Forced and self-excited oscillations of an optomechanical cavity

We experimentally study forced and self-excited oscillations of an optomechanical cavity, which is formed between a fiber Bragg grating that serves as a static mirror and a freely suspended metallic mechanical resonator that serves as a moving mirror. In the domain of small amplitude mechanical oscillations, we find that the optomechanical coupling is manifested as changes in the effective resonance frequency, damping rate, and cubic nonlinearity of the mechanical resonator. Moreover, self-excited oscillations of the micromechanical mirror are observed above a certain optical power threshold. A comparison between the experimental results and a theoretical model that we have recently derived and analyzed yields a good agreement. The comparison also indicates that the dominant optomechanical coupling mechanism is the heating of the metallic mirror due to optical absorption.

Figure 1

(a) The experimental system. In addition to the network analyzer, a number of measuring and excitation schemes can be used, such as homodyne vibration detection with a lock-in amplifier and direct time sampling of the reflected optical power. A tunable infrared laser is used, with a maximum output power of $2\mathrm{mW}$. A single mode optical fiber is used to transmit light to and from the sample, allowing reflection measurements (with the aid of a circulator to separate the incident and the reflected beams). A graded index lens at the end of the optical fiber is used to focus the light on the mirror. The focus distance is $\approx 40\mu$m. A FBG located inside the fiber can serve as a second mirror to create a relatively high-finesse cavity at wavelengths that fall inside the Bragg region. The micromechanical mirror can be capacitively actuated by applying a voltage between the mirror itself and a ground plate located $500\mu$m below it (the ground plate is not shown). Panels (b) and (c) exhibit the top views of the micromechanical mirrors employed in the experiments. The thickness of the metal layers is 300 nm for AuPd samples and 200 nm for Al samples.

Figure 2

Optical reflection in the vicinity of an optical resonance. The incident optical power is $I_{\mathrm{pump}}=3\mu \mathrm{W}$. Blue circles denote the measured reflected power. the Solid black line denotes theoretical reflection values [see Eq. (2)]. The dashed red line represents the theoretical values of the optical power $I_0$ incident on the micromechanical mirror [see Eq. (3)]. In this case, $\Gamma =134.5$ nm, and the finesse is 5.8. In the inset, two time domain traces of the photodetector output signal are presented, corresponding to a steady oscillation of the micromechanical mirror with a large (thin black line) and a small (bold blue line) amplitude.

Figure 3

Changes in the effective quality factor $Q_{\mathrm{eff}}$ [see Eq. (16)] as a function of the optical detuning $x_0$ and the optical power incident on the cavity $I_{\mathrm{pump}}$. Panel (a) shows the theoretical value of $Q/Q_{\mathrm{eff}}$, whereas panel (b) shows the measured data. In the case of a rectangular AuPd mirror presented here, ${\omega }_0=2\pi \times 160.088\mathrm{kHz}$, $Q=2.43\times {10}^5$, and the cavity finesse is 7.3. Panels (c), (d), and (e) show cross sections of the top color maps at different values of $I_{\mathrm{pump}}$. Blue dots represent the experimental values of $Q/Q_{\mathrm{eff}}$, while solid black lines represent the theoretical values. The values of all system parameters used in the fit are similar to those given in Sec. 4. The temperature is 77 K. The estimated uncertainty in the measured results is $5\%$.

Figure 4

Changes in the small amplitude nonlinear behavior as a function of the optical detuning $x_0$. It follows from Eqs. (15) that the micromechanical mirror should exhibit hardening ($q>0$) behavior [42] in the region $|x_0|<\Gamma /2\sqrt{3}=0.289\Gamma$, and softening behavior outside this region. This effect is illustrated for two different samples. (a), (b) AuPd rectangular mirror measurements. (c), (d) Al doubly clamped beam mirror measurements. In both cases, the micromechanical mirrors are excited capacitively and the frequency response is measured when sweeping the excitation frequency up (red dots) and down (blue diamonds). As expected, the frequency responses (a) and (c), for which $x_0>\Gamma /2\sqrt{3}$, exhibit softening elastic nonlinearity, while the frequency responses (b) and (d), corresponding to $x_0<\Gamma /2\sqrt{3}$, show hardening. In addition, the theoretical results are shown (black solid lines). The values of all the system parameters used in the fit are similar to those given in Sec. 4, except for $\beta =4.8\times {10}^4\mathrm{rad}/\mathrm{s}\mathrm{K}$ for the Sample II Au mirror (see text for more details). The optical power incident on the cavity and the external excitation amplitudes are $I_{\mathrm{pump}}=14\mu \mathrm{W}$ and $f_m=0.16\mathrm{N}/\mathrm{kg}$ for AuPd rectangular mirror, and $I_{\mathrm{pump}}=2.5\mu \mathrm{W}$ and $f_m=7\times {10}^{-3}\mathrm{N}/\mathrm{kg}$ for Al doubly clamped beam mirror, respectively.

Figure 5

The self-oscillation amplitude $A_{\mathrm{LC}}$ as a function of the optical detuning $x_0$ and optical power incident on the cavity $I_{\mathrm{pump}}$. Panel (a) shows theoretically predicted amplitude values, whereas panel (b) shows measured data. The color represents the value $A_{\mathrm{LC}}/\Gamma$. The dashed bold green line in the experimental color map (b) represents the theoretical threshold of self-oscillations. In the case of a rectangular AuPd mirror presented here, ${\omega }_0=2\pi \times 160.088\mathrm{kHz}$, $Q=2.43\times {10}^5$, and the cavity finesse is 5.9. Panels (c), (d), and (e) show cross sections of the top color maps at different values of $I_{\mathrm{pump}}$. Blue dots represent the experimental values of $A_{\mathrm{LC}}/\Gamma$, while solid black lines represent the theoretical values. The values of all the system parameters used in the fit are similar to those given in Sec. 4. The temperature is 77 K. The estimated uncertainty in the measured results is $8\%$.

Figure 6

Spectral power density of the reflected optical power as a function of frequency and optical detuning $x_0$. The mechanical resonance frequency is $148.495\mathrm{kHz}$. The incident optical power is $I_{\mathrm{pump}}=140\mu \mathrm{W}$. The color represents the spectral power density in arbitrary logarithmic units. The region in which self-oscillations occur is denoted by dashed red lines. The thermal motion peak can be readily recognized outside this region. The abundance of additional peaks in the self-excited oscillations domain may be attributed to the nonlinearity of the detection system and to mixing between higher mechanical modes. In the inset, the experimental values of frequency shift (blue circles) both for thermal peak and for self-oscillations are plotted together with the theoretical predictions (solid black line) given by Eqs. (14d) and (18).