Phonon focusing of dispersive phonons in Ge

A recent phonon-imaging experiment measuring ballistic propagation of high-frequency acoustic phonons in Ge has shown a spatial shift, due to dispersion, in phonon-focusing singularity patterns. Existing phonon-focusing theory is based upon continuum elasticity and is unable to describe this shift. In this paper the concept of the phonon-focusing enhancement factor is extended from the two-dimensional continuum limit to the three-dimensional dispersive case. Singularity lines for a range of frequencies are shown to form a singularity surface. With the use of an extended Born---von K\'arm\'an model for the lattice dynamics of Ge, the lines of singular-phonon-flux enhancement are calculated for several frequencies between 0 and 1500 GHz. The results for 0 GHz and 700-1000 GHz compare favorably with the results in those experimental regimes.