Compensation of front-end and modulation delays in phase and ranging measurements for time-delay interferometry

In the context of the Laser Interferometer Space Antenna (LISA), the laser subsystems exhibit frequency fluctuations that introduce significant levels of noise into the measurements, surpassing the gravitational wave signal by several orders of magnitude. Mitigation is achieved by means of time-shifting individual measurements in a data processing step known as time-delay interferometry (TDI). The suppression performance of TDI relies on accurate knowledge and consideration of the delays experienced by the interfering lasers. While considerable efforts have been dedicated to the accurate determination of inter-spacecraft ranging delays, the sources for delays onboard the spacecraft have been either neglected during TDI processing or assumed to be known. Contrary to these assumptions, analog delays of the phasemeter front end and the laser modulation are not only large but also prone to change with temperature and heterodyne frequency. This motivates our proposal for a novel method enabling a calibration of these delays on-ground and in-space, based on minimal functional additions to the receiver architecture. Specifically, we establish a set of calibration measurements and elucidate how these measurements are utilized in data processing, leading to the mitigation of the delays in the TDI Michelson variables. Following a performance analysis of the calibration measurements, the proposed calibration scheme is assessed through numerical simulations. We find that in the absence of the calibration scheme, the assumed drifts of the analog delays increase residual laser noise at high frequencies of the LISA measurement band. A single, on-ground calibration of the analog delays leads to an improvement by roughly one order of magnitude, while recalibration in space may improve performance by yet another order of magnitude. Towards lower frequencies, ranging error is always found to be the limiting factor for which countermeasures are discussed.

Figure 1

The LISA constellation consists of three $\mathrm{S}/\mathrm{C}\mathrm{s}$ each hosting two OBs, highlighted in light blue. The light travel times among these OBs are indicated via the parameter ${\tau }_{\mathrm{R}}$.

Figure 2

A simplified sketch indicating the interferometers present at $\mathrm{S}/\mathrm{C}$ 1 is depicted. The $\mathrm{S}/\mathrm{C}$ hosts two OBs. Each OB is equipped with three interferometers: the long-arm interferometer (L), the reference interferometer (R), and the test-mass interferometer (T). The photodiodes of these interferometers have been labeled with the respective letter. The OBs are connected via a fiber backlink highlighted in purple. For the sake of clarity, the beams of each laser have been distinctly color-coded. At the EOM lasers are modulated with a chip sequence supplied by the code generation unit, indicated via “Code Gen.”.

Figure 3

Schematic indicating the delays affecting the carrier phase and code read-out at the long-arm interferometer of OB 12. Individual phase and code delays are separated by thick black bars and labeled at the end of the individual delays with symbols $\delta$ and $\tau$, respectively.

Figure 4

Schematic indicating the routing of the (nested) delay operators applied to the individual $\eta$ terms composing the TDI $X$ variable. The letters refer to the subscripts of the delay operators as given in Eq. (17) and indicate the start of the respective (nested) delay operator. The $\eta$ terms entering with a positive and negative sign have been separated in panels (a) and (b), respectively.

Figure 5

Schematic indicating via red frames the changes to the current baseline, necessary for the implementation of the proposed calibration scheme. The calibrations are controlled by an operation logic, which specifies the prn code used in the reference code generator (“REF Code Gen.”) and the error channel (“Q”) that is routed to the DLL via a multiplexer. The processing electronic, TIA and ADC, is summarized as signal conditioning (“SC”).

Figure 6

Residual laser noise in the TDI $X$ variable for three different scenarios. Panels (a) and (b) illustrate the results for the baseline ranging scheme and an improved ranging scheme, respectively.