Abstract
Control of sound in phononic bandgap structures promises novel control and guiding mechanisms. Designs in photonic systems were quickly matched in phononics and rows of defects in phononic crystals were shown to guide sound waves effectively. The vast majority of work in such phononic guiding has been in the frequency domain, because of the importance of the phononic dispersion relation in governing acoustic confinement in waveguides. However, frequencydomain studies miss vital information concerning the phase of the acoustic field and eigenstate coupling. Using a wide range of wavevectors k, we implement an ultrafast technique to probe the wave field evolution in straight and Lshaped phononic crystal surfacephonon waveguides in real and kspace in two spatial dimensions, thus revealing the eigenstateenergy redistribution processes and the coupling between different frequencydegenerate eigenstates. Such use of kt space is a first in acoustics and should have other interesting applications such as acousticmetamaterial characterization.
Introduction
Wavepackets can be tightly controlled by periodic media, such as electrons in solids, light in photonic crystals^{1}, sound in phononic crystals^{2} and coastal water waves in periodic sandbars^{3}. Interferences between waves reflected at each unit cell result in complex frequencywavevector (fk) relations characterized by frequency ranges of dispersive propagation separated by band gaps in which undamped eigenstates cannot exist. Bandgap physics and technology is the origin of revolutionary advances such as solidstate electronics and, more recently, the science of metamaterials, the latter spawning new fields such as negative refraction, superresolution focusing, slow light and cloaking^{4,5,6}. Momentumspace, i.e. kspace, monitoring of eigenstates directly reveals the physical processes at work in such systems: just as kspace can be used to follow the evolution of quantum eigenstates in atomic^{7}, solidstate^{8} and matterwave systems^{9}, it can also be used to follow the progression of quasimonochromatic wavepackets in photonic crystal waveguides^{10}. However, such kspace temporal imaging has not been applied to phononic eigenstates. In this paper we harness ultrafast imaging to achieve this in phononic crystal waveguides, which have never been mapped in the time domain, thus revealing their broadband propagation characteristics in real and kspaces, as well as in the frequency domain. Applications of our approach include acousticmetamaterial characterization.
We investigate structures based on defects in periodic elastic bandgap materials in the form of twodimensional (2D) surface phononic crystals (PCs)^{11,12,13} formed using microscopic holes in crystalline silicon. Our waveguides support surfacephonon wavepackets near ~1 GHz and are representative structures in the burgeoning field of phononic crystals^{13,14,15,16,17,18,19,20,21,22}.
Results
Circular holes with spacing a = 6.2 μm are milled as square lattices on (100) silicon substrates by dry reactive ion etching, leaving single rows of holes unopened to produce straight or Lshaped waveguides (Fig. 1a and b). The hole top diameter is 5.6 μm and the depth is ~100 μm. The 50 nm polycrystalline chromium coating does not significantly affect surface phonon propagation. The walls of the holes taper outwards and intersect with neighbouring holes at a depth of 3 μm (see Fig. 1d), helping to localize the acoustic energy near the surface. This implies that the surface waves inside the PC regions of the sample (as opposed to inside the waveguide regions) are more accurately described by Lamb waves rather than Rayleigh waves. The taper is determined from oblique scanning electron microscope (SEM) images (see Fig. 1c).
The 2D acoustic wave field is obtained by the use of an optical pumpandprobe technique combined with a commonpath interferometer^{23,24}. We capture a series of images with increasing pumpprobe delays to build animations (see the Supplementary Information) of the outofplane (zdirected) surface velocity over the 12.4ns laser repetition period at 0.41ns intervals. Figures 2a–c show a selection of frames (every 10th) from the straight waveguide. The optical pump beam is focused at a point ℓ = 14 μm outside the lefthand entrance of the waveguide (taking the lefthand edge of the first column of holes as a reference point). The first frame, Fig. 2a, shows the moment (t = 0) the pump pulse arrives at the surface. The following frames, Fig. 2b and c at 4.1 and 8.2 ns, respectively, show moments close to when the phonon pulses enter and leave the waveguide. Figures 2d–f show a similar series for the L waveguide (ℓ = 3 μm).
Timeresolved images superficially reveal the phonon dynamics, but the interaction between the eigenstates remains obscure. When the full complex wave field (i.e. separate amplitude and phase) is available, the temporal evolution in kspace can be directly obtained, as in the photonic case^{10}. However, for our real sound field f(r, t) a standard spatial Fourier transform is insufficient to resolve the signs of the wavevectors. This problem can be circumvented by calculating the ktspace amplitude F_{k}(k, t) using the socalled analytic signal^{25}:
F(k, ω) is the spatiotemporal Fourier transform of f(r, t) and ω the angular frequency. A direct spatial Fourier transform only yields , hence we apply Eq. (1) (see also the Supplementary Information). At a single angular frequency ω for wave propagation in two dimensions, there are usually an infinite number of degenerate states which satisfy the dispersion relations ω = ω(k). So the Fourier amplitude F_{k}(k, t) contains more information than an analogous function such as F(r, ω, t) (that could be obtained through wavelet analysis) and the dispersion relations. Even in the 1D case, one can discuss the scattering between a state k and a state −k from a knowledge of F_{k}(k, t). Here we show how in our 2D PC waveguides the function F_{k}(k, t) reveals the scattering dynamics.
The insets to the images in Fig. 2a–c show F_{k}(k, t) obtained from Eq. (1) for the straight waveguide by integrating to find F(k, ω) over the PC region, a 92 μm × 92 μm square, as well as over the region corresponding to the waveguide alone, a 92 μm × 7 μm rectangle in the waveguide (see Fig. 1a). The horizontal curves show the profiles for k_{y} = 0, dominated by waveguide contributions. Interestingly, interference fringes (dependent on the phase), analogous and related to those seen in real space, are evident in the integrations over the PC regions. In the following we shall see that kt space gives unprecedented information on the phonon dynamics that is impossible to gauge from real space. In Fig. 2a, at 0 ns, the newlygenerated phonon pulse is yet to enter the waveguide, but there is a small residual peak at k_{x}a/π = 0.8 –this being the main transmission wave number–arising from the previous phonon pulse. In Fig. 2b, at 4.1 ns, a phonon pulse is just entering the waveguide, producing a large broad peak centred at k_{x}a/π ≈ +2. The k_{y} = 0 profile at this point roughly corresponds to the broadband spectrum of phonon wavevectors initially excited (k_{x}a/π≈0.2–3 or wavelength λ = 2π/k_{x}≈4–60 μm). (The broadband acoustic frequency spectrum measured in a region not influenced by the PC waveguide is given in the Supplementary information.) As the wave front moves through the PC and waveguide, the height of this initial peak decreases (Fig. 2c) and transmission peaks emerge and decay, particularly in the waveguide at k_{x}a/π≈0.8, 1.6 and 2.2. A relatively strong peak, ~45% of that at k_{x}a/π = +1, is also evident in Fig. 2b for the guide near k_{x}a/π = −1, a wave number very close to the Bragg scattering condition from the phononic lattice: the shift from +k_{x} to −k_{x} is a reciprocal lattice vector G, indicating that this peak is a Bloch harmonic^{10,24,26} introduced by the periodic waveguide boundaries^{14}. The temporal dynamics of these peaks is closely related to the spatial decay of the corresponding waveguide eigenstates, as shown below quantitatively.
The equivalent series for F_{k}(k, t) for the L waveguide is shown in Fig. 2d–f. Regions corresponding to the two sections of the waveguide are treated separately (see Fig. 1b and Supplementary Information). The curves are the profiles for k_{x} = 0 (vertical curves–with y axis on the left) and k_{y} = 0 (horizontal curves). The peak near k_{y}a/π = +0.8 at 4.1 ns (see Fig. 2d–f and the animation–see Supplementary Information) shows that a significant acoustic amplitude, ~20%, is reflected at the bend. So in the horizontal section of the waveguide the eigenstate with k_{x}a/π = +0.8 has a reduced amplitude, ~40% of that of the equivalent eigenstate in the vertical section; the transmission around the bend is similar to that calculated for surface phonon waveguides of related design^{11} and losses can be explained by scattering to the bulk.
The ktspace data is related to two other representations, constantfrequency (rω) and frequency vs wavevector (ω vs k), which complement and validate kt space. Frequencyfiltered images of A = F_{ω}(r, ω) and Re[F_{ω}(r, ω)] = A cos ϕ (ϕ the phase) can be derived from temporal Fourier transforms^{24}. Figure 3a and b compare such experimental images of A and A cos ϕ at 322 and 482 MHz. The curves represent the profiles along the centre lines of the waveguides. At 322 MHz both waveguides transmit efficiently; a plausible explanation is the presence of a band gap in the PC, i.e. a stop or a deaf band^{27}, where the mode represented by the ω, k combination in the waveguide is evanescent in the PC or forbidden by the excitation symmetry, respectively. In contrast, at 482 MHz (and at other frequencies–see Supplementary Information) where we presume a pass band exists in the PC, there is strong attenuation inside both waveguides. (Waves diffracted round and reentering the waveguides from the right are estimated to have a small influence^{28}.)
Columns 2 and 4 of Fig. 3 show corresponding results from 3D finite element time domain numerical simulations using the geometry of Fig. 1a–f and impulse forces as periodic sources (see Supplementary Information). The agreement with experiment is good. Subsurface cross sections (column 4) confirm that at distances 2λ–3λ from the source, the waveguide eigenstates are concentrated within ~λ from the surface, as expected for Rayleighlike surface waves. At shorter distances scattering to bulk waves is evident.
To quantify the waveguiding efficiency, we fit a spatiallydecaying exponential function to the acoustic amplitude inside the straight waveguide. Figure 4a shows the frequency dependence of the derived decay constant (in ). At 322, 643 and 964 MHz, shows dips, possibly corresponding to frequencies inside PC band gaps^{14,15,27}. Similarly, frequencies with large should correspond to pass bands. Further evidence for efficient waveguiding can be obtained by considering the wave amplitude A at four different points on the Lwaveguide structure^{27} shown in Fig. 1b. Figure 4b shows that at points 1, 3 and 4 in the waveguide there are clear peaks at 322 MHz, as expected. Point 1 also has a distinct peak at 643 MHz (the second guiding region of the waveguide). The curve for point 2 shows that acoustic energy does not transmit easily into the PC structure, the highest amplitudes appearing at ~100 MHz.
We now turn to the ω vs k representation, accessible by a dual spacetime Fourier transform F(k, ω). We extract F(k, ω) for the same waveguide regions, as shown in Fig. 4d–f for both experiment and simulation, which largely agree. For the straight waveguide (Fig. 4d) the +k_{x}directed eigenstates are dominant. The slope is nearly linear, as previously predicted in simulations of surface waves in PC waveguides^{11,12} and corresponds to a phase velocity, v_{p}≈5 km/s, governed by Rayleighlike waves on the Si substrate^{29}. The previouslynoted Bloch harmonic, shifted by −2.0 in k_{x}a/π, is also visible at 402 MHz. Figure 4g shows the amplitude profiles along the ±k_{x} branches of the corresponding dispersion relations, allowing the quantitative comparison of counterpropagating eigenstates. Waveguiding peaks occur at k_{x}a/π≈0.8, 1.6 and 2.2 (322, 643 and 964 MHz), confirming our previous identifications. The higher frequencies have a greater amplitude, characteristic of the source. As explained later, the reflection coefficients corresponding to these transmission peaks are all very small, ~10%.
For the L waveguide (Fig. 4e, f and h, i), we divide into vertical (before the bend) and horizontal (after the bend) sections. Only one frequency, 322 MHz, is significantly maintained after the bend, as previously noted in our ktspace analysis and evident in the profiles in Fig. 3. The amplitude transmission and reflection coefficients at the bend agree with those measured in kt space.
One can now see how kt space is related to the spatial decay of eigenstates. Figure 4c shows the measured evolution of the kspace amplitude in the straight waveguide at three frequencies, making use of the linear waveguide dispersion relation to convert k to ω; at 322 MHz, in the first guiding window, the temporal decay constant is smaller than at the nonguiding frequencies 563 and 804 MHz. Converting the temporal decay constants to spatial ones using the derived v_{p} gives the crosses in Fig. 4a, in good agreement with from rω space. (Unlike the latter results, these points are derived without the approximation of ignoring counterpropagating waves–an advantage of using kt space.) The full temporal evolution of the eigenstateenergy (E) distribution F(E, t), sampled by the plots in Fig. 4c, can be viewed as animations in the Supplementary Material together with equivalent simulations.
Discussion
Acoustic energy transmits in and out of the waveguides very easily, as previously noted in numerical calculations^{12}. To better understand the origin of this effect we simulated a straight “slab” waveguide of fixed width w equal to that of the straight PC waveguide. Animations in kt space show a single peak at positive k_{x} with negligible reflection (see Supplementary Information). In addition, the slab dispersion is found to be linear–evidently a consequence of –and the attenuation practically zero and frequencyindependent. This reveals the physics behind the PC surfacephonon waveguiding: the strong frequency dependences we observe in transmission arise from the periodic nature of the containing medium and the weak endreflections and concomitant ease of incoupling, as well as the low surfacephonon dispersion regions–very different from the highlydispersive behaviour of bulk waves in PC waveguides exhibiting lateral resonances^{13,15}–are inherent in solid PC waveguide structures. In contrast to the strong impedance mismatch at the end of an organ pipe, in which ~100% of the acoustic energy is reflected^{30}, in such solid waveguides the surfacewave end reflection is strikingly small owing to a relatively low effective impedance mismatch.
In conclusion, by use of the raw wave field we have followed the evolution of a phononic system in both 2D real and kspaces for the first time. The acoustic propagation, damping, reflection and the in and outcoupling for phononic crystal waveguides are elucidated and the data corroborated in rω and ω vs k spaces. To our knowledge, animating kspace is new to acoustics and opens up fascinating possibilities. In addition to the characterization of surface acoustic wave devices in telecommunications, one could, for example, monitor acoustic pulses passing through new classes of acoustic metamaterials such as acoustic cloaks^{31} or negativeindex prisms^{20}. Our broadband approach to kt space should also be widely applicable in other fields: extension to ultrafast photonic imaging with attosecond pulses is one possible avenue^{32}; modifying angleresolved photoelectron spectroscopy^{8,33} for timedomain probing of electronic states in 2D kspace is another.
Methods
Surface phonons are thermoelastically excited by frequencydoubled optical pulses from a modelocked Ti:sapphire laser at a wavelength of 415 nm, repetition rate 80.4 MHz, duration ~200 fs, energy per pulse of ~0.15 nJ and a spot diameter at fullwidthhalfmaximum (FWHM) of 1.9 μm on the sample. Outofplane motion is detected interferometrically^{34} by a pair of 830 nm optical pulses that are separated by a time interval of τ ≈300 ps and are focused at normal incidence on the surface to a 1.5μm FWHM diameter spot. We monitor the optical phase difference between two consecutive probe pulses, proportional to the outward surface velocity of a surface particle. (τ is short compared with the minimum phonon period ~1 ns.) Chopping the pump beam at 1 MHz and using a lockin amplifier allows displacements ~10 pm to be resolved with a precision of ~0.1 pm at 100 Hz bandwidth. Each phonon image is built up by scanning the probe beam relative to the sample while keeping the pump focused on a fixed point at a fixed pumpprobe delay time. The continuous repetition of the pump laser source pulses induces a nondestructive local steadystate heating of ~4 K and a transient rise of ~250 K.
References
Joannopoulos, J. D., Villeneuve, P. R. & Fan, S. Photonic crystals: putting a new twist on light. Nature 386, 143–149 (1997).
Vasseur, J. O. et al. Experimental and theoretical evidence for the existence of absolute acoustic band gaps in twodimensional solid phononic crystals. Phys. Rev. Lett. 86, 3012–3015 (2001).
Mei, C. C. Resonant reflection of surface water waves by periodic sandbars. J. Fluid Mech. 152, 315–335 (1985).
Veselago, V. G. & Narimanov, E. E. The left hand of brightness: past, present and future of negative index materials. Nature Mater. 5, 759–762 (2006).
Zhang, X. & Liu, Z. Superlenses to overcome the diffraction limit. Nature Mater. 7, 435–441 (2008).
Ergin, T., Stenger, N., Brenner, P., Pendry, J. B. & Wegener, M. Threedimensional invisibility cloak at optical wavelengths. Science 328, 337–339 (2010).
Gopal, R. et al. Threedimensional momentum imaging of electron wave packet interference in fewcycle laser pulses. Phys. Rev. Lett. 103, 053001 (2009).
Rohwer, T. et al. Collapse of longrange charge order tracked by timeresolved photoemission at high momenta. Nature 471, 490–493 (2011).
Mark, M. J. et al. Demonstration of the temporal matterwave talbot effect for trapped matter waves. New J. Phys. 13, 085008 (2011).
Engelen, R. J. P. et al. Ultrafast evolution of photonic eigenstates in kspace. Nature Phys. 3, 401–405 (2007).
Sun, J.H. & Wu, T.T. Propagation of surface acoustic waves through sharply bent twodimensional phononic crystal waveguides using a finitedifference timedomain method. Phys. Rev. B 74, 174305 (2006).
Tanaka, Y., Yano, T. & Tamura, S. Surface guided waves in twodimensional phononic crystals. Wave Motion 44, 501–512 (2007).
Pennec, Y., Vasseur, J. O., DjafariRouhani, B., Dobrzyński, L. & Deymier, P. A. Twodimensional phononic crystals: Examples and applications. Surf. Sci. Rep. 65, 229–291 (2010).
Kafesaki, M., Sigalas, M. M. & Garcia, N. Frequency modulation in the transmittivity of wave guides in elasticwave bandgap materials. Phys. Rev. Lett. 85, 4044–4047 (2000).
Khelif, A., DjafariRouhani, B., Vasseur, J. O. & Deymier, P. A. Transmission and dispersion relations of perfect and defectcontaining waveguide structures in phononic band gap materials. Phys. Rev. B 68, 024302 (2003).
Yang, S. et al. Focusing of sound in a 3d phononic crystal. Phys. Rev. Lett. 93, 24301 (2004).
Gorishnyy, T., Ullal, C., Maldovan, M., Fytas, G. & Thomas, E. Hypersonic phononic crystals. Phys. Rev. Lett. 94, 115501 (2005).
Cheng, W., Wang, J., Ulrich, J., Fytas, G. & Stefanou, N. Observation and tuning of hypersonic bandgaps in colloidal crystals. Nat. Mater. 5, 830 (2006).
Ke, M. et al. Surface resonantstatesenhanced acoustic wave tunneling in twodimensional phononic crystals. Phys. Rev. Lett. 99, 44301 (2007).
Lu, M. et al. Negative birefraction of acoustic waves in a sonic crystal. Nat. Mater. 6, 744–748 (2007).
Still, T. et al. Simultaneous occurrence of structuredirected and particleresonanceinduced phononic gaps in colloidal films. Phys. Rev. Lett. 100, 194301 (2008).
Boechler, N., Theocharis, G. & Daraio, C. Bifurcationbased acoustic switching and rectification. Nat. Mater. 10, 665–668 (2011).
Sugawara, Y. et al. Watching ripples on crystals. Phys. Rev. Lett. 88, 185504 (2002).
Profunser, D. M., Muramoto, E., Matsuda, O., Wright, O. B. & Lang, U. Dynamic visualization of surface acoustic waves on a twodimensional phononic crystal. Phys. Rev. B 80, 014301–7 (2009).
Smith, J. Mathematics of the Discrete Fourier Transform (DFT) with Audio Applications (W3K Publishing, http://books.w3k.org/, 2007).
Profunser, D. M., Wright, O. B. & Matsuda, O. Imaging ripples on phononic crystals reveals acoustic band structure and bloch harmonics. Phys. Rev. Lett. 97, 055502 (2006).
Wu, T.C., Wu, T.T. & Hsu, J.C. Waveguiding and frequency selection of lamb waves in a plate with a periodic stubbed surface. Phys. Rev. B 79, 104306 (2009).
At 322 MHz these waves contribute to the −kx amplitude in the L waveguide.
Vines, R. E., Hauser, M. R. & Wolfe, J. P. Imaging of surface acoustic waves. Z. Phys. B 98, 255–271 (1995).
Blackstock, D. T. Fundamentals of Physical Acoustics (Wiley, New York, 2000).
Stenger, N., Wilhelm, M. & Wegener, M. Experiments on elastic cloaking in thin plates. Phys. Rev. Lett. 108, 14301 (2012).
Cavalieri, A. L. et al. Attosecond spectroscopy in condensed matter. Nature 449, 1029–1032 (2007).
Krömker, B. et al. Development of a momentum microscope for time resolved band structure imaging. Rev. Sci. Instrum. 79, 053702–053702 (2008).
Tachizaki, T. et al. Scanning ultrafast sagnac interferometry for imaging twodimensional surface wave propagation. Rev. Sci. Instrum. 77, 043713 (2006).
Acknowledgements
We thank Alex Maznev for stimulating discussions.
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P.H.O. and O.B.W. wrote the main manuscript text. K.N., M.T., D.P. and S.D. contributed to the measurements. I.A.V. and P.H.O. made the simulations. S.B. and K.N. prepared the samples. A.K., V.L., O.B.W. and O.M. provided theoretical guidance. All authors reviewed the manuscript.
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Supplementary Information
StraightWaveguide Experimental Movies
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StraightWaveguide Simulation Movies
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LWaveguide Experimental Movies
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LWaveguide Simulation Movies
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StraightWaveguide Experimental and Simulation EigenstateEnergy Animations
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Slab waveguide Simulation Movies
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Otsuka, P., Nanri, K., Matsuda, O. et al. Broadband evolution of phononiccrystalwaveguide eigenstates in real and kspaces. Sci Rep 3, 3351 (2013). https://doi.org/10.1038/srep03351
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