Nonlinear Phonon Focusing

  • D. Armbruster
  • G. Dangelmayr
  • W. Güttinger
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 51)


Acoustic phonon propagation in a cold anisotropic crystal is dominated by focusing which typically occurs on structurally stable caustics [1]. By applying singularity theory, the forms of these caustics and the associated high-intensity diffraction patterns can be classified into a few topological types [2], [3]. Suppose a monochromatic point source of frequency ω generates phonons with wave vectors k that propagate ballistically in a crystal whose anisotropy is described by a dispersion relation ω=Ω(k). Then only those k contribute to the phonon field u(r,ω) at a point r in space which make up the constant-frequency surface S:ω=Ω(k)=const, i.e.,
$$ u(\underline r ,\omega ) \propto \int {d\underline k \delta } (\omega - \Omega (k)){{e}^{{i\underline k \underline r }}} = \int {dS\frac{{{{e}^{{ir\phi }}}}}{{\left| {\nabla \Omega (k)} \right|}}} $$
for a given polarization mode. Here, \( \underline r = r\underline {\hat{r}} \) with unit vector \( \underline {\hat{r}} ,\phi = \underline {\hat{r}} \cdot \underline k \) and the second integral is taken over S. Suppose first that the phonon’s group velocity v=▽Ω(k), with \( \underline {\hat{v}} = \underline v /\left| {\underline v } \right| = \underline n \) the unit normal to S, has no zeros. Then the phonon flux is in the directions \( \underline {\hat{r}} = \underline {\hat{v}} (\underline k ) \). The corresponding wave vectors k on S are (for large r) those for which ⌽ is stationary, t•▽k⌽=0 for vectors t tangent to S. Phonon focusing directions, i.e., angular caustics, come from the inflection points of S along a principal curvature line where the Gaussian curvature vanishes. These are the stationary points where the Hessian determinant of ⌽ vanishes. Since the caustics are structurally stable, i.e., insensitive to small perturbations, ⌽ is equivalent to a Thom catastrophe polynomial ⌽=⌽T.


Dispersion Relation Gaussian Curvature Umbilic Point Frequency Surface Detailed Experimental Investigation 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • D. Armbruster
    • 1
  • G. Dangelmayr
    • 1
  • W. Güttinger
    • 1
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenFed. Rep. of Germany

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