Transient analysis of the arm locking controller

Arm locking is one of the key technologies to suppress the laser phase noise in spaced-based gravitational waves observatories. Since arm locking was proposed, the phase margin criterion was always used as the fundamental design strategy for the controller development. In this paper, we find that this empirical method from engineering actually cannot guarantee the arm locking stability. Therefore, most of the advanced arm locking controllers reported so far may have stability problems. After comprehensive analysis of the single arm locking's transient responses, strict analytical stability criterions are summarized for the first time. These criterions are then generalized to dual arm locking, modified-dual arm locking, and common arm locking, and special considerations for the design of arm locking controllers in different architectures are also discussed. It is found that proportional integral controllers can easily meet our stability criterions in most of the arm locking systems. Using a simple high gain proportional integral controller, it is possible to suppress the laser phase noise by 5 orders of magnitude within the science band. Our stability criterions can also be used in other feedback systems, where several modules with different delays are connected in parallel.