Phonon magnification and the Gaussian curvature of the slowness surface in anisotropic media: Detector shape effects with application to GaAs

A new short derivation is given for the magnification of photons, phonons, and other wave phenomena in anisotropic media. The results, when specialized to the phonon case, agree with those of Maris and of Philip and Viswanathan. At nondegenerate points the magnification is shown, regardless of the nature of the wave phenomenon, to be expressible in terms of the Gaussian curvature $K$ at the wave vector $\overrightarrow{\mathrm{q}}$ of the surface $\omega \left(\overrightarrow{\mathrm{q}}\right)=\omega$. A representation of $K$, free of surface coordinates, is given as the cofactor of a 3×3 curvature tensor. The curvature tensor contains information beyond the Jacobian, such as the two principal magnifications for a given mode. Along degenerate directions (such as the [100] direction in cubic crystals) the results, even in the infinitesimal limit, are sensitive to the shape of the detector. Explicit expressions for circular, rectangular and square detectors are given and applied to the case of ballistic phonons in GaAs. Numerical calculation of finite aperture effects indicates significant qualitative as well as quantitative differences with infinitesimal aperture results.