## 3 Optimization of Distribution Scheme
### Optimization of Frequency Domain Selection Extension  
The three spacecraft in Taiji totally have six sets of interferometric links [17]. After determining the fixed frequency for the main laser, five offset frequencies are added to correct for the Doppler shift and to lock the other five lasers sequentially. The setting of its locking order is the phase-locking scheme. Different locking schemes have different effects on the beat note. The resulting distribution scheme will be also distinct. The common phase-locking scheme in the current Taiji is shown as scheme 1 in Figure 2[13]. Laser A onboard spacecraft S1 is chosen to be the main laser and the frequency is constant at f0. Laser B on board the same spacecraft and laser F onboard spacecraft S3 is locked to A. Laser E is locked to laser F. Laser C onboard spacecraft S2 locks through laser B, and laser D on the same spacecraft S2 is locked to C. That completes the locking of the six lasers. The locked lines on both sides of the phase-locked scheme 1 are completely symmetric and do not affect each other and are, therefore, the preferred scheme for each gravitational wave detection program. Scheme 1 is suitable for most reasonable frequency domains. However, it may not be satisfied when the domain of the system is set more strictly. In the [7 MHz, 21 MHz] region, a suitable offset frequency will not be found after 1581 days to keep the beat frequencies within the setting domain. This is the time that the phase-locking scheme needs to be modified. Figure 2 also shows the other different locking sequences for gravitational wave detection. Three options all use laser A as the master laser, with adaptive offset frequencies added at various points to complete the locking [9]. We can calculate the beat frequency received by each interference system according to the order of the locking scheme. All locking processes in the figure have their own adaptive offset frequencies from \(\Delta f_{11}\) to \(\Delta f_{33}\). The complete setting of these five parameter values over the five years of observatory operation is the offset frequency distribution scheme. \(f_{d1}\) to \(f_{d3}\) is the Doppler shift. The beat frequencies received by each interferometric platform  
Figure 2: Phase-locking Schemes. In addition to the above schemes, the different choices of master laser and different orbital positions of the spacecraft can also lead to the generation of different distribution schemes, named the orbital alignment pattern of the detection system. A complete orbital alignment pattern consists of the following six types from A1 to E2. A indicates that laser A on spacecraft S1 is the main laser with a constant frequency. The other lasers are thus locked to the frequency of laser A. C is dominated by laser C on spacecraft S2, and E is dominated by laser E on spacecraft S3. Figure 3 shows the complete orbital alignment pattern when laser A, C and E are the main lasers. The offset frequency distribution schemes generated by different phase-locking schemes in the [5 MHz, 25 MHz] frequency domain and alignment pattern A1 setting are shown in Figure 4. The distribution schemes given in each phase-locking scheme in the figure are all different from each other. Scheme 1 requires only three frequency switches. The other three schemes switch at roughly the same time and number of times. They all require frequency switching five times, and the switching nodes are only two to three days apart. All four schemes can generate the complete frequency scheme under the current setting, and all of them can be considered for use. However, scheme 1 has the highest priority. Figure 3: Complete Alignment Pattern. Figure 5 shows the distribution scheme generated by the six alignment patterns in the [5 MHz, 25 MHz] frequency domain and the phase-locking scheme 1 setting. The five lines indicate the five offset frequencies. The complete generation of the scheme demonstrates the feasibility of the six alignment patterns and the underlying genetic algorithm.