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1 Ultrasonics 5 (21) Contents lists available at ScienceDirect Ultrasonics journal homepage: Solitary surface acoustic waves and bulk solitons in nanosecond and picosecond laser ultrasonics Peter Hess *, Alexey M. Lomonosov Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253, D-6912 Heidelberg, Germany article info abstract Article history: Received 2 July 29 Received in revised form 13 July 29 Accepted 4 August 29 Available online 9 August 29 PACS: y b d Ja Recent achievements of nonlinear acoustics concerning the realization of solitons and solitary waves in crystals and their surfaces attained by nanosecond and picosecond laser ultrasonics are discussed and compared. The corresponding pump probe setups are described, which allow an all-optical contact-free excitation and detection of short strain pulses in the broad frequency range between 1 MHz and about 3 GHz. The formation of solitons in the propagating longitudinal strain pulses is investigated for nonlinear media with intrinsic lattice-based dispersion. The excitation of solitary surface acoustic waves is realized by a geometric film-based dispersion effect. Future developments and potential applications of nonlinear nanosecond and picosecond ultrasonics are discussed. Ó 29 Published by Elsevier B.V. Keywords: Nonlinear acoustics Nano- and picosecond ultrasonics Solitons Shock waves Laser ultrasonics 1. Introduction The invention of lasers in 196 had and is still having a strong impact on the fast development of nonlinear sciences. This is true not only for the field of nonlinear optics but also for continued progress in nonlinear acoustics with its long tradition. In fact, nonlinear effects seem to be even more pronounced in acoustics than in optics [1]. Laser techniques allow the efficient contact-free excitation and detection of coherent pulses of longitudinal acoustic bulk waves and surface acoustic waves (SAWs) with shocks in nonlinear media. After a strain pulse with finite amplitude is launched with a short pump-laser pulse, the developing shock front becomes steeper and steeper with propagation distance due to the elastic nonlinearity of the material, caused by the anharmonicity of the interatomic potential. This process can be described as a resonant coupling of higher harmonics with the fundamental wave, yielding an increasingly nonlinear pulse profile. If the effects of nonlinearity are balanced by intrinsic or deliberately introduced geometric dispersion, solitons or solitary pulses can be formed. In this situation the dispersion effect limits the * Corresponding author. Tel.: ; fax: address: peter.hess@urz.uni-heidelberg.de (P. Hess). growth of higher harmonics, generating a localized stable wavepacket that keeps its shape while traveling. This is remarkable since a constant shape is usually observed only in systems that are both linear and non-dispersive. The strain pulses are detected at the surface by applying an optical probe, which measures the change in optical reflectivity or the transient surface displacement or its velocity. Here two laser-based approaches to nonlinear acoustics will be discussed, the generation of nonlinear longitudinal bulk waves using picosecond ultrasonics, and studies of nonlinear SAW pulses employing nanosecond lasers. Note that the different time scales involved (nanosecond versus picosecond) and types of acoustic waves excited (bulk versus surface wave) provide complementary insight into the nature of moderate and strongly nonlinear elastic wavepackets. Only longitudinal pulses propagating in crystals and Rayleigh waves launched on the plane surface of an elastic half space will be considered. 2. Experimental 2.1. Excitation and detection of longitudinal acoustic strain pulses In the last few years laser-based picosecond ultrasonics has made tremendous progress in the excitation of nonlinear ultrashort strain pulses and their optical detection by the changes X/$ - see front matter Ó 29 Published by Elsevier B.V. doi:1.116/j.ultras

2 168 P. Hess, A.M. Lomonosov / Ultrasonics 5 (21) induced in the reflectivity of the laser probe [2,3] or by using Brillouin scattering [4]. The so-called low and high power laser sources employed were a mode-locked Ti:sapphire laser radiating at 8 nm with nanojoule pulse energy and 2 fs pulse duration at 8 MHz [2] and an amplified Ti:sapphire laser system radiating at 8 nm with millijoule pulse energy and 1 fs pulse duration at 1 khz [4]. The scheme of a typical picosecond pump probe setup is presented in Fig. 1. The optical pump pulses are strongly absorbed in a 15 5 nm thick metal film (e.g. Al, Ti, Cr) within a depth of tens of nanometers and launch two counterpropagating strain pulses by thermoelastic emission. The absorbed photons excite conduction electrons, which diffuse through the thin metal film and equilibrate later with the phonon system, heating the lattice by a few degrees. With the sample configuration of Fig. 1a bipolar strain pulse profile is launched. The elastic nonlinearity, caused by the anharmonicity of the intermolecular potential, decreases in the order MgO, sapphire, a-quartz, and silicon for the materials investigated in the first detection of bulk solitons [2]. The intrinsic phonon dispersion, separating higher and lower frequencies, is caused by the discrete spacing of the lattice atoms in the crystal. The shape of an emitted bipolar pulse changes during propagation in a nonlinear and dispersive medium, forming a coherent train of solitons and an oscillating tail, as a result of the compensation of phonon dispersion and elastic nonlinearity. The modified strain pulse hits the detector, consisting of a 15 8 nm thick metal film (e.g. Al) on the opposite side of the sample. The resulting change in reflectivity is monitored as a function of time by shifting the probe pulse with respect to the pump pulse by a suitable delay line. Calculation of the strain from the measured optical reflectivity change yields the shape of the strain pulse [5] Excitation and detection of nonlinear surface acoustic wave pulses SAWs are guided waves, which penetrate approximately one wavelength deep into the solid. It is important to note that the main part of the pulse energy stays within this depth during wave propagation along the surface. This particular property reduces diffraction losses as compared with acoustic bulk waves. To launch strongly nonlinear SAW pulses no sophisticated laser system is required. Typically, Nd:YAG lasers radiating at 1.64 lm with 3 16 mj pulse energy and 1 8 ns pulse duration are used in single-pulse experiments [6]. As depicted in Fig. 2, the explosive evaporation of a thin layer of a highly absorbing carbon suspension, deposited in the source region, is employed to excite SAWs with finite amplitude. By sharply focusing the pump-laser pulse with a cylindrical lens into a line a plane surface wave propagating Pump 2-5 nm Transducer (Al,Ti,Cr) ~mm Nonlinear medium (MgO, sapphire) 3-8 nm Probe Metal detector (Al) Fig. 1. Schematic diagram of a typical picosecond laser pump probe setup for generating longitudinal strain pulses in a thin metal film propagating in the sample to the metal detector. in a well-defined direction is obtained [7,8]. During propagation the shape of the elastic pulse changes due to nonlinear elasticity, which in the frequency domain is visible as both frequency-up and frequency-down conversion that steepens the shocks and changes the pulse length. Since Rayleigh waves are non-dispersive in homogeneous media, a layer needs to be added to introduce geometric dispersion caused by the finite width of the layer. The transient surface displacement can be detected at a distance of several millimeters from the source with a stabilized Michelson interferometer [6]. Frequently, however, another method is applied that uses transient deflection of a cw probe-laser beam monitored by a position-sensitive detector. This delivers the vertical surface velocity or shear displacement gradient. This detection method is substantially simpler to use. In the two-point-probe method SAWs are registered at 1 2 mm and 15 2 mm from the line source, which enables one to evaluate the nonlinear evolution. The pulse shape of the first probe spot is inserted as initial condition in the nonlinear evolution equation to simulate the nonlinear pulse development. The profile measured at the second probe spot allows a comparison between theory and experiment [6]. 3. Result Laser pulse CW probe beams Fig. 2. Schematic diagram of a nanosecond pump probe setup for generating SAW pulses with a pulsed laser and two-point laser probe-beam deflection detected with a position-sensitive detector Bulk solitons and formation of shocks in picosecond ultrasonics Longitudinal bulk solitons have been excited and detected at low temperatures around K in the crystalline materials silicon, a-quartz, sapphire, and MgO, with strongly increasing nonlinear parameters (C 3 ) from silicon to MgO, using the mode-locked low-power Ti:sapphire laser [2]. Fig. 3 illustrates the typical shape of an excited linear strain pulse, the development of nonlinear profiles, and of solitons if intrinsic dispersion is included. In the latter case the leading compressive part of the bipolar pulse transforms into a soliton, whereas the trailing rarefaction part forms an oscillating radiative tail [9]. Note that the compressive part, consisting of one or several half-cycle strain pulses, is traveling slightly faster and the rarefaction part slightly slower than the linear sound velocity. While for silicon (C 3 = gcm 1 s 2 ) only one soliton has been separated from the compressive phase (see Fig. 4), for MgO (C 3 = gcm 1 s 2 ) the development of four solitons could be observed at a laser-pulse energy of 1 2 nj [2,1]. As expected from theory, the velocity of solitons increased with their magnitude. At higher temperatures, e.g., 3 K, the strain pulse is not able to travel far enough to form a soliton due to strong attenuation that is mainly caused by the interaction with thermal phonons. At very low temperatures, however, attenuation and diffraction of the strain pulse may be negligible. In fact, simulations of soliton pulse shapes by a one-dimensional (1D) treatment, based

3 P. Hess, A.M. Lomonosov / Ultrasonics 5 (21) (a) 8 x1.22 mj/cm 2 (arb.units) Strain (b) (c) Retarted time (arb.units) Fig. 3. Illustration of the development of an initial strain pulse (a) into a nonlinear strain pulse with shocks (b) and into a bulk soliton in the presence of dispersion (c) (see [5] for details). 3 1 Strain x mj/cm mj/cm mj/cm 2 Reflectivity change (arb.units) Time (ps) 12 Fig. 5. Formation of strain pulses calculated from measured reflectivities as a function of time. Formation of shock fronts in the propagating strain pulses in sapphire as a function of laser fluence by picosecond ultrasonics; dashed curve: initial pulse shape (see [5] for details). Time (arb.units) Fig. 4. Change of reflectivity versus time showing the separation of a soliton and an oscillatory tail from the propagating picosecond strain pulse in silicon; measurement: solid line, simulation with KdV equation: dashed line (see [1] for details). on the Korteweg de Vries (KdV) equation, yielded good agreement with the experiments [2,9]. As an example, Fig. 4 displays the reflectivity change measured and simulated for a soliton generated in silicon [1]. Solitons have also been studied in GaAs at 15 K by interferometric detection [11]. With the high-power Ti:sapphire laser, strongly nonlinear picosecond strain pulses could be generated for the first time at 3 K [5]. For laser fluencies of mj/cm 2 the bipolar strain pulse observed at the lowest fluence was continuously stretched, extending by about a factor of three at the highest fluence studied. Besides the increase of pulse length and amplitude with fluence, the extending pulse profile developed shock fronts. An extremely short rise time of 1.2 ps of the supersonic leading edge and subsonic trailing edge of the strain pulse has been reported (see Fig. 5). Due to elastic nonlinearity the acoustic frequency spectrum increased substantially from 1 GHz up to 5 GHz by frequencyup conversion. Simulations of wavepackets propagating in a viscous and nonlinear medium, based on the Burgers equation, allowed a description of the nonlinear pulse shapes [5] Solitary surface acoustic waves in nanosecond ultrasonics Nonlinear SAW pulses with spikes and shocks were excited in isotropic fused silica and anisotropic silicon with a nanosecond Nd:YAG laser at 1.64 lm and pulse energies of <13 mj, using the absorption-layer technique [6 8]. Since SAWs show no dispersion in a homogeneous half-space, a film was deposited on the substrate to cause formation of solitary pulses. In a layered system the film thickness introduces a new length scale and the medium becomes dispersive. This length scale is much larger than the interatomic spacing. In fact, intrinsic dispersion usually is neglected. During propagation in the nonlinear and dispersive medium stable bipolar or Mexican hat shaped solitary pulses are formed, which travel faster than the linear Rayleigh velocity, whereas the small oscillatory tail ( radiation ) moves with lower velocity for a loaded system, or vice versa for the stiffening case [12 14]. Fig. 6a and b shows the two stationary solutions of the nonlinear SAW evolution equation with a linear dispersion term, namely a bipolar and Mexican hat profile. A bipolar pulse was observed for a NiCr film loading the silica substrate, leading to normal dispersion as shown in Fig. 6c. An added TiN film stiffens silica, leading to anomalous dispersion that changes the polarity of the solitary pulse. In this latter case the solitary pulse travels with a speed lower than the linear Rayleigh velocity [12]. A Mexican hat shaped solitary pulse was found by covering the Si(1 1 1) surface with a thermally grown 11 nm thick silicon-dioxide layer, responsible for normal dispersion [12] (see Fig. 6d). In this system the theoretically predicted Mexican hat shaped profile [13] could be seen for the first time. The observed bipolar and Mexican hat shaped pulses agree well with numerical solutions of the nonlinear evolution equation with an essentially nonlocal character of nonlinearity and a linear dispersion term of the Benjamin Ono type. In this respect the nonlinear evolution of surface waves differs from that in fluids, which can be described by the KdV equation [14]. 4. Discussion and conclusions 4.1. Comparison of time and length scales The two ultrasound methods presented above for studying the effects of nonlinear elasticity differ substantially in their frequency bandwidth with strain pulses on the nanosecond and picosecond time scale. Accordingly, the bandwidth reaches several hundred gigahertz in picosecond ultrasonics, whereas the highest frequencies in the nanosecond regime were 1 GHz. This implies wavelengths in the micrometer range in the latter case, but nanometer wavelengths in picosecond ultrasonics.

4 17 P. Hess, A.M. Lomonosov / Ultrasonics 5 (21) Shear displacement gradient (arb.units) a) b) c) d) Retarded time (ns) Fig. 6. Theoretical bipolar (a) and Mexican hat (b) shaped solitary surface wave pulses calculated by using a nonlinear evolution equation with linear dispersion term. Measured solitary pulses for the systems silica/nicr film (c) and silicon/sio 2 film (d). The frequency range has a strong influence on the attenuation encountered by the strain pulse. Typically, the evolution of nonlinear pulses occurs over length scales of hundreds of micrometers, which is large compared to the characteristic wavelength in picosecond ultrasonics. For example, the propagation distance needed for the development of shocks in sapphire was about.1 mm for the highest fluencies studied at 3 K [5], whereas in nanosecond ultrasonics less than 1 mm was needed to form steep shock fronts that even fracture silicon in easy-cracking geometries [15 18]. Note that silicon has a substantially lower nonlinear parameter than sapphire [2] Differences between nonlinear bulk and surface waves While in nanosecond ultrasonics frequency-up conversion concentrates the energy in an even smaller depth from the surface, diffraction is an important issue in longitudinal strain pulses. Geometrical focusing has been suggested as a means to increase the strain in longitudinal bulk pulses to approach the elastic limit [9]. Another important point is the laser-pulse energy and power employed for excitation. The pulse energies applied were much higher in nanosecond ultrasonics. This fact is partly connected with the availability of higher pulse energies but also with the irreversible destruction of the sample by high-power laser pulses. With the two-point-probe detection the SAW experiments were independent of any local damage occurring at the source line [7]. In principle, it is very difficult to estimate accurately the stress exerted in the surface region by explosive evaporation of the absorption layer and to compare this process with the thermoelastic generation of strain pulses in a metal transducer. The absorption-layer method seems to be almost two orders of magnitude more efficient than the thermoelastic effect. A rough estimate yields a stress of 61 GPa exerted by explosive evaporation, which explains the destruction seen along the source line in silicon in agreement with its mechanical strength [18]. Unfortunately, no information on potential damage caused by the highest acoustic strain of 4 1 3, realized with the metal film, has been reported for picosecond experiments [9]. The accurate resolution of the rise time of laser-generated shock fronts is a serious experimental problem. Of course, picosecond ultrasonics provides a substantially better time resolution in this respect, with the fastest reported rise time of the shock front of 1.2 ps [5]. In any case, the accurate determination of the steepness of shock fronts remains a challenge and effective rise times are much more difficult to ascertain than pulse elongation Future applications To the extent nonlinear picosecond ultrasonics is confined to very low temperatures, practical applications will be limited. In fundamental research the extension of studies from 1D to higher-dimensional solitons can be expected in the near future. The 1D equation of KdV has already be combined with a 3D analysis of diffraction effects in the far field for crystalline silicon [19]. Intense ultrashort acoustic wave pulses with their highly localized energy content may find applications in investigations of vibrational dynamics, including chemical bond breaking, with nanometer spatial resolution in the picosecond time scale. Owing to their lower attenuation, the availability of much higher laser pulse energies, and highly efficient excitation nonlinear SAWs can be easily realized at 3 K. In addition, the setup needed is relatively simple. These features open up new areas of fundamental research and practical applications. It can be expected that in the near future, theory will be extended to 2D and 3D treatments taking into account linear and nonlinear substrate and coating materials [2]. An important topic will be the development of new theoretical concepts going beyond weak or moderate nonlinearity up to the limit of elasticity inducing bond breaking. In the near future, nanosecond ultrasonics can be expected to play an important role in the investigation of nonlinear elastic properties of layered film-substrate systems, opening the door to a variety of possibilities (e.g. isotropic and anisotropic materials, quadratic and cubic nonlinearity). Guided SAWs provide an ideal tool for such studies due to the relatively low power needed to achieve strongly nonlinear behavior. While theory currently is under intense investigation, experiments are still widely missing. The theoretical concepts of envelope solitons in layered systems, for example, have been presented already [21]. References [1] C. Flytzanis, Merging nonlinear acoustics and optics: light driven large amplitude short acoustic pulses and breakdown in dielectrics, in: B.O. Enflo, C.M. Hedberg, L. Kari (Eds.), Nonlinear Acoustics-Fundamentals and Applications (ISNA 18), vol. CP122, American Institute of Physics, Melville, 28, pp

5 P. Hess, A.M. Lomonosov / Ultrasonics 5 (21) [2] H.Y. Hao, H.J. Maris, Experiments with acoustic solitons in crystalline solids, Phys. Rev. B 64 (21) [3] H.J. Maris, W. Singhsomroje, Picosecond ultrasonics experiments to study phonon solitons, Phys. Status Solidi 241 (24) [4] O.L. Muskens, J.I. Dijkhuis, High amplitude, ultrashort, longitudinal strain solitons in sapphire, Phys. Rev. Lett. 89 (22) [5] P.J.S. van Capel, J.I. Dijkhuis, Optical generation and detection of shock waves in sapphire at room temperature, Appl. Phys. Lett. 88 (26) [6] A.M. Lomonosov, P. Hess, A.P. Mayer, Laser-based surface acoustic waves in materials science, in: M. Levy, H.E. Bass, R. Stern (Eds.), Modern Acoustical techniques for the Measurement of Mechanical Properties, vol. 39, Academic Press, San Diego, 21, pp [7] Al.A. Kolomenskii, A.M. Lomonosov, R. Kuschnereit, P. Hess, V.E. Gusev, Laser generation and detection of strongly nonlinear elastic surfaces pulses, Phys. Rev. Lett. 79 (1997) [8] A.M. Lomonosov, P. Hess, Effects of nonlinear elastic surface pulses in anisotropic silicon crystals, Phys. Rev. Lett. 83 (1999) [9] O.L. Muskens, J.I. Dijkhuis, High amplitude, ultrashort strain solitons in solids, in: K.-T. Tsen (Ed.), Non-Equilibrium Dynamics of Semiconductors and Nanostructures, CRC Press, Boca Raton, 25, pp [1] H.Y. Hao, W. Singhsomroje, H.J. Maris, Studies of soliton formation of longitudinal acoustic phonons in crystalline solids, Physica B (22) [11] E. Péronne, B. Perrin, Generation and detection of acoustic solitons in crystalline slabs by laser ultrasonics, Ultrasonics 44 (26) e123 e127. [12] A.M. Lomonosov, P. Hess, A.P. Mayer, Observation of solitary surface pulses, Phys. Rev. Lett. 88 (22) [13] C. Eckl, A.P. Mayer, A.S. Kovalev, Phys. Rev. Lett. 81 (1998) [14] C. Eckl, A.S. Kovalev, A.P. Mayer, A.M. Lomonosov, P. Hess, Solitary surface acoustic waves, Phys. Rev. E 7 (24) [15] A.M. Lomonosov, P. Hess, Impulsive fracture of silicon by elastic surface pulses with shocks, Phys. Rev. Lett. 89 (22) [16] V.V. Kozhushko, A.M. Lomonosov, P. Hess, Intrinsic strength of silicon crystals in pure- and combined-mode fracture without precrack, Phys. Rev. Lett. 98 (27) [17] V.V. Kozhushko, P. Hess, Anisotropy of the strength of Si studied by a laserbased contact-free method, Phys. Rev. B 76 (27) [18] A.M. Lomonosov, P. Hess, Nonlinear surface acoustic waves: realization of solitary pulses and fracture, Ultrasonics 48 (28) [19] B.C. Daly, T.B. Norris, J. Chen, J.B. Khurgin, Picosecond acoustic phonon propagation in silicon, Phys. Rev. B 7 (24) [2] G.A. Maugin, Nonlinear surface waves and solitons, Eur. Phys. J. 147 (27) (special topics). [21] A. Kovalev, O. Sokolova, Rayleigh envelope solitons near the surface of a linear half-space covered with a nonlinear film, J. Phys.: Condens. Matter 2 (28)