Nonlinear Phonon Focusing

Abstract

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., $$ u(\underline r ,\omega ) \propto \int {d\underline k \delta } (\omega - \Omega (k)){{e}^{{i\underline k \underline r }}} = \int {dS\frac{{{{e}^{{ir\phi }}}}}{{\left| {\nabla \Omega (k)} \right|}}} $$ (1) 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.KeywordsDispersion RelationGaussian CurvatureUmbilic PointFrequency SurfaceDetailed Experimental InvestigationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.