## Abstract

We analyze cloaking of transverse electric (TE) fields through homogenization of radially symmetric metallic structures. The two-dimensional circular cloak consists of concentric layers cut into a large number of small infinitely conducting sectors which is equivalent to a highly anisotropic permittivity. We find that a wave radiated by a magnetic line current source located a couple of wavelengths away from the cloak is almost unperturbed in magnitude but not in phase. Our structured cloak is shown to work for different wavelengths provided they are ten times larger than the outermost sectors.

© 2008 Optical Society of America

## 1. Introduction

Over a year ago, Pendry, Schurig and Smith theorized that a finite size object surrounded by a coating consisting of a meta-material might become invisible for electromagnetic waves [1]. The cornerstone of this work is the geometric transformation

$$\theta \prime =\theta ,0<\theta \le 2\pi ,$$

$${x}_{3}^{\prime}={x}_{3},{x}_{3}\in IR,$$

where *r*′, *θ*′, *x*′_{3} are radially contracted cylindrical coordinates *r*, *θ*, *x*
_{3}. The former transformation maps the disk
${D}_{{R}_{2}}$
with radius *R*
_{2} onto an annulus with outer radius *R*
_{2} and inner radius *R*
_{1}. In other words, if a source located outside the disk
${D}_{{R}_{2}}$
radiates in vacuum, the electromagnetic field cannot reach the disk
${D}_{{R}_{1}}$
.

In terms of electromagnetic parameters, one should replace the material in the annulus by an equivalent one that is inhomogeneous and anisotropic [2]. The diagonalized form of the permittivity tensor $\left(\underset{=}{\epsilon \prime}\right)$ and the permeability tensor $\left(\underset{=}{\mu \prime}\right)$ was given by Pendry, Schurig and Smith [1]

Note that there is no change in the impedance of the media since the permittivity and permeability undergo the same transformation.

An international team involving these authors subsequently implemented this idea using a meta-material consisting of concentric layers of SRR [3], which makes a copper cylinder invisible to an incident plane wave at a specific microwave frequency (8.5 GHz). The smooth behaviour of the electromagnetic field in the *far field* limit could be expected in view of the numerical evidence provided in [1] using a geometrical optics based software. This is also in agreement with the work of Leonhardt who independently studied conformal invisibility devices using the stationary Schrödinger equation [4]. To date, the only evidence that invisibility is preserved in the intense *near field* limit is purely numerical [5].

A very different route to invisibility is proposed by McPhedran, Nicorovici and Milton who studied a countable set of line sources using anomalous resonance when it lies in the close neighborhood of a cylindrical coating filled with negative permittivity material, which is nothing but a cylindrical version of the celebrated perfect lens of Sir John Pendry [6, 7, 8, 9, 10]. The former researchers attribute this cloaking phenomenon to anomalous localized resonances [11, 12].

Last, but not least, a team led by Shalaev [13] has recently shown the possibility to make an object nearly invisible in TE polarization using radially symmetric locally resonant inclusions at optical frequencies. The key point in their proposal is that when the magnetic field is polarized along the *x*
_{3}-axis only two entries of
$\underset{\xaf}{\underset{\xaf}{\epsilon \prime}}$
must satisfy the requirements in equation (2) (*ε _{r}* and

*ε*). Moreover

_{θ}*µ*

_{3}is the only entry of $\underset{\xaf}{\underset{\xaf}{\mu \prime}}$ which is involved in equation (2) and it can therefore be normalized to 1. The reduced set of material parameters is obtained by multiplying

*ε*and

_{r}*ε*by

_{θ}*µ*

_{3}in (2) [13]:

The electromagnetic parameters of the cloak (2) and (3) provide the same wave trajectory, but the latter one leads to impedance mismatch on the outer boundary *r*=*R*
_{2} of the cloak. Importantly, we note that the value of *ε _{θ}* increases with decreasing values of

*r*in the annulus until it eventually becomes infinite on the inner boundary

*r*=

*R*

_{1}of the cloak in (2). The cloak proposed by Cai

*et al.*does not exhibit a very large azimuthal anisotropy as it is clear that all parameters in (3) remain bounded (unlike the cloak proposed by Pendry

*et al.*). To meet the criterion of (3), Cai

*et al.*propose to use a locally resonant structure consisting of concentric layers of thin elongated ellipsoidal wires elongated along the radial direction. A derivation based on effective medium theory shows that such a cloak displays the prerequisite electromagnetic parameters around a given frequency (in the visible spectrum) at which the local resonators are excited by the TE wave.

In the current paper, we propose an alternative route to invisibility taking advantage of designs studied in [3] and [13]: we model a cloak consisting of concentric layers of increasing thickness, each of them being cut in a large number of small sectors. These inclusions are assumed to be infinitely conducting, which is an accurate model for waves ranging from Gigahertz to Terahertz frequencies.

## 2. Homogenization of the micro-structured cloak at fixed frequency

A rigorous and detailed derivation of the homogenized permittivity tensor is beyond the scope of this paper and involves subtle mathematics from the limit analysis domain. Thus, we will give here only the main homogenization results and the proof will be published elsewhere.

Homogenization theory predicts that this kind of ‘approximate invisibility’ is applicable for any TE incident wave (possibly out-of-plane) whose wavelength *λ* is large compared with the characteristic size *d* of sectors *i.e.* when *η*=*λ*/*d*≪1, unlike for the proposals in [3] (involving C-shaped Split ring Resonators) and [13] (involving elongated ellipsoidal wires). In the numerical experiments, we will see that the shadow region of the ‘F’-shaped object is already noticeably reduced when the parameter *η*~1/3. We notice that our micro-structured cloak which works through field averaging does not suffer from any blow-up of the electromagnetic field, unlike its locally resonant counterpart [14].

Throughout this paper, we work in cylindrical coordinates (*r*, *θ*, *x*
_{3}), thereby taking into account the axi-symmetric geometry of the structure. From a two-scale model of the problem, assuming that the cloak lies in vacuum, we found that the homogenized permittivity tensor of the proposed structure is given by:

where *Y** denotes the elementary area around each scatterer *B* and *ϕ _{ij}* represent corrective terms defined by:

the brackets denoting averaging over *Y**. Hence, thanks to the symmetry of the right matrix above (*ϕ _{ij}*=

*ϕ*), the homogenized permittivity is given by the knowledge of three terms

_{ji}*ϕ*. One can note that

_{ij}*V*is the unique

_{j}*Y*-periodic solution with null mean of the following system, where derivatives are taken in the usual sense:

where ∂*B* denotes the boundary of *B*, and *n _{j}*,

*j*∈ {

*r*,

*θ*}, denotes the projection on the axis

**e**

*of a unit outward normal to ∂*

_{j}*B*. Although these results might at first glance look similar to those reported in [15], we emphasize that they differ since the earlier paper considered merely dielectric structures.

## 3. Numerical analysis of electromagnetic cloaking

In this section, we give some numerical illustrative examples of cloaking of an electric field coming from a source located closeby the obstacle through the infinitely conducting cloak. We replace the structured cloak by a cylinder surrounded by an effective coating whose homogenized permittivity is deduced from the numerical solution of the annex problems given by Eq. (6). This provides us with a qualitative picture of the cloaking mechanism (see Fig. 1). We then compare this asymptotic theory against the numerical solutions of the same scattering problems when we model the complete structured cloak.

#### 3.1. The homogenized matrix

Using finite elements, we numerically solved the annex problems *𝓚 _{r}* and

*𝓚*, for the geometry shown in Fig. 1 which provided us with two electrostatic potentials

_{θ}*V*and

_{r}*V*. Using (5) we found that

_{θ}*ϕ*and

_{rθ}*ϕ*vanish. From

_{θr}*ϕ*and

_{θθ}*ϕ*we deduced the following homogenized permittivity $\underset{\xaf}{\underset{\xaf}{{\epsilon}_{hom}}}=\mathrm{diag}(1.7,8.2,\infty )$ which displays a strong azimuthal anisotropy and moreover an infinite longitudinal anisotropy.

_{rr}The corresponding scattering problem for a magnetic line current source is reported on Fig. 1 right. We notice that although very few back reflection takes place on the cloak, the shadow region behind the F-shaped obstacle has been nearly removed (compare with Fig. 3). This clearly demonstrates that the field follows an optical path surrounding the central region, thus making the object located inside invisible.

#### 3.2. Structured infinitely conducting cloak

On Figure 2, we display the real part ℜ*e*(*H*
_{3}) of the longitudinal component *H*
_{3} of the magnetic field along the *x*
_{2}-axis. The black curve represents ℜ*e*(*H*
_{3}) in free space; the red curve represents ℜ*e*(*H*
_{3}) with a F-shaped infinitely conducting obstacle; the blue curve represents ℜ*e*(*H*
_{3}) with a F-shaped infinitely conducting obstacle surrounded by the cloak. We notice that the black and blue curves have nearly the same amplitude outside the cloak, but experience a phase shift. This phase shift can be attributed to the change in the optical path followed by the waves around the cloak: the path is longer than that of rays going straightforwardly in free space.

Figure 3 shows maps of the real part ℜ*e*(*H*
_{3}) of the longitudinal component *H*
_{3} of the magnetic field when the F-shaped scatterer is coated (right) and uncoated (left) and for frequencies ν ranging from 3.5 to 5.5. Note that the modelling takes into account the actual structure of the infinitely conducting cloak. The diffraction is clearly reduced by the structure especially for the lowest frequencies. Moreover, the numerical results show that although not perfect the cloak allows us to reduce the scattering for frequencies in a relatively large domain thanks to the fact that the physical principle is not based on any resonant property of the structure. Of course, when the frequency increases the typical size of the sectors becomes significant when compared to the wavelength and thus the homogenized model is not longer valid.

Importantly, we observe a strong penetration of the field within the annular shell, similarly to recently reported simulations for annular cloaks, but in contrast with numerical simulations reported in [16] for a ferro-magnetic cloaking device whereby the electric field was mostly concentrated within the first concentric outer-ring.

These diagrams numerically show that the propagation of the longitudinal magnetic field *H*
_{3} (and hence the transverse electric field) is controlled by the cloak.

## 4. Conclusion

Invisibility was numerically studied using modelling of both the complete metallic structured device displaying around two hundred sub-wavelength sectors and its strongly anisotropic homogenized counterpart.

We numerically checked that a structured device with a large number of thin metallic sectors is already a reasonable cloak. Almost no back-scattering is observed, the field is well reconstructed behind the cloak with a much reduced shadow zone.

All our results are in dimensionless physical units, but for a micro-wave design, it is enough to take the size of the sector *d* to be of the order of magnitude of a millimeter, in which case results should hold for TE micro-waves propagating out-of-plane provided their propagation constant γ is such that γ*d* ≪1. This suggests potential applications lie in improved telecommunication lines, for instance to transfer secure information. Interestingly, our cloak works well for the near field, and it is broad band to certain extent. Our design might prove useful for instance in certain problems of electromagnetic noise insulation for antennas applications. The influence of the absorption remains to be evaluated if the visible spectrum is aimed at.

## Acknowledgments

ABM and SG acknowledge funding from EPSRC under grant EP/F027125/1.

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