Cavity-stabilized laser with acceleration sensitivity below ${10}^{-12}$ g${}^{-1}$

We characterize the frequency sensitivity of a cavity-stabilized laser to inertial forces and temperature fluctuations, and perform real-time feedforward to correct for these sources of noise. We measure the sensitivity of the cavity to linear accelerations, rotational accelerations, and rotational velocities by rotating it about three axes with accelerometers and gyroscopes positioned around the cavity. The worst-direction linear acceleration sensitivity of the cavity is $2\left(1\right)\times {10}^{-11}$ g${}^{-1}$ measured over 0–50 Hz, which is reduced by a factor of 50 to below ${10}^{-12}$ g${}^{-1}$ for low-frequency accelerations by real-time feedforward corrections of all of the aforementioned inertial forces. A similar idea is demonstrated in which laser frequency drift due to temperature fluctuations is reduced by a factor of 70 via real-time feedforward from a temperature sensor located on the outer wall of the cavity vacuum chamber.

Figure 1

(a) Computer-aided design (CAD) drawing of the mounted Fabry-Pérot cavity. The spherical cavity spacer is 50 mm in diameter. (b) Cross-sectional view showing the details of the contact between the cavity and the mount. A Torlon ball is compressed between the flexure spring and the vacuum pump-out hole in the cavity. FS: fused silica.

Figure 2

Schematic of the fiber-optic frequency servo and feedforward. A laser is stabilized to the length of a spherical Fabry-Pérot cavity by the PDH method. Environmental sensors detect inertial forces and temperature drift that perturb the length of the cavity; the resulting frequency perturbations of the laser are corrected by feedforward to the frequency of an AOM. The stability of the laser is measured by comparing it with an independent laser. Note that both the frequency servo and the feedforward are implemented entirely digitally in FPGAs. Black lines denote optical fiber, red lines denote free-space laser propagation, and blue dashed lines denote electrical signals. VOA: variable optical attenuator; OI: optical isolator; PBS: polarizing beam splitter; $\lambda /4$: quarter-wave plate; DET: detector; BN: beat note; TX: transmission; PDH: Pound-Drever-Hall; THMS: thermistor; ACC: accelerometer; GYRO: gyroscope. See text for other abbreviations.

Figure 3

Allan deviation of the laser frequency when the physics package is located on a passively isolated optical table, with real-time inertial force feedforward (FF) off and on. The data with feedforward on is a 120 s data set with no drift removal. Also shown are the expected contribution of accelerometer noise to laser frequency noise and estimates of the laser stability if a vibration-insensitive laser were to be operated on several types of moving vehicles [32]. For these estimates, we assume the use of a hypothetical passive vibration isolation system that attenuates accelerations above 30 Hz by 40 dB per decade. To reach this performance, it will be necessary to extend the feedforward system, so that the low-frequency acceleration sensitivity of $4\times {10}^{-13}$ g${}^{-1}$ applies to the entire band of vehicle acceleration frequencies (0–2 kHz). The expected contribution of gyroscope noise to laser frequency noise, ${10}^{-19}{\tau }^{-1}$, where $\tau$ is the averaging time in seconds, is not visible on this plot.

Figure 4

Laser frequency power spectral density with the physics package located on a passively isolated optical table, with real-time inertial force feedforward on. The resolution bandwidth is 1 Hz. Dashed lines denote the calculated frequencies of rigid-body motional resonances of the cavity within its mount.

Figure 5

Linear acceleration sensitivity of the cavity versus frequency, without inertial force feedforward. These data were taken with the physics package sitting on a commercial active-vibration isolation (AVI) platform operated in reverse as a shaker. The measurement sensitivity shown in black is limited by the amplitude of vibrations that could be driven by the AVI platform. With the real-time inertial force feedforward on, the linear acceleration sensitivity was measured to be consistent with the measurement sensitivity over a frequency range of 0–30 Hz.

Figure 6

Fractional frequency shift of the output light versus time while rotating the laser and cavity about all three axes, with real-time inertial force feedforward on. Also shown from top to bottom are the linear acceleration, angular acceleration, and angular velocity experienced by the laser and cavity. The data are averaged in 10 ms bins. The root mean square (rms) fractional frequency noise of the laser is $4.0\times {10}^{-13}$ and that of the feedforward corrections is $2.0\times {10}^{-11}$, so inertial force feedforward has reduced the laser frequency noise amplitude by a factor of 50 in this measurement. An estimate of the linear acceleration sensitivity with real-time inertial force feedforward can be obtained by dividing the rms fractional frequency noise by the rms acceleration (0.33, 0.47, and 0.73 g along the optical axis, mounting axis, and orthogonal axis) to obtain $4\times {10}^{-13}$ g${}^{-1}$. The relative contributions of the different types of inertial forces to the feedforward frequency correction power are 96.8$\%$ first-order linear acceleration, 2.2$\%$ second-order linear acceleration, 0.1$\%$ first-order rotational acceleration, and 0.9$\%$ second-order rotational velocity.

Figure 7

Fractional frequency shift of the laser (green line) while the temperature of the vacuum chamber which holds the cavity (blue line) is stepped by 1 K. The fit (black dashed line) is a second-order low-pass filter of the measured temperature with time constants of 1.1 and 6.1 h, scaled by $7.3\times {10}^{-8}$ K${}^{-1}$.

Figure 8

Temperature of the vacuum chamber that holds the cavity (blue line) and the corresponding fractional frequency shift of the laser (green line) while the temperature of the vacuum chamber is measured and thermal length changes of the cavity are compensated by feedforward to the frequency of an acousto-optic modulator (AOM). Temperature control of the vacuum chamber is disabled during this measurement, so the vacuum chamber temperature drifts with the temperature of the laboratory. The root mean square (rms) fractional frequency shift of the laser is $4.5\times {10}^{-11}$ and that of the feedforward corrections is $3.0\times {10}^{-9}$, so temperature feedforward has reduced the laser frequency noise by a factor of 70 in this measurement.

Figure 9

Temperature of the vacuum chamber that holds the cavity (blue line) and the corresponding fractional frequency shift of the laser (green line) while the temperature of the vacuum chamber is stabilized. The root mean square (rms) fractional frequency shift of the laser is $1.0\times {10}^{-11}$.