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Time Reversal of Electromagnetic Waves

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink

Laboratoire Ondes et Acoustique, ESPCI, UniversiteParis VII, U MR 7587, 10 r ue Vauquelin, 75005 Paris France(Received 9 December 2003; published 14 May 2004)

We report the first experimental demonstration of time-reversal focusing with electromagnetic waves.An antenna transmits a 1-s electromagnetic pulse at a central frequency of 2.45 GHz in a high-Q

cavity. Another antenna records the strongly reverberated signal. The time-reversed wave is built andtransmitted back by the same antenna acting now as a time-reversal mirror. The wave is found toconverge to its initial source and is compressed in time. The quality of focusing is determined by thefrequency bandwidth a nd the spectral correlations of the field with in the cavity.

DOI: 10.1103/PhysRevLett.92.193904 PACS numbers: 41. 20. Jb, 42.65. Hw, 84. 40.Ua

In acoustics, time-reversal experiments can be ca rriedout with broadband wave forms [1]. In such experiments,a source sends a short pulse that propagates through amore or less complex (but ideally nondissipative) mediumand is captured by a transducer array, termed a time-reversal mirror (TRM). The recorded signals are digi-

tized, stored in electronic memories, time reversed,reanalogized, and finally transmitted back by the TRM.The time-reversed wave is found to converge back to itssource al l t he more accurately when the medium is com-plex and the frequency bandwidth is larger [2]. This isvery appealing for applications such as subsurface detec-tion [3], scatterer analysis [4], and telecommunications.Indeed, it was recently shown with ultrasonic waves thatit is possible to take advantage of the complexity of amedium to convey more information through it by meansof a TRM [5]. From a practical point of view, a TRM wasused to focus a random series of bits simultaneously todifferent receivers which were only a few wavelengths

apart. In the language of communication, it correspondsto a MIMO-MU (multiple input, multiple outputmul-tiple users) configuration. While the transmission was freeof error when strong multiple scattering occurred in thepropagation medium, the error rate was huge in thehomogeneous medium (free space) due to cross talk be-tween receivers. Indeed, the spatial resolution of a TRMcan be much thinner in a multiple scattering medium thanin free space. It is the well-known super-resolution thathas been experimentally highlighted [2,6] and theoreti-cally discussed [2,7] in the past.

Is it possible to transpose this idea to the electromag-netic case? It is a challenging question because in manyreal environments (buildings or cities), microwaves withwavelengths between 10 and 30 cm are scattered byobjects such as walls, desks, cars, etc., which produces amultitude of paths from the transmitter to the receiver.In such situations, a time-reversal antenna should be ablenot only to compensate for these multipaths, but alsoincrease the information transfer rate thanks to themany reflections or reverberations, as it was alreadyshown with ultrasound [5]. However, the first step is to

prove the feasibility of a time-reversal experiment withelectromagnetic waves in the GHz range. This is the aimof this Letter. To that goal, we present the first one-channel electromagnetic time-reversal mir ror workingaround 2.45 GHz.

From a practical point of view, the main difficulty to

transpose the time-reversal technique developed for ul-trasound directly to the electromagnetic case lies in themuch higher sampling frequencies that are needed todigitize radio frequency signals. One way to overcomethis l imitation is to work only with quasimonochromaticsignals and to do a phase conjugation using the so-calledthree-wave or four-wave mixing in a nonlinear materialin order to naturally produce the analogic phase-conjugated wave [8]. Here, we want to perform a trulybroadband time reversal experiment for an electromag-netic signal mt cos20t t, or equivalentlymIt cos20t mQt sin20t. The car rier f re-quency is0,mItand mQtare the baseband signals.

All the time-reversal operations are performed on thebaseband signals. The advantage is that the samplingfrequency can be much lower than 20.

The experimental setup is the following (Fig. 1). Weuse two omnidirectional antennas working around0 2:45 GHz and two transceiver circuit boards. Onthe transmit side, they permit one to encode the in-phase (cos) and quadrature (sin) components of a base-band signal (labeled I and Q, respectively) onto a2.45 GHz wave carrier that can be radiated by thetransmit antenna. On the receive side, the circuitboards demodulate the radio frequency signal back tothe baseband.

The experiment takes place in a strongly reverberantcavity with dimensions 3:08 m 1:84 m2:44 m. Usingan arbitrary waveform generator, we deliver a short pulsemIt (central frequency 3 MHz, 6 dB band-width 2 MHz) to t heIanalog input [cf. Fig. 2(a)] of thetransmit board. No signal is delivered to the Q analoginput [Fig. 2(a0)]. A mixer up converts this signal to theGHz band and delivers et mIt cos20t. Then thewaveform et is transmitted by antenna A. After

P H Y S I C A L R E V I E W L E T T E R S week ending14 MAY 2004VOLUME92, NUMBER19

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propagation, the signal st m0It cos20t m0Qt sin20t is recorded by antenna B and down-converted to produce the I and Q components of the

output signal m0It and m0Qt that can be observed atthe oscilloscope [Figs. 2(b) and 2(b0)]. The received sig-nals last more than 8 s, i.e., 8 times longer than theinitial baseband pulse, or19 600periods of the rf wave.The rf wave has travelled more t han 2 km in the 14-m3

cavity, i.e., undergone 6000 reflections.Next, our goal is to time reverse the received radio

signalst. To that end, the baseband Ia ndQ signalsm0Iand m0Q are digitized by the oscilloscope at a 40-MHzsampling rate, sent to a computer, and time reversed. Thewave carr ier has to be conjugated, too. The following stepconsists in reanalogizing the t ime-reversedIa ndQ sig-nals and encoding them on the phase-conjugated wavecarrier: the resulting rf signal writes m0It cos20tm0Qt sin20t st. It is then transmitted backby antennaB. After propagation, the rf signal received onantennaA is down-converted to baseband. As can be seenin Fig. 3(a), the received signal on channel I is com-pressed in time and recovers its initial duration. It shouldbe noticed that the acquisition window has been arbi-trarily chosen in order to center the focused pulse withinthe figure.

Actually, since the reverberated wave field has beencaptured by a single antenna, the time-reversal operationis not perfect. The waveform that is recreated is not theexact replica of the initial pulse: there are sidelobesaround the peak on channel I, a nd a signal is measuredon channel Q [Fig. 3(a0)] even though nothing was sent onthat channel. Similar effects have been observed for time

FIG. 2. (a),(a0) Baseband representation [mIt a nd mQt] ofthe signal transmitted by antenna A. (b),(b0) Baseband repre-sentation [i.e., m0It and m0Qt] of the signal reverberatedinside the cavity and received by antenna B.

FIG. 3. (a),(a0) Baseband representation of the signal receivedby antenna A after time reversal. (b),(b0) Baseband represen-tation of the signal received several wavelengths away fromantenna Aafter time reversal.

FIG. 1. Sketch of a transmit or receive experiment. The base-

band signal mIt

[mQt] is fed into a mixer that multiplies itby cos20t [ sin20t], with 0 2:45 GHz. After am-

plification, the output rf signal is sent by antenna A, which islocated i n the reverberating room. The reverberated field isrecorded by antenna B. Its spectrum is down-shifted to thebaseband by a multiplier and a low-pass filter. The resultingsignalsm0Itand m

0Qtare sampled and stored in the computer

memory. Then they can be time reversed, reanalogized, and,following the same principle, sent back by antenna B whileantenna Aacts now as a receiver.


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reversal of ultrasonic waves: it can be shown that thepeak-to-noise ratio in a one-channel time-reversal experi-ment varies as

=

p , where is the available band-

width and defines the correlation frequency of thereverberated field [2]. is the characteristic width ofthe field-field correlation function

Rh00 id0,

withthe scattered electromagnetic field. Therefore, onecould expect an even stronger pulse compression if the

bandwidth was larger, or the correlation frequencysmaller. Given the dimensions of the cavity, the Heisen-berg time (i.e., the inverse of the mean distance betweeneigenmodes) is tH 80 s. But the characteristic absorp-tion time is ta 3:6 s because of the attenuation dueto the skin effect on the walls; therefore, the modes arenot resolved and the correlation frequency is deter-mined by ta rather than tH. Taking 2 MHz and 1=ta 280 kHzleads to a predicted pea k-to-noiseratio roughly equal to 3, comparable to our experimentalresults.

The experiment shows that time reversal is able tocompensate for multiple reverberation and recreate a

short electromagnetic pulse at the source. But t here ismore to it: we have also verified that the amplitude ofthe recreated signal is stronger at antenna A than any-where else in the cavity [Figs. 3(b) and 3(b0)]; i.e., thetime-reversed wave is spatially focused. Th is is consistentwith past experiments using ultrasound in reverberant orstrongly scattering media that proved that even with aone-channel t ime-reversal device, the pulse is sharplyfocused at the source and only at the source, the noiselevel surrounding t he source being, once again, controlledby

=

p [2]. Here, the role of reverberation as well as

the large frequency band must be emphasized. Since wehave only a one-channel t ime reversal, no focusing would

occur in free space no matter how large the bandwidth; ina reverberating medium, no focusing would occur eitherif the bandwidth was too narrow, as is the case in classicalphase-conjugation techniques. Indeed, imagine a singletime-reversal antenna trying to focus a pure monochro-matic wave on some receiver; the phase-conjugated wavehas no reason at all to be focused on this point since theantenna sends back only a sinusoidal spherical wavethrough the medium. But if the frequency bandwidth is much larger than the correlation frequency ,the spectral components of the reverberated field attwo frequencies apar t by more t han a re decorrelated:there are roughly =decorrelated components in thespectrum of the reverberated signals. W hen we timereverse (i.e., phase conjugate coherently a ll along thebandwidth, and not just at one frequency) all these com-ponents, they add up in phase at t 0 at the receiverposition, because all the phases have been set back tozero a ll a long the bandwidth. Thus, the a mplitude at thisposition and at this time increases as =, whereasoutside the receiver position (and outside t 0) the vari-

ous frequency components add up incoherently and their

sum rises as=

p . On the whole, the peak-to-noise

ratio increases as=

p as the bandwidth is enlarged.

It explains why a time-reversal experiment is pos-sible even with one single time-reversal channel [2,7,9],as long as the frequency bandwidth is sufficientlylarger than the correlation frequency of the propagationmedium.

We have presented the first time-reversal experimentfor electromagnetic waves in the GHz domain. The ex-periments were carried out in a reverberating cavity. Toavoid digitizing the radio signals at GHz frequencies, wehave time reversed the baseband signals and phase con-

jugated the wave car rier. Than ks to reverberation, theresulting signal at the source is compressed in time andin space with a signal-to-noise ratio depending on theratio of the bandwidth to the frequency correlation of themedium. In the future, a larger bandwidth has to be usedto improve both spatial and temporal focusing. Thisseems to be possible with the emergence of ultrawidebandelectromagnetics components. Thus time-reversal focus-

ing could have promising applications in the field ofwideband wireless communications in complex reverber-ant environments.

The authors wish to acknowledge the Departement deRecherche en Electromagnetisme, Laboratoire Signaux etSystemes, Supelec, Gif-sur-Yvette, France (www.lss.supelec.fr), and particularly A. Azoulay and V.Monebhurrun for having let us use their reverberantcavity, as well as the Conseil Regional dIle deFrance and the Conseil Departemental de lEssonnewho financed the cavity. This work is a part of theresearch projects developed within the Groupement deRecherche ImCoDe (GDR 2253, CNRS, http://lpm2c.grenoble.cnrs.fr/IMCODE/IMCODE.html).

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Geosci. Remote Sens. 41, 1813 (2003).[4] D. H. Chambers and J. G. Berry man, Phys. Rev. Lett.92,

023902 (2004).[5] A. Derode, A. Tourin, J. de Rosny, M. Tanter, S. Yon, a nd

M. Fink, Phys. Rev. Lett. 90, 014301 (2003).[6] A. Derode, P. Roux, and M. Fink, Phys. Rev. Lett. 75,

4206 (1995).[7] P. Blomgren, G. Papanicolaou, a nd H. Zhao, J. Acoust.

Soc. Am. 11, 230 (2002).[8] D. M. Pepper, Opt. Eng. (Bellingham, Wash.), 21, 156

(1982); Y. Chang, H. R. Fetterman, I. L. Newberg, andS. K. Panaretos, Appl. Phys. Let t.72 , 745 (1998).

[9] C. Draeger and M. Fink, Phys. Rev. Let t.79, 407 (1997).


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