1. Introduction

The design and realization of invisibility cloaks in the infrared and visible domains represent today a great challenge for the scientific community of electromagnetism and optics regarding both fundamental aspects and potential applications. An invisibility cloak is expected to expel the electromagnetic field from a given region of space while preserving the field lines outside this concealment region. The design process for the cloak thus involves a coordinate transformation that squeezes space from the concealment region into a shell surrounding this region [1–3]. Typically, a spherical (or cylindrical) shape is chosen for the concealment region defined by: r<b, and the space transformation compressed it into a spherical (or cylindrical) shell defined by: a<r<b (see for instance Fig. 2(b) in [1,2]). This in turn allows determining the tensor components associated to either the dielectric permittivity or magnetic permeability of the cloak shell. However, the practical implementation of an electromagnetic cloak defined in this way still remains out of reach due to the fact that some of the tensor components vary in a rather complex manner with space coordinates [1,2]. Simplified versions of electromagnetic cloaks can actually be found that preserve dispersion relations and light trajectory in the cloak despite of a small mismatch at the outer cloak boundary [4,5]. For a cylindrical geometry and a polarization of the incident light either parallel or perpendicular to the cylinder axis, it has been shown that the electromagnetic tensors could be reduced to a set of only three components: ε_z, µ_r and µ_θ for an incident field polarized parallel to the cylinder axis or µ_z, ε_r and ε_θ for an incident field polarized perpendicular to the cylinder axis. Such an approximate version of electromagnetic cloak has been experimentally demonstrated in the microwave range for the first polarization case [4]. For this purpose, a structure of metallic split ring resonators (SRR) [6] was designed to provide a magnetic response µ_r(r) that varied in an appropriate manner with the radial coordinate. More recently, an alternative route for terahertz cloaking has been theoretically proposed [7] using the magnetic response of micro-cut BST (Barium Strontium Titanate) rods [8]. Regarding the second polarization case (H field parallel to the cylinder axis), a structure made of metallic ellipsoids has also been proposed [5] for the design of a non-magnetic cloak at optical frequencies. One advantage in this latter work stems from the absence of magnetic materials which still remain hard to obtain in the optical regime. The main drawback is the difficulty to design precisely metallic ellipsoid at infrared wavelengths.

In the present work, we propose an electromagnetic cloak, which exploits the electric response of SRRs instead of their magnetic response. Numerical simulations performed at infrared frequencies (~100 THz) reveal low loss and weak impedance mismatch, thereby proving the interest in using SRRs as “universal” atoms in the design of metamaterials. We also show that SRRs can be ultimately replaced by simple cut wires for the construction of “simplified” electromagnetic cloaks whose dielectric permittivity is the only parameter varying with space coordinates.

2. Artificial control of the permittivity of split ring resonators

Split ring resonators have been introduced in the context of artificial magnetic materials by J. Pendry and co-workers [5]. One important objective was to achieve negative permeability and, ultimately, negative refraction by combining these materials with artificial dielectrics possessing negative permittivity [9,10]. Actually, a strong magnetic response of SRRs possibly leading to negative permeability can be obtained when only the magnetic field couples to the SRR resonance. This implies that the main component of the magnetic field is along the SRR axis, and that the electromagnetic wave illuminates the SRR plane at an oblique or grazing incidence. In these conditions, the magnetic response of SRRs is superposed with a weak electric anti-resonant response [11,12]. Artificial magnetism for an oblique incidence of the electromagnetic wave has been claimed in recent works on SRR structures at near-infrared and visible wavelengths [13]. The situation is different for a normal incidence where only the electric field couples to the SRR resonances [14]. Contrary to the preceding case, a strong electric response of SRRs is obtained together with a weak magnetic anti-resonant response. Calculated values of the effective ε and µ parameters for this configuration can be found in [12,14,15]. Despite the weakness of the magnetic response, the normal incidence configuration is of practical interest for sensing SRR resonances at optical wavelengths [12,13, 16], which are but plasmon resonances [17].

The SRR geometry and field polarization used in this work are shown in Fig. 1(a). The magnetic field is parallel to the SRR legs. The electric field has one component parallel to the SRR gap (E_r), and the other parallel to the SRR axis (E_θ). Following the considerations above, the electric response of the SRR is therefore the dominant one [15]. This will remain true even for a stack of SRRs provided that there is no coupling between adjacent metallic elements in the direction of propagation [18]. For the purpose of practical realization, the SRRs are supposed to be made of gold wires embedded in a host medium of dielectric constant ε_m=4. A Drude model is used to simulate the permittivity and loss tangent of the gold wires:

where ω_p and ω_c are the plasma frequency and collision frequency of the gold film, respectively. The values of ω_p and ω_c chosen in the simulations are: ω_p=1.367×10¹⁶ s^-1 (f_p=2176 THz) and ω_c=6.478×10¹³ s^-1 (f_c=10.3 THz). Both these values have been validated from previous measurements of SRR resonances in the near-infrared region [19]. The collision frequency is 2.6 times larger than in bulk gold, which is supposed to account for additional scattering experienced by electrons at the metal surfaces.

Calculated transmission and reflection spectra of a two-dimensional array of SRRs around the first SRR resonance (the so-called LC resonance [14]) are presented in Fig. 1(b) for normal incidence of the electromagnetic wave (E_θ=H_r=H_θ=0). The geometrical parameters of the SRRs are chosen in such a way as to obtain the first SRR resonance near 60 THz, and calculations are performed from a finite element HFSS software. Figure 1(c) shows the effective ε and µ parameters (real and imaginary parts) retrieved from the complex transmission and reflection amplitudes by using the inversion method described in [20,21]. It is confirmed that the SRR essentially exhibits an electric response to the incident wave with a slight magnetic anti-resonance. The permittivity (ε_r) associated to the field component E_r covers the full range of values in the interval [0, 1] for frequencies (wavelengths) comprised between ~68 and 100 THz (~3 and 4.4 µm). As is seen in Fig. 1(b), the minimum reflection is reached near 100 THz with a value lower than -35 dB. It is also worthwhile noticing that the SRR does not exhibit any electric response in the azimuthal direction (E_θ ≠ 0, E_r=0) within the frequency range of interest. In other words, the dielectric tensor component in this direction is simply a constant equal to the permittivity of the host medium (ε_θ=ε_m=4). As will be seen in the next section, all these conditions are very favourable for the design of an invisibility cap with weak impedance mismatch at 100 THz.

 

Fig. 1. (a) Unit cell and field polarization used in calculations. (b) Transmission and reflection of gold SRRs calculated at normal incidence (E_θ=H_r=H_θ=0) in the frequency range from 20 to 120 THz. The SRRs are supposed to be embedded in a dielectric with ε_m=4 permittivity. The lengths of the SRR gap, base and legs are 120, 360 and 260 nm, respectively. The width and thickness of the gold wires are w=50 nm and h=40 nm. The dimensions of the unit cell used in calculations are: l_θ=350 nm, l_r=570 nm, l_z=580 nm. (c) Effective permittivity (ε_r) and permeability (µ) of the SRRs for the field polarization considered in Fig. 1(b).

Download Full Size | PDF

3. Design and simulation of the invisibility cloak

The space transformation used in the proposed cloak is similar to that in [4], by which the cylindrical region r<b is compressed into the annular region a<r<b. This transformation leads to the following expression for the permittivity and permeability tensor components:

We will further restrict the problem to electromagnetic waves with the magnetic field polarized along the z axis, thereby benefiting from a significant simplification in that only µ_z, ε_r and ε_θ are relevant [5]. Since µ_z is the only component of interest in the permeability tensor, we can multiply ε_r and ε_θ by the value of µ_z to obtain the following reduced set of parameters:

which provides the same wave trajectory as the original set of Eq. (2). The only penalty for choosing the reduced parameters is an imperfect impedance matching at the outer boundary of the cloak.

 

Fig. 2. (a) Values of the SRR parameters used in the different annular regions of the invisibility cloak. The regions are numbered from 1 to 20 going from the outer to the inner boundary of the cloak. (b) Schematic view of a cloak sector with an enlarged view in the insert. The elementary cells within a given annular region are identical. In contrast, the cell width l _θ linearly increases with the radial position of the cell: $l_{\theta }=\frac{{\mathrm{rl}}_{{\theta }_1}}{b}$ , where $l_{{\theta }_1}=500$ nm is the width of the elementary cells in the annular region at the outer cloak boundary (r=b). (c) Evolutions of the three tensor components (ε_r, ε_θ, µ_z) with the radial position of the elementary cell (dotted curves).

Download Full Size | PDF

The metamaterial cloak proposed in this work consists of an ensemble of elementary cells similar to that of Fig. 1(a) and arranged in a cylindrical geometry (Fig. 2). The inner and outer radii of the cloak are a=11.4 µm and b=22.8 µm, respectively. The cloak is comprised of 20 annular regions which themselves are composed of 300 elementary cells. The elementary cells within a given annular region are identical. In contrast, the cell width and the SRR dimensions vary with the radial position. The cell width l _θ linearly increases with the radial position: $l_{\theta }=\frac{{\mathrm{rl}}_{{\theta }_1}}{b}$ , where $l_{{\theta }_1}=500$ nm is the width of the elementary cells in the annular region at the cloak periphery (r=b). The other cell dimensions l_r and l_z as well as the thickness h and width w of the gold wires are kept constant. In principle, one should consider elementary cells with a trapezoidal base and curved boundaries (Fig. 2(b)) instead of the parallelepiped shown in Figure 1(a). For the sake of simplicity, parallelepipeds are used in the calculations since the low curvature of the cell boundaries has actually a negligible impact on the permittivity tensor components associated to the cell [5]. The SRR dimensions are varied from the outer region to the inner region of the cloak (Table 2b) to produce the radial dependence ε_r(r) given in Eq. (2bis) at the fixed frequency of 100 THz. An increase of the SRR gap or a decrease of the SRR legs allows shifting the SRR resonance to higher frequencies, which in turn allows decreasing the value of ε_r from the outer region to the inner region of the cloak. The values of SRR gap and legs reported in Table 1a actually allow covering the whole range of dielectric constants from 1 (region n=1) to 0 (region n=20) at 100 THz (Fig. 2(c)). It is worthwhile noticing that this range of values is wider than that required for the µ_r parameter in the case of a magnetic cloak with the same dimensions (E parallel to the cylinder axis). The magnetic permeability would actually vary from 0 to 0.25 going from the inner region to the outer region of the cloak. The larger range of permittivities required in the present case imposes in turn a larger number of annular regions to correctly reproduce the radial dependence of ε_r.

 

Fig. 3. Finite element simulations showing the magnetic field (H_z) distribution around a metallic cylinder illuminated by a plane wave at 100 THz. The wave propagates in the air from the left to the right, and the magnetic field is polarized parallel to the cylinder axis. (a) In the absence of invisibility cloak, wave scattering and shadowing effects are clearly evidenced. (b) In the presence of the invisibility cloak as described in Fig. 2, the wave travels as if there was no bulky obstacle. The wave fronts and field amplitudes behind the cloak are essentially the same as in the absence of the metallic cylinder, thus demonstrating the effectiveness of the cloak.

Download Full Size | PDF

 

Fig. 4. (a) Unit cell including a metallic cut wire instead of a split ring resonator. The field polarization is the same as in Fig. 1(a). (b) Effective permittivity (ε) and permeability (µ) of the cutwire based metamaterial for the field polarization considered in Fig. 4(a). The length, width and thickness of the gold wire are: l=400 nm, w=50 nm and h=40 nm. The dimensions of the unit cell are: l_θ=200 nm, l_r=570 nm and l_z=580 nm.

Download Full Size | PDF

This being, the length of elementary cells in the radial direction, l_r=570 nm, remains sufficiently long so as to prevent coupling effects between adjacent SRRs. It can also be verified that the permittivity chosen for the host material in Section 2 (ε_m=4) simply identifies to the tensor component ε_θ expressed as a function of the outer and inner diameters of the cloak in Eq. (2bis). The inner cloak diameter (i.e. the diameter of the concealment region) is presently 7.6 times larger than the working wavelength (λ ~3 µm). To the best of our knowledge, this is the largest concealment region ever reported in the literature in wavelength units [4,5,7]. This characteristic is achieved with a quite reasonable ratio between the outer diameter and the inner diameter of the cloak (b/a=2), i.e. with a quite reasonable “thickness” of the cloak. In contrast, the very large number of elementary cells contained in the entire cloak (6000) makes a full electromagnetic simulation of the structure very difficult. For the purpose of demonstrating the effectiveness of the cloak design, we restricted ourselves to field mapping simulations [4,5] where each elementary cell is simply represented by a homogeneous medium with the ε_r(r), ε_θ, and µ_z tensor components. Calculations were performed using the finite element package “Comsol Multiphysics” [22].

Results of our calculations are shown in Fig. 3, which shows the magnetic field (H_z) distribution around a metallic cylinder illuminated by a plane wave at 100 THz. The dimensions of the simulation domain are: 91.5×58 µm². Two cases are considered depending whether an invisibility cloak is present or not. In the first case where the cylinder is surrounded by air, wave scattering and shadowing effects are clearly evidenced (Fig. 3(a)). In the second case where the invisibility cloak of Fig. 2 is used, the wave travels as if there was no bulky obstacle (Fig. 3(b)). The wave fronts and field amplitudes behind the cloak are essentially the same as in the absence of the metallic cylinder, thus demonstrating the effectiveness of the cloak. Residual reflections and minute perturbations of the wave fronts can be readily attributed to the small impedance mismatch of the cloak.

4. From split ring resonators to metallic cut wires

The simulation results of Fig. 3 demonstrate the interest of using SRRs not only for their magnetic response but also for their electric response. Split ring resonators can indeed be seen as “universal” atoms for the construction of two- and three-dimensional photonic metamaterials at optical frequencies. However, the fabrication of SRR based cloaks as the one proposed in Fig. 2 represents a very difficult challenge at infrared and visible frequencies. Modern micro- nano-fabrication technologies are for the most part based on thin-film depositions and processes. Even though complex structuring of these films or layers is not excluded for the realization of innovative two- and three-dimensional photonics, the feasibility of an invisibility cloak at optical frequencies must be first examined in the context of standard photonics and planar optics. Regarding this point, it is important to note that the electric response of SRRs used for the proposed cloak can also be obtained from simpler metallic nanostructures derived from SRRs. In particular, recent works have shown numerically [17] and experimentally [19] that a gradual shift in the electric SRR resonances was obtained by reducing the length of SRR legs to the point where the SRR was transformed into a single wire piece. A simplified version of optical cloak could be constructed, for instance, by using an ensemble of elementary cells as the one shown in Fig. 4(a) where the SRR is replaced by a simple metallic cut wire. This cloak version would be also somewhat simpler than that recently proposed using metal ellipsoids [5], and its fabrication should be indeed achievable with standard thin film deposition and nanolithography techniques.

In order to prove the interest of such an optical cloak, we determined the effective parameters of a cutwire based metamaterial in the same way as it was done for the SRR based metamaterial (see Section 2). Results are presented in Fig. 4(b), which shows the effective ε and µ parameters (real and imaginary parts) retrieved from the complex transmission and reflection amplitudes by using the inversion method described in [20,21]. The field polarization (E_r ≠ 0, E_θ=0) and the dielectric permittivity of the host material of the cell are the same as in Fig. 1. The width and thickness of the metallic wire are also the same as for the SRR in Fig. 1. The wire length (l=400 nm) is chosen to be sufficiently long so as to maintain the first cutwire resonance in the infrared domain while simultaneously keeping reasonable dimensions for the elementary cell. The resonance frequency (here ~110 THz) is approximately two times larger than that of the SRR represented in Fig. 1, whose total length is 930 nm (unfolded SRR). The length l_r and height l_z of the elementary cell in Fig. 4(a) are the same as in Fig. 1(a) while its width l_θ is 1.75 times smaller (200 nm instead of 350 nm). As seen in Fig. 4(b), these dimensions allow covering the full range of ε values in the interval [0, 1] for frequencies (wavelengths) comprised between ~135 and 285 THz (~1.06 and 2.2 µm). All the evolutions reported in Fig. 4(b) are actually quite similar to those of Fig. 1(c). This demonstrates that an invisibility cloak can be obtained with the same performances as those of the cloak described in Fig. 3 but with the use of metallic cut wires instead of SRRs. The only difference stems from the narrower width of the elementary cells and therefore from the larger number of cells needed to construct the cloak.

5. Conclusion

We have proposed an electromagnetic cloak, which exploits the electric response of metallic split ring resonators instead of their magnetic response used so far. This electric response has been engineered in the infrared part of the spectrum to achieve a gradual variation of the relative permittivity in the interval [0,1] with very low absorption loss. Numerical simulations have shown the effectiveness of the proposed cloak, which provides an almost perfect cloaking and preserves the wave fronts in the surrounding region with negligible impedance mismatch. Incidentally, this is, to our knowledge, the first time that the diameter of the cloaked region is as large as about eight times the working wavelength. These results confirm the high potential of SRRs for the design of metamaterials where their electric and magnetic responses can be used either independently or in conjunction. Other cloak configurations than the one studied in this work could be envisaged using a different field polarization with, for instance, the electric field polarized perpendicular to the SRR gap, and/or higher-order resonant modes of the SRRs, and/or other cloak shapes. We have also shown that SRRs could be ultimately replaced by simple cut wires for the construction of non-magnetic cloaking whose fabrication would be highly compatible with the thin film deposition and processing techniques currently used for photonics at infrared and visible wavelengths.