virtual hyperbolic metamaterials for manipulating radar … kudys… · virtual hyperbolic...

ARTICLE

Received 2 Jul 2013 | Accepted 5 Sep 2013 | Published 2 Oct 2013

Virtual hyperbolic metamaterials for manipulatingradar signals in airZhaxylyk A. Kudyshev1, Martin C. Richardson2 & Natalia M. Litchinitser1

Microwave beam transmission and manipulation in the atmosphere is an important but

difficult task. One of the major challenges in transmitting and routing microwaves in air is

unavoidable divergence because of diffraction. Here we introduce and design virtual hyper-

bolic metamaterials (VHMMs) formed by an array of plasma channels in air as a result of

self-focusing of an intense laser pulse, and show that such structure can be used to

manipulate microwave beams in air. Hyperbolic, or indefinite, metamaterials are photonic

structures that possess permittivity and/or permeability tensor elements of opposite sign

with respect to one another along principal axes, resulting in a strong anisotropy. Our proof-

of-concept results confirm that the proposed virtual hyperbolic metamaterial structure can be

used for efficient beam collimation and for guiding radar signals around obstacles, opening a

new paradigm for electromagnetic wave manipulation in air.

DOI: 10.1038/ncomms3557

1 Department of Electrical Engineeering, University at Buffalo, The State University of New York, Buffalo, New York 14260, USA. 2 The College of Optics andPhotonics, Townes Laser Institute, University of Central Florida, Orlando, Florida 32816, USA. Correspondence and requests for materials should be addressedto N.M.L. (email: [email protected]).

NATURE COMMUNICATIONS | 4:2557 | DOI: 10.1038/ncomms3557 | www.nature.com/naturecommunications 1

& 2013 Macmillan Publishers Limited. All rights reserved.

mailto:[email protected]

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When microwaves or terahertz beams propagate in air,they diffract, as does any electromagnetic beam.Therefore, to be able to receive a signal of appreciable

intensity at a distance, one has to either increase the intensity ofthe incident beam or counteract this beam-broadening diffractioneffect. Moreover, detection of radar signals in a realisticenvironment may be negatively affected by the divergence ofthe beam resulting from distortions by natural obstacles, forexample, by non-planar landscapes. To overcome the distortionof beams propagating in such environments, some method ofguiding the beam around obstacles (or ‘cloaking’ the obstacles) isdesirable.

The development of ordered structures of laser-induced plasmafilaments opens the possibility of creating conductive filamentarystructures capable of guiding microwaves or terahertz frequencyradiation. In the last few years, various types of filament-basedwaveguides (conducting or dielectric like) have been proposed forchannelling microwaves1–5. Guiding of 10 GHz radiation withfilaments was experimentally demonstrated by Chateauneufet al.3 using a 100 TW laser system to produce a cylindricalarray 45 mm in diameter composed of B1,000 filaments.A 10-GHz signal was coupled into the induced plasmawaveguide and guided over a distance of 16 cm. The intensityof the guided signal detected at the exit of the cylindrical arraywas increased by a factor greater than six compared with the freepropagation of the microwave radiation. Although this approachoffers a way to counteract the diffraction of a radar signal oversome distance, the issues of beam steering and couplingmicrowave signals into such waveguides are likely to bechallenging. Finally, very recently, Marian et al.6 reportedparametric studies of interactions of microwaves withsubwavelength arrays of parallel plasma filaments. Althoughthese studies elucidated reflection and transmission properties ofvarious periodically and randomly arranged filament arrays, nospecific physical mechanism enabling guiding, focusing orsteering of the microwave beam was discussed.

In this Article, we propose a fundamentally new approachbased on unique physical properties of virtual hyperbolicmetamaterials (VHMMs) formed of plasma filaments in air. Weshow that VHMMs can be used for focusing, guiding and steeringradar signals to facilitate new degrees of freedom in the detectionof such signals in realistic environments. Further, we show that byvariation of realistically available parameters of VHMM, such asfill-fraction of plasma filaments and recombination rate of freeelectrons with positive ions inside the filament channels (that canbe controlled by heat), it is possible to control the performance ofthe VHMM structure.

ResultsMathematical model of VHMMs. Consider a microwave beamincident on a two-dimensional rectangular array of filaments ormicroscopic wires of conductivity, as shown in Fig. 1a. Figure 1bshows isofrequency curves and relative directions of the Poyntingvector S and wave vector k. In the isotropic case, the isofrequencysurfaces are circles and, therefore, Sprko(k)pk (that is, vectorsS and k are collinear), whereas in anisotropic materials the wavevector surfaces become either ellipsoidal or hyperbolic, dependingon the signs of e> and e||. Taking y as the direction of the opticalaxis, we can characterize the extraordinary wave in a uniaxialanisotropic structure by the dispersion relation

k2k

e?þ k2

?ek¼ o2

c2; ð1Þ

where e|| and e>are the components of the dielectric permittivitytensor parallel to the wires and perpendicular to the wires,

respectively, k|| and k> are corresponding components of thewave vector, o is the angular frequency and c is the speed of light.In the case of hyperbolic metamaterials, the components ofdielectric permittivity satisfy the following condition: e>40,e||o0 and the signs of the projections of St and kt on to the x axisare opposite. As a result, such structures support negativerefraction and can be used for focusing and steering electro-magnetic radiation7–15.

Conventionally, hyperbolic metamaterials are realized usingeither metal/dielectric multilayers or an array of metal wiresin a dielectric matrix10–15. Here we show that an array ofplasma filaments formed by the propagation of intense shortpulses in air is able to form a VHMM structure. Indeed, plasmafilaments have the role of metal wires and the air has the role of adielectric host. We propose to use hyperbolic metamaterial-likestructures formed from filaments in air, with approximatediameters dplE50B120 mm (ref. 1) and average distancebetween filaments aplE450B650mm for microwave andterahertz guiding (Fig. 1a). If the wavelength of the incidentelectromagnetic (EM) field linc is much larger than the diameterdpl, the EM wave cannot resolve individual filaments, andthe composite medium can be considered as an effectivemedium. The two components of the permittivity tensor,perpendicular to direction of propagation (e>) and parallel

z,

4k||

St

kt

kinc

k

2

0

–2

–4–2 –1 0 1 2

dpl

ap|

x,

y,

∋

∋

∋

Figure 1 | Concept of the VHMM. (a) Schematic illustration of the

filament-based VHMM formed of laser-induced plasma filaments (red

wires) in air for microwave signal guiding (schematically shown as yellow

trace of the beam). The lower inset shows the orientation of the coordinate

system and corresponding component of the permittivity tensor e|| and e>.

The upper inset outlines two characteristic geometrical parameters of

VHMM diameter of plasma channel (dpl) and distance between filaments

(apl); (b) Isofrequency contour for free space (circular) and VHMM

(hyperbolic).

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3557

2 NATURE COMMUNICATIONS | 4:2557 | DOI: 10.1038/ncomms3557 | www.nature.com/naturecommunications



to the direction of propagation (e||), can be calculated using themixing rule as12

e? ¼ eairð1þ f ÞefilðoÞþ ð1� f Þeair

ð1� f ÞefilðoÞþ ð1þ f Þeair;

ek ¼ f efilðoÞþ ð1� f Þeair;

ð2Þ

where f is the fill-fraction of filaments (f¼ dpl/apl in the two-dimensional case and f ¼ pd2

pl=ð4a2plÞin the three-dimensional

case); eair and efil(o) are the permittivities of the air and filamentchannel, respectively. The permittivity of the filament channelcan be described by the Drude model as

efilðoÞ ¼ 1�o2

pl

o2þ ion; ð3Þ

where opl ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiNee2=ðe0meÞ

pis the plasma frequency, e and me are

the electron charge and mass, respectively, e0 is the permittivity offree space, Ne is the electron density and n is the collisionfrequency. The concentration of the electrons inside the laser-induced filament channel in air varies between 1015 and1017 cm� 3 (refs 1,5) and the corresponding plasma frequencyrange is 300–3,000 GHz (ref. 5). Generally, for plasma filamentsin the air, where the gas is not fully ionized, the collisionfrequency is given by

n ¼ neiþ nen; ð4Þwhere nei ¼ 2:91�10� 6ðlnLÞNeT � 3=2

e is an electron-ioncollision frequency, Te is the electron temperature, (ln L) isCoulomb logarithm, nen ¼ 3:91�10� 8NaT1=2

e is an electron-neutral collision frequency and Na is the density of the atoms. Inthe case of weakly ionized gas at a pressure of 1 atm,Ne¼ 1017 cm� 3, Te¼ 1 eV, Na¼ 2.7� 1019 cm� 3, the collisionfrequency4 is neiE7.61� 1010 s� 1 and nenE1.05� 1012 s� 1.

The dynamics of the electron density (Ne), the density ofpositive (Nþ ) and negative ions (N� ) inside each filament can becalculated using rate equations16,17

dNe

dt¼ aNe� ZNe�beþNeNþ ;

dNþdt¼ aNe�beþNeNþ �b� þN�Nþ ;

dN�dt¼ ZNe�b� þN�Nþ ;

ð5Þ

where a is the collisional ionization rate, Z is the attachment rate,beþ is the electron-positive ion recombination rate coefficientand b� þ is the recombination rate coefficient between negativeions and positive ions.

In the case of atmospheric pressure and in the absence of anyexternal fields, the collisional ionization rate (a) is much smallerthan the attachment rate coefficient (Z), and thus can be neglected(aE0). At the same time, electrons participate in three-bodyreaction with rate Z3 and two-body reaction with rate Z2, whichleads to the creation of negative molecular ions (O�2 ) andnegative ions (O� ), respectively. Therefore, the total attachmentrate coefficient18,19 is Z¼ Z2þ Z3E2.5� 107 s� 1.

Another important phenomenon, which takes place inside thefilament channel, is the recombination between electrons andpositive ions, and similarly between negative ions with positive ions,with corresponding rate coefficients beþ and b� þ . Recombinationrate coefficients are defined as beþ ¼ beþ ½Oþ2 � þ beþ ½Nþ2 � �1:2�10� 13 m3 s� 1 and b� þE2.2� 10� 13 m3 s� 1(ref. 19).

The system of rate equation (5) is solved numerically with theassumptions that initial Ne¼NþE2� 1017 cm� 3, N�E0.Figure 2a shows the evolution of electron, positive and negativeions densities inside filament channels with time. Reduction of theelectron density of filaments with time results in the VHMM

medium having time-dependent effective permittivity. Figure 2bshows the real part of e|| as a function of time for the fill-fraction offilaments, which varies from 0.08 to 0.2. It is noteworthy that the realpart of e|| changes sign when a particular moment in time is reached,whereas the real part of e> remains positive and, for example, forf¼ 0.1 equals Re(e>)¼ 1.2. As a result, the dispersion relation of theVHMM changes from hyperbolic to elliptical when e|| becomespositive. As long as the dispersion relation remains hyperbolic, thediverging Gaussian beam entering from air to VHMM is negativelyrefracted and converging/collimated inside the effective medium,assuming it did not reach its focus point. Figure 2b indicates that thehigher the fill-fraction, the longer e|| stays negative, that is, thedispersion relation remains hyperbolic.

Microwave signal guiding and steering using VHMM. Oncethe effective permittivity tensor is determined using equations

2.0a

b

c

1.5

1.0

0.5

0.00.0 0.1 0.2 0.3 0.4

t (ns)

Ne×

1017

(cm

–3)

N–×

1014

(cm

–3)

Re

(�||)

Re

(�||)

0.5 0.60.0

0.5

1.0

1.5

2.0

1

0

–1

–2

–3

–40.0

1.0

0.5

0.0

–0.5

–1.0

–1.50 1 2 3

t (ns)4 5

0.2 0.4t (ns)

0.6 0.8 1.0

Figure 2 | Density and permittivity of the VHMM. (a) Evolution of

electron (dashed blue curve) and negative ion (solid black curve) densities

inside filament channels; (b) Real part of e|| as a function of time for

different fill-fraction of filaments f¼0.08 (blue curve), f¼0.1 (black curve),

f¼0.15 (red curve) and f¼0.2 (yellow curve), and fixed recombination

rate coefficient beþ ¼ 1.2� 10� 13 m3 s� 1; (c) Real part of e|| as a

function of time for different recombination rate coefficients beþ ¼ 1.2�10� 13 m3 s� 1(dotted blue curve), beþ ¼0.6� 10� 13 m3 s� 1(dashed red

curve) and beþ ¼0.12� 10� 13 m3 s� 1(solid black curve), and fixed

fill-fraction f¼0.1.

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3557 ARTICLE




(2)–(5), we investigate the propagation of the microwave beaminside the VHMM medium using Maxwell’s equations

r�E ¼ � 1c@H@t

;r�H ¼ 1c@D@t;

~DðoÞ ¼e? 0 0

0 ek 0

0 0 e?

0B@

1CA~EðoÞ:

ð6Þ

The relative permeability of the medium m is assumed to be equalto one. Noting that the incident field is assumed to be a trans-verse-magnetic (TM) polarized continuous wave Gaussian beamwith a wavelength linc¼ 5 cm, the components of the incidentEM wave are chosen as H¼ {0,0,Hz} and E¼ {Ex,Ey,0}.

Here we summarize several assumptions that have been made.As we consider the VHMM structure made of plasma filamentswith initial free electron density 2� 1017 cm� 3 and fill-fractionf¼ 0.1, in reality it is difficult to achieve this configurationwithout any background plasma between the filaments. Weneglect the background plasma in the simulations, and assumethat VHMM consists of plasma filaments and air (see equation(2)). Besides, it is a well-known fact that it is difficult to achieveplasma filaments with such configuration (electron density andfill-fraction) longer than few centimetres in air. Within this work,we make an assumption that VHMM structure can be of arbitrarylength and the guiding of the signal wave is limited only by thelifetime of the plasma filaments inside VHMM medium.

On the basis of developed finite-difference time domain (FDTD)code20, we have considered the case when microwave signalpropagates in the free space (negative values of y axis) and launchesto VHMM medium (positive values of y axis), while the air-VHMMboundary is set to be at the point y¼ 0 (see Fig. 3). Besides this, weneed to mention that at the initial moment of time (first 0.3 ns)while the microwave signal propagates inside the free space, thecomponents of the dielectric permittivity tensor(e> and e||) areassumed to be constant and determined by initial concentration offree electrons inside plasma filaments Ne(0). The components of thedielectric permittivity tensor (e> and e||) are considered to be timedependent once the microwave beam reaches the air-VHMMboundary. This assumption implies negligibly small time delay

between high-power excitation laser, which forms VHMMstructure, and microwave signal.

Figure 3 shows the intensity distribution for three cases: air(Fig. 3a), VHMM with fill-fraction f¼ 0.1 (Fig. 3b) and VHMMwith fill-fraction f¼ 0.2 (Fig. 3c) at t¼ 1.2 ns moment of time.Figures 3b,c confirm the prediction that the VHMM structure canbe used to compensate for strong diffraction broadening of amicrowave beam propagating in a free space. It clearly shows thata larger fill-fraction of filaments leads to a stronger focusing effectin the beam. At the same time, it should be noted that amicrowave beam propagating in a VHMM experiences lossesassociated with the metallic nature of plasma filaments, and as aresult, the beam intensity decays with propagation distance.

It should also be noted that the decay of plasmas in the wake ofthe laser pulse limits the lifetime of plasma-induced filament-array waveguides. Fortunately, it was proposed and demonstratedthat the lifetime of the filaments can be increased by heatingplasma filaments with a second, delayed nanosecond pulse21,22 orby employing a system composed of a dual femtosecond/nanosecond laser pulse23,24. Such additional delayed laser pulsesefficiently increase electron temperature, and as a result, suppressdissociative recombination. Therefore, we performed a study ofmicrowave propagation in a VHMM for different recombinationrate coefficient beþ , as shown in Fig. 3d–f. Figure 3d showsmicrowave beam propagation corresponding to the no-heatingcase with beþ ¼ 1.2� 10� 13 m3 s� 1 and f¼ 0.1. Figure 3e,fshows the cases corresponding to beþ ¼ 0.6� 10� 13 m3 s� 1 andbeþ ¼ 0.12� 10� 13 m3 s� 1, respectively, for the same fill-fraction f¼ 0.1. These results clearly indicate that decreasingthe recombination rate would significantly increase the distanceof microwave beam guiding by VHMM.

To quantify the improvement in microwave guidance enabledby VHMM, we calculate transmission enhancement factorz¼ IVHMM/Iair as a ratio of on-axis peak time average intensitiesof the signal transmitted through focusing VHMM medium(IVHMM) and of the signal propagating in a free space (Iair).Transmission enhancement factor is calculated and studied onlyfor positive values of y. Figure 4a shows the transmissionenhancement factor z as a function of propagation distance fordifferent fill-fractions of filaments. As discussed above, inside the

1

0

Inte

nsity

(a.

u.)

1

0

Inte

nsity

(a.

u.)

x (cm)10 20 30

25 1

0

Inte

nsity

a.u

.

1

0

Inte

nsity

(a.

u.)

20

15

10

y (c

m)

5

0

–5

–10

25

20

15

10

y (c

m)

5

0

–5

–10

25

20

15

10

y (c

m)

5

0

–5

–10

25

20

15

10

y (c

m)

5

0

–5

–10

25

20

15

10

y (c

m)

5

0

–5

–10

25

20

15

10

y (c

m)

5

0

–5

–10

x (cm)10 20 30

1

0

Inte

nsity

(a.

u.)

1

0

Inte

nsity

(a.

u.)

x (cm)10 20 30

x (cm)10 20 30

x (cm)10 20 30

x (cm)10 20 30

Figure 3 | Intensity distributions for k¼ 5 cm. The field intensity distribution for TM-polarized continuous wave Gaussian beam at wavelength

l¼ 5 cm propagating in (a) air, (b) in a VHMM with fill-fraction f¼0.1 and (c) in a VHMM with f¼0.2, and for fixed fill-fraction f¼0.1 but different

recombination rate coefficients (d) beþ ¼ 1.2� 10� 13 m3 s� 1, (e) beþ ¼0.6� 10� 13 m3 s� 1 and (f) beþ ¼0.12� 10� 13 m3 s� 1.





VHMM, an initially diverging microwave beam experiences afocusing effect, whereas the same beam propagating in a freespace diverges because of diffraction. Accordingly, the enhance-ment factor z increases. However, it can be seen in Fig. 4a thatafter a certain propagation distance, losses inside the VHMMbecome significant, leading to reduction of the transmissionenhancement factor. This suggests that the design of the VHMMcan be optimized for the desired propagation distance andenhancement factor. For example, propagation distance corre-sponding to z¼ 1 and to IVHMM¼ Iair ranges from 0.3 to 0.5 mfor different fill-fraction parameters. Figure 4b shows thetransmission enhancement factor z as a function of propagationdistance for different recombination rates, demonstrating thatboth the enhancement factor and the distance corresponding tothe maximum enhancement increase as recombination ratecoefficient decreases. These results are in agreement with theprediction of Fig. 3d,f.

Finally, Fig. 4c demonstrates another potentially usefulfunctionality—microwave beam steering enabled by filament-basedVHMM in air. In this case, the VHMM structure is formed suchthat the microwave beam enters the VHMM at an angle. The angleof refraction can be controlled by changing the fill-fraction of thefilaments. It is noteworthy that the microwave beam refractsnegatively with respect to the normal to the air-VHMM interface.Such negative refraction is again due to hyperbolic dispersionrelations. Indeed, it can be seen in Fig. 1b that the projections of thePoynting vector S and wave vector k on the horizontal axis (k>) areantiparallel, resulting in such negative refraction, as shown inFig. 4c. We envision that such reconfigurable engineered VHMM-based beam steering ‘devices’ will be useful in natural environmentswhere the detection of radar signals may be complicated bydivergence of the beam during its free space propagation or becauseof distortions by various obstacles.

DiscussionIt is noteworthy that the proposed approach of guiding microwavebeams using the VHMMs is robust with respect to changes in thefilaments parameters, and the wavelength of the microwaveradiation and in general does not require rather high free electrondensities and it can realized at lower free electron densities thanthose considered above. Here we show that similar performancecan be achieved at lower free electron densities, for example,ne¼ 5� 1016 cm� 3, and at a different microwave wavelengthl¼ 7.5 cm. Figure 5a shows real part of e|| as a function of time fordifferent fill-fraction of filaments at a fixed recombination ratecoefficient. Figure 5b shows real part of e|| as a function of time for

different recombination rate coefficients at a fixed fill-fractionf¼ 0.3. Finally, Fig. 6 shows field intensity distributions for aTM-polarized continuous wave Gaussian beam at wavelengthl¼ 7.5 cm in VHMMs with different fill-fraction for givenrecombination rate coefficient beþ ¼ 0.12� 10� 13 m3 s� 1.

In summary, in this paper we proposed to use VHMMs formedby an array of plasma channels in air to manipulate microwaveradar signals. We performed a comprehensive study ofthe performance of such VHMMs and demonstrated that usingvarious realistically available degrees of freedom such as

1.8a b c1.6

1.4

1.2

1.0

0.8

0.6

Enh

ance

men

t fac

tor �

0.4 Enh

ance

men

t fac

tor �

0.2

0.0

30

1

Inte

nsity

(a.

u.)20

0

–10

20 30 40x (cm)

500

10

y (c

m)

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.00.0 0.1 0.2 0.3 0.4

y (m) y (m)0.6 0.7 0.80.5 0.0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2.01.0

Figure 4 | Enhancement factors. Transmission enhancement factor z¼ IVHMM/Iair as a function of propagation distance for different fill-fractions:

f¼0.1 (red curve), f¼0.15 (green curve) and f¼0.2 (blue curve; a). Transmission enhancement factor z¼ IVHMM/Iair as a function of propagation distance

for different recombination rates: beþ ¼ 1.2� 10� 13 m3 s� 1 (green curve), beþ ¼0.6� 10� 13 m3 s� 1 (blue curve) and beþ ¼0.12� 10� 13 m3 s� 1

(red curve; b). (c) Beam steering using VHMMs (f¼0.1 and beþ ¼0.12� 10� 13 m3 s� 1).

0 1 2 3t (ns)

t (ns)

Re(

� ||)

Re(

� ||)

4

0 1 2 3 4 5

–1.5

1.0b

a

0.5

0.0

–0.5

–1.0

–0.5

0.0

0.5

Figure 5 | Permittivity behaviour. (a) Real part of e|| as a function of time

for different fill-fraction of filaments f¼0.2 (dotted blue curve), f¼0.3

(solid black curve) and f¼0.4 (solid red curve), and fixed recombination

rate coefficient beþ ¼ 1.2� 10� 13 m3 s� 1. (b) Real part of e|| as a

function of time for different recombination rate coefficients beþ ¼ 1.2�10� 13 m3 s� 1 (dotted blue curve), beþ ¼0.6� 10� 13 m3 s� 1 (dashed red

curve) and beþ ¼0.12� 10� 13 m3 s� 1 (solid black curve) and fixed

fill-fraction f¼0.3.





fill-fraction of plasma filaments and recombination rate (that canbe controlled by heat), one can remotely focus, guide and steerradar signals to facilitate new degrees of freedom in the detectionof such signals in realistic environments.

MethodsSolution of the Maxwell equation for VHMM structure. The system of Max-well’s equations:

r�E ¼ � 1c@H@t

;r�H ¼ 1c@D@t;

~DðoÞ ¼e? 0 0

0 ek 0

0 0 e?

0B@

1CA~EðoÞ:

ð7Þ

is solved numerically using the FDTD method within the auxiliary differentialequation (ADE) technique.

By taking into account equation(2), the material equation for electricdisplacement field D for different component of the field can be rewritten in thefollowing way:

~DxðoÞ ¼ eairð1þ f ÞefilðoÞþ ð1� f Þeair

ð1� f ÞefilðoÞþ ð1þ f Þeair

~ExðoÞ;

~DyðoÞ ¼ ðf efilðoÞþ ð1� f ÞeairÞ~EyðoÞ;ð8Þ

where permittivity of the filaments is determined by the Drude model.By implementing inverse Fourier transform of equation (8), the ADEs, which

determine relation between components of electric field E and correspondingcomponents of electric displacement field D in time domain, can be derived. TheADE in the case under consideration takes the following form:

g1@2Dx

@t2þ n

@Dx

@t

� �þð1� f Þo2

pðtÞDx ¼ g2@2Ex

@t2þ n

@Ex

@t

� �þ eairð1þ f Þo2

pðtÞEx ;

@2Dy

@t2þ n

@Dy

@t¼ ð1� g2Þ

@2Ey

@t2þ n

@Ey

@t

� �þ fo2

pðtÞEy :

ð9Þ

here g1¼ (1–f)þ (1þ f)eair, g2¼ 1þ fþ (1–f)eair.To determine spatial distribution of the E and H field at the given moment of

time, the system of curl equations from equation (7) and one of the ADE (equation(9)) for corresponding field component should be solved simultaneously.

For the case of TM incident wave for each time step, the following equationsshould be solved:

To determine Ex:

ðr�HÞx ¼1c@Dx

@t;

g1@2Dx

@t2þ n

@Dx

@t

� �þð1� f Þo2

pðtÞDx ¼ g2@2Ex

@t2þ n

@Ex

@t

� �þ eairð1þ f Þo2

pðtÞEx :

ð10ÞTo determine Ey:

ðr�HÞy ¼1c@Dy

@t;

@2Dy

@t2þ n

@Dy

@t¼ ð1� g2Þ

@2Ey

@t2þ n

@Ey

@t

� �þ fo2

pðtÞEy :

ð11Þ

For determine Hz:

ðr�EÞz ¼1c@Hz

@t: ð12Þ

It is noteworthy that plasma frequency op in equation (9) in the case underconsideration is a time-dependent parameter because of evolution of the freeelectron concentration inside the filaments and need to be updated at every timestep.

The system equations(10)–(12) is solved with help of leapfrog scheme withinFDTD technique.20,25

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Figure 6 | Intensity distributions for k¼ 7.5 cm. Field intensity distributions for a TM-polarized continuous wave Gaussian beam at wavelength l¼ 7.5 cm

in (a) vHMM with fill-fraction f¼0.2; (b) VHMM with fill-fraction f¼0.3; and (c) vHMM with fill-fraction f¼0.4 for given recombination rate coefficient

beþ ¼0.12� 10� 13 m3 s� 1.





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AcknowledgementsWe acknowledge useful discussions with Magali Durand and Scott Will. This work wassupported by US ARO under Awards # W911NF-11-1-0333 and MURI Grant W911NF-11-0297.

Author contributionsAll authors contributed equally to the study, discussed the results and commented on themanuscript at all stages. N.M.L. and Z.A.K. designed and optimized the VHMM struc-ture for guiding and routing microwave signals based on realistic experimentally rea-lizable parameters. Z.A.K. developed and operated the FDTD-based code that couplesMaxwell’s equations and rate equations to study microwave beam transmission inVHMMs. N.M.L. and M.C.R. initially developed the concept of microwave guiding usingphotonic structures in air, identified realistic experimental conditions for the realizationof the proposed approach and identified the ways to extend the lifetime of the proposedvirtual metamaterial structure. M.C.R. evaluated the data and N.M.L. supervised thework.

Additional informationCompeting financial interests: The authors declare no competing financial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

How to cite this article: Kudyshev, Z. A. et al. Virtual hyperbolic metamaterials formanipulating radar signals in air. Nat. Commun. 4:2557 doi: 10.1038/ncomms3557(2013).




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