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Abstract
Active optics is an important technology for improving the observation performance of large telescopes. For active optics with hard points, the hard points determine a reference plane for aberration correction. Relative movement to the reference plane will cause dents at the hard points, thus degrading the active correction effect. Herein, a compensation plane is proposed to solve the nonzero problem at hard points for surface detection and surface fitting. A series of experiments are designed on a 4-m thin mirror to illustrate the function of the compensation plane for active optics with hard points. The results show that the method is effective in avoiding the relative movement to the reference plane. With the compensation plane, the measured influence functions are more reasonable, and the dents after correction can be eliminated, which significantly reduces the fitting error of the 4-m thin mirror.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Active optics technology was first applied to the ESO New Technology Telescope (NTT) and has become one of the key technologies in telescopes [1]. By actively correcting the primary mirror surface, active optics can significantly improve the observation performance of large telescopes [2,3].
The active support system for the primary mirror of a large-aperture telescope can be divided into two parts: the supporting system and the positioning system [4,5]. The supporting mechanism is used to ensure the surface precision of the primary mirror shape. The positioning mechanism is used to constrain rigid-body motion of the primary mirror relative to the cell. Hard points are commonly used for positioning constraints in many active support systems of large telescopes, such as the LBT (Large Binocular Telescope) [6], Subaru telescopes [7], etc. The hard points have the characteristic of constant displacement, which determines a reference plane for aberration correction [8]. During active correction, the relative movement to the reference plane will cause dents at the hard points. These dents are high-frequency distortions that cannot be compensated easily, thus degrading the active correction effect. Moreover, in the process of surface detection and surface fitting, the reference plane cannot be discarded as the piston, tilt-x, and tilt-y aberrations of the standard Zernike distribution. Therefore, the implementation of active correction under a reference plane is a significant problem.
In this paper, a compensation plane is proposed to solve the nonzero problem at hard points for surface detection and surface fitting. A series of experiments are designed on a 4-m thin mirror to illustrate the function of the compensation plane for active support system with hard points. In Section 2, we describe the dents caused by the relative movement to the reference plane and propose a method to establish the compensation plane. In Section 3, the experimental platform of the 4-m thin mirror with hardpoints is introduced. In Section 4, we illustrate the function of the compensation plane in influence functions measurement and surface fitting, and then complete the gravity deformation correction of the 4-m thin mirror. The results demonstrate that the method is effective in avoiding the relative movement to the reference plane for active optics with hard points. With the compensation plane, the measured influence functions are more reasonable, and the dents are eliminated after correction, so that the fitting error is significantly reduced.
2. Principle of active support system
The task of an active support system for large telescope mirrors is the effective elastic removal of mirror deformations [9–11].
2.1 Active points and hard points of 4-m primary mirror
In this study, the primary mirror is a thin mirror with the diameter of 4,060 mm and thickness of 103 mm. The mirror surface is aspheric with the apical curvature radius of 12,084 mm. The material of the primary mirror is ZERODUR, with a mass of approximately 3,500 kg. As shown in Fig. 1(a), the primary mirror is supported with 90 pneumatic actuators placed in parallel and 20 actuators in lateral. A finite element model of the 4-m primary mirror was established using ANSYS. Apart from friction effects introduced by the lateral supports, the edges of the 4-m thin mirror will generally be free in the direction parallel to the optical axis. The axial support points were evenly placed on five support rings, which were arranged as close to the triangle to improve support efficiency. For the 4-m thin mirror, all 90 axial supports bear their axial gravitational loads together. The radii and supporting forces were optimized using finite element analysis (FEA).
Fig. 1. (a) Distribution of pneumatic actuators of the primary mirror. (b) Constraints of 4-m thin mirror.
There are three passive actuators in the axial direction and four passive actuators in the lateral direction, which are used as hard points to constrain the rigid-body motion of the mirror relative to the cell. The three axial hard points can constrain the following three directions (Fig. 1(b)): Δz, direction perpendicular to the middle surface; Δθx, rotation in the x-direction; and Δθy, rotation in the y-direction. Meanwhile, the lateral hard points constrain the other three degrees of freedom, denoted as Δθz, Δx, and Δy. Theoretically, three lateral hard points are sufficient. However, owing to the distribution of the lateral active points, the even placement of three lateral hard points is difficult. Therefore, a lateral positioning system with three main lateral hard points and one auxiliary hard point was designed. The function of the auxiliary hard point is to ensure that the forces on the three main hard points are uniform during the primary mirror assembly.
The structure of axial actuators is shown in Fig. 2. The actuators are push-pull, consisting of invar, electromagnet, ball decoupler, load cell, and pneumatic cylinder. The actuator load cell measures the axial force applied by the pneumatic cylinder. This force is regulated by the inner control loop by changing the cylinder pressure. The ball decoupler allows the mirror to move in the lateral and cross lateral directions without appreciably changing the load applied to the mirror. The structure of axial hard points is similar to that of the active actuators. The difference is that it has a restriction structure to fix the motion of the hard point (Fig. 2).
2.2 Basic formula
For a continuous primary mirror, the mirror deformation is considered linear with the force on the active support point [12,13]. The influence matrix was used to characterize the impact of the active forces on the mirror shape, and is determined by the surface deformation caused by a unit force exerted by each actuator on every axial support point [14,15]. Mathematically, the 4-m primary mirror active optics obeys:
where Δw is the sampling result of mirror deformation in the z-direction, which is stacked into M2×1 vectors containing the M×M samples of the degraded image; Δf is the vector of the active forces exerted by the actuators, with size N×1; N is the number of active actuators; and C is the influence matrix of the deformation in the z-direction caused by the unit force applied on the primary mirror. The active forces for correction Δw can be estimated as follows: where C+ denotes the pseudo inverse of matrix C, and $\Delta \hat{f}$ is the estimated force vector to correct surface deformation Δw. Because the positioning system of the 4-m primary mirror contains three axial hard points, it is believed that the mirror deformation at the hard points is always zero, which determines a reference plane for aberration correction [7,8]. The relative movement to the reference plane will cause dents at the hardpoints. Figure 3 shows the dents created by the relative movement to the reference plane.Fig. 3. (a) The dents created by piston movement to the reference plane. (b) The dents created by tilt-y movement to the reference plane. (c) The dents created by tilt-x movement to the reference plane.
Although the three hard points are circumferentially distributed on the primary mirror, the relative movements to reference plane are different from the piston, tilt-x and tilt-y aberrations of the standard Zernike distribution. Taking astigmatism (the 5th Zernike aberration) as an example, the aberration of one wavelength ($\lambda $=632.8 nm) is shown in Fig. 4, where the displacements at the three hard points are 279, -843, and 585 nm. There are no piston and tilt aberrations, but the displacements at the three hard points are not zero, and differ from each other.
2.3 Compensation plane for active support system with hard points
Active corrections involve two important steps: influence functions measurement and correction force calculation. To avoid relative movement to the reference plane during active correction, it is necessary to accurately calibrate the influence shape relative to the reference plane. For the active optics with hard points, the theoretical analysis of finite element shows that: the deformation magnitudes at the hard points are zero. However, when we measure the influence functions or detect the distorted surfaces, the detected surface can’t guarantee that.
A 2D case for two hard points is presented in Fig. 5 to illustrate the difference between the actual mirror surface and the detected mirror surface. Curve 1 is the cross-section through two hard points on the detected surface. Curve 2 is the cross-section through two hard points on the actual surface. The detected displacements at the hard points are s1 and s2, while the displacements of the actual mirror surface are 0. In fact, there is a piston and a tilt/tip between the two surfaces. Thus, a compensation plane is proposed to solve the nonzero problem at hard points for surface detection. After compensation, the actual mirror surface was obtained.
The compensation plane includes the piston, tilt, and tip terms. wcorrection represents the compensation plane, represented by wcorrection = p + t1x + t2y, where p is the piston term between the two surfaces, and t1 and t2 are the tilt/tip terms. Accordingly:
Subsequently, the actual mirror surface w′ can be calculated as:
Thus, w′ is the actual surface. The compensation plane is established by detected displacements at the hard points.
Similarly, when we need to obtain the surface w with the detection system, the mirror should generate the distortion surface w′ instead. In Section 4, we illustrate the function of the compensation plane in the influence functions measurement and surface fitting through experiments on the 4-m thin mirror.
3. Active correction platform
An active correction platform based on the 4-m thin mirror was established to verify the performance of active support system. The mirror shape detection system detects deformation of the mirror surface via an interferometer sensor.
All measurements of the figure were obtained through phase-shifting interferometry from the center of curvature. The mirror was positioned on the base of a vibration-isolated 12-m test tower, and the interferometer was mounted on a five-dimensional adjustable platform. The detection optical path is displayed in Fig. 6(a).
4. Active experiments on the 4-m thin primary mirror
4.1 Measurement of the actuators’ influence functions
There are 90 axial supports on the back of the 4-m thin mirror, among which the 25th, 55th, and 85th are hard points and the rest are active points. The influence functions of the primary mirror are different from those of the deformable mirror. Each influence function of the deformable mirror can be approximately expressed by a two-dimensional Gaussian model [16]. For a primary mirror, the deformation degree and the shape of the influence functions are nonuniform. In general, the unit force applied to the outer ring has a greater influence than that at the inner ring.
To reduce the effect of measurement noise, 87 active points were applied with support forces of different amplitudes. Through the repeatability experiment, the support force applied by each actuator is related to the normalized radius. Smaller forces were applied at the points with a larger normalized radius. The force distribution for the influence functions measurement on active points is shown in Fig. 7. With these variations, the influence functions vary in the range of 60–200 nm RMS (Root mean square) surface deflection. When measuring the influence functions, the 87 active supports successively apply a pull-push force to the mirror, which can remove the linear component of varying thermal deformations and misalignment of the test optics.
Part of the influence functions of the actuators are shown in Fig. 8. The red dots indicate the positions of the active points whose influence functions were measured. The black dots indicate the positions of the hard points on the mirror. The influence functions of the first column were directly measured using the detection system. The influence functions of the second column were obtained by superimposing the influence functions of the first line on the compensation plane provided in Section 2.3. The influence functions of the third column were obtained via FEA. These were normalized to a force of 1 N. With the compensation plane added, the deformations at the hard points (second column) are zero, which is consistent with the simulation results. The influence functions after compensation are similar in shape to those obtained via FEA, but differ in amplitudes. The FEA results underestimate the amplitudes, indicating that the mirror appears stiffer in the FEA than in practice. We haven’t delved into the reasons for the discrepancy in amplitudes and we will try to figure out that in the future.
Fig. 8. Influence functions of the 1st, 3rd, 5th, 9th, and 14th actuators. The red dots indicate the positions of the active points whose influence functions were measured. The black dots indicate the positions of the hard points on the mirror. The influence functions in the first column were directly measured by the detection system. The influence functions in the second column were obtained by superimposing the influence functions in the first line on the compensation plane provided in Section 2.3. The influence functions in the third column were obtained via finite element analysis (FEA).
Among the actuators, the 1st, 3rd, 5th, 9th, and 14th actuators are located at normalized radii of 0.2025, 0.3789, 0.5640, 0.7495, and 0.9394, respectively. The shapes of the influence functions differ from one another. The influence function of the 1st actuator is close to the two-dimensional Gaussian distribution. Meanwhile, the influence function of the 14th actuator is closer to the fan shape. The amplitudes of the influence functions also vary greatly, and are described as the RMS of the surface; from inside to outside, these are 1.4, 3.3, 5.3, 8.2, and 13.0 nm.
The 7th, 24th, and 54th actuators are the actuators close to the hard points. Affected by the hard points, the main influence regions of the functions are in the symmetrical region, not the region near the actuators (Fig. 9).
4.2 Fitting results for the Zernike aberrations
After obtaining the influence functions of the active points, the fitting capability of the 4-m thin mirror for standard Zernike aberrations was analyzed.
The fifth Zernike aberration is taken as an example to illustrate the elimination of dents with a compensation plane. The deformations at hard points are nonzero in the target surface, which is shown in Fig. 10(a). If the surface is directly fitted with influence functions, because the hard points cannot provide motions, there are dents at the hard points, as shown in Fig. 10(b). Therefore, the surface was firstly compensated with the plane provided in Section 2.3 to guarantee zero values at the hard points. The surface after compensation is shown in Fig. 10(c), which only differs in the piston and tilt/tip from the initial surface. Then, the surface (Fig. 10(c)) is set as the target surface. After correction, the dents at the hard points disappear (Fig. 10(d)).
Astigmatism and trefoil are the most frequently aberrations corrected by the primary mirror [17,18]. The fitting errors before and after compensation via LSE (Least-squares estimation) for astigmatism and trefoil are listed in Table 1. The fitting errors have been normalized. The results show that the fitting errors are significantly reduced with compensated target surface.
Table 1. Fitting Error for Astigmatism and Trefoil
We completed fitting experiments for astigmatism and trefoil aberration on the 4-m primary mirror. With the fitting method mentioned above, the support forces were calculated to fit the aberration of the 1/3 wavelength. Then, forces were applied to the back of the primary mirror, and the fitting results were obtained. The fitting results, force distributions, and Zernike distribution of the fitting results are presented in Fig. 11. The coefficients of the first ten Zernike modes are shown in the third column of Fig. 11. Fitting astigmatism requires only a small range of forces. Meanwhile, fitting trefoil requires a large force, and produces a small amount of astigmatism. A larger variation range of the support force is required to fit the high-order aberration [18].
4.3 Active correction of the mirror’s gravitational deformations
During the process of active optics correction, the primary mirror mainly corrects the low-order aberrations caused by gravity and temperature changes [8,13]. We completed the active correction of gravitational deformation for the 4-m thin mirror with the compensation plane.
The RMS of the mirror figure at the completion of polishing detected by the interferometer is 191.8 nm. During the correction process, we mainly correct the aberrations of astigmatism and trefoil. The correction process is shown in Fig. 12. It can be seen from the figure that after three iterative corrections, the low-order aberrations are basically corrected. The RMS of the optimized figure is 45 nm. The Zernike distributions of the initial and optimized surfaces are presented in Fig. 13. Before correction, the Zernike coefficients of astigmatism and trefoil were 0.235λ, 0.148λ, 0.006λ, and 0.018λ. After correction, the Zernike coefficients significantly reduced to 0.009λ, 0.009λ, 0.001λ, and 0.002λ.
5. Conclusion
In this study, we designed and established a series of active optics experiments based on a 4-m thin primary mirror. Firstly, because of the dents caused by hard points, we proposed a compensation plane to solve the nonzero problem at hard points for surface detection and surface fitting. Secondly, the influence functions of the active points were measured. The influence functions are more reasonable after compensation. Then, we illustrated the function of the compensation plane for fitting Zernike aberrations. With the compensation plane, the dents are eliminated, and the fitting error is significantly reduced. Finally, gravitational deformations for the 4-m thin mirror were corrected. The results show that, for active optics with hard points, the method is effective in avoiding the relative movement to the reference plane during active correction.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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