Observation of three-mode parametric instability

Three-mode parametric interactions occur in triply resonant optomechanical systems: Photons from an optical pump mode are coherently scattered to a high-order mode by mechanical motion of the cavity mirrors, and these modes resonantly interact via radiation pressure force when some conditions are met. They can either pump energy into acoustic modes, leading to parametric instability, or extract mechanical energy, leading to optomechanical cooling. Such effects are predicted to occur in long baseline advanced gravitational-wave detectors, possibly jeopardizing their stable operation. We have demonstrated both three-mode cooling and amplification in two different three-mode optomechanical systems. We report an observation of the three-mode parametric instability in a free-space Fabry-Perot cavity, with ring-up amplitude saturation.

Figure 1

Radiation-pressure amplification seen as a macroscopic scattering process. (a) Two-mode amplification. The nonresonant incident photon at ${\omega }_0$ is scattered to the same transverse optical mode, with a frequency offset ${\Omega }_{\mathrm{m}}$ with respect to the incident photon. The cavity is detuned in order to favor the Stokes process, and the pump laser is far detuned from the resonance. (b) Three-mode amplification. The resonant photon is scattered to a different optical mode of the cavity. Both the pump laser and the Stokes band can be simultaneously resonant with the cavity. If the vibration profile of the mechanical mode matches the mode shape of the output optical mode, the system can achieve very strong coupling.

Figure 2

Cavity tuning close to the three-mode resonance condition, for the two experimental setups envisioned. (a) Mode spacing $\Delta \omega /2\pi$ for the case of a ${\mathrm{TEM}}_{00}$ and a ${\mathrm{TEM}}_{04}$ mode, as a function of cavity length. The curves are computed for a plano-concave cavity, with a radius of curvature $R=50$ mm. Insert shows a closeup of the degeneracy point. (b) Optical resonance frequencies of the compound cavity as a function of the membrane position $z_0$. For simplicity, only the relevant transverse modes ${\mathrm{TEM}}_{00}$ and ${\mathrm{TEM}}_{02}$ are shown. (Insert) Closeup of the avoided crossing near Crossover 1 (with relative membrane position). Note the different horizontal and vertical scales.

Figure 3

Membrane-in-the-middle configuration for the three-mode interaction experiment and illustration of the overlap between the membrane $(2,6)$ mechanical mode and ${\mathrm{TEM}}_{00}$ and ${\mathrm{TEM}}_{20}$ optical modes. (a) Cavity configuration. (b) Mode shape of the membrane (2,6) mode. (c) Mode shape of the ${\mathrm{TEM}}_{20}$ cavity mode. (d) Product of the mode shapes, with the optical mode correctly located on the membrane for optimized overlap.

Figure 4

Experimental demonstration of the resonant character of the three-mode anti-Stokes process. Mechanical dampings of the (1,7) [green (middle)], (5,5) [black (left)], and (3,7) [red (right)] mechanical modes. The damping increases substantially close to the three-mode resonance condition. Dots are experimental values of the effective damping, while the lines are fits to Eq. (4). Inserts display the simulated mode shape profiles of the mechanical modes.

Figure 5

Experimental demonstration of the parametric instability. The dots are experimental data of the beat-note signal between the cavity ${\mathrm{TEM}}_{00}$ mode and ${\mathrm{TEM}}_{20}$ mode at mechanical frequency, as a function of time for different input powers. The curves display exponential ring-up before reaching saturation. The solid curves are simulation results for the corresponding input power. The input power is $26\mu \mathrm{W}$ (black), $15\mu \mathrm{W}$ [orange (dark gray)], and $7\mu \mathrm{W}$ [yellow (light gray)].

Figure 6

Measured steady-state beat-note signal amplitudes (solid squares) and ring-up times (hollow circles) as a function of the input power. The dashed blue line fit for the ring-up time is from Eq. (A2), and yields a value of $G_{\mathrm{ps}}=2\pi \times 0.10$ Hz. The full red line fit for the beat-note amplitude is from Eqs. (A5) and (A6). The vertical line indicates the parametric instability threshold of $3.92\mu \mathrm{W}$.

Figure 7

Steady-state behavior of three-mode parametric instability. As soon as the cavity pump circulating power exceeds the instability threshold, the power remains constant for all input powers, while the transverse mode and the mechanical mode amplitude increase monotonically.

Figure 8

Simulation results for the ${\mathrm{TEM}}_{00}$ mode amplitude $\left(a_{\mathrm{p}}\right)$, the ${\mathrm{TEM}}_{20}$ scattered mode amplitude $\left(a_{\mathrm{s}}\right)$, and the mechanical motion amplitude $\left(b_{\mathrm{m}}\right)$. The input powers used here are $5\mu \mathrm{W}$ [yellow (light gray curves)], $15\mu \mathrm{W}$ [orange (dark gray curves)], and $25\mu \mathrm{W}$ (black curves). The system acts to control the pump power in the cavity to a value which is independent of the incident power, by scattering into the mechanical and transverse modes.

Figure 9

Three-mode cooling. Displacement thermal noise spectra observed close to the (1,7) mechanical resonance frequency, for different frequency offsets $\Delta \omega$ between ${\mathrm{TEM}}_{00}$ and ${\mathrm{TEM}}_{04}$ modes of the cavity. Due to the three-mode cooling, the thermal noise spectrum is both widened and reduced. Curves (a)–(d) are measured for different detunings, (a) being the furthest from resonance, and (d) the closest.

Figure 10

Experimental demonstration of the Stokes mechanism. Thermal noise spectra observed close to the Stokes process resonance. The curves are for different detunings (a)–(d); see insert. (Insert) Evolution of the effective mechanical damping of the (1,7) mechanical mode close to the Stokes resonance condition, together with a fit with theory.

Figure 11

Cavity spectrum when the mode spacing is tuned close to the mechanical mode frequency. A fit with a double Lorentzian allows the frequency difference $\Delta \omega$ and the cavity losses for both optical modes to be determined.