Abstract
As a critical payload of the gravitational wave detection interferometry system, the tilt-to-length (TTL) noise has a significant influence on the detection accuracy of the interferometry system. The non-geometric TTL (NG-TTL) noise, which is the main component of the TTL noise within the telescope, is closely related to the sensitive aberrations of the system’s small pupil. Due to the different environments of the earth and space, structural deformations of the telescope can cause significantly unfavorable changes in the sensitive aberrations at the small pupil. And its negative influence must be considered carefully during the whole structural design phase of the gravitational wave telescope. In this paper, the theoretical model of the NG-TTL noise within the telescope based on the Fringe Zernike polynomials has been summarized. An evaluation function of the NG-TTL noise within the telescope was proposed. A desensitization design method for the high-stability structure of the gravitational wave telescope was developed. The results show that under the same disturbances, the NG-TTL noise within the telescope was reduced to 0.1 pm·Hz− 1/2 from 2 pm·Hz− 1/2, taking the primary mirror (PM) as an example.
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Introduction
In 2016, LIGO announced the detection of gravitational waves, marking the beginning of the era of gravitational wave for humanity1. Space-based gravitational wave detection programs can effectively avoid the impact of ground vibrations and gravity gradient noise. These programs focus on the low-frequency band (0.1 mHz ~ 1 Hz), covering various sources such as supermassive black hole binaries. They complement well with ground-based gravitational wave detection. TianQin and LISA are representatives of the space gravitational wave detection mission in geocentric orbit and heliocentric orbit, respectively2,3. These detectors are composed of an equilateral triangle formation of three drag-free satellites. Each satellite contains two test masses. The high-precision inter-satellite laser interferometry system records the distance changes between the test masses in different satellites, thereby detecting gravitational wave signals. TianQin employs a truss-based telescope structural design4. Therefore, a stable structure design is necessary to minimize the thermal deformation of the telescope structure.
The telescope is a key component of the space gravitational wave detection interferometry system5,6. Its function is to achieve the laser beam expansion and contraction of inter-satellite lasers. Due to the influence of non-conservative forces in space, the spacecraft will experience jitter. Consequently, the received laser beam passing through the telescope will also tilt. The optical path error introduced into the interferometric measurement due to this reason is referred to as TTL noise7.
Research indicates that TTL noise is closely related to the wavefront aberration RMS value of the telescope system. Since 1998, Caldwell M, Bender P L, and others have established an initial analytical model of TTL noise based on primary aberrations8,9. Based on this foundation, C P Sasso et al. have conducted studies using a Gaussian beam model10,11,12, while Zhao et al. introduced higher-order Zernike aberrations into the model7. The wavefront aberration RMS value requirement of the telescope is increased from λ/30 to λ/500, which is not possible with the most advanced technology13,14. Without increasing the wavefront aberration requirements, other researchers have begun to seek methods to suppress TTL noise. Zhao et al. analytically derived that the coupling of TTL noise and aberrations, especially coma and clover aberrations, can be partially offset by appropriate transverse displacement of the interference beam relative to the detector14. HONGAN LIN et al. investigated the theory associated with the cancellation and superposition after the coupling of optical aberration with TTL noise and relaxed the wavefront aberration RMS value to λ/2015. Chen et al. evaluated the impact of various primary aberrations on TTL noise16. By suppressing sensitive aberrations such as defocus and astigmatism, they were able to effectively reduce the telescope’s TTL noise. A. Weaver et al. analyzed the effects of surface shape errors and pointing jitter on TTL, expressing the TTL coupling noise coefficient as a first-order function of the Zernike terms in the WEF, enabling rapid analysis of TTL noise17,18,19. In 2023, WEN TONG FAN et al. established an optimization model for TTL noise within the telescope. By reducing the proportion of sensitive aberrations, they increased the probability of the wavefront aberration in the same system meeting the noise requirements20. Currently, research on TTL noise mainly focuses on optical design, there have been no published studies on the impact of telescope structural deformation on TTL noise.
The wavefront aberration of the orbiting telescope system is not only related to the optical design, but also to the structural design. Currently, the design of the telescope’s support structure primarily requires an root mean square (RMS) value of λ/3021,22,23,24. No research has been found that specifically addresses the changes in TTL noise caused by structural deformation. In this paper, the influence of mirror shape variation and rigid body displacement on the sensitivity aberration at the small pupil is analyzed. A sensitivity matrix of TTL noise within the telescope is established. Using this as an evaluation function, the support structure was optimized. A primary mirror assembly with low TTL noise is designed. This provides valuable insights for the suppression of TTL noise and the design of telescope structures in gravitational wave detection.
Theoretical models of optical systems
TTL noise theoretical model
The orientation of the spacecraft slowly deteriorate due to factors such as gravitational perturbations from celestial bodies. This causes a change in the direction of the light beam received by the telescope Consequently, it results in changes of the optical path within the telescope, which constitutes the TTL noise within the telescope. The principle of TTL noise generation within the telescope is shown in Fig. 1. The direction of the received beam changes, resulting in a variation of the chief ray’s path transmitted through the telescope, which is called the geometric TTL coupled noise inside the telescope. Simultaneously, beam pointing jitter will cause the change of the wavefront aberrations at the small pupil. Due to interference between the local reference beam and received beams with different wavefront aberrations, the phase information measured by the QPD (Quadrant Photodiode) is different, which is referred to as NG-TTL noise within the telescope. At present, TianQin’s index for these two noises are both \(\:\text{2}\sqrt{\text{2}}\) pm·Hz− 1/2.
The laser emitted from the remote spacecraft’s telescope, after being collimated and expanded, is received by the local spacecraft after traveling 100,000 km. The beam arriving at the large pupil of the telescope can be approximated as a flat-top beam. After being focused by the telescope, it becomes a flat-top beam with wavefront aberrations at the small pupil position. The measurement beam with a certain wavefront aberration can be represented as:
In the equation, k is the wave number, k = 2π/λ; W(ρ,θ) is the wavefront aberration of the measurement beam at the small pupil, represented in the form of Zernike polynomials;
In the equation, n represents the order of the Zernike polynomial; Ai represents the coefficient of the polynomial; Zi(ρ,θ) is the expression of the Zernike polynomial.
In order to facilitate the correlation between the simulation analysis results and the actual system detection results, the cosine and sine terms of the Zernike aberrations in Eq. 2 are combined. The wavefront aberration of the telescope’s small pupil is represented by the combined 22 Zernike aberration terms.
The local reference Gaussian beam can be expressed as:
In the equation, P0 is the output laser power, ω0 is the waist radius of the outgoing laser beam.
The measurement beam with wavefront aberration interferes with the local reference beam at the position of the detector, and the phase information can be calculated through complex amplitude. Assuming the measurement beam is aligned with the center of the reference beam, the complex amplitude of the interfered beam can be represented as:
The phase of the interference light field can be expressed as φ = arg(E). Phase information can be converted into the measured optical path difference20,26:
In the case of an incident light pointing deviation of α, the NG-TTL noise δZNG within the telescope is expressed as:
In the equation, β represents the angular deviation of the outgoing light from the telescope, which is related to the incident angle α and the telescope magnification M, with β = Mα.
Telescope optical system
The off-axis four-mirror optical design is the baseline of the TianQin’s telescope. The primary mirror is a paraboloid, the secondary mirror is a hyperboloid, and both the tertiary and quaternary mirrors are spherical. Each of the four designed mirrors has a certain degree of off-axis displacement, with specific parameters shown in Table 125.
As shown in Fig. 2a, the current wavefront aberration of the telescope’s small pupil is 0.0035λ (λ = 1064 nm) As shown in Fig. 2b, the maximum NG-TTL noise within the telescope is 0.15 pm/Hz1/2, meeting TianQin’s requirement of 0.2\(\:\sqrt{\text{2\:}}\) pm/Hz1/2.
In the telescope system, the primary mirror has the largest volume and the most complex support structure. The following will analyze the impact of telescope structural deformation on the TTL noise within the telescope using the primary mirror as the research subject. The vertex position of the primary mirror is taken as the coordinate origin, the optical axis direction is the Z axis, and the off-axis direction of the primary mirror is the Y axis. The coordinate system is established by the right hand rule, as shown in Fig. 3.
Suppression methods
Suppression method of NG-TTL coupled noise within the telescope
The NG-TTL noise within the telescope is related to the wavefront aberration at the small pupil and pointing jitter. As shown in Fig. 4, previous research has found that defocus, astigmatism, primary spherical aberration, and secondary astigmatism contribute the most to NG-TTL noise26.
To establish the relationship between the wavefront aberration at the small pupil and the NG-TTL noise within the telescope, the following function is constructed.
In the equation, CNG27 is the coefficient matrix, with units of pm Hz-1/2 nm-1/2.
Zpupil is the sensitive aberration at the small pupil.
In the off-axis four-mirror telescope, variations of the surface shape and rigid body displacement of all mirrors can affect the wavefront aberration of the telescope’s small pupil. The surface shape variations and system wavefront aberration can both be represented in the form of Zernike polynomials. The deformation of the telescope structure due to gravity and temperature is very small, with rigid body displacements on the order of microns or micro radians (µm, µrad) and surface shape changes on the order of nanometers (nm). It can be assumed that the small pupil wavefront aberration is related to the rigid body displacements and surface shape variations through a polynomial relationship. Thus, the wavefront aberration of the telescope’s small pupil can be expressed as:
Where Z1pupil represents the Fringe Zernike coefficients of the wavefront aberration at the small pupil position after structural deformation; Z0pupil represents the Fringe Zernike coefficients of the wavefront aberration at the small pupil in the ideal state; Cpupil represents the transfer coefficient matrix from structural deformation to the wavefront aberration of the telescope’s small pupil; Rmirro represents the rigid body displacement of mirror; Wmirro represents the surface shape variations of mirror.
In the equation, DX, DY, and DZ represent the rigid body displacements of the mirror in three translational directions, while RX, RY, and RZ represent the rigid body displacements in three rotational directions.
The final sensitive aberration at the telescope’s small pupil position can be expressed as:
The evaluation function for the NG-TTL noise of the gravitational wave telescope can be defined as:
In the equation, fi indicates the influence of the structural deformation of the i-th mirror on the final sensitive aberration at the telescope’s small pupil position.
In Zemax software, different rigid body displacements and surface shape variations are applied to the primary mirror, and the transfer coefficient matrix Cpupil is calculated by analyzing the variations of sensitive aberration at the small pupil. The rigid body displacement of the primary mirror affects the wavefront aberration at the small pupil position. Rigid body displacements of 1 μm (µrad) were applied to the primary mirror in six degrees of freedom, respectively. Figure 5a,b,c,d show the variations of defocus, astigmatism, primary spherical aberration, and secondary astigmatism with the rigid body displacement. By analyzing Fig. 5a,c, it can be seen that the defocus and primary spherical aberration at the small pupil are mainly related to the Y-direction translation (DY), Z-direction translation (DZ), and X-direction rotation (RX) of the primary mirror. And all exhibit linear relationships. By analyzing Fig. 5,b,d, it can be seen that astigmatism and secondary astigmatism at the small pupil are related to rigid body displacements in five directions except for Z-direction rotation (RZ). And they are linearly related to DY, DZ, and RX, and proportional to the square of X-direction translation (DX) and Y-direction rotation (RY). Through the analysis of Fig. 5, it is evident that the impact of the primary mirror’s rigid body displacement on defocus and astigmatism in the sensitive aberrations is far greater than that on primary spherical aberration and secondary astigmatism.
The transfer coefficient matrix from the primary mirror’s rigid body displacement to the wavefront aberration at the small pupil position is shown in Table 2. The transfer coefficient matrix obtained is substituted into Eq. 11, resulting in the conclusion: The change in the NG-TTL noise of the system caused by a 1 μm (µrad) rigid body displacement of the primary mirror is on the order of 0.1 pm·Hz-1/2.
The surface shape variations of the primary mirror can also affect the wavefront aberrations at the small pupil. Figure 6a,b,c,d respectively show the impact of the primary mirror’s 21st-order fitting aberrations on the defocus, astigmatism, first-order spherical aberration, and second-order astigmatism at the small pupil position. By analyzing Fig. 6a, it can be seen that the defocus at the small pupil is mainly related to the defocus, first-order spherical aberration, and second-order spherical aberration among the high-order spherical aberrations in the primary mirror’s surface shape fitting aberrations. By analyzing Fig. 6b, it can be found that the astigmatism at the small pupil is mainly related to astigmatism, first-order astigmatism, and second-order astigmatism among the high-order astigmatisms in the primary mirror’s surface shape-fitting aberrations. As shown in Fig. 6c, the primary spherical aberration at the exit pupil position is mainly influenced by the higher-order spherical aberration in the primary mirror’s surface shape-fitting aberrations. As shown in Fig. 6d, the secondary astigmatism at the exit pupil position is primarily affected by the higher-order astigmatism. Subsequent analysis will focus on the spherical aberrations and astigmatism aberrations in the primary mirror’s surface shape-fitting aberrations.
The transfer coefficient matrix from the primary mirror’s surface shape variation to the wavefront aberration at the small pupil position is shown in Table 3. The transfer coefficient matrix obtained is substituted into Eq. 11, resulting in the conclusion: The NG-TTL noise variation caused by a 1 nm surface shape variation of the primary mirror is on the order of 0.1 pm·Hz-1/2. This is in the same order as the NG-TTL noise impact caused by rigid body displacement.
Under the same environmental disturbances, changing the parameters of the support structure can control the rigid body displacement and surface shape variations of the mirror. Different rigid body displacements and surface shape variations of the mirror will result in different wavefront aberrations at the small pupil, which correspond to different NG-TTL noises within the telescope. Therefore, by adjusting the parameters of the mirror support structure, the NG-TTL noise within the telescope can be controlled.
Desensitization design method flow
Based on the analysis in Section 3.1, a design method for structure of a gravitational wave detection telescope with low TTL noise sensitivity is proposed. As shown in Fig. 7, the design process consists of four main steps:
-
(1)
Construct the initial model of the support structure component according to the design constraints.
-
(2)
RMS optimization. According to the error distribution, the tolerance requirements for each mirror component are obtained. Based on the working environment requirements of the structure, the performance parameters of the designed support structure are calculated using finite element software. By optimizing the structural parameters, a support structure design scheme that meets the system’s RMS requirements is obtained.
-
(3)
TTL sensitive parameter calculation. According to Section 3.1, the transfer coefficient matrix from rigid body displacement and surface shape variations to NG-TTL noise is obtained. Establish a sensitivity evaluation function for NG-TTL noise. Through experimental design, complete the sensitivity analysis of the evaluation function with respect to various structural parameters.
-
(4)
Noise sensitivity optimization. According to the TTL sensitivity function established above, the size parameters of the support structure are optimized and iterated, and the structural design scheme that can meet the performance requirements is finally obtained.
Optimization results
The primary mirror has a diameter of 330 mm and is supported by three points on its back. The preliminary design of the primary mirror assembly is shown in Fig. 8, which includes a lightweight primary mirror, a backplate, three flexible supports, and three sleeves. Table 4 provides the main material parameters for the primary mirror assembly.
Among the three flexible supports, those at the thicker end of the primary mirror have identical structures. Their installation angles(Φ) are symmetric relative to the center line. The primary parameters of the flexure mainly include the cutting width (c), height (h), interval (k), distance (l) from the cutting to the top, the wall thickness (bh) of the flexible support structure, and the length (L). The detailed geometric parameters and analysis outcomes for the supporting structure are presented in Table 5. The conditions for the static analysis of the primary mirror are X-direction gravity and a uniform temperature rise of 5 K.
The parameters in the first row of Table 5 represent the dimensions and installation parameters of the flexres. The unit of phi is °, while the remaining parameters are measured in mm. The parameter subscripts indicate different flexures. As shown in Fig. 8, Flexure 1 is installed on the thinner side of the primary mirror, while Flexures 2 and 3, which have the same parameters, are paired and installed on the thicker side of the primary mirror.
In the ISIGHT software, an optimal Latin hypercube experimental design method was used to complete the sensitivity analysis of the noise evaluation function concerning various structural parameters. As shown in Fig. 9, blue and red represent the positive and negative effects of structural parameters on the noise evaluation function, respectively. The results show that Φ, c1, l1, k1, k2, and h2 have the greatest impact on the system’s NG-TTL noise. and these parameters should be the focus of subsequent optimization processes.
Under gravity and temperature conditions, the structural parameters of the flexible support and the backplate structure are optimized with the NG-TTL noise evaluation function as the objective. When optimizing the primary mirror structure, it is necessary to control the surface shape changes when gravity acts on the primary mirror in three different directions. The optimization problem can be expressed as:
In the equation, xn represents the structure parameter; wfe_gx, wfe_gy, and wfe_gz represent surface shape variation of the primary mirror under gravity in the X, Y, and Z directions, respectively; on is the minimum value of each design parameter; un is the maximum value of each design parameter.
The optimization problem was solved using the Pointer optimizer in Isight. The optimization process involved nearly 200 iterations, and the iteration curve of the objective function is shown in Fig. 10. The structural parameters and analysis results of the new primary mirror components are shown in Table 6.
A comparison of the telescope system performance before and after primary mirror optimization is shown: (a), (b), and (c) respectively represent the surface shape variation of the primary mirror, the telescope wavefront aberration, and the NG-TTL noise before optimization; (d), (e), and (f) represent the corresponding parameters after optimization. The red line represents is contour line where the NG-TTL coupling noise is zero.
Figure 11 shows the comparison of fitting aberrations of various orders of the telescope’s small pupil before and after optimization. The wavefront aberrations of the telescope system after optimization have significantly improved compared to the initial structure. Specifically, the defocus has decreased from − 0.0482λ to 0.0053λ, the astigmatism has decreased from 0.0605λ to 0.0091λ, the spherical aberration has decreased from − 0.0077λ to 0.0011λ, and the coma has decreased from 0.0078λ to 0.0013λ. Figure 12 shows that, after optimization, the RMS value of the primary mirror surface shape variations decreased from 9.43 nm to 0.88 nm, and the wavefront aberration at the pupil position decreased from 0.039λ to 0.008λ. After optimization, the maximum NG-TTL noise within the telescope is 0.1 pm·Hz-1/2, which is significantly lower than the previous value of 2 pm·Hz-1/2.
Conclusion
TTL noise is a very important noise in space gravitational wave detection. It is closely related to the optical design and structural design of the telescope system. Given the optical design, reducing the impact of structural deformation caused by environmental disturbances on the telescope’s TTL noise is of great significance for gravitational wave detection. This paper established a suppression model for the TTL noise within the telescope, simulating the effects of primary mirror shape variations and rigid body displacements on TTL noise. The research shows that structural deformations significantly impact the system’s NG-TTL noise. Under gravitational and thermal conditions, the primary mirror support structure was redesigned to minimize the telescope’s NG-TTL noise. The optimized NG-TTL noise within the telescope was reduced to 0.1 pm·Hz-1/2 from 2 pm·Hz-1/2. Ensuring that the system’s output wavefront meets λ/30, further reductions in TTL noise can be achieved through the optimization of mirror support structure parameters, providing a valuable reference for subsequent telescope support structure design optimizations.
Data availability
The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.
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S.F., W.F., B.L. and R.Z. planned and carried out the computational modeling and simulations. S.F., Y.Y. and L.F. wrote the main manuscript text. H.H., J.L. and Y.C. drafted the manuscript and designed the figures. All authors reviewed the manuscript.
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Fang, S., Fan, W., Zhang, R. et al. Design method for the structure of a gravitational wave detection telescope with low TTL noise. Sci Rep 14, 29575 (2024). https://doi.org/10.1038/s41598-024-80692-9
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DOI: https://doi.org/10.1038/s41598-024-80692-9