Phonon focusing in reflection and transmission

A unified treatment of phonon focusing in single-layered and multilayered solids is provided. A Jacobian is established that determines the position dependence of the phonon flux transmitted, via any mode sequence which may include backreflections, through a solid consisting of any number of parallel anisotropic layers. The Jacobian is related in a simple way to the curvature of an effective slowness surface whose equation is the weighted mean of the equations of the slowness surfaces of the individual layers. Two illustrative numerical examples are provided, one showing how the reflection focusing pattern of silicon depends on the mode sequence, and the other showing the evolution of the transverse phonon focusing pattern of a LiF-${\mathrm{ThO}}_2$ bilayer with variation in the ratio of the thicknesses of the two layers.