Abstract
The realization of phase discontinuities across metasurfaces has led to a new class of reflection and refraction. Here we present theory and experiment on the discontinuous propagation of wavepackets across subwavelengththickness metaatoms. Using acoustic waves, we observe the process of wavepackets traversing a metaatom with abrupt displacements, which appear as path discontinuities on a spacetime diagram. We construct a tunable metaatom from two coupled resonators at ~500 Hz, map the spatiotemporal trajectories of individual sonic pulses and reveal discontinuities at the metaatom where the pulses exit at a time ~50 ms ahead or behind their arrivals. Applications include thin acoustic metasurface lenses.
Introduction
Metamaterials are artificial structures engineered on a subwavelength level to produce exotic wave propagation characteristics^{1,2,3,4,5}. Ranges of constitutive parameters exhibited by their bulk forms have been extended to beyond classically available ranges in both electromagnetism and acoustics: single and double negativity as well as a wide range of doublepositive parameters have brought many remarkable phenomena into reality, including negative refraction, reverse Doppler effect, superlensing and cloaking^{6,7,8,9}.
Recently a new direction for metamaterials research has emerged by the introduction of metasurfaces^{10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}. Holloway et al.^{10} define a metasurface as any array of metamaterial unitcells (metaatoms) on a surface for which the thickness and periodicity are small compared to the wavelength in the surrounding media. For the simplest case of onedimensional (1D) waves propagating in a waveguide, a metasurface is represented by a dot, or single metaatom^{10}. With significantly smaller loss, metasurfaces produce abrupt phase and amplitude changes, enabling new mechanisms for refraction, polarization conversion and beam shaping^{15,16,17,18,19,20,21}.
Despite intensive investigations on abrupt phase shifts, few studies have been reported on the explicit action of metasurfaces on wavepackets, there having been only one reported case of the direct observation of wavepackets passing through a metasurface: Dolling et al.^{26} measured an abrupt advancement of the group timeshift across an electromagnetic metasurface in the nearinfrared. However, a detailed picture of the transmission process in the vicinity of the metasurface was not obtained. Here we present the first spatiotemporal measurement of a wavepacket as it propagates across a metaatom. To achieve this we use an acoustic system consisting of a tunable metaatom made up of two coupled resonators.
Geometry and theory of the acoustic metaatom
We start by briefly reviewing pulse transmission through a dispersive medium or an impedance boundary with dispersive transmission characteristics^{27}, showing analytically that metasurfaces producing a phase discontinuity for monochromatic waves also engender an abrupt displacement for passing wavepackets. This analysis applies to both electromagnetic and acoustic waves. The phase shift δ across a metasurface exhibits a strong dispersion. Therefore, the beat pattern resulting from the generic case of the superposition of two monochromatic waves of nearby frequency is altered by the difference in phase shift as they traverse the metasurface. Figure 1 shows schematically the case of a higher frequency wave at angular frequency ω_{h} encountering greater phase advancement δ than a lower frequency wave at ω_{l} [see Fig. 1a]. In this case it is straightforward to show that the temporal advance of the envelope τ_{g} [see Fig. 1b] is given by τ_{g} = Δδ/Δω.
For the more general case of a wavepacket of central frequency ω_{c} and amplitude p_{I} (t) = m (t) cos (ω_{c}t) = m(t)Re{exp(−iω_{c}t)} at the position of the metasurface, where m(t) is a slowly varying envelope function, the complex form of the transmitted wavepacket (t) is
where P_{I} (ω) is the Fourier transform of (t) and T(ω) is the complex transmission coefficient. If the spectral width of the wavepacket Δω is sufficiently narrow, T(ω) can be approximated to first order as T(ω) ≈ T(ω_{c}) exp(iτ_{p}ω_{c} − iτ_{g}(ω − ω_{c})), where τ_{p} = δ(ω_{c})/ω_{c} and τ_{g} = dδ/dω are respectively the phase shift and the group timeshift evaluated at the frequency ω_{c}. Here we are assuming that the transmission amplitude does not vary much over the spectral width of the pulse, Δω. Substituting for T(ω) in the above equation, we have
The transmitted wavepacket is therefore given by
Clearly, the envelope of the exiting wavepacket has the same shape and is temporally advanced by the amount
If the slope is positive, the wavepacket peak transmits in advance of its arrival, whereas if the slope is negative, it transmits with a delay.
As the wavepacket experiences a sudden temporal shift τ_{g}, it also experiences a spatial jump, the position of the exiting wavepacket being shifted by the amount η = c_{0}τ_{g}, where c_{0} is the phase velocity in the surrounding medium. Here we present observations of the process of wavepackets passing through a metaatom with a spatiotemporal jump at ~500 Hz by constructing the acoustic metaatom shown in Fig. 2a. Two acoustic resonators A and B are coupled by a cavity C, a system designed to emulate an atomic system exhibiting electromagnetically induced transparency (EIT)^{28,29}.
Resonator A is made from an 80 mm section of 15.5 mm innerdiameter tube with a thin (~10 μm thick) tensioned polyethylene membrane at one end. This end of the tube is connected to the coupling cavity C of length 51.8 mm and inner diameter 88 mm, whereas the other end is open to a one dimensional acoustic medium–a duct of diameter 30 mm–so that it interacts with external acoustic waves. The quality factor for resonator A, ~45, is comparable to that of typical Helmholtz resonators^{30}. Resonator B consists of a cavity of tunable length ~200 ± 50 mm and diameter 32.7 mm and a massloaded membrane of the same type as in A. Tunable metasurfaces mimicking EITlike effects have only previously been demonstrated for electromagnetic systems^{31,32,33}. The Qfactor of the resonator B is ~140, which is significantly better than that of typical acoustic resonators such as resonating ducts and Helmholtz resonators^{30}.
We shall derive the dynamics of this system with a lumpedelement approximation^{34}, appropriate here since the characteristic geometrical parameters are small compared with our acoustic wavelengths (λ~0.68 m). The air in tube A can be considered to move uniformly and its displacement ξ_{A}(t) in the direction of the open end is given by
where p(t) and p_{C} are the pressures at the tubeA mouth and in the cavity C, respectively and the effective mass m_{A} is the sum of the masses of the air in the tube and the membrane A. Here, ρ_{0}, S_{A} and are the density of air, the cross section and the effective length of the tube, respectively. (See the Supplementary Information for further experimental details.) Also, k_{A0} and b_{A} are the spring constant of the membrane and the drag coefficient of the air in the tube, respectively. Resonator A emits acoustic waves to the duct and so represents a “radiative” resonator^{28}. For convenience we shall include the radiation damping into the drag coefficient b_{A}. Resonator B has spring constant k_{memB}^{5} and thus its net spring constant is k_{B0} = k_{memB} + k_{air}. The contribution of the elasticity of the air in the tube B is , where V_{B} is the volume of this tube. The displacement ξ_{B} (t) is given by
where the effective mass m_{B} is the sum of the loaded mass, the membrane and the air moving with the membrane, b_{B} is the drag coefficient and S_{B} is the cross section of the tube. Here m_{A} and m_{B} are estimated to be 0.0233 and 0.56 g, respectively. The loaded mass on the resonator B membrane is estimated to be 0.34 g. Resonator B is not directly coupled to the duct and so represents a “dark” resonator^{28}. The pressure p_{C} is related to ξ_{A} and ξ_{B} as follows:
where B_{0} is the bulk modulus of air.
Equations (2)–(4) can be rewritten in the form
where , are acoustic inertances, , , k_{C} = B_{0}/V_{C} are acoustic stiffnesses, , are acoustic drag coefficients of the resonators A and B and V_{C} is the cavity volume C. These equations have the classical form of an EIT system^{28}. If, in the uncoupled state, A and B are set to have the same resonance frequency, the coupling through the cavity C removes the degeneracy and results in a split resonance response and a consequent sharp phase dispersion. In practice A and B are set to have similar uncoupled resonant frequencies.
The pressure should be continuous, so p_{i} + p_{r} = p_{t}, where p_{i}, p_{r} and p_{t} are the pressures of the incident, reflected and transmitted waves, respectively. Using the reflection and transmission coefficients R = p_{r}/p_{i} and T = p_{t}/p_{i}, one obtains the relation 1 + R = T. Also, the particlevelocity boundary condition is determined by equating the volume velocities into and out of the metaatom:
where S_{duct} is the crosssectional area of the duct and the particle velocities of the incident, reflected and transmitted waves are defined as u_{i}, u_{r} and u_{t}, respectively. From the acoustic impedance relations, u_{i} = p_{i}/Z_{0}, u_{r} = −p_{r}/Z_{0} and u_{t} = p_{t}/Z_{0}, Eq. (6) becomes
where is the impedance of air. These relations yield the acousticamplitude transmission coefficient:
An analytical expression for T can be obtained from the particle velocity by solving the coupled equations of motion for the system, Eqs. (5), under sinusoidal excitation conditions (see the Supplementary Information).
Results
Equation (8) predicts a phase discontinuity across the metaatom. To investigate this experimentally we measured T(ω) = T(ω)exp(−iδ (ω)) for monochromatic waves, where T(ω) and δ(ω) are the magnitude and phase advance, respectively, over the frequency range 400 to 600 Hz using the setup shown in Fig. 2b, with microphones placed 160 mm away either side of the metaatom. The data for l_{B} = 195 mm (V_{B} = 163.8 cm^{3}) are shown in Fig. 3a by the dots. The solid (red) theoretical curves are obtained from the solutions of Eq. (8). These curves, based on measured and fitted parameters (see Supplementary Information), agree with the experiment with only minor differences, the differences presumably stemming from the approximate nature of the lumpedelement approximation and the effective lengths.
It is evident that our system can produce phase discontinuities ~±1 radian for a single metaatom, analogous to those seen (up to ±π rad^{11,17,21}) in electromagnetic metasurfaces or atomic systems. Our system provides a grouptime shift, which is important in broadband lensing applications^{35}. However, for full control of the waves, additional measures may be needed because the span of the phase shift δ in our system is not optimal for redirecting beams. Group shifts were measured by comparing the time differences of the peaks for the incident and transmitted pulses. Figure 3b shows the results of experiments with Gaussian wavepackets of spectral width 3 Hz for respectively, the group shift as a function of frequency and the pressure amplitude as a function of time at the central frequencies 506 and 512 Hz, corresponding respectively to the the green solid circles p and q in Fig. 3a. The incident and transmitted wavepackets are shown in blue (larger pulses) and red (smaller pulses), respectively. At 506 Hz, the peak of the transmitted pulse is delayed by 54.0 ms (having the same shape but with reduced height), whereas at 512 Hz it is advanced by 40.3 ms.
To access the full spatiotemporal field, we measured the time evolution of the pressure distribution in the tube near the metaatom before and after the wavepacket passes through it. The results are shown in the normalized spacetime diagrams of Fig. 3c. A discontinuity is clearly seen at the position z = 0 of the metaatom. The trajectories for incident, reflected and transmitted pulses far (>λ) from the metaatom were not measured, but should be straight lines with slopes ±c_{0}, where c_{0} = 340 m/s is the speed of sound. At the position of the metaatom, the spacetime trajectory makes temporal steps up to −54 ms and 36 ms. These steps occur over a distance ~0.02 m (~0.03λ), governed by the diameter (15.5 mm) of the radiative resonator. This spatiotemporal observation of a positive temporal step provides a first detailed view of the “supersonic” transmission of pulses through a subwavelengththickness metaatom, analogous to that of “superluminal” propagation of light through a microscopic interface. Negative group delay here does not imply causality violation. It just means that the peak of the transmitted wavepacket appears before the arrival of the peak of the incident one. As demonstrated in earlier works^{36}, this is because the peak of the transmitted wavepacket is not causally connected to that of the incident one: more energy of the wavepacket is absorbed from the rear part than from the front part.
Group shifts in metasurfaces consisting of resonating metaatoms result from rapid changes in phase shift δ near the resonance frequencies. Clearly, for maximum group shift the metaatoms should be designed for maximum dδ/dω. Our structure exhibits a maximum group shift τ_{g} ≈54 ms, corresponding to the distance c_{0}τ_{g} ≈18 m, or 27λ. Our work greatly improves on the magnitude of the discontinuity compared to the value of 4λ reported by the timedomain measurements of Dolling et al.^{26}.
For the case of optical pulses used for digital communications, the pulse widths are approximately equal to the coherence length. When metasurface lenses are used to refocus such broadband pulses, the path lengths for rays passing through different parts of the metasurface are different. To compensate the path differences it is necessary to provide appropriate group shifts τ_{g} which vary with position over a wide tuning range. We tuned the group shift of our metaatom for a fixed central frequency of 500 Hz by adjusting the resonance frequency of the dark oscillator, i.e., resonator B, via the cavity volume V_{B}, as shown by the results of Fig. 4a. The measured incident (larger, blue) and transmitted (smaller, red) wavepackets are plotted in Fig. 4b and are marked by the green solid squares in Fig. 4a. Here the group shift is tunable from −55 ms (slightly better than our previous −54 ms) to 29 ms, corresponding to a path difference of ~42λ. This range is ample for applications to metasurfaces functioning as thin sonic lenses. 3D plots of T and δ as a function of frequency and V_{B} are given in Fig. 4c. The curves for a fixed frequency of 500 Hz are redrawn in the insets.
Discussion
As previously mentioned, negative group delay in electromagnetism or acoustics ushers in the possibility of “superluminal” or “supersonic” propagation. For electromagnetic pulses this is experimentally well established^{37,38,39,40}. Researchers sometimes account for such effects in metasurfaces from the effective bulk constitutive parameters, by regarding singlelayer metasurfaces as slabs of metamaterials with defined thicknesses. However, problems arise from the artificial character of the definition of such thicknesses that obscure, at a fundamental level, the localized effects at work at the sample boundaries^{10}. Instead of bulk parameters such as permittivity, permeability, density or modulus, all that needs to be known to characterize a metasurface are the delay and attenuation engendered: in Ref. 10 it was shown that metasurfaces are well described by modelling them as sheets of zero thicknesses that abruptly change the phases and amplitudes of the reflected and transmitted waves owing to the scattering characteristics of the resonant metaatoms. Our spatiotemporal pressure measurements clearly support this picture. It would be interesting to extend this research to acousticfield spatiotemporal imaging of waves passing through metasurfaces and to electromagneticfield spatiotemporal imaging in analogous EITlike systems; several reported electromagnetic metamaterials mimicking EIT are of planar form^{31,32,33,41,42}. Therefore, this electromagneticfield imaging should be feasible using similar structures.
In conclusion, we have analytically and experimentally demonstrated that metaatoms producing phase discontinuities for monochromagnetic waves also produce discontinuities in the spacetime trajectories of wavepackets. Our tunable experimental design, based on coupled radiative and dark acoustic oscillators mimicking an EIT system, allows us to observe transmitted pulses at ~500 Hz that leave the metaatom with group shifts in the range −55 ms to 40 ms, corresponding to record values of the normalized timeshift −28 < c_{0}τ_{g}/λ < 20 for a roomtemperature metamaterial system. A lumpedelement model accurately describes the underlying dispersion. Observation of the entire process of the wavepackets exiting the metaatom both ahead and behind of their arrivals with spacetime diagrams accurately reveals the discontinuities represented by the group timeshift τ_{g} = dδ/dω. This work opens the way to practical focusing devices for acoustic waves based on 2D arrays^{22,23} of our metaatom and to novel applications in beam steering and wavefront shaping not only in the audio range but also for analogous systems at ultrasonic frequencies.
Methods
A 10 W loudspeaker driven by a function generator provides sinusoidal or pulsed excitation at 1 m distance from the metaatom. For the anechoic termination we used 60 m lengths of 30 mm diameter corrugated tubes attached to both ends of the duct, each closed with a commercial air filter for automobiles (Hyundai 28113 37101) to prevent residual reflections.
As shown in Fig. 2b, the acoustic metaatom is positioned in the center of the duct. Resonators A and B and cavity C are fashioned from plastic tubes and resonator B is fitted with a plunger to allow its volume to be tuned. The effective masses m_{A} and m_{B} of the resonators A and B are estimated to be 0.0233 and 0.56 g, respectively. The loaded mass on the resonator B membrane is estimated to be 0.34 g.
For the frequency domain measurements, probe microphones are set halfway up the duct, as shown in Fig. 2b. For the spatiotemporal measurements these probe microphones are scanned along the tube length to produce the spacetime plots.
The envelopes of the Gaussian wavepackets of spectral width 3 Hz (full width at half maximum in the power spectrum) were extracted by taking the magnitudes of the Hilbert transform of the waveforms, a widely used demodulation scheme for amplitudemodulated signals^{43}. Then the curves were fitted using a leastsquares algorithm to find the peaks and the group shifts were then obtained from the differences.
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Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (NRF2013K2A2A4003469 and NRF2013R1A5A1A95042044). We also acknowledge Grantsinaid for Scientific Research from the Japan Society for the Promotion of Education, Science, Sports and Culture.
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S.L. and O.B.W. wrote the main manuscript text. I.Y., S.L. and C.K.H. contributed to the measurements. S.L., I.Y., K.J.B.L., J.W.W., D.S.S., H.S.M. and O.B.W. contributed to the theory. All authors reviewed the manuscript.
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Supplementary Information
Spatiotemporal path discontinuities of wavepackets propagating across a metaatom
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Yoo, I., Han, C., Shin, DS. et al. Spatiotemporal path discontinuities of wavepackets propagating across a metaatom. Sci Rep 4, 4634 (2014). https://doi.org/10.1038/srep04634
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