Symmetry of phonon transmission and reflection images in (111)-oriented cubic crystals

The ballistic focusing patterns of phonons in a cubic crystal specularly reflected from the (111) surface and of those transmitted through identical (111) slabs twist bonded at a 60° angle have the same caustic distributions. In addition, they exhibit a sixfold symmetry rather than the threefold symmetry expected from the crystal symmetry. However, it is not obvious if the intensity distributions in their focusing patterns also exhibit the sixfold symmetry. We study the conditions under which those phonon intensities possess sixfold symmetry. It is shown that this symmetry is realized if the occupation number of phonons excited depends only on frequency and is independent of the phonon polarizations and wave-vector directions, just like the case of Planckian distribution for equilibrium phonons. We illustrate the results for the ballistic phonon images in (111)-oriented silicons.