Robustness Analysis and Controller Design of Arm-locking System in Space-based Gravitational Wave Detectors

TL;DR

This paper addresses the robustness of arm-locking in space-based gravitational-wave detectors by developing a parametric stability framework that blends D-subdivision theory with Semi-Discretization to map stability regions under inter-module perturbations. It then designs a Nyquist-aware, two-stage robust controller (high-pass filtering plus phase-lead compensation) for dual-arm locking to preserve closed-loop stability, even as multiplicative perturbations shift the operating point near stability boundaries. The approach is validated through frequency-domain analyses (Nyquist plots and stability boundaries) and time-domain simulations showing 3–4 orders of magnitude suppression in laser-frequency noise from 0.1 mHz to 1 Hz, with stability maintained under time-varying perturbations. The methodology provides a practical framework for ensuring robust laser frequency stabilization in large-scale space-based GW detectors such as LISA, Taiji, and TianQin.

Abstract

Arm-locking frequency stabilization is a key technique for suppressing laser frequency noise in space-based gravitational-wave detectors. The robustness of the arm-locking control loop is crucial for maintaining laser frequency stability, which directly impacts the accuracy of gravitational-wave measurements. In this work, a parametric stability analysis framework is developed by combining the D-subdivision theory with the Semi-Discretization method to map the stability regions of arm-locking systems in the parameter space and identify their critical stability boundaries. Based on the frequency-domain characteristics, a robust arm-locking controller is designed to enhance loop stability under parameter perturbations. Theoretical analysis and time-domain simulations confirm that the proposed controller maintains closed-loop stability and realize suppression of laser frequency noise against parameter perturbation.

Figures (11)

• Figure 1: Conceptual diagram of the arm-locking technique. The key factor of arm-locking is the laser phase measurement, which is directly influenced by functional modules such as the phase measurement modules and the phase-locked loop (PLL) module. Other functional modules, including the telescope pointing module, can indirectly affect the laser phase measurement.
• Figure 2: Arm-locking technology (a) Conceptual diagram of the arm-locking technique. The arm-locking system involves several functional modules, including Laser frequency control, Optical Bench, and Phasemeter. The red lines represent the optical phase of the laser beams, while the blue lines denote digital control signals. Symbols $\delta$ and $\eta$ indicate potential multiplicative and additive disturbances, respectively. (b) Control block diagram of the single arm-locking system. Considering multiplicative disturbances, the single arm-locking control diagram includes the laser frequency actuator transfer function $G_a$ and the arm-locking controller $G(s)$.
• Figure 3: Stability chart of the Hayes equation. the red line (slope -1 through the origin) marks the ideal phase-locking condition $(i.e.,k2=1)$, corresponding to the single arm-locking system.
• Figure 4: Common arm system: (a) $c_1$-$c_2$ plane D-curves: the gray curves represent the D-curves plotted from the corresponding equations. The regions separated by these curves have different numbers of unstable roots (roots with positive real parts). (b) Stability region obtained using the Semi-Discretization method: the dark gray area indicates the stable region computed by the Semi-Discretization method, and the blue curve represents the corresponding stable boundary. The step size $h=0.05$. (c) Local stability boundary of the common-mode arm system: the red straight line is given by $c_1 + c_2 = 2$, and the dark gray shaded area denotes the stable region.
• Figure 5: Stability chart of the dual arm locking system when $g = 10$. The gray-shaded region (Area1) represents the stable domain, while the Point A(1,1) indicates the location corresponding to the ideal dual arm-locking system. The red line represents the stability boundary obtained from the proposed method.