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Abstract
A stable and reliable support system for large aperture wedge prisms is the priority of the atmospheric dispersion corrector (ADC). The prism is not a rotationally symmetric component, and the stress distribution on large aperture wedge prisms caused by the support system is different compared with the rotationally symmetric mirror. A scheme of support forces passing through the prism center of gravity (COG) is proposed in this paper. Comparing with the scheme of support force passing through the prism geometry center of rotation (COR) under the same conditions, the root-mean-square (RMS) value of the optical surface shape error shows that the proposed scheme obtains better optical surface quality when the prism rotates from 0° to 360° under the conditions of gravity coupling at 2°C and 42°C. In addition, based on the proposed scheme, a multi-island genetic algorithm (MIGA) is used to optimize the position parameters of the supports. The results show that the RMS value of the optical surface deformation of the wedge prism decreases effectively. Under the conditions of gravity coupling at temperatures of 2°C and 42°C, the RMS value decreases from 260.7 nm to 107.8 nm with 58.6% and from 108.6 nm to 69.5 nm with 36.0%, respectively.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As the aperture of the telescope increases, the aperture of the scientific instruments it carries increases. ADC is a kind of instrument that works with spectrometers, correcting the dispersion caused by the atmosphere before the light enters the spectrometer [1,2]. ADCs have been used to successfully improve image quality in several large telescopes [3–5]. The ADC of the thirty-meter telescope (TMT) adopts the structure in which the two prisms move toward or away from each other along the optical axis to produce dispersion compensation. Furthermore, the two prisms should be able to rotate independently to track the parallax angle when the telescope pointing directions change across the sky [6]. The prisms keep the optical axis horizontal during operation.
The key components of ADC are a pair of large aperture wedge-shaped prisms. In general, the larger the aperture of the optical mirror is, the more flexible it is, so it is easily affected by working conditions such as temperature gradients, inhomogeneity of thermal properties of materials, internal stresses, forces applied by supports, acceleration, gravity, etc.
Some examples and basic design principles are presented for reliable supports of large diameter mirrors [7,8]. Topology optimization of large aperture reflective mirror always aims to reduce mirror mass and improve mirror stiffness to reduce the mirror deformation caused by gravity [9–11]. Adding a flexible structure to the supports of large aperture mirrors can also improve the surface quality of mirrors by isolating optical elements from the mechanical and thermal effects of the support system [12–14]. In the tight specification accuracy requirement of large telescope mirrors, an active support method is used by applying opposing force at the support position to correct the figure error caused by force [15–17]. Another way to improve the support effect is to optimize the support dimension and position parameters [18–20], especially for large aperture prisms with horizontal axis. When the optical axis remains horizontal, the distance between the supports and the neutral plane will greatly affect the support effects [7].
As a multidisciplinary problem, large aperture mirror support system design work is necessary to use a multidisciplinary analysis method to evaluate the performance from the starting point of mechanical design to the end point of optical design. The integrated opto-mechanical analysis method is generally used to evaluate the optical surface and support system. [21–23].
Adequate design and optimization work of the support scheme for large reflective mirrors have been performed. However, support schemes for large aperture wedge-shaped prisms are rarely reported. Reflectors can be supported at the back, while the prism, as a transmissive component, can only be supported along the outer cylinder surface [24], and the main supports are located along the radial direction. In this paper, we propose a scheme in which the radial support forces pass through the center of gravity of the prism in the principal plane. It can reduce the degradation of the optical surface by the contact forces. Based on the proposed scheme, we also perform optimization work, and the support position parameters along the optical axis are optimized to improve the optical performance. Based on the integrated optimization platform, the optimization process is executed automatically driven by the optimization algorithm, which improves the efficiency of the opto-mechanical analysis work.
2. Design and analysis of the wedge-shaped prism support scheme
2.1 Stress analysis of prisms
Since the wedge prism is not rotationally symmetric, the mass center does not coincide with the rotational center, as shown in Fig. 1.
Fig. 1. The change in the arm of force as the prism rotates: (a) start position of the prism, $\theta = 0^\circ $, (b) prism in rotation, 0 $< \theta < $90°, (c) position at which the arm of force obtains the maximum value, $\theta = 90^\circ $
When the COR does not coincide with the COG, the distance between the COG and COR generates the arm of force $l$, and gravity will apply torque ${M_z}$ to the prism, which can deteriorate the optical surface of the prism. The torque ${M_z}$ can be expressed as:
where d is the distance between the COG and the COR.$\theta$ is the angle at which the prism rotates counterclockwise from the vertical upward position.$l$ is the arm of force. When the prism rotates 90°, the force arm is the longest, and the torque reaches the maximum value.Therefore, in Fig. 1, if the support force passes through the COG, it will avoid the surface degradation caused by the coupled moment of gravity and support forces.
When the prism is supported by contact forces along its cylindrical boundary, the distribution of the forces usually depends on the specific support method to result in different deformations of the optical surface [25,26]. Regardless of the support method, the resulting boundary stresses can be decomposed into two force systems [27]. The first force system varies along the circumference; however, it is uniformly distributed across the cylindrical boundary shown in Fig. 2(a). In the main section of the prism, these normal boundary forces are transmitted through the mirror body to balance the weight of each volume element. This first force system causes the deformation of the optical surface, and the deformations can be decomposed into radial deformation ${d_r}$, tangential deformation ${d_t}$, and axial deformations ${d_a}$, as shown in Fig. 2(a) and Fig. 2(b).
In the direction perpendicular to the optical axis, there is a neutral plane to which the gravity torques of the volume elements are equal, so the neutral plane is not subjected to pressure or tension. If the material on both sides of the neutral plane is symmetrically distributed, the neutral plane can remain flat, as shown in Fig. 2(c). However, for a wedge prism, the section is not symmetrical, and the force transfer process of the first system is accompanied by bending moments. This is shown in Fig. 2(d), an element of the prism of thickness h with the transmitted stresses on the two opposite faces. The upward and downward stress ranges differ $1/2dh$, thus forming bending couple elements. Integrating the volume element over the whole prism surface produces a total bending moment $GS$. $G$ is the weight of the prism, and $S$ is the distance between the mass center and neutral plane, as shown in Fig. 2(b). The moment $GS$ is balanced by the bending moments t distributed on the cylindrical boundary. The moment $t$ is the second boundary force system. This causes the prism to bend so that the neutral plane is wavy, resulting in the displacement of the optical surface along the optical axis shown in Fig. 2(c).
The mirror has a uniform cross section, as shown with dashed line in Fig. 2(c), in which the material is symmetrically distributed on both sides of the neutral plane; then, the supports can be placed on the neutral plane or placed on the two sides of the neutral plane symmetrically. This will not cause torque in the main section, and the neutral plane will remain flat. For the prism, the material distribution on the section is not uniform, the boundary stress results in the wavy natural plane, and gravity does not coincide with the wavy natural plane. In the upper space of gravity, there is no space to put supports on the outer cylindrical surface of the prism, so we cannot place all the supports aligned with gravity in the section plane. However, we can find some positions to put the supports within a reasonable range that has relatively little effect on the optical surface deformation by optimization method.
2.2 Support scheme of the prism
To reduce the influence caused by boundary stress on the prism surface deformation, we use the COG scheme to improve the optical surface quality of the prism.
The larger one of the two prisms, as shown in Fig. 3, characterized by a diameter of 1460 mm, wedge angle of 7.6°, and weight of 560 kg, e and q mark the position of the gravity center along the z-axis and the y-axis. $p$ is the width of the thicker edges of the prism. The positive direction of the z-axis represents the direction of light. The light comes in from Optical Surface 1 (OS1) and comes out from Optical Surface 2 (OS2).
With the aim of obtaining better optical surface quality of the prism, we propose a support scheme based on the characteristics of the wedge prism, as shown in Fig. 4.
The prism is supported by six polytetrafluoroethylene (PTFE) pads, and its thermal expansion property can coordinate thermal deformation caused by the inconsistent thermal expansion rate of the prism material and the cell materials to reduce thermal stress and improve the optical performance of the prism surface. Screws are used to connect pads and prism cells, and the contact force at the support position passes through the COG rather than the COR by adjusting the directions of the axis of the screw and the axis of the threaded hole on the prism cell. The prism cell also acts as a supporting frame for the prism and the interface to the bearing assembly.
2.3 Integrated opto-mechanical analysis method
The objective of support scheme design is to make the prism have good surface quality and optical performance. The evaluation index is the deformation of the optical surface, and the starting point of ADC mechanical work is the structure design, including the scheme of supports, cell, frame and motion parts. Factors affecting the whole optical system of ADC include mechanical load, temperature gradient, inertia, pressure, vibration and so on. The design chain needs to be integrated with opto-mechanical and thermal analysis. The connection between different disciplines can be expressed by Eq. (2) to Eq. (4). Equation (2) and Eq. (3) illustrates the connection between the mechanical discipline and the thermal discipline, Eq. (4) illustrates the connection between the mechanical discipline and the optical discipline.
where ${\textbf{u}_t}$ is the node displacements caused by temperature change, ${\textbf{u}_g}$ is the node displacements caused by gravity, $\textbf{P}$ represent the node forces of discrete points on optical surfaces, $\boldsymbol{\alpha}$ is the linear expansion coefficient, $\textbf{T}$ is temperature change value, and $\textbf{K}$ is stiffness. The deformations caused by the temperature gradient and gravity are calculated and outputted by FEA software. The output node displacement data are fitted by Zernike polynomials and the rigid body displacements are removed.The whole integrated opto-mechanical and thermal analysis process is shown in Fig. 5. We build 3D models in 3D model creation software and transfer the models to finite element analysis software. Loads and constraints are applied, and the stress and displacements of the optical surface nodes are calculated by the finite element method. Then, we obtain the optical surface displacement data, input them into data process scripts, remove the rigid body displacements, and finally use Zernike polynomials to fit the optical surface. By calculating the RMS value of the fitted optical surface, the optical surface quality can be evaluated, and the support schemes can be evaluated.
2.4 Analysis result of the COG scheme
The wedge-shaped prism is subjected to statics analysis using the finite element method. The thin end of the prism along the positive direction of the Y-axis and the gravity along the negative direction of the Y-axis; the material properties involved in the scheme are shown in Table 1. In the finite element model, we set all the connections to be bonded.
Table 1. Material Properties of Prism and Supports
Figure 6(a) and Fig. 6(b) show the deformation of OS1 and OS2 along the optical axis, respectively. Both the deformation of OS1 and OS2 are severe near the upper edge of the prism where the supports are located. Figure 6(c) and Fig. 6(d) show the maximum principal stress of OS1 and OS2, respectively, indicating that the stress concentration occurs at the support position, especially in the thin edge where the supports are located. We can see that the area with a high stress value on the surface of the prism also has large surface deformation. The maximum stress on the prism surface is 0.56 MPa which is much less than the allowable stress of fused silica which is 50 MPa for pressure and 1150 MPa for tension.
Fig. 6. Finite element analysis result: (a) the deformation along the Z direction of OS1, (b) the deformation along the Z direction of OS2, (c) the maximum principal stress of OS1, (d) the maximum principal stress of OS2. Both the deformation and maximum principal stress of OS1 and OS2 are severe in the upper end of the prism, which are thinner.
Figure 7(a) and Fig. 7(b) show the fitting surfaces of OS1 and OS2 fitted by Zernike polynomials, respectively. The deformation of OS1 and OS2 is basically the same. In the thin edge of the prism, the deformation is larger, while in the thick edge of the prism, the deformation is smaller, which is consistent with the surface deformation results of the finite element analysis.
Fig. 7. Zernike fitting result, (a) Zernike fitting surface of OS1, (b) Zernike fitting surface of OS2
2.5 Analysis result of the COG scheme and the COR scheme during prism rotation
The proposed scheme of supporting forces passing through the prism COG shown in Fig. 4 can make the optical surface have better performance than that of supporting force passing through the prism COR when the prism rotates. This can be illustrated by the change in the RMS value as the prism rotation angles change from 0° to 360°.
The gravity and temperature gradient seriously influence the optical systems, especially for ADC prisms with large masses and wide ranges of operating temperatures. We study the deformation of the prism optical surface under gravity coupling with temperatures of 2°C and 42°C, which are the starting point and end point of the temperature range, respectively. As shown in Fig. 8. Figure 8(a) and Fig. 8(b) show the RMS of the prism surface of OS1 and OS2 under gravity and 2°C, the blue curves represent the results of the COG scheme, and the red curves represent the results of the COR scheme. The two curves have the same change trend with the change in the prism rotation angle. The red curves obtain the minimums near 0° and 325° and obtain the maximum near 140°, and the blue curves obtain the minimums near 0° and 345° and obtain the maximum near 170°. The red curves are below the blue curves in the full range, which means that the optical surface RMS of the COG scheme always induces a smaller surface shape error than the COR scheme. Figure 8(c) and Fig. 8(d) show the RMS of the prism surface shape error of OS1 and OS2 under gravity and 42°C. Under the same gravity and temperature conditions, regardless of the COG scheme or COR scheme, the curves of OS1 and OS2 have the same change trend, and the blue curves are always below the red curves, which also means that the COG scheme is always better than the COR scheme under gravity and 42°C conditions at the full rotation range [0°,360°]. Table 2 lists the maximum and minimum values of Fig. 8(a) to Fig. 8(d).
Fig. 8. RMS value change with the change of rotation angle, blue curve is the RMS value of scheme COG, red curve is the RMS value of scheme COR, (a) RMS value change with rotation angle under gravity and 2 °C of OS1,(b) RMS value change with rotation angle under gravity and 2 °C of OS2,(c) RMS value change with rotation angle under gravity and 42°C of OS1,(d) RMS value change with rotation angle under gravity and 42°C of OS2, RMS values of COG scheme are smaller than COR scheme.
Table 2. Maximum and minimum RMS under two conditions
The analysis results show that the proposed COG scheme induces a smaller RMS than the COR scheme under gravity and temperature conditions when the prism rotates from 0° to 360°.
The optical surfaces of the prism are affected not only by the support schemes but also by the positions of the support pads. In the support scheme of COG and COR, the position parameters of the support pads can be divided into position parameters along the radial direction and position parameters along the optical axis in the main section.
First, we fix the parameters along the radial direction, change the parameter values in the main section, and study the optical surface shape changes under the two support schemes when the prism rotates. There are three parameters in the main section, combined with the changed angles of gravity when the prism rotates. The four parameters are sampled uniformly in the parameter space, and the RMS values of the optical surface error are calculated for every sample. We obtain the RMS value under changed gravity in different directions coupled with temperature at 2°C and 42°C for OS1 and OS2, and the results are shown in Fig. 9. In Fig. 9(a) and Fig. 9(b), there are three blue points above the red points, and in Fig. 9(c) and Fig. 9(d), there are four blue points above the red points. This means that in most of the samples, the COG scheme performs better than the COR scheme under gravity in different directions coupled with 2 °C and 42 °C.
Fig. 9. The RMS value changes with the change in the rotation angle and position parameters in the main section. blue curve is the RMS value of scheme COG, red curve is the RMS value of scheme COR, (a) RMS value change with rotation angle and different main section parameters under gravity and 2 °C of OS1,(b) RMS value change with rotation angle and different main section parameters under gravity and 2 °C of OS2,(c) RMS value change with rotation angle and different main section parameters under gravity and 42°C of OS1,(d) RMS value change with rotation angle and different main section parameters under gravity and 42°C of OS2.
2.6 Optimization of the COG scheme
We can see that in Fig. 9, in the design space, within the value range of each parameter, there are some specific parameter combinations that can obtain better support effects. We can also see from Fig. 8 that under gravity coupling with 2 °C and 42 °C, with the rotation of the prism, the RMS value has the opposite trend. We would like to determine a set of parameters to minimize the RMS value at 2°C and 42 °C when the prism rotates, and we can realize the aim by an optimization method. To close the opto-mechanical and thermal integrated analysis loop, we write some scripts to deal with the data transfer problem between different analysis software in integrated analysis and add the optimization algorithm into the whole analysis loop.
A genetic algorithm (GA) is an optimization method to find the optimal solution of a problem by mimicking biological evolution [30]. It codes the individuals in the solution space of the optimization problem and then selects, crosses, and mutates the coding individuals and searches for the combination that contains the optimal solution from the new population generation after generation. The traditional genetic algorithm has some problems, such as poor local search, premature convergence and slow convergence. To improve the performance of the traditional GA, a multi-island genetic algorithm (MIGA) based on a parallel distributed genetic algorithm is proposed [31]. MIGA divides an initial population into several subpopulations, and each subpopulation is called an island. A certain percentage of individuals from different islands will be moved to other islands for population exchange to increase the number of individuals every certain algebraic to avoid local solutions and premature convergence. At the same time, the traditional GA method is applied on each island to conduct subpopulation evolution in MIGA. The flow of MIGA is shown in Fig. 10.
The optimization model is illustrated in Fig. 11. The prism structure is symmetrical along the Y axis. In the principal plane, the positions of the six supports are also symmetric with respect to the Y-axis. ${d_{1/6}}$, ${d_{2/5}}$, and ${d_{3/4}}$ are the distances from the three pairs of pads to the origin of coordinates along the optical axis, and their values are set variably. ${\beta _1}$, ${\beta _2}$, and ${\beta _3}$ are the position parameters of the pads in the principal plane, whose values are 30°, 90°, 30°. The position parameters and their value ranges are listed in Table 3.
Fig. 11. Position parameters of pads, fix the parameters in the main plane, and change the parameters in the main section.
Table 3. Value range of the input parameters
The mathematical model of the optimization problem can be expressed as:
We use the integrated optimization platform to build the entire optimization process. First, we build parametric modeling in 3D modeling software and then pass the input parameters to finite element analysis software through the platform’s integration interface. Second, the scripts written by the APDL language automatically output the optical surface node displacement data to the specified position, and then the scripts written by the data processing software read the optical surface node displacement data from the position and processed them to calculate the RMS value. These operations are carried out in a cycle driven by the optimization algorithm. In the design space, the optimal solution is automatically searched with the aim of minimizing the RMS value.
We use MIGA to set 3 islands, with 6 subpopulations on each island, and evolve 6 generations. A total of 108 calculations are run, and the optimum solution is obtained at the 94th run. In the optimization processes, the two optimization objectives are given equal weight, and the iterative processes are shown in Fig. 12. Figure 12(a) shows the iterative process with the RMS value as the objective function under gravity and 2°C. Figure 12(b) shows the iterative process with the RMS value as the objective function under gravity and 42°C.
Fig. 12. The iterative processes: (a) iterative process with the RMS value as the objective function under gravity and 2°C, (b) iterative process with the RMS value as the objective function under gravity and 42°C.
The optimum results are shown in Fig. 13. Figure 13(a) shows the pads’ initial positions, and Fig. 13(b) shows the optimized positions. The detailed values of the parameters are listed in Table 4. The RMS of the optical surface error is calculated, and the results show that the RMS value decreases from 260.7 nm to 107.8 nm under gravity and 2 °C with 58.6%, and the RMS value decreases from 108.6 nm to 69.5 nm under gravity and 42 °C with 36.0%. We can see that the optimized support positions significantly improve the support effect under both conditions.
Fig. 13. Pad positions along the optical axis in the main section of the original model and optimized model. (a) Original position, (b) optimized position.
Table 4. Comparison between optimization results and initial values
2.7 Dynamics analysis under the COG scheme
After the statics analysis of the prism assembly, it is necessary to predict its dynamic performance. The modal analysis of the prism, supports and cell based on the optimized COG scheme is carried out. The vibration modal analysis results of the first six orders are shown in Fig. 14. The first-order frequency of the prism is 69.61 Hz, and the corresponding vibration mode is the prism translation along the X axis. The second- to sixth-order frequencies are 69.68 Hz, 94.16 Hz, 121.45 Hz, 126.79 Hz and 163.27 Hz, and the corresponding vibration modes are translation along the Y axis, rotation around the Z axis, rotation around the X axis, translation along the Z axis and rotation around the Y axis. With the increase in mode order, some local responses appear in the cell in the same position from the fourth mode to the sixth mode, but they are not easily excited by the system because of their high frequency.
3. Conclusions
To meet the support requirements of a large aperture prism with a diameter of 1460 mm, we propose a COG support scheme, in which the support forces pass through the center of gravity of the prism. Then, we use the opto-mechanical and thermal integration analysis method and optimization method to evaluate and improve the optical performance.
Based on the opto-mechanical and thermal integration analysis, the support effects of the COG scheme and COR scheme are compared when the prism rotates from 0° to 360° under gravity and different temperature conditions. First, all the position parameters of the pads are fixed when the prism rotates from 0° to 360°, and the results show that the RMS value of the optical surfaces of the COG scheme is smaller than that of the COR scheme. Second, we fix the position parameters in the principal plane and change the position parameters in the main section at different rotation angles. The results show that the COG scheme still performs better than the COR scheme.
We further optimize the COG support position parameters to obtain a better support effect under the COG scheme. In the gravity and temperature environment, taking the prism optical surface minimum RMS value as the objective function, a multi-island genetic algorithm is used to optimize the position parameters of the support pads in the main section of the prism. Under the conditions of gravity coupling at temperatures of 2°C and 42°C, the RMS value decreases from 260.7 nm to 107.8 nm with 58.6% and from 108.6 nm to 69.5 nm with 36.0%, respectively. Finally, we calculate the first six vibration modes of the prism cell, and the first-order frequency is 69.61 Hz, it is within the acceptable frequency range. Taken together, both the RMS of optical surface deformation and the dynamic performance meet the current ADC conceptual design requirements. The support scheme, the opto-mechanical and thermal integration analysis method and optimization method are valuable as a reference for other large aperture optical component design work.
Funding
National Natural Science Foundation of China (12103081).
Disclosures
The authors declare no conflicts of interest.
Data availability
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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