Enhanced Detection Precision of the Taiji Program by Frequency Setting Strategy Based on a Hierarchical Optimization Algorithm

For space-based gravitational wave detection, a laser interferometric measurement system composed of a three-spacecraft formation offers the most rewarding bandwidth of astrophysical sources. There are no oscillators available that are stable enough so that each spacecraft could use its own reference frequency. The conversion between reference frequencies and their distribution between all spacecrafts for the synchronization of the different metrology systems is the job of the inter-spacecraft frequency setting strategy, which is important for continuously acquiring scientific data and suppressing measurement noise. We propose a hierarchical optimization algorithm to solve the frequency setting strategy. The optimization objectives are minimum total readout displacement noise and maximum beat-note frequency feasible range. Multiple feasible parameter combinations were obtained for the Taiji program. These optimized parameters include lower and upper bounds of the beat note, sampling frequency, pilot tone signal frequency, ultrastable clock frequencies, and modulation depth. Among the 20 Pareto optimal solutions, the minimum total readout displacement noise was 4.12 pm/Hz, and the maximum feasible beat-note frequency range was 23 MHz. By adjusting the upper bound of beat-note frequency and laser power transmitted by the telescope, we explored the effects of these parameters on the minimum total readout displacement noise and optimal local laser power in greater depth. Our results may serve as a reference for the optimal design of laser interferometry system instrument parameters and may ultimately improve the detection performance and continuous detection time of the Taiji program.


Introduction
In 2016, the Laser Interferometer Gravitational-Wave Observatory (LIGO) [1] successfully detected gravitational wave phenomena that proved the existence of gravitational waves.Space-based gravitational wave detection missions have been proposed and performed in recent years because of the surface vibrations of the Earth, the noise of the gravitational gradient, and limitations of the ground test baseline length [2].These missions include the Laser Interferometer Space Antenna (LISA) mission [3,4] and the Taiji [5][6][7] and Tianqin programs [8].Space-based gravitational-wave detection usually uses the laser interferometry principle [9] and employs a three-spacecraft constellation placed in an equilateral triangle [3][4][5][6]8]; many functions are achieved by laser beams exchanged between the spacecrafts (S/Cs), such as scientific interferometry, absolute inter-spacecraft distance measurements, digital data communication, and clock-noise transfer [10][11][12].
In a three-spacecraft constellation, the laser on one S/C interferes with the laser received from another S/C.The beat-note signal of two laser is digitized by an analog-todigital converter (ADC) and analyzed by a high-precision phasemeter (PM).Both the ADC and the PM are triggered by an ultrastable oscillator (USO) that provides a time reference.The frequency instability of the trigger signal introduces additional ranging noise, affecting the arm length measurements.Additionally, the inherent jitter of the ADC distorts the sampling process.A pilot tone (i.e., a stable sinusoidal reference signal derived from the USO) has been inserted to correct the clock and ADC noise [6,13,14].Considering the differential jitter and relative drift of three onboard USOs on different S/Cs, a clock-tone transfer chain has been proposed via an electro-optic modulator (EOM) to modulate the USO signal to sidebands on the outgoing beam to remove clock noise and correct the relative clock drift by using postprocessing [10][11][12].To realize the functions listed above, the frequencies of the USO, sideband, pilot tone, ADC sampling, and beat note must be comprehensively considered and optimized.Moreover, total laser power is a limited resource used to ensure the effectiveness of scientific signal interference, and the readout displacement noise of the signal is affected by the power ratio of the sideband and carrier.The power ratio needs to be reasonably designed to minimize readout displacement noise.
Different frequency-setting schemes have been established for the LISA mission to address different concerns.For example, Kullmann [15] conducted a detailed, in-depth device experiment for the setting of the ADC sampling and pilot-tone frequencies involved in the LISA mission.They concluded that when the pilot-tone frequency was 72 MHz and the ADC sampling frequency was 50 MHz, sampling the beat-note frequency from 2 to 20 MHz could meet the noise requirements of this component of LISA.Barke [13] designed an inter-satellite frequency distribution scheme for the LISA program for beatnote frequencies in the range of [7 MHz,23 MHz]; the ADC sampling and pilot-tone frequencies were 80 and 75 MHz, respectively.Zhang [16] analyzed the Taiji mission orbit and a possible phase-locking scheme and developed an offset frequency-setting scheme for beat-note frequencies in the range of [5 MHz,25 MHz].Although all these schemes have considered the problem of frequency setting in space-based gravitationalwave detection missions from different perspectives, they have not provided complete constraints or optimal setting schemes that include the beat-note frequency range as well as the frequencies of USO, ADC sampling, pilot tone, and sideband, and none of them consider the coupling relationship between frequency parameters.The unified consideration of these factors avoids the one-sidedness of individual parameter settings and reduces unnecessary redundancy between parameters, further improving the detection capability.
In this study, a frequency distribution scheme using a hierarchical optimization algorithm is introduced for the Taiji program by considering the frequency bands or frequencies for each function or device to ensure an accurate readout of the inter-spacecraft heterodyne signal, synchronize the onboard clock with those on other S/Cs, and generate an ADC sampling frequency and pilot tone signal.
The remainder of this paper is organized as follows.Section 2 introduces frequency factors and constraints, such as sideband frequency, ADC sampling, and pilot-tone frequencies.The optimization algorithm is introduced in Section 3. Section 4 describes the solution for the frequency setting scheme for each function or device.The effects of key parameters and total readout displacement noise are also analyzed.Finally, Section 5 concludes this study.

Frequency Factors and Constraints
In space-borne, gravitational-wave detection missions, laser links need to accomplish ultralong-range laser interference and various auxiliary functions, such as clock-noise transfer, pseudorandom noise (PRN) ranging, and information transfer [10,17].The Doppler shift affects the beat-note signal generated by heterodyne interference; therefore, offset Sensors 2023, 23, 9431 3 of 14 frequencies need to be added to the phase-locking process to make them fall into a reasonable range.The signal is sampled and analyzed by the ADC and PM and triggered by the USO onboard.The USO signal is first multiplied to the GHz level and imprinted on the sidebands of the outgoing laser beam by EOM to perform inter-spacecraft clock transfer.Setting the sideband frequency and power ratio will affect the size of the readout displacement noise.The USO is then multiplied to the MHz level to provide the internal clock signal for the ADC and PM operations and construct the pilot tone to eliminate ADC sampling jitter and clock noise.In this process, some parameters have complex mutual constraints and need to be optimized systematically.

Readout Displacement Noise and Sideband Frequency Constraints
The Taiji program requires that the total readout displacement noise be as small as possible, which includes carrier and first-order sideband readout displacement noise.Moreover, based on the experience of the LISA mission [13,14,18], the first-order sideband readout displacement noise cannot be higher than 1/10 of the total readout displacement noise.The power ratio between the first-order sidebands and carrier could be expressed by the ratio of the squares of first-and zero-orders of the Bessel functions of the first kind: where J 0 (m) and J 1 (m) denote the zero-and first-orders of the Bessel functions of the first kind, respectively, and m is the modulation depth.Research on the LISA mission [14,18] indicates that this power ratio should be in the range of 5% to 10%, and that the corresponding modulation depth m is in the range of [0.44, 0.61], as shown in Figure 1.
ultralong-range laser interference and various auxiliary functions, such as clock-noise transfer, pseudorandom noise (PRN) ranging, and information transfer [10,17].The Doppler shift affects the beat-note signal generated by heterodyne interference; therefore, offset frequencies need to be added to the phase-locking process to make them fall into a reasonable range.The signal is sampled and analyzed by the ADC and PM and triggered by the USO onboard.The USO signal is first multiplied to the GHz level and imprinted on the sidebands of the outgoing laser beam by EOM to perform inter-spacecraft clock transfer.Setting the sideband frequency and power ratio will affect the size of the readout displacement noise.The USO is then multiplied to the MHz level to provide the internal clock signal for the ADC and PM operations and construct the pilot tone to eliminate ADC sampling jitter and clock noise.In this process, some parameters have complex mutual constraints and need to be optimized systematically.

Readout Displacement Noise and Sideband Frequency Constraints
The Taiji program requires that the total readout displacement noise be as small as possible, which includes carrier and first-order sideband readout displacement noise.Moreover, based on the experience of the LISA mission [13,14,18], the first-order sideband readout displacement noise cannot be higher than 1/10 of the total readout displacement noise.The power ratio between the first-order sidebands and carrier could be expressed by the ratio of the squares of first-and zero-orders of the Bessel functions of the first kind: where J0(m) and J1(m) denote the zero-and first-orders of the Bessel functions of the first kind, respectively, and m is the modulation depth.Research on the LISA mission [14,18] indicates that this power ratio should be in the range of 5% to 10%, and that the corresponding modulation depth m is in the range of [0.44, 0.61], as shown in Figure 1.Therefore, the optimization objective and constraints are where δx sideband and δx total indicate the sideband and total readout displacement noise, respectively.The expressions of the readout displacement noise of the carrier and first-order sideband and the total readout displacement noise are [13] respectively, where λ is the laser wavelength, which is 1064 nm in the Taiji program; f het is the beat-note frequency; and f USO is the USO frequency.Given that two identical sidebands are generated on the left-and right-hand sides of the carrier after modulation by EOM, a factor of 1/ √ 2 must be added to the calculation of the sideband frequency noise.δφ total is the total readout phase noise and contains three components, which is shown as follows [13,19]: where δφ SN , δφ RIN , and δφ EN are the shot, RIN, and electronic noise, respectively; and P local and P receive represent the local laser power and laser power received by the telescope, respectively.P receive [12] is calculated as For the Taiji program, the explanation and values of other parameters in Equations ( 4) and ( 5) are listed in Table 1.In Equation ( 3), δx sideband is positively correlated with f het and negatively correlated with f USO .Because the beat-note frequency signal f het varies with time, δx sideband is usually calculated by replacing f het with f upper .From Equations ( 3) and ( 4), the total Sensors 2023, 23, 9431 5 of 14 readout displacement noise may be minimized by reasonably setting f upper , f USO , m, P receive , and P local .

Constraints of the ADC Sampling and Pilot-Tone Frequencies with the Beat-Note Frequency
According to Nyquist's theorem, the sampling frequency of the ADC needs to be greater than at least two times the beat-note frequency.That is, The USO is used to control the ADC sampling frequency and construct the pilot-tone signal; the divider or synthesizer can realize this process.According to Heinzel [10], the noise introduced by the dividers is much smaller than that introduced by the synthesizers, and, therefore, the frequency division approach is typically used.In the actual application process, integer frequency division is usually chosen.
The aliasing signal generated by the ADC sampling of the pilot-tone signal interferes with the beat-note frequency measurement.For example, when the sampling frequency of the ADC is 82 MHz and the frequency of the pilot-tone signal is 80 MHz, the frequency of the aliased signal generated by the ADC in the under-sampled pilot-tone signal will be 2 MHz.Therefore, the frequency of the aliased signal must not overlap the beat-note frequency range.This constraint can be expressed by the following equation: Therefore, setting the pilot-tone signal frequency f PT and ADC sampling frequency f ADC imposes the following constraints: where the value of 5 GHz is the artificial upper bound set for f USO to constrain it to finite values.Based on Kullmann's [15] study, the upper bound of f PT is set to 98 MHz to avoid poor performance of the pilot tone correction.

Hierarchical Optimization Algorithm
In this section, the optimization model is introduced in Section 3.1, and then the hierarchical optimization algorithm for this optimization model is introduced in Section 3.2.

Optimization Model
According to the analysis in Section 2, the frequency of the mission operation involves four terms: the beat-note f het , the USO f USO , the pilot-tone signal f PT , and the ADC sampling f ADC frequencies.The variable f het is affected by the Doppler shift and changes dynamically over time, while the remaining three terms remain unchanged.The variation range of the inter-satellite beat-note frequency f het is determined by the lower and upper bounds of the beat-note frequency, namely f lower, and f upper .Therefore, the optimization goal is to make the sideband readout displacement noise meet the mission requirement, minimize the total readout displacement noise, and maximize the feasible range of beat-note frequency by reasonably allocating the values of each frequency or frequency band, and modulation depth m, respectively.The optimization model is as follows: where Γ 1 and Γ 2 represent the two objective functions.

Optimization Process
Owing to the multiple objectives and parameters involved in the optimization solution, a hierarchical optimization approach is used for this optimization model based on computational efficiency considerations when selecting f USO , f PT , f ADC , f upper , f lower , and m.The optimization process is as follows.
Step 1: A multi-objective optimization algorithm is used to set the lower and upper bounds of the beat-note frequency (f lower , f upper ), USO frequency f USO , and sideband modulation depth m.The multi-objective optimization model for this step is expressed by Equation (10).
In Equation (10), Γ x denotes the x-th optimization objective.In Γ 1 , where f PT , f ADC and P local denote that the variables are taken as constant.Γ 2 and Γ 3 in Equation (10) are the two objective functions derived from Γ 1 in Equation ( 9) according to Equations ( 3) and ( 8), respectively.Γ 2 aims to expand the selection spaces of f ADC and f PT , and Γ 3 aims to reduce the sideband readout displacement noise, which is proportional to f upper .The objective function Γ 4 aims to maximize the feasible range of the beat-note frequency, which is the opposite of Γ 2 and Γ 3 .In actual mission operations, f upper , f lower , and f USO are commonly set as integers.To reduce the complexity of the optimization problem while determining the solution, f upper , f lower , and f USO are not constrained as integers in this step.
In the current Taiji program, the upper bound of the beat-note frequency is 25 MHz.Using the parameter values listed in Table 1, P receive = 2.154 × 10 −9 W. Therefore, the values of all parameters in Equation ( 4), except P local , are obtained.Because P local appears in both the numerator and denominator, the optimal value of P local , which is 2.06 × 10 −3 W in the current parameter settings, can be derived by simply minimizing.When calculating δx total in this step, the values of P local = 2.06 × 10 −3 W and the parameters in Table 1 are used by default.
Step 2: The values of f lower , f upper , and f USO obtained in Step 1 are adjusted to be integers: where f new lower , f new upper , and f new USO represent the adjusted values of f lower , f upper , and f USO , respectively.
Step 3: Exhaustive enumeration is used to search for possible [ f USO , f PT , f ADC ] combinations.The values of f PT and f ADC need to satisfy the following constraints: Suppose there are n possible combinations stored in the following matrix: The combination such that f j USO , j = 1, 2, . . ., n is closest to The minimum value of the total readout displacement noise after updating [ f new upper , f opt USO ] is obtained by adjusting the parameter m in the range [0.44, 0.61].
Step 5: Backtracking mechanism.Since the default P local value is obtained by assuming f upper = 25 MHz, it may change after completing Steps 1-4.Therefore, if P local changed, then update P local and return to Step 1 until no new P local appears.The flowchart of the algorithm as shown in Figure 2.
The minimum value of the total readout displacement noise after updating The flowchart of the algorithm as shown in Figure 2. Step 5

Optimization Results
The mathematical model presented in Section 3 was solved using an AMD Ryzen 9 3900X 12-core processor.The time consumption of each step is listed in Table 2. To reduce the complexity associated with the solution of multi-objective problems and improve the convergence efficiency of the solution set, we used one of the most popular multi-objective optimization algorithms: the nondominated sorting genetic algorithm II (NSGA-II) [24].As a characteristic of multi-objective optimization algorithms, an optimal solution set is usually obtained instead of a single optimal solution to balance the degree of optimization of each objective.
The Pareto-optimal solution plane obtained in Step 1 of Section 3.2 is shown in Figure 3.

Optimization Results
The mathematical model presented in Section 3 was solved using an AMD Ryzen 9 3900X 12-core processor.The time consumption of each step is listed in Table 2. To reduce the complexity associated with the solution of multi-objective problems and improve the convergence efficiency of the solution set, we used one of the most popular multi-objective optimization algorithms: the nondominated sorting genetic algorithm II (NSGA-II) [24].As a characteristic of multi-objective optimization algorithms, an optimal solution set is usually obtained instead of a single optimal solution to balance the degree of optimization of each objective.
The Pareto-optimal solution plane obtained in Step 1 of Section 3.2 is shown in Figure 3.
During the operation of a mission, a smaller total readout displacement noise δx total indicates a better detection of signals, and a larger feasible range of the beat-note frequency, which exists between f upper and f lower , indicates a better offset frequency setting.In Figure 3a, the warmer the color of the scatter plot, the higher the value of δx total .Hence, the smaller the value of the beat-note frequency range, the smaller the value of the modulation depth; a larger value of f USO corresponds to a smaller value of δx total .In Figure 3b, a Sensors 2023, 23, 9431 9 of 14 warmer scatter color indicates a larger beat-note frequency interval; a larger beat-note frequency range corresponds to a larger δx total value.Based on Figure 3, we can conclude that the value of the beat-note frequency range is inversely related to the magnitude of δx total .The two targets need to be reasonably balanced via optimization to increase the maximum feasible range of the beat-note frequency and decrease δx total .This planning problem was solved based on the optimization algorithm introduced in Section 3.2.The results of the 20 Pareto-optimal (feasible) solutions obtained from the final solution are listed in Table 3.The units of f lower , f upper , f ADC , f PT , and f USO are MHz, and the total readout displacement noise δx total has units of pm/ √ Hz.It is worth noting that these are the optimal 20 solutions by balancing the two optimization objectives.The results in Table 3 show that, after utilizing the proposed algorithm, the maximum beat-note frequency feasible range is 23 MHz and yields a total readout displacement noise of 4.21 pm/ √ Hz, which corresponds to the parameter combination (f lower , f upper , f ADC , f PT , f USO , m) = (3 MHz, 25 MHz, 92 MHz, 90 MHz, 4140 MHz, 0.44).Additionally, the smallest total readout displacement noise was 4.12 pm/ √ Hz, which corresponds to (f lower , According to Taiji mission budget, the position noise is 8 pm/ √ Hz [6], which includes laser frequency noise, readout displacement noise, laser pointing noise, tilt-tolength noise, and so on.Among them, the frequency stability is 30 Hz/ √ Hz, the limit of is laser-pointing noise and tile-to-length noise is 1 pm/ √ Hz [25], and the readout displacement noise is about 7.5 pm/ √ Hz [26].After parameter optimization, the total readout displacement noise is reduced to 4.12 pm/ √ Hz.The sensitivity curve of Taiji program detection limit and after optimization with other noise budgets the same, is shown

Experimental Adjustment of Ptel and fupper
According to Equation ( 5 3, while in the second experiment, we set Ptel = 2 W. In Figure 5, as Ptel increases, Preceive increases linearly, and total x δ decreases approximately linearly.In Equation (4) and Step 5 of the optimization algorithm proposed in this study, the variation of Preceive may cause a change in the optimal Plocal value.However, in practice, the variation of Preceive is so small that it barely affects the optimal value of Plocal.Taiji detection limit After optimization

Experimental Adjustment of P tel and f upper
According to Equation ( 5), the value of P receive is positively proportional to P tel .The optimal value of P local is directly influenced by f upper .To describe the effect of the values of P tel and f upper on P receive and P local in a more intuitive manner, we conducted the following experiments: (1) vary the value of P tel in the range [2 W, 3 W] with an interval of 0.1 W and observe the variations of P receive and total readout displacement noise; and (2) vary the value of f upper in the range [20 MHz, 30 MHz] with an interval of 1 MHz and observe the variations of the optimal P local value and minimum total readout displacement noise δx total .The results are shown in Figures 5 and 6.It should be noted that in the first experiment, other parameters such as P local , m, and f upper were set according to the first optimization results shown in Table 3, while in the second experiment, we set P tel = 2 W. The optimal value of P local The minimum total readout displacement noise   In actual mission operations, a larger value of Preceive can not only reduce total x δ , but also reduce the difficulty of weak-light phase-locked loops to some extent [27], which means that the power of Ptel must be increased.However, increasing Ptel undoubtedly increases the difficulty of the design of the devices associated with this system, and therefore, a trade-off needs to be made considering the practical applications of this system.In addition, different upper bounds of the beat-note frequency correspond to different optimal values of Plocal, and different minimum total readout displacement noise values.The value of Plocal can be set by referring to Figure 6.

Conclusions
In this study, a hierarchical optimization algorithm is proposed to solve the Taiji program's system-level frequency setting scheme.The optimization model considered the effects of six main factors, namely flower, fupper, fADC, fPT, fUSO, and m.Two optimization objectives were used, including minimizing the total readout displacement noise and  The optimal value of P local The minimum total readout displacement noise Figure 6.Variations of the optimal P local value and the minimum total readout displacement noise with f upper .The x-, left-hand y-, and right-hand y-axes show f upper , the optimal P local , and the minimum total readout displacement noise, respectively.
In Figure 5, as P tel increases, P receive increases linearly, and δx total decreases approximately linearly.In Equation ( 4) and Step 5 of the optimization algorithm proposed in this study, the variation of P receive may cause a change in the optimal P local value.However, in practice, the variation of P receive is so small that it barely affects the optimal value of P local .Figure 6 shows the variations of P local and δx total with f upper .The optimal value of P local increases as f upper increases, while δx total first decreases and then increases.When f upper = 27 MHz, the optimal P local values is 2.175 mW, and δx total has its smallest value of 4.2078 pm/ √ Hz.In actual mission operations, a larger value of P receive can not only reduce δx total , but also reduce the difficulty of weak-light phase-locked loops to some extent [27], which means that the power of P tel must be increased.However, increasing P tel undoubtedly increases the difficulty of the design of the devices associated with this system, and therefore, a trade-off needs to be made considering the practical applications of this system.In addition, different upper bounds of the beat-note frequency correspond to different optimal values of P local , and different minimum total readout displacement noise values.The value of P local can be set by referring to Figure 6.

Conclusions
In this study, a hierarchical optimization algorithm is proposed to solve the Taiji program's system-level frequency setting scheme.The optimization model considered the effects of six main factors, namely f lower , f upper , f ADC , f PT , f USO , and m.Two optimization objectives were used, including minimizing the total readout displacement noise and maximizing the feasible beat-note frequency range.Considering the characteristics involved in solving multi-objective optimization problems, 20 Pareto-optimal solutions were obtained.The minimum total readout displacement noise was 4.12 pm/ √ Hz, which corresponded to a beat-note frequency feasible range of 21 MHz, with (f lower , f upper , f ADC , f PT , f USO , m) = (3 MHz, 23 MHz, 96 MHz, 94 MHz, 4512 MHz, 0.44).The maximum feasible range of the beat-note frequency was 23 MHz with a total readout displacement noise of 4.21 pm/ √ Hz, with (f lower , f upper , f ADC , f PT , f USO , m) = (3 MHz, 25 MHz, 92 MHz, 90 MHz, 4140 MHz, 0.44).Hence, different values of the parameters f upper , f ADC , f PT , and f USO result in different final optimization results.Therefore, these two objectives were not simultaneously optimized, and a trade-off between these two objectives needs to be made in practical applications of this system.Moreover, we analyzed the effects of P tel and f upper on P receive and P local , and then explored the effects of these two factors on the total readout displacement noise.The results provide a reference for setting the frequency setting strategy during laser transmission and readout, determining the power ratio between the sidebands and carrier and selecting the relevant equipment parameters of laser interferometry systems in the Taiji program.

Figure 1 . 2 Figure 1 .
Figure 1.Carrier, first-order sideband (normalized power over the modulation depth m), and the ratio between the first-order sideband and carrier.

Step 5 :
adjusting the parameter m in the range [0.44, 0.61].Backtracking mechanism.Since the default Plocal value is obtained by assuming fupper = 25 MHz, it may change after completing Steps 1-4.Therefore, if Plocal changed, then update Plocal and return to Step 1 until no new Plocal appears.

Figure 3 .
Figure 3. Pareto-optimal solution plane: (a) Scatter plot of the total readout displacement noise with the modulation depth; USO frequency; and beat-note frequency in a feasible range along the x-, y-, and z-axes, respectively.The total readout displacement noise total x δ is represented by the colored bar.(b) Scatter plot of the beat-note frequency in a feasible range with the modulation depth; USO frequency; and total readout displacement noise along the x-, y-, and z-axes, respectively.The beatnote frequency in a feasible range is represented by the colored bar.

Figure 3 .
Figure 3. Pareto-optimal solution plane: (a) Scatter plot of the total readout displacement noise with the modulation depth; USO frequency; and beat-note frequency in a feasible range along the x-, y-, and z-axes, respectively.The total readout displacement noise δx total is represented by the colored bar.(b) Scatter plot of the beat-note frequency in a feasible range with the modulation depth; USO frequency; and total readout displacement noise along the x-, y-, and z-axes, respectively.The beat-note frequency in a feasible range is represented by the colored bar.

Figure 4 .
It can be seen from the figure that the optimized parameters have improved the sensitivity in the range of 10 mHz-1 Hz.

Figure 4 .
Figure 4.The sensitivity curve of the Taiji program detection limit (red) and after optimization (blue).
), the value of Preceive is positively proportional to Ptel.The optimal value of Plocal is directly influenced by fupper.To describe the effect of the values of Ptel and fupper on Preceive and Plocal in a more intuitive manner, we conducted the following experiments: (1) vary the value of Ptel in the range [2 W, 3 W] with an interval of 0.1 W and observe the variations of Preceive and total readout displacement noise; and (2) vary the value of fupper in the range [20 MHz, 30 MHz] with an interval of 1 MHz and observe the variations of the optimal Plocal value and minimum total readout displacement noise total x δ .The results are shown in Figures 5 and 6.It should be noted that in the first experiment, other parameters such as Plocal, m, and fupper were set according to the first optimization results shown in Table

Figure 6
shows the variations of Plocal and total x δ with fupper.The optimal value of Plocal increases as fupper increases, while total x δ first decreases and then increases.When fupper = 27 MHz, the optimal Plocal values is 2.175 mW, and total x δ has its smallest value of 4.2078 pm / Hz .

Figure 4 .
Figure 4.The sensitivity curve of the Taiji program detection limit (red) and after optimization (blue).

Sensors 2023 , 14 Figure 5 .
Figure 5. Variation of Preceive and the total readout displacement noise as a function of Ptel.The x-, left-hand y-, and right-hand y-axes show Ptel, Preceive, and the total readout displacement noise, respectively.

Figure 5 .
Figure 5. Variation of P receive and the total readout displacement noise as a function of P tel .The x-, left-hand y-, and right-hand y-axes show P tel , P receive , and the total readout displacement noise, respectively.

Figure 5 .
Figure 5. Variation of Preceive and the total readout displacement noise as a function of Ptel.The x-, left-hand y-, and right-hand y-axes show Ptel, Preceive, and the total readout displacement noise, respectively.

Figure 6 .
Figure 6.Variations of the optimal Plocal value and the minimum total readout displacement noise with fupper.The x-, left-hand y-, and right-hand y-axes show fupper, the optimal Plocal, and the minimum total readout displacement noise, respectively.

Table 2 .
Time consumption of each step.

Table 2 .
Time consumption of each step.

Table 3 .
Pareto-optimal feasible solutions obtained from hierarchical optimization algorithm.