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PHYS I CS

1Daniel Guggenheim School of Aerospace Engineering, Georgia Institute of Tech-nology, Atlanta, GA 30332, USA. 2Department of Electrical and Computer Engi-neering, The University of Texas at Austin, Austin, TX 78712, USA. 3PhotonicsInitiative, Advanced Science Research Center, City University of New York, NewYork, NY 10031, USA. 4George W. Woodruff School of Mechanical Engineering,Georgia Institute of Technology, Atlanta, GA 30332, USA. 5Physics Program, Grad-uate Center, City University of New York, New York, NY 10016, USA. 6Departmentof Electrical Engineering, City College of The City University of New York, NewYork, NY 10031, USA.*These authors contributed equally to this work.†Corresponding author. Email: aalu@gc.cuny.edu

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Coherent virtual absorption of elastodynamic wavesG. Trainiti1*, Y. Ra'di2,3*, M. Ruzzene1,4, A. Alù2,3,5,6†

Absorbers suppress reflection and scattering of an incident wave by dissipating its energy into heat. As materialabsorption goes to zero, the energy impinging on an object is necessarily transmitted or scattered away. Specificforms of temporal modulation of the impinging signal can suppress wave scattering and transmission in the tran-sient regime, mimicking the response of a perfect absorber without relying onmaterial loss. This virtual absorptioncan store energy with large efficiency in a lossless material and then release it on demand. Here, we extend thisconcept to elastodynamics and experimentally show that longitudinal motion can be perfectly absorbed using alossless elastic cavity. This energy is then released symmetrically or asymmetrically by controlling the relative phaseof the impinging signals. Our work opens previously unexplored pathways for elastodynamic wave control andenergy storage, which may be translated to other phononic and photonic systems of technological relevance.

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INTRODUCTIONEfficient absorbers are of great importance in a wide variety of tech-nological fields, from energy harvesting and radar detection in electro-magnetics to sound proofing in acoustics and vibration isolation inmechanical systems (1–4). Common to these systems is the notionthat efficient absorption can be achieved when the material loss isbalanced by the impedance of the impinging wave. In other words,a proper amount of material loss can push one of the complex scat-tering zeros of the system onto the real frequency axis (5, 6), and, as aresult, the impinging energy at this frequency is all lost into heat orother chemical processes. Therefore, the system is not conservative,and its scatteringmatrix is not unitary (for amultiport linear network,the scattering and transmission toward the ports are governed by thescattering matrix, which maps the incident fields to the outgoingfields).

Considering simultaneous excitation provides an additional degreeof freedom to control the location of the scattering zeros of the systemand to move one of them onto the real frequency axis. Coherent per-fect absorption (CPA) is achieved through the interference ofmultipleincident waves impinging on the absorber, enabling a way to controlthe absorption mechanism in real time through the proper choice ofthe relative intensities and phases of the input beams. The dependenceof CPAs on the input waveforms therefore provides the opportunityto flexibly control light scattering and absorption (7). Thedemonstra-tion of an acoustic CPA has been presented in (8), opening a pathtoward several applications of practical interest, including highly sensi-tive detection and amplification of small variations in the incidentsignals or in the properties of the involved materials to realize massor temperature transducers and for the efficient control and conver-sion of energy in harvesting applications (9, 10). These opportunitiessuggest that coherent absorption can also be of great interest in thecontext of elastodynamic waves.

In lossless systems, the scatteringmatrix cannot admit zeros on thereal frequency axis, given its unitarity; therefore, the impinging waveneeds to be transmitted, reflected, or scattered at all real frequencies.The zeros are necessarily confined to the upper half of the complexfrequency plane (5, 6, 11, 12), above the real axis. It has been recentlysuggested that the time evolution of an incoming signal can be tailoredto efficiently engage these complex zeros, implying that a specificchoice of nonmonochromatic signals oscillating at a complex fre-quency can totally eliminate transmission, reflections, and scattering,thus realizing a virtual absorber with zero material loss (13–15). Aslong as the input signal illuminates the structure with the right evolu-tion in time, the impinging energy is neither scattered nor transmitted,but instead, it is captured and stored within the system with unitaryefficiency. By varying the impinging signals, the stored energy canthen be released through its scattering channels in a controllable fash-ion. Having such a coherent control over the stored energy in the cav-ity enables unprecedented functionalities, such as flexible energystorage and memory.

This concept is in some ways connected with the time-reversalexcitation technique (16, 17), based on which, under the assumptionsof linearity and time-reversal symmetry, the excitation of a cavity witha time-reversed replica of its decaying fields should be accepted by thecavity without scattering or reflections. However, one cannot draw astrict analogy between the two phenomena because they deal with dif-ferent objectives. The time-reversal technique is based on the idea thatan arbitrarily radiated pulse can be made to converge toward thesource, provided that an array of sensors can time reverse it with therequired accuracy. In contrast to coherent virtual absorption, the time-reversal approach does not therefore deal with eigenmodes of a struc-ture. In the complex zero approach, the incident pulse is determinedby the open cavity geometry. Exciting the structure with any of thecomplex zeros of the structure enables zero scattering and ideal wavecapturing, which is a nontrivial conclusion. In analogy with the CPAoperation, in the following, we experimentally demonstrate coherentvirtual absorption of elastodynamic waves traveling along a solid bar,controlling storage and release of impinging longitudinal waves bytailoring their time evolution.

RESULTSConsider a two-port lossless elastic waveguide with a circular crosssection supporting longitudinal motion (Fig. 1A). The system can bedivided into three domains, with stepwise constant cross section

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Aj ¼ pr2j for each j-th domain. A cavity of length L connects twoidentical side channels, excited by input signals I

þð0;wÞ andI�ðL;wÞ, as shown in Fig 1A. Because of the mechanical impedancemismatch of the central section, such a structure produces scatteredfields O

þðL;wÞ and O�ð0;wÞ at the ports, linearly related to the

input fields through the frequency-based scattering matrix (18)

OþðL;wÞ

O�ð0;wÞ

� �¼ S11ðwÞ S12ðwÞ

S21ðwÞ S22ðwÞ� �

⋅ Iþð0;wÞI�ðL;wÞ

� �ð1Þ

Here and in the following, the hat symbol denotes a Fourier trans-form, while w is the radial frequency. We assume that the waveguideis made of a single material of wave velocity c ¼ ffiffiffiffiffiffiffiffi

E=rp

, where E andr are the material modulus of elasticity and density, respectively.Note that the impedance mismatch at the boundary of the outerand inner cores not only relates to the impedance mismatch of thematerials on different sides of each interface but also is imparted bythe cross-sectional areas of the rods on different sides of the interface(see the Supplementary Materials). Because of symmetry, S11ðwÞ ¼S22ðwÞ ¼ RðwÞ, and because of time-reversal symmetry, S12ðwÞ ¼

Trainiti et al., Sci. Adv. 2019;5 : eaaw3255 30 August 2019

S21ðwÞ ¼ TðwÞ. As a result, the components of the scattering matrixcan be derived as (see the Supplementary Materials)

RðwÞ ¼ R0 þ R1ei2wLc

1þ R0R1ei2wLc

; TðwÞ ¼ T0T1eiwLc

1þ R0R1ei2wLc

ð2Þ

where R0 ¼ �R1 ¼ ðr20 � r21Þ=ðr20 þ r21Þ and T0 ¼ r20r21T1 ¼ 2r20=ðr20 þ

r21Þ are the local reflection and transmission coefficients at the two in-terfaces. Here, r0 and r1 are the radius of the outer and inner rods,respectively. By analyzing the scattering matrix, we investigate theconditions under which the system can efficiently absorb the inci-dent energy impinging at its interfaces at x = 0 and x = L. For sym-metric or antisymmetric excitations I

þð0;wÞ ¼ ±I�ðL;wÞ, because of

symmetry, the outputs follow OþðL;wÞ ¼ ±O

�ð0;wÞ ¼ lðwÞIþð0;wÞ.The eigenvalue l(w) for the symmetric case is lSðwÞ ¼ TðwÞ þ RðwÞ,while that for the antisymmetric case is lAðwÞ ¼ TðwÞ � RðwÞ. Zerosof the scattering matrix, associated to perfectly absorbing modes, arefound when lS(w) = 0 or lA(w) = 0, at frequencies

wz ¼ cL

pnþ i lnr21 þ r20r21 � r20

� �� �; n ¼ 1; 2; 3;… ð3Þ

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Fig. 1. Elastic coherent virtual absorber. (A) Illustration of a mirror-symmetric waveguide with stepwise constant cross-sectional area A and wave velocity c. Thesystem, whose central domain has length L, supports incoming fields I±(x, t) and outgoing fields O±(x, t). (B) The scattering properties of the waveguide are described bythe scattering matrix SðWÞ. When W is analytically continued to the complex plane as W = WRE + iWIM, SðWÞ has a countable infinite set of zeros, divided into symmetricand antisymmetric ones. The contour plot shows the quantity lAðWÞ ¼ TðWÞ � RðWÞ and the location of the antisymmetric zeros (black dots), as well as the location ofthe symmetric zero at WRE = 2p (blue dot). The inputs of the system can be designed to be equal to one of those zeros, thus canceling the scattered fields. (C) Incidentand scattered fields for W = p + i0.51 such that lA(W) = 0. In this case, the scattered fields are identically zero for t < 0. a.u., arbitrary unit. (D) Incident and scattered fieldsfor W = 2p + i0.51 such that lA(W) ≠ 0. In this case, the scattered fields appear as soon as the excitations hit the structure.

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where even and odd n values correspond respectively to symmetricand antisymmetric excitations. In lossy systems, these zeros can cor-respond to real frequencies, yielding what is known in the optics lit-erature as CPA (6, 7). This corresponds to a system with loss balancedto the outer impedance, for which coherent excitation on both sideswith same or opposite phase is fully absorbed without transmission orscattering. If the system is lossless, however, then all these zeros lie inthe upper half of the complex frequency plane, with wz = wRE + iwIM,wIM > 0 (throughout the paper we assume an e−iwt time convention).In this case, no real frequency excitation can be absorbed in the system,as expected from energy conservation. If insteadwe excite the structure

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coherently from the two input ports with time-growing waves oscillat-ing at the complex frequencywz and the proper relative phase, thenweengage the corresponding complex zero of the system and achieve co-herent virtual absorption, i.e., absence of scattering and transmission,and energy storage in the cavity with unitary efficiency. In practice,these exponentially growing inputs cannot be sustained indefinite-ly, and the stored energy is released once the excitation is stoppedor modified.While the signals are growing at the required rate ewIMt,the system stores energy at a rate proportional to e2wIMtcos2(2wREt),which can be tailored with large flexibility by controlling the posi-tion of the complex zero in the frequency plane.

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A

DCB Experiment (incident waves)Simulation (incident waves)

Experiment (scattered waves)Simulation (scattered waves)

Fig. 2. Experimental wavefield measurements. (A) Waveguide with resonator and side channels of length L = 0.2 m and circular cross section with radii r0 = 0.005 mand r1 = 0.010 m. The excitation is provided by piezoelectric actuators (PZT) placed at the two ends of the system. (B) Radial velocity fields simulated with the finite-difference time-domain (FDTD) method: incident waves (top) and scattered waves (bottom). (C) Similarly, the measured radial velocity: incident waves (top) andscattered waves (bottom). The incoming energy is first stored in the resonator (i.e., 0.2 m < x < 0.4 m) and then released through scattering roughly at t = 270 ms.(D) Time history for both numerical and experimental fields at x = 0.5 m [red dashed line in (B) and (C)] shows that the incoming energy is released through scatteringonly after the incident fields stop growing exponentially. [Photo credit for (A): Giuseppe Trainiti, Georgia Institute of Technology].

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To investigate the dynamics of the virtual absorption process, con-sider a coherent antisymmetric excitation where I+(0, t) = − I−(L, t) =f(t), where t = t/tL and tL = L/c is the time needed for the wave totravel through the cavity at speed c. To determine the required exci-tation signal f(t), we first find the complex zeros for antisymmetricexcitation by setting lA(W) = 0, whereW = wL/c =WRE + iWIM. Figure1B shows two of these zeros in the upper half of the complex frequencyplane, indicated by black dots. We choose to engage the first zero, atW = p + i0.51. We therefore excite the two ports with input signalsf(t) to oscillate at the complex frequency W for t < 0 and modulatedby a fast-decaying exponential for t > 0

f ðtÞ ¼ eWIMtQð�tÞ þ eð�DtÞ2QðtÞh i

cosðWREtÞ ð4Þ

whereQ(t) is the step function andD is a decay factor of choice. Theexcitation signals at the two ports are shown in the upper panel ofFig. 1C, while the lower panel shows the time domain output signals.As long as the input signals engage the complex zero of the system,for t < 0, all the impinging energy is virtually absorbed and stored inthe system. As soon as the input signals diverge from the virtual ab-sorption condition, for t > 0, the system releases its stored energy. Ingeneral, the system can release the stored energy at t = 0 through allits complex poles, which are symmetrically located to its zeros in thecomplex frequency plane in the case of lossless systems because oftime-reversal symmetry. Depending on the transient region aroundt = 0, different eigenmodes of the system may be excited with differ-ent amplitudes. In the example at hand, the system releases its storedenergy mostly into the first (dominant) eigenmode, which is con-sistent with a time-reversed replica of the input signals, but, for dif-ferent truncation schemes of the input signal, the outgoing fieldsmay be substantially different than the input signals. Figure 1Dshows a scenario in which we excite the system antisymmetricallybut at the complex frequency W = 2p + i0.51, which would corre-spond to a zero for even excitation. In this case, the incorrect relativephase of the incoming signals produces strong reflections at theport, and virtual absorption is not achieved. Simply flipping thephase of one of the two input signals would completely suppressall output fields for any t < 0, underlining the importance of the co-herent excitation at the two ports to achieve this phenomenon.

To validate our theoretical results, we performed a proof-of-principle experiment in the setup shown in Fig. 2A. The geometryconsists of a 0.6-m aluminum bar with a circular cross section anddesign parameters L = 0.2 m and r1 = 2r0 = 0.01 m. The system is ex-cited antisymmetrically with piezoelectric actuators [lead-zirconate-titanate (PZT)] at its fifth zero [i.e., f = w/2p = (64.85 + i2.11) kHz].A scanning laser Doppler vibrometer (see Materials and Methods)measures the radial velocity field vr(x, t) along thewaveguide in responseto the external excitation. The radial contraction of the waveguide dueto the Poisson effect, albeit not accounted for in our waveguide model,may slightly affect the dynamic properties of the system for slenderstructures at low frequencies; however, this effect is negligible here(see the Supplementary Materials). In parallel, to validate our experi-mental results, we developed a finite‐difference time‐domain (FDTD)–based tool to perform realistic numerical simulations of the samegeometry (seeMaterials andMethods). Numerical simulations shownin Fig. 2B represent the radial velocity of incident (top) and scattered(bottom) fields, each normalized with respect to the peak velocityvalue of the incident fields. As it is seen from these figures, the inci-

Trainiti et al., Sci. Adv. 2019;5 : eaaw3255 30 August 2019

dent energy is perfectly absorbed and stored in the middle portion ofthe bar, which acts as the resonating cavity, as long as the incidentwaveform is tailored to excite the complex zero of the system (in thisexample, this is the case for t < 270 ms). For the example consideredhere, the excited zero is not close to the real frequency axis; hence, thecorresponding Q-factor is limited. For this reason, the stored energyinside the cavity is not much larger than the incident one. For acomplex zero closer to the real frequency axis, i.e., for a largerQ-factorcavity, at any instant in time, the stored energy may be much larger,roughly equal to Q times the instantaneous impinging energy.

As soon as the incident wave starts decaying (i.e., for t > 270 ms),the cavity releases its energy. The standing wave pattern inside thecavity (middle rod) is due to its resonance. The experimental resultsshown in Fig. 2C are in very good agreement with our simulationresults. Figure 2D shows a cut of the results shown in panels B andC of Fig. 2 at x = 0.5 m. The top panel in Fig. 2D compares the in-cident signals from numerical simulations and experiments, whilethe output signals are compared in the lower panel, confirming theevidence of coherent virtual absorption in the rod. In principle, onecan excite the resonator with any of the complex zeros of the system.In our realization, we chose the fifth zero because of the limiteddimensions of the symmetric side channels of length L. If we usedone of the first zeros, then the incident pulseswould bewider,meaningthat after releasing the energy from the resonator, the signal will reachthe end of the side rods (i.e., the position of source) and reflect backtoward the middle resonator. This reflected signal would distort themeasured signal at the probe point. However, by choosing longer siderods, one may avoid this issue and also excite the system with lower-order zeros. On the other hand, zeros of very high order may become

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Fig. 3. Control of scattering and energy storage by changing the complexfrequency of the excitation signals. (A) Excitation signals. (B) Outgoing signals.(C) Stored energy in the system.

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challenging, as the slender rod assumption may not hold for waveshaving wavelengths comparable to the rod cross section.

Having verified that the incident energy can be virtually absorbedand stored in a lossless resonator, we explore the degree of controlover the release of stored energy, exploiting the coherence of thetwo impinging signals. Figure 3 illustrates a scenario in which we re-peatedly excite the system at the complex zero, release its energy, andpump and release it again. The top panel shows the input signal, whilethe middle and bottom panels show the output signals and the storedenergy in themiddle section of the rod, respectively. The structure cancoherently capture the impinging pulses with high efficiency and thenrelease it at will as the exciting pulses are stopped. The system is thenready to store the next pulse. The release of stored energy can also becontrolled by changing the relative phase f of the excitation signals atthe opposite ports, exploiting the coherence of the storage process. Toinvestigate this scenario, we consider an exponentially modulated var-iation of the relative phase between the two input signals, f(t) =p[ehWIMtQ(−t) + Q(t)], with h being a control parameter. We com-pare the case of ideal excitation of the complex zero (Fig. 4A) to thecase in which the relative phase is slowly changed as f(t) (Fig. 4B).The middle panels show the instantaneous power at the input andoutput ports and the net power flow into the resonator [i.e., PR(t) =

Trainiti et al., Sci. Adv. 2019;5 : eaaw3255 30 August 2019

PI+(0, t) + PI−(L, t) − PO−(L, t) − PO+(0, t)]. We also integrate thesequantities in time to obtain a measure of the total energy enteringand exiting the resonator, as well as the net energy stored in the sys-tem up to time t [i.e., ER(t) = EI(t) − EO(t)]. Figure 4B shows that, assoon as we start deviating the relative phase from the required value(in this example, we assume that h = 5), the resonator starts releasingits stored energy, again highlighting the effect of coherence in thestorage process. In the case without phase variation (Fig. 4A), becauseof mirror symmetry, the resonator releases its energy equally throughits scattering channels [PO−(L, t) = PO+(0, t)]. On the contrary, in Fig.4B, the asymmetry in excitation enables additional control on the portthrough which the stored energy is released. Note that the transitiontime over which the relative phase change is applied can markedly af-fect the redistribution of released energy between outputs. Figure 5shows how the amount of released energy difference between outputsDE0 can be controlled by howquickly the relative phase f(t) varies from0 to p, changing the value of h. For relatively small values of h, we breakthe symmetry between scattering ports and maximize DE0.

In practice, growing the input signals for a long period of timemaybe impractical, and most of the stored energy at any given instantwould, in any case, be contributed by the last part of the excitationtransient, inversely proportional to the quality factor of the cavity. In

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Fig. 4. Scattering and energy storage control through input relative phase variation. (A) Response for zero relative phase f(t) = 0 (top) between the inputs at W =5p + i0.51 is represented in terms of the normalized power inputs PI+(0, t) and PI−(L, t), the power outputs PO−(L, t) and PO+(0, t), and the power stored into the resonatorPR(t) (middle), as well as the associated integrals evaluated between −∞ and t, with EI(t) and EO(t) as the energy that entered and exited the system up to time t,respectively (bottom). (B) Imposing an exponentially increasing relative phase law f = f(t) between the inputs (top) enables the dynamic control of the scatteringprocess, with scattering onset anticipated at t < 0 (middle), a different stored energy profile (bottom). The imposed relative phase also induces different energyredistribution between the two outputs of the system, with EO+(L, t) − EO−(0, t) = DE0 ≠ 0.

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this sense, cavities with higher-quality factors have complex zeros closerto the real frequency axis, implying that in this case, the excitationbecomes quasi-harmonic. As the intensity grows, we may also incurinto nonlinearities, which change the picture and break the temporalsymmetry between the storage and release processes. Suitably tailorednonlinear cavities may be envisioned to accept an incoming signalwith unitary efficiency but then trap it in an embedded eigenstate, asrecently envisioned in the context of quantum optics (19).

DISCUSSIONHere, we have introduced and experimentally demonstrated the con-cept of coherent virtual absorption, storage, and release of energyon demand in elastodynamics. We have shown that the impingingdisplacement energy can be absorbed and stored in a lossless reso-nator with unitary efficiency for a desired period of time.We have alsoshown that we can control the release of the elastic energy with largeflexibility and its directionality, exploiting the coherence of the storageprocess. Our results open interesting opportunities for applicationsin elastodynamics and structural mechanics, including sensors thatmay be able detect small changes in the resonance characteristics of acavity, for example, as a result of appliedmechanical strain, tempera-

Trainiti et al., Sci. Adv. 2019;5 : eaaw3255 30 August 2019

ture, or material changes. These may also be of interest for energystorage and release. In addition, the control of the rate and time evo-lution of energy release may be beneficial for efficient conversion ofmechanical energy into electrical or for the implementation of a broadrange of memory, amplification, and computational functionalitieswithin mechanical substrates. Efficient excitation of an elastodynamicor acoustic cavity; a phase-dependent, nonlinear amplification of theinput signals; and controlled storage and release of acoustic energy arealso relevant for focused sound generation as part of arrays, loudspea-kers, and ultrasonic transducers. More broadly, this proof of conceptmay be directly translated to other phononic or photonic setups,enabling a large degree of control of phonons and photons using thecoherence of specifically tailored nonmonochromatic signals.

MATERIALS AND METHODSFor the experimental validation of the concept of coherent virtual ab-sorption in elastodynamics, the elastic waveguides were realized by aslender 1566 carbon steel rod (E = 210 GPa and r = 7800 kg/m3) withstepwise constant circular cross section (Fig. 6). The rod is 600 mmlong, and it is made of a 200-mm resonant element with sectional ra-dius r0 = 10mmand two symmetric side channelswith radii r0 = 5mm.The slenderness of the system guaranteed waves below approximately118 kHz to be considered purely longitudinal (20). Elastic waves wereexcited at the ends of the system by two separate cylindrical piezo-electric actuators, which were glued to the rod through a thin epoxylayer. The excitation signal was first provided by a function generator(i.e., Agilent 33220A), then sent to an amplifier (E&I 1040L), and, lastly,sent to the actuators. The transient response of the system is measuredas the radial component of the velocity field through a scanning laserDoppler vibrometer (Polytec PSV-400M2). A grid of 648 points wasdefined across the entire length of the system. The data collection pro-cess consisted of exciting the system and then collecting the time re-sponse one grid point at a time. This approach assumed that theexperiment is repeatable and required that both the system’s excitationand its response measurement are repeated for each of the grid points.

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Fig. 5. Output energy redistribution due to input relative phase variation. (A) Exponential relative phase law f(t) = p[ehWIMtQ(−t) + Q(t)] for different values of theparameter h. (B) Effect of the relative phase variation on the total output energies EO+(L, t) and EO−(0, t) and their difference DE0.

Fig. 6. Schematic of the experimental setup.

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Once all the individual grid point’s time domain responses werecollected, they were combined to produce a representation of the entiresystem’s response in the space-time domain as a transient wavefield.The data sampling was performed at 512 kHz for 1 ms. For each pointin the measurement grid, 10 averages were performed to improve thesignal-to-noise ratio. Upon acquisition, the collected data were filteredin the frequency domainwith a band-pass filter between 30 and 90 kHz.The incident and scattered fields were identified on the basis of theirpropagation direction. The forward and backward propagatingwaves were isolated by Fourier-transforming themeasured wavefieldfrom the space-time domain to the frequency-wavenumber domain.Here, the frequency axis divides the domain into two subdomains,corresponding the components of the wavefield traveling in eitherthe forward or backward directions. By setting one of the two com-ponents to zero and inverse Fourier–transforming the information inthe space-time domain, it was possible to isolate the other component.On the basis of which component was filtered, we retained either theincident or the scattered field.

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SUPPLEMENTARY MATERIALSSupplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/8/eaaw3255/DC1Scattering matrixZeros of the scattering matrix

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AcknowledgmentsFunding: This work was supported by the Air Force Office of Scientific Research through MURIgrant No. FA9550-17-1-0002, the National Science Foundation through EFRI grant 1641069and EFRI grant 1741685 and the Simons Foundation. Author contributions: G.T. conductedthe experiments and developed the numerical codes in collaboration with Y.R., who alsocontributed to the formulation of the concept. A.A. formulated the idea and supervised theproject, while M.R. contributed to the theoretical formulation and supervised the experimentaldemonstrations. All authors contributed to writing the paper. Competing interests: The authorsdeclare that they have no competing interests. Data and materials availability: All dataneeded to evaluate the conclusions in the paper are present in the paper and/or theSupplementary Materials. Additional data related to this paper may be requested fromthe authors.

Submitted 9 December 2018Accepted 19 July 2019Published 30 August 201910.1126/sciadv.aaw3255

Citation: G. Trainiti, Y. Ra'di, M. Ruzzene, A. Alù, Coherent virtual absorption of elastodynamicwaves. Sci. Adv. 5, eaaw3255 (2019).

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Coherent virtual absorption of elastodynamic wavesG. Trainiti, Y. Ra'di, M. Ruzzene and A. Alù

DOI: 10.1126/sciadv.aaw3255 (8), eaaw3255.5Sci Adv 

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