Arm locking for space-based laser interferometry gravitational wave observatories

Laser frequency stabilization is a critical part of the interferometry measurement system of space-based gravitational wave observatories such as the Laser Interferometer Space Antenna (LISA). Arm locking as a proposed frequency stabilization technique transfers the stability of the long arm lengths to the laser frequency. The arm locking sensor synthesizes an adequately filtered linear combination of the interspacecraft phase measurements to estimate the laser frequency noise, which can be used to control the laser frequency. At the University of Florida we developed the hardware-based University of Florida LISA Interferometer Simulator to study and verify laser frequency noise reduction and suppression techniques under realistic LISA-like conditions. These conditions include the variable Doppler shifts among the spacecraft, LISA-like signal travel times, optical transponders, realistic laser frequency, and timing noise. We review the different types of arm locking sensors and discuss their expected performance in LISA. The presented results are supported by results obtained during experimental studies of arm locking under relevant LISA-like conditions. We measured the noise suppression as well as initial transients and frequency pulling in the presence of Doppler frequency errors. This work has demonstrated the validity and feasibility of arm locking in LISA.

Figure 1

The heliocentric orbit of the LISA constellation. The constellation trails the Earth by 20°, and the plane of it is inclined with respect to the elliptic by 60°. The arm length between each two spacecraft is generally 5 Gm.

Figure 2

The baseline design of arm locking. The phase meter on the far spacecraft $S{C}_i$ ($i=2,3$) measures the phase noise of the beat signal between the far laser $L_i$ and the incoming laser $L_1$. The measured phase difference is then used to phase lock the frequency of the laser $L_i$, as required by the optical transponders. The frequency noise of the $L_i$ is transmitted back to the master spacecraft $S{C}_1$ and then superimposed onto the instantaneous frequency noise of $L_1$. The phase meters on the master spacecraft measure the two beat signals individually. Arm locking linearly combines these two measurements to control the frequency of the laser $L_1$. In this configuration the arm locking system is integrated with the laser $L_1$ prestabilized in a Pound-Drever-Hall setup with a length-tunable cavity.

Figure 3

The magnitude and phase response of the single arm locking sensor (${\tau }_2=33\mathrm{s}$).

Figure 4

The magnitude and phase response of the common arm locking sensor in the case of $\sim 1\%$ arm length mismatch ($\overline{\tau }=33\mathrm{s},\mathrm{\Delta }\tau =0.16\mathrm{s}$).

Figure 5

The magnitude and phase response of the dual arm locking sensor in the case of $\sim 1\%$ arm length mismatch ($\overline{\tau }=33\mathrm{s},\mathrm{\Delta }\tau =0.16\mathrm{s}$).

Figure 6

The magnitude response of $h_{+}(s)$ and $h_{-}(s)$ with $\overline{\tau }=33\mathrm{s}$ and $\mathrm{\Delta }\tau =0.16\mathrm{s}$ for modified dual arm locking.

Figure 7

The magnitude and phase response of the dual and common arm components that constitute the modified dual arm locking sensor ($\sim 1\%$ arm length mismatch). The dual arm component is high-pass filtered and the common arm component is low-pass filtered. Therefore, the common arm component will dominate at the frequencies below $1/\overline{\tau }$, and the dual arm component will dominate at the frequencies above it. The phase shift of the common arm component in the low frequency region needs to be attenuated to zero to maintain the overall transfer function.

Figure 8

The noise floors in dual arm locking are sensitive to the arm length mismatch. Here we assume a relatively short arm length mismatch of 0.1% and the noise floors are significantly higher than initially recommended for LISA using arm locking. As the noise floor in dual arm locking is inversely proportional to the arm length mismatch, the performance of dual arm locking is insufficient to meet the TDI capability when the arm length mismatch is less than about 60 km.

Figure 9

The noise floors in modified dual arm locking are less dependent on the arm length mismatch due to the low-frequency filtering scheme. The noise floors are effectively suppressed by the high-pass filter at frequencies below $1/\overline{\tau }$. With a more carefully designed high-pass filter, the noise floors will be further reduced and asymptotically approach the noise floors determined in the common arm locking configuration.

Figure 10

The implementation of the phase meter on the FPGA. The phase meter is essentially a “frequency meter” that tracks the frequency fluctuations of the input signal.

Figure 11

The experimental setup of the single arm locking experiment using an additional heterodyne phase-locked laser. In this setup the reference laser $RL$ and the master laser $L_1$ are cavity stabilized via the Pound-Drever-Hall technique. The slave laser $L_2$ is phase locked to $L_1$ with a frequency offset, which is driven by the NCO in the arm locking controller. Therefore, within the PLL bandwidth $L_2$ faithfully reproduces the laser frequency noise of $L_1$, both referenced to the “optical clock” $RL$. Note that the PLL bandwidth is about 20 kHz, which is much larger than the arm locking bandwidth ($\sim 1\mathrm{kHz}$); therefore it will not limit our arm locking performance, and a direct feedback of the arm locking control signal to the laser is not necessary.

Figure 12

The noise spectra of the initial $RL-L_1$ beat signal (blue line) and the stabilized $RL-L_2$ beat signal (red line). Compared with the frequency noise spectrum of the initial $RL-L_1$ beat signal, the frequency noise spectrum of the stabilized $RL-L_2$ beat signal is suppressed by the four-stage integrators by 5 to 6 orders of magnitude from 0.1 mHz to 10 mHz.

Figure 13

The experimental setup of common arm locking using an NCO to track the input noise. The delay times are 2.1 s and 1.9 s.

Figure 14

The noise spectra of the free-running VCO signal (blue lines) and the VCO-NCO beat signal (red lines). The noise spectrum of the residual frequency noise is limited by the 32-bit digitization noise floor, which is given by the integrated quadrature sum of the 32-bit digitization noise from the two independent phase meters.

Figure 15

Experimental setup of dual arm locking (the solid path) and modified dual arm locking (the dashed path) with cavity prestabilized lasers using an auxiliary phase-locked laser. We measured the frequency noise of the arm locking stabilized $RL-L_2$ beat signal and cavity prestabilized $RL-L_1$ beat signal.

Figure 16

The noise spectra of the cavity stabilized beat signal $RL-L_1$ (blue line) and the beat signal $RL-L_2$ further stabilized by dual arm locking. Compared to the $RL-L_1$ beat signal, the frequency noise of $RL-L_2$ is suppressed by 7 to 8 orders of magnitude in the noise limited region, where the dominant noise floor comes from the phase meter ADCs. Also we investigate the frequency noise spectra of $RL-L_2$ in the presence of various differential delay times. These measurements simulate the inversely proportional change of the limiting noise floor in accordance with the change of the arm length mismatch in dual arm locking.

Figure 17

Measured frequency noise of the cavity stabilized laser and the modified dual arm locking stabilized laser in the presence of a 0.15% arm length mismatch. Compared with the previous dual arm locking experiment, the noise suppression performance has been significantly improved. The $1/f^{1/2}$ slope ADC noise floor in Fig. 16 has been surpassed by a $f^{1/2}$ slope, which is determined by the transfer function of the modified dual arm locking sensor as well as the arm length mismatch. The noise floor is given by Eq. (32).

Figure 18

Experimental setup of modified dual arm locking with an optical transponder. The delay line to simulate the LISA arm 1-2 is divided equally into two, representing the outgoing and return travel time individually. Another beat signal $RL-L_3$ is phase locked to the $RL-L_2$ delayed by the outgoing travel time with an offset frequency of 2 MHz and then delayed by the return travel time to have the heterodyne interference with the prompt beam. In this setup the voltage controlled oscillator demodulating the heterodyne frequency of the PLL has an equivalent function as the clock on far spacecraft, where the clock noise enters the phase measurement of the optical phase-locked loop in a similar way.

Figure 19

Noise spectra of the laser beat frequency stabilized by modified dual arm locking in the presence of transponder noise. In this figure the stabilized frequency noise is represented by the red curve, which is limited by the transponder noise floor. The yellow curve represents the transponder noise floor given by the combined noise multiplied with the differential arm gain $\overline{\tau }/(2\pi \mathrm{\Delta }\tau ·H)$. As the PLL noise increases or decreases, the transponder noise floor in the stabilized laser frequency also tracks this change, which is shown by the green (a higher PLL gain) and purple (a lower PLL gain) curves.

Figure 20

Observed frequency pulling of dual/modified dual arm locking with Doppler frequency errors. For a dual arm locking and a modified dual arm locking configuration, the observed frequency pulling rate is $64.2\mathrm{Hz}/\mathrm{s}$ and $0.245\mathrm{Hz}/\mathrm{s}$, respectively. These measurement results are consistent with the theoretical predictions and demonstrate that the modified dual arm locking sensor is capable of alleviating the Doppler issue substantially.