Electromagnetically induced invisibility cloaking

Darran F. Milne and Natalia KorolkovaSchool of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK

(Dated: June 19, 2012)

Invisibility cloaking imposes strict conditions on the refractive index profiles of cloaking mediathat must be satisfied to successfully hide an object. The first experimental demonstrations ofcloaking used artificial metamaterials to respond to this challenge. In this work we show howa much simpler technique of electromagnetically induced transparency can be used to achieve apartial, carpet cloaking at optical frequencies in atomic vapours or solids. To generate a desiredcombination of low absorption with strong modifications of the refractive index, we use chiral mediawith an induced magneto-electrical cross-coupling. We demonstrate that high-contrast positiverefractive indices can be attained by fine tuning the material with a gradient magnetic field andcalculate the parameters required to construct a carpet cloak.

PACS numbers: 42.79.-e, 42.50.Gy

In electromagnetically induced transparency (EIT) [1]opaque substances are made transparent by external elec-tromagnetic fields. The fields prepare the substance indark states [1] that are decoupled from light of a cer-tain frequency and polarization, thus making the mate-rial transparent for such light. EIT has been the basis forslow light in atomic vapors [2, 3] or solids [4]. Here we putforward a method of electromagnetically induced cloak-ing where external fields manipulate the refractive indexof a specially designed EIT material to guide light aroundan object without making the object itself transparent.Electromagnetically induced transparency and slow lighthave inspired applications in metamaterials [5, 6]; herewe adopt a concept that originated in the area of meta-materials — cloaking [7–11] — and show how it can beimplemented in atomic vapors and solids.

A cloaking device consists of a transparent mantel thatencloses the object to be hidden. The mantel refractslight around its hidden interior such that light exits asif it had traversed empty or uniform space, thus hidingthe object and the act of hiding itself. For this, certainrefractive-index profiles are required that are difficult toimplement in ordinary optical materials but are (barely)feasible in metamaterials [12]. In most cases, these shouldbe optically anisotropic materials, but it is also possibleto cloak with isotropic materials [8]. Electromagneticallyinduced transparency is able to create index profiles suit-able for lensing [13] but with an index contrast that istoo low for cloaking. Here we point out that anothermethod [14, 15] borrowed from metamaterials research— the implementation of chiral materials [16] — is capa-ble of cloaking. The currently achievable index contrastis only sufficient for partial cloaking, also known as “car-pet cloaking” [17], where a curved object is made to ap-pear flat. The flattened object can then be camouflagedagainst the background to render it effectively invisible.

The index profiles for cloaking are often designed bythe optical implementation of coordinate transformations[11]. Here, the cloaking device creates the illusion that

physical space is transformed into a virtual space, whichis most easily explained by a two-dimensional example[8] that also serves to introduce carpet cloaking. Sup-pose that light propagates in a planar material. We de-scribe the Cartesian coordinates x and y of the physicalpropagation plane with the complex number z = x+ iy.Imagine we transform z to the new complex coordinatez′ of a virtual plane where z′ is only a function of zand not of z∗ (such a function z′(z) is analytic and de-fines a conformal map [18]). Suppose that the virtualplane is empty. In this case light experiences a virtualgeometry described by the line element dl′. In the phys-ical plane, dl′ appears as |dz′/dz| dl. Now, according toFermat’s principle [11], in an optically isotropic mediumwith refractive-index profile n light rays follow geodesicswith the path length measured by ndl. Therefore, if weidentify n with |dz′/dz| dl, the medium implements thecoordinate transformation z′(z). Consider, for examplethe Zhukowski transform

z′ = z +a2

z, z =

z′ ±√z′2 − 4a2

2(1)

with constant a (defining the characteristic scale of thedevice) that is implemented by the refractive-index pro-file

n =

∣∣∣∣dz′dz∣∣∣∣ =

∣∣∣∣1− a2

z2

∣∣∣∣ . (2)

Fig. 1(a) shows that straight lines in virtual space arecurved lines in physical space, in particular the linesz′ = x+ iη with η = const. As light experiences the ge-ometry of the virtual space, such curved lines appear tobe flat and so does any object behind the curve z(x+ iη).An optical material with the index profile of Eq. (2) abovethe curve z = (x + iη) thus optically flattens this curveand anything underneath, which is the defining propertyof carpet cloaking [17]. Instead of conformal maps onecould use quasi-conformal transformations [17] that havethe advantage of acting in a finite domain. The strongest

arX

iv:1

206.

3944

v1 [

phys

ics.

optic

s] 1

8 Ju

n 20

12

2

FIG. 1: (a) The grid shows the curved path of light in thephysical space z due to the transformation in Eq.(1). (b) Thevariation of refractive index at the boundary for some valuesof η.

deformations of the conformal map occur at the bound-aries and thus require the highest refractive-index con-trast there. Figure 1(b) shows the index profiles at theboundaries z = (x + iη) for various values of η. In thefollowing we discuss how such optical properties can beelectromagnetically induced.

We want to take advantage of the quantum coherenceproperties of atomic media exploiting electromagneticallyinduced transparency to manage the refractive index pro-file and, at the same time, to profit from the low ab-sorption window. Inspired by the lensing effects demon-strated in standard EIT [13], we seek to enhance the (pos-itive) refractive index contrast by introducing magneto-electric cross-coupling. Recently, [14, 16] have shownthat such chiral media can be manipulated to attain neg-ative refractive indices. These schemes have been furtherimproved upon in [15] using off resonant Raman tran-sitions. To avoid the inherent experimental challengesassociated with decoherence in atomic vapors we suggesta solid state implementation of chiral EIT media [4].

EIT in chiral media. The electromagnetic constitu-tive equation relations between medium polarization Por magnetization M and electromagnetic fields E and Hare usually expressed in terms of permittivity and perme-ability only. They can however be generalized to includethe chirality effects, i.e., the cross coupling between theelectric and magnetic fields:

P = χeE +ξEH4π

H, (3)

M =ξHE4π

E + χmH, (4)

where ε = 1 + 4πχe and µ = 1 + 4πχm are the com-plex valued permittivity and permeability tensors. Thesegeneralized constitutive equations describe media with

FIG. 2: A 5-level scheme for the implementation of tunable re-fraction via electromagnetically induced cross coupling. Themagnetic dipole transition |2〉 - |1〉 and the electric dipoletransition |3〉-|4〉 are coupled by Ωc to induce chirality. The”ground” state of the system is formed by the dark state |D〉of the subsystem |1〉, |4〉, |5〉. For details see [14].

magneto-electric cross-coupling which are also known asbianisotropic media. ξEH and ξHE denote tensorialcoupling coefficients between the electric and magneticdegrees of freedom. We can imagine a simple chiralmedium to be composed of three level atomic systems,with ground states |g〉 and two excited states |1〉 and |2〉.If we choose |g〉 → |1〉 to be a magnetic dipole transitionand |g〉 → |2〉 to be an electric dipole transition, we caninduce cross coupling through a strong resonant couplingof the upper two states. Then the magnetization givenby the transition from the ground state to |1〉 will containan additional contribution from the electric field and viceversa for the polarization. Note, that such a simple 3-level system is resemblant of a typical λ-scheme for EIT.Such a medium can be realized by inducing chirality ina solid state EIT system, using doped bulk crystals [4].In practice, an extended 5-level atomic system should beused (Fig. 2), which will be discussed later.

To find the refractive index, we examine the propa-gation properties of the electromagnetic waves in suchmedia as governed by the Helmholtz equation:

[ε+ (ξEH +c

ωk×)µ−1(

c

ωk×−ξHE)]E = 0. (5)

The solution to this equation for the wavevector k is ingeneral very complicated [19]. The analysis can be sim-plified by assuming the permittivity and the permeabilityto be isotropic, ε = ε1, µ = µ1. Further, we confine ouranalysis to the 1D theory with the wave propagating inthe z-direction. Finally, we only consider media which al-low for conservation of the photonic angular momentumat their interfaces. This leads to the following expression

3

for the refractive index (see also [14]):

n+ =

√εµ−

(ξ+EH + ξ+HE)2

4+i

2(ξ+EH − ξ

+HE). (6)

Note that Eq.(6) has more degrees of freedom than thenon-chiral version and allows for a tunable refractive in-dex that can even by allowed to go negative without re-quiring a negative permeability. For ξ > 0, Eq. (6) canbe further simplified by setting ξ+EH = −ξ+HE = iξ thusgiving n =

√εµ − ξ. Then adaptive changes to the re-

fractive index in a chiral media are performed by alteringthe magnitudes of the coupling coefficients ξEH and ξHE .These are dependent on the strength of the coupling fieldΩC and the detuning from the magnetic and electric en-ergy levels ∆ [14, 16].

Managing the refractive index of a chiral medium. Us-ing the theory of the chiral electromagnetic media de-veloped in [14], we have identified the parameters of thesystem to facilitate the refractive index profile desired forinvisibility cloaking. Our goal is to tailor the refractiveindex of such a material in acordance with Fig. 1 (b), con-trolling the electric and magnetic dipole levels ∆E and∆B and the cross coupling Rabi frequency Ωc.

Consider the system modeled by the 5-level atomic sys-tem interacting with the strong laser beams Ω1,2 andthe cross-coupling light field Ωc as presented in Figure 2.The transition |2〉 − |1〉 is a magnetic dipole transitionin nature. All others are electric dipole transitions. TheHamiltonian H = Hatom− d.E(t)−µ.B(t) of the systemcan be expressed in a convenient matrix form:

H =

~ω1 − 1

2µ21Beiωpt 0 0 −~

2Ω1eiω1t

− 12µ21Be

−iωpt ~ω2 −~2Ω∗ce

iωct 0 00 −~

2Ωce−iωct ~ω3 − 1

2d34Ee−iωpt 0

0 0 − 12d34Ee

iωpt ~ω4 −~2Ω2e

iω2t

−~2Ω1e

−iω1t 0 0 ~2Ω2e

−iω2t ~ω5

(7)

with the electric and magnetic dipole moments

d34 = 〈3|er.eE |4〉, µ21 = 〈2|µ.eB |1〉. (8)

E and B are the electric and magnetic components ofthe weak probe field which oscillates at a frequency ωp.The Rabi frequencies Ω1, Ω2 and Ωc belong to strongcoupling lasers which oscillate at frequencies ω1, ω2 andωc respectively. We choose d34, µ21, Ω1 and Ω2 to bereal but allow the strong field Ωc to stay complex. Usingthe Liouville equation, we have found the equations ofmotion for all twenty five density matrix elements. Thissystem of equations is simplified considerably by focusingon just the linear response so we treat probe fields E andB as weak fields which allows us to neglect the upperstate populations ρ33 and ρ22.

The true index of refraction which takes the full ten-sor form i.e. the angle dependent correction to ε and µ,gets very complicated. However under the assumptionof isotropic permittivity and permeability we can neglectthe tensor form for the electric and magnetic polarizabil-ities αEE and αBB and find the refractive index for theleft n+ and right n− circular polarizations by choosingthe top left and middle tensor elements respectively:

n+ = n− =

√εµ− ξEHξHE −

(ξEH − ξHE)2 cos2 θ

4

+i

2(ξEH − ξHE) cos θ. (9)

Thus even in the ideal case ε ≈ µ ≈ 1 we do not havean isotropic index of refraction. In fact, we see that thecoupling effect disappears if the fields are orthogonal (θ =π/2) to each other.

From this analysis we see that there are two physicalparameters that can be readily manipulated to achievethe refractive indices required to implement electromag-netically induced carpet cloaking.

Setting I: Refractive index n as a function of cross-coupling Ωc. The first of our proposals to implementthe carpet cloak refractive index profile relies on di-rectly varying the coupling field Ωc for constant detuning∆ = −∆E = −∆B = 1000kHz and constant number ofscatterers, % = 1.5623. The refractive index in a chiralmedium is dependent on the relative directions of the in-cident laser fields [14]. However, as long as we keep thesedirections constant, we have a simple formula, Eq. (9), fordetermining n. Hence we can arrange our system to keepthe fields at a constant relative angle of θ = π/4 which isconsiderably easier to realize experimentally than if theywere required to be collinear. The variation of n with thecoupling field is given in Figure 3. Using numerical anal-ysis, we find that, at a constant detuning, a maximumrefractive index n = 3.17 for a coupling of Ωc = 44MHzcan be reached. This range of n is more than sufficientto implement the conformal transform given in Eq. (1).Note, however, that such a scheme requires continuousvariation of the coupling field over the device which could

4

FIG. 3: Real part of the refractive index as a function of thecoupling Rabi frequency |Ωc| relative to γ3, the radiative de-cay rate of the electric dipole transition, for % = 1.561017cm−3

and detuning ∆ = 1.5γp.

FIG. 4: An incident magnetic field B is applied to control thedetuning ∆ of the chiral media implemented by a magneto-electric cross-coupling in a solid. The coupling field strengthis kept at a constant value across the media. The refractiveindex can be tailored as in Fig. 5 to achieve the profiles de-picted in Fig. 1b. The insert shows the actual cloaking, thecurved physical space as in Fig. 1a, delivering desired straightlines in virtual space.

be challenging experimentally. To avoid this, we presentan alternative scheme, where Ωc is kept constant acrossthe cloak.

Setting II: Refractive index n as a function of detun-ing ∆. Here we manipulate the detuning ∆ to controlthe refractive index of the chiral media required for agiven conformal transform. The detuning is controlledby an incident gradient magnetic field that varies overthe length of the device. This scheme is depicted in Fig-ure 4. We choose a refractive index range by setting aparticular value of the cross-coupling Ωc. We then varythe detuning via a magnetic field to fine tune n withinthe selected range along the length of the cloak. Thevariation of refractive index with the detuning for someconstant values of the cross-coupling Ωc is plotted in Fig-ure 5. For a coupling of Ωc = 15.54γ2e

iπ/2 as in Fig. 5(a),we find that the refractive index achieves a maximum ofn = 1.83 for a detuning ∆ = 0.63kHz. Comparing withthe refractive index requirements of the carpet cloak asin Fig.1, we see that our scheme provides sufficient tun-ability to implement the Jukowski transform of Eq. (1).This set up is experimentally more feasible as it only re-lies on precise control of the magnetic field gradient (anestablished technique) and allows us to keep the coupling

FIG. 5: Real parts of the refractive index including local fieldeffects as a function of detuning ∆ relative of γ2, the radiativedecay rate of the magnetic dipole transition, for two couplingfield strengths: (a) Ωc = 164γ2e

iπ/2 and (b) Ωc = 125γ2eiπ/2.

We note that for increasing coupling strengths we can access alarger range of refractive indices but at the expense of requir-ing large detunings, which, at a certain magnitude becomeunfeasible.

fields at a constant value.So far we have considered a very smooth conformal co-

ordinate transform to implement carpet cloaking. How-ever, such a transform limits the dimensions of the objectwe wish to hide. Recently, other conformal transformshave been suggested that yield larger and more irregu-lar cloaked areas [20]. However, these transforms havemore stringent demands on the refractive index rangeof the cloaking media. To test whether the chiral mediaare suitable to perform more demanding conformal trans-forms we increase the cross-coupling field (Fig. 5(b)). Wefind that at higher field strengths we access larger refrac-tive indices (n = 6.61 for detuning of ∆ = 7.935kHz).However, these require ever larger detunings which canthen become unphysical. We must also limit the cou-pling strength of the field to avoid thermal effects. Inthe ranges we have plotted, the thermal effects are ex-tremely small as the coupling fields correspond to mWlaser powers.

We have presented a scheme for electromagnetically-induced invisibility using a carpet cloak. The scheme isbased on coherent magneto-electric cross-coupling in amedium modeled by the 5-level atom interacting witha weak probe and the three control light fields (Fig. 2).We obtained the steady state solutions of the correspond-ing Liouville equation for the density matrix elements toshow that such chiral media allow for a precise controlof the refractive index over a wide range, while simulta-neously suppressing absorption due to quantum interfer-ence effects similar to EIT. We have devised the confor-mal transformation that is suitable for this system andfound the physical parameters required to accomplish thecloaking.

The research leading to these results has received fund-ing from the European Community’s Seventh FrameworkProgramme (FP7/2007-2013) under grant agreement no

270843 (iQIT) and has also been supported by the Scot-tish Universities Physics Alliance (SUPA) and by the En-

5

gineering and Physical Sciences Research Council (EP-SRC). The authors acknowledge invaluable discussionswith Ulf Leonhardt and are grateful for his continuoussupport during this research project, as well as assistancein technical matters.

[1] M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev.Mod. Phys. 77, 633 (2005).

[2] D. F. Phillips, A. Fleischhauer, A. Mair, R. L.Walsworth, and M. D. Lukin, Phys. Rev. Lett. 86, 783(2001).

[3] D. Hckel and O. Benson, Phys. Rev. Lett. 105, 153605(2010).

[4] J. J. Longdell, E. Fraval, M. J. Sellars, and N. B. Manson,Phys. Rev. Lett. 95, 063601 (2005).

[5] S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang,Phys. Rev. Lett. 101, 047401 (2008); N. Liu, Nat. Mater.8, 758 (2009).

[6] K. L. Tsakmakidis, A. D. Boardman, and O. Hess, Na-ture, (London) 450, 397 (2007).

[7] A. Greenleaf, M. Lassas, and G. Uhlmann, Math. Res.Lett. 10, 685 (2003).

[8] U. Leonhardt, Science 312, 1777 (2006).

[9] J.B. Pendry, D. Schurig, and D.R. Smith, Science 312,1780 (2006).

[10] H. Chen, C.T. Chan, and P. Sheng, Nature Materials 9,387 (2010).

[11] U. Leonhardt and T.G. Philbin,Geometry and Light: TheScience of Invisibility (Dover, Mineola, 2010).

[12] D. Schurig, J.J. Mock, B.J. Justice, S.A. Cummer, J.B.Pendry, A.F. Starr, and D.R. Smith, Science 314, 977(2006).

[13] R. R. Moseley, S. Shepherd, D. J. Fulton, B. D. Sinclair,and M. H. Dunn, Phys. Rev. A 53, 408 (1996).

[14] J. Kstel, M. Fleischhauer, S. F. Yelin, and R. L.Walsworth, Phys. Rev. A 79, 063818 (2009).

[15] D. E. Sikes and D. D. Yavuz, Phys. Rev. A 82, 011806(R)(2010).

[16] J. B. Pendry, Science 306, 5700 (2004); S. A. Tretyakov,C. R. Simovski, and M. Hudlicka, Phys. Rev. B 75,153104 (2007),

[17] J.S. Li and J. B. Pendry, Phys. Rev. Lett. 101, 203901(2008).

[18] T. Needham, Visual Complex Analysis (Clarendon, Ox-ford, 2002).

[19] T.H. O’Dell, The electrodynamics of magneto-electricmedia, North- Holland Publishing Group (1970)

[20] T. Ochiai, J. C. Nacher, arXiv:1202.4447v1,[physics.optics] (2012)