Parametric Resonance of Optically Trapped Aerosols

The Brownian dynamics of an optically trapped water droplet are investigated across the transition from over- to underdamped oscillations. The spectrum of position fluctuations evolves from a Lorentzian shape typical of overdamped systems (beads in liquid solvents) to a damped harmonic oscillator spectrum showing a resonance peak. In this later underdamped regime, we excite parametric resonance by periodically modulating the trapping power at twice the resonant frequency. The power spectra of position fluctuations are in excellent agreement with the obtained analytical solutions of a parametrically modulated Langevin equation.

Figure 1

The measured power spectra of trapped aerosol particle at two different powers. At lower power (black circles), it is overdamped, and the mean squared amplitude of the high frequency motion decays as ${\omega }^{-2}$. At higher powers (white circles), the aerosol is underdamped, and the mean squared amplitude decays as ${\omega }^{-4}$. Gray lines show the calculated slopes $-2$ and $-4$. The inset shows an optical image of a trapped aerosol particle.

Figure 2

The measure power spectrum of a trapped water droplet for no modulation of the laser power (white circles) and modulation at 3.9 kHz (${\Omega }_1\simeq 2{\Omega }_0$) (black circles). The peak is higher and narrower on the resonant condition, thus indicating parametric excitation. The solid line below the black circles is the predicted spectrum from (11).

Figure 3

Evolution of position power spectra on varying the modulation frequency ${\Omega }_1$. Parametric excitation of oscillations is evident at the parametric resonance condition ${\Omega }_1/{\Omega }_0=2$. The solid lines are the theoretical predictions from (11). The vertical gray line indicates ${\Omega }_0/2\pi =2.3\mathrm{kHz}$.

Figure 4

(a) Peak full width at half maximum as a function of modulation frequency. The solid lines are the theoretical predictions from (11). (b) Peak position shift as a function of modulation frequency. ${\Omega }_{p0}$ is the peak position in the absence of modulation.