Geometrical approach for phase-locking time-delay interferometry

Phase-locking configuration was proposed as a baseline optical implementation for the space-borne gravitational wave detectors. By assigning one laser as the reference, termed the master, the remaining five lasers, dubbed slaves, are no longer free-running but phase-locked to form a master-slave arrangement. As a baseline optical implementation for the space-borne gravitational wave detectors, such an approach addresses the relative central frequency drift occurring to individual lasers over time. It is crucial for the effectiveness of the interference measurements, as the resultant beat note frequencies are guaranteed to remain in the phasemeter range. Moreover, it aims to reduce hardware redundancy and effectively decrease the total number of independent laser noise and data streams. The present study focuses on the implication of the time-delay interferometry (TDI) algorithm when implementing the master-slave configuration governed by phase-locking. We examine the modifications to the data streams' structure and the reduction in the number of independent optical links compared to the standard TDI formalism. From a geometric TDI perspective, we analyze the valid TDI combinations by exhausting the solution space up to sixteen links. We show that a geometric TDI combination can always be reiterated regarding the modified data streams associated with the phase-locking scheme, resulting in a size equal to or shorter than the original TDI solution. Notably, for a specific phase-locking scheme, it is demonstrated that an originally sixteen-link TDI combination typically shrinks down to its counterpart with a smaller number of links, effectively simplifying the solutions' forms. The analysis can be extended to all six distinct phase-locking schemes, and the aim is to identify TDI combinations that offer enhanced performance.