The detection of gravitational waves in space has gained significant international attention, driven by its ability to identify a wide range of detectable wave source frequencies and diverse wave sources. Several representative missions, such as Laser Interferometer Space Antenna (LISA)^[1], Taiji^[2], and Tianqin^[3], are expected to launch in the next decade and achieve remarkable detection sensitivities. In those space-based gravitational wave detection missions, split heterodyne interferometry^[4] has been adopted, enabling the measurement of distance variations caused by gravitational waves between two test masses separated by millions of kilometers. The so-called science interferometers are utilized for measuring distance variations between satellites, while test mass interferometers are employed for measuring distance variations between the optical bench with respect to the test mass^[4]. Notably, both types of interferometers are susceptible to a specific type of optical noise known as tilt-to-length (TTL) coupling noise, which arises due to beam tilt during measurement. The beam jitter received by the telescope in the interferometer and the test mass’s rotation in the test mass interferometer can lead to TTL coupling noise, for instance^[5-6]. Presently, it stands as the second most significant entry in the metrology error budget of LISA-like missions, following shot noise^[7]. Hence, fully comprehending this noise’s characteristics is paramount for effectively recognizing, suppressing, and subtracting the noise as needed.

Presently, researchers focus on two primary areas in investigating TTL coupling noise. The first aspect involves analyzing the factors contributing to the noise, employing theoretical analysis or numerical simulation methods. Theoretical analysis of TTL coupling noise involves deriving analytical expressions of path length signals under various influential factors. Previous studies have provided qualitative and quantitative explanations of both geometric^[8-9] and non-geometric mechanisms^[10] of TTL coupling. Some research focused on specific conditions related to the cancellation of TTL and found that geometric optics alone cannot accurately describe the characteristics of TTL coupling noise^[8-9]. Further studies explored the impacts of beam parameters^{[10, 11]}, signal definition^[12], and wavefront aberrations^[13-16] on TTL coupling noise. Requirements for telescope wavefront aberration^[13-15] and the effect of low-order aberrations were also analyzed^[16]. The second aspect entails exploring strategies to suppress the TTL coupling noise through optical design or data processing techniques for noise subtraction. Imaging systems have been effectively utilized to suppress TTL coupling noise. Studies have demonstrated significant noise suppression using dual-lens or four-lens imaging systems^{[6-7, 17]}. Additionally, simulations have provided insights into subtracting TTL coupling noise through linear regression methods and Bayesian inference^[18-21].

The above research mainly focused on deriving analytical expressions for analyzing TTL coupling noise induced by individual factors, developing imaging systems aimed at noise suppression, and utilizing data processing techniques for noise extraction. The innovative contribution of this paper lies in its introduction of a methodology capable of analyzing multiple influential factors and discerning their individual impact by assigning a ranking. As a demonstration, we implemented this methodology in the test mass interferometer of a LISA-like mission.

Following this introduction, Section 2 introduces the definition of TTL coupling. Section 3 presents the configuration for simulating TTL coupling and compares the simulated results with data obtained from an experiment. Subsequently, Section 4 presents the methodology for analyzing multiple influential factors. Based on this, Section 5 delves into an analysis incorporating multiple factors using the simulation results. Finally, Section 6 summarizes the primary contributions of this paper and outlines future work.

A fundamental feature of TTL coupling is the additional path length readout signal generated in the interferometer due to the beam’s jitter. Under the assumption of geometrical optics, where both interfering beams are considered geometric rays, the noise can be defined by the optical path length difference (OPD) value. However, the interfering beams are not simple geometric rays in actual interferometers. This implies that non-geometric effects cannot be disregarded. In such instances, the interferometer will use the longitudinal path length signal (LPS) to detect any additional changes in path length. In space-based gravitational wave detection missions, the quadrant photodiodes (QPD) are used as detectors. These have different combinations of signals from its four segments, resulting in various definitions for the LPS. Notably, the LPF (LISA pathfinder) and AP (averaged phase) definitions are widely employed^[8]:

$ {{LPS}_{{\text{LPF}}}} = \frac{1}{k}\left( {{C_{{A}}} + {C_{{B}}} + {C_{{C}}} + {C_{{D}}}} \right)\quad, $ (1)

$ \begin{split} {{LPS}_{{\text{AP}}}} =& \frac{1}{{4k}}( \arg ({C_{{A}}}) + \arg ({C_{{B}}}) +\\ &\arg ({C_{{C}}}) + \arg ({C_{{D}}}) )\quad, \end{split} $ (2)

where k represents the wavenumber (2π/λ), λ denotes the beam's wavelength. Meanwhile, the variables C_A, C_B, C_C, and C_D refer to the complex electric field on each quadrant of the QPD.

To analyze TTL coupling noise, numerical simulations were implemented first and subsequently applied to the test mass interferometer. The results were then validated through an experiment. This section primarily concentrates on the simulation of TTL coupling noise within the test mass interferometer. It is divided into two subsections. The first subsection outlines the simulation configuration utilized in this paper to simulate TTL coupling noise. The second subsection compares the simulation results and experimental data, demonstrating the accuracy of the simulation.

The software employed in this study for simulation purposes is one that we developed in Ref. [22], which addresses geometric and non-geometric TTL coupling. The reference and measurement beams are treated as fundamental Gaussian beams for the test mass interferometer simulation. Their propagation can be easily described using the q-parameter and ray transfer matrix, which facilitates the computation of the complex electric field detected by the QPD, allowing for the determination of the LPS, i.e., Eq. (1) and Eq. (2).

Based on the concept of split heterodyne interferometry and the simulation software we developed, an equivalent simulated optical bench for the test mass interferometer has been designed in this paper, taking into account the motion of the freely suspended test mass, schematically shown in Fig. 1 (color online). The beam from the laser is split into two beams by a beam splitter (BS). The reference beam is transmitted to the mirror (M) for reflection, while the other is directed towards the test mass and reflected off its surface. The reference and measurement beams are recombined at the beam splitter (BS) and interfere at the QPD.

In Fig. 1, the left upper corner shows the coordinates of the optical bench, with the detailed coordinates and normal vectors of every optical component in Table 1.

The use of this optical configuration enables the simulation of various factors affecting TTL coupling noise in the test mass interferometer. The influential factors have been classified into four distinct categories: positional factors, beam parameters, detector parameters, and signal definition. Positional factors include beam offset, piston effect, test mass shift, and test mass rotation, as depicted in Fig. 2. The beam offset refers to the displacement between the measurement beam and the reference beam caused by misalignment. When the center of rotation of the test mass does not coincide with the reflection point of the beam on the test mass surface, the piston effect arises, which can be characterized by two parameters: the lateral parameter and the longitudinal parameter. In space-based gravitational wave detection, changes in the position and orientation of the test mass lead to TTL coupling. Therefore, these changes are considered as test mass shift and rotation angle. Beam parameters account for variations in beam waist and the distance from the waist of the measurement fundamental Gaussian beam. Regarding the detector parameter factor, TTL coupling noise arises because the signal cannot be fully detected due to the presence of slits in the QPD. This analysis focuses on the TTL coupling caused by QPD slits. The presence of QPD slits alters the definition of LPS. Unlike positional factors, beam parameters, and detector parameters, this type of factor affects TTL coupling not through input parameters but rather through how the detected signals are combined, as shown in Eq. (1) and Eq. (2).

The simulation software developed and employed in this study has undergone verification with IfoCAD, a toolkit created by the Albert Einstein Institute, as discussed in Ref. [22]. To gain deeper insight into its accuracy, this subsection undertakes a comparative analysis between the simulation outcomes and experimental data.

Figure 3 (color online) depicts the schematic illustration of the experimental setup. The measurement and reference beams, denoted by blue and red lines, respectively, share identical beam parameters since they originate from the same laser. After passing through acousto-optical modulators (AOMs), their frequencies diverge. As shown in Figure 3, the reference beam is reflected by a stationary mirror (labeled "Ref M"), whereas the measurement beam is reflected by a fine steering mirror (FSM). The FSM is intentionally rotated to achieve the rotation of a test mass, where the center of rotation does not coincide with the beam's reflection point on the FSM surface. The detectors employed in this setup are four-channel phasemeters. Consequently, this experimental configuration results in a TTL signal arising from the piston effect when the FSM rotates.

The simulation utilized the experimental parameters to compute the LPS signal, adopting the same rotation angles as those used in the experiment. Table 2 outlines the physical parameters involved. Subsequently, the experimental and simulation data are graphically represented in Figure 4 (color online), and Table 3 provides the relative error of the LPS for various rotation angles.

The average relative error between the simulation results and experimental data has been calculated to be 3.14%. The discrepancies observed in TTL coupling noise between the simulated and experimental data can be attributed to several factors. First, the simulation model may incorporate assumptions and simplifications that fail to fully encapsulate the intricacies and non-ideal aspects of the actual system. Second, unaccounted-for experimental conditions, such as environmental disturbances, may have introduced measurement errors. Additionally, the system parameters’ precision and the simulation model’s accuracy can further affect the consistency between the two datasets.

In practical applications, TTL coupling noise typically arises due to the combined influence of multiple factors. Analyzing the LPS becomes complex and challenging when these factors act simultaneously. In such cases, numerical simulation offers a viable alternative approach. Through employing the numerical simulation introduced in Section 3, it is possible to comprehensively analyze the effects of various factors on TTL coupling noise and obtain accurate estimations of the LPS. This simulation-based approach provides valuable insights into the behavior and characteristics of TTL coupling noise in real-world scenarios, enabling researchers to develop effective strategies for its recognition, suppression, and subtraction in practical applications.

The method used to analyze the multiple factors involved is built on the fundamental concept of employing random parameters within a defined and reasonable range of those that affect the TTL coupling. By generating a diverse set of random parameter combinations, the simulation captures the variability of real-world scenarios. Based on the simulation, the variance-based global sensitivity analysis method was used to assess the sensitivity of the TTL coupling noise to each parameter. The analysis calculates the main effect index S₁ and the total effect index S_T for each parameter, providing quantitative measurements of their influences on the TTL coupling noise. Therefore, this method enables the identification of the key factors that significantly impact TTL coupling noise when multiple factors operate simultaneously. By systematically examining the main and total effect indices, researchers can prioritize the factors that contribute the most to TTL coupling noise and focus their efforts on mitigating their impact.

The main effect index S₁ quantifies the extent to which an individual influential factor, x_i contributes to the variation in the output variance of the system. A higher S₁ value indicates a stronger influence of that particular factor on the TTL coupling noise. On the other hand, the total effect index S_T assesses the collective contribution of all influential factors to the overall variation in the output variance. This index considers the factor's direct effect and its interactions with other factors. A larger S_T value indicates a more significant combined influence of all the factors on the TTL coupling noise. By evaluating both S₁ and S_T, researchers can gain a comprehensive understanding of each influential factor’s relative importance and impact, enabling them to prioritize and address the critical factors in their efforts to suppress TTL coupling noise. It should be noted that S₁ and S_T serve as indicators of the relative importance of various factors affecting TTL coupling noise rather than direct measures of the impact on the interferometer’s final design accuracy.

The calculations of S₁ and S_T are calculated as follows^[23-24]:

$ {S _1} = \frac{{{V_{{x_i}}}\left( {{E_{{x_i}}}(y|{x_i})} \right)}}{{V(y)}}\quad, $ (3)

$ {S _{\mathrm{T}}} = \frac{{{E_{{x_{\sim i}}}}\left( {{V_{{x_i}}}(y|{x_{\sim i}})} \right)}}{{V(y)}} \quad,$ (4)

where V represents variance, E represents mathematical expectation, and x_irepresents the i-th influential factor, with ~ indicating all influential factors except the i-th. y represents the corresponding LPS.

To address the challenges in directly calculating equations Eq. (3) and Eq. (4), commonly employed approximation methods are utilized^[25-26]:

$ {V_{{x_i}}}\left( {{E_{{x_i}}}(y|{x_i})} \right) = \frac{1}{N}\sum\limits_{j = 1}^N {f{{(B)}_j}} \left( {f{{(A{B_i})}_j} - f{{(A)}_j}} \right) ,$ (5)

$ {E_{{x_{\sim i}}}}\left( {{V_{{x_i}}}(y|{x_{\sim i}})} \right) = \frac{1}{{2N}}{\sum\limits_{j = 1}^N {\left( {f{{(A)}_j} - f{{(A{B_i})}_j}} \right)} ^2} \quad.$ (6)

The step-by-step algorithm for analyzing multiple factors is as follows:

Step 1: Randomly generate a sampling matrix of size N×2n in the parameter space, where N is the number of sampling points and n is the number of parameters that influence the TTL coupling noise.

Step 2: Using the first n columns of the N×2n matrix as matrix A and the last n columns as matrix B. The parameter order of matrix B must be consistent with that of matrix A.

Step 3: Construct a matrix AB_i of size N×n, where i = 1, 2,···, n. The i-th column of AB_i is the i-th column of matrix B, and the other columns are the columns of matrix A except for the i-th column.

Step 4: Calculate the corresponding LPS for random parameter matrices A, B, and AB_i.

Step 5: Calculate the main and total effect indexes based on Eq. (5) and Eq. (6).

In this section, the method introduced in Section 4, is employed to analyze the characteristics of TTL coupling noise when multiple factors act simultaneously. The simulated optical setup presented in this section is illustrated in Fig. 1.

First, a sampling matrix containing parameters influencing TTL coupling was generated. For convenience, Table 4 presents the parameter space, detailing the range of each parameter, which satisfies the requirements of Taiji.

For 10,000 simulations, parameters were randomly chosen from the parameter space with a uniform distribution, and the corresponding LPS were computed using both LPF and AP definitions. Their respective histograms are illustrated in Fig. 5.

Based on the simulation results, the LPS using the AP definition spans a range from −9.1285×10⁵ pm to 7.6925×10⁵ pm, while the range is −7.9348×10⁵ pm to 9.6939×10⁵ pm with the LPF definition. Fig. 3. also indicates that the highest range of LPS signal with the LPF definition is from −1×10⁵ pm to 0 pm (1033 occurrences), whereas, for the AP definition, this range is from 0 to 1×10⁵ pm (1,088 occurrences). Within the 10,000 simulations, the maximum absolute value and minimum absolute value for both signal definitions are detailed in Table 5.

Table 5 shows that the minimum absolute value of LPS is approximately 10 pm under both signal definitions. However, while setting parameter values may theoretically result in minimal noise, the practical feasibility of achieving such conditions remains uncertain.

Subsequently, the simulation data can be utilized to compute the main effect index S₁ and the total effect index S_T for both signal definitions. The results are presented in Tables 6 and 7, respectively, and are also depicted visually in Figures 6 and 7 to provide a clear and intuitive representation.

As previously mentioned, the role of S₁ is to rank the importance of input variables. When the LPF definition is used, the ranking of S₁ in descending order of importance is: rotation angle, piston effect (lateral), test mass lateral shift, measurement beam waist radius, measurement beam offset, QPD slit, measurement beam waist position, piston effect (longitudinal), and test mass longitudinal shift. As for the AP definition, the ranking of S₁ in descending order of importance is: rotation angle, piston effect (lateral), test mass lateral shift, measurement beam offset, measurement beam waist radius, QPD slit, measurement beam waist position, piston effect (longitudinal), and test mass longitudinal shift. It can be seen that there is no significant difference in the order of S₁ between the two definitions.

On the other hand, S_T serves to simplify the model parameters. When the LPF definition is applied, the value of S_T in descending order is: rotation angle, piston effect (lateral), measurement beam waist radius, test mass lateral shift, QPD slit, measurement beam offset, measurement beam waist position, piston effect (longitudinal), and test mass longitudinal shift. For the AP definition, the ranking of S_T in descending order is: rotation angle, piston effect (lateral), test mass lateral shift, measurement beam waist radius, measurement beam offset, QPD slit, measurement beam waist position, piston effect (longitudinal), and test mass longitudinal shift. It can be seen that S₁ and S_T of the LPF definition have slightly different rankings, but the overall trend is roughly the same, while for the AP definition, the S₁ and S_T rankings defined by AP are exactly the same.

According to the analysis of S₁ and S_T, it can be concluded that regardless of which signal definition is used, the S₁ and S_T values of rotation angle and lateral piston effect are much larger than those of other factors. Therefore, these two factors have the greatest impact on LPS. Additionally, the effects of measurement beam waist position, longitudinal piston effect, and testing mass longitudinal shift on LPS are minor. For further analysis, the parameter space was expanded by 50% and reduced by 50%, and the main effect index S₁ and the total effect index S_T were calculated again. The conclusion remained the same. This means that in practice, efforts should be made to control the rotation of the test mass and keep its center of mass as close as possible to its geometric center to reduce the piston effect.

This paper proposes a method to analyze the impacts of multiple factors on TTL coupling noise using random parameters. A simulated optical bench was constructed based on the principle of test mass interferometer, supporting the simulation of various noise-affecting factors, and validated with experimental data. The simulation and analysis cover positional factors, beam parameters, detector parameters, and signal definition. With parameters were randomly drawn from reasonable ranges, 10000 simulations were carried out. Then, the simulation data were used for the variance-based global sensitivity analysis, where sensitivity indices were calculated to identify the key factors impacting TTL coupling noise, namely, test mass rotation angle and lateral piston effect. This conclusion offers key knowledge for system design and optimization. The forthcoming research will primarily concentrate on the practical application of the proposed method to study TTL coupling noise in scientific interferometers. Additionally, further experiments will be conducted to verify the accuracy of the analysis results obtained through this method.