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Curr. Opt. Photon. 2022; 6(3): 236-243

Published online June 25, 2022 https://doi.org/10.3807/COPP.2022.6.3.236

Copyright © Optical Society of Korea.

Opto-mechanical Design of Monocrystalline Silicon Mirror for a Reflective Imaging Optical System

Xiaofeng Liu1 , Xin Zhang1, Fuxiang Tian2

1State Key Laboratory of Optics System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China
2Space Optics Department Ⅱ, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

Corresponding author: liuxiaofengshen@163.com, ORCID 0000-0002-7940-2357

Received: December 27, 2021; Revised: April 7, 2022; Accepted: April 19, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Monocrystalline silicon has excellent properties, but it is difficult to design and manufacture siliconbased mirrors that can meet engineering applications because of its hard and brittle properties. This paper used monocrystalline silicon as the main mirror material in an imaging system to carry out a feasibility study. The lightweight design of the mirror is completed by the method of center support and edge cutting. The support structure of the mirror was designed to meet the conditions of wide temperature applications. Isight software was used to optimize the feasibility sample, and the optimized results are that the root mean square error of the mirror surface is 3.6 nm, the rigid body displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″ under the conditions of a temperature of Δ20 °C and a gravity load of 1 g. The optimized result show that the silicon-based mirror developed in this paper can meet the requirements of engineering applications. This research on silicon-based mirrors can provide guidance for the application of other silicon-based mirrors.

Keywords: Imaging optical system, Mirror, Monocrystalline silicon, Silicon-based mirror

OCIS codes: (000.3110) Instruments, apparatus, and components common to the sciences; (350.4600) Optical engineering

In the application of a reflective imaging optical system, it has always been the goal of opto-mechanical engineers to find a kind of mirror material that is low cost and has excellent performance and a short processing time. However, these three attributes are often not found together, and how to balance the relationship between material properties and processing cycle is very important and difficult work.

The mirror materials commonly used are SiC, ultra-low expansion glass (ULE), zerodur, fused quartz, beryllium, aluminum, etc. Silicon carbide has a higher specific hardness than other materials, but the silicon carbide mirror manufacturing process is complicated, and a series of processes such as blank sintering, back lightweight processing, aspheric processing, modification, and aspheric polishing are required. Each link may decide the success or failure of the whole mirror. In addition, the manufacturing cycle of a silicon carbide mirror is extremely long, ranging from three months to several years. Zerodur, fused quartz and ULE materials have low specific hardness and poor thermal conductivity. Beryllium aluminum alloy also has high specific hardness, but it is expensive and toxic, which makes the manufacturing cost of a beryllium aluminum alloy mirror very high. Due to the high temperature expansion coefficient of aluminum, an aluminum mirror does not have good temperature stability.

With the rise of high-precision turning and polishing technology, the application of monocrystalline silicon materials has been given more and more attention over the years. Monocrystalline silicon material has low density, high homogeneity, no internal stress, and high thermal conductivity. The single-point diamond turning process can quickly obtain a high-precision monocrystalline silicon mirror, and its manufacturing cost is far lower than a silicon carbide mirror and beryllium aluminum alloy mirror. Also, the specific hardness and thermal conductivity are better than fused quartz, zerodur and ULE glass mirrors, and temperature stability is better than aluminum mirrors.

Although monocrystalline silicon has excellent properties, it is very difficult to design and manufacture silicon-based mirrors that can meet actual needs due to the hard and brittle characteristics of monocrystalline silicon materials. It is seldom selected as the preferred material in engineering applications. In order to give full play to the performance advantages of monocrystalline silicon mirrors and accumulate engineering experience, the application of φ200 mm diameter monocrystalline silicon mirrors was studied in this paper. An objective evaluation of silicon mirrors and other common mirrors is given in this paper. The lightweight design of the mirror and the optimization of the support structure are completed. The results show that the application of a monocrystalline silicon mirror in imaging optical system is feasible.

2.1. Optical System

The optical system consists of a Schmidt-Cassegrain and corrective mirror group, which consists of a primary mirror, secondary mirror, Schmidt board and corrective lens group. The spherical aberration is corrected by the Schmidt plate, primary and secondary mirrors, and other off-axis aberrations are corrected by the rectification mirror group. The structural form of the optical system is shown in Fig. 1.

Figure 1.The structural form of the optical system.

2.2. Material Selection

The optical system has extremely high requirements for the accuracy and surface and position of the mirror, which should have good stability to ensure the imaging performance of the system. The mechanical and thermal stability of the mirror play an extremely important role in the selection of materials. In addition, cycle, cost and risk are also not negligible parts.

Newswander et al. [15] proposed a relatively objective material selection evaluation method. The specific indicators that affect the selection of materials are divided into five categories, which are optical performance, hardness and mass, thermal stability, metering structure and programmatic. Each indicator is assigned a different weight according to its importance. All indicators are multiplied by weights and then summed to obtain the overall evaluation value of the material. A single indicator also contains different elements, and the evaluation method of a single indicator is similar to the overall evaluation.

According to the actual needs of the engineering project, the factors that need to be considered in the selection of mirror materials in this article mainly include self-weight deformation, dynamic characteristics, thermal stability, thermal distortion, optical performance, and programmatic. The self-weight deformation is inversely proportional to the specific hardness E/ρ of the material, E is the Young’s modulus of elasticity, and ρ is the density. The thermal stability of the material is proportional to the expansion coefficient α; the dynamic characteristics are proportional to (E/ρ) (0.5) Thermal distortion is proportional to the thermal distortion factor α/λ, λ is the thermal conductivity. Silicon carbide, beryllium aluminum alloy and aluminum cannot obtain good optical surfaces by themselves, so the surface needs to be modified, and it is easy to cause a bimetal effect in the modified layer. Optical performance is measured by bimetal deformation, which is proportional to (Ef df / Es ds)(αs − αf )∆T [6]. ∆T is temperature difference. Ef, Es are the Young’s modulus of the modified layer and the substrate, respectively; αf, αs are the linear expansion coefficients of the modified layer and the substrate. df, ds are the thickness of the modified layer and the substrate, respectively. Programmatic is reflected by the processing time and cost. Common material parameters of mirrors are shown in Table 1.

TABLE 1 Mirror material properties at room temperature

ParametersSiCBerylliumSiAlULEFused QuartzZerodurAstrosital
ρ (103 Kg/m3)3.052.12.32.72.22.22.52.46
E (GPa)4002301576967709291
λ (W/mK)1851951692201.31.381.21.18
α (10−6 K−1)2.514.52.623.90.030.550.10.15
E/ρ (107 Nm/kg)1310.96.82.73.13.23.683.69
α/λ (10−8 m/W)1.47.41.510.82.3408.312.7


Programmatic is an index that is difficult to quantify. According to past project experience, silicon carbide is used as a reference basis, and the processing cycle of silicon carbide is about 1.5 times that of glass materials such as ULE, fused quartz, Zerodur and Astrosital, four times that of beryllium aluminum, four times that of silicon, and three times that of aluminum. The cost is two times that of glass materials, one time that of beryllium aluminum alloy, three times that of silicon, and two times that of aluminum.

The diameter of the mirror studied in this paper is 200 mm, and the weight of each element is equal. Comprehensive evaluation results are shown in Fig. 2. Single crystal silicon has the highest score. It was selected as the material of the mirror.

Figure 2.Category scores and combined scores.

2.3. Lightweight Design

A reasonable lightweight design can ensure that the mirror has sufficient rigidity under the premise of meeting the requirements of quality constraints.

For the lightweight design of the mirror, the mirror thickness ratio needs to be determined first. Yoder [7] gave an empirical formula for calculating the thickness ratio of cylindrical mirrors:

δ=3ρgdr2D2256E

δ is the self-weight deformation; g is gravitational acceleration; dr is the mirror thickness ratio; D is the mirror diameter. According to the empirical formula, the thickness ratio of different material mirrors varies somewhat, and the value is generally between 1/5 and 1/12. According to the deformation requirements of the mirror, the thickness of the mirror is calculated to be about 24 mm.

The commonly used support methods for small and medium-diameter mirrors are three-point back support, back central support, peripheral bonding, and axial crimping. The back three-point support is the most commonly used form of support for mirrors, but the process is complicated, and the space occupied by the three-point support structure is very unfavorable to the structural layout of the lens group. Peripheral bonding and axial crimping require a certain amount of space around the mirror, which is very difficult for the light weight of the mirror and support structure. The back center support has a simple structure and takes up a small space, which is helpful for the lightweight design of the mirror and support structure. Therefore, this study chose the back center support program. For the central support, the edge of the mirror can be regarded as the cantilever end of the cantilever beam. The lightweight design idea of the central support should be to reduce the structural quality of the edge of the mirror, and balance the axial and radial stiffness of the mirror. The weight reduction scheme with a lightweight slot on the back is prone to edge breakage, forming microcracks and reducing the strength of the mirror. Considered comprehensively, the lightweight scheme chooses the form of a solid mirror with a cone at the back of the mirror. The lightweight model is shown in Fig. 3.

Figure 3.Lightweight mirror.

As shown in Fig. 3, the conical angle θ determines the weight of the mirror and the root mean square (RMS) error of the surface shape. Constrained by the size of the structure, θ ranges from 5° to 33°. As θ increases, The RMS of the mirror first increases, reaches a maximum value at 25°, and then decreases, reaches the bottom at 30°, and then increases again. The design goal of the mirror is to be as light as possible on the basis of satisfying RMS. The RMS of a single lens is not greater than 4 nm, and the mass is not greater than 1.2 kg. From the above analysis, it can be seen that the optimized result of θ is 30°, the RMS of the mirror is 4.0, and the mass is 1.03 kg.

There are processing stress, assembly stress and thermal stress in the process of processing and assembly of mirror components, which would lead to the deformation of the mirror, and then deformation would cause the deterioration of optical imaging quality if it was not well compensated or removed. Therefore, effective heat treatment is needed to eliminate residual stress during machining. Obviously, this point alone is far from enough. Optimizing the support structure of the mirror to release the stress on the mirror is an important step to ensure the optical performance of the mirror [8].

3.1. Bonding Ring Structure

Due to the characteristics of hard and brittle of monocrystalline silicon, it can’t be installed with a screw connection with a mechanical structure, and a glue connection was chosen for the fixation of the silicon mirror. Generally, the temperature expansion coefficient of the adhesive is one to two orders of magnitude of that of single crystal silicon. If the hardness of the connecting structure of the mirror is too high, when the temperature changes, the deformation stress of the glue will be transferred to the mirror, causing the surface of the mirror to change or even break. Therefore, to reduce the influence of temperature changes on the performance of the mirror, a flexible bonding ring structure was designed.

The outer cylindrical surface of the bonding ring is bonded to the mirror, and this part was designed as a discontinuous structure to break the continuity of the hardness of the closed ring structure. Each segment of the discontinuous structure is a circular arc plate, and the center of the circular arc plate coincides with the center of the mounting hole of the mirror. The inner layer corresponding to the circular arc plate is a flexible reed, and the circular arc plate and the flexible reed are connected by a short rib. The short ribs are located in the middle of the circular arc plate. The arc plate provides rotational freedom around the short rib, and the flexible reed can provide radial freedom of movement. The flexible structure composed of the arc plates and reeds can release temperature stress, and the reeds are connected to the rigid disc inside. The disc is provided with a mounting hole connected to the outside, and the structure of the mirror connection ring is shown in Fig. 4. The design of the bonding ring structure refers to the flexible structure of the secondary mirror of giant Magellan telescope (GMT) [9].

Figure 4.Bonding ring structure.

The thickness of the flexible reed can reflect the ability of the bonding ring to adapt to stress and temperature changes. In this paper, the thickness of the flexible reed was analyzed for different loads. The loads considered are lateral gravity and temperature changes. The constraints are that the RMS of the mirror is not greater than 4 nm, the rigid body displacement of the mirror is not greater than 3 μm, and the angular displacement of the mirror is not greater than 3″. The working temperature range of the mirror is 20 ℃ ± 20 ℃. Figure 5 and Table 2 show the calculation results for different load conditions.

TABLE 2 Calculation results for the surface shape accuracy of the mirror under. t = 0.75 mm, t = 1 mm and t = 2 mm. The optical axis of the mirror is horizontal; The x axis is in the same direction as gravity; The z axis is optical axis; The y axis direction is horizontal, and along the mirror radius direction; δx, δy, and δz represent the displacement of the mirror in the x, y, and z directions, and θx, θy, and θz represent the rotation angles of the mirror around the x, y, and z axes, respectively

Namet = 0.75 mmt = 1 mmt = 2 mm
PV/nm12.433512.48811.7393
RMS/nm2.614282.542.52081
δx/μm0.20660.190.136
δy/μm−9.65E-048.42E-041.23E-03
δz/μm−3.82E-042.96E-031.86E-04
θx/″−0.0118460.0190230.00649
θy/″−1.32362−1.23261−0.91044
θz/″−0.000994−0.01463−0.0089


Figure 5.Calculation results for the surface shape accuracy of the mirror at the temperature change of 20 ℃. The thickness of the flexible reed is t.

The results in Fig. 5 and Table 2 show that all three reed thicknesses can meet the requirements. Considering the feasibility of machining, the thickness of the reed is set to t = 2 mm, and meanwhile the RMS of the mirror is 2.2 nm under the condition of temperature change of 20 ℃.

3.2. Mounting Bracket

A bipod is a commonly used form in a mirror support structure [1013]. The design of the mounting bracket in this paper refers to the bipod form. The mounting bracket consists of three bipods, and each leg of the bipod is a flexible rod or a flexible beam that can be equivalent to a flexible rod. Homogeneous flexible beam units are relatively easy to process. When the width of the beam is much larger than the thickness, the flexible beam is equivalent to a flexible plate. A flexible plate has 3 degrees of freedom. Two cross plates have 5 degrees of freedom, which restricts the degree of freedom of tension and compression in the longitudinal direction. It can be equivalent to a flexible rod. Two feet with a cross flexible plate feature can be combined into a bipod leg.

In order to compress the size of the optical axis, an improved bipod structure was designed in this paper. The tangential flexible plate and the cross flexible plate partially overlap in space. One end of the three bipod legs is fixed to the connecting ring of the mirror, and the other end sits on the base together. The base is designed with three radial flexible leaf springs to relieve the installation stress of the mirror assembly. Figure 6 shows the mounting bracket of the mirror. The stiffness of the bipod is related to the six parameters of the thickness t1, t2, t3 and the length parameters l1, l2, and l3.

Figure 6.Mounting bracket of the mirror.

The slits in the mounting bracket seem complicated, but they can be manufactured with a single piece of metal. Figure 7 shows the path of the wire-electrode cutting.

Figure 7.Wire-electrode cutting of slits.

3.3. Mounting Bracket Parameter Optimization

According to the spatial layout of the optical system, the height of the bipod support structure is 20 mm. The angle between the two legs is 60 degrees, and the intersection of the two crossed feet is on the neutral plane of the mirror. The length of the radial flexible board is 4 mm. The material selected for the support structure was an invar alloy that matches the temperature expansion coefficient of single crystal silicon.

The support structure was optimized and analyzed based on the working conditions of the mirror assembly. The working temperature range of the mirror is 0 ℃– 40 ℃, and the stress state is 1 g gravity load. To simplify optimization, let l3 be constant, and its value is 4 mm. The optimization variable is the surface accuracy RMS. The RMS value of the surface accuracy of the mirror includes two parts, namely, the RMS value under gravity and the RMS value under temperature conditions. When the mirror works, the optical axis of the mirror is horizontal, so when the effect of gravity was evaluated, we only considered the RMS when the optical axis is horizontal. Assuming that the contribution of gravity and temperature to RMS is equal, the optimization problem can be described as

FindX={t1,l1,t2,l2,t3}T

MinPX=12RMSG+RMST

RMSG and RMST are the surface accuracy under gravity and temperature, respectively. The constraints are that the rigid body displacements of the mirror are less than 0.01 mm, and the angular displacement is less than 10″. Based on structural size constraints and design experience, the optimized parameters of the mounting bracket are determined as shown in Table 2.

RMSG and RMST are the surface accuracy under gravity and temperature, respectively. The constraints are that the rigid body displacements of the mirror are less than 0.01 mm, and the angular displacement is less than 10″. Based on structural size constraints and design experience, the optimized parameters of the mounting bracket are determined as shown in Table 3.

TABLE 3 Design parameters of the mirror and flexure support

NameParameterRanges (mm)Step (mm)
Tangential Flexible Plate Thicknesst10.5–30.1
Tangential Flexible Plate Lengthl14–100.5
Cross Flexible Plate Thicknesst20.5–30.1
Cross Flexible Plate Lengthl24–100.5
Radial Flexible Plate Thicknesst30.5–40.1
Radial Flexible Plate Lengthl34-


The parameter optimization of the mounting bracket is a process of opto-mechanical thermal integration analysis. In order to avoid the huge workload of pure manual work, Isight software (Dassault Systèmes Simulia Co., RI, USA) is often used to optimize and analyze the opto-mechanical structure in engineering. The process of integrated analysis is shown in Fig. 8.

Figure 8.Integrated analysis process.

The method of integrated analysis was adopted to calculate the sample set with Isight software. Considering the convenience of manufacturing, the lengths of l1 and l2 are set to be equal. The optimal solution was selected from all the results as shown in Table 4.

TABLE 4 Optimization results of mounting bracket

NameParameterOptimization Results
Tangential Flexible Plate Thicknesst10.8
Tangential Flexible Plate Lengthl17
Cross Flexible Plate Thicknesst21.8
Cross Flexible Plate Lengthl27
Radial Flexible Plate Thicknesst32.1
Radial Flexible Plate Lengthl3Const


In order to find whether the optimization results meet the requirements of use, the mirror assembly was analyzed and verified. Under the action of a temperature of Δ20 ℃ and a gravity load of 1 g, the RMS error of the mirror is 3.6 nm, the linear displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″. The calculation results meet the constraints. The surface fringes of the mirror under the actions of gravity and temperature load are shown in Figs. 9 and 10.

Figure 9.Mirror surface accuracy root mean square (RMS) under gravity conditions.

Figure 10.Mirror surface accuracy root mean square (RMS) under temperature condition.

Because of a lack of engineering experience in the development of single crystal silicon mirrors, this study carried out the design of a single crystal silicon mirror. The lightweight design of the monocrystalline silicon mirror fully considers the processing technology of monocrystalline silicon material. The support scheme adopted a central support, and the lightweight scheme was a lightweight form with a conical profile on the back. In order to be able to meet the temperature environment requirements of practical applications, the bipod support structure model of the reflector was built based on the principle of precise positioning, and the structural parameters of the reflector were optimized. The optimization results were calculated and analyzed. Under the actions of a temperature of Δ20 ℃ and a gravity load of 1 g, the RMS error of the mirror surface is 3.6 nm, the rigid body displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″. The results meet the requirements of the constraints.

Data underlying the results presented in this paper are not publicly available at the time of publication, which may be obtained from the authors upon reasonable request.

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Article

Article

Curr. Opt. Photon. 2022; 6(3): 236-243

Published online June 25, 2022 https://doi.org/10.3807/COPP.2022.6.3.236

Copyright © Optical Society of Korea.

Opto-mechanical Design of Monocrystalline Silicon Mirror for a Reflective Imaging Optical System

Xiaofeng Liu1 , Xin Zhang1, Fuxiang Tian2

1State Key Laboratory of Optics System Advanced Manufacturing Technology, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China
2Space Optics Department Ⅱ, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

Correspondence to:liuxiaofengshen@163.com, ORCID 0000-0002-7940-2357

Received: December 27, 2021; Revised: April 7, 2022; Accepted: April 19, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/4.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Monocrystalline silicon has excellent properties, but it is difficult to design and manufacture siliconbased mirrors that can meet engineering applications because of its hard and brittle properties. This paper used monocrystalline silicon as the main mirror material in an imaging system to carry out a feasibility study. The lightweight design of the mirror is completed by the method of center support and edge cutting. The support structure of the mirror was designed to meet the conditions of wide temperature applications. Isight software was used to optimize the feasibility sample, and the optimized results are that the root mean square error of the mirror surface is 3.6 nm, the rigid body displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″ under the conditions of a temperature of Δ20 °C and a gravity load of 1 g. The optimized result show that the silicon-based mirror developed in this paper can meet the requirements of engineering applications. This research on silicon-based mirrors can provide guidance for the application of other silicon-based mirrors.

Keywords: Imaging optical system, Mirror, Monocrystalline silicon, Silicon-based mirror

I. INTRODUCTION

In the application of a reflective imaging optical system, it has always been the goal of opto-mechanical engineers to find a kind of mirror material that is low cost and has excellent performance and a short processing time. However, these three attributes are often not found together, and how to balance the relationship between material properties and processing cycle is very important and difficult work.

The mirror materials commonly used are SiC, ultra-low expansion glass (ULE), zerodur, fused quartz, beryllium, aluminum, etc. Silicon carbide has a higher specific hardness than other materials, but the silicon carbide mirror manufacturing process is complicated, and a series of processes such as blank sintering, back lightweight processing, aspheric processing, modification, and aspheric polishing are required. Each link may decide the success or failure of the whole mirror. In addition, the manufacturing cycle of a silicon carbide mirror is extremely long, ranging from three months to several years. Zerodur, fused quartz and ULE materials have low specific hardness and poor thermal conductivity. Beryllium aluminum alloy also has high specific hardness, but it is expensive and toxic, which makes the manufacturing cost of a beryllium aluminum alloy mirror very high. Due to the high temperature expansion coefficient of aluminum, an aluminum mirror does not have good temperature stability.

With the rise of high-precision turning and polishing technology, the application of monocrystalline silicon materials has been given more and more attention over the years. Monocrystalline silicon material has low density, high homogeneity, no internal stress, and high thermal conductivity. The single-point diamond turning process can quickly obtain a high-precision monocrystalline silicon mirror, and its manufacturing cost is far lower than a silicon carbide mirror and beryllium aluminum alloy mirror. Also, the specific hardness and thermal conductivity are better than fused quartz, zerodur and ULE glass mirrors, and temperature stability is better than aluminum mirrors.

Although monocrystalline silicon has excellent properties, it is very difficult to design and manufacture silicon-based mirrors that can meet actual needs due to the hard and brittle characteristics of monocrystalline silicon materials. It is seldom selected as the preferred material in engineering applications. In order to give full play to the performance advantages of monocrystalline silicon mirrors and accumulate engineering experience, the application of φ200 mm diameter monocrystalline silicon mirrors was studied in this paper. An objective evaluation of silicon mirrors and other common mirrors is given in this paper. The lightweight design of the mirror and the optimization of the support structure are completed. The results show that the application of a monocrystalline silicon mirror in imaging optical system is feasible.

Ⅱ. Mirror structure design

2.1. Optical System

The optical system consists of a Schmidt-Cassegrain and corrective mirror group, which consists of a primary mirror, secondary mirror, Schmidt board and corrective lens group. The spherical aberration is corrected by the Schmidt plate, primary and secondary mirrors, and other off-axis aberrations are corrected by the rectification mirror group. The structural form of the optical system is shown in Fig. 1.

Figure 1. The structural form of the optical system.

2.2. Material Selection

The optical system has extremely high requirements for the accuracy and surface and position of the mirror, which should have good stability to ensure the imaging performance of the system. The mechanical and thermal stability of the mirror play an extremely important role in the selection of materials. In addition, cycle, cost and risk are also not negligible parts.

Newswander et al. [15] proposed a relatively objective material selection evaluation method. The specific indicators that affect the selection of materials are divided into five categories, which are optical performance, hardness and mass, thermal stability, metering structure and programmatic. Each indicator is assigned a different weight according to its importance. All indicators are multiplied by weights and then summed to obtain the overall evaluation value of the material. A single indicator also contains different elements, and the evaluation method of a single indicator is similar to the overall evaluation.

According to the actual needs of the engineering project, the factors that need to be considered in the selection of mirror materials in this article mainly include self-weight deformation, dynamic characteristics, thermal stability, thermal distortion, optical performance, and programmatic. The self-weight deformation is inversely proportional to the specific hardness E/ρ of the material, E is the Young’s modulus of elasticity, and ρ is the density. The thermal stability of the material is proportional to the expansion coefficient α; the dynamic characteristics are proportional to (E/ρ) (0.5) Thermal distortion is proportional to the thermal distortion factor α/λ, λ is the thermal conductivity. Silicon carbide, beryllium aluminum alloy and aluminum cannot obtain good optical surfaces by themselves, so the surface needs to be modified, and it is easy to cause a bimetal effect in the modified layer. Optical performance is measured by bimetal deformation, which is proportional to (Ef df / Es ds)(αs − αf )∆T [6]. ∆T is temperature difference. Ef, Es are the Young’s modulus of the modified layer and the substrate, respectively; αf, αs are the linear expansion coefficients of the modified layer and the substrate. df, ds are the thickness of the modified layer and the substrate, respectively. Programmatic is reflected by the processing time and cost. Common material parameters of mirrors are shown in Table 1.

TABLE 1. Mirror material properties at room temperature.

ParametersSiCBerylliumSiAlULEFused QuartzZerodurAstrosital
ρ (103 Kg/m3)3.052.12.32.72.22.22.52.46
E (GPa)4002301576967709291
λ (W/mK)1851951692201.31.381.21.18
α (10−6 K−1)2.514.52.623.90.030.550.10.15
E/ρ (107 Nm/kg)1310.96.82.73.13.23.683.69
α/λ (10−8 m/W)1.47.41.510.82.3408.312.7


Programmatic is an index that is difficult to quantify. According to past project experience, silicon carbide is used as a reference basis, and the processing cycle of silicon carbide is about 1.5 times that of glass materials such as ULE, fused quartz, Zerodur and Astrosital, four times that of beryllium aluminum, four times that of silicon, and three times that of aluminum. The cost is two times that of glass materials, one time that of beryllium aluminum alloy, three times that of silicon, and two times that of aluminum.

The diameter of the mirror studied in this paper is 200 mm, and the weight of each element is equal. Comprehensive evaluation results are shown in Fig. 2. Single crystal silicon has the highest score. It was selected as the material of the mirror.

Figure 2. Category scores and combined scores.

2.3. Lightweight Design

A reasonable lightweight design can ensure that the mirror has sufficient rigidity under the premise of meeting the requirements of quality constraints.

For the lightweight design of the mirror, the mirror thickness ratio needs to be determined first. Yoder [7] gave an empirical formula for calculating the thickness ratio of cylindrical mirrors:

δ=3ρgdr2D2256E

δ is the self-weight deformation; g is gravitational acceleration; dr is the mirror thickness ratio; D is the mirror diameter. According to the empirical formula, the thickness ratio of different material mirrors varies somewhat, and the value is generally between 1/5 and 1/12. According to the deformation requirements of the mirror, the thickness of the mirror is calculated to be about 24 mm.

The commonly used support methods for small and medium-diameter mirrors are three-point back support, back central support, peripheral bonding, and axial crimping. The back three-point support is the most commonly used form of support for mirrors, but the process is complicated, and the space occupied by the three-point support structure is very unfavorable to the structural layout of the lens group. Peripheral bonding and axial crimping require a certain amount of space around the mirror, which is very difficult for the light weight of the mirror and support structure. The back center support has a simple structure and takes up a small space, which is helpful for the lightweight design of the mirror and support structure. Therefore, this study chose the back center support program. For the central support, the edge of the mirror can be regarded as the cantilever end of the cantilever beam. The lightweight design idea of the central support should be to reduce the structural quality of the edge of the mirror, and balance the axial and radial stiffness of the mirror. The weight reduction scheme with a lightweight slot on the back is prone to edge breakage, forming microcracks and reducing the strength of the mirror. Considered comprehensively, the lightweight scheme chooses the form of a solid mirror with a cone at the back of the mirror. The lightweight model is shown in Fig. 3.

Figure 3. Lightweight mirror.

As shown in Fig. 3, the conical angle θ determines the weight of the mirror and the root mean square (RMS) error of the surface shape. Constrained by the size of the structure, θ ranges from 5° to 33°. As θ increases, The RMS of the mirror first increases, reaches a maximum value at 25°, and then decreases, reaches the bottom at 30°, and then increases again. The design goal of the mirror is to be as light as possible on the basis of satisfying RMS. The RMS of a single lens is not greater than 4 nm, and the mass is not greater than 1.2 kg. From the above analysis, it can be seen that the optimized result of θ is 30°, the RMS of the mirror is 4.0, and the mass is 1.03 kg.

III. Flexible support components

There are processing stress, assembly stress and thermal stress in the process of processing and assembly of mirror components, which would lead to the deformation of the mirror, and then deformation would cause the deterioration of optical imaging quality if it was not well compensated or removed. Therefore, effective heat treatment is needed to eliminate residual stress during machining. Obviously, this point alone is far from enough. Optimizing the support structure of the mirror to release the stress on the mirror is an important step to ensure the optical performance of the mirror [8].

3.1. Bonding Ring Structure

Due to the characteristics of hard and brittle of monocrystalline silicon, it can’t be installed with a screw connection with a mechanical structure, and a glue connection was chosen for the fixation of the silicon mirror. Generally, the temperature expansion coefficient of the adhesive is one to two orders of magnitude of that of single crystal silicon. If the hardness of the connecting structure of the mirror is too high, when the temperature changes, the deformation stress of the glue will be transferred to the mirror, causing the surface of the mirror to change or even break. Therefore, to reduce the influence of temperature changes on the performance of the mirror, a flexible bonding ring structure was designed.

The outer cylindrical surface of the bonding ring is bonded to the mirror, and this part was designed as a discontinuous structure to break the continuity of the hardness of the closed ring structure. Each segment of the discontinuous structure is a circular arc plate, and the center of the circular arc plate coincides with the center of the mounting hole of the mirror. The inner layer corresponding to the circular arc plate is a flexible reed, and the circular arc plate and the flexible reed are connected by a short rib. The short ribs are located in the middle of the circular arc plate. The arc plate provides rotational freedom around the short rib, and the flexible reed can provide radial freedom of movement. The flexible structure composed of the arc plates and reeds can release temperature stress, and the reeds are connected to the rigid disc inside. The disc is provided with a mounting hole connected to the outside, and the structure of the mirror connection ring is shown in Fig. 4. The design of the bonding ring structure refers to the flexible structure of the secondary mirror of giant Magellan telescope (GMT) [9].

Figure 4. Bonding ring structure.

The thickness of the flexible reed can reflect the ability of the bonding ring to adapt to stress and temperature changes. In this paper, the thickness of the flexible reed was analyzed for different loads. The loads considered are lateral gravity and temperature changes. The constraints are that the RMS of the mirror is not greater than 4 nm, the rigid body displacement of the mirror is not greater than 3 μm, and the angular displacement of the mirror is not greater than 3″. The working temperature range of the mirror is 20 ℃ ± 20 ℃. Figure 5 and Table 2 show the calculation results for different load conditions.

TABLE 2. Calculation results for the surface shape accuracy of the mirror under. t = 0.75 mm, t = 1 mm and t = 2 mm. The optical axis of the mirror is horizontal; The x axis is in the same direction as gravity; The z axis is optical axis; The y axis direction is horizontal, and along the mirror radius direction; δx, δy, and δz represent the displacement of the mirror in the x, y, and z directions, and θx, θy, and θz represent the rotation angles of the mirror around the x, y, and z axes, respectively.

Namet = 0.75 mmt = 1 mmt = 2 mm
PV/nm12.433512.48811.7393
RMS/nm2.614282.542.52081
δx/μm0.20660.190.136
δy/μm−9.65E-048.42E-041.23E-03
δz/μm−3.82E-042.96E-031.86E-04
θx/″−0.0118460.0190230.00649
θy/″−1.32362−1.23261−0.91044
θz/″−0.000994−0.01463−0.0089


Figure 5. Calculation results for the surface shape accuracy of the mirror at the temperature change of 20 ℃. The thickness of the flexible reed is t.

The results in Fig. 5 and Table 2 show that all three reed thicknesses can meet the requirements. Considering the feasibility of machining, the thickness of the reed is set to t = 2 mm, and meanwhile the RMS of the mirror is 2.2 nm under the condition of temperature change of 20 ℃.

3.2. Mounting Bracket

A bipod is a commonly used form in a mirror support structure [1013]. The design of the mounting bracket in this paper refers to the bipod form. The mounting bracket consists of three bipods, and each leg of the bipod is a flexible rod or a flexible beam that can be equivalent to a flexible rod. Homogeneous flexible beam units are relatively easy to process. When the width of the beam is much larger than the thickness, the flexible beam is equivalent to a flexible plate. A flexible plate has 3 degrees of freedom. Two cross plates have 5 degrees of freedom, which restricts the degree of freedom of tension and compression in the longitudinal direction. It can be equivalent to a flexible rod. Two feet with a cross flexible plate feature can be combined into a bipod leg.

In order to compress the size of the optical axis, an improved bipod structure was designed in this paper. The tangential flexible plate and the cross flexible plate partially overlap in space. One end of the three bipod legs is fixed to the connecting ring of the mirror, and the other end sits on the base together. The base is designed with three radial flexible leaf springs to relieve the installation stress of the mirror assembly. Figure 6 shows the mounting bracket of the mirror. The stiffness of the bipod is related to the six parameters of the thickness t1, t2, t3 and the length parameters l1, l2, and l3.

Figure 6. Mounting bracket of the mirror.

The slits in the mounting bracket seem complicated, but they can be manufactured with a single piece of metal. Figure 7 shows the path of the wire-electrode cutting.

Figure 7. Wire-electrode cutting of slits.

3.3. Mounting Bracket Parameter Optimization

According to the spatial layout of the optical system, the height of the bipod support structure is 20 mm. The angle between the two legs is 60 degrees, and the intersection of the two crossed feet is on the neutral plane of the mirror. The length of the radial flexible board is 4 mm. The material selected for the support structure was an invar alloy that matches the temperature expansion coefficient of single crystal silicon.

The support structure was optimized and analyzed based on the working conditions of the mirror assembly. The working temperature range of the mirror is 0 ℃– 40 ℃, and the stress state is 1 g gravity load. To simplify optimization, let l3 be constant, and its value is 4 mm. The optimization variable is the surface accuracy RMS. The RMS value of the surface accuracy of the mirror includes two parts, namely, the RMS value under gravity and the RMS value under temperature conditions. When the mirror works, the optical axis of the mirror is horizontal, so when the effect of gravity was evaluated, we only considered the RMS when the optical axis is horizontal. Assuming that the contribution of gravity and temperature to RMS is equal, the optimization problem can be described as

FindX={t1,l1,t2,l2,t3}T

MinPX=12RMSG+RMST

RMSG and RMST are the surface accuracy under gravity and temperature, respectively. The constraints are that the rigid body displacements of the mirror are less than 0.01 mm, and the angular displacement is less than 10″. Based on structural size constraints and design experience, the optimized parameters of the mounting bracket are determined as shown in Table 2.

RMSG and RMST are the surface accuracy under gravity and temperature, respectively. The constraints are that the rigid body displacements of the mirror are less than 0.01 mm, and the angular displacement is less than 10″. Based on structural size constraints and design experience, the optimized parameters of the mounting bracket are determined as shown in Table 3.

TABLE 3. Design parameters of the mirror and flexure support.

NameParameterRanges (mm)Step (mm)
Tangential Flexible Plate Thicknesst10.5–30.1
Tangential Flexible Plate Lengthl14–100.5
Cross Flexible Plate Thicknesst20.5–30.1
Cross Flexible Plate Lengthl24–100.5
Radial Flexible Plate Thicknesst30.5–40.1
Radial Flexible Plate Lengthl34-


The parameter optimization of the mounting bracket is a process of opto-mechanical thermal integration analysis. In order to avoid the huge workload of pure manual work, Isight software (Dassault Systèmes Simulia Co., RI, USA) is often used to optimize and analyze the opto-mechanical structure in engineering. The process of integrated analysis is shown in Fig. 8.

Figure 8. Integrated analysis process.

The method of integrated analysis was adopted to calculate the sample set with Isight software. Considering the convenience of manufacturing, the lengths of l1 and l2 are set to be equal. The optimal solution was selected from all the results as shown in Table 4.

TABLE 4. Optimization results of mounting bracket.

NameParameterOptimization Results
Tangential Flexible Plate Thicknesst10.8
Tangential Flexible Plate Lengthl17
Cross Flexible Plate Thicknesst21.8
Cross Flexible Plate Lengthl27
Radial Flexible Plate Thicknesst32.1
Radial Flexible Plate Lengthl3Const


In order to find whether the optimization results meet the requirements of use, the mirror assembly was analyzed and verified. Under the action of a temperature of Δ20 ℃ and a gravity load of 1 g, the RMS error of the mirror is 3.6 nm, the linear displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″. The calculation results meet the constraints. The surface fringes of the mirror under the actions of gravity and temperature load are shown in Figs. 9 and 10.

Figure 9. Mirror surface accuracy root mean square (RMS) under gravity conditions.

Figure 10. Mirror surface accuracy root mean square (RMS) under temperature condition.

Ⅳ. Conclusion

Because of a lack of engineering experience in the development of single crystal silicon mirrors, this study carried out the design of a single crystal silicon mirror. The lightweight design of the monocrystalline silicon mirror fully considers the processing technology of monocrystalline silicon material. The support scheme adopted a central support, and the lightweight scheme was a lightweight form with a conical profile on the back. In order to be able to meet the temperature environment requirements of practical applications, the bipod support structure model of the reflector was built based on the principle of precise positioning, and the structural parameters of the reflector were optimized. The optimization results were calculated and analyzed. Under the actions of a temperature of Δ20 ℃ and a gravity load of 1 g, the RMS error of the mirror surface is 3.6 nm, the rigid body displacement of the mirror is 2.1 μm, and the angular displacement is 2.5″. The results meet the requirements of the constraints.

DISCLOSURES

The authors declare no conflicts of interest.

DATA AVAILABILITY

Data underlying the results presented in this paper are not publicly available at the time of publication, which may be obtained from the authors upon reasonable request.

FUNDING

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Fig 1.

Figure 1.The structural form of the optical system.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 2.

Figure 2.Category scores and combined scores.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 3.

Figure 3.Lightweight mirror.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 4.

Figure 4.Bonding ring structure.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 5.

Figure 5.Calculation results for the surface shape accuracy of the mirror at the temperature change of 20 ℃. The thickness of the flexible reed is t.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 6.

Figure 6.Mounting bracket of the mirror.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 7.

Figure 7.Wire-electrode cutting of slits.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 8.

Figure 8.Integrated analysis process.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 9.

Figure 9.Mirror surface accuracy root mean square (RMS) under gravity conditions.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

Fig 10.

Figure 10.Mirror surface accuracy root mean square (RMS) under temperature condition.
Current Optics and Photonics 2022; 6: 236-243https://doi.org/10.3807/COPP.2022.6.3.236

TABLE 1 Mirror material properties at room temperature

ParametersSiCBerylliumSiAlULEFused QuartzZerodurAstrosital
ρ (103 Kg/m3)3.052.12.32.72.22.22.52.46
E (GPa)4002301576967709291
λ (W/mK)1851951692201.31.381.21.18
α (10−6 K−1)2.514.52.623.90.030.550.10.15
E/ρ (107 Nm/kg)1310.96.82.73.13.23.683.69
α/λ (10−8 m/W)1.47.41.510.82.3408.312.7

TABLE 2 Calculation results for the surface shape accuracy of the mirror under. t = 0.75 mm, t = 1 mm and t = 2 mm. The optical axis of the mirror is horizontal; The x axis is in the same direction as gravity; The z axis is optical axis; The y axis direction is horizontal, and along the mirror radius direction; δx, δy, and δz represent the displacement of the mirror in the x, y, and z directions, and θx, θy, and θz represent the rotation angles of the mirror around the x, y, and z axes, respectively

Namet = 0.75 mmt = 1 mmt = 2 mm
PV/nm12.433512.48811.7393
RMS/nm2.614282.542.52081
δx/μm0.20660.190.136
δy/μm−9.65E-048.42E-041.23E-03
δz/μm−3.82E-042.96E-031.86E-04
θx/″−0.0118460.0190230.00649
θy/″−1.32362−1.23261−0.91044
θz/″−0.000994−0.01463−0.0089

TABLE 3 Design parameters of the mirror and flexure support

NameParameterRanges (mm)Step (mm)
Tangential Flexible Plate Thicknesst10.5–30.1
Tangential Flexible Plate Lengthl14–100.5
Cross Flexible Plate Thicknesst20.5–30.1
Cross Flexible Plate Lengthl24–100.5
Radial Flexible Plate Thicknesst30.5–40.1
Radial Flexible Plate Lengthl34-

TABLE 4 Optimization results of mounting bracket

NameParameterOptimization Results
Tangential Flexible Plate Thicknesst10.8
Tangential Flexible Plate Lengthl17
Cross Flexible Plate Thicknesst21.8
Cross Flexible Plate Lengthl27
Radial Flexible Plate Thicknesst32.1
Radial Flexible Plate Lengthl3Const

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