Particle-like wave packets in complex scattering systems

Benoît Gérardin,1 Jérôme Laurent,1 Philipp Ambichl,2 Claire Prada,1 Stefan Rotter,2 and Alexandre Aubry1, ∗

1ESPCI Paris, PSL Research University, CNRS, Institut Langevin, UMR 7587, 1 rue Jussieu, F-75005 Paris, France2Institute for Theoretical Physics, Vienna University of Technology (TU Wien),

Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria(Dated: September 17, 2018)

A wave packet undergoes a strong spatial and temporal dispersion while propagating through a complexmedium. This wave scattering is often seen as a nightmare in wave physics whether it be for focusing, imagingor communication purposes. Controlling wave propagation through complex systems is thus of fundamentalinterest in many areas, ranging from optics or acoustics to medical imaging or telecommunications. Here, westudy the propagation of elastic waves in a cavity and a disordered waveguide by means of laser interferometry.From the direct experimental access to the time-delay matrix of these systems, we demonstrate the existence ofparticle-like wave packets that remain focused in time and space throughout their complex trajectory. Due totheir limited dispersion, their selective excitation will be crucially relevant for all applications involving selectivewave focusing and efficient information transfer through complex media.

I. INTRODUCTION

Waves propagating in complex media typically undergodiffraction and multiple scattering at all the inhomogeneitiesthey encounter. As a consequence, a wave packet suffersfrom strong temporal and spatial dispersion while propagatingthrough a scattering medium. Eventually, the incident wave isconverted into a diffuse halo that gives rise to a complicatedinterference pattern (speckle) at the output of the medium. Al-beit complex, this wave-field remains, however, deterministic.By actively shaping the wave-field at the input, one can ma-nipulate the interference between all the scattering paths thatthe wave can follow. On the one hand, this insight has givenrise to spectacular focusing schemes in which scattering en-ables - rather than impedes - wave focusing and pulse com-pression [1–9]. On the other hand, it can lead to an optimizedcontrol of wave transport [10–13]. A designed wave-front can,e.g., be completely transmitted/reflected at will [14, 15] as aresult of a multiple scattering interference that is intrinsicallynarrowband [16]. Here we will aim at the more challenginggoal to generate states that are fully transmitted/reflected, yetvery robust in a broadband spectral range. As we will demon-strate explicitly, this goal can be reached by way of highlycollimated scattering states that are concentrated along indi-vidual particle-like bouncing patterns inside the medium [17].By avoiding the multi-path interference associated with con-ventional scattering states, these wave beams also avoid thefrequency sensitivity associated with this interference. As weshall see, particle-like scattering states give rise, in the timedomain, to wave packets that remain focused in time andspace throughout their trajectory within the medium. Thiscrucial feature makes these states uniquely suited for manyapplications in a variety of fields, ranging from high intensityfocused ultrasound [18, 19] or underwater acoustics [20, 21]to endoscopic microscopy [22–24], fibre optics [25–29] ortelecommunications [30–32].

∗ alexandre.aubry@espci.fr

The key aspect of our experimental study is to demonstratethat these particle-like wave packets can be created just basedon the information stored in the scattering matrix [17]. Thishighly dimension S-matrix relates any arbitrary wave-field atthe input to the output of the scattering medium, and in prin-ciple, allows the reconstruction or prediction of either. It fullydescribes wave propagation across a scattering medium andcan meanwhile be routinely measured not only in acoustics[33, 34], but also in microwave technology [13, 35] and optics[6, 12]. The sub-blocks of the scattering matrix contain thecomplex-valued transmission (t, t′) and reflection (r, r′) ma-trices with a certain number N of input and output channels,

S =

(r t′

t r′

). (1)

To describe the statistical properties of S for wave trans-port through complex media, random matrix theory (RMT)has been very successful [36]. One of the striking result ofRMT is the universal bimodal distribution followed by thetransmission eigenvalues T of tt† [37] through diffusive me-dia [36, 38, 39] or chaotic cavities [40, 41]. In contradic-tion with a classical diffusion or chaotic picture, a substantialfraction of propagation channels are found to be essentiallyclosed (T ∼ 0) or open (T ∼ 1). Going beyond such astatistical approach, Rotter et al. [17] recently showed howa system-specific combination of fully open or fully closedchannels may lead to scattering states that follow the particle-like bouncing pattern of a classical trajectory throughout theentire scattering process. Such particle-like scattering stateswith transmission close to 1 or 0 are eigenstates of the Wigner-Smith time-delay matrix:

Q = − i

2πS†∂fS, (2)

where ∂f denotes the derivative with respect to the frequencyf . Originally introduced by Wigner in nuclear scattering the-ory [42] and extended by Smith to multichannel scatteringproblems [43], the Q-matrix generally describes the time thatthe incoming wave accumulates due to the scattering process:each eigenvalue yields the time delay of the associated scat-tering state. Compared to a mere study of the S-matrix, the

arX

iv:1

602.

0581

2v2

[ph

ysic

s.op

tics]

26

Jul 2

016

2

Q-matrix provides an elegant and powerful tool to harnessthe dispersion properties of a complex medium. In this arti-cle, we show, in particular, how a time-delay eigenstate canbe engineered to be “particle-like” not only in its station-ary wave function patterns [17], but also in the sense thata non-dispersive wave packet can be propagated along thecorresponding particle-like bouncing pattern. The associatedeigenvalue of Q then corresponds to the propagation-time ofthis wave packet.

Our experimental setup consists of an elastic cavity and adisordered elastic wave guide at ultrasound frequencies [seeFig. 1]. In a first step we measure the entries of the S- and Q-matrices over a large bandwidth using laser-ultrasonic tech-niques. The eigenvalues of the transmission matrix are shownto follow the expected bimodal distributions in both configura-tions. The wave-fields associated with the open/closed chan-nels are monitored within each system in the time domain bylaser interferometry. Not surprisingly, they are shown to bestrongly dispersive as they combine various path trajectoriesand thus many interfering scattering phases. To reduce thisdispersion and to lift the degeneracy among the open/closedchannels, we consider the eigenstates of the Q-matrix thathave a well-defined time-delay, corresponding to a wave thatfollows a single path trajectory. In transmission, a one-to-oneassociation is found between time-delay eigenstates and ray-path trajectories. The corresponding wave functions are im-aged in the time domain by laser interferometry. The synthe-sized wave packets are shown to follow particle-like trajecto-ries along which the temporal spreading of the incident pulseis minimal. In reflection, the Q-matrix yields the collimatedwave-fronts that focus selectively on each scatterer of a multi-target medium. Contrary to alternative approaches based ontime-reversal techniques [44–47], the discrimination betweenseveral targets is not based on their reflectivity but on their po-sition. The eigenvalues of Q directly yield the time-of-flightof the pulsed echoes reflected by each scatterer.

II. EXPERIMENTAL RESULTS

A. Revealing the open and closed channels in a cavity

The waves we excite and measure are flexural waves ina duralumin plate of dimension 500× 40× 0.5 mm3 [seeFig. 1].The frequency range of interest spans from 0.23 to0.37 MHz (∆f = 0.14 MHz) with a corresponding aver-age wavelength λ of 3.5 mm. We thus have access to N =2W/λ ∼ 22 independent channels, W being the width of theelastic plate. Two complex scattering systems are built fromthe homogeneous plate: (i) a regular cavity formed by cuttingthe plate over 20 mm on both sides of the plate [see Fig. 1(a)]and (ii) a scattering slab obtained by drilling several circularholes in the plate [see Fig. 1(b)]. The thickness L of eachsystem is 45 mm and 52 mm, respectively.

The S-matrix is measured for each system with the laser-ultrasonic set-up described in Fig. 1, following the procedureexplained in the Appendix A. Transmission and reflection ma-trices are expressed in the basis of the modes of the homoge-

(a)

(b)

yxz

yxz

20 ns

Lasersource

Heterodyneinterferometert

L

L

W

W

FIG. 1. The two systems under investigation consist (a) of a regularcavity and (b) of a disordered slab machined in an elastic plate. Inboth configurations, the S-matrix is measured in the time-domainbetween two arrays of points placed on the left and right sides ofthe system (see Appendix A). Flexural waves are generated on eachpoint by a pulsed laser via thermo-elastic conversion over a focal spotof 1 mm2. The normal component of the plate vibration is measuredwith an interferometric optical probe. The laser source and the probeare both mounted on 2D translation stages.

(a)

(d)

(b)

(c)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

10 20 10 20

10

20

10

20 −1

−0.5

0

0.5

1

Left channel Right channelEmission

Rec

eptio

nRig

ht c

hann

elLe

ft c

hann

el

evenmodes

0 0.5 1T

0

1

2

3

(T)

oddmodes

FIG. 2. (a), Real part of the S-matrix measured in the cavity[Fig. 1(a)] at f0 = 0.30 MHz. The black lines delimit transmis-sion and reflection matrices as depicted in Eq. (1). (b), Transmissioneigenvalue histogram, ρ (T ), averaged over the frequency bandwidthf = 0.23− 0.37 MHz. The distribution is compared to the bimodallaw ρb (T ) [red continuous line, Eq. (3)]. (c)-(d), Absolute values ofthe monochromatic wave-fields (f0 = 0.30 MHz) associated withthe two most open channels.

3

neous plate [15]. These eigenmodes and their eigenfrequen-cies have been determined theoretically using the thin elasticplate theory[48, 49]. They are renormalized such that eachof them carries unit energy flux across the plate section[15].Figure 2(a) displays an example of an S-matrix recorded atthe central frequency f0 = 0.30 MHz for the cavity. Mostof the energy emerges along the main diagonal and two sub-diagonals [50] of the reflection/transmission matrices. Thesereflection and transmission matrix elements correspond tospecular reflection of each mode on the cavity boundaries andto the ballistic transmission of the incident wave-front, respec-tively.

We first focus on the statistics of the transmission eigenval-ues Tl computed from the measured t-matrix (see AppendixB). Their distribution, ρ(T ) is estimated by averaging the cor-responding histograms over the frequency bandwidth. Fig-ure 2(b) shows the comparison between the distribution mea-sured in the rectangular cavity and the bimodal law ρb whichis theoretically expected in the chaotic regime[40, 41],

ρb (T ) =1

π√T (1− T )

(3)

Even though our system is not chaotic, but exhibits regulardynamics, a good agreement is found between the measuredeigenvalue distribution and the RMT predictions, confirmingprevious numerical studies [51]. A similar bimodal distribu-tion of transmission eigenvalues is obtained in the disorderedplate, as shown in the Supplemental Material [52].

Whereas the eigenvalues Tl of tt† yield the transmissioncoefficients of each eigenchannel, the corresponding eigen-vectors provide the combination of incident modes that allowto excite this specific channel. Hence, the wave-field associ-ated with each eigenchannel can be measured by propagatingthe corresponding eigenvector. To that aim, the whole systemis scanned with the interferometric optical probe. A set of im-pulse responses is measured between the line of sources at theinput and a grid of points that maps the medium. The wavefunction associated with a scattering state is then deduced bya coherent superposition of these responses weighted by theamplitude and phase of the eigenvector at the input (see Ap-pendix C). Hence all the wave functions displayed in this ar-ticle are only composed of experimentally measured data anddo not imply any theoretical calculation or numerical simula-tion.

Figures 2(c) and 2(d) display the wave-field associated withthe two most open eigenchannels (Tl ∼ 1) of the cavity. Al-though such open channels allow a full transmission of the in-cident energy, they do not show a clear correspondence witha particular path trajectory. The same observation holds inthe disordered plate [52]. As a consequence, the associatedscattering state undergoes multiple scattering when passingthrough the cavity. Figures 3(a) and 3(b) illustrate this dis-persion by displaying the output temporal signal associatedwith the two open channels shown in Figs. 2(c) and 2(d) (seeAppendix B). Both signals contain several peaks occurring ataltogether three different times of flight. As we will see fur-ther, each peak is associated with a particular path trajectory

and can be addressed independently by means of the Wigner-Smith time-delay matrix.

Nor

mal

ized

inte

nsity

Nor

mal

ized

inte

nsity

Nor

mal

ized

inte

nsity

Nor

mal

ized

inte

nsity 5,7 cm

Nor

mal

ized

inte

nsity

(a)

(b)

(c)

(d)

(e)

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

0 10 20 30 40 50 60 70 80 90 100 1100

0.5

1

τ [µs]

τ [µs]

τ [µs]

τ [µs]

τ [µs]

3

2

1

1 32

2 3

FIG. 3. The time trace of a scattering or time-delay eigenstates iscomputed from the set of t-matrices measured over the frequencyrange f = 0.23 − 0.37 MHz (see Appendix B). The output inten-sity is displayed versus time for, (a)-(b), the two first open channelsdisplayed in Figs. 2(c) and 2(d) and for, (c)-(e), the three time-delayeigenstates displayed in Fig. 4. For each time trace, the differentechoes are labeled with a number 1, 2 or 3 that corresponds respec-tively, to the direct, double or quadruple scattering paths highlightedin Fig. 4.

B. Addressing particle-like scattering states in a cavity

The Wigner-Smith time-delay matrix Q is now investi-gated to generate coherent scattering states from the set ofopen channels. Since Q is Hermitian when derived froma unitary S-matrix [Eq. (2)], the time-delay eigenstates qin

m

form an orthogonal and complete set of states, to each ofwhich a real proper delay time τm can be assigned, such thatQqin

m = τmqinm . In general, qin

m is a 2N -dimensional eigen-vector which implies an injection from both the left and theright leads of the system. However, among this set of time-delay eigenstates, a subset features an incoming flux fromonly one lead that also exits through just one of the leads.These are exactly the desired states that belong to the sub-space of open or closed channels and display trajectory-likewave function patterns.

As was shown by Rotter et al. [17], the above argumentscan be translated into a straightforward operational proce-dure (see Appendix B), which we apply here to identify theparticle-like scattering states among the time-delay eigen-states of the measured Q-matrix. The litmus test for this pro-cedure in the present context will be to show that the three

4

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

(a)

(c)(b) 0

5

10

15

20

25

30

35

40

45

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIG. 4. Absolute value of the wave-field associated with the threeparticle-like scattering states derived from the matrix Q. The corre-sponding time delays, (a), τm = 20 µs, (b), τm = 30 µs and, (c),τm = 59 µs, nicely match with the run-times of the correspondingclassical trajectories shown in the inset (top left).

different time traces that are identifiable in the open transmis-sion channels [see Figs. 3(a) and 3(b)] can now be individu-ally addressed through an associated particle-like state. Theresults we obtain for the cavity geometry [Fig. 1(a)] fully con-firm our first successful implementation of particle-like scat-tering states: The propagation of the states we obtain from ourprocedure yields monochromatic wave states that are clearlyconcentrated on individual bouncing patterns [see Fig. 4].Whereas Fig. 4(a) corresponds to the direct path between theinput and output leads, Figs. 4(b) and 4(c) display a more com-plex trajectory with two and four reflections on the boundariesof the cavity, respectively. The associated time-delays τm docorrespond to the run-times of a particle that would follow thesame trajectory at the group velocity vg ∼ 2.6 mm/µs of theflexural wave [53]. Their transmission coefficients |tm| areequal to 0.90, 0.95 and 0.85, respectively, meaning that theyare almost fully transmitted through the cavity.

The time trace associated with each particle-like scatteringstate is computed from the frequency-dependent t-matrix (seeAppendix B). The result is displayed in Figs. 3(c), 3(d) and3(e). Unlike the open transmission channels studied above,each particle-like state gives rise to a single pulse that arrivesat the output temporally unscattered at time τ = τm. Fig-ure 3 also shows that each temporal peak in the time trace ofthe open channels can be attributed to a particular path trajec-tory. We may thus conclude that the open channel displayedin Fig. 2(c) is mainly associated with the double and quadru-ple scattering paths displayed in Figs. 4(b) and 4(c). The openchannel displayed in Fig. 2(d) consists of a linear combinationof the paths displayed in Fig.4. This association is also con-firmed by explicitly analyzing the vectorial decomposition ofthe particle-like state in terms of the transmission eigenchan-nel basis.

The frequency dependence of the particle-like scatteringstates is investigated in the Supplemental Material [52]. Theyare shown to be stable over the frequency ranges f = 0.2−0.6MHz [Fig. 4(a)], f = 0.3 − 0.6 MHz [Fig. 4(b)] and f =0.2− 0.4 MHz [Fig. 4(c)]. The corresponding bandwidths areat least one order of magnitude larger than the frequency cor-relation width of the transmission matrix coefficients whichis equal to 0.02 MHz [52]. This proves the robustness ofparticle-like states over a broadband spectral range. Given thisnon-dispersive feature, they turn out to be perfect candidatesalso for the formation of minimally dispersive wave packetsin the time domain. To check this conjecture, we investigatehere the spatio-temporal wave functions of these states overthe aforementioned bandwidths (see Appendix B). The prop-agation of the particle-like wave-packets through the cavitycan be visualized in the three first movies of the SupplementalMaterial [52]. Figure 5 displays successive snapshots of thewave-packet synthesized from the particle-like scattering statedisplayed in Fig. 4(c). Quite remarkably, the spatio-temporalfocus of the incident wave-packet is maintained throughout itstrajectory despite the multiple scattering events it undergoes inthe cavity.

C. Lifting the degeneracy of particle-like scattering states

In a next step, we investigate particle-like scattering statesin the disordered wave guide [Fig. 1(b)]. The correspond-ing Q-matrix is measured at the central frequency f0 =0.3 MHz (see Appendix A). Figures 6(a) and 6(b) displaythe monochromatic wave functions associated with two fullytransmitted time-delay eigenstates. Whereas Fig. 6(a) dis-plays the typical features of a particle-like scattering state thatwinds its way through the scatterers, the time-delay eigenstateof Fig. 6(b) is clearly associated with two scattering paths ofidentical length. We thus encounter here a degeneracy in thetime-delay eigenvalues that needs to be lifted by an additionalcriterion, such as by considering well-defined subspaces of themeasured S-matrix [54]. In this instance, the two ray pathscan be discriminated by their different angles of incidence.Correspondingly, we consider two subspaces S′ of the orig-inal S-matrix by keeping either positive or negative anglesof incidence from the left lead (see Appendix D). The cor-responding time-delay matrices lead to two distinct particle-like scattering states displayed in Figs. 6(c) and 6(d). Thetwo scattering paths that were previously mixed in the orig-inal time-delay eigenstate [Fig. 6(b)] are now clearly sepa-rated. The frequency stability of these states is investigated inthe Supplemental Material [52]. They are shown to be stableover the frequency ranges f = 0.2− 0.4 MHz [Fig. 6(c)] andf = 0.2 − 0.5 MHz [Fig. 6(d)]. The corresponding particle-like wave packets are shown in the two last movies of Supple-mental Material [52] where the high quality of their focus inspace and time is immediately apparent.

5

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40X [mm]

Y [m

m]

0 5 10 15 20 25 30 35 40

0.9

1

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

(a) (b) (c) (d)

(e) (f) (g) (h)0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

0

5

10

15

20

25

30

35

40

45

τ=0 µs τ=8 µs τ=16 µs τ=24 µs

τ=32 µs τ=40 µs τ=48 µs τ=56 µs

FIG. 5. A spatio-temporal wave-packet is synthesized from the particle-like scattering state displayed in Fig. 4(c) over the frequency bandwidth0.2− 0.4 MHz. The different subsets (a)-(h) display successive snapshots of its propagation across the cavity versus time.

(a) (b) (c) (d)

(e) (f) (g) (h)

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50Y

[mm

]0 5 10 15 20 25 30 35 40

X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

0.9

1

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

FIG. 6. Absolute value of the wave-field associated with several time-delay/scattering eigenstates of the disordered system at the centralfrequency f0. (a), Transmitted particle-like scattering state (τm = 29 µs, |tm| = 0.84). (b) Degenerate time-delay eigenstate mixing twoparticle-like trajectories (τm = 26 µs, |tm| = 0.91). (c)-(d) Unmixing of the time-delay eigenstate displayed in (b) by considering differentsubspaces of the S-matrix (see Appendix D). Their transmission coefficients |tm| are equal to 0.94 and 0.98, respectively. (e)-(f) Closedchannels deduced from the S-matrix (Tl ∼ 0) showing simultaneous reflections from the scatterers of the disordered wave guide. (g) Reflectedtime-delay eigenstate showing selective focusing on the scatterer no2 (time delay τm = 11 µs, reflection coefficient |rm| = 0.94). (h)Reflected time-delay eigenstate showing selective focusing on the scatterer no3 (time delay τm = 30 µs, reflection coefficient |rm| = 0.88).

D. Revealing time-delay eigenstates in reflection.

Time-delay eigenstates can also result from a suitable com-bination of closed channels. Figures 6(e) and 6(f) displaytwo such closed channels derived from the S-matrix of thedisordered slab. The closed channels combine multiple re-

flections from the holes of the scattering layer. However, theclosed channel displayed in Fig. 6(e) mixes the contributionsfrom the scatterers labeled 1 and 2. Also the closed channeldisplayed in Fig. 6(f) is associated with reflections from al-together three scatterers (2, 3 and 4). A simple eigenvaluedecomposition of rr† or tt† does not allow a discrimination

6

between the scatterers. On the contrary, the analysis of theQ-matrix allows a one-to-one correspondence with each scat-terer based on a time-of-flight discrimination. Figures 6(g)and 6(h) actually display the wave functions associated withtwo reflected time-delay eigenstates. Each of these eigenstatesis associated with a reflection from a single scatterer (2 and 3,respectively). The corresponding time delays τm given in thecaption of Fig. 6 are directly related to the depth z of eachscatterer, such that τm ∼ 2z/vg .

III. DISCUSSION

The first point we would like to emphasize is the relevanceof the time-delay matrix for selective focusing and imaging inmulti-target media. The state-of-the-art approach is the DORTmethod (French acronym for Decomposition of the Time Re-versal Operator). Initially developed for ultrasound [44, 45]and more recently extended to optics [46, 47], this widelyused approach takes advantage of the reflection matrix r tofocus iteratively by time reversal processing on each scattererof a multi-target medium. Mathematically, the time-reversalinvariants can be deduced from the eigenvalue decompositionof the time reversal operator rr† or, equivalently, from the sin-gular value decomposition of r. On the one hand, the eigen-vectors of r should, in principle, allow selective focusing andimaging of each scatterer. On the other hand, the associatedeigenvalue directly yields the scatterer reflectivity. However, aone-to-one association between each eigenstate of r and eachscatterer only exists under a single-scattering approximationand if the scatterers exhibit sufficiently different reflectivities.Figures 6(e) and 6(f) illustrate this limit. Because the holesdisplay similar scattering cross-sections in the disordered slab,closed channels are associated with several scatterers at once.On the contrary, the time-delay matrix allows a discriminationbetween scatterers based on the time-of-flight of the reflectedechoes [see Figs. 6(g) and 6(h)]. Moreover, unlike the DORTmethod, a time-delay analysis also allows to discriminate thesingle scattering paths from multiple scattering events, the lat-ter ones corresponding to longer time-of-flights. Hence, thetime-delay matrix provides an alternative and promising routefor selective focusing and imaging in multi-target media.

A second relevant point to discuss is the nature of trans-mitted time-delay eigenstates in other complex systems. Re-cently, Carpenter et al. [27] and Xiong et al. [28] inves-tigated the group delay operator, i/(2π)t−1∂ft, in a multi-mode optical fiber. The eigenstates of this operator are knownas the principal modes in fiber optics [25, 26]. For a principalmode input, a pulse that is sufficiently narrow-band reachesthe output temporally un-distorted, although it may have beenstrongly scattered and dispersed along the length of propaga-tion. On the contrary, for a particle-like scattering state in-put, the focus of the pulse is not only retrieved at the outputof the scattering medium but maintained throughout its en-tire propagation through the system. This crucial differenceprovides particle-like states with a much broader frequencystability than principal modes, which translates into the possi-bility to send much shorter pulses through these particle-like

scattering channels. Last but not least, we also emphasize thateven though particle-like states are unlikely to occur in diffu-sive scattering media, the time-delay eigenstates are still veryrelevant also in such a strongly disordered context. Considerhere, e.g., that the eigenstates with the longest time delay canbe of interest for energy storage, coherent absorption [55] orlasing [56, 57] purposes. From a more fundamental point-of-view, the trace of the Q- matrix directly provides the densityof states of the scattering medium [58], a quantity that turnsout to be entirely independent of the mean free path in a dis-ordered system [59].

Finally, we would like to stress the impact of our study onother fields of wave physics and its extension to more com-plex geometries. For this experimental proof-of-concept, ameasurement of the wave-field inside the medium was re-quired in order to image the wave functions and prove theirparticle-like feature. However, such a sophisticated protocolis not needed to physically address particle-like wave packets.A simple measurement of the scattering matrix (or a subpartof it) at neighboring frequencies [27] yields the time-delaymatrix from which particle-like state inputs can be extracted.Such a measurement can be routinely performed through 3Dscattering media whether it be in optics [6, 12], in the mi-crowave regime [13, 35] or in acoustics [33, 34]. As to thegeneration of particle-like wave packets, multi-element tech-nology is a powerful tool for the coherent control of acous-tic waves and electromagnetic waves [60]. Moreover, recentprogress in optical manipulation techniques now allows for aprecise spatial and temporal control of light at the input ofa complex medium [60]. Hence there is no obstacle for theexperimental implementation of particle-like wave packets inother fields of wave physics. At last, we would like to stressthe fact that in our experimental implementation we are inthe limit of only a few participating modes with a wavelengththat is comparable to the spatial scales of the system. In thislimit the implementation of particle-like states is truly non-trivial since interference and diffraction dominates the scat-tering process as a whole. When transferring the concept tothe optical domain one may easily reach the geometric opticslimit where the wavelength is much shorter than most spa-tial scales of the system and particle-like states may in fact bemuch easier to implement in corresponding complex opticalmedia.

IV. CONCLUSION

In summary, we experimentally implemented particle-likescattering states in complex scattering systems. Based onan experimentally determined time-delay matrix, we havedemonstrated the existence of wave packets that followparticle-like bouncing patterns in transmission through or inreflection from a complex scattering landscape. Strikingly,these wave packets have been shown to remain focused in timeand space throughout their trajectory within the medium. Weare convinced that the superior properties of these states interms of frequency stability and spatial focus will make themvery attractive for many applications of wave physics, rang-

7

ing from focusing to imaging or communication purposes. Intransmission, the efficiency of these states in terms of infor-mation transfer as well as their focused in- and output pro-file will be relevant. In reflection, selective focusing basedon a time-of-flight discrimination will be a powerful tool toovercome aberration and multiple scattering in detection andimaging problems.

V. ACKNOWLEDGMENTS

The authors wish to thank A. Derode for fruitful discus-sions and advices. The authors are grateful for funding pro-vided by LABEX WIFI (Laboratory of Excellence within theFrench Program Investments for the Future, ANR-10-LABX-24 and ANR-10-IDEX-0001-02 PSL*). B.G. acknowledgesfinancial support from the French “Direction Générale del’Armement”(DGA). P.A. and S.R. are supported by the Aus-trian Science Fund (FWF) through Projects NextLite F49-10and I 1142-N27 (GePartWave).

Appendix A: Experimental procedure

The first step of the experiment consists in measuring theimpulse responses between two arrays of points placed on theleft and right sides of the disordered slab [see Fig. 1]. Thesetwo arrays are placed 5 mm away from the disordered slab.The array pitch is 0.8 mm (i.e < λ/2) which guarantees asatisfying spatial sampling of the wave field. Flexural wavesare generated in the thermoelastic regime by a pumped diodeNd:YAG laser (THALES Diva II) providing pulses having a20 ns duration and 2.5 mJ of energy. The out-of-plane com-ponent of the local vibration of the plate is measured witha heterodyne interferometer. This probe is sensitive to thephase shift along the path of the optical probe beam. Thecalibration factor for mechanical displacement normal to thesurface (100 mV/nm) was constant over the detection band-width (100 kHz - 400 kHz). Signals detected by the opti-cal probe were fed into a digital sampling oscilloscope andtransferred to a computer. The impulse responses betweeneach point of the same array (left and right) form the time-dependent reflection matrices (r and r′, respectively). The setof impulse responses between the two arrays yield the time-dependent transmission matrices t (from left to right) and t′

(from right to left). From these four matrices, one can buildthe S-matrix in a point-to-point basis [Eq. (1)]. A discreteFourier transform (DFT) of S is then performed over a timerange ∆t = 120 µs that excludes the echoes due to reflec-tions on the ends of the plate and ensures that most of theenergy has escaped from the sample when the measurementis stopped. The next step of the experimental procedure con-sists in decomposing the S-matrices in the basis of the flexuralmodes of the homogeneous plate. These eigenmodes and theireigenfrequencies have been determined theoretically using thethin elastic plate theory[15, 48, 49]. They are normalized suchthat each of them carries unit energy flux across the plate sec-tion. Theoretically, energy conservation would imply that S is

unitary. In other words, its eigenvalues should be distributedalong the unit circle in the complex plane. However, as shownin a previous work [15], this unitarity is not retrieved experi-mentally because of experimental noise. A dispersion of theeigenvalues si of the S−matrix is observed around the unitcircle. We compensate for this undesirable effect by consider-ing a normalized scattering matrix with the same eigenspacesbut with normalized eigenvalues [15]. The Q−matrix is thendeduced from S using Eq. (2). The frequency-derivative of Sat f = f0 is estimated from the centered finite difference,

∂fS (f0) =S (f0 + δf)− S (f0 − δf)

2δf,

with δf = 3 kHz.

Appendix B: Revealing transmission/time-delay eigenchannelsand their temporal/spectral features

The transmission and time-delay eigenchannels are derivedfrom the matrices S and Q measured at the central frequencyf0. The transmission matrix t(f0) (from the left to the rightlead) is extracted from S(f0) [Eq. (1)]. The output and in-put transmission eigenvectors, ul and vl, are derived from thesingular value decomposition of t(f0):

t(f0) =∑l

√Tl(f0)ul(f0)v†l (f0)

with Tl(f0) the intensity transmission coefficient associatedwith the lth scattering eigenstate at the central frequency.The frequency-dependent amplitude transmission coefficienttl (f) of this eigenstate can be obtained from the set of trans-mission matrices t (f) measured over the whole frequencybandwidth, such that

tl (f) = u†l (f0)t (f) vl(f0)

An inverse DFT of tl (f) finally yields the time-dependentamplitude transmission coefficient tl(τ) of the lth scatteringeigenstate measured at the central frequency f0. The timetraces displayed in Figs. 3(a) and 3(b) correspond to the squarenorm of this quantity.

The time-delay eigenchannels at the central frequency arederived from the eigenvalue decomposition of Q(f0)

Q(f0) =∑m

τmqinm (f0)

[qinm (f0)

]†The time-delay eigenvector qin

m is a 2N -dimensional columnvector that can be decomposed as

qin(f0) =

(qinm,L(f0)

qinm,R(f0)

)where qin

m,L(f0) and qinm,R(f0) contain the complex coeffi-

cients of qinm (f0) in the basis of the N incoming modes in

8

the left and right leads, respectively. Among the set of time-delay eigenstates, particle-like scattering states injected fromthe left lead should fulfill the following condition [17]:

||qinm,L(f0)||2 >> ||qin

m,R(f0)||2 ' 0.

A particle-like scattering state is thus associated with a N -dimensional input eigenvector qin

m,L(f0). The correspond-ing output eigenvector qout

m,R(f0) and transmission coefficient

t(q)m (f0) can be deduced from the t-matrix:

t(f0)qinm,L(f0) = t(q)m (f0) qout

m,R(f0).

The frequency-dependent amplitude transmission coefficientt(q)m (f) of this time-delay eigenstate can be obtained from the

set of transmission matrices t (f) measured over the wholefrequency bandwidth, such that

t(q)m (f) =[qoutm,L(f0)

]†t (f) qin

m,R(f0).

An inverse DFT of t(q)m (f) finally yields the time-dependentamplitude transmission coefficient t(q)m (τ) of the mth time-delay eigenstate measured at the central frequency f0. Thetime traces displayed in Figs. 3(c), 3(d) and 3(e) correspondto the square norm of this quantity.

Appendix C: Imaging spatio-temporal wave functions oftransmission/reflection and time-delay eigenchannels

Impulse responses are measured between the line of sources(denoted by the index i) and a grid of points (denoted by theindex j) that maps the medium, following the same proce-dure as the one described above. The grid pitch is 1.3 mm.

This set of impulse responses forms a transmission matrixk(τ) = [kji(τ)]. A discrete Fourier transform (DFT) ofk(τ) yields a set of frequency-dependent transmission ma-trices k(f). The lines of k(f) are then decomposed in thebasis of the plate modes. The monochromatic wave-fieldΨ(f) = [ψj(f)] associated with a transmission/reflection ortime-delay eigenchannel is provided by the product betweenthe matrix k(f) and the corresponding eigenvector (vl(f0)or qin

m,L(f0), respectively). The time-dependent wave-fieldΨ(τ) = [ψj(τ)] is deduced by an inverse DFT over a fre-quency bandwidth of our choice. Note that a Hann windowfunction is priorly applied to k(f) to limit side lobes in thetime-domain.

Appendix D: Unmixing degenerated time-delay eigenstates.

Depending on the geometry of the scattering medium, thedifferent scattering paths involved in a degenerated time-delayeigenstate can be discriminated either in the real space or inthe spatial frequency domain [54]. Here, the time-delay eigen-state displayed in Fig. 6(b) shows two scattering paths withopposite angles of incidence. Hence, they can be discrimi-nated by analyzing each block of the S-matrix in the spatialfrequency domain. The left lead of each block is decomposedover the positive or negative angles of incidence. This sub-space of the S-matrix, refered to as S′, is then used to computea reduced time-delay matrix Q′ [54] such that

Q′ = − i

2πS′−1∂fS′.

Note that the transpose conjugate operation of Eq. (2) is herereplaced by an inversion of S′ because of its non-unitarity[54]. Depending on the sign of the angle of incidence cho-sen for the left lead, the reduced matrix Q′ provides the time-delay eigenstates displayed in Figs. 6(c) and 6(d).

[1] A. Derode, P. Roux, and M. Fink, “Robust acoustic time re-versal with high-order multiple scattering,” Phys. Rev. Lett. 75,4206–4209 (1995).

[2] M. Tanter, J.-L. Thomas, and M. Fink, “Time reversal and theinverse filter,” J. Acoust. Soc. Am. 108, 223–234 (2000).

[3] G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Mon-taldo, and M. Fink, “Time reversal of electromagnetic waves,”Phys.Rev. Lett. 92, 193904 (2004).

[4] G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing be-yond the diffraction limit with far-field time reversal,” Science315, 1120–1122 (2007).

[5] I. M. Vellekoop and A. P. Mosk, “Focusing coherent lightthrough opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).

[6] S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara,and S. Gigan, “Measuring the Transmission Matrix in Optics:An Approach to the Study and Control of Light Propagation inDisordered Media,” Phys. Rev. Lett. 104, 100601 (2010).

[7] J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. La-gendijk, “Control of light transmission through opaque scatter-ing media in space and time,” Phys. Rev. Lett. 106, 103901(2011).

[8] D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A.Walmsley, S. Gigan, and B. Chatel, “Spatio-temporal focussingof an ultrafast pulse through a multiply scattering medium,” Na-ture Commun. 2, 447 (2011).

[9] O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusingand compression of ultrashort pulses through scattering media,”Nature Photon. 5, 371–377 (2011).

[10] I. M. Vellekoop and A. P. Mosk, “Universal optimal transmis-sion of light through disordered materials,” Phys. Rev. Lett.101, 120601 (2008).

[11] J. B. Pendry, “Light finds a way through the maze,” Physics 1,20 (2008).

[12] M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q-H. Park, andW. Choi, “Maximal energy transport through disordered mediawith the implementation of transmission eigenchannels,” Na-

9

ture Photon. 6, 583–587 (2012).[13] Z. Shi and A. Genack, “Transmission eigenvalues and the bare

conductance in the crossover to anderson localization,” Phys.Rev. Lett. 108, 043901 (2012).

[14] W. Choi, A. P. Mosk, Q.-H. Park, and W. Choi, “Transmis-sion eigenchannels in a disordered medium,” Phys. Rev. B 83,134207 (2011).

[15] B. Gérardin, J. Laurent, A. Derode, C. Prada, and A. Aubry,“Full transmission and reflection of waves propagating througha maze of disorder,” Phys. Rev. Lett. 113, 173901 (2014).

[16] C. W. Hsu, A. Goestschy, Y. Bromberg, A. D. Stone, andH. Cao, “Broadband coherent enhancement of transmission andabsorption in disordered media,” Phys. Rev. Lett. 115, 223901(2015).

[17] S. Rotter, P. Ambichl, and F. Libisch, “Generating particle-like scattering states in wave transport,” Phys. Rev. Lett. 106,120602 (2011).

[18] J.-F. Aubry, M. Pernot, F. Marquet, M. Tanter, and M. Fink,“Transcostal high-intensity-focused ultrasound: ex vivo adap-tive focusing feasibility study,” Phys. Med. Biol. 53, 2937–2951(2008).

[19] E. Cochard, C. Prada, J.-F. Aubry, and M. Fink, “Ultrasonicfocusing through the ribs using the dort method,” Med. Phys.36, 3495–3503 (2009).

[20] G. F. Edelmann, T. Akal, W. S. Hodgkiss, S. Kim, W. A. Kuper-man, and H. C. Song, “An initial demonstration of underwateracoustic communication using time reversal,” IEEE J. Ocean.Eng. 27, 602–609 (2002).

[21] C. Prada, J. de Rosny, D. Clorennec, J.-G. Minonzio, A. Aubry,M. Fink, L. Berniere, P. Billand, S. Hibral, and T. Folegot,“Experimental detection and focusing in shallow water by de-composition of the time reversal operator,” J. Acoust. Soc. Am.122, 761–768 (2007).

[22] T. Cizmar and K. Dholakia, “Exploiting multimode waveguidesfor pure fibre-based imaging,” Nature Comm. 3, 1027 (2012).

[23] I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Fo-cusing and scanning light through a multimode optical fiberusing digital phase conjugation,” Opt. Exp. 20, 10583–10590(2012).

[24] M. Plöschner, T. Tyc, and T. Cizmàr, “Seeing through chaos inmultimode fibres,” Nature Photon. 9, 529–535 (2015).

[25] S. Fan and J. M. Kahn, “Principal modes in multimode waveg-uides,” Opt. Lett. 30, 135–137 (2005).

[26] A. A. Juarez, C. A. Bunge, S. Warm, and K. Petermann,“Perspectives of principal mode transmission in mode-division-multiplex operation,” Opt. Exp. 20, 13810–13823 (2012).

[27] J. Carpenter, B. J. Eggleton, and J. Schröder, “Observation ofeisenbud–wigner–smith states as principal modes in multimodefibre,” Nature Photon. 9, 751–757 (2015).

[28] W. Xiong, P. Ambichl, Y. Bromberg, S. Rotter, and H. Cao,“Spatio-temporal control of light transmission through a mul-timode fiber with strong mode coupling,” arXiv:1601.04646(2016) [Phys. Rev. Lett., in press].

[29] B. C. Carpenter, J. Thomsen and T. D. Wilkinson, “Degener-ate mode-group division multiplexing,” J. Lightw. Technol. 30,3946–3952 (2012).

[30] J. Salz, “Digital transmission over cross-coupled linear chan-nels,” At&T Technical Journal 64, 1147–1159 (1985).

[31] G. G. Raleigh and J. M. Cioffi, “Spatio-temporal coding forwireless communication,” Communications, IEEE Transactionson 46, 357–366 (1998).

[32] A. Tulino and S. Verdù, “Random matrix theory and wirelesscommunications,” Fundations and Trends in Communicationsand Information Theory 1, 1–182 (2004).

[33] R. Sprik, A. Tourin, J. de Rosny, and M. Fink, “Eigenvaluedistributions of correlated multichannel transfer matrices instrongly scattering systems,” Phys. Rev. B 78, 012202 (2008).

[34] A. Aubry and A. Derode, “Random Matrix Theory Appliedto Acoustic Backscattering and Imaging In Complex Media,”Phys. Rev. Lett. 102, 084301 (2009).

[35] O. Dietz, H.-J. Stöckmann, U. Kuhl, F. M. Izrailev, N. M.Makarov, J. Doppler, F. Libisch, and S. Rotter, “Surface scatter-ing and band gaps in rough waveguides and nanowires,” Phys.Rev. B 86, 201106 (2012).

[36] C. W. J. Beenakker, “Random-matrix theory of quantum trans-port,” Rev. Mod. Phys. 69, 731–808 (1997).

[37] The symbol † here stands for transpose conjugate.[38] O. N. Dorokhov, “On the coexistence of localized and extended

electronic states in the metallic phase,” Sol. St. Comm. 51, 381–384 (1984).

[39] Y. Imry, “Active transmission channels and universal conduc-tance fluctuations,” Europhys. Lett. 1, 249–256 (1986).

[40] H. U. Baranger and P. A. Mello, “Mesoscopic transport throughchaotic cavities: A random s-matrix theory approach,” Phys.Rev. Lett. 73, 142–145 (1994).

[41] R. A. Jalabert, J.-L. Pichard, and C. W. J. Beenakker, “Uni-versal quantum signatures of chaos in ballistic transport,” Euro-phys. Lett. 27, 255–260 (1994).

[42] E. P. Wigner, “Lower limit for the energy derivative of the scat-tering phase shift,” Phys. Rev. 98, 145–147 (1955).

[43] F. T. Smith, “Lifetime matrix in collision theory,” Phys. Rev.118, 349–356 (1960).

[44] C. Prada and M. Fink, “Eigenmodes of the time reversal opera-tor: A solution to selective focusing in multiple-target media,”Wave Motion 20, 151–163 (1994).

[45] C. Prada, S. Manneville, D. Spoliansky, and M. Fink, “Decom-position of the time reversal operator: Detection and selectivefocusing on two scatterers,” J. Acoust. Soc. Am. 99, 2067–2076(1996).

[46] S. M. Popoff, A. Aubry, G. Lerosey, M. Fink, A. C. Boccara,and S. Gigan, “Exploiting the time-reversal operator for adap-tive optics, selective focusing, and scattering pattern analysis,”Phys. Rev. Lett. 107, 263901 (2011).

[47] A. Badon, D. Li, G. Lerosey, A. C. Boccara, M. Fink, andA. Aubry, “Smart optical coherence tomography for ultra-deepimaging through highly scattering media,” arXiv:1510.08613(2015).

[48] M. C. Cross and R. Lifshitz, “Elastic wave transmission at anabrupt junction in a thin plate with application to heat trans-port and vibrations in mesoscopic systems,” Phys. Rev. B 64,085324 (2001).

[49] D. H. Santamore and M. C. Cross, “Surface scattering analy-sis of phonon transport in the quantum limit using an elasticmodel,” Phys. Rev. B 66, 144302 (2002).

[50] The more energetic sub-diagonals of the transmission/reflectionmatrices account for the conversion between even and oddmodes of similar momentum.

[51] F. Aigner, S. Rotter, and J. Burgdörfer, “Shot noise inthe chaotic-to-regular crossover regime,” Phys. Rev. Lett. 94,216801 (2005).

[52] See Supplemental Material which includes the scattering matrixanalysis of the disordered waveguide, the frequency study ofparticlelike scattering states and five movies showing particle-like wave packets propagating through the cavity and the disor-dered waveguide.

[53] E. Dieulesaint and D. Royer, Elastic waves in solids I (Springer-Verlag, Berlin, 2000).

10

[54] Andre Brandstötter, Master’s thesis, Institute for TheoreticalPhysics - Vienna University of Technology (2016).

[55] Y. D. Chong and A. D. Stone, “Hidden black: Coherent en-hancement of absorption in strongly scattering media,” Phys.Rev. Lett. 107, 163901 (2011).

[56] N. Bachelard, J. Andreasen, S. Gigan, and P. Sebbah, “Tamingrandom lasers through active spatial control of the pump,” Phys.Rev. Lett. 109, 033903 (2012).

[57] N. Bachelard, S. Gigan, X. Noblin, and P. Sebbah, “Adaptivepumping for spectral control of random lasers,” Nature Phys.10, 426–431 (2014).

[58] M. Davy, Z. Shi, J. Wang, X. Cheng, and A. Z. Genack, “Trans-mission eigenchannels and the densities of states of random me-dia,” Phys. Rev. Lett. 114, 033901 (2015).

[59] R. Pierrat, P. Ambichl, S. Gigan, A. Haber, R. Carminati,and S. Rotter, “Invariance property of wave scattering throughdisordered media,” Proc. Nat. Acad. Sci. U.S.A. 111, 17765–17770 (2014).

[60] A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Con-trolling waves in space and time for imaging and focusing incomplex media,” Nature Photon. 6, 283–292 (2012).

Supplemental MaterialThis document provides further information on the scatter-

ing matrix measured in the disordered wave guide and the sta-bility of particle-like states in the frequency domain. The cap-tions of the Supplementary Movies are also listed at the endof the document.

I. SCATTERING MATRIX ANALYSIS IN THEDISORDERED WAVEGUIDE

Figure S1(a) displays an example of S−matrix recordedin the disordered wave guide depicted in Fig. 1(a) of the ac-companying paper. Despite a moderate level of disorder, theS−matrix exhibits an overall random appearance. Neverthe-less, a residual ballistic wave-front slightly emerges along thediagonal of the transmission matrices. The statistics of the

a

d

b

c

10 20 10 20

10

20

10

20 −1

−0.5

0

0.5

1

Left channel Right channelEmission

Rec

eptio

nRig

ht c

hann

elLe

ft c

hann

el

0 5 10 15 20 25 30 35 40X [mm]

0

5

10

15

20

25

30

35

40

45

50

Y[m

m]

0 0.5 1T

0

1

2

3

(T)

0 5 10 15 20 25 30 35 40X [mm]

Y[m

m]

0

5

10

15

20

25

30

35

40

45

50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

evenmodes

oddmodes

FIG. S1. (a), Amplitude of the S−matrix measured in the scatteringslab at f0 = 0.30 MHz. The black lines delimit transmission andreflection matrices. (b), Transmission eigenvalue histograms, ρ (T ),averaged over the frequency bandwidth. (c)-(d), Absolute value ofthe monochromatic wave-fields associated with two open channelsat the central frequency.

transmission eigenvalues Tl computed from the t−matrix isnow investigated. As for the regular cavity, their distributionρ (T ) is estimated by averaging their histograms over the fre-quency bandwidth. Fig. S1(b) displays the estimator of thisdistribution. Most of the transmission coefficients are foundto be either close to zero or one, accounting respectively forclosed and open channels. Figures S1(c) and S1(d) display thewave-fields associated with two open channels at the centralfrequency (see Methods). As in the cavity, the open chan-nels combine multiple path trajectories. As a consequence,

11

they undergo a strong spatial and temporal dispersion whilepropagating through the scattering slab. As shown in Fig.6 ofthe accompanying paper, the study of the time-delay matrixallows to lift this degeneracy.

II. STABILITY OF THE PARTICLE-LIKE STATES IN THEFREQUENCY DOMAIN

δf [MHz]

spec

tral

cor

rela

toin

fu

nctio

n

(d)

tran

smis

sion

coe�

cien

t

(a)

δf [MHz]

(e)

δf [MHz]

(f)

0.2 0.3 0.4 0.5 0.6−1

0

1

f [MHz]

0.2 0.3 0.4 0.5 0.6−1

0

1

f [MHz]

(b)

(c)

0.2 0.3 0.4 0.5 0.6−1

0

1

f [MHz]

spec

tral

cor

rela

toin

fu

nctio

nsp

ectr

al c

orre

lato

in

func

tion

tran

smis

sion

coe�

cien

ttr

ansm

issi

on c

oe�

cien

t

−0.1 −0.05 0 0.05 0.10

0.5

1

−0.1 −0.05 0 0.05 0.10

0.5

1

−0.1 −0.05 0 0.05 0.10

0.5

1

FIG. S2. (a)-(b)-(c), Frequency dependence of the transmission co-efficients t(q)m (f) for the three cavity particle-like states displayed inFig. 4(a)-(b)-(c), respectively (blue line: real part, red line: imag-inary part). The black vertical dashed lines delimit the bandwidthof each state. (d)-(e)-(f), Spectral correlation function |Cm(δf)|(blue line) of the transmission coefficient t(q)m (f) displayed in (a)-(b)-(c), respectively. Cm(δf) is computed by considering the centralfrequency f0 of each state. Each curve is compared with the spec-tral correlation function C(δf) of the transmission matrix elements(black dashed line).

Figures S2(a) and S2(c) display the frequency dependenceof the transmission coefficients t(q)m (f) for the three cavityparticle-like states displayed in Fig. 4. The frequency evo-lution of t(q)m (f) allows to delimit the frequency range overwhich each particle-like state remains stable: f = 0.2 − 0.6MHz [Fig. S2(a)], f = 0.3 − 0.6 MHz [Fig. S2(b)] andf = 0.2 − 0.4 MHz [Fig. S2(c)]. To be more quantitative,we have estimated the spectral correlation function of eachstate,

Cm(δf) = t(q)m (f0)t(q)∗m (f0 + δf),

with f0 the associated central frequency. The result is dis-played in Figs. S2(d), S2(e) and S2(f), and compared to themean spectral correlation function,

C(δf) =⟨tij(f)t∗ij(f + δf)

⟩,

of the measured transmission matrix coefficients tij(f). Thesymbol 〈.〉 here denotes an average over the frequency band-width. The FWHM of Cm(δf) yields the spectral correlationwidth δfm of each state displayed in Fig. 4. We find δfm =0.16 MHz [Fig. S2(d)], 0.17 MHz [Fig. S2(e)] and 0.12 MHz

[Fig. S2(f)]. These values should be compared to the fre-quency correlation width δfc = 0.02 MHz of the transmissionmatrix coefficients. We see that the spectral correlation widthδfm of particle-like states is 8, 8.5 and 6 times larger than δfc.This illustrates the stability of the cavity particle-like states inthe frequency domain and accounts for the spatio-temporal fo-cusing of particle-like wave packets in the time domain (seeSupplementary Movies 1, 2 and 3).

(c)(a)

(d)(b)

0.2 0.3 0.4 0.5 0.6−1

0

1

f [MHz]

0.2 0.3 0.4 0.5 0.6−1

0

1

f [MHz]

tran

smis

sion

coe�

cien

ttr

ansm

issi

onco

e�ci

ent

δf [MHz]

spec

tral

cor

rela

toin

fu

nctio

nsp

ectr

al c

orre

lato

in

func

tion

δf [MHz]

−0.1 −0.05 0 0.05 0.10

0.5

1

−0.1 −0.05 0 0.05 0.10

0.5

1

FIG. S3. (a)-(b), Frequency dependence of the transmission coeffi-cients tqm(f) for the two particle-like states in the disordered waveguide, displayed in Figs. 6(c) and 6(d), respectively (blue line: realpart, red line: imaginary part). The black vertical dashed lines de-limit the bandwidth of each state. (c)-(d), Spectral correlation func-tion, |Cm(δf )|, for the transmission coefficient tqm(f) displayed in(a)-(b), respectively. Cm(δf) is computed by considering the centralfrequency f0 of each state. Each curve is compared with the spec-tral correlation function |C(δf)| of the transmission matrix elements(black dashed line).

The same analysis can be performed for the disorderedwave guide. As an example, we investigate the frequencystability of the particle-like states displayed in Figs.6(c) and6(d). Figures S3(a) and S3(b) display the frequency depen-dence of the corresponding transmission coefficients t(q)m (f).Each state remains stable over a frequency range f = 0.2−0.4MHz [Fig.S3(a)] and f = 0.2 − 0.5 MHz [Fig.S3(b)]. Thecorresponding spectral correlation functions Cm(δf) are dis-played in Figs.S3(c) and S3(d). We measure a frequency cor-relation width δfm of 0.13 and 0.12 MHz. Contrastingly, thetransmission matrix elements exhibit a frequency correlationwidth δfc = 0.025 MHz. δfm is thus 5 times higher thanδfc. This demonstrates the frequency stability of these twoparticle-like states and accounts for the spatio-temporal fo-cusing of the corresponding particle-like wave packets in thetime domain (see Supplementary Movies 4 and 5).

III. CAPTIONS OF THE SUPPLEMENTARY MOVIES

Movie 1: Particle-like wave-packet in the cavity synthesizedfrom the scattering state displayed in Fig. 4(a) over the fre-quency range f = 0.2− 0.6 MHz.Movie 2: Particle-like wave-packet in the cavity synthesizedfrom the scattering state displayed in Fig. 4(b) over the fre-quency range f = 0.3− 0.6 MHz.Movie 3: Particle-like wave-packet in the cavity synthesized

12

from the scattering state displayed in Fig. 4(c) over the fre-quency range f = 0.2− 0.4 MHz.Movie 4: Particle-like wave-packet in the disordered waveguide synthesized from the scattering state displayed in Fig.

6(c) over the frequency range f = 0.2− 0.4 MHz.Movie 5: Particle-like wave-packet in the disordered waveguide synthesized from the scattering state displayed in Fig.6(d) over the frequency range f = 0.2− 0.5 MHz.