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Achieving invisibility over a finite range

of frequencies.

M. Farhat1, S. Guenneau2, A.B. Movchan2, S. Enoch1

1Institut Fresnel, CNRS, Aix-Marseille Universit´ e, 13013 Marseille, France

2Department of Mathematical Sciences, Peach Street, Liverpool L69 3BX, UK

guenneau@liverpool.ac.uk

Abstract:

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.

We analyze cloaking of transverse electric (TE) fields

© 2008 Optical Society of America

OCIS codes: (230.3990) Microstructure device; (260.2110) Electromagnetic theory

References and links

1. J.B. Pendry, D. Shurig, D.R. Smith, “Controlling electromagnetic fields,” Science 312 1780-1782 (2006).

2. U. Leonhardt and T. G. Philbin, “General relativity in electrical engineering, New J. Phys.” 8, 247 (2006).

3. D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B. Pendry, A.F. Starr, D.R. Smith, “Metamaterial electro-

magnetic cloak at microwave frequencies,” Science 314 977-980 (2006).

4. U. Leonhardt, “Optical conformal mapping,” Science 312 1777-1780 (2006).

5. F. Zolla, S. Guenneau, A. Nicolet and J.B. Pendry, “Electromagnetic analysis of cylindrical invisibility cloaks

and the mirage effect,” Opt. Lett. 32, 1069-1071 (2007).

6. R.C. McPhedran, N.A. Nicorovici, G.W. Milton,“Optical and dielectric properties of partially resonant compos-

ites,” Phys. Rev. B 49, 8479-8482 (1994).

7. J.B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).

8. J.B. Pendry, S.A. Ramakrishna, “Focussing light using negative refraction,” J. Phys. Cond. Matter 15, 6345-6364

(2003)

9. D. Maystre, S. Enoch, “Perfect lenses with left-handed material: Alice’s mirror?,” J. Opt. Soc. Am. A 21, 122

(2004)

10. S.A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449-521 (2005)

11. G.W. Milton, N.A. Nicorovici, “On the cloaking effects associated with anomalous localised resonance,” Proc.

Roy. Lond. A 462, 3027-3059 (2006).

12. N.A.P. Nicorovici, G.W. Milton, R.C. McPhedran, L.C. Botten, “Quasistatic cloaking of two-dimensional polar-

izable discrete systems by anomalous resonance,” Opt. Express 15, 6314-6323 (2007)

13. W. Cai, U.K. Chettiar, A.V. Kildiev, V.M. Shalaev, “Optical Cloaking with metamaterials,” Nature 1, 224-227

(2007).

14. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Improvement of cylindrical cloaking with the SHS

lining,” Opt. Express 15, 12717-12734 (2007)

15. S. Guenneau, F. Zolla, “Homogenization of three-dimensional finite photonic crystals,” JEWA 14, 529-530

(2000) & Progress In Electromagnetics Research 27, 91-127 (2000).

16. D.P.Gaillot, C. Croenne, D. Lippens, “An all-dielectric route for terahertz cloaking,” Opt. Express 16, 3986-3992

(2008).

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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

r?

θ?

x?

= R1+r(R2−R1)/R2, 0 ≤ r ≤ R2,

= θ , 0 < θ ≤ 2π ,

= x3, x3∈ IR ,

3are radially contracted cylindrical coordinates r, θ, x3. The former transforma-

tion maps the disk DR2with radius R2onto an annulus with outer radius R2and inner radius R1.

In other words, if a source located outside the disk DR2radiates in vacuum, the electromagnetic

field cannot reach the disk DR1.

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 (ε?) and the permeability tensor (μ?) was given by Pendry, Schurig and

Smith [1]

3

(1)

where r?, θ?, x?

εr= μr=r−R1

r

, εθ= μθ=

r

r−R1

, ε3= μ3=

?

R2

R2−R1

?2r−R1

r

.

(2)

Note that there is no change in the impedance of the media since the permittivity and perme-

ability 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 in-

visible 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 de-

vices using the stationary Schr¨ odinger 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 noth-

ing 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

objectnearlyinvisibleinTEpolarizationusingradiallysymmetriclocallyresonantinclusionsat

optical frequencies. The key point in their proposal is that when the magnetic field is polarized

along the x3-axis only two entries of ε?must satisfy the requirements in equation (2) ( εrand

εθ). Moreover μ3is the only entry of μ?which is involved in equation (2) and it can therefore

be normalized to 1. The reduced set of material parameters is obtained by multiplying εrand

εθby μ3in (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 = R2of the cloak. Impor-

tantly, we note that the value of εθincreases with decreasing values of r in the annulus until it

εr=

R2

R2−R1

?2?r−R1

r

?2

, εθ=

?

R2

R2−R1

?2

, μ3= 1 .

(3)

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eventuallybecomesinfiniteontheinnerboundaryr=R1ofthecloakin(2).Thecloakproposed

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

studiedin[3]and[13]:wemodelacloakconsistingofconcentriclayersofincreasingthickness,

each of them being cut in a large number of small sectors. These inclusions are assumed to be

infinitelyconducting,whichisanaccuratemodelforwavesrangingfromGigahertztoTerahertz

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 homogenizationresults 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 electro-

magnetic field, unlike its locally resonant counterpart [14].

Throughout this paper, we work in cylindrical coordinates (r,θ,x3), thereby taking into ac-

count 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:

⎛

00

εhom=

1

area(Y∗)

⎝

area(Y∗)

0

00

0

∞

area(Y∗)

⎞

⎠−

⎛

⎝

φrr

φθr

0

φrθ

φθθ

0

0

0

0

⎞

⎠,

(4)

whereY∗denotesthe elementaryareaaroundeachscatterer B andφijrepresentcorrectiveterms

defined by:

∀i, j ∈ {r,θ}, φij=<∂Vj

∂yi

>Y∗=<∂Vi

∂yj

>Y∗= − <∇Vi·∇Vj>Y∗ ,

(5)

the brackets denoting averaging over Y∗. Hence, thanks to the symmetry of the right matrix

above (φij= φji), the homogenized permittivity is given by the knowledge of three terms φij.

One can note that Vjis the uniqueY-periodic solution with null mean of the following system,

where derivatives are taken in the usual sense:

Kj: ΔVj= 0 ,inY∗=Y \B ,and∂Vj

where ∂B denotes the boundary of B, and nj, j ∈ {r,θ}, denotes the projection on the axis

ejof a unit outward normal to ∂B. Although these results might at first glance look similar to

∂n= −nj,on ∂B ,

(6)

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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 homoge-

nized 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

comparethis asymptotictheoryagainst the numericalsolutions of the same scattering problems

when we model the complete structured cloak.

3.1. The homogenized matrix

Using finite elements, we numerically solved the annexproblems Krand Kθ, for the geometry

shown in Fig. 1 which provided us with two electrostatic potentials Vrand Vθ. Using (5) we

found that φrθand φθrvanish. From φθθand φrrwe deduced the following homogenized per-

mittivity εhom= diag(1.7,8.2,∞) which displays a strong azimuthal anisotropy and moreover

an infinite longitudinal anisotropy.

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.

Fig. 1. Left: Geometry of the structured infinitely conducting cloak. The structure consists

of 256 angular sectors. Right: Diffraction of the field radiated by a magnetic wire source

by an infinitely conducting F-shaped obstacle surrounded by a homogeneous dielectric

anisotropic coating with effective permittivity εhom= diag(1.7,8.2,∞) and effective trans-

?

and outer boundaries (r = R1= 0.144 and r = R2= 0.4). The frequency ν is equal to 3.5.

mission conditions n∧

εhom−1(∇+iγ)∧Hhom

?

|r−=n∧(∇+iγ)∧Hhom|r+ on its inner

3.2.

On Figure 2, we display the real part ℜe(H3) of the longitudinal component H3of the mag-

netic field along the x2−axis. The black curve represents ℜe(H3) in free space; the red curve

represents ℜe(H3) with a F-shaped infinitely conducting obstacle; the blue curve represents

ℜe(H3) with a F-shaped infinitely conductingobstacle surroundedby the cloak. We notice that

the black and blue curves have nearly the same amplitude outside the cloak, but experience

Structured infinitely conducting cloak

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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.

012345

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Re(H3)

(x1=x2)-axis

Free space

F not cloaked

F cloaked

Fig. 2. Real part ℜe(H3) of the longitudinal component H3of the magnetic field along the

x2−axis. The origin of the x2axis is taken at point (4,4) of Fig. 3 and it ends at point (0,0).

A magnetic line source of wavelength λ = c/ν = c/3.5 is located at point (2.3,2.3). The

vertical thick bold lines represent the outer boundary of the cloak.

Figure 3 shows maps of the real part ℜe(H3) of the longitudinal component H3of the mag-

netic 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 propagationof the longitudinal magnetic field H3

(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 ho-

mogenized 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 recon-

structed 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

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0

1

-1

?=3.5

?=3.5

?=4.5

?=4.5

?=5.5

?=5.5

P M L

P M L

source

P

M

L

P

M

L

P M L

source

P

M

L

P

M

L

P M L

Fig. 3. 2D plot of the real part ℜe(H3) of the longitudinal component H3of the magnetic

field radiated by a harmonic line current source of wavelength λ =c/ν. Left panel: Diffrac-

tion by an infinitely conducting F-shaped obstacle; Right panel: Diffraction by an infinitely

conducting F-shaped obstacle surrounded by the structured cloak. When the frequency ν

increases, the diffraction worsens and cloaking becomes less effective.

results should hold for TE micro-waves propagating out-of-plane provided their propagation

constant γ is such that γd ?1. This suggests potential applications lie in improvedtelecommu-

nication 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 in-

stance 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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