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We monitor how destructive interference of undesired phonon frequency components shapes a quasi-monochromatic hypersound wavepacket spectrum during its local real-time preparation by a nanometric transducer and follow the subsequent decay by nonlinear coupling. We prove each frequency component of an optical supercontinuum probe to be sensitive to one particular phonon wavevector in bulk material and cross-check this by ultrafast x-ray diffraction experiments with direct access to the lattice dynamics. Establishing reliable experimental techniques with direct access to the transient spectrum of the excitation is crucial for the interpretation in strongly nonlinear regimes, such as soliton formation.
© 2013 Optical Society of America
1. Introduction
Brillouin scattering describes the inelastic interaction of photons with acoustic phonons, which can be excited or detected by this process. [1] It has important applications in the determination of elastic and photoelastic properties [2, 3] and is used for optical amplifiers or phase conjugation [4]. Stimulated Brillouin scattering can create intense hypersound waves [5], where the wavevector Q⃗ can be selected by tuning the transient grating induced by two intersecting laser-pulses. The diffraction of a probe pulse then senses the presence of phonons with the imprinted phonon wavevector Q⃗ [6]. Ultrashort laser pulse excitation of strongly absorbing materials or transducers on transparent substrates generates even larger phonon amplitudes up to 1% strain and can be detected by picosecond acoustics, the time-domain analog of Brillouin scattering [7,8]. This proved advantageous for investigating anharmonic propagation and damping of phonons and has generated excitement about ultrashort acoustic solitons [9–13]. Similar to transient gratings, optical multipulse excitation enhances a certain wavelength in the hypersound wave [14, 15]. Recently optical broadband probe pulses were used to access many phonon wavevectors simultaneously in picosecond acoustics experiments. [13, 16, 17]
Visible light only interacts with bulk phonons near the Brillouin-zone center, unless impurities are used to enhance the spectroscopy [18]. Optical phonons with high wavevectors in Bismuth have been accessed by microstructuring the film under investigation [19]. The back-folding of the phonon dispersion relation in superlattices can convert acoustic phonons with large wavevector into quasi-optical phonons with detectably small wavevector [20–22]. Alternatively, the acoustic reflection from the backside of the sample or from an interface propagating back to the transducer yields reflectivity modulations of the transducer which can be detected by a probe pulse [23]. The pulse-echo technique was refined by introducing additional thin detector films and combined with acoustic pulse shaping [24] and was extended to the detection of shear waves in glycerol [14]. The fourier-transform of real-time signals due to pulse-echoes at the transducer or from Brillouin scattering in the bulk yields the spectrum of acoustic phonons. The analysis of the reflection coefficient of such acoustic perturbations was recently discussed in detail [25].
In order to observe high phonon wavevectors in a bulk material directly, shorter probe wavelengths are needed. The phonon dispersion for larger wavevectors is measured by inelastic X-ray scattering, which is essentially Brillouin scattering of X-rays [26,27]. The new millennium came along with the rapidly developing scientific field of ultrafast X-ray diffraction (UXRD), which allows direct measurement of the lattice oscillation amplitude associated with propagating strain pulses. [28–32] Very recently optically synthesized quasi-monochromatic phonon wavepackets in the 100 GHz range were clearly detected by UXRD as sidebands to bulk Bragg reflections [15].
In this paper we present a unifying view on UXRD and optical picosecond acoustics as two types of Brillouin scattering. The presence of monochromatic phonons in bulk SrTiO3 is directly evidenced by the scattering of photons. The conceptually simple analysis provides a real-time perspective on the spectral shaping of high-frequency phonon wavepackets by tailored multipulse excitation. In particular, we demonstrate how the optical supercontinuum probe accomplishes a versatile simultaneous broadband sensing of bulk phonon wavevectors constituting a large amplitude phonon wavepacket that decays by anharmonic interactions. We believe that this combination of ultrafast X-ray- and optical broadband-detection of phonons will significantly enhance the confidence regarding interpretations of future optical picosecond ultrasound experiments. The supercontinuum detection scheme is also applicable to other quasiparticles such as magnons or polaritons which are generally identified by their dispersion relation ω(k). The wave-particle duality requires such wave-packet description, in which only a coherent excitation of a broad wavevector-spectrum allows for localization of quasiparticles. Real-time preparation and detection of such wavepackets will aid the understanding and controlling such excitations.
2. Synthesizing quasi-monochromatic phonons
We first discuss how to synthesize coherent quasi-monochromatic phonon wavepackets in the GHz - THz range, using a thin metal transducer of SrRuO3 (SRO) or La0.7Sr0.3MnO3 (LSMO) on the material of interest SrTiO3 (STO). The absorption of an ultrashort light pulse leads to rapid expansion of the metal film. The good acoustic impedance matching of the SRO/LSMO transducer and the STO substrate suppresses reflections at the interface. [13, 33] Consequently, clean bipolar strain pulses without unintended replica are sent into the substrate. [7] The spatio-temporal dynamics can be simulated by a linear-chain model which includes the anharmonicity in the interatomic potentials [13, 33]. In the present simulations we neglected the anharmonic terms according to the moderate excitation fluence. Due to the very fast electron phonon coupling of SRO and LSMO the optically excited electrons are rapidly localized and consequently the spatial profile of the exciting stress corresponds to the absorption of light according to Lambert-Beer’s law [34]. A single ultrashort laser pulse generates a broad phonon spectrum in the substrate. The red line in Fig. 1 shows the Fourier-transformed spatial strain pattern of the substrate only. If the heated transducer layer with thickness d is added to the analysis, a strong Fourier-component at k = 0 emerges because of the thermally expanded absorbing region. If the pump penetration depth dabs≫ d the calculated excited strain pattern of the substrate yields a bipolar strain pulse with a frequency spectrum centered around Q = 0.74 ·πvt/vsd whereas for dabs≪ d the exponential shaped profile of the induced strain pulse have a spectral maximum at Q = vt/vsdabs. Here d, vt and vs are the thickness of the transducer film and the sound velocities of transducer and substrate in [001] direction, respectively. Our conditions are well described by the first limit value because dabs > d. For longer excitation pulses, the phonon spectrum is suppressed for wavevectors Q corresponding to phonon frequencies ω > 2π/τpulse where τpulse is the temporal duration of the pump pulse.
Fig. 1 Calculated phonon spectra. Spectral phonon amplitude present in the STO substrate after excitation of a 35 nm LSMO transducer by 1, 2, 4 and 8 pulses with a pulse spacing of ΔT = 15.4 ps. All pulse sequences have the same integrated pulse energy.
Successive excitation of the metal transducer with a sequence of light pulses equally spaced by ΔT in time generates a strain wave traveling into the substrate with a fundamental frequency ν = 1/ΔT and contributions of its higher harmonics due to the sharp edges of the strain pulses. As an example we simulate the phonon spectra for a transducer thickness d = 35 nm and a pulse spacing of ΔT = 15.4 ps. Figure 1 shows how additional pulses in the pump sequence sharpen the spectrum around Q = 2π/ΔTvs and its higher harmonics by canceling the amplitude of other wavevectors. Note, that in Fig. 1 the integral pump energy is kept fixed in the simulation. As a consequence, the different pulse trains induce a constant phonon amplitude for the constructive interference, whereas the suppressed modes interfere destructively. Hence, for multipulse excitation, less energy is deposited in coherent phonons, compared to single pulse excitation with the same integral fluence. [33]
3. Inelastic light scattering from directed phonons
The energy quantum h̄ωs of such a strain wave is the longitudinal acoustic phonon with a magnitude of the wavevector |Q⃗| = ωs/vs. Photons with wavevector k⃗ are scattered by phonons with wavevector Q⃗ only in accordance with the energy and momentum conservation. In a crystal with reciprocal lattice vectors G⃗, the equation for momentum conservation with k⃗′ as the wavevector of the scattered photon reads
Here we discuss the situation where ΔQ⃗ = ±Q⃗ is the momentum added to the scattering photon by the creation or annihilation of the particular phonon with wavevector Q⃗ which was synthesized into the crystal. It is important to see that generating phonons with an optical transducer thin film breaks the symmetry, and only phonons with wavevector Q⃗ directed into the crystal are generated. Figure 3(a) schematically shows how the creation of an additional phonon with wavevector Q⃗ leads to a scattering with momentum transfer G⃗ + Q⃗. In the geometry depicted in Fig. 3(a), i.e. for a phonon propagating into the crystal, the energy conservation imposes a constraint on the angular frequencies ω′ of the scattered and ω of the incident photons: because the stimulated emission of a phonon with wavevector Q⃗ directed into the substrate acquires its energy from the scattering photon, whereas the annihilation of a phonon with the same Q⃗ adds energy to the scattering photon and corresponds to a positive wavevector transfer ΔQ⃗ = +Q⃗.Fig. 2 Birth and decay of phonon-wavepackets observed by UXRD. (a) Calculated X-ray diffraction pattern for an STO substrate with a coherent phonon spectrum excited by 1 to 8 pulses. (b) UXRD data demonstrating the successive sharpening of the diffraction pattern with 8 excitation pulses separated by 7.2 ps. (c) Same UXRD data for larger delay time t showing the decay of the coherent phonons.
Fig. 3 Schematics of Brillouin scattering and picosecond acoustics. (a) Schematic of the inelastic X-ray scattering with creation of a phonon with wavevector +Q⃗. (b) Schematic of the Brillouin scattering with creation of a phonon with wavevector +Q⃗. (c) Schematic showing the interference of waves which is used for a time-domain explanation of the observed oscillations (see text).
These equations describe an inelastic scattering process which generally leads to an asymmetric scattering geometry, where the incoming and outgoing photon do not have the same angle with respect to the surface. However, the vast difference of the light and sound velocities implies a very small length change of the scattered photon wavevector with respect to the incident one. This quasi-elastic condition leads to a nearly symmetric scattering geometry with Δk⃗ = G⃗ ± Q⃗.
4. Brillouin scattering of X-rays
We first discuss Brillouin scattering in the hard X-ray range [27]. The simulation of the UXRD signal is shown in Fig. 2(a) for the excitation with an increasing number of pump-pulses. All strain pulses have fully entered the substrate and in contrast to Fig. 1 the deposited fluence rises with each absorbed pulse. Figure 2(a) directly shows how the diffraction feature at G⃗ + Q⃗ sharpens as more and more bipolar strain pulses are sculptured into the crystal.We performed the corresponding UXRD experiment at the ID09B beamline at the synchrotron source ESRF which provides ∼ 100 ps hard X-ray pulses. The SRO transducer film was excited with eight pulses spaced by ΔT = 7.2 ps, consistent with the calculation of Fig. 2(a). Details of the setup were discussed previously [15]. Here, we show an angular resolved analysis of the data recorded with 100 ps time resolution. The spectral narrowing of the feature around G⃗ +Q⃗, is qualitatively confirmed in the experiment. When the synthesis of the quasi-monochromatic phonon wavepacket (Fig. 2(b)) was stopped after eight pulses, we observed that this sideband decayed (c.f. Fig. 2(c)) within 130 ps. This is not much longer than it took to create the wavepacket. The corresponding phonon decay is due to anharmonic phonon-phonon scattering processes within the STO substrate [35–39]. Up to now we have shown that the interference of sound waves (phonons) in the crystal created by multiple pulses in fact leads to the predicted quasi-monochromatic phonon wavepacket. UXRD directly measures the shaping of the quasi-monochromatic phonon spectrum in real time. Moreover, we showed that this wavepacket decays on a 100 ps timescale.
5. Brillouin scattering and picosecond acoustics
Now we turn to Brillouin scattering of optical photons from similar phonon wavepackets. In a classical interpretation of Brillouin scattering, the incident electromagnetic wave with wavelength λm = λ/n(λ) in the material is diffracted from the Bragg grating given by the strain-induced refractive index modulation according to the photoelastic effect. Here λ and n(λ) are the wavelength of the incident electromagnetic wave in vacuum and the wavelength dependent refractive index of the material, respectively. For visible light STO has a refractive index of n ≈ 2.4. [40] The scattering angle is given by Bragg’s law [1]
implying that for a given scattering angle θ an optical photon with wavelength λm = 2π/|k⃗| is diffracted from a refractive index grating induced by phonons with wavelength λs = 2π/|Q⃗| as schematically shown in Fig. 3(b). For a phonon propagating perpendicular to the sample surface the scattering angle θ is equal to the angle between the incident photon wavevector in the sample and the sample surface. This angle can be easily calculated by Snell’s law. Given a strain wave propagating into the crystal, the scattered light undergoes a tiny Doppler-red-shift corresponding to the frequency of the moving sound wave, which can be detected by high-resolution Brillouin scattering experiments. [41, 42]Eq. (3) is a direct result of the optical Laue-condition k⃗′ − k⃗ = ±Q⃗, (Fig. 3(b)) under quasi-elastic scattering conditions (|k⃗′| ≈ |k⃗|), and is in fact Eq. (1) describing Brillouin scattering with G⃗ = 0. For a fixed angle θ Eq. (3) implies that an optical photon with wavelength λ specifically scatters from phonons with the wavevector magnitude
For n(λ) = 1 Eq. (4) defines the well known scattering vector in elastic X-ray scattering theory.Up to now we have adopted a perspective which is suitable for conventional Brillouin scattering experiments detecting the frequency shift of a narrow-band-laser. In optical picosecond acoustics experiments a laser pulse excites a short bipolar strain pulse with a broad spectrum (Fig. 1, red line). The detection by a time-delayed laser pulse is explained as follows: A part of the supercontinuum probe pulse is diffracted by the refractive index modulation associated with the propagating sound pulse fulfilling Eq. (4) and interferes with the reflection of the probe pulse at the sample surface (Fig. 3(c)). The moving sound pulse leads to a phase change of the diffracted wave which depends on the pump-probe delay t. The resulting intensity of the interfering electric fields is proportional to cos(ωst) = cos(2πt/Ts) with the period
For normal incidence Ts corresponds to the time a soundwave with wavevector Q⃗ perpendicular to the surface needs to propagate one half of the optical wavelength λm. Combining Eqs. (4) and (5) leads to vs· Q = 2π/Ts = ωs. Hence ωs is the angular frequency of the phonon with the wavevector Q⃗. Such oscillations are in fact observed in all-optical reflectivity experiments [13,16] using a single optical pump pulse and an ultrashort broadband probe pulse. Figure 4(a) shows the recorded reflectivity change for the LSMO transducer on STO after subtraction of a slowly varying electronic and thermal contribution to the signal. The observed oscillation period depends on the probe wavelength as predicted by Eq. (5) and extends across the entire visible spectrum according to the broad spectrum of the excited coherent phonons calculated to obtain the red line in Fig. 1. Here, Eq. (4) is applied to translate the optical probe wavelength λ into the phonon wavevector Q⃗.Fig. 4 Experimental proof for the wavevector selectivity of supercontinuum probe pulses. (a) Measured transient relative reflectivity change for single-pulse excitation as a function of the phonon wavevector given by Eq. (4). Slowly varying background is subtracted. (b) Same for an excitation with 8 pulses. (c) The dashed lines show the calculated spectral amplitude of the excited phonons for 1 (red dashed) and 8 (black dashed) excitations pulses (reproduced from Fig. 1). The solid lines show short-time Fourier transform data of Fig. 4(b). Each of the extracted datasets was multiplied with the probe wavelength to obtain a quantity proportional to the spectral amplitude of the coherent phonons.
6. Selectivity of the supercontinuum probe
Finally, we show that Brillouin scattering with broadband optical probe pulses not only has a mathematical one-to-one correspondence between the optical probe wavelength and a specific phonon with wavevector Q⃗ (Eq. (4)) but that in fact individual wavevector components can be experimentally discriminated. Similar to the X-ray experiments shown in Fig. 2, we synthesized a monochromatic phonon via excitation of an LSMO transducer by a train of eight optical pump pulses (λ = 800 nm) with the pulse duration τpulse ∼ 100 fs and separation ΔT = 15.4 ps. Figure 4(b) shows the relative reflectivity change of the sample. After t ≈ 150 ps oscillations are mainly visible in the vicinity of |Q⃗| = 0.05 rad/nm. The black dashed line in Fig. 4(c), reproduced from Fig. 1, shows the idealized simulation of the coherent phonon spectrum in the substrate after 150 ps when all bipolar strain pulses have entered the substrate. The excellent agreement of the measured suppression of the reflectivity oscillations in certain wavelength regions of Fig. 4(b) with the simulated destructive interference in the phonon spectrum confirms both the wavevector-selective probing process and the shaping of quasi-monochromatic phonon wavepackets by tailored destructive interference. To support this argument we have extracted the spectral amplitude of oscillations observed in Fig. 4(b) by short-time Fourier-transform and multiplied the spectra by the wavelength of the probe light to get a quantity which is in first approximation proportional to the spectral amplitude of the occupied phonons. [7] The results plotted in Fig. 4(c) show good quantitative agreement of the measured spectra with the predicted spectrum (black dashed line). In particular, the measured spectral width of the main spectral component of the wavepacket exactly matches the prediction, demonstrating the simultaneous spatial and temporal resolution of the supercontinuum pump-probe setup. The higher amplitude of the secondary maximum is due to slight imperfections in the experimental optical pump pulse train, which leads to an imperfect destructive interference of phonon modes. Furthermore, Figure 4(c) directly measures the damping of the phonon amplitude in time. This is analogous to the measured phonon attenuation using UXRD (Fig. 2(c)). In particular, both experiments show that high-frequency components of the wavepackets undergo stronger damping.
Scattering experiments using synthesized monochromatic phonon wavepackets thus show directly that both UXRD and picosecond ultrasonics are wavevector-selective probes of ultrafast phonon dynamics, which are both described by Eq. (1). The difference of the two methods is that in the X-ray range we have selected the detected phonon wavevector Q⃗ by tuning the Bragg angle, keeping the light wavelength fixed, whereas for the optical experiment we kept the angle of incidence fixed and measured a broad frequency range using a spectrometer. In principle, the experiments could also be carried out vice versa. More importantly, the absorption of the probe pulses is different. STO has essentially no optical absorption, while the penetration depth for hard X-rays is not larger than 10 μm. Generally, for very short sound wavelengths hard X-rays are definitely the only suitable choice, since the required VUV and XUV photons undergo very strong absorption.
7. Conclusions
In conclusion, we have given a unifying interpretation of ultrafast versions of Brillouin scattering in the ranges of optical and X-ray photon energies. We have synthesized large-amplitude quasi-monochromatic phonon wavepackets and proved that their spectrum and their anharmonic decay can be directly observed in both types of experiments. In particular, we showed that the optical supercontinuum is a direct, simultaneous and nonetheless selective real-time probe of the spectra composing a phonon wavepacket in bulk material. We confirmed this interpretation by comparison to UXRD experiments for which the direct access to the spectrum of the lattice strain is obvious. We think that this experimental confirmation will be essential for the interpretation of future experiments where wavevector-selective excitation and probing will be used to measure anharmonic and nonlinear phonon interactions in condensed matter. In particular, we envision experiments on nonlinear phononics as an analog of nonlinear optics, in which we observe sum and difference frequency mixing of synthesized phonons.
Acknowledgments
We thank M. Wulff and D. Kakhulin for their valuable contributions to the synchrotron experiments at the ESRF and I. Vrejoiu for sample growth. A.B. thanks the Leibnitz graduate school ”Dynamics in new Light” for financial support. This research was made possible through the funding by BMBF via 05K10IP1.
References and links
1. H. J. Eichler, P. Günter, and D. W. Pohl, Laser induced dynamic gratings, vol. 50 of Springer series in optical sciences(Springer, Berlin [u.a.], 1986). [CrossRef]
2. R. Vacher and L. Boyer, “Brillouin scattering: A tool for the measurement of elastic and photoelastic constants,” Phys. Rev. B 6, 639–673 (1972). [CrossRef]
3. J. A. Rogers, A. A. Maznev, M. J. Banet, and K. A. Nelson, “Optical generation and characterization of acoustic waves in thin films: Fundamentals and applications,” Annual Review of Materials Science 30, 117–157 (2000). [CrossRef]
4. V. Wang and C. R. Giuliano, “Correction of phase aberrations via stimulated brillouin scattering,” Opt. Lett. 2, 4–6 (1978). [CrossRef] [PubMed]
5. R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 592–595 (1964). [CrossRef]
6. K. A. Nelson, R. Casalegno, R. J. D. Miller, and M. D. Fayer, “Laser-induced excited state and ultrasonic wave gratings: Amplitude and phase grating contributions to diffraction,” J. Chem. Phys. 77, 1144–1152 (1982). [CrossRef]
7. C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986). [CrossRef]
8. H. N. Lin, R. J. Stoner, H. J. Maris, and J. Tauc, “Phonon attenuation and velocity measurements in transparent materials by picosecond acoustic interferometry,” J. Appl. Phys. 69, 3816–3822 (1991). [CrossRef]
9. P. J. S. van Capel and J. I. Dijkhuis, “Optical generation and detection of shock waves in sapphire at room temperature,” Appl. Phys. Lett. 88, 151910 (2006). [CrossRef]
10. P. J. S. van Capel, H. P. Porte, G. van der Star, and J. I. Dijkhuis, “Interferometric detection of acoustic shock waves,” J. Phys.: Conf. Ser. 92, 012092 (2007). [CrossRef]
11. O. L. Muskens and J. I. Dijkhuis, “High amplitude, ultrashort, longitudinal strain solitons in sapphire,” Phys. Rev. Lett. 89, 285504 (2002). [CrossRef]
12. P. J. S. van Capel and J. I. Dijkhuis, “Time-resolved interferometric detection of ultrashort strain solitons in sapphire,” Phys. Rev. B 81, 144106 (2010). [CrossRef]
13. A. Bojahr, M. Herzog, D. Schick, I. Vrejoiu, and M. Bargheer, “Calibrated real-time detection of nonlinearly propagating strain waves,” Phys. Rev. B 86, 144306 (2012). [CrossRef]
14. T. Pezeril, C. Klieber, S. Andrieu, and K. A. Nelson, “Optical generation of gigahertz-frequency shear acoustic waves in liquid glycerol,” Phys. Rev. Lett. 102, 107402 (2009). [CrossRef] [PubMed]
15. M. Herzog, A. Bojahr, J. Goldshteyn, W. Leitenberger, I. Vrejoiu, D. Khakhulin, M. Wulff, R. Shayduk, P. Gaal, and M. Bargheer, “Detecting optically synthesized quasi-monochromatic sub-terahertz phonon wavepackets by ultrafast x-ray diffraction,” Appl. Phys. Lett. 100, 094101 (2012). [CrossRef]
16. S. Brivio, D. Polli, A. Crespi, R. Osellame, G. Cerullo, and R. Bertacco, “Observation of anomalous acoustic phonon dispersion in SrTiO3by broadband stimulated brillouin scattering,” Appl. Phys. Lett. 98, 211907 (2011). [CrossRef]
17. E. Pontecorvo, M. Ortolani, D. Polli, M. Ferretti, G. Ruocco, G. Cerullo, and T. Scopigno, “Visualizing coherent phonon propagation in the 100 GHz range: A broadband picosecond acoustics approach,” Appl. Phys. Lett. 98, 011901 (2011). [CrossRef]
18. W. E. Bron, “Spectroscopy of high-frequency phonons,” Reports on Progress in Physics 43, 301 (1980). [CrossRef]
19. Z. Chen, B. C. Minch, and M. F. DeCamp, “High wavevector optical phonons in microstructured bismuth films,” Opt. Express 18, 4365–4370 (2010). [CrossRef] [PubMed]
20. G.-W. Chern, K.-H. Lin, Y.-K. Huang, and C.-K. Sun, “Spectral analysis of high-harmonic coherent acoustic phonons in piezoelectric semiconductor multiple quantum wells,” Phys. Rev. B 67, 121303 (2003). [CrossRef]
21. N. M. Stanton, R. N. Kini, A. J. Kent, M. Henini, and D. Lehmann, “Terahertz phonon optics in gaas/alas superlattice structures,” Phys. Rev. B 68, 113302 (2003). [CrossRef]
22. M. F. Pascual-Winter, A. Fainstein, B. Jusserand, B. Perrin, and A. Lemaître, “Spectral responses of phonon optical generation and detection in superlattices,” Phys. Rev. B 85, 235443 (2012). [CrossRef]
23. T. C. Zhu, H. J. Maris, and J. Tauc, “Attenuation of longitudinal-acoustic phonons in amorphous SiO2at frequencies up to 440 GHz,” Phys. Rev. B 44, 4281–4289 (1991). [CrossRef]
24. C. Klieber, E. Peronne, K. Katayama, J. Choi, M. Yamaguchi, T. Pezeril, and K. A. Nelson, “Narrow-band acoustic attenuation measurements in vitreous silica at frequencies between 20 and 400 GHz,” Appl. Phys. Lett. 98, 211908 (2011). [CrossRef]
25. S. Ayrinhac, M. Foret, A. Devos, B. Rufflé, E. Courtens, and R. Vacher, “Subterahertz hypersound attenuation in silica glass studied via picosecond acoustics,” Phys. Rev. B 83, 014204 (2011). [CrossRef]
26. E. Burkel, “Phonon spectroscopy by inelastic x-ray scattering,” Rep. Prog. Phys. 63, 171 (2000). [CrossRef]
27. P. Eisenberger, N. G. Alexandropoulos, and P. M. Platzman, “X-ray brillouin scattering,” Phys. Rev. Lett. 28, 1519–1522 (1972). [CrossRef]
28. A. M. Lindenberg, I. Kang, S. L. Johnson, T. Missalla, P. A. Heimann, Z. Chang, J. Larsson, P. H. Bucksbaum, H. C. Kapteyn, H. A. Padmore, R. W. Lee, J. S. Wark, and R. W. Falcone, “Time-resolved x-ray diffraction from coherent phonons during a laser-induced phase transition,” Phys. Rev. Lett. 84, 111–114 (2000). [CrossRef] [PubMed]
29. D. A. Reis, M. F. DeCamp, P. H. Bucksbaum, R. Clarke, E. Dufresne, M. Hertlein, R. Merlin, R. Falcone, H. Kapteyn, M. M. Murnane, J. Larsson, T. Missalla, and J. S. Wark, “Probing impulsive strain propagation with x-ray pulses,” Phys. Rev. Lett. 86, 3072–3075 (2001). [CrossRef] [PubMed]
30. K. Sokolowski-Tinten, C. Blome, C. Dietrich, A. Tarasevitch, M. Horn von Hoegen, D. von der Linde, A. Cavalieri, J. Squier, and M. Kammler, “Femtosecond x-ray measurement of ultrafast melting and large acoustic transients,” Phys. Rev. Lett. 87, 225701 (2001). [CrossRef] [PubMed]
31. M. Bargheer, N. Zhavoronkov, Y. Gritsai, J. C. Woo, D. S. Kim, M. Woerner, and T. Elsaesser, “Coherent atomic motions in a nanostructure studied by femtosecond x-ray diffraction,” Science 306, 1771–1773 (2004). [CrossRef] [PubMed]
32. M. Trigo, Y. M. Sheu, D. A. Arms, J. Chen, S. Ghimire, R. S. Goldman, E. Landahl, R. Merlin, E. Peterson, M. Reason, and D. A. Reis, “Probing unfolded acoustic phonons with x rays,” Phys. Rev. Lett. 101, 025505 (2008). [CrossRef] [PubMed]
33. M. Herzog, D. Schick, P. Gaal, R. Shayduk, C. von Korff Schmising, and M. Bargheer, “Analysis of ultrafast x-ray diffraction data in a linear-chain model of the lattice dynamics,” Appl. Phys. A 106, 489–499 (2012). [CrossRef]
34. A. Bojahr, D. Schick, L. Maerten, M. Herzog, I. Vrejoiu, C. von Korff Schmising, C. J. Milne, S. L. Johnson, and M. Bargheer, “Comparing the oscillation phase in optical pump-probe spectra to ultrafast x-ray diffraction in the metal-dielectric SrRuO3/SrTiO3 superlattice,” Phys. Rev. B 85, 224302 (2012). [CrossRef]
35. H. J. Maris, “6 - interaction of sound waves with thermal phonons in dielectric crystals,” in “Principles and Methods,” vol. 8 of Physical Acoustics, W. P. Mason and R. N. Thurston, eds. (Academic Press, 1971), pp. 279–345.
36. A. Akhieser, “On the absorption of sound in solids,” J. Phys. (USSR) 1, 277 (1939).
37. C. Herring, “Role of low-energy phonons in thermal conduction,” Phys. Rev. 95, 954–965 (1954). [CrossRef]
38. A. Koreeda, T. Nagano, S. Ohno, and S. Saikan, “Quasielastic light scattering in Rutile, ZnSe, Silicon, and SrTiO3,” Phys. Rev. B 73, 024303 (2006). [CrossRef]
39. B. C. Daly, K. Kang, Y. Wang, and D. G. Cahill, “Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in Silicon,” Phys. Rev. B 80, 174112 (2009). [CrossRef]
40. M. Cardona, “Optical properties and band structure of SrTiO3and BaTiO3,” Phys. Rev. 140, A651–A655 (1965). [CrossRef]
41. V. Mahajan and J. Gaskill, “Doppler interpretation of the frequency shifts of light diffracted by sound waves,” J. Appl. Phys. 45, 2799 (1976). [CrossRef]
42. O. L. Muskens and J. I. Dijkhuis, “Inelastic light scattering by trains of ultrashort acoustic solitons in sapphire,” Phys. Rev. B 70, 104301 (2004). [CrossRef]
