Radiation-pressure-mediated control of an optomechanical cavity

We describe and demonstrate a method to control a detuned movable-mirror Fabry-Pérot cavity using radiation pressure in the presence of a strong optical spring. At frequencies below the optical spring resonance, self-locking of the cavity is achieved intrinsically by the optomechanical (OM) interaction between the cavity field and the movable end mirror. The OM interaction results in a high rigidity and reduced susceptibility of the mirror to external forces. However, due to a finite delay time in the cavity, this enhanced rigidity is accompanied by an antidamping force, which destabilizes the cavity. The cavity is stabilized by applying external feedback in a frequency band around the optical spring resonance. The error signal is sensed in the amplitude quadrature of the transmitted beam with a photodetector. An amplitude modulator in the input path to the cavity modulates the light intensity to provide the stabilizing radiation pressure force.

Figure 1

Schematic of the experimental setup. The laser is passed through an amplitude modulator (AM), which modulates the intensity of the field before entering the detuned optomechanical cavity. The light is detected in transmission of the cavity and passed through a high-pass filter and servo controller containing a gain and low-pass filter component to obtain the error signal, which is fed back to the AM. The inset shows the scale of the movable mirror that forms one of the cavity mirrors.

Figure 2

Detailed loop diagram for the cavity's transfer function. The amplitude modulator adds to the intracavity power $P_{\mathrm{cav}}$. The cavity power converts into radiation pressure force $F$, which then converts into cantilever displacement $x$ via its mechanical susceptibility ${\chi }_{\mathrm{m}}$. The displacement causes a length change for the cavity, leading again to a change in the intracavity power via the cavity response $C$. This forms a closed-loop system.

Figure 3

Loop diagram for the feedback $G_{\mathrm{f}}$. $H$ is the response of the high-pass filter and servo controller, and $\beta$ is the response of the amplitude modulator. $\frac{1}{1+G_{\mathrm{os}}}$ is the closed-loop response of the cavity's optical spring system, as shown in Fig. 2. $G_{\mathrm{in}}=\frac{4T_1}{T^2}\frac{1}{1+{\delta }_{\gamma }^2}$ is the transfer function of the power input to power inside the cavity with the effect of the detuning taken into account, and similarly $G_{\mathrm{out}}=T_2$ is the transfer function from cavity power to transmitted power that is measured on the PD. Here $T_1$ is the transmission of the input mirror, $T_2$ is the transmission of the microresonator, and $T$ is the total loss of power from the cavity, in the form of transmission, scattering, absorption, etc. $G_{\mathrm{f}}=G_{\mathrm{out}}\times \mathrm{PD}\times H\times \beta \times G_{\mathrm{in}}$ is calculated by using the measurements of the individual transfer functions.

Figure 4

Transfer function measurements of the plant, $G_{\mathrm{CL}}=\frac{1}{1+G_{\mathrm{os}}}$, shown in blue, the feedback, $G_{\mathrm{f}}$, shown in red, and the open-loop optical spring, $G_{\mathrm{os}}$, shown in pink. A model for the plant, shown in dashed cyan, is calculated with Eq. (2) using the measured values for ${\mathrm{\Omega }}_{\mathrm{os}}$, ${\mathrm{\Omega }}_{\mathrm{m}}$, and ${\mathrm{\Gamma }}_{\mathrm{m}}$ and setting ${\mathrm{\Gamma }}_{\mathrm{os}}$ so that the peak height and width match the measured data. $G_{\mathrm{cl}}$ is obtained using the open-loop gain, $\frac{G_{\mathrm{f}}}{1+G_{\mathrm{os}}}$, shown in Fig. 5 and dividing out the measured $G_{\mathrm{f}}$. The effect of the optical spring is visible with the peak at 75 kHz and a rise in phase of the plant transfer function. The measurement begins to flatten out below 5 kHz due to other circuitous signal couplings (e.g., scattered light). $G_{\mathrm{os}}$ is then obtained from $G_{\mathrm{cl}}$. The applied feedback loop is $G_{\mathrm{f}}$. At 75 kHz, $G_{\mathrm{f}}$ has a magnitude of 0.53 and a phase of $-{80}^{\circ }$.

Figure 5

Measurement of the open-loop gain or $\frac{G_{\mathrm{f}}}{1+G_{\mathrm{os}}}$ taken by injecting a signal before $H$, done with a circulating power of 0.2 W. Higher-order mechanical modes are visible at 1.4 kHz (yaw), 4.3 kHz (pitch), and 54 kHz (translation and yaw) are shown in the inset. Unity-gain crossings are at 61 and 93 kHz with phases ${109}^{\circ }$ and $-{115}^{\circ }$. The gain is 0.34 at 250 kHz where the phase crosses $-{180}^{\circ }$. Thus, the system is stable with phase margins of ${71}^{\circ }$ and ${65}^{\circ }$, respectively, and a gain margin of 9.4 dB.

Figure 6

Measurement of the closed-loop response performed by modulating the laser frequency. This plot shows the suppression of low-frequency frequency noise below the optical spring frequency at $\approx 75$ kHz. The amount of suppression is calculated by taking the ratio of the measurement above the optical spring and at a low frequency below the optical spring. Using the values at 100 kHz and 500 Hz, noise is suppressed by a factor of at least 50 000. Higher order mechanical modes are again visible at 1.4, 4.3, and 61 kHz.