Acoustic whispering gallery modes within the theory of elasticity

Abstract

Investigations of nonlinear phenomena in optical whispering gallery mode (WGM) microresonators are booming because of rich physics and applications. Stimulated Brillouin scattering is one of the strongest processes in these devices. Here, the optical WGMs interact with acoustic counterparts. The acoustic WGMs are well known for resonators based on liquids and gases, where the sound waves are longitudinal. The situation with solid-state resonators is different because of the presence of the longitudinal (l) and transverse (t) sound waves with substantially different velocities vl,t. Moreover, the l- and t-parts of the acoustic displacement are coupled at the resonator surface breaking the separation of modes into longitudinal and transverse. Investigation of the acoustic WGMs is of high priority. Here, analytically and numerically we investigate the resonant frequencies and the eigenfunctions (displacement vector distributions) for acoustic WGMs in microresonators made of isotropic solid-state materials. Cylindrical and spherical resonators are considered. Each mode has the azimuth, radial, and orbital (for sphere) numbers m, q, and ℓ; its properties are controlled also by the ratio vl/vt. All modes are either transverse (t) or hybrid transverse-longitudinal (tl). Pure l-modes, providing the strongest interaction with optical modes in fibers and bulk crystals, are absent. The tl-modes include distorted Rayleigh waves, the modes with q∼1 and dominating t-part, and pseudo-longitudinal modes with q≫1, closely spaced frequencies, and weakly localized t-part. They have no analogies to the optical WGMs and are of high relevance for Brillouin lasing in optical microresonators. The actual values of ℓ and m are 102−105, and the lasing thresholds lie in the μW range. Our findings include exact dispersion equations for acoustic WGMs, which can be solved numerically for ℓ,m≲104, asymptotic tools for ℓ,m≳103, and particular illustrations.