## 2 Modulation Phase Noise Analysis  
Currently, the inter-satellite absolute position measurement and data communication systems of programs like LISA and Taiji are integrated into the overall laser interferometry system, as shown in Figure 1[14]. This system exhibits symmetry and is capable of performing functions such as interferometric measurements, inter-satellite communication, and clock noise transfer. The main principle is as follows: independent lasers (referred to as Laser 1 and Laser 2) are installed on the designated master and slave satellites. The master satellite sends a laser to the slave satellite, and after it is reflected by test mass 2 on the slave satellite, it interferes with Laser 2. The resulting signal is then transmitted through a Trans Impedance Amplifier to a Quadrant Photoelectric Detector (QPD), which converts it into an electrical signal. The phase difference between Laser 2 and the received Laser 1 is measured by a phase meter. Weak-light phase locking is used for differential frequency locking, allowing Laser 2 to carry the phase information of the received Laser 1. The phase information is then encoded into the PRN sequence via an Electro-Optic Modulator and modulated onto the laser phase. The locked Laser 2 is then transmitted back to the master satellite. After passing through the same process at the master satellite, the laser interferometric signal containing the inter-satellite test mass distance variation information is obtained. Finally, the phase meter on the master satellite measures the phase change, which is used to calculate the distance variation between the test masses caused by the gravitational wave. Figure 1: Space Gravitational Wave Inter-Satellite Ranging and Communication System. In the above process, the laser heterodyne interference signal is finally input into a single quadrant of the QPD on the master satellite. Under ideal conditions, it can be represented as [15]:  
\[ \begin{array}{c}V=G_{t}J_{0}^{2}(m_{sb})\cos[2\pi(f_{main}+f_{D})t+\varphi_{ main}+m_{prn}\sum\limits_{-\infty}^{\infty}c_{n}p(t-nT_{c})]\\ +G_{t}J_{1}^{2}(m_{sb})\cos[2\pi(f_{up}+f_{D})t+\varphi_{up}]\\ +G_{t}J_{1}^{2}(m_{sb})\cos[2\pi(f_{down}+f_{D})t+\varphi_{down}]\end{array}  \tag{1}\]  
\(G_{t}=\frac{\eta CT_{IA}\sqrt{\gamma_{P}S_{P}}}{N}\) represents the total gain of the Trans-Impedance Amplifier (TIA), where \(G_{TIA}\) is the TIA gain, \(P_{S}\) and \(P_{L}\) are the received signal power, and the local laser power, \(\gamma\) is the heterodyne detection efficiency, and \(N\) is the number of photodiodes (QPD: \(N\) = 4). \(J_{n}(m)\) denotes the nth-order Bessel function and \(m_{sb}\) is the frequency modulation index, approximately 0.45 rad. The control index under this modulation function takes the \(J_{0}\) and \(J_{1}\) terms. \(f_{main}\), \(f_{up}\) and \(f_{down}\) represent the main frequency and the upper and lower frequencies, respectively. \(f_{D}\) refers to the Doppler frequency shift, and \(\varphi_{main}\), \(\varphi_{up}\) and \(\varphi_{down}\) correspond to the phase shifts of the main, upper, and lower frequencies, respectively. The PRN code modulation index \(m_{prn}\) is approximately 0.1 rad, and \(c_{n}\) represents the PRN code in binary form, its pulse shape as \(p(t)\), and \(T_{c}\) is the period. From Equation (1), it can be seen that under ideal conditions, the interference signal is only affected by the Doppler shift. However, in actual PRN modulation, BPSK phase modulation generates additive noise in the main and side lobes, which is superimposed on the phase of the main wave beat frequency signal, thus affecting the phase-tracking and locking accuracy. The main lobe noise is caused by the DC component generated by an unbalanced PRN code (with unequal numbers of 1 s and \(-\)1 s) during PRN modulation, while the sidelobe noise arises from the incomplete energy concentration in the main lobe and the clock sideband modulation during PRN modulation.