Tone-assisted time delay interferometry on GRACE Follow-On

We have demonstrated the viability of using the Laser Ranging Interferometer on the Gravity Recovery and Climate Experiment Follow-On (GRACE-FO) space mission to test key aspects of the interspacecraft interferometry proposed for detecting gravitational waves. The Laser Ranging Interferometer on GRACE-FO will be the first demonstration of interspacecraft interferometry. GRACE-FO shares many similarities with proposed space-based gravitational wave detectors based on the Laser Interferometer Space Antenna (LISA) concept. Given these similarities, GRACE-FO provides a unique opportunity to test novel interspacecraft interferometry techniques that a LISA-like mission will use. The LISA Experience from GRACE-FO Optical Payload (LEGOP) is a project developing tests of arm locking and time delay interferometry (TDI), two frequency stabilization techniques, that could be performed on GRACE-FO. In the proposed LEGOP TDI demonstration one GRACE-FO spacecraft will have a free-running laser while the laser on the other spacecraft will be locked to a cavity. It is proposed that two one-way interspacecraft phase measurements will be combined with an appropriate delay in order to produce a round-trip, dual one-way ranging (DOWR) measurement independent of the frequency noise of the free-running laser. This paper describes simulated and experimental tests of a tone-assisted TDI ranging (TDIR) technique that uses a least-squares fitting algorithm and fractional-delay interpolation to find and implement the delays needed to form the DOWR TDI combination. The simulation verifies tone-assisted TDIR works under GRACE-FO conditions. Using simulated GRACE-FO signals the tone-assisted TDIR algorithm estimates the time-varying interspacecraft range with a rms error of $\pm 0.2\mathrm{m}$, suppressing the free-running laser frequency noise by 8 orders of magnitude. The experimental results demonstrate the practicability of the technique, measuring the delay at the 6 ns level in the presence of a significant displacement signal.

Figure 1

LISA will combine the six one-way displacement measurements between spacecraft, shown on the left, to synthesize interferometer configurations with arms of equal length, such as the configuration shown on the right. This combination will cancel laser frequency noise in the gravitational wave measurement. TDI must account for all optical and clock-derived delays in order to correctly “relay” the laser frequency noise of spacecraft 1 (${\mathrm{SC}}_1$) around the constellation.

Figure 2

GRACE-FO LRI performs a heterodyne laser interferometry displacement measurement between each spacecraft’s center of mass (c.m.). Onboard each spacecraft, as shown on the left, the received laser is routed through the triple mirror assembly [23] and is combined with a local laser for measurement upon a photodiode (PD) where the phase of the resulting beat note is measured with a phase meter (PM). The local laser can be either prestabilized to a reference cavity or phase locked with an offset frequency via the phase meter. Phase measurements are made between the two identical spacecraft, allowing the laser onboard one slave spacecraft to be phase locked to the laser on the other master laser, as shown in the lower image.

Figure 3

Simulated GRACE-FO displacement noise spectrum. The signals are (a) free-running laser phase noise; (b) GRACE range; (c) simulated GRACE range; (d) stabilized laser phase noise; (e) residual laser phase noise; and (f) single link shot noise. With one laser free running and the other stabilized the TDI combination, assuming zero error in delay, would suppress the laser noise down to the residual laser phase noise curve. The single link shot noise is quantum noise in both measurements due to the random arrival time of photons on the detector. We have assumed $(3\times {10}^4)\times (1\mathrm{Hz}/f)\mathrm{Hz}/\sqrt{\mathrm{Hz}}$ free-running laser frequency noise and a stabilized laser frequency noise of $30\mathrm{Hz}/\sqrt{\mathrm{Hz}}$. The tone used in the tone-assisted TDIR is shown on the free-running laser phase noise curve. The shot noise is assumed to be $1\mathrm{pm}/\sqrt{\mathrm{Hz}}$. The GRACE range trace shows four days of GRACE microwave displacement data from early 2012, demonstrating the expected signal spectra for the LRI.

Figure 4

Tone-assisted TDIR uses an iterative process to find the best guess of the time-varying interspacecraft delay $\tau$. The delay is initially estimated using ${\varphi }_{\text{basic}}(t)$, a TDI combination proportional to the range ${\varphi }_{\text{path}}(t)$, that is however biased by the unknown initial spacecraft separation. The range ambiguity is compensated for using an offset, $d_0$, a parameter that is updated using a least-squares fitting algorithm to minimize a tone in the TDI combination, ${\varphi }_{\mathrm{DOWR}}(t)$. The loop iterates until the least-squares fitting algorithm converges producing a best guess of the time-varying spacecraft delay.

Figure 5

Root-power spectral density plot of simulated displacement measurements and TDI combinations from a $5\times {10}^5\mathrm{s}$ simulation sampled at 200 mHz. The traces are (a) ${\mathrm{SC}}_1$ displacement measurement $x_1(f)$; (b) ${\mathrm{SC}}_2$ displacement measurement $x_2(f)$; (c) basic DOWR combination $x_{\mathrm{DOWR},\text{basic}}(f)$; (d) residual laser displacement noise $x_{\mathrm{DOWR},\text{laser}\text{residual}}(f)$; and (e) phase-locking residual displacement noise $x_{\mathrm{DOWR},\text{phase}\text{−}\text{locking}\text{residual}}(f)$. The individual spacecraft displacement measurements are dominated by the free-running laser phase noise from ${\mathrm{SC}}_1$ with the tone-assisted TDIR tone visible at 90 mHz. The basic DOWR combination suppresses some laser phase noise. Using a best guess of the range delay the tone is suppressed further. The known range signal is subtracted, to reveal the residual laser displacement noise. The trace shows the residual free-running laser noise in the DOWR TDI combination is down to the target level of $30\mathrm{nm}/\sqrt{\mathrm{Hz}}$. To verify the phase-locking requirement is met, the residual laser noise in a true DOWR is subtracted to give the phase-locking residual displacement noise which itself meets the $1\mathrm{nm}/\sqrt{\mathrm{Hz}}$ phase-locking requirement. The lower panel shows a zoomed in portion of the root-power spectral density to demonstrate the suppression of the TDIR tone.

Figure 6

The best guess of the interspacecraft range used to form the TDI combinations in Fig. 5. The simulated data used to find the best guess was from a $5\times {10}^5\mathrm{s}$ simulation sampled at 200 mHz. The range error, found by subtracting the known range from the fitted range, is plotted in the lower panel. The range error is an order of magnitude below the $\pm 2\mathrm{m}$ of the known range and therefore meets the timing requirements of the proposed GRACE-FO TDI experiment.

Figure 7

The effect of varying data length and modulation depth in the tone-assisted TDIR algorithm. The rms range error can be seen to decrease as the data length increases or as the modulation depth is increased. For reference the 2 m rms ranging requirement needed to meet the timing requirements for a GRACE-FO TDI demonstration is shown. It can be seen that it is possible to meet this requirement for a range of data lengths and modulation depths.

Figure 8

Schematic layout of the TDIR experiment. The details of the injected displacement signal can be found in Fig 9.

Figure 9

Schematic layout showing the injection of the interspacecraft displacement signal. Three optical phase measurements are combined via two phase-locking algorithms to implement the required feedback combinations.

Figure 10

The measurement phase noise root-power spectral density for various TDI data combinations. The traces are (a) ${\mathrm{SC}}_1$ displacement measurement $x_1(f)$; (b) basic DOWR combination $x_{\mathrm{DOWR},\text{basic}}(f)$; (c) DOWR combination $x_{\mathrm{DOWR}}(f)$; and (d) corrected alpha combination ${\alpha }_{\text{corr}}(f)$. While the 0.85 Hz TDIR tone is suppressed in $x_{\mathrm{DOWR}}(f)$, the TDI combination is dominated by path-length noise and the displacement signal s(t). The corrected alpha combination shows however that the residual laser frequency noise in the DOWR combination is at the level of the $1\mathrm{nm}/\sqrt{\mathrm{Hz}}$ GRACE-FO phase-locking requirement. The contribution of laser frequency noise is suppressed by ${10}^9$ between the ${\mathrm{SC}}_1$ displacement measurement and the corrected alpha TDI combination.