Phase Space Distribution Near the Self-Excited Oscillation Threshold

We study the phase space distribution of an optomechanical cavity near the threshold of self-excited oscillation. A fully on-fiber optomechanical cavity is fabricated by patterning a suspended metallic mirror on the tip of the fiber. Optically induced self-excited oscillation of the suspended mirror is observed above a threshold value of the injected laser power. A theoretical analysis based on the Fokker-Planck equation evaluates the expected phase space distribution near threshold. A tomography technique is employed for extracting phase space distribution from the measured reflected optical power vs time in steady state. Comparison between theory and experimental results allows the extraction of the device parameters.

Figure 1

A schematic drawing of sample $A$ and the experimental setup. An on-fiber optomechanical cavity is excited by a laser. The reflected light intensity is measured and analyzed. The inset shows a typical trace of the normalized displacement $(x-x_0)/\lambda$ vs time measured by the oscilloscope above the self-excited oscillation threshold with $\mathrm{\Delta }{P}_L/P_{\mathrm{LC}}=0.034$. The photodetector signal is translated to displacement by measuring the thermomechanical noise and employing the calibration method that is described in Ref. [58].

Figure 2

The dependence on laser power $P_L$ for sample $A$. (a) Phase space distribution extracted from the measured probability distribution function $w(X_{\varphi }^{\prime })$ using Eq. (6). (b) Phase space distribution calculated using Eq. (5). The following device parameters have been employed for generating the plot in panel (b): ${\omega }_0=2\pi \times 144\mathrm{kHz}$, $\lambda =1.55\mu \mathrm{m}$, $T_{\mathrm{eff}}=300\mathrm{K}$, $s_D=0.8$, and ${\gamma }_2{\lambda }^2/{\gamma }_0=8\times {10}^4$.

Figure 3

The dependence on wavelength $\lambda$ for sample $B$. The static mirror of the optomechanical cavity is provided by a fiber Bragg grating mirror (made using a standard phase mask technique [59], grating period of $0.527\mu \mathrm{m}$ and length $\approx 8\mathrm{mm}$) with the reflectivity band of 0.4 nm full width at half maximum centered at 1550 nm. The length of the optical cavity was 10 mm. (a) Phase space distribution extracted from the measured probability distribution function $w(X_{\varphi }^{\prime })$ using Eq. (6). The laser wavelength is varied from the cavity resonance value of ${\lambda }_R=1.545135\mu \mathrm{m}$, for which the detuning factor $s_D$ vanishes, to $1.545158\mu \mathrm{m}$, for which $s_D=1.3$. (b) Phase space distribution calculated using Eq. (5). The following device parameters have been employed for generating the plot in panel (b): ${\omega }_0=2\pi \times 225\mathrm{kHz}$, $m=1.1\times {10}^{-12}\mathrm{kg}$, ${\beta }_{+}=0.68$, $T_{\mathrm{eff}}=300\mathrm{K}$, and ${\gamma }_2{\lambda }^2/{\gamma }_0=5\times {10}^5$. Panels (c)–(f) show cross sections of the top color maps at different values of the detuning factor $s_D$.