Quasinormal-mode theory of elastic Purcell factors and Fano resonances of optomechanical beams

Al-Waleed El-Sayed,Stephen Hughes

doi:10.1103/physrevresearch.2.043290

Abstract

We introduce a quasinormal mode theory of mechanical open-cavity modes for optomechanical resonators, and demonstrate the importance of using a generalized (complex) effective mode volume and the phase of the quasinormal mode. We first generalize and fix the normal mode theories of the elastic Purcell factor, and then show a striking example of coupled quasinormal modes yielding a pronounced Fano resonance. Our theory is exemplified and confirmed by full three-dimensional calculations on optomechanical beams, but the general findings apply to a wide range of mechanical cavity modes. This quasinormal mechanical mode formalism, when also coupled with a quasinormal theory of optical cavities, offers a unified framework for describing a wide range of optomechanical structures where dissipation is an inherent part of the resonator modes.

Highlights

The ability to describe optical cavities in terms of normalized modes and cavity figures of merit has played a significant role in laser optics and cavity quantum electrodynamics
We have introduced a quasinormal modes (QNMs) formalism for mechanical cavity modes and shown that the commonly used normal-mode description is problematic for cavity resonances with finite loss
We have presented and employed a complex, position-dependent effective mode volume for mechanical modes using a QNM normalization that can be used to solve a range of force-displacement problems

Summary

INTRODUCTION

The ability to describe optical cavities in terms of normalized modes and cavity figures of merit has played a significant role in laser optics and cavity quantum electrodynamics (cavity-QED). Optomechanical structures can support coupling between mechanical and optical modes [11], offering a wide range of applications in optomechanics [12,13,14,15,16,17] Despite these similarities, the common use of cavity mode theories in optics is far less developed in elastics, and, for example, one rarely talks about “mechanical” effective mode volumes. III, we present numerical calculations for a fully threedimensional (3D) optomechanical beam, first for a single QNM design, and for coupled QNMs that show a striking Fano resonance In both cases, we highlight the failure of using a normal-mode theory, which is shown to be drastic in.

THEORY

Complex mode volumes and the elastic Purcell formula in terms of QNMs

Mechanical mode effective mode volumes and Purcell factors

Coupled mechanical quasinormal modes

CONCLUSIONS