## 2 Principles of Inter-Satellite Laser Communication for Space-Based Gravitational Wave Detection  
In space-based gravitational wave-detection missions, the detection satellites form a three-satellite constellation configured in an equilateral triangular formation. Each arm of the triangle is equipped with bidirectional laser links, enabling the exchange of laser beams between satellites, as illustrated in Figure 1.  
In the satellite constellation, individual satellites are represented by circular shapes with dashed lines. The laser emitters on the satellites are represented by rectangular boxes with solid lines. The laser emitted by satellite 1 (S1) is indicated by a blue arrow. The laser emitted by satellite 2 (S2) is indicated by a red arrow. The laser emitted by satellite 3 (S3) is indicated by a yellow arrow. In this equilateral triangular constellation, each satellite houses two square test masses that are completely isolated from external disturbances and suspended internally. Gravitational waves induce variations in the optical path length between two points in space. By employing a responder-based inter-satellite laser interferometry technique, interferometric measurements are conducted on the test masses located within different satellites. This process enables the detection of phase variations in the laser signals between the test masses, thereby allowing precise measurement of optical path changes. To illustrate,  
Figure 1: Schematic diagram of the three-satellite laser links for space–based gravitational wave detection. Figure 2 demonstrates the principle of gravitational wave detection using two satellites as an example. Each satellite in the constellation is equipped with an independent frequency-stabilized laser. Taking S1 as the transmitting terminal, the laser emitted by the frequency-stabilized laser onboard S1 is expanded through a telescope and transmitted over hundreds of thousands of kilometers to S2. At the receiving terminal, the incoming weak laser is first heterodyned with the local laser emitted by the frequency-stabilized laser onboard S2 on its optical platform. The resulting interference signal is then detected by a quadrant photodetector (QPD), which converts the optical signal into an electrical signal. This electrical signal is processed using a digital phase-locked loop (DPLL) to extract the phase information of the interferometric signal. Subsequently, a weak-light phase-locking technique is employed to control the frequency-stabilized laser on S2, enabling it to emit a laser containing the phase variation information of the laser from S1. This laser is then transmitted back to the transmitting satellite, S1. At S1, the returning laser interferes with the local oscillator laser emitted by its onboard frequency-stabilized laser. The DPLL at S1 processes the resulting signal to extract the final phase information. This phase information contains the relative displacement variations of the test masses induced by gravitational waves, thereby enabling the detection of space-based gravitational waves. In addition to its primary function of relative displacement measurement, the gravitational wave-detection constellation incorporates DS/SS techniques within its existing architecture. Using an electro-optic modulator (EOM), PRN code and data code are phase-modulated onto the laser via low-depth BPSK modulation. This approach facilitates absolute distance measurement and data transmission without compromising the picometer-level precision required for relative displacement measurements. The DPLL and DLL (Delay locked loop), as key modules for implementing communication demodulation and data recovery, are designed and operate as follows:  
(1) DPLL  
For the DPLL, it consists of a phase detector, a PI controller and a numerically controlled oscillator (NCO). The structural schematic is presented in Figure 3.  
Figure 2: Schematic diagram of the principle of space–based gravitational wave detection.