Scaling rules in optomechanical semiconductor micropillars

Abstract

Semiconductor pillar microcavities have recently emerged as a promising optomechanical platform in the unprecedented 20-GHz frequency range. Currently established models for the mechanical behavior of micropillars, however, rely on complete numerical simulations or semianalytical approaches, which makes their application to experiments notoriously difficult. Here we overcome this challenge with an effective model by reducing the full, hybridized mechanical mode picture of a micropillar to an approach that captures the observed global trends. We show experimentally the validity of this approach by studying the lateral size dependence of the frequency, amplitude, and lifetime of the mechanical modes of square-section pillar microcavities, using room-temperature pump-probe microscopy. General scaling rules for these quantities are found and explained through simple phenomenological models of the physical phenomena involved. We show that the energy shift $\mathrm{\ensuremath{\Delta}}{\ensuremath{\omega}}_{m}$ of the modes due to confinement is dependent on the inverse of their frequency ${\ensuremath{\omega}}_{0}$ and lateral size $L$ ($\mathrm{\ensuremath{\Delta}}{\ensuremath{\omega}}_{m}\ensuremath{\propto}1/{\ensuremath{\omega}}_{0}{L}^{2}$) and that the mode lifetime $\ensuremath{\tau}$ is linear with pillar size and inversely proportional to their frequency ($\ensuremath{\tau}\ensuremath{\propto}L/{\ensuremath{\omega}}_{0}$). The mode amplitude is in turn inversely proportional to the lateral size of the considered resonators. This is related to the dependence of the optomechanical coupling rate (${g}_{0}\ensuremath{\propto}1/L$) with the spatial extent of the confined electromagnetic and mechanical fields. Using a numerical model based on the finite-element method, we determine the magnitude and size dependence of ${g}_{0}$ and, by combining the results with the experimental data, we discuss the attainable single-photon cooperativity in these systems. The effective models proposed and the scaling rules found constitute an important tool in micropillar optomechanics and in the future development of more complex micropillar based devices.