Abstract

We present an investigation of the effect of the collecting lens aperture on the line shape of phonon peaks observed in surface Brillouin light scattering (SBLS) from surfaces of opaque materials and transparent thin films. In general, the broadening that is due to the aperture is asymmetric and can be as large as 60% of the peak frequency shift in the case of a f/1.4 aperture with an angle of incidence θi = 30°. We calculated SBLS spectra accounting for the spread in scattering wave vectors across the collecting lens aperture, the polarization and angular dependence of the scattering, and the spectrometer instrumental function. By performing a detailed comparison between measured and calculated SBLS spectra for Si(001), we identified a set of simple rules for the placement of a rectangular slit in the collecting lens aperture to reduce the effects of aperture broadening. By use of a slit, the peak linewidths can be reduced substantially, without reducing the peak heights significantly, while eliminating false shifts in the measured frequency values.

© 1998 Optical Society of America

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References

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  1. H. Z. Cummins, “Brillouin scattering studies of phase transitions in crystals,” in Light Scattering Near Phase Transitions, H. Z. Cummins, A. P. Levanyuk, eds. (North-Holland, Amsterdam, 1983), pp. 359–447.
    [CrossRef]
  2. H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
    [CrossRef]
  3. J. R. Sandercock, “Trends in Brillouin light scattering: studies of opaque materials, supported films and central modes,” in Light Scattering in Solids III, M. Cardona, G. Güntherodt, eds. (Springer-Verlag, Berlin, 1982), pp. 173–206.
    [CrossRef]
  4. J. R. Sandercock, W. Wettling, “Light scattering from surface and bulk thermal magnons in iron and nickel,” J. Appl. Phys. 50, 7784–7789 (1979).
    [CrossRef]
  5. W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
    [CrossRef]
  6. H. G. Danielmeyer, “Aperture corrections for sound-absorption measurements with light scattering,” J. Acoust. Soc. Am. 47, 151–154 (1969).
    [CrossRef]
  7. S. Lee, “Elastic properties of polymeric Langmuir-Blodgett studied using Brillouin light scattering,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991).
  8. J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
    [CrossRef] [PubMed]
  9. J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
    [CrossRef] [PubMed]
  10. P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
    [CrossRef]
  11. L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
    [CrossRef] [PubMed]
  12. G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
    [CrossRef]
  13. R. Loudon, J. R. Sandercock, “Analysis of the light-scattering cross-section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
    [CrossRef]
  14. G. W. Farnell, “Properties of elastic surface waves,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1970), Vol. 6, pp. 109–166.
  15. E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic, New York, 1985), Vol. 1, p. 564.
  16. J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
    [CrossRef]
  17. G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. IX, pp. 35–127.

1996

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

1994

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

1992

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
[CrossRef] [PubMed]

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

1989

J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
[CrossRef]

1980

R. Loudon, J. R. Sandercock, “Analysis of the light-scattering cross-section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

1979

J. R. Sandercock, W. Wettling, “Light scattering from surface and bulk thermal magnons in iron and nickel,” J. Appl. Phys. 50, 7784–7789 (1979).
[CrossRef]

1977

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

1969

H. G. Danielmeyer, “Aperture corrections for sound-absorption measurements with light scattering,” J. Acoust. Soc. Am. 47, 151–154 (1969).
[CrossRef]

Adler, E. L.

G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. IX, pp. 35–127.

Agarwal, G. S.

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Beghi, M.

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

Bottani, C. E.

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

Briggs, G. A. D.

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

Cummins, H. Z.

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

H. Z. Cummins, “Brillouin scattering studies of phase transitions in crystals,” in Light Scattering Near Phase Transitions, H. Z. Cummins, A. P. Levanyuk, eds. (North-Holland, Amsterdam, 1983), pp. 359–447.
[CrossRef]

Dalnoki-Veress, K.

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

Danielmeyer, H. G.

H. G. Danielmeyer, “Aperture corrections for sound-absorption measurements with light scattering,” J. Acoust. Soc. Am. 47, 151–154 (1969).
[CrossRef]

Du, W. M.

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

Dutcher, J. R.

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

Farnell, G. W.

G. W. Farnell, “Properties of elastic surface waves,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1970), Vol. 6, pp. 109–166.

G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. IX, pp. 35–127.

Forrest, J. A.

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

Ghislotti, G.

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

Giovannini, L.

L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
[CrossRef] [PubMed]

Herbst, C. A.

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

Hernandez, J.

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

Hillebrands, B.

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

Karanikas, J. M.

J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
[CrossRef]

Lee, S.

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

S. Lee, “Elastic properties of polymeric Langmuir-Blodgett studied using Brillouin light scattering,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991).

Li, G.

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

Lindsay, S. M.

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

Loudon, R.

R. Loudon, J. R. Sandercock, “Analysis of the light-scattering cross-section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

Marvin, A. M.

L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
[CrossRef] [PubMed]

McLaughlin, G. J.

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

Mutti, P.

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

Nickel, B. G.

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

Nizzoli, F.

L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
[CrossRef] [PubMed]

Oliver, W. F.

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

Phillips, J. M.

J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
[CrossRef]

Sandercock, J. R.

R. Loudon, J. R. Sandercock, “Analysis of the light-scattering cross-section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

J. R. Sandercock, W. Wettling, “Light scattering from surface and bulk thermal magnons in iron and nickel,” J. Appl. Phys. 50, 7784–7789 (1979).
[CrossRef]

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

J. R. Sandercock, “Trends in Brillouin light scattering: studies of opaque materials, supported films and central modes,” in Light Scattering in Solids III, M. Cardona, G. Güntherodt, eds. (Springer-Verlag, Berlin, 1982), pp. 173–206.
[CrossRef]

Sooryakumar, S.

J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
[CrossRef]

Stegeman, G. I.

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

Stevens, J. R.

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

Tao, N. J.

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

Wettling, W.

J. R. Sandercock, W. Wettling, “Light scattering from surface and bulk thermal magnons in iron and nickel,” J. Appl. Phys. 50, 7784–7789 (1979).
[CrossRef]

Wolf, G. H.

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

J. Acoust. Soc. Am.

H. G. Danielmeyer, “Aperture corrections for sound-absorption measurements with light scattering,” J. Acoust. Soc. Am. 47, 151–154 (1969).
[CrossRef]

J. Appl. Phys.

J. R. Sandercock, W. Wettling, “Light scattering from surface and bulk thermal magnons in iron and nickel,” J. Appl. Phys. 50, 7784–7789 (1979).
[CrossRef]

J. M. Karanikas, S. Sooryakumar, J. M. Phillips, “Dispersion of elastic waves in supported CaF2 films,” J. Appl. Phys. 65, 3407–3410 (1989).
[CrossRef]

J. Phys. C

R. Loudon, J. R. Sandercock, “Analysis of the light-scattering cross-section for surface ripples on solids,” J. Phys. C 13, 2609–2622 (1980).
[CrossRef]

J. Phys.: Condens. Matter

H. Z. Cummins, G. Li, W. M. Du, J. Hernandez, N. J. Tao, “Light scattering spectroscopy of the liquid-glass transition,” J. Phys.: Condens. Matter 6, A51–A62 (1994).
[CrossRef]

Phys. Rev. B

G. S. Agarwal, “Interaction of electromagnetic waves at rough dielectric surfaces,” Phys. Rev. B 15, 2371–2383 (1977).
[CrossRef]

Phys. Rev. Lett.

L. Giovannini, F. Nizzoli, A. M. Marvin, “Theory of surface acoustic phonon normal modes and light scattering cross section in a periodically corrugated surface,” Phys. Rev. Lett. 69, 1572–1575 (1992).
[CrossRef] [PubMed]

J. R. Dutcher, S. Lee, B. Hillebrands, G. J. McLaughlin, B. G. Nickel, G. I. Stegeman, “Surface-grating-induced zone folding and hybridization of surface acoustic modes,” Phys. Rev. Lett. 68, 2464–2467 (1992).
[CrossRef] [PubMed]

J. A. Forrest, K. Dalnoki-Veress, J. R. Stevens, J. R. Dutcher, “Effect of free surfaces on the glass transition temperature of thin polymer films,” Phys. Rev. Lett. 77, 2002–2005 (1996).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

W. F. Oliver, C. A. Herbst, S. M. Lindsay, G. H. Wolf, “A general method for determination of Brillouin linewidths by correction for instrumental effects and aperture broadening: application to high-pressure diamond anvil cell experiments,” Rev. Sci. Instrum. 63, 1884–1895 (1992).
[CrossRef]

Other

J. R. Sandercock, “Trends in Brillouin light scattering: studies of opaque materials, supported films and central modes,” in Light Scattering in Solids III, M. Cardona, G. Güntherodt, eds. (Springer-Verlag, Berlin, 1982), pp. 173–206.
[CrossRef]

P. Mutti, C. E. Bottani, G. Ghislotti, M. Beghi, G. A. D. Briggs, J. R. Sandercock, “Surface Brillouin scattering-extending surface wave measurements to 20 GHz,” in Advances in Acoustic Microscopy 1, G. A. D. Briggs, eds. (Plenum, New York, 1995), pp. 249–300.
[CrossRef]

S. Lee, “Elastic properties of polymeric Langmuir-Blodgett studied using Brillouin light scattering,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1991).

H. Z. Cummins, “Brillouin scattering studies of phase transitions in crystals,” in Light Scattering Near Phase Transitions, H. Z. Cummins, A. P. Levanyuk, eds. (North-Holland, Amsterdam, 1983), pp. 359–447.
[CrossRef]

G. W. Farnell, “Properties of elastic surface waves,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1970), Vol. 6, pp. 109–166.

E. D. Palik, ed., Handbook of Optical Constants of Solids, (Academic, New York, 1985), Vol. 1, p. 564.

G. W. Farnell, E. L. Adler, “Elastic wave propagation in thin layers,” in Physical Acoustics, Principles and Methods, W. P. Mason, R. N. Thurston, eds. (Academic, New York, 1972), Vol. IX, pp. 35–127.

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Figures (5)

Fig. 1
Fig. 1

Surface Brillouin light scattering geometry. p and s refer to the polarization of the light. The plane of the figure is the plane of incidence. The inset shows the scattered wave vector k s , which is not necessarily in the plane of incidence. The scattered angle θ s is measured between k s and the sample normal n.

Fig. 2
Fig. 2

(a) Contour plot of the magnitude of Q for θ i = 30°. The circles represent lens apertures and the inset shows the orientation of the phonon wave vector with respect to the sample surface plane x′, y′, corresponding to points a, b, c, and d. ΔQ between curves is 0.05 × k i . The cross-hatched areas are described in the text. (b) Same as in (a) but for θ i = 70° and ΔQ between curves is 0.02 × k i .

Fig. 3
Fig. 3

Measured and calculated spectra for the Rayleigh mode of Si(001). For the measured peaks: θ i = 30°, lens f/2, [110] direction of sample along β direction, FSR = 30 GHz. Calculated curves: v R = 5080 m/s,14 ∊ = 17.8 + 0.506i.15 The vertical dashed line represents the position of the peak for an infinitesimal lens aperture and the horizontal dashed lines represent zero-scattered intensity for each spectrum. Collection times are 2.76, 1.69, and 1.48 s/data point for the No Slit, Slit A, and Slit B spectra, respectively.

Fig. 4
Fig. 4

Same as Fig. 3, but for θ i = 70°. Collection times are 1.25, 1.23, and 1.35 s/data point for the No Slit, Slit A, and Slit B spectra, respectively.

Fig. 5
Fig. 5

BLS spectra for a freely standing polystyrene film, 75 nm thick, θ i = 45°, FSR = 10 GHz, and f/2 lens. The instrumental FWHM was 0.4 GHz. The horizontal dashed lines represent zero-scattered intensity for each spectrum. Collection times are 4.38 and 1.68 s/data point for the No slit and Slit A spectra, respectively.

Tables (2)

Tables Icon

Table 1 Measured and Calculated Peak Frequency and FWHM of Phonon Peaks Presented in Fig. 3 i = 30°)

Tables Icon

Table 2 Measured and Calculated Peak Frequency and FWHM of Phonon Peaks Presented in Fig. 4 i = 70°)

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

k i = k i sin   θ i j ,
k s = k x s i + cos   θ i k y s -   sin   θ i k z s j ,
Q = ± k i   sin   θ   cos   ϕ i + cos   θ i   sin   θ   sin   ϕ - sin   θ i × 1 + cos   θ j ,
d σ d Ω     F θ i ,   θ ,   ϕ cos 2   ψ   cos 2   θ s cos   θ i Q / k i - 1 cos   θ s + - sin 2   θ s 1 / 2 cos   θ i + - sin 2   θ i 1 / 2 2 ,   s s sin 2   ψ   cos 2   θ s cos   θ i Q / k i - 1 - sin 2   θ s 1 / 2   cos   θ s + - sin 2   θ s 1 / 2 cos   θ i + - sin 2   θ i 1 / 2 2 ,   s p sin 2   ψ   cos 2   θ s cos   θ i Q / k i - 1 - sin 2   θ i 1 / 2 cos   θ s + - sin 2   θ s 1 / 2   cos   θ i + - sin 2   θ i 1 / 2 2 ,   p s cos 2   θ s cos   θ i Q / k i - 1 cos   ψ - sin 2   θ s 1 / 2 - sin 2   θ i 1 / 2 -   sin   θ i sin   θ s   cos   θ s + - sin 2   θ s 1 / 2   cos   θ i + - sin 2   θ i 1 / 2 2 ,   p p
I c ω = 0 θ max sin   θ d θ   0 2 π d ϕ I s Q θ ,   ϕ ,   θ i ,   ω ,
I s Q θ ,   ϕ ,   θ i ,   ω     F θ ,   ϕ ,   θ i × δ ω - Q v R ,
Q = k i | sin   θ i + sin θ i - δ | ,

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