Nonlinear dynamics of a microelectromechanical mirror in an optical resonance cavity

Abstract

The nonlinear dynamical behavior of a micromechanical resonator acting as one of the mirrors in an optical resonance cavity is investigated. The mechanical motion is coupled to the optical power circulating inside the cavity both directly through the radiation pressure and indirectly through heating that gives rise to a frequency shift in the mechanical resonance and to thermal deformation. The energy stored in the optical cavity is assumed to follow the mirror displacement without any lag. In contrast, a finite thermal relaxation rate introduces retardation effects into the mechanical equation of motion through temperature dependent terms. Using a combined harmonic balance and averaging technique, slow envelope evolution equations are derived. In the limit of small mechanical vibrations, the micromechanical system can be described as a nonlinear Duffing-like oscillator. Coupling to the optical cavity is shown to introduce corrections to the linear dissipation, the nonlinear dissipation and the nonlinear elastic constants of the micromechanical mirror. The magnitude and the sign of these corrections depend on the exact position of the mirror and on the optical power incident on the cavity. In particular, the effective linear dissipation can become negative, causing self-excited mechanical oscillations to occur as a result of either a subcritical or supercritical Hopf bifurcation. The full slow envelope evolution equations are used to derive the amplitudes and the corresponding oscillation frequencies of different limit cycles, and the bifurcation behavior is analyzed in detail. Finally, the theoretical results are compared to numerical simulations using realistic values of various physical parameters, showing a very good correspondence.

Equations of a moving mirror and the electromagnetic field

“…Afterwards, we deduce equations that are correct up to first order in both the velocity and acceleration of the slab. With these results at hand, we conclude that the motion of the mirror and its interaction with the field give rise to a position-and time-dependent mass related to the effective mass taken in phenomenological treatments of this type of systems [4,10,13] and to a velocity-dependent force that is related to the cooling of mechanical objects [4,5]. Moreover, the field must satisfy a wave equation that depends on the slab's position, velocity, and acceleration.…”

“…Therefore, the motion of the mirror and the coupling to the field give rise to two effects. The first one is a position-and time-dependent mass related to the effective mass taken in phenomenological treatments of this type of systems [4,10,13]. The second one is a velocity-dependent force that can give rise to friction and that is related to the cooling of mechanical objects [4,5].…”

Controllable chaotic dynamics in a nonlinear fiber ring resonators with balanced gain and loss

“…It implies that two trajectories emerging out from two distinct but nearby initial conditions diverge exponentially from each other with the passage of time. The idea of chaos is studied quite extensively in natural sciences, especially in chemistry [1][2][3], electronics [4][5], fluid dynamics [6] and nonlinear optics [7][8][9][10][11]. In this work, we study a single fiber resonator (SFR) system with balanced gain and loss modeled by the so-called Ikeda map [12].…”

Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

“…The nonlinearity of the optomechanical interaction gives rise to nonlinear drift and diffusion coefficients in the FPE that describe, respectively, the (classical) nonlinear physics of limit cycles [19,20] and the impact of quantum noise of the cavity. The approach taken here permits us to work in a picture that interpolates between the dressed-state picture introduced in Refs.…”