## 1 Introduction  
In 1915, [PERSON] proposed the theory of general relativity, which soon led to the theoretical prediction of gravitational waves [1]. To validate this prediction, physicists and astronomers worldwide began exploring methods to detect these ripples in spacetime. After decades of relentless effort, the LIGO achieved the first direct detection of gravitational waves in 2015 [2], capturing signals from a binary black hole merger event. This milestone marked the dawn of a new era in gravitational wave astronomy. Direct detection of gravitational waves relies on laser interferometers [3], which are broadly categorized into ground-based and space-based detection systems. In the realm of ground-based gravitational wave detection, the primary observatories currently include LIGO [2], Virgo [4], GEO600 [5] and KAGRA [6]. Ground-based detectors face limitations due to restricted arm lengths, ground vibrations and gravitational gradient noise, making them less effective at detecting low-frequency gravitational waves below 1 Hz [7]. However, the frequency range of 0.1 mHz to 1 Hz harbors a rich diversity of gravitational wave sources, promising valuable insights into astrophysical and cosmological phenomena [8]. To probe this low-frequency band, space-based gravitational wave detection has become a critical area of research. Currently, the major space-based gravitational wave-detection missions include LISA [9], TianQin [10], Taiji [11], ASTROD-GW [12] and DECIGO [13]. From the perspective of the positioning of gravitational wave observatories, these space-based missions can generally be categorized into heliocentric, geocentric and libration configurations [14; 15]. For heliocentric orbits, the relevant missions include LISA, Taiji and DECIGO. The LISA mission will deploy an equilateral triangular constellation with an arm length of 2.5 million km in a heliocentric orbit, approximately 20\({}^{\circ}\) ahead of Earth, as the gravitational wave observatory [16]. The Taiji mission will use a similar triangular constellation as the gravitational wave observatory, with a slightly larger arm length of 3 million km, positioned in a heliocentric orbit approximately 20\({}^{\circ}\) behind Earth [17]. The DECIGO mission will deploy four equilateral triangular constellations with the arm length of 1000 km in a heliocentric orbit as the gravitational wave observatory [18]. As for geocentric orbits, the TianQin mission is involved. The TianQin mission will place its gravitational wave observatory, a triangular constellation with an arm length of approximately 170,000 km, in a geocentric orbit at an altitude of about 100,000 km [19]. In terms of libration configuration, the ASTROD-GW mission is involved. In the ASTROD-GW mission, an equilateral triangle constellation with a side length of approximately 260 million km will be deployed at the Sun-Earth Lagrange points (L3, L4 and L5) as the gravitational wave observatory [20]. In space-based gravitational wave detection, the relative orbital motion of satellites, driven by orbital dynamics, causes continuous variations in the interferometer arm lengths, resulting in unequal arm configurations [21]. In laser interferometers, the frequency instability of the laser source constitutes a primary noise source, with its amplitude proportional to the arm length mismatch [22]. Taking LISA as an example, the interferometer arm length is \(2.5\times 10^{9}\) m, while relative arm length variations due to orbital drift can reach up to \(10^{5}\) km. To mitigate the laser frequency noise induced by these arm length variations, several stabilization techniques are employed. First, the laser frequency noise is suppressed to a pre-stabilized level of 30 Hz/Hz\({}^{1/2}\) using the Pound-Drever-Hall (PDH) stabilization technique. Subsequently, it is further reduced to the \(10^{-6}\) Hz/Hz\({}^{1/2}\) level through the combined application of the Arm-Locking Technique and Time-Delay Interferometry (TDI) [23].