for a rotator:

The transformation matrix T^♯ of the resulting rotator is defined through its inverse:

Permittivity and permeability tensors as in (8) where the entries are given by (12) and (15) define a unique bicephalous metamaterial which rotates the electromagnetic field in p-polarization and concentrates it in s-polarization. The numerical computation for the bifunctional metamaterial which is a rotator in p-polarization and a concentrator in s-polarization is given for the telecommunication wavelength 1.4μm in Fig. 1.

 

Fig. 1 Bicephalous metamaterial rotating E_z field and concentrating H_z field: A plane wave of wavelength 1.4μm incident from the top rotates the longitudinal electric field E_z in p-polarization and concentrates the longitudinal magnetic field H_z in s-polarization. The difference in color scale is due to the large field inside the rotator in (a).

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4. Metamaterial cloaking in p polarization and super-scattering in s polarization

Let us now consider two radially symmetric geometric transforms f and g from (r′, θ, z) to (r, θ, z) such that r = f (r′) and r = g(r′) which leads us to the transformation matrices T^★ = Diag(f(r′)/(f′(r′)r′),r′f′(r′)/f(r′),f(r′)/(r′f′(r′)) and T^♯ = Diag(g(r′)/(g′(r′)r′),r′g′(r′)/g(r′),g(r′)/(r′g′(r′)).

Let us write the tensors of permittivity and permeability of (8) in the stretched polar coordinates as $\underset{̳}{{\epsilon }^{″}}={\epsilon }_r\mathbf{Diag}\left({\epsilon }_{\text{r}\prime {\text{r}}^{\prime }}^{★},{\epsilon }_{\theta \theta }^{★},{\epsilon }_{\text{zz}}^{♯}\right)={\epsilon }_{\text{r}}\mathbf{Diag}(\text{f}({\text{r}}^{\prime })/\left({\text{f}}^{\prime }({\text{r}}^{\prime }){\text{r}}^{\prime }\right),{\text{r}}^{\prime }{\text{f}}^{\prime }({\text{r}}^{\prime })/\text{f}({\text{r}}^{\prime }),\text{g}({\text{r}}^{\prime })/\left({\text{r}}^{\prime }{\text{g}}^{\prime }\left({\text{r}}^{\prime }\right)\right)$ and $\underset{̳}{{\mu }^{″}}={\mu }_r\mathbf{Diag}\left({\mu }_{\text{r}\prime {\text{r}}^{\prime }}^{★},{\mu }_{\theta \theta }^{★},{\mu }_{\text{zz}}^{♯}\right)={\mu }_{\text{r}}\mathbf{Diag}\left(\text{g}({\text{r}}^{\prime })/\left({\text{g}}^{\prime }({\text{r}}^{\prime }){\text{r}}^{\prime }\right),{\text{r}}^{\prime }{\text{g}}^{\prime }({\text{r}}^{\prime })/\text{g}({\text{r}}^{\prime }),\text{f}({\text{r}}^{\prime })/({\text{r}}^{\prime }{\text{f}}^{\prime }\left({\text{r}}^{\prime }\right)\right)$,. The p and s polarized electromagnetic fields satisfy the two equations

To design the superscatterer, we take the following transform:

The bicephalous cloak/superscatterer is thus described by:

Let us now illustrate numerically such a bicephalous cloak. We show in Fig. 2 some finite element computations for this metamaterial which demonstrate that it works as a superscatterer in E_z polarization and as a cloak in H_z polarization. Moreover, if we now interchange the roles of permittivity and permeability, we design a second bicephalous cloak which works as a superscatterer in H_z polarization and as a cloak in E_z polarization, as shown in Fig. 3.

 

Fig. 2 Superscatterer in E_z and cloak in H_z with inner radius 2.5μm and outer radius 4μm:: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal electric field E_z in p-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This first bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal magnetic field H_z in s-polarization (d).

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Fig. 3 Superscatterer in H_z and cloak in E_z with inner radius 2.5μm and outer radius 4μm: A plane wave of wavelength 1.4μm incident from the top enhances the scattering by an infinite conducting obstacle of radius 2.5μm for the longitudinal magnetic field H_z in s-polarization (b), which scatters like an infinite conducting obstacle of radius 6.4μm (a). This second bicephalous cloak/superscatterer makes an infinite conducting obstacle of radius 2.5μm (c) invisible for the longitudinal electric field E_z in p-polarization (d).

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5. Conclusion

We have introduced the notion of electromagnetic bicephalous metamaterials, which have one functionality (e.g. rotator or superscatterer) in one polarization and a second one (e.g. rotator or cloak) in the other polarization. We stress that such designs only work for cylindrical meta-materials as they heavily rely upon the property stated in section 2 (decoupling E_z/H_z). Finally, we note that some birefringent dielectric metamaterials were introduced in [11], which also perform two optical functions at the same time, depending on polarization, as experimentally demonstrated in metal-dielectric waveguides in [12]. Although [11] is an inspirational work, we note that it relies upon manipulations of Maxwell’s equations in a curved virtual space in the geometrical optics limit, which is not the case for our bicephalous transformed media.