Research ArticlePHYSICS

Coherent virtual absorption of elastodynamic waves

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Science Advances  30 Aug 2019:
Vol. 5, no. 8, eaaw3255
DOI: 10.1126/sciadv.aaw3255

Abstract

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

INTRODUCTION

Efficient absorbers are of great importance in a wide variety of technological fields, from energy harvesting and radar detection in electromagnetics to sound proofing in acoustics and vibration isolation in mechanical systems (14). Common to these systems is the notion that efficient absorption can be achieved when the material loss is balanced by the impedance of the impinging wave. In other words, a proper amount of material loss can push one of the complex scattering zeros of the system onto the real frequency axis (5, 6), and, as a result, the impinging energy at this frequency is all lost into heat or other chemical processes. Therefore, the system is not conservative, and its scattering matrix is not unitary (for a multiport linear network, the scattering and transmission toward the ports are governed by the scattering matrix, which maps the incident fields to the outgoing fields).

Considering simultaneous excitation provides an additional degree of freedom to control the location of the scattering zeros of the system and to move one of them onto the real frequency axis. Coherent perfect absorption (CPA) is achieved through the interference of multiple incident waves impinging on the absorber, enabling a way to control the absorption mechanism in real time through the proper choice of the relative intensities and phases of the input beams. The dependence of CPAs on the input waveforms therefore provides the opportunity to flexibly control light scattering and absorption (7). The demonstration of an acoustic CPA has been presented in (8), opening a path toward several applications of practical interest, including highly sensitive detection and amplification of small variations in the incident signals or in the properties of the involved materials to realize mass or temperature transducers and for the efficient control and conversion of energy in harvesting applications (9, 10). These opportunities suggest that coherent absorption can also be of great interest in the context of elastodynamic waves.

In lossless systems, the scattering matrix cannot admit zeros on the real frequency axis, given its unitarity; therefore, the impinging wave needs to be transmitted, reflected, or scattered at all real frequencies. The zeros are necessarily confined to the upper half of the complex frequency plane (5, 6, 11, 12), above the real axis. It has been recently suggested that the time evolution of an incoming signal can be tailored to efficiently engage these complex zeros, implying that a specific choice of nonmonochromatic signals oscillating at a complex frequency can totally eliminate transmission, reflections, and scattering, thus realizing a virtual absorber with zero material loss (1315). As long as the input signal illuminates the structure with the right evolution in time, the impinging energy is neither scattered nor transmitted, but instead, it is captured and stored within the system with unitary efficiency. By varying the impinging signals, the stored energy can then be released through its scattering channels in a controllable fashion. Having such a coherent control over the stored energy in the cavity enables unprecedented functionalities, such as flexible energy storage and memory.

This concept is in some ways connected with the time-reversal excitation technique (16, 17), based on which, under the assumptions of linearity and time-reversal symmetry, the excitation of a cavity with a time-reversed replica of its decaying fields should be accepted by the cavity without scattering or reflections. However, one cannot draw a strict analogy between the two phenomena because they deal with different objectives. The time-reversal technique is based on the idea that an arbitrarily radiated pulse can be made to converge toward the source, provided that an array of sensors can time reverse it with the required accuracy. In contrast to coherent virtual absorption, the time-reversal approach does not therefore deal with eigenmodes of a structure. In the complex zero approach, the incident pulse is determined by the open cavity geometry. Exciting the structure with any of the complex zeros of the structure enables zero scattering and ideal wave capturing, which is a nontrivial conclusion. In analogy with the CPA operation, in the following, we experimentally demonstrate coherent virtual absorption of elastodynamic waves traveling along a solid bar, controlling storage and release of impinging longitudinal waves by tailoring their time evolution.

RESULTS

Consider a two-port lossless elastic waveguide with a circular cross section supporting longitudinal motion (Fig. 1A). The system can be divided into three domains, with stepwise constant cross section Aj=πrj2 for each j-th domain. A cavity of length L connects two identical side channels, excited by input signals Î+(0,ω) and Î(L,ω), as shown in Fig 1A. Because of the mechanical impedance mismatch of the central section, such a structure produces scattered fields Ô+(L,ω) and Ô(0,ω) at the ports, linearly related to the input fields through the frequency-based scattering matrix (18)[Ô+(L,ω)Ô(0,ω)]=[Ŝ11(ω)Ŝ12(ω)Ŝ21(ω)Ŝ22(ω)][Î+(0,ω)Î(L,ω)](1)

Fig. 1 Elastic coherent virtual absorber.

(A) Illustration of a mirror-symmetric waveguide with stepwise constant cross-sectional area A and wave velocity c. The system, 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 by the scattering matrix Ŝ(Ω). When Ω is analytically continued to the complex plane as Ω = ΩRE + iΩIM, Ŝ(Ω) has a countable infinite set of zeros, divided into symmetric and antisymmetric ones. The contour plot shows the quantity λA(Ω)=T̂(Ω)R̂(Ω) and the location of the antisymmetric zeros (black dots), as well as the location of the symmetric zero at ΩRE = 2π (blue dot). The inputs of the system can be designed to be equal to one of those zeros, thus canceling the scattered fields. (C) Incident and scattered fields for Ω = π + i0.51 such that λA(Ω) = 0. In this case, the scattered fields are identically zero for τ < 0. a.u., arbitrary unit. (D) Incident and scattered fields for Ω = 2π + i0.51 such that λA(Ω) ≠ 0. In this case, the scattered fields appear as soon as the excitations hit the structure.

Here and in the following, the hat symbol denotes a Fourier transform, while ω is the radial frequency. We assume that the waveguide is made of a single material of wave velocity c=E/ρ, where E and ρ are the material modulus of elasticity and density, respectively. Note that the impedance mismatch at the boundary of the outer and inner cores not only relates to the impedance mismatch of the materials on different sides of each interface but also is imparted by the cross-sectional areas of the rods on different sides of the interface (see the Supplementary Materials). Because of symmetry, Ŝ11(ω)=Ŝ22(ω)=R̂(ω), and because of time-reversal symmetry, Ŝ12(ω)=Ŝ21(ω)=T̂(ω). As a result, the components of the scattering matrix can be derived as (see the Supplementary Materials)R̂(ω)=R0+R1ei2ωLc1+R0R1ei2ωLc, T̂(ω)=T0T1eiωLc1+R0R1ei2ωLc(2)where R0=R1=(r02r12)/(r02+r12) and T0=r02r12T1=2r02/(r02+r12) are the local reflection and transmission coefficients at the two interfaces. Here, r0 and r1 are the radius of the outer and inner rods, respectively. By analyzing the scattering matrix, we investigate the conditions under which the system can efficiently absorb the incident energy impinging at its interfaces at x = 0 and x = L. For symmetric or antisymmetric excitations Î+(0,ω)=±Î(L,ω), because of symmetry, the outputs follow Ô+(L,ω)=±Ô(0,ω)=λ(ω)Î+(0,ω). The eigenvalue λ(ω) for the symmetric case is λS(ω)=T̂(ω)+R̂(ω), while that for the antisymmetric case is λA(ω)=T̂(ω)R̂(ω). Zeros of the scattering matrix, associated to perfectly absorbing modes, are found when λS(ω) = 0 or λA(ω) = 0, at frequenciesωz=cL[πn+iln(r12+r02r12r02)], n=1,2,3,(3)where even and odd n values correspond respectively to symmetric and antisymmetric excitations. In lossy systems, these zeros can correspond to real frequencies, yielding what is known in the optics literature as CPA (6, 7). This corresponds to a system with loss balanced to the outer impedance, for which coherent excitation on both sides with same or opposite phase is fully absorbed without transmission or scattering. If the system is lossless, however, then all these zeros lie in the upper half of the complex frequency plane, with ωz = ωRE + iωIM, ωIM > 0 (throughout the paper we assume an eiωt time convention). In this case, no real frequency excitation can be absorbed in the system, as expected from energy conservation. If instead we excite the structure coherently from the two input ports with time-growing waves oscillating at the complex frequency ωz and the proper relative phase, then we engage the corresponding complex zero of the system and achieve coherent 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 indefinitely, and the stored energy is released once the excitation is stopped or modified. While the signals are growing at the required rate eωIMt, the system stores energy at a rate proportional to eIMtcos2(2ωREt), which can be tailored with large flexibility by controlling the position of the complex zero in the frequency plane.

To investigate the dynamics of the virtual absorption process, consider a coherent antisymmetric excitation where I+(0, τ) = − I(L, τ) = f(τ), where τ = t/tL and tL = L/c is the time needed for the wave to travel through the cavity at speed c. To determine the required excitation signal f(τ), we first find the complex zeros for antisymmetric excitation by setting λA(Ω) = 0, where Ω = ωL/c = ΩRE + iΩIM. Figure 1B shows two of these zeros in the upper half of the complex frequency plane, indicated by black dots. We choose to engage the first zero, at Ω = π + i0.51. We therefore excite the two ports with input signals f(τ) to oscillate at the complex frequency Ω for τ < 0 and modulated by a fast-decaying exponential for τ > 0f(τ)=[eΩIMτΘ(τ)+e(Dτ)2Θ(τ)]cos(ΩREτ)(4)where Θ(τ) is the step function and D is a decay factor of choice. The excitation signals at the two ports are shown in the upper panel of Fig. 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 τ < 0, all the impinging energy is virtually absorbed and stored in the system. As soon as the input signals diverge from the virtual absorption condition, for τ > 0, the system releases its stored energy. In general, the system can release the stored energy at τ = 0 through all its complex poles, which are symmetrically located to its zeros in the complex frequency plane in the case of lossless systems because of time-reversal symmetry. Depending on the transient region around τ = 0, different eigenmodes of the system may be excited with different amplitudes. In the example at hand, the system releases its stored energy mostly into the first (dominant) eigenmode, which is consistent with a time-reversed replica of the input signals, but, for different truncation schemes of the input signal, the outgoing fields may be substantially different than the input signals. Figure 1D shows a scenario in which we excite the system antisymmetrically but at the complex frequency Ω = 2π + i0.51, which would correspond to a zero for even excitation. In this case, the incorrect relative phase of the incoming signals produces strong reflections at the port, and virtual absorption is not achieved. Simply flipping the phase of one of the two input signals would completely suppress all output fields for any τ < 0, underlining the importance of the coherent 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 geometry consists of a 0.6-m aluminum bar with a circular cross section and design parameters L = 0.2 m and r1 = 2r0 = 0.01 m. The system is excited antisymmetrically with piezoelectric actuators [lead-zirconate-titanate (PZT)] at its fifth zero [i.e., f = ω/2π = (64.85 + i2.11) kHz]. A scanning laser Doppler vibrometer (see Materials and Methods) measures the radial velocity field vr(x, t) along the waveguide in response to the external excitation. The radial contraction of the waveguide due to the Poisson effect, albeit not accounted for in our waveguide model, may slightly affect the dynamic properties of the system for slender structures at low frequencies; however, this effect is negligible here (see the Supplementary Materials). In parallel, to validate our experimental results, we developed a finite-difference time-domain (FDTD)–based tool to perform realistic numerical simulations of the same geometry (see Materials and Methods). Numerical simulations shown in Fig. 2B represent the radial velocity of incident (top) and scattered (bottom) fields, each normalized with respect to the peak velocity value of the incident fields. As it is seen from these figures, the incident energy is perfectly absorbed and stored in the middle portion of the bar, which acts as the resonating cavity, as long as the incident waveform is tailored to excite the complex zero of the system (in this example, this is the case for t < 270 μs). For the example considered here, the excited zero is not close to the real frequency axis; hence, the corresponding Q-factor is limited. For this reason, the stored energy inside the cavity is not much larger than the incident one. For a complex zero closer to the real frequency axis, i.e., for a larger Q-factor cavity, at any instant in time, the stored energy may be much larger, roughly equal to Q times the instantaneous impinging energy.

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 m and 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) and scattered 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 μs. (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 scattering only after the incident fields stop growing exponentially. [Photo credit for (A): Giuseppe Trainiti, Georgia Institute of Technology].

As soon as the incident wave starts decaying (i.e., for t > 270 μs), the cavity releases its energy. The standing wave pattern inside the cavity (middle rod) is due to its resonance. The experimental results shown in Fig. 2C are in very good agreement with our simulation results. Figure 2D shows a cut of the results shown in panels B and C of Fig. 2 at x = 0.5 m. The top panel in Fig. 2D compares the incident signals from numerical simulations and experiments, while the output signals are compared in the lower panel, confirming the evidence of coherent virtual absorption in the rod. In principle, one can excite the resonator with any of the complex zeros of the system. In our realization, we chose the fifth zero because of the limited dimensions of the symmetric side channels of length L. If we used one of the first zeros, then the incident pulses would be wider, meaning that after releasing the energy from the resonator, the signal will reach the end of the side rods (i.e., the position of source) and reflect back toward the middle resonator. This reflected signal would distort the measured signal at the probe point. However, by choosing longer side rods, 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 challenging, as the slender rod assumption may not hold for waves having wavelengths comparable to the rod cross section.

Having verified that the incident energy can be virtually absorbed and stored in a lossless resonator, we explore the degree of control over the release of stored energy, exploiting the coherence of the two impinging signals. Figure 3 illustrates a scenario in which we repeatedly excite the system at the complex zero, release its energy, and pump and release it again. The top panel shows the input signal, while the middle and bottom panels show the output signals and the stored energy in the middle section of the rod, respectively. The structure can coherently capture the impinging pulses with high efficiency and then release it at will as the exciting pulses are stopped. The system is then ready to store the next pulse. The release of stored energy can also be controlled by changing the relative phase ϕ of the excitation signals at the opposite ports, exploiting the coherence of the storage process. To investigate this scenario, we consider an exponentially modulated variation of the relative phase between the two input signals, ϕ(τ) = π[ehΩIMτΘ(−τ) + Θ(τ)], with h being a control parameter. We compare the case of ideal excitation of the complex zero (Fig. 4A) to the case in which the relative phase is slowly changed as ϕ(τ) (Fig. 4B). The middle panels show the instantaneous power at the input and output ports and the net power flow into the resonator [i.e., PR(τ) = PI+(0, τ) + PI(L, τ) − PO(L, τ) − PO+(0, τ)]. We also integrate these quantities in time to obtain a measure of the total energy entering and exiting the resonator, as well as the net energy stored in the system up to time τ [i.e., ER(τ) = EI(τ) − EO(τ)]. Figure 4B shows that, as soon as we start deviating the relative phase from the required value (in this example, we assume that h = 5), the resonator starts releasing its stored energy, again highlighting the effect of coherence in the storage process. In the case without phase variation (Fig. 4A), because of mirror symmetry, the resonator releases its energy equally through its scattering channels [PO(L, τ) = PO+(0, τ)]. On the contrary, in Fig. 4B, the asymmetry in excitation enables additional control on the port through which the stored energy is released. Note that the transition time over which the relative phase change is applied can markedly affect the redistribution of released energy between outputs. Figure 5 shows how the amount of released energy difference between outputs ΔE0 can be controlled by how quickly the relative phase ϕ(τ) varies from 0 to π, changing the value of h. For relatively small values of h, we break the symmetry between scattering ports and maximize ΔE0.

Fig. 3 Control of scattering and energy storage by changing the complex frequency of the excitation signals.

(A) Excitation signals. (B) Outgoing signals. (C) Stored energy in the system.

Fig. 4 Scattering and energy storage control through input relative phase variation.

(A) Response for zero relative phase ϕ(τ) = 0 (top) between the inputs at Ω = 5π + i0.51 is represented in terms of the normalized power inputs PI+(0, τ) and PI(L, τ), the power outputs PO(L, τ) and PO+(0, τ), and the power stored into the resonator PR(τ) (middle), as well as the associated integrals evaluated between −∞ and τ, with EI(τ) and EO(τ) as the energy that entered and exited the system up to time τ, respectively (bottom). (B) Imposing an exponentially increasing relative phase law ϕ = ϕ(τ) between the inputs (top) enables the dynamic control of the scattering process, with scattering onset anticipated at τ < 0 (middle), a different stored energy profile (bottom). The imposed relative phase also induces different energy redistribution between the two outputs of the system, with EO+(L, τ) − EO(0, τ) = ΔE0 ≠ 0.

Fig. 5 Output energy redistribution due to input relative phase variation.

(A) Exponential relative phase law ϕ(τ) = π[ehΩIMτΘ(−τ) + Θ(τ)] for different values of the parameter h. (B) Effect of the relative phase variation on the total output energies EO+(L, τ) and EO(0, τ) and their difference ΔE0.

In practice, growing the input signals for a long period of time may be impractical, and most of the stored energy at any given instant would, in any case, be contributed by the last part of the excitation transient, inversely proportional to the quality factor of the cavity. In this sense, cavities with higher-quality factors have complex zeros closer to the real frequency axis, implying that in this case, the excitation becomes quasi-harmonic. As the intensity grows, we may also incur into nonlinearities, which change the picture and break the temporal symmetry between the storage and release processes. Suitably tailored nonlinear cavities may be envisioned to accept an incoming signal with unitary efficiency but then trap it in an embedded eigenstate, as recently envisioned in the context of quantum optics (19).

DISCUSSION

Here, we have introduced and experimentally demonstrated the concept of coherent virtual absorption, storage, and release of energy on demand in elastodynamics. We have shown that the impinging displacement energy can be absorbed and stored in a lossless resonator with unitary efficiency for a desired period of time. We have also shown that we can control the release of the elastic energy with large flexibility and its directionality, exploiting the coherence of the storage process. Our results open interesting opportunities for applications in elastodynamics and structural mechanics, including sensors that may be able detect small changes in the resonance characteristics of a cavity, for example, as a result of applied mechanical strain, temperature, or material changes. These may also be of interest for energy storage and release. In addition, the control of the rate and time evolution of energy release may be beneficial for efficient conversion of mechanical energy into electrical or for the implementation of a broad range of memory, amplification, and computational functionalities within mechanical substrates. Efficient excitation of an elastodynamic or acoustic cavity; a phase-dependent, nonlinear amplification of the input signals; and controlled storage and release of acoustic energy are also relevant for focused sound generation as part of arrays, loudspeakers, and ultrasonic transducers. More broadly, this proof of concept may be directly translated to other phononic or photonic setups, enabling a large degree of control of phonons and photons using the coherence of specifically tailored nonmonochromatic signals.

MATERIALS AND METHODS

For the experimental validation of the concept of coherent virtual absorption in elastodynamics, the elastic waveguides were realized by a slender 1566 carbon steel rod (E = 210 GPa and ρ = 7800 kg/m3) with stepwise constant circular cross section (Fig. 6). The rod is 600 mm long, and it is made of a 200-mm resonant element with sectional radius r0 = 10 mm and two symmetric side channels with radii r0 = 5 mm. The slenderness of the system guaranteed waves below approximately 118 kHz to be considered purely longitudinal (20). Elastic waves were excited at the ends of the system by two separate cylindrical piezoelectric actuators, which were glued to the rod through a thin epoxy layer. 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 measured as the radial component of the velocity field through a scanning laser Doppler vibrometer (Polytec PSV-400M2). A grid of 648 points was defined across the entire length of the system. The data collection process consisted of exciting the system and then collecting the time response one grid point at a time. This approach assumed that the experiment is repeatable and required that both the system’s excitation and its response measurement are repeated for each of the grid points. Once all the individual grid point’s time domain responses were collected, they were combined to produce a representation of the entire system’s response in the space-time domain as a transient wavefield. The data sampling was performed at 512 kHz for 1 ms. For each point in the measurement grid, 10 averages were performed to improve the signal-to-noise ratio. Upon acquisition, the collected data were filtered in the frequency domain with a band-pass filter between 30 and 90 kHz. The incident and scattered fields were identified on the basis of their propagation direction. The forward and backward propagating waves were isolated by Fourier-transforming the measured wavefield from 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 either the forward or backward directions. By setting one of the two components to zero and inverse Fourier–transforming the information in the space-time domain, it was possible to isolate the other component. On the basis of which component was filtered, we retained either the incident or the scattered field.

Fig. 6 Schematic of the experimental setup.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/8/eaaw3255/DC1

Scattering matrix

Zeros of the scattering matrix

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REFERENCES AND NOTES

Acknowledgments: Funding: This work was supported by the Air Force Office of Scientific Research through MURI grant No. FA9550-17-1-0002, the National Science Foundation through EFRI grant 1641069 and EFRI grant 1741685 and the Simons Foundation. Author contributions: G.T. conducted the experiments and developed the numerical codes in collaboration with Y.R., who also contributed to the formulation of the concept. A.A. formulated the idea and supervised the project, while M.R. contributed to the theoretical formulation and supervised the experimental demonstrations. All authors contributed to writing the paper. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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