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Full transmission and reflection of waves propagating through a maze of disorder

Benoît Gérardin, Jérôme Laurent, Arnaud Derode, Claire Prada, and Alexandre AubryInstitut Langevin, ESPCI ParisTech, PSL Research University, CNRS UMR 7587,

Université Denis Diderot - Paris 7, 1 rue Jussieu, 75005 Paris, France(Dated: April 9, 2014)

Multiple scattering of waves in disordered media is often seen as a nightmare whether it be for detec-tion, imaging or communication purposes. Thus, controlling wave propagation through scattering mediais of fundamental interest in many areas, ranging from optics1,2 or acoustics3,4 to medical imaging5,6 ortelecommunications7. Thirty years ago, theorists showed that a properly designed combination of incidentwaves could be fully transmitted through (or reflected by) a disordered medium, based on the existence ofpropagation channels which are essentially either closed or open (bimodal law)8–11. Although this spectac-ular prediction has attracted a great deal of attention, it has never been confirmed experimentally12–16. Inthis letter, we study elastic waves in a disordered waveguide and present the first experimental evidenceof the bimodal law. Full transmission and reflection are achieved. The wave-field is monitored by laserinterferometry and highlights spectacular interference effects within the medium.

Light travelling through thick clouds, electrons conduct-ing through metals or seismic waves in the earth crust areall examples of waves propagating through disordered mate-rials. Energy transport by waves undergoing strong scatteringis usually well described by diffusion theory. However, thisclassical picture neglects interference effects that may resistthe influence of disorder. Interference gives rise to fascinat-ing phenomena in mesoscopic physics. On the one hand, itcan slow down and eventually stop the diffusion process, giv-ing rise to Anderson localization17,18. On the other hand, itcan also help waves to find a way through a maze of disor-der19. Actually, a properly designed combination of incidentwaves can be completely transmitted through a strongly scat-tering medium, as suggested by Dorokhov and others morethan twenty years ago8–11. This prediction has recently re-ceived a great deal of attention mostly due to the emergenceof wave-front shaping techniques in optics2.

In order to address the open channels (i.e., to achieve fullenergy transmission) across a disordered wave guide, one hasto perform a complete measurement of the scattering matrixS [11]. The S-matrix relates the input and output of themedium11. It fully describes wave propagation across a scat-tering medium. It can be generally divided into blocks con-taining transmission and reflection matrices,t andr, with acertain numberN of input and output channels. Initially, ran-dom matrix theory (RMT) has been successfully applied tothe transport of electrons through chaotic systems and disor-dered wires11. However, the confrontation between theory andexperiment has remained quite restrictive since specific inputelectron states cannot be addressed in practice. On the con-trary, a coherent control of the incident wave-field is possiblein classical wave physics. Several works have demontrated theability of measuring the S-matrix, or at least some of its sub-spaces, in disordered media, whether it be in acoutics13,14,20,electromagnetism12,21 or optics15,16,22,23.

The existence of open channels has been revealed by inves-tigating the eigenvaluesTi of the Hermitian matrixtt†. The-oretically, their distribution should follow a bimodal law8,9,11,exhibiting two peaks. The highest one, aroundTi ∼ 0, corre-spond to closed (i.e strongly reflected) eigenchannels. At theother end of the spectrum (Ti ∼ 1), there areg open eigen-

channels.g = Nl∗/L is the dimensionless conductance24,L is the sample thickness, andl∗ is the transport mean freepath. By exciting selectively open or closed channels, a nearlycomplete transmission25, reflection or absorption26 of wavescan be achieved. It literally means that a designed wave-frontcan be fully transmitted or, on the contrary, fully reflectedbya scattering medium, which is in total contradiction with theclassical diffusion picture. Nevertheless, these remarkable in-terference effects have never been observed experimentally sofar. Indeed, the bimodal distribution relies on the conservationof energy (i.e S is a unitary matrix). In other words, all thechannels should be addressed at the input and measured at theoutput27. Optical experiments are limited by the finite numer-ical aperture of the illumination and detection systems whichimplies an incomplete angular coverage of the input and out-put channels28. In acoustics or electromagnetism, the spatialsampling of measurements has not been sufficient to have ac-cess to the fullS−matrix so far12–14. In this Letter, we presentexperimental measurements of the fullS-matrix across a dis-ordered elastic wave guide. To that aim, laser-ultrasonic tech-niques have been used in order to obtain a satisfying spatialsampling of the field at the input and output of the scatteringmedium. The unitarity of theS−matrix is investigated andthe eigenvalues of the transmission matrix are shown to followthe expected bimodal distribution. Moreover, full experimen-tal transmission and reflection of waves propagating throughdisorder are achieved for the first time. The wave-fields asso-ciated to the open and closed channels are monitored withinthe scattering medium by laser interferometry to highlightthespectacular interference effects operating in each case.

We study here the propagation of flexural waves acrossa duralumin plate of dimension500 × 40 × 0.5 mm3 (seeFig. 1). Duralumin is an aluminum alloy known for its weakabsorption at ultrasonic frequencies (α < 0.1 dB.m−1). Thefrequency range of interest spans from 0.32 to 0.37 MHz(∆f = 0.05 MHz). The corresponding wavelengthλ is 3.5mm. Therefore, we have access toN = 2W/λ ∼ 22 inde-pendent channels, withW the width of the elastic plate. Thehomogeneous plate is a waveguide in which randomness is in-troduced by drilling circular holes with diameter 1.5 mm andconcentration 11 cm−2, distributed over a thicknessL = 20

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FIG. 1: Measuring the scattering matrix with laser-ultrasonic techniques. TheS−matrix is measured in the time-domain between twoarrays of points placed on the left and right sides of the disordered slab. The array pitch is0.8 mm (i.e< λ/2) which guarantees a satisfyingspatial sampling of the wave field. Flexural waves are generated on each point by a pulsed laservia thermo-elastic conversion over a focal spotof 1 mm2. The normal component of the plate vibration is measured with an interferometric optical probe. The laser source and theprobe areboth mounted on 2D translation stages.

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FIG. 2: Revealing the existence of open and closed eigenchannels. a, Real part of theS-matrix measured atf = 0.36 MHz. The blacklines delimit transmission and reflection matrices as depicted in Eq. 1.b, Eigenvaluessi (red dots) andsi (blue dots) of the measured andnormalized scattering matrices,S andS, respectively. The eigenvalues are displayed in the complex plane. The black continuous line denotesthe unit circle. c-d, Transmission eigenvalue histograms,ρ(T ) and ρ(T ), averaged over the frequency bandwidth. Both distributions arecompared to the bimodal lawρb(T ) (red continuous line, Eq. 3).

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mm. The dimensionless conductanceg of the disordered slabis 7.5 which yields a ratioL/l∗ ∼ 3.

TheS-matrix associated to the disordered slab is measuredwith the laser-ultrasonic set-up described in Fig. 1 and thepro-cedure explained in the Method. TheS-matrix has the follow-ing block structure

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(

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(1)

with N × N reflection matricesr andr′ (reflection from leftto left and from right to right) and transmission matricest andt′ (transmission from left to right and from right to left). Each

matrix is expressed in the basis of the modes of the homo-geneous plate. These eigenmodes and their eigenfrequencieshave been determined theoretically using the thin elastic platetheory29,30. They are renormalized such that each of them car-ries unit energy flux across the plate section.

Fig. 2(a) displays an example of matrixS recorded at fre-quencyf = 0.36 MHz. Despite its overall random appear-ance, one can see the residual ballistic wave-front that slightlyemerges along the diagonal of the transmission matrices. Thematrix S also exhibits long-range correlations that will ac-count for the bimodal behavior of the transmission/reflectionmatrices. Theoretically, energy conservation would implythatS is unitary. In other words, its eigenvalues should be dis-tributed along the unit circle in the complex plane. The eigen-valuessi of theS-matrix atf = 0.36 MHz are displayed inthe complex plane in Fig. 2(b). The dispersion of these eigen-values around the unit circle questions the validity of the en-ergy conservation assumption. Temperature fluctuations dur-ing the experiment and experimental noise mainly account forthe non-unitarity ofS. Absorption and radiative losses (∼ 1dB.m−1) can be actually neglected compared to those mea-surement issues. In the following, we compensate for theseundesirable effects by building a virtual scattering matrix S

with the same eigenspaces asS but with normalized eigenval-ues (see Fig. 2(b)), such that

si = si/ |si| , for i = 1, · · · , N (2)

The phase information is thus unchanged and only the ampli-tude is normalized to meet the energy conservation require-ment. We will show that this normalization based on energyconservation is essential to observe the open channelsi.e., apeak close to 1 in the distribution of theTi. We first focus onthe statistics of the transmission eigenvaluesT computed fromS and S. Their distributions,ρ(T ) and ρ(T ), are estimatedby averaging their histograms over the frequency bandwidth.Figs. 2(c)-(d) show the comparison between these distribu-tions and the bimodal lawρb which is theoretically expectedin the diffusive regime11,

ρb(T ) =g

2T√1− T

(3)

The measured eigenvalue distributionρ(T ) is in correct agree-ment with the bimodal lawρb(T ) with a large peak aroundT = 0 associated to the closed channels and a smaller peak

aroundT = 1 associated to the open channels (Fig. 2(c)).Nevertheless, some channels exhibit a transmission coefficientsuperior to 1, which violates energy conservation. This isexplained by the non-unitarity ofS pointed out in Fig. 2(b).On the contrary, after normalization ofS (Eq. 2), all trans-mission eigenvalues are repelled below 1 and the eigenvaluedistribution ρ(T ) closely follows the expected bimodal law(Fig. 2(d)). This confirms that the unitarity ofS is decisiveand that experimental noise can prevent from recovering thebimodal law experimentally.

Whereas the eigenvalues oftt† yield the transmission coef-

ficients of each eigenchannel, the corresponding eigenvectorprovide the combination of incident modes that allow to excitethis specific channel. Hence, the wave-field associated to eacheigenchannel can be measured by backpropagating the cor-responding eigenvector. To that aim, the scattering mediumis scanned with the interferometric optical probe. As a ref-erence, the wave-field induced by an incident plane-wave isshown in Fig. 3(a). Fig. 3(e) plots the corresponding inten-sity averaged along the plate section (y−axis) as a function ofthe depthx. Fig. 3(b) displays the wave-field associated to aclosed eigenchannel (T ∼ 0). The wave is fully reflected backto the left. This shows that the incoming wavefield has beensuccessfully tailored to fit the randomness of the waveguide,making it a nearly perfectly reflecting medium. Beyond onetransport mean-free path, the intensity decreases very rapidlyin agreement with numerical simulations16. Figs. 3(c)-(d) dis-play the propagation of open eigenchannels (T ∼ 1) deducedfrom the measured and normalized matricesS andS, respec-tively. Both wave-fields clearly show the contructive inter-ferences that help the wave to find its way through the mazeof disorder. The corresponding intensity profiles are showninFigs.3(g)-(h). In both cases, the field intensity increasesinsidethe medium. For the eigenchannel derived from theS-matrix,the intensity measured at the output of the scattering mediumis smaller than at the input (Fig. 3(g)). Experimental noiseintheS−matrix prevents from addressing a fully opened chan-nel. On the contrary, Fig. 3(h) illustrates how nicely the nor-malizedS-matrix gives access to a fully open eigenchannelwith equal intensities at the input and output of the scatter-ing medium. The incoming wavefield has been successfullydesigned to make the scattering slab completely transparent.

This experimental study demonstrates for the first time theexistence of fully open and closed eigenchannels through ascattering medium. In agreement with theoretical predic-tions, transmission eigenvalues across a disordered system areshown to follow a bimodal law. The wave-field associated toeach eigenchannel is measured and illustrates the remarkableinterference mechanisms induced by disorder. From a funda-mental point of view, such measurements will allow to checka whole set of RMT predictions that could not have been con-fronted to experiment so far. The transition towards Andersonlocalization should lead to an extinction of fully opened eigen-channels8,9,11 and the occurence of necklace states19. From amore practical point of view, this work paves the way towardsan optimized control of wave-fronts through scattering mediain all fields of wave physics, whether it be for communication,imaging, focusing, absorbing, or lasing purposes.

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FIG. 3: Probing the wave-fields of open and closed eigenchannels within the scattering medium. Absolute value of the wave-field in thescattering medium atf = 0.36 MHz associated to,a, an incident plane wave,b, a closed eigenchannel,c, an open eigenchannel deduced fromthe measuredS-matrix and,d, an open eigenchannel deduced from the normalizedS-matrix. The corresponding intensities averaged over thewave guide section (y-axis) are shown versus depthx in bottom panelse-h. They are all normalized by the intensity at the plane of sources(x = 0).

Methods

S-matrix measurement: Experimental procedure. The firststep of the experiment consists in measuring the impulse responsesbetween two arrays of points placed on the left and right sides ofthe disordered slab (see Fig. 1). These two arrays are placed5mm away from the disordered slab. The array pitch is0.8 mm (i.e< λ/2) which guarantees a satisfying spatial sampling of the wavefield. Flexural waves are generated in the thermoelastic regime bya pumped diode Nd:YAG laser (THALES Diva II) providing pulseshaving a20 ns duration and2.5 mJ of energy. The out-of-planecomponent of the local vibration of the plate is measured with aheterodyne interferometer. This probe is sensitive to the phase shiftalong the path of the optical probe beam. The calibration factor formechanical displacement normal to the surface (10 nm/V) was con-stant over the detection bandwidth (100 kHz - 400 kHz). Signalsdetected by the optical probe were fed into a digital sampling oscillo-scope and transferred to a computer. The impulse responses betweeneach point of the same array (left and right) form the time-dependentreflection matrices (r andr′, respectively). The set of impulse re-sponses between the two arrays yield the time-dependent transmis-sion matricest (from left to right) andt′ (from right to left). Fromthese four matrices, one can build theS−matrix in apoint-to-pointbasis (Eq. 1). A temporal Fourier transform ofS is then performedover a time range∆t = 120 µs that excludes the echoes due to reflec-tions on the ends of the plate. Frequency components that arespacedby more than the correlation frequencyδf give rise to uncorrelatedspeckle patterns. In our disordered sample, we have measured δf ∼

0.017 MHz. Hence, the numberNf = ∆f/δf of independent scat-tering matricesS that are obtained over the frequency bandwidth is3. The correlation frequency also yields an estimation for the Thou-less time (τD ∼ 1/δf ∼ 60 µs), i.e the mean time it takes for awave to cross the sample through its zigzag motion. We check thatτD ∼ ∆t/2 and that most of the energy has escaped from the samplewhen the measurement is stopped. The next step of the experimen-tal procedure consists in decomposing theS-matrices in the basis ofthe flexural modes of the homogeneous plate. These eigenmodes andtheir eigenfrequencies have been determined theoretically using thethin elastic plate theory29,30. They are normalized such that each ofthem carries unit energy flux across the plate section. At last, theS−matrix is normalized by the quadratic mean of its eigenvalues.

Measurement of the wave-field associated to each eigenchan-nel. Impulse responses are measured between the line of sourcesand a grid of points that maps the scattering medium, following thesame procedure as described previously for theS-matrix measure-ment. The grid pitch is 1 mm. The whole set of impulse responsesforms a transmission matrixk(t). A temporal Fourier transform ofkis performed. The wave-field associated to an incident planewave isobtained by performing the product between the eigenvector[1 · · · 1]

and the matrixk(f). For the wave-field associated to each eigen-channel, the columns ofk are first decomposed in the basis of theplate modes. The wave-field associated to an eigenchannel corre-sponds to the product between the corresponding eigenvector ui oftt

† and the matrixk.

5

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Author contributions

A.A. initiated and supervised the project. B.G., J.L. andC.P. conceived the experiments. B.G. performed the experi-ments. B.G. and A.A. analyzed the experiments. All authorsdiscussed the results and wrote the manuscript.

Acknowledgements

The authors wish to thank M. Davy, G. Lerosey, S. Rotter,S. Popoff, and A. Goetschy for fruitful discussions, as wellas A. Souilah for his technical help. The authors are grate-ful for funding provided by LABEX WIFI (Laboratory of Ex-cellence within the French Program Investments for the Fu-ture, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*).B.G. acknowledges financial support from the French “Direc-tion Générale de l’Armement”(DGA).