Influence of Fabrication and Assembly Errors on the Root Mean Square Surface Distortion of a 2 m Lightweight Mirror and Its Correction

Abstract

The influence of fabrication and assembly errors on the surface distortion due to gravity of a 2 m primary mirror and its correction method are presented. The effect of fabrication errors on the surface distortion is verified by Monte Carlo analysis. The results show that, within the 46.3% confidence interval, the surface accuracy root mean square (RMS) caused by fabrication errors is more than 5.0 nm (indicator requirement). The sensitivity of mirror surface accuracy to the matching relationship between the flexible support axial assembly position and the inherent properties (neutral surface and center of gravity) of the mirror were analyzed. Then, the correction principle of the RMS was proposed based on the analysis result. The surface accuracy RMS of surface gravity distortion is sensitive to fabrication and assembly errors, which can be effectively corrected using a flexible support mounting technique. This new flexible support mounting technique replaces the conventional method with flexible supports having shims so that adjustments can be made during testing to counteract the gravitational distortion of the mirror surface. Astigmatic aberration due to gravitational changes is effectively reduced by selecting a suitable thickness of shim, and the relationship between the astigmatism and the thickness of shim was investigated using the finite element analysis method. Finally, the finite element analysis results showed that the optimal surface gravity accuracy of the mirror assembly could be obtained by adjusting the shim, while the other performance of the assembly was not affected.

1. Introduction

Large-aperture primary mirrors have emerged as a popular way to meet the increasing system angular resolution requirements for space telescope systems [1]. The main task of primary mirror assembly design is to rationally design the mirror and flexible support structure to ensure that the surface accuracy meets the requirements according to the optical system design and technical specifications. The two most important inputs in the design of mirror assembly are performance indicators and constraints. The former are used to describe the performance of mirror assembly in normal operation, while the latter are used to determine the constraints of the structure, such as size, weight, and layout. Once the structure design is determined, the theoretical values of the structural parameters that meet the requirements are determined. However, fabrication and assembly errors are inevitable in a space-based telescope system. The fabrication error of the lightweight pattern at the back surface of a SiC mirror is relatively large due to inherent manufacturing limitations, and the fabrication errors are usually a few hundred microns. The influence of error on the performance of mirror assembly cannot be ignored. Slight deviation of the mirror’s dimensions leads to a change in center of gravity, especially for large-aperture mirrors. The mirror’s surface distortion cannot be as small as expected due to the combined influence of assembly and fabrication errors of the mirror. The optical performance of mirrors employed in space telescopes can be severely degraded or, worse, a new set of flexible supports could be made to compensate for the impact of the mirror’s fabrication errors on the surface accuracy if the deviation is too large to be acceptable.

In general, to minimize the surface gravity distortion when the mirror is measured horizontally, the pivot center of the flexible support is located on the neutral surface of the mirror in the direction of the mirror’s optical axis [2,3,4,5]. When the mirror has a relatively small surface curvature and a simple lightweight structure, the position of neutral surface can be calculated through empirical formula or finite element analysis. Usually, the mirror’s center of gravity is used as a reference to describe the position of the neutral surface in the direction of the mirror’s optical axis. In practical engineering, the flexible support is assembled according to the position relationship between the flexible support and the mirror’s center of gravity in the initial design. The relative position relationship between the neutral surface and the center of gravity is changed due to fabrication errors, resulting in the optimal axial position of the flexure support deviating from the initial design position in practice. The RMS (root mean square) surface accuracy will degenerate significantly when the flexible support has deviations from the optimum axial mount location. It is not possible to directly infer the optimal axial position of the flexure by the variation of the center of gravity of the actual mirror relative to the initial design. As a result, the traditional assembly method of the mirror cannot obtain the optimal surface gravity, or even exceed the requirements of the indicators. Even if the mount position of the flexible support is directly obtained by measuring the axial position of the mirror centroid, and the support processing is carried out after the mirror centroid is measured after milling, the impact of this processing cycle on the project’s progress cannot be ignored.

Much research has been devoted to investigating the influences of fabrication error, assembly error, material properties, temperature, and fabrication residual stresses on surface distortion, and their correction methods are presented in [6,7,8,9]. An adjustable bipod flexure technique for a large-aperture mirror of a space telescope has been studied by Lan [10]. The proposed flexible support can reduce the surface distortions caused by the fabrication and assembly error of the mirror assembly in horizontal optical testing. A mirror mounting technique that replaced the traditional method with bipod flexures having mechanical shims was proposed by Kihm [11], which can counteract the effect of gravitational distortion of the mirror surface caused by fabrication and assembly errors in horizontal optical testing. However, the influence of mirror fabrication and assembly errors on RMS has been observed but not studied deeply and systematically. Moreover, these studies mainly focus on the assembly of mirror supports through bipod flexure. Few studies have been carried out on large-aperture mirrors with three support points on the back.

In this paper, the influence of fabrication and assembly errors of 2 m SiC primary mirrors on surface gravity distortions was studied by Monte Carlo analysis. Furthermore, the sensitivity of mirror surface gravity accuracy to the matching relationship between the flexible support axial position and the inherent properties (neutral surface and center of gravity) of the mirror was analyzed. Then, the principle of the adjustable flexible support was proposed, based on the analysis result. We describe a new flexible support mounting technique using shims to compensate for the impact of the mirror’s fabrication errors on the surface accuracy. Even when there are inevitable fabrication and assembly errors deviating from ideal design values, the mirror’s gravitational distortion can be adjusted or minimized by replacing three shims of a suitable thickness. At the same time, the compensation method combined with simulation analysis can determine the size of the flexible support and cone sleeve after the mirror design has been completed, and it can be put into production for processing, which is conducive to shortening the development cycle of mirror-assembly.

2. Mechanical Structures of Mirrors and Mountings

A partially closed 265 kg lightweight structure SiC primary mirror with 2 m aperture and 11.5 m radius of curvature is presented in this study. Compared with the solid mirror, the lightweight ratio reached 86.8%. The mirror was supported by three flexures through the supporting holes located on its back, as shown in Figure 1a. To further reduce the self-weight surface distortion, each semi-kinematic flexure was designed to take a third of the mirror’s weight. The three Invar sleeves were bonded to the internal surface of supporting holes by using an epoxy adhesive (GHJ-01(Z)) in a 120-deg interval for improving thermal stability. The titanium alloy (TC4) flexure was attached to sleeves and the optical bench by screws, respectively. The lightweight pockets on the back of the mirror were perpendicular to the direction of gravity. Also, ground testing with the optical horizontal axes can result in less distortion than in the vertical orientation.

The optimum surface gravity accuracy of mirror assembly is shown in Figure 1b, with the optical horizontal axes. The surface accuracy RMS of gravity was 4.61 nm. Meanwhile, when the ambient temperature varies in the range of 4 °C, the surface accuracy RMS was 3.46 nm. And the RMS of resisting 0.1 mm disturbance caused by the installation surface deformation was 5.29 nm. Finally, the performance degradation of the mirror assembly under various working conditions is shown in

Table 1. The performance degradation of the mirror assembly under various working conditions.

Operating Conditions	Initial Design Result	Indicator Requirement
1 g gravity	4.61 nm	5.0 nm
4 °C thermal change	3.46 nm	5.0 nm
Forced displacement of 0.1 mm	5.29 nm	5.5 nm
Frequency	130 Hz	100 Hz

.

3. Influence of Fabrication Errors on the RMS Surface Distortion

Due to the fabrication errors of the mirror and the assembly errors of flexible supports, the mirror’s surface gravity distortion can hardly be minimized at once. The fabrication errors of SiC mirrors are in the range of 0~1/0.5 mm, which will cause a change in the center of gravity that cannot be ignored. Moreover, the error is randomly distributed and cannot be accurately estimated. The Monte Carlo method can randomly combine the effects of different errors on surface accuracy. The effect of fabrication error on the surface gravity distortion was investigated by Monte Carlo analysis. For better manufacturing and testing, the mirror is usually designed as a centrally symmetric structure, and the lightweight ribs distributed on the back have a distinct geometric distribution. Thus, the ribs can be grouped according to their geometric position characteristics and marked with the same color, as shown in Figure 2. Each rib is a separate variable in Monte Carlo analysis. The variation of the mirror panel thickness is non-uniform, considering the actual processing conditions. Due to the fabrication limit of our optical shop, the tolerance range of each parameter is shown in

Table 2. The tolerance range of each parameter.

	Structural Parameters/mm	Parameter Value	Tolerance Range
1	1#Rib group	4 mm	0–0.5 mm
2	Outer ring thickness	5 mm	0–0.5 mm
3	2#Rib group	4 mm	0–0.5 mm
4	Back panel thickness	7.5 mm	0–0.5 mm
5	3#Rib group	5 mm	0–0.5 mm
6	Supporting hole thickness	12 mm	0–0.5 mm
7	Front panel thickness	5 mm	0–1 mm
8	Front panel thickness (supporting hole)	8 mm	0–0.5 mm
9	Sidewall thickness (lifting area)	12 mm	0–0.5 mm

.

A Monte Carlo simulation was performed to verify the RMS of the surface gravity caused by the fabrication error [11]. Siemens NX 8.0 (UG8.0), HyperMesh 2022 [12], Ansys Mechanical, and Matlab 2023 were integrated into the Isight 5.8 [13] software to perform the Monte Carlo simulation, as outlined in Figure 3. The simulation proceeds as follows: the UG8.0 is responsible for building three-dimensional models, and the HyperMesh 2022 module is responsible for pre-processing the finite elements. The Ansys Mechanical module and Matlab 2023 are used for finite element analysis and surface fitting to obtain the RMS, respectively. Monte Carlo analysis is used to generate 1000 sets of random fabrication errors, and the probability distribution of RMS values is obtained as shown in Figure 4. The RMS caused by fabrication errors are all greater than 5.0 nm within the confidence interval of 46.3% (qualification rate 53.7%), and the mean of RMS is 5.37 nm, as shown in

Table 3. Geometric parameter of the lightweight mirror.

RMS-g/nm	Value/nm
Initial design	4.61
Standard Deviation	1.231
Minimum	3.972
Maximum	8.864
Mean	5.366
Probability less than upper limit 5.0 nm (qualification rate)	53.7%

.

As we all know, there are two factors that influence the surface gravity caused by fabrication error: one is that the stiffness of the mirror is changed; the other is that the fabrication error changes the mass distribution of the mirror, resulting in the change of the flexible support’s optimal axial position. However, as shown in Table 2, this means that the stiffness of the mirror will be increased due to its positive machining tolerance. Therefore, the surface degradation is mainly due to fabrication error, which results in the change of mirror mass distribution.

According to the Monte Carlo analysis results, ten groups of parameters were randomly selected from the combination of parameters, whose surface accuracy RMS exceeded the surface requirements, to study the influence of fabrication errors on the surface, centroid and neutral surface. As shown in Figure 5, the green line represents the optimal RMS at the initial design of mirror assembly. The blue line indicates the variation of RMS when the mass distribution of mirror is changed due to fabrication errors. The blue line in Figure 5 represents the RMS obtained by installing the flexible support in the initial design‘s axial support position, when fabrication errors with different parameters were introduced. The red line in Figure 5 represents the RMS obtained by changing the support position of the flexure according to the axial variation and direction of the center of gravity when different fabrication errors were introduced. The black line represents the optimal RMS obtained according to the neutral surface. The change of the two lines (red and black) means that it was difficult to obtain the optimal RMS by measuring the center of gravity’s position when assembling the flexure in practical engineering. However, the RMS obtained by installing flexible supports based on the position of the center of gravity of the reflector is close to the optimal surface shape accuracy. Therefore, based on the method of assembling flexible supports by measuring the center of gravity, even if there are inevitable fabrication errors deviating from nominal design values, the mirror’s gravitational distortion can be adjusted or minimized by the flexure mounting technique.

4. Principle and Method of Surface Distortion Correction

In general, in order to minimize the surface gravity distortion when the mirror is measured horizontally, the pivot center of the flexible support is located on the neutral surface of the mirror in the direction of the mirror’s optical axis. The neutral surface is the collection of points where the elastic bending properties of the mirror substrate and the moments due to reaction forces balance to minimize optical distortions. Since the reaction force and gravity are acting collinearly, the self-deflection of the mirror will not be introduced when the pivot center of the flexible support is aligned with the neutral surface. As usual, the large-aperture mirror and the flexible supports are designed optimally with an optimization process. However, because of the fabrication errors of the SiC mirror, misalignment between the neutral surface of the mirror and the pivot center of the flexible support would be inevitable, leading to mirror-surface distortion.

The sensitivity of surface accuracy RMS to the support position of flexible support leads to surface gravity degradation when fabrication and assembly errors are introduced. At the same time, this is the basis for reducing surface error through the mounting technique in practical engineering. The influence of the flexible support position on the RMS of gravity was studied, as shown in Figure 6. We found that the optimal RMS could be obtained by changing the flexible support position in the direction of the mirror’s optical axis when the structure of the mirror and flexible support were determined. Therefore, we propose a flexure mounting method based on a three-point back support structure. This new flexible support mounting technique replaces the conventional method with flexible supports using shims so that adjustments can be made during testing to counteract the gravitational distortion of the mirror’s surface.

The RMS can be effectively adjusted by adjusting the flexible support position according to the change in the mirror’s center of gravity under the influence of fabrication error. In the initial design, the distance between the top surface of the sleeve and the adhesive tape was L = L0, as shown in Figure 7. When the fabrication error was introduced, the optimal axial position of the flexure installation changed when the optimum surface accuracy was obtained, then L = L1. The center of gravity of the mirror was determined after the milling and grinding were completed, then L1 could initially be determined. Thereafter, it would take more than six months for the mirror to be completed, and there would be enough time to finish the processing of the invar sleeve.

Further, the RMS of gravity is accurately adjusted by the titanium alloy (TC4) mechanical shim located between the flexible support and the invar sleeve. Finally, considering the 1 mm error caused by fabrication and the 1 mm error caused by assembly, the distance between the top surface of the invar sleeve and the adhesive tape was L = L1 + 2 mm. The flexure had a two-way 2 mm adjustment in axial direction (and the flexure had a bidirectional adjustment of 2 mm), and the thickness of shim was 4 mm.

The detection process of mirror-assembly was carried out on the ground with the optical horizontal axis. Its framework is illustrated in Figure 8. Firstly, the mirror’s center of gravity was measured when the milling and grinding were complete. Then, the rationality of tolerance range of the Monte Carlo analysis was proved by the measured results of mirror parameters after processing. Secondly, the length of invar sleeve (L) was determined by finite element analysis combined with actual measurement. When the mirror, invar sleeves and TC4 flexible support were processed, the mirror assembly was ready to be assembled. Then the invar sleeve was uniformly assembled on the mirror’s bonding zone in support holes through epoxy adhesive. Despite the possible misalignment of the assembly, the flexible support missed the optimal support position as the initial design. Then, the adjustable shim and flexible support were attached to sleeves by screws, as shown in Figure 9.

Finally, to measure the optical distortion due to gravity, the mirror was measured horizontally with an optical interferometer. If the measurement results of the surface distortion cannot satisfy the design requirement, the screws for fixing shims, invar sleeve and flexures are removed, and the thickness of shim should be adjusted by grinding to improve the optical performance until the results reach the predictive accuracy requirement. The surface distortions caused by the misalignment between the flexure and the neutral surface of the mirror would then not exist. Therefore, the thickness of shim based on the optimal regulating variable Δξ 1, according to the measurement results, was determined.

5. Optical Performance

It is critical to verify the optical performance of mirror assembly on the ground before launching. The initial optical performance design of 2 m mirror assembly is presented in Section 2. In order to verify the feasibility of the adjustable flexible support with shims, a finite element simulation is carried out on the mirror assembly. The influence of the shims’ thickness on the RMS of gravity was studied, as shown in Figure 10. The shim thickness can be changed from zero, which means no shim, to 4 mm, which is the maximum thickness. When the deviation between the neutral surface of the mirror and the pivot center of flexure is 2 mm, the shim with 2 mm thickness can compensate for the surface distortion caused by the misalignment between the mirror’s center of gravity and the pivot center of the flexible support.

The Zernike term Z5 is the astigmatism in x and has a linear relationship with the shim thickness, as shown in

Table 4. Optical distortions and selected Zernike term of the mirror assembly supported by the flexure with shims.

Shim Thickness (mm)	0.5	1	2	2.5	4
PV (nm)	40.1	35.4	30.8	35.6	57.8
RMS (nm)	7.12	5.74	4.27	4.46	9.33
Z5	12.01	7.73	−0.73	−4.77	−21.78

. Table 4 summarizes the optical performance of the mirror assembly through the regulation of the shim thickness. Z5 was 12.01 when shim thickness was 0.5 mm, Z5 was 0.73 nm when shim thickness was 2 mm, Z5 was −4.77 when shim thickness was 2.5 mm, and Z5 was −21.78 nm when shim thickness was 4 mm. Figure 11 shows that the astigmatism was the main component of the optical distortions of the mirror when the misalignment occurred. With the regulation of the shim thickness, the astigmatism weakened gradually. When the regulating variable of the shim thickness reached 2 mm, it showed that the Zernike fitting surface deformation graphs were smoother in Figure 11c, and the absolute value of astigmatism decreased from −21.78 to 0.73. This relation can be used as a reference table for finding optimum shim thickness to minimize the mirror’s distortion due to gravity. The minimum surface distortion due to gravity in Figure 10 depended on the mounting configuration with flexible support. Finally, the finite element analysis results showed that the optimal surface gravity could be obtained by adjusting the shim thickness, while other assembly performance was not affected, as shown in

Table 5. Finite element analysis results of shim effect on mirror assembly performance.

Operating Conditions	Without Shims	With Shims	Constraints
1 g gravity	4.61 nm	4.57 nm	5.0 nm
4 °C temperature change	3.46 nm	3.50 nm	5.0 nm
Forced displacement of 0.1 mm	5.29 nm	5.28 nm	5.5 nm
Frequency	123.6 Hz	123.6 Hz	100 Hz

. On the premise that the relative position between the pivot center of the flexible support and the neutral surface of the mirror remained unchanged, there was only a slight difference in performance between components with and without gaskets. It can be seen that the shim itself had no negative impact on the performance of the mirror assembly, as the optimal surface accuracy could be obtained by increasing the adjustment method of the shim. Compared to the initial design, the frequency of the mirror assembly decreased slightly due to fabrication errors, while the quality of the mirror increased.

6. Conclusions

An adjustable flexible support with shims for a 2 m SiC mirror of a space camera was presented in this paper with the aim of decreasing optical distortion caused by the fabrication errors of the mirror assembly due to gravity. The influences of fabrication errors on the surface distortion due to gravity were verified by Monte Carlo analysis. The results showed that the influence of fabrication and assembly errors on surface accuracy could not be ignored. Through finite element analysis, the positional relationship between the mirror’s center of gravity and the neutral surface in the initial design was investigated. Then, the principle of the adjustable flexible support was proposed, based on the analysis result. Optical distortion due to gravity, which is mostly astigmatism, was reduced effectively by adjusting the shim thickness. The adjustable flexible support with shims not only fully considers the optical performances of the mirror’s assembly, but also avoids multiple physical testings, reduces time and cost consumption, and offers a good reference for structural design in moving opto-mechanical systems.