Abstract

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.

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  13. A. Bojahr, M. Herzog, D. Schick, I. Vrejoiu, and M. Bargheer, “Calibrated real-time detection of nonlinearly propagating strain waves,” Phys. Rev. B86, 144306 (2012).
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  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]
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  26. E. Burkel, “Phonon spectroscopy by inelastic x-ray scattering,” Rep. Prog. Phys.63, 171 (2000).
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  27. P. Eisenberger, N. G. Alexandropoulos, and P. M. Platzman, “X-ray brillouin scattering,” Phys. Rev. Lett.28, 1519–1522 (1972).
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  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,” Science306, 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. A106, 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. B85, 224302 (2012).
    [CrossRef]
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  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. B73, 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. B80, 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. B70, 104301 (2004).
    [CrossRef]

2012

A. Bojahr, M. Herzog, D. Schick, I. Vrejoiu, and M. Bargheer, “Calibrated real-time detection of nonlinearly propagating strain waves,” Phys. Rev. B86, 144306 (2012).
[CrossRef]

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]

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. B85, 235443 (2012).
[CrossRef]

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. A106, 489–499 (2012).
[CrossRef]

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. B85, 224302 (2012).
[CrossRef]

2011

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]

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. B83, 014204 (2011).
[CrossRef]

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]

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]

2010

P. J. S. van Capel and J. I. Dijkhuis, “Time-resolved interferometric detection of ultrashort strain solitons in sapphire,” Phys. Rev. B81, 144106 (2010).
[CrossRef]

Z. Chen, B. C. Minch, and M. F. DeCamp, “High wavevector optical phonons in microstructured bismuth films,” Opt. Express18, 4365–4370 (2010).
[CrossRef] [PubMed]

2009

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]

B. C. Daly, K. Kang, Y. Wang, and D. G. Cahill, “Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in Silicon,” Phys. Rev. B80, 174112 (2009).
[CrossRef]

2008

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]

2007

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]

2006

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]

A. Koreeda, T. Nagano, S. Ohno, and S. Saikan, “Quasielastic light scattering in Rutile, ZnSe, Silicon, and SrTiO3,” Phys. Rev. B73, 024303 (2006).
[CrossRef]

2004

O. L. Muskens and J. I. Dijkhuis, “Inelastic light scattering by trains of ultrashort acoustic solitons in sapphire,” Phys. Rev. B70, 104301 (2004).
[CrossRef]

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,” Science306, 1771–1773 (2004).
[CrossRef] [PubMed]

2003

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. B67, 121303 (2003).
[CrossRef]

N. M. Stanton, R. N. Kini, A. J. Kent, M. Henini, and D. Lehmann, “Terahertz phonon optics in gaas/alas superlattice structures,” Phys. Rev. B68, 113302 (2003).
[CrossRef]

2002

O. L. Muskens and J. I. Dijkhuis, “High amplitude, ultrashort, longitudinal strain solitons in sapphire,” Phys. Rev. Lett.89, 285504 (2002).
[CrossRef]

2001

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]

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]

2000

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]

E. Burkel, “Phonon spectroscopy by inelastic x-ray scattering,” Rep. Prog. Phys.63, 171 (2000).
[CrossRef]

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 Science30, 117–157 (2000).
[CrossRef]

1991

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]

T. C. Zhu, H. J. Maris, and J. Tauc, “Attenuation of longitudinal-acoustic phonons in amorphous SiO2at frequencies up to 440 GHz,” Phys. Rev. B44, 4281–4289 (1991).
[CrossRef]

1986

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B34, 4129–4138 (1986).
[CrossRef]

1982

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]

1980

W. E. Bron, “Spectroscopy of high-frequency phonons,” Reports on Progress in Physics43, 301 (1980).
[CrossRef]

1978

1976

V. Mahajan and J. Gaskill, “Doppler interpretation of the frequency shifts of light diffracted by sound waves,” J. Appl. Phys.45, 2799 (1976).
[CrossRef]

1972

R. Vacher and L. Boyer, “Brillouin scattering: A tool for the measurement of elastic and photoelastic constants,” Phys. Rev. B6, 639–673 (1972).
[CrossRef]

P. Eisenberger, N. G. Alexandropoulos, and P. M. Platzman, “X-ray brillouin scattering,” Phys. Rev. Lett.28, 1519–1522 (1972).
[CrossRef]

1965

M. Cardona, “Optical properties and band structure of SrTiO3and BaTiO3,” Phys. Rev.140, A651–A655 (1965).
[CrossRef]

1964

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]

1954

C. Herring, “Role of low-energy phonons in thermal conduction,” Phys. Rev.95, 954–965 (1954).
[CrossRef]

1939

A. Akhieser, “On the absorption of sound in solids,” J. Phys. (USSR)1, 277 (1939).

Akhieser, A.

A. Akhieser, “On the absorption of sound in solids,” J. Phys. (USSR)1, 277 (1939).

Alexandropoulos, N. G.

P. Eisenberger, N. G. Alexandropoulos, and P. M. Platzman, “X-ray brillouin scattering,” Phys. Rev. Lett.28, 1519–1522 (1972).
[CrossRef]

Andrieu, S.

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]

Arms, D. A.

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]

Ayrinhac, S.

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. B83, 014204 (2011).
[CrossRef]

Banet, M. J.

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 Science30, 117–157 (2000).
[CrossRef]

Bargheer, M.

A. Bojahr, M. Herzog, D. Schick, I. Vrejoiu, and M. Bargheer, “Calibrated real-time detection of nonlinearly propagating strain waves,” Phys. Rev. B86, 144306 (2012).
[CrossRef]

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]

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. A106, 489–499 (2012).
[CrossRef]

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. B85, 224302 (2012).
[CrossRef]

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,” Science306, 1771–1773 (2004).
[CrossRef] [PubMed]

Bertacco, R.

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]

Blome, C.

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]

Bojahr, A.

A. Bojahr, M. Herzog, D. Schick, I. Vrejoiu, and M. Bargheer, “Calibrated real-time detection of nonlinearly propagating strain waves,” Phys. Rev. B86, 144306 (2012).
[CrossRef]

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]

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. B85, 224302 (2012).
[CrossRef]

Boyer, L.

R. Vacher and L. Boyer, “Brillouin scattering: A tool for the measurement of elastic and photoelastic constants,” Phys. Rev. B6, 639–673 (1972).
[CrossRef]

Brivio, S.

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]

Bron, W. E.

W. E. Bron, “Spectroscopy of high-frequency phonons,” Reports on Progress in Physics43, 301 (1980).
[CrossRef]

Bucksbaum, P. H.

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]

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]

Burkel, E.

E. Burkel, “Phonon spectroscopy by inelastic x-ray scattering,” Rep. Prog. Phys.63, 171 (2000).
[CrossRef]

Cahill, D. G.

B. C. Daly, K. Kang, Y. Wang, and D. G. Cahill, “Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in Silicon,” Phys. Rev. B80, 174112 (2009).
[CrossRef]

Cardona, M.

M. Cardona, “Optical properties and band structure of SrTiO3and BaTiO3,” Phys. Rev.140, A651–A655 (1965).
[CrossRef]

Casalegno, R.

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]

Cavalieri, A.

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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).
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B. C. Daly, K. Kang, Y. Wang, and D. G. Cahill, “Picosecond ultrasonic measurements of attenuation of longitudinal acoustic phonons in Silicon,” Phys. Rev. B80, 174112 (2009).
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Figures (4)

Fig. 1
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.

Fig. 2
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
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).

Fig. 4
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.

Equations (5)

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Δ k = k k = G + Δ Q .
ω ω = { ω s , if Δ Q = + Q ; phonon annihilation ω s , if Δ Q = Q ; phonon creation ,
λ m = 2 λ s sin θ ,
Q ( λ ) = 4 π λ n ( λ ) sin θ .
T s = λ 2 v s n ( λ ) sin θ .

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