Topological features of the polarization fields of plane acoustic waves in anisotropic media

Abstract

Crystal anisotropy entails an intricate distribution of polarization vectors of bulk acoustic waves dependent on their propagation directions. It is characterized in particular by bifurcations arriving near the directions of phase–speed degeneracy. A theory is envisaged which classifies various patterns of these polarization bifurcations in terms of the topological charge of singular points of plane vector fields. The global outlook of the polarization distribution in an anisotropic medium may be understood by projecting polarization vectors onto the unit sphere of wave normals and by appealing to the fruitful concepts lent by topology with respect to tangential vector fields. It allows the recognition of the role of longitudinally polarized modes, which, along with degenerate modes, stipulate singularities of these tangential fields and may be also attributed by the appropriately defined topological charge. A specific set of singularities largely predetermines the configuration of a given polarization field. Individual (local) topological charges are incorporated by using the concept of the global topological charge. This is introduced as a sum of local topological charges of both degenerate and longitudinal modes occurring in a wave branch specified by the condition of simple connectivity of the slowness–surface sheets. This sum is equal to two, being a topological invariant for arbitrary anisotropic media with any admissible distribution of degenerate and longitudinal modes between the wave branches. Complementary invariant equalities hold for the sums of local topological charges related to either types of modes. The validity of the basic conclusions is confirmed in the more general setting for piezoelectric crystals as well. Examples are discussed which reveal the intrinsic relation between degenerate and longitudinal modes propagating in different directions but belonging to the same wave branch of a given anisotropic body.