1. Introduction

The space-based gravitational wave detection mission involves a configuration of three spacecraft (SC), positioned in the formation of an equilateral triangle [1–3]. Spanning arm lengths that vary from tens of thousands to millions of kilometers, notable examples of such missions include TianQin, LISA (Laser Interferometer Space Antenna), and Taiji. Due to significant power reduction during long-distance transmission, each spacecraft is equipped with two telescopes for expanding and converging the laser beam. The telescope plays a crucial role in the laser interferometry system and requires low measurement noise. The interaction between wavefront aberrations and pointing jitter leads to the generation of non-geometric TTL coupling noise [4–7]. In optical systems, inherent manufacturing and installation errors inevitably lead to wavefront distortion at the exit pupil of the telescope. For the transmitting telescope, the wavefront aberrations at the exit pupil (large pupil) couple with pointing jitter after spatial propagation, resulting in TTL coupling noise. For the receiving telescope, the wavefront aberrations at its exit pupil (small pupil) couple with pointing jitter within the telescope, generating TTL coupling noise.

Extensive research has been conducted on TTL coupling noise over the years. Bender, Sasso, and other researchers established initial computational models to analyze the phase noise of far-field wavefronts caused by primary aberrations, providing valuable insights and methodologies for subsequent studies [6]. Sasso et al. incorporated a Gaussian beam model and considered the first 14 terms of Zernike aberrations in their calculations, suggesting that adjusting defocus terms could reduce phase noise [8]. As research progressed, analysis models became more accurate and comprehensive. Zhao et al. utilized the first 25 terms of Fringe Zernike polynomials to fit wavefront aberrations at the small pupil of the telescope and proposed that the wavefront error RMS value should be below λ/50 [9,10].

In recent years, researchers have not only developed analysis models for TTL coupling noise but also proposed methods to mitigate it. Kenny et al. found that only even-order terms of the fitted wavefront aberrations contribute to the noise, and reducing these terms can decrease TTL coupling noise [11]. Ming et al. found that a large defocus and a small coma can achieve better alignment between the stationary pointing direction and the line of sight to the satellite, thereby reducing TTL coupling noise [12]. Xiao et al. suggested adding TTL error as a piston term to the far-field phase during the pointing process, which helps bring the stationary point closer to the z-axis and reduces phase noise detected at the stationary point [13]. Lin et al. proposed requirements for controlling aberrations in the optical system of gravitational wave telescopes, such as the requirement for the ratio of magnitudes between two aberrations when eliminating noise and the requirement for controlling the magnitudes of aberrations when superimposing noise [14]. Li proposed using TTL coupling coefficients as control requirements and determining the telescope's tolerance allocation based on the sensitivity matrix of the coupling coefficients [15].

The aforementioned research primarily concentrates on modeling and suppressing TTL coupling noise in space transmission and optical platforms. However, the published research has shown a lack of theoretical analysis and corresponding proposed suppression methods specifically for non-geometric TTL coupling noise inside the telescope. In the TianQin project, this paper introduces noise suppression methods that specifically target the control of coupled aberrations and optimization of the Gaussian beam waist radius. These methods aim to meet the benchmark requirement of ensuring that the non-geometric TTL coupling noise inside the telescope remains below 0.2$\sqrt 2 $ pm/Hz^1/2. By simultaneously using two noise suppression methods, the wavefront aberration at the small pupil of the telescope was reduced from 0.0055λ to 0.033 λ. This offers valuable insights and techniques for mitigating TTL coupling noise in gravitational wave detection projects.

2. Optical model

The TTL coupling noise occurs exclusively within the receiving telescope, and it is caused by the jitter of the spacecraft, which result in variations in the optical path length of the received beam within the telescope. The measurement principle is depicted in Fig. 1 and can be further classified into geometric and non-geometric TTL coupling noise. Geometric TTL coupling noise arises due to the distinct propagation paths of incident beams with different incident angles inside the telescope, leading to changes in the optical path length. On the other hand, non-geometric TTL coupling noise refers to the measurement noise caused by the coupling between wavefront aberrations at the small pupil within different fields of view and the spacecraft jitter. Taking the TianQin project as an example, the requirement for TTL coupling noise inside the telescope is set to be better than 0.4 pm/Hz^1/2. Currently, both geometric and non-geometric TTL-coupled noise need to be controlled at a level of 0.2$\sqrt 2 \; $ pm/Hz^1/2. The primary focus of this paper revolves around addressing the non-geometric TTL coupling noise aspect.

 

Fig. 1. Schematic diagram of TTL coupling noise measurement inside the telescope.

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The gravitational wave detection telescope is designed with the capability of expanding the beam during emission and converging the beam during reception. After the transmission of the Gaussian beam emitted from the transmitting telescope over a distance of 170,000 kilometers, the receiving telescope at the receiving end captures a flattened beam. Due to inherent constraints in the telescope's design, manufacturing and installation, the wavefront at the small pupil of the telescope will inevitably undergo distortion. The schematic diagram in Fig. 2 illustrates the interference between the flat-top beam with wavefront error and the Gaussian beam emitted by the local laser on the QPD (quadrant photodiode). The light spots created by rays from different fields of view at the telescope's small pupil remain unchanged. The phase information of the interfering beam on the QPD, positioned at the small pupil location of the telescope, can be mathematically represented as:

 

Fig. 2. Principle of non-geometric TTL coupling noise measurement in the internal of a telescope

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In the equation, $W({\rho ,\theta } )= \mathop \sum \nolimits_{i = 1}^{37} {a_i}{z_i}({\rho ,\theta } )$, where ${z_i}$ represents the aberration term, ${a_i}$ represents the coefficient corresponding to ${z_i}$, w represents the waist radius of the fundamental Gaussian beam, and a represents the small pupil radius of the telescope. ${z_1}$ represents the piston term which does not cause any phase variation. Therefore, it can be directly eliminated in this case.

Due to the spacecraft's jitter at the transmitting end, the beam received by the telescope at the receiving end is not stationary. Its propagation direction will oscillate with the shaking of the spacecraft. The coefficients of Zernike x and y tilt aberrations, expressed as directional deviations ${\alpha _x}$ and ${\alpha _y}$ in terms of x and y.

To streamline the computation, the cosine and sine components of the wave aberration W(ρ,θ) are consolidated through equation $m\cos \theta + n\sin \theta = \sqrt {({{m^2} + {n^2}} )} \cos ({\theta - {\theta_x}} )\textrm{,}\; {\theta _x} = \arctan (n/m)$. Finally, the 21 combined terms of Zernike aberrations can be obtained. The resulting wavefront distortion of the system can be expressed as $W({\rho ,\theta } )= a\alpha \rho \cos ({\theta - {\theta_{\textrm{Ti}}}} )+ \mathop \sum \nolimits_{j = 2}^{21} {A_j}{Z_j}({\rho ,\theta } )$. Therefore, Eq. (1) can be rewritten as:

In this equation, ${Z_j}$ represents the combined aberration terms, ${A_i} $ represents the coefficient corresponding to ${Z_j}$. In this case, the combined terms are referred to as ${A_i} = \sqrt {a_i^2 + a_{i + 1}^2} $, while the terms that are not combined are referred to as A_j = a_i. α represents the pointing deviation of the spacecraft.

The measured optical path difference can be expressed as:

In this context, k is equal to 2π/λ, $E = {E_R} + i{E_I}$ represents the complex form of the total electric field, where E_R and E_I respectively represent the real and imaginary parts of E.

The relationship between the incident angle α and the exit angle β of the incoming light in a telescope is related to the magnification of the telescope. If the magnification factor of the telescope is M, then:

For the TianQin gravitational wave detection telescope, it has a magnification factor of 75. The interstellar pointing deviation is 10 nrad, and the pointing jitter is 10 nrad/Hz^1/2[16]. This means that the outgoing light pointing deviation of the receiving end telescope is 750 nrad, and the pointing jitter is 750 nrad/Hz^1/2.

Expanding Eq. (3) using Taylor series and neglecting terms that are independent of the pointing angle and higher-order terms of the pointing angle. When the receiving telescope at the large pupil has a pointing deviation angle of α, the introduced optical path variation at the small pupil of the telescope due to the pointing deviation angle β is [2][3]:

In the equation, ${C_1}$ represents the coefficient of $\beta $², and ${C_2}$ represents the coefficient of $\beta $.

The coupling coefficient of non-geometrical TTL coupling noise inside the telescope can be represented as:

Based on the aforementioned theoretical model, the initial step is to assess the magnitude of the RMS value which can meet the criteria for non-geometric TTL coupling noise inside the telescope, without any optimization. The research method involves utilizing the Monte Carlo method to analyze the likelihood of satisfying non-geometric TTL coupling noise under varying wavefront aberrations RMS of the telescope. For each fixed RMS value obtained from fitting, we generated 100,000 random samples, wherein each sample had distinct weighting of Zernike coefficients while maintaining the same RMS value constraint. The range of RMS values was defined from λ/200 to λ/50 with an incremental step size of 0.0015 λ. In total, 11 RMS values were chosen for analysis. Figure 3 illustrates the proportion of RMS values meeting the noise requirements as the RMS value ranges from 0.005 λ to 0.020 λ. Therefore, we conducted an analysis to determine the probability of meeting the noise requirements for each case using 100,000 samples under different RMS values, namely 0.0050 λ, 0.0065 λ, 0.0080 λ, 0.0095 λ, 0.0110 λ, 0.0125 λ, 0.0140 λ, 0.0155 λ, 0.0170 λ, 0.0185 λ, and 0.0200 λ. By analyzing the data presented in Fig. 3, we can observe that when the wavefront aberration RMS value at the small pupil of the telescope is 0.005 λ, the proportion of samples meeting the noise requirements is 100%. However, as the RMS value increases to 0.0065 λ, the proportion decreases to 83.32%. Further increasing the RMS value of the wavefront aberration at the small pupil of the telescope causes the proportion of samples meeting the noise requirements to drop below 80%. This indicates that without suppressing the non-geometric TTL coupling noise inside the telescope, the manufacturing and installation demands for the telescope are extremely stringent. Considering the current level, it is highly probable that the non-geometric TTL coupling noise within the telescope does not meet the specified requirements. Hence, it is crucial to devise techniques for mitigating the non-geometric TTL coupling noise within the telescope, thereby alleviating the demands on wavefront aberrations at the telescope's small pupil.

3. Suppression methods

3.1 Control the proportion of coupling aberrations

To enhance research convenience, the initial 20 Zernike polynomial terms are utilized and subsequently amalgamated into a set of 12 terms through merging similar ones. After excluding the piston term, which does not contribute to tilt aberration coupling, a total of 10 aberration terms were identified as being significant in the non-geometric TTL coupling noise. Let's define the aberration vectors as b₁ and b₂, represented as follows:

 

Fig. 3. Proportion of different RMS values at the small pupil of the telescope that meet the noise requirements.

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The coefficients C₁ and C₂ can be expressed as:

M₁ and M₂ are coefficient matrices for non-geometric TTL coupling noise. M₁ is a 1 × 6 coefficient matrix, while M₂ is a 4 × 6 coefficient matrix.

According to Eq. (6), the coupling coefficient of non-geometric TTL coupling noise can be separated into two terms. One term is associated with the pointing deviation angle β and corresponds to individual aberration terms, as indicated in Table 1. Let ${M_1} = [{m_{i,j}^{{M_1}}} ]$ be a 1 × 6 matrix, where the element in the i-th row and j-th column of ${M_1}$ is denoted as $m_{i,j}^{{M_1}}$. The coupling coefficient of non-geometric TTL coupling noise for each individual aberration term can be expressed as follows:

Table 1. Coupling coefficients of individual aberration terms (w = 2mm)

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Among them, ${A_4}$ represents defocus, ${A_{5,6}}$ represents primary coma, ${A_9}$ represents primary spherical aberration, and ${A_{16}}$ represents secondary spherical aberration. $m_{1,1}^{{M_1}}$, $m_{1,2}^{{M_1}}$, $m_{1,3}^{{M_1}}$, $m_{1,4}^{{M_1}}$, and $m_{1,5}^{{M_1}}$ represent the noise coefficients corresponding to ${A_4}$, ${A_{5,6}}$, ${A_9}$, ${A_{12,13}}$, and ${A_{16}}$, respectively. $\; $Due to the noise coefficient $m_{1,6}^{{M_1}} $ corresponding to A_17,18 being 0, this item has been omitted.

Another term is the coupling term between pairwise aberrations that are independent of the pointing deviation angle β, as shown in Table 2. Let ${M_2} = [{m_{i,j}^{{M_2}}} ]$ be a 4 × 6 matrix, where the element in the i-th row and j-th column of ${M_2}$ is denoted as $m_{i,j}^{{M_2}}$. The noise coupling coefficient for the non-geometric TTL coupling of pairwise coupled aberrations can be represented as:

Table 2. Noise coefficients of coupled aberration terms (w = 2 mm, absolute sum of coefficients = 216.73)

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In this context, the coefficients $m_{1,1}^{{M_2}}$, $m_{1,2}^{{M_2}}$, $m_{1,3}^{{M_2}}$, $\; m_{1,4}^{{M_2}}$ , $\; m_{1,5}^{{M_2}}$, $\; \; m_{2,1}^{{M_2}}$, $\; \; m_{2,2}^{{M_2}}$, $m_{2,3}^{{M_2}}$, $m_{2,4}^{{M_2}}$, $\; m_{2,5}^{{M_2}}$, $\; m_{3,2}^{{M_2}}$, $m_{3,4}^{{M_2}}$, $m_{3,6}^{{M_2}}$, $m_{4,2}^{{M_2}}$, $m_{4,4}^{{M_2}}$ and $m_{4,6}^{{M_2}}$ represent the coupling noise coefficients for ${A_{7,8}}{A_4}$, ${A_{7,8}}{A_{5,6}}$, ${A_{7,8}}{A_9}$, ${A_{7,8}}{A_{12,13}}$, ${A_{7,8}}{A_{16}}$, ${A_{14,15}}{A_4}$, ${A_{14,15}}{A_{5,6}}$, ${A_{14,15}}{A_9}$, ${A_{14,15}}{A_{12,13}}$, ${A_{14,15}}{A_{16}}$, ${A_{10,11}}{A_{5,6}}$, ${A_{10,11}}{A_{12,13}}$, ${A_{10,11}}{A_{17,18}}$, ${A_{19,20}}{A_{5,6}}$, ${A_{19,20}}{A_{12,13}}$ and ${A_{19,20}}{A_{17,18}}$, respectively.

To compare and analyze the effects of different aberrations on non-geometric TTL coupling noise, the aberration coefficient terms were set to 1, and the azimuth angle of the combined aberration was set to 0. Table 1 presents the aberration terms in single aberrations that contribute to the noise, along with their respective actual contribution and normalized contribution. Comparative analysis shows that among the single aberrations, defocus, astigmatism, and spherical aberration contribute to the noise. Among them, astigmatism has the highest contribution to the noise, followed by defocus, and finally spherical aberration. Additionally, primary aberrations have a greater contribution to the noise compared to higher-order aberrations.

Table 2 illustrates the contribution of coupling aberrations to noise for a Gaussian beam with a waist radius of w = 2 mm. Through analysis, it can be concluded that the aberration terms coupled with coma (${A_{7,8}}$ and ${A_{14,15}}$) include defocus (${A_4}$), astigmatism (${A_{5,6}}$ and ${A_{12,13}}$), and spherical aberration (${A_9}$ and ${A_{16}}$). The aberrations coupled with trefoil (${A_{10,11}}$ and ${A_{19,20}}$) include astigmatism (${A_{5,6}}$ and ${A_{12,13}}$) and 4-foil (${A_{17,18}}$). By comparing and analyzing the noise coupling coefficients of single aberrations and coupled aberrations, it can be concluded that the coupling coefficient of single aberrations is four orders of magnitude larger than that of coupled aberrations. However, according to Eq. (11), it is known that single aberrations need to be multiplied by the pointing deviation angle β (with a maximum value of 750 nrad). This implies that the final impact of single aberrations on noise can be negligible compared to coupled aberrations. Therefore, the primary focus is on optimizing the coupled aberrations to reduce non-geometric TTL coupling noise inside the telescope.

Analysis reveals that the aberration types coupled with the same aberration type are consistent. For instance, coma terms (first-order coma ${A_{7,8}}$ and second-order coma ${A_{14,15}}$) are coupled with defocus (${A_4}$), astigmatism (first-order astigmatism ${A_{5,6}}$ and second-order astigmatism ${A_{12,13}}$), and spherical aberration (first-order spherical aberration ${A_9}$ and second-order spherical aberration ${A_{16}}$). The noise coefficients of coupled aberrations do not exhibit a clear pattern, and the coupling and coefficient distribution between each type of aberration and other aberrations are relatively uniform. This indicates that the optimization of coupling coefficients for coupled aberration noise cannot be achieved by reducing specific sensitive aberration terms.

Furthermore, taking into consideration, the RMS value of coupled aberration terms can be expressed as follows:

Among them, ${\sigma _{b1}}$ represents the RMS value of the aberrations contained in the aberration matrix b₁, ${\sigma _{b2}}$ represents the RMS value of the aberrations contained in the aberration matrix b₂, and σ represents the RMS value of the aberrations contained in both aberration matrices b₁ and b₂.

When σ is constant, the noise coupling coefficients of coupled aberrations in Eq. (12) can only be reduced by decreasing the value of one of the aberration coefficients in the coupled aberration terms. In other words, when σ is constant, the greater the difference between ${\sigma _{b1}}$ and ${\sigma _{b2}}$, the smaller the noise coupling coefficients of the coupled aberrations. The ${P_{b2}}$ can be defined as follows:

We set the values of ${P_{b2}}$ to be 1%, 20%, 50%, 80%, and 99%, respectively. The range of RMS values is from λ/200 to λ/50. We select a total of 11 RMS values with an interval of 0.0015 λ and collect 100,000 random samples for each RMS value. We utilize the Monte Carlo method to analyze the proportion of meeting the noise requirements for varying values of ${P_{b2}}$. This analysis is presented in Fig. 4. Through the analysis, it has been found that when ${P_{b2}}$ is set to 50%, the proportion of 100,000 samples that can satisfy the noise requirements is the lowest among the same RMS values. In this case, if the requirement is to have a proportion of noise satisfaction higher than 80%, the RMS value of wavefront aberration at the small pupil position of the telescope needs to be lower than 0.0055 λ. When ${P_{b2}}$ is set to 99%, the proportion of 100,000 samples that can satisfy the noise requirements is the highest among the same RMS values. If the requirement is to have a proportion of noise satisfaction higher than 80%, an RMS value of wavefront aberration lower than 0.0140 λ at the small pupil position of the telescope can satisfy the noise requirements. This indicates that the non-geometric TTL coupling noise within the telescope can be optimized by adjusting the value of ${P_{b2}}$, specifically by reducing the proportion of coma and astigmatism to lower the required RMS value of wavefront aberration at the small pupil position of the telescope.

After completing the optical design of the off-axis four-reflecting telescope, fine-tuning the manufacturing and adjustment processes is essential to suppress non-geometric TTL coupling noise. This involves controlling specific combinations of aberrations to achieve the desired proportion of ${P_{b2}}$. While it's possible to control coefficients for individual aberrations, controlling combinations of aberrations directly is challenging. To overcome this, measurement of wavefront error coefficients using an interferometer or profilometer is conducted during the mirror manufacturing and optical system adjustment processes. Then, each aberration term in b₁ is individually controlled to meet the system requirements. Despite the complexity of this process, our proposed approach, which centers on controlling the coupling aberration proportion, effectively reduces non-geometric TTL coupling noise within the telescope and is viable in practical engineering.

 

Fig. 4. The proportion of noise requirements satisfied for different values of ${P_{b2}}$

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3.2 Optimization of the waist radius of a Gaussian beam

The Gaussian beam emitted by the local laser on the QPD exhibits a waist radius of w. This coupling between the waist radius (w) and aberration terms will have an impact on the non-geometric TTL coupling noise within the telescope. For instance, the noise coupling coefficient $\delta OP{E_{{A_4}}}$ associated with the defocus aberration A₄ can be expressed as follows:

The small pupil radius of the TianQin gravitational wave detection telescope is 2.0 mm. The waist radius of the Gaussian beam can be precisely controlled by utilizing lenses. Figure 5 illustrates a plot depicting the changes in defocus A₄, astigmatism (A_5,6 and A_12,13), spherical aberration (A₉ and A₁₆), and the coupling coefficient ${\delta _1}$ of single aberration noise as a function of w. After analyzing the curves presented in the plot, it can be observed that the coupling coefficients of first-order and second-order aberrations belonging to the same type exhibit opposite signs. This leads to a partial cancellation effect. Defocus aberration does not possess higher-order terms, meaning that the coupling coefficients cannot be nullified or canceled out. The red curve in the plot represents the relationship between the variation of ${\delta _1}$ and the change in w. As w increases, the absolute value of the coupling coefficient initially increases slowly, then slightly decreases, and finally sharply increases. In general, when the value of w is below 1 mm, ${\delta _1}$ tends to be small, suggesting a relatively minimal impact on the non-geometric TTL coupling noise within the telescope.

 

Fig. 5. Variation of the coupling coefficient of single aberration noise with respect to the waist radius (w) of a Gaussian beam.

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The coupled aberrations play a crucial role in determining the non-geometric TTL coupling noise within the telescope. Figure 6(a), (b), (c), and (d) showcase the change in the noise coefficients of the coupled aberrations for ${A_{7,8}}$, ${A_{14,15}}$, ${A_{10,11}}$, and ${A_{19,20}}$, as they vary with w. Upon analyzing the curve in Fig. 6(a), it becomes evident that the noise coefficients of A_5,6 and A_12,13, which are coupled with ${A_{7,8}}$, exhibit a partial cancellation effect at smaller values of w. Nevertheless, as w increases, these coefficients gradually accumulate and overlap with one another. The noise coefficients of A₉ and A₁₆, which are coupled with ${A_{7,8}}$, consistently demonstrate a notable degree of cancellation. This cancellation effect is more pronounced when w is below 1 mm, but gradually diminishes as w increases beyond 1 mm. When examining Fig. 6(b), it becomes evident that the noise coefficients of A₉ and A₁₆, which are coupled with ${A_{14,15}}$, exhibit a partial cancellation effect when w is smaller. Nonetheless, as w increases, these coefficients start to accumulate and overlap with each other. The noise coefficients of A_5,6 and A_12,13, which are coupled with ${A_{14,15}}$, consistently exhibit a partial cancellation effect. The noise coefficients of A_5,6 and A_12,13, which are coupled with ${A_{10,11}}$, demonstrate a similar pattern of change in relation to was depicted in Fig. 6(c). However, the trend of variation in the aberration noise coefficients coupled with ${A_{19,20}}$ in Fig. 6(d) is completely opposite to that shown in Fig. 6(c). Upon careful comparison and analysis, it becomes evident that the noise coupling coefficients exhibited in Fig. 6(a) and Fig. 6(b), as well as in Fig. 6(c) and Fig. 6(d), display a partial cancellation effect. This observation suggests that among the noise coefficients coupled with the same type of aberration, especially at lower values of w, there exists a partial mutual offsetting mechanism, which leads to the reduction in aberration noise coefficients.

 

Fig. 6. Variation of the coupling coefficient of coupled aberration noise with respect to the waist radius (w) of a Gaussian beam

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In Fig. 7, the change in the coupling aberration noise coupling coefficient ${\delta _2}$ is depicted as a function of w. Before reaching a value of 0.50 mm, the coupling aberration noise coupling coefficient ${\delta _2}$ shows a gradual increase. However, once w exceeds 0.50 mm, the rate of increase noticeably accelerates. Hence, when w is less than or equal to 0.50 mm, the magnitude of ${\delta _2}$ is smaller, indicating a stronger ability to suppress non-geometric TTL coupling noise. Table 3 provides the noise coefficients for coupling aberrations when w is 0.5 mm. In comparison to the data in Table 2, most of the coupling aberration noise coefficients exhibit a decrease in absolute values, barring a slight increase in the ${A_{7,8}}$ and ${A_{16}}$ terms. Moreover, the absolute value of ${\delta _2}$ is reduced by a factor of 4.5 in comparison to when w was 2.0 mm.

 

Fig. 7. The variation of the sum of coupling coefficients of coupled aberration noise with respect to the waist radius w of a Gaussian beam.

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Table 3. the noise coefficients of the coupled aberration terms (w = 0.50 mm, sum of coefficients is 48.11)

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Figure 8 illustrates the percentage of wavefront aberrations at the telescope's small pupil position that meet the noise requirements. It shows the variations as the RMS value ranges from 0.005 λ to 0.33 λ for different values of w, namely 0.5 mm, 1.0 mm, 1.5 mm, 2.0 mm, and 2.5 mm. Upon analyzing the curves in the graph, it becomes evident that when w is set to 0.5 mm, a significantly higher proportion of 100,000 samples meets the noise requirements for the same wavefront aberration RMS value compared to other values of w. At this specific point, when the wavefront aberration RMS value at the telescope's small pupil position is below 0.016 λ, approximately 85.41% of the samples meet the noise requirements. This suggests that by optimizing the value of w, it is possible to effectively decrease non-geometric TTL coupling noise within the telescope.

 

Fig. 8. The proportion of meeting the noise requirement for different waist radius w of the Gaussian beam is varied.

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4. Comprehensive optimization results

By adjusting the proportion of coupled aberration, it is possible to achieve a relaxation of the wavefront aberration RMS at the telescope small pupil to a value of 0.014 λ. By optimizing the waist radius w of the local laser beam, the wavefront aberration RMS value at the telescope small pupil can be relaxed to 0.016 λ. The aforementioned results were obtained through separate optimization methods. Therefore, the purpose of the following is to simultaneously consider both optimization methods in order to suppress non-geometrical TTL coupling noise within the telescope.

Figure 9 illustrates the proportions of meeting the noise requirements in four different scenarios: without any optimization (w = 2.0 mm with uniform distribution of aberrations), optimization of the Gaussian beam waist radius (w = 0.5 mm with uniform distribution of aberrations), controlling the proportion of coupling aberrations (w = 2.0 mm with ${P_{b2}}$=99%), and simultaneous optimization of both methods (w = 0.5 mm with ${P_{b2}}$=99%). Based on the criterion of achieving noise requirements by more than 80%, the results suggest the following: when no optimization measures are applied, the RMS value should be below 0.0065 λ; by employing two optimization methods simultaneously, it is possible to effectively reduce the wavefront aberration at the exit pupil of the telescope to 0.033 λ.

 

Fig. 9. The proportion of meeting the noise requirements by employing various optimization methods.

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To validate the simulation results and provide a more comprehensive understanding of the optimization effects, Fig. 10 showcases the non-geometric TTL coupling noise within the internal components of the telescope. The figure presents different proportions of ${P_{b2}}$ and utilizes various optimization methods, all while maintaining a constant wavefront aberration of λ/30. In Fig. 10(a), the wavefront aberration Zernike components are evenly distributed, resulting in an RMS wavefront aberration of RMS = 0.033 λ. In this particular scenario, without any optimization (w = 2.0 mm), the maximum level of non-geometric TTL coupling noise within the internal components of the telescope amounts to 15 pm/Hz^1/2, as depicted in Fig. 10(b). By optimizing the waist radius of the local laser beam (setting w = 0.5 mm), the maximum level of non-geometric TTL coupling noise within the internal components of the telescope is significantly reduced to 2 pm/Hz^1/2. This improvement is demonstrated in Fig. 10(c). In Fig. 10(d), a wavefront aberration is depicted with a ${P_{b2}}$ value of 99%, and an RMS wavefront aberration of RMS=λ/30. When the waist radius is set to w = 2.0 mm, the maximum level of non-geometric TTL coupling noise within the internal components of the telescope is effectively reduced to 1 pm/Hz^1/2. The corresponding outcomes are illustrated in Fig. 10(e). By simultaneously implementing both optimization methods, the maximum internal non-geometric TTL coupling noise in the telescope can be reduced to 0.2 pm/Hz^1/2, as depicted in Fig. 10(f). These results confirm the significant improvement effects of both optimization methods. It is worth noting that relying solely on a single optimization method is insufficient to meet the noise requirements in the case of RMS = 0.033 λ. Therefore, it is necessary to simultaneously optimize both the coupling aberrations and the waist radius w of the local laser beam for achieving the desired noise reduction.

 

Fig. 10. The wavefront aberration at the small pupil position of the telescope and the TTL coupling noise in four different scenarios

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5. Conclusion

The TianQin project has set a requirement for the non-geometric TTL coupling noise inside the telescope to be below $0.2\sqrt {2} $ pm/Hz^1/2. This paper presents an optical calculation model, based on the principle of interference, that has been developed to address this requirement. The fundamental requirement for all analyses is to ensure that the noise ratio exceeds 80%. The results suggest that, in the absence of any optimization, the RMS value of wavefront aberration at the small pupil of the telescope should not exceed 0.0055 λ. The demand for wavefront aberration at the small pupil of the telescope is highly stringent. Consequently, noise suppression methods have been proposed to regulate the proportion of coupled aberrations and optimize the waist radius of the beam. By adjusting the value of ${P_{b2}}$, it becomes possible to effectively decrease the coupling coefficient of noise aberrations. When setting ${P_{b2}}$ to 99%, the RMS value of wavefront aberration at the small pupil of the telescope is below 0.014 λ, which satisfies the noise requirements. By fine-tuning the waist radius (w) of the laser beam, it is necessary for the RMS value of wavefront aberration at the small pupil of the telescope to be below 0.016 λ. By employing two optimization methods simultaneously, it is possible to relax the wavefront aberration at the telescope's small pupil to 0.033 λ, thereby achieving a significantly high proportion of 91.15% that meets the noise requirements. Moreover, when the RMS is set to 0.033 λ, employing two optimization methods simultaneously can effectively reduce the non-geometric TTL coupling noise inside the telescope from 15 pm/Hz^1/2 to 0.2 pm/Hz^1/2. These research findings can serve as a valuable reference for studies focusing on the suppression of TTL coupling noise within gravitational wave detection telescopes, as well as other relevant research areas.