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Abstract
As an essential part of optical telescope, the secondary mirror is subject to the influence of ambient temperature, which leads to temperature-induced distortion on the surface shape. A hybrid ball-hinged secondary mirror assembly (HSMA) is proposed to achieve thermal adaptation over a wide range of temperature. Simulation investigation on the temperature-induced surface shape distortion of the HSMA were carried out by using the finite element model. Simulation results show that the change of secondary mirror surface distortions over a wide range of temperature are minimal and negligible. For the wide ambient temperature range from -30°C to 70°C, the PV and RMS values of the maximum residual distortions can reach as small as 16.31 nm and 3.005 nm, respectively. Furthermore, the influence of gravity-induced distortion on the surface shape is also carried out. Both simulation and experiment results show that the HSMA is able to maintain high-precision surface shape of the secondary mirror over a wide range of temperature and at different attitudes from 0 to 90$^\circ $.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
As high-precision observation equipment, optical telescope is widely used in various fields, such as astronomical observation [1–4], laser communication [5,6], laser ranging [7–11], dynamic target tracking [12,13] and space debris detection [14–18]. Among these optical telescopes, the 1-meter class optical telescopes play important roles and have very widespread applications due to its well balance of scientific observation, construction cost, and excellent mobility. For example, the Nanjing Institute of Astronomical Optics and Technology developed the SONG 1-meter telescope to carry out studies on the internal structure of stars [19]. Yunnan Observatories of the Chinese Academy of Sciences uses the 1-meter New Vacuum Sloar Telescope (NVST) to achieve high-resolution observations of the Sun near the diffraction limit [20]. The Laser Ranging Station of the Tianqin Project has successfully measured the echo signals of five retro-reflectors arrays installed on the surface of the lunar using a 1.2-meter laser ranging telescope [21]. Performance of 1-meter optical telescope mainly depends on its adaptability in complex environment, especially the thermal adaptability over a wide range of temperature in the observatory. In the 1-meter optical telescope, the secondary mirror is an essential element mounted on a mechanical support structure. The surface shape of the secondary mirror needs to be maintained in high precision to ensure that wavefront distortion is well depressed and imaging quality could meet system requirements. Along with the support structures, the secondary mirror is directly exposed to the external environment during operation. The change of ambient temperature will directly lead to surface shape deterioration of the secondary mirror due to thermal stress caused by the thermal expansion coefficient difference between the mirror and its support structure. Therefore, the athermal design of the secondary mirror assembly, which is to maintain the high-precision surface shape over a wide temperature range, is a critical challenge in the design and development of 1-meter class optical telescopes.
Numerous studies have been conducted on the athermal design of the secondary mirror assembly to depress the surface shape thermal distortion. Among these athermal designs, multi-point adhesive schemes and flexible support schemes are mostly adopted. The adhesive method refers to filling adhesive between secondary mirror and support structure to achieve bonding and fixation [22–27]. Doyle et al. carried out athermal design research on near-incompressible adhesives and obtained the formula for calculating the thickness of the near-incompressible adhesive [22]. Vyacheslav et al. pointed out that thermal stress largely depends on bonding thickness and can be reduced or even eliminated by adjusting the bonding thickness. A comparative study found that three-point adhesives usually produce less thermal stress than continuous adhesives [24]. Wang et al. analyzed the influence factors in the process of mirror bonding to eliminate or reduce the curing shrinkage stress of the adhesive, including area and distribution of the adhesive spot, thickness of the adhesive layer, curing time, and curing temperature of the adhesive layer [23]. As mentioned above, the adhesive method could theoretically meet the athermal design requirements of the secondary mirror assembly. However, the design and bonding process is complicated in practice. And the adhesive’s moisture absorption/mass loss will cause the mirror to produce shrinkage stress and unwanted micro-radian tilts [28–30]. Therefore, it is difficult to achieve high-precision surface shape of the secondary mirror after bonding. Flexible support is another classic structure to realize the athermal design of the secondary mirror [31–39]. It provides flexible isolation for the secondary mirror to resist external thermal disturbances while maintaining the mirror in a precise position. Li et al. proposed a novel flexure mounting structure based on the cartwheel flexural hinge for a lightweight SiC mirror [34]. The design of a flexible composite structure with cross-distributed bushing is presented by Liu et al. [33], composed of outer ring, inner ring and driver ring. Vukobratovich et al. studied a single-point back flexible structure, which has multiple radial flexible metal domes and each dome is bonded to the back boss of the mirror [32]. Yu et al. presented a new type of three-leaf flexible structure to achieve high-precision surface shape, which includes a taper bushing, a flexible structure and triangular backplate [31]. Although the flexible support can improve thermal adaptability of the secondary mirror assembly, it will reduce the stiffness of the support structure and lead to misalignment of the secondary mirror. Therefore, flexible structures are often used to achieve the athermal design over a limited range of temperature variations. In summary, the mentioned typical methods are not fully applicable to maintaining high-precision surface shape for the secondary mirror over a wide temperature range.
A hybrid ball-hinged secondary mirror assembly (HSMA) maintaining high-precision surface shape over a wide range of temperature is proposed in this paper. This paper is organized as follows. In section 2, the configuration and principle of the HSMA are illustrated in details. In section 3, the corresponding finite element model of the HSMA is presented. The capabilities of the HSMA to maintain high-precision surface shape of the secondary mirror at different ambient temperature (-30°${C}$ to 70°${C}$) are analyzed and studied. Since the secondary mirror is also subject to the influence of gravity at different attitudes, the capabilities of the HSMA to maintain high-precision surface shape of the secondary mirror at different attitudes (0$^\circ $ and 90$^\circ $) are investigated in section 4. In section 5, an experiment based on a lab-manufactured HSMA is carried out. Experimental results show that the HSMA has excellent ability of maintaining high-precision surface shape and accurate position of the secondary mirror at different temperatures and attitudes.
2. Configuration and principle of the HSMA
Shown in Fig. 1 is the configuration of the hybrid ball-hinged secondary mirror assembly HSMA. As seen from Figs. 1(a) and (b), the HSMA mainly consists of a secondary mirror, a triangular backplate and a hybrid ball hinge unit. The hybrid ball hinge unit includes a central positioning ball hinge, three sets of axial positioning ball hinges and three sets of axial preload spring units. As shown in Fig. 1(a) and Table 1, a blind hole is set in the center and three grooves are evenly distributed along the outer circumference of the mirror. The external diameter of the secondary mirror is 185 mm, while the central thickness is 30 mm and the vertex radius of curvature is 1008.4 mm. The central positioning ball hinge structure is machined with a ball at the end to achieve radial positioning and radial support of the secondary mirror.
Fig. 1. Configuration of the HSMA. (a) Section diagram of the HSMA. (b) Top view of the HSMA. (c) Partial view of the axial positioning ball hinge and axial preload spring unit.
Table 1. Structural parameters of the HSMA
In order to achieve high radial positioning accuracy of the mirror during operation, the stiffness of central positioning ball hinge and triangular backplate are ensured, while a grinding process is applied on the ball surface to reduce the mating gap with the central blind hole. It is also necessary to ensure the center of the ball coinciding with the plane of mass center of the mirror [Fig. 2(a)]. Otherwise, additional bending moment will generate and exert on the secondary mirror. In the axial direction, the secondary mirror uses three sets of axial positioning ball hinges and three sets of axial preload spring units to realize the 3-point positioning and axial support of the secondary mirror. Three sets of axial positioning ball hinges are evenly distributed between the secondary mirror and the triangular backplate [Figs. 1(a) and (b)], which can achieve axial constraint and two-dimensional angular constraint of the secondary mirror. Each of the ball hinge mainly consists of a ball-head adjusting screw rod, a tapered pad and a PTFE pad. As shown in Fig. 1(c), the axial preload spring unit mainly includes a claw, a steel ball, a tapered pad, a PTFE pad and two sets of spring element. One end of the axial preload spring unit connects with the secondary mirror, while the other connects with the triangular backplate. Through three sets of axial preload spring units, the secondary mirror is reliably connected with three sets of axial positioning ball hinges in the axial direction. The rotation about the central axis of the mirror is constrained by the adhesive force provided by the RTV adhesive between the central positioning ball hinge and the central hole of the secondary mirror. From described above, a semi-kinematic type of support [37,40,41] is actually adopted in the proposed HSMA. Compared with conventional semi-kinematic supports, the hybrid ball hinge structure in the HSMA could help the secondary mirror improve thermal adaptation in a wide temperature range and maintain high-precision surface shape.
Fig. 2. Schematic of the secondary mirror. (a) Geometric parameters of the secondary mirror, including the external diameter (D), the diameter of the central hole (d), the radius of vertex curvature (R), the thickness of the center area (t) and the thickness between the groove and the back (L). (b) External load and constraint of the secondary mirror. ${F_{pd}}$ is the axial preload force and $\Delta {F_{pd}}$ is the force variation due to temperature variation. ${F_f}$ is the radial friction force.
To achieve reliable positioning of the mirror in the axial direction, the axial preload spring unit needs to apply a specific preload ${F_{pd}}$ at the groove. As shown in Fig. 1, in the compact space between the claw and the triangular backplate, each axial preload spring unit consist of two piece of spring element with the mean diameter of 8 mm and free length of 18 mm, which could endure the maximum load force of 70N. In order to provide the highest possible preload force and keep the spring at a high safety rate, the preload force applied on the spring is set 35N (i.e., safety rate 2.0). Thus, the total preload force ${F_{pd}}$ is 105N, which is five times the weight of the secondary mirror (i.e., 21N). The long-term stability of the secondary mirror is mainly affected by the stress relaxation behavior in the axial preload spring unit, which is closely related to the material and initial stress of the spring element. In order to achieve long-term stability of the secondary mirror, the spring element in the HSMA is designed using the material of alloy spring steel (60Si2Mn) with high stress relaxation resistance. The high safety rate 2.0 of the spring element at the axial preload force could help decrease the initial stress and improve the stress relaxation resistance. In practice, in order to furtherly ensure the long-term stability of the HSMA, all the screw connections are bonded with screw glue to prevent the screws from loosening. In order to avoid additional bending moments on the mirror, the axial preload should be applied in strict correspondence with the three sets of axial positioning ball hinges. Due to the different thermal expansion coefficients ${\alpha _1}$ and ${\alpha _2}$, the secondary mirror and support structure will produce different thermal deformations in the axial and radial directions when the ambient temperature changes. The central positioning ball hinge can achieve free thermal deformation of the secondary mirror in the radial direction and avoid the generation of thermal stress on the mirror. However, the thermal deformation in the radial direction could not be avoided by using the ball hinge, and the friction forces ${F_f}$ [Eq. (1)] between the secondary mirror and the axial positioning ball hinge will also generate at the contact position [Fig. 2(b)]. In order to effectively reduce the friction forces ${F_f}$, PTFE pads with low friction coefficient ${\mu _f}$ are added to the contact position of the axial positioning ball hinge and the secondary mirror [Fig. 1(c)]. Moreover, to furtherly depress the thermal stress, the center positioning ball hinge is made of invar with low thermal expansion coefficient (0.5 × 10−6), which is almost the same with the secondary mirror.
In the axial direction, the axial preload force (${F_{pd}}$) is applied to the secondary mirror by using an axial preload spring unit (stiffness factor $K$). The preload force ${F_{pd}}$ is an elastic force. When the secondary mirror and support structure are not aligned due to axial thermal deformation, relative deformation ($\Delta L$) will causes force variation ($\Delta {F_{pd}}$) on the axial preload force ${F_{pd}}$ [Eqs. (1)-(3)]. Since the preload force ${F_{pd}}$ is an elastic force, the stiffness factor ($K$) can be optimized to substantially reduce the force variation $\Delta {F_{pd}}$. Furthermore, in order to effectively reduce the influence of thermal deformation and depress the relative deformation $\Delta L$, the grooves are located far away from the mirror surface [Fig. 2(a)]. Thus, the HSMA could realize fine isolation from external thermal disturbances and be able to maintain high-precision surface shape of the secondary mirror under different ambient temperatures.
3. Temperature-induced distortion of the HSMA
3.1 Simulation model
In order to carry out investigation on the thermal characteristics of the HSMA in a wide range of temperature, a finite element model is constructed, as shown in Fig. 3. In the finite element model, the HSMA consists of a secondary mirror, a triangular backplate, and the hybrid ball-hinged support structure. Invar and carbon steel are chosen as the materials of the central positioning ball hinge and all other structural elements, respectively. The Zerodur with low thermal expansion coefficient is chosen as the material of the secondary mirror. Detailed geometric dimensions and material parameters of the HSMA are listed in Tables 1 and 2.
Fig. 3. Finite element model of the HSMA. $\theta $ is the zenith angle of the secondary mirror. When the optical axis of the secondary mirror points to the zenith, $\theta = 0^\circ $. When the optical axis points to the horizon, $\theta = 90^\circ $.
Table 2. Material parameters of the HSMA.
In the simulation, the radial constraint is applied on the neutral surface of the axial preload spring unit, which can reduce the influence of the boundary constraint on the thermal investigation of the HSMA. The axial degree of freedom constraints is applied to the triangular backplane. The axial preload spring unit is simulated by using a one-dimensional spring element. The preload force ${F_{pd}}$, five times the weight of the secondary mirror, is applied on the spring unit. The initial temperature is set as $20^{\circ}{\mathrm C}$ and the operating temperature range is set as -30°${C}$ to 70°${C}$ (i.e., the temperature variation range $\Delta T$ is ±50°${C}$). The angle $\theta $ [Fig. 3] between the secondary mirror optical axis and the gravity direction is set as the zenith angle.
3.2 Temperature-induced stress and surface shape distortion
In practical operation of the optical telescope, the secondary mirror is subjected to the combined effects of gravity G, temperature variation $\Delta T$, and preload force ${F_{pd}}$. In order to simulate the actual conditions, the effects of multiple loads (i.e., gravity and preload force) are taken into account when thermal characteristics investigation of the HSMA is conducted. As shown in Fig. 4, the combined stress of the HSMA under practical multiple loads is calculated first when ambient temperature is -30°${C}$ and zenith angle is 0$^\circ $. The maximum combined stress of the HSMA occurs in the axial preload spring unit with the value of 23.2 MPa, shown in Fig. 4(a). Since the point contact between the axial positioning ball hinge and the secondary mirror is replaced by a small area contact, the combined stress value of the secondary mirror is significantly reduced. And the effect of thermal deformation on the combined stress of the secondary mirror can be reduced by controlling the magnitude of the stiffness factor K, as the preload ${F_{pd}}$ is an elastic force. Thus, the maximum combined stress value of the secondary mirror is reduced to 7.54 MPa under practical multiple loads, as shown in Fig. 4(b). Moreover, the position of the secondary mirror groove where the external load acts on is far away from the mirror surface. The combined stress on the secondary mirror surface can be significantly reduced to 9.08 × 10−3 MPa [Fig. 4(c)]. Thus, the HSMA has an excellent isolation effect for the secondary mirror surface against the external complex environmental disturbances.
Fig. 4. (a) Combined stress of the HSMA under multiple loads at -30°${C}$ temperature and 0$^\circ $ zenith angle. (b) Combined stress of the secondary mirror. (c) Combined stress of the mirror surface.
In the simulation, the FEM data of the mirror surface, including the node number, the node location, deviations of the node in each coordinate directions, are extracted from the FEA analysis results and sent to the analysis software (e.g., MATLAB). Based on the extracted data, the deformation of the surface shape is calculated and the corresponding RMS/PV values and Zernike terms are obtained. Figure 5 shows the surface shape distortions of the secondary mirror due to temperature variation, gravity and preload force. Figures 5(a1) and (a2) show surface shape distortions of the secondary mirror when temperature variation $\Delta T$ is -50°C and 50°C, respectively. The PV values of the distortions are both 11.46 nm, which are equal in amplitude and opposite in direction. The corresponding Zernike coefficients of the distorted surface shapes are shown in Figs. 5(b1) and (b2). It can be seen that in the HSMA, the temperature-induced surface shape distortion is mainly the defocus aberration. The effect of gravity on the surface distortions of the secondary mirror at different attitudes are shown in Figs. 5(a3) and (a4). The gravity-induced distortions are mainly the oblique trefoil and defocus at zenith angle of 0$^\circ $ [PV value of 17.42 nm, Fig. 5(a3)] and astigmatism aberration at 90$^\circ $ [PV value of 5.6 nm, Fig. 5(a4)], respectively. Figures 5(b3) and (b4) show the corresponding Zernike coefficients of the distorted surface shapes. Surface shapes and Zernike coefficients affected by preload force ${F_{pd}}$ are displayed in Figs. 5(a5) and (b5). It should be noted that the preload force is about five times the weight of the secondary mirror, as described in section 2. The distortion is mainly oblique trefoil and defocus aberration with the PV value of 1.54 nm. It can be seen that the effects of temperature variation, gravity and preload force on the surface shape distortion of the secondary mirror are minimal and negligible. The HSMA has good isolation capability against complex external environment and could maintain high-precision surface shape of the secondary mirror.
Fig. 5. Surface shape distortions of the secondary mirror due to temperature variation, gravity and preload force. (a1) and (a2) are temperature-induced distortions at the ambient temperature of -30°${C}\; (\Delta T$ of -5$0^{\circ}{\mathrm C})$ and 70°${C}\; (\Delta T$ of 5$0^{\circ}{\mathrm C})$, respectively. (a3) and (a4) are gravity-induced distortions at zenith angle of 0$^\circ $ and 90$^\circ $, respectively. (a5) is the distortion caused by the preload force. (b1) -(b5) are the corresponding Zernike coefficients of (a1)-(a5).
Figures 6(a1)-(a5) furtherly show the temperature-induced surface shape distortions under multiple loads at temperature ranging from -30°${C}$ to 70°${C}$ when the zenith angle $\theta $ is 0$^\circ $, while Figs. 6(b1)-(b5) show the corresponding Zernike coefficients. As shown in Figs. 6(a3) and (b3), the HSMA is only affected by gravity and preload force in this operating condition at the initial ambient temperature 20°${C}$ ($\Delta T$ of $0^{\circ}{\mathrm C}$). The surface shape distortion is mainly oblique trefoil and defocus aberration with the PV value of 17.42 nm and RMS value of 4.186 nm (Table 3). Figures 6(a4) and (a5) show the temperature-induced surface distortions when the ambient temperature increases to 45°${C}$ and 70°${C}$, respectively, which are mainly the variation of the defocus term [Figs. 6(b4) and (b5)] compare with the initial state. Especially, when the ambient temperature reaches 70°${C}$ ($\Delta T$ of 50°${C}$), the surface shape distortion is mainly the defocus aberration with the PV value of 28.47 nm and RMS value of 7.178 nm. It is worthy to mention that the temperature-induced defocus is in the same direction as that induced by gravity when the ambient temperature increases. Figures 6(a1) and (a2) show the temperature-induced surface distortions when the ambient temperature decreases to -30°${C}$ and -5°${C}$, respectively. The temperature-induced surface distortions mainly exhibit the variation of the defocus as shown in Figs. 6(b1) and (b2). This type of defocus can be compensated by the gravity induced defocus to a certain extent as they are right in the opposite direction. When the ambient temperature is -30°${C}$ ($\Delta T$ of -50°${C}$), the surface shape distortion is mainly the oblique trefoil aberration with the PV value of 16.31 nm and RMS value of 3.067 nm (Table 3). Figure 7 shows the temperature-induced surface shape distortions when optical axis of the HSMA points to the horizon (i.e., zenith angle $\theta $ is 90$^\circ $). The initial aberration [Fig. 7(a3)] caused by the gravity at the ambient temperature 20°${C}$, mainly astigmatism aberration with the PV value of 6.76 nm (Table 3), is even slighter than that at the zenith angle of 0$^\circ $ as the direction of gravity is perpendicular to the optical axis of the HSMA. As shown in Fig. 7, the surface shape changes along with the of the ambient temperature. When ambient temperatures are 70°${C}$ and -30°${C}\; $($\Delta T$ of ±50°${C}$), the surface shape distortions are mainly defocus aberration relative to the initial state with the PV value of 14.43 nm and 15.16 nm, RMS value of 3.431 nm and 4.053 nm, respectively. Table 3 and Fig. 8 show the PV and RMS values and the variation curves of the surface shape at different ambient temperatures and two zenith angles.
Fig. 6. Temperature-induced surface distortions of the secondary mirror under multiple loads at 0$^\circ $ zenith angle. (a1)-(a5) are the surface shape distortions at the ambient temperature of -30°${C}$, -5°${C}$, 20°${C}$, 45°${C}$ and 70°${C}$, respectively. (b1)-(b5) are the corresponding Zernike decomposition coefficients, respectively. (c1) and (c2) are the residual surface shape distortion and Zernike coefficients without defocus at the temperature of -30°${C}$. (c3) and (c4) are the residual surface shape distortion and Zernike coefficients without defocus at the temperature of $70^{\circ}{\mathrm C}$.
Fig. 7. Temperature-induced surface distortions of the secondary mirror under multiple loads at 90$^\circ $ zenith angle. (a1)-(a5) are the surface shape distortions at the ambient temperature of -30°${C}$, -5°${C}$, 20°${C}$, 45°${C}$ and 70°${C}$, respectively. (b1)-(b5) are the corresponding Zernike decomposition coefficients, respectively. (c1) and (c2) are the residual surface shape distortion and Zernike coefficients without defocus at the temperature of -30°${C}$. (c3) and (c4) are the residual surface shape distortion and Zernike coefficients without defocus at the temperature of $70^{\circ}{\mathrm C}$.
Fig. 8. Variation curves of the PV and RMS values of the secondary mirror with ambient temperature. (a) shows the variation curves of the PV and RMS values when zenith angle $\theta $ is 0$^\circ $ and 90$^\circ $ respectively. (b) shows the corresponding residual distortion without defocus term.
Table 3. Surface shape distortion of the secondary mirror with the temperature ranging from -30°$\mathrm{{C}}$ to 70°$\mathrm{{C}}$
In practical operation of the telescope, the defocus term of the surface shape distortion can be eliminated by axial alignment of the secondary mirror. Thus, more attention should be focused on the residual distortion without defocus term. Figures 6(c1)-(c4) present the residual surface shape distortions of the secondary mirror at extreme ambient temperatures of -30°${C}$ and 70°${C}$, respectively, when the zenith angle $\theta $ is 0$^\circ $. The residual distortions are both mainly oblique trefoil aberration with the PV value of 16.31 nm and 15.49 nm (Table 3) at the ambient temperature of -30°${C}$ and 70°${C}$, respectively. As shown in Fig. 8(b), the variation rates of PV and RMS values are 8.2 × 10−3 $nm/^{\circ}{\mathrm{C}}$ and 1.4 × 10−3 $nm/^{\circ}{\mathrm{C}}$, respectively. Figures 7(c1)-(c4) represent the residual distortions corresponding to the zenith angle $\theta $ of 90$^\circ $. The residual distortion is mainly astigmatism aberration. As shown in Fig. 8(b), the corresponding variation rate of the PV value is 6.8 × 10−3 $nm/^{\circ}{\mathrm{C}}$ and the rate of RMS value is 0.38 × 10−3$nm/^{\circ}{\mathrm{C}}$. From Fig. 8 and Table 3, It can be seen that the surface shape distortion of the secondary mirror is well depressed at the zenith angle from 0$^\circ $ to 90$^\circ $ even when the ambient temperature varies by 50°${C}$. The proposed HSMA has excellent adaptability to a wide range of temperature variations.
4. Gravity-induced distortion of the HSMA
Since the secondary mirror is also subject to the influence of gravity at different attitudes during optical telescope operation, the capabilities of the HSMA to maintain high-precision surface shape of the secondary mirror at different attitudes (0$^\circ $ and 90$^\circ $) are analyzed. In the investigation of temperature-induced distortion above, the surface shape distortions at two limit attitude angles of 0$^\circ $ and 90$^\circ $ have been analyzed. In this section, the gravity-induced surface shape distortion and gravity-induced mirror misalignment of the HSMA will be analyzed in detail for different zenith angles.
In the simulation, the ambient temperature is set as -30°${C}$ and the zenith angle ranges from 0$^\circ $ to 90$^\circ $, while the axial preload is five times the weight of the mirror. Figures 9(a1) and (b1) display the surface shape distortion with corresponding Zernike coefficients at the zenith angle of 0$^\circ $ (i.e., optical axis of the secondary mirror orients to the direction of gravity). The surface shape distortion of the secondary mirror is mainly the oblique trefoil aberration with the PV and RMS values of 16.31 nm and 3.067 nm, respectively shown in Table 4. Figures 9(b1)-(b4) show that the oblique trefoil term decreases with the increase of the zenith angle, while the defocus and astigmatism terms gradually increase. As shown in Figs. 9(c1)-(c4), when defocus term is removed, the surface shape distortions are mainly the oblique trefoil and astigmatism aberration for the zenith angle of 0$^\circ $ and 90$^\circ $, respectively. Figure 10 shows the variation curve of gravity-induced surface shape distortion with the zenith angle. The PV value of the distortion decreases gradually with the increase of the zenith angle, while the RMS value is almost invariant. As shown in Table 4, the RMS values are 3.005 nm and 1.050 nm for the zenith angle of 0$^\circ $ and 90$^\circ $, respectively. Table 4 also shows the misalignment of the secondary mirror at the zenith angle of 90$^\circ $. In Table 4, $\Delta x$ and $\Delta y$ in the table represent the lateral displacements, while $\Delta z$ denotes the axial displacement (as show in Fig. 3). Tilt X and Tilt Y respectively represent the rotations of the secondary mirror around x-axis and y-axis. Taking the zenith angle of 0$^\circ $ as the reference, the lateral displacement of the secondary mirror at the zenith angle of 90$^\circ $ is 0.41 nm, while the axial displacement is 0.29 nm and the rotation angle is 0.042′′. Thus, the misalignment of the secondary mirror is very slight and could be ignored. From Figs. 9, 10 and Table 4, it can be seen that the gravity-induced surface shape distortion is well depressed in the zenith angle range of 0$^\circ $ to 90$^\circ $ and the HSMA is able to maintain high-precision surface shape and accurate alignment of the secondary mirror under different attitudes.
Fig. 9. Surface shape distortions of the secondary mirror under multiple loads at different zenith angle. (a1)-(a4) show the distortions when the zenith angle is 0$^\circ $, 30$^\circ $, 60$^\circ $ and 90$^\circ $, respectively. (b1)-(b4) show the corresponding Zernike decomposition coefficients of (a1)-(a4). (c1) and (c2) represent the residual distortions and Zernike coefficients of (a1) after removing the defocus term. (c3) and (c4) are the residual distortions and Zernike coefficients of (a4).
Table 4. Surface shape distortion and misalignment of the secondary mirror at zenith angle of 0$^\circ $, 30$^\circ $, 60$^\circ $, and 90$^\circ $
5. Experimental investigation of the HSMA
In the experiment, a HSMA fabricated in our lab at the room temperature of 25°$\mathrm{{C}}$ is adopted to evaluate the surface shape maintenance under critical conditions. As shown in Fig. 11, the HSMA mainly consists of a secondary mirror, a triangular backplate, and a hybrid ball hinge unit. Here, the secondary mirror is a convex hyperboloid mirror with a curvature radius of 1008.4 mm and the conic constant of -3.4624 mm. To decrease the thermal effect under a wide range of environmental temperatures, the Zerodur with the thermal expansion coefficient close to zero is chosen as the material of the secondary mirror.
Fig. 11. Lab-manufactured HSMA. (a) and (b) show the top view and side view of the HSMA with the uncoated secondary mirror, respectively. (c) shows the HSMA with the coated secondary mirror.
In order to measure the secondary mirror surface shape under a wide range of ambient temperature, a test optical path is built as shown in Fig. 12. It mainly consists of a HSMA, a 4D interferometer (PhaseCam 6010), a compensator assembly, an adjustment mount and a temperature control box. In the test optical path, a meniscus compensator is adopted as a null lens and the design residual PV/RMS values of the optical path could reach as low as 1.71 nm and 0.32 nm respectively. The effective diameter of the compensator is set 220 mm, while the curvature radii of the front and rear surfaces are set 409.87 mm and 394.5 mm respectively. In the experiment, due to the limitation of the temperature control range of the temperature control box, the experimental temperature is set ranging from 5°${C}$ to 45°${C}$ (i.e., $\Delta T$ of ±20°${C}$ based on the initial temperature of 25°${C}$). The wavelength of the test optical beam employed in the interferometer is 632.8 nm.
Fig. 12. Experiment setup for the temperature-induced distortion measurement of the HSMA at the zenith angle $\theta $ of 90$^\circ $. (a) shows the schematic diagram of the test optical path, where L1 = 33.472 mm and L2 = 1301.339 mm. (b) shows the compensator assembly and the HSMA in the temperature control box.
First, the surface shape of the secondary mirror is measured as a reference at the room temperature of 25°${C}$. Then, the ambient temperature is changed to 5°${C}$ ($\Delta T$ of -20°${C}$) and the surface shape is measured by the 4D interferometer after the temperature of the HSMA reaching balance. After that, the ambient temperature slowly raises to 45°${C}$ ($\Delta T$ of 20°${C}$) and the surface shape is measured again after temperature balance. Figures 13(a1)-(a3) show the measured surface shapes of the secondary mirror at the ambient temperature of 45°${C}$, 25°${C}$ and 5°${C}$, respectively. Figures 13(b1)-(b3) display the interference fringes. Corresponding Zernike coefficients of the measured surface shape are exhibited in Fig. 13(c1)-(c3). Taking the surface shape at room temperature as the reference, the variations of Zernike coefficients at the temperature variation of ±20°${C}$ are shown in Figs. 13(d) and (e), respectively. It can be seen that the surface distortion of the secondary mirror is mainly the change of oblique trefoil and astigmatism terms, which is consistent with the simulation results. When the ambient temperature is 25°${C}$, as shown in Fig. 13(a2), the surface shape mainly represents the initial distortion with the RMS value of 10.315 nm. When the ambient temperature varies ±20°$\mathrm{{C}}$ from the initial temperature, the surface shape of the secondary mirror almost remains unchanged, while the RMS values are respectively 10.680 nm and 10.606 nm [Fig. 13(a1), Fig. 13(a3)] and very close to the initial value. The corresponding variation rates of the RMS value are as low as 5.95 × 10−3$nm/^{\circ}{\mathrm{C}}$ and 7.7 × 10−3 $nm/^{\circ}{\mathrm{C}}$, respectively. It should be noted that the experiment results are not strictly the same with the simulation due to practical processing and mounting of the HSMA, experiment environment and measurement error. Based on the above experiment results, the change of the secondary mirror surface shape at a wide range of ambient temperature is very small. The effect of the slight change of surface shape on the performance of the telescope could be ignored. It can be concluded that the proposed HSMA can significantly reduce the effect of thermal stress on the mirror surface caused by the temperature change. The HSMA is able to maintain high-precision surface shape of the secondary mirror over a wide range of temperature.
Fig. 13. Experimental results of the surface shape of the secondary mirror at a wide range of ambient temperature. (a1) is the measured surface shape at the temperature of 45°${C}$ ($\Delta T$ of 20°${C}$) with PV value of 0.127 µm and RMS value of 10.606 nm, (a2) is the measured surface shape at the initial room temperature of 25°${C}$ with PV value of 0.109 µm and RMS value of 10.315 nm, (a3) is the measured surface shape at the temperature of $5^{\circ}{\mathrm C}$ ($\Delta T$ of -20°${C}$) with PV value of 0.135 µm and RMS value of 10.680 nm. (b1)-(b3) are the interference fringe patterns of (a1)-(a3), respectively. (c1)-(c3) are the Zernike decomposition coefficients of (a1)-(a3), respectively. (d) and (e) are the variations of Zernike coefficients for the temperature of 45°${C}$ and 5°${C}$, taking the Zernike coefficients of temperature of 25°${C}$ (c2) as the reference.
To furtherly investigate thermal performance of the HSMA under complex ambient environment, a cycling temperature change test is carried out by using the same experimental setup as shown in Fig. 12. Before the temperature change test, the surface shape and the corresponding interference fringe are recorded. During one thermal cycle, the temperature inside the control box raises from the original 25°$\mathrm{{C}}$ to 45°$\mathrm{{C}}$, then decreases to 5°$\mathrm{{C}}$, and finally raises back to 25°$\mathrm{{C}}$ at the rate of 0.8°$\mathrm{{C}}$/minute. At the highest and lowest temperatures of 45°$\mathrm{{C}}$ and 5°$\mathrm{{C}}$, the temperature is controlled to maintained 4 hours to ensure temperature balance. After one thermal cycle, the surface shape and interference fringe of the secondary mirror are measured at the temperature of 25°$\mathrm{{C}}$ after temperature balance. In the experiment, the cycling temperature change test are repeated five times and the measurement results are listed in Table 5. In the five times temperature change test, the variation of the PV value of the secondary mirror is as small as 17.8 nm, while that of the RMS value is 0.271 nm. Moreover, misalignment of the mirror, deriving from the interference fringe, can reach high positioning accuracy with maximum value of 0.84′′ and average value of 0.74′′. These results indicate that the HSMA has good surface shape stability and positioning accuracy under complex ambient temperature.
Table 5. Experiment data of the cycling temperature change test
In order to measure the gravity-induced surface shape distortion at the zenith angle θ of 0°, an experiment setup was also built in the laboratory, as shown in Fig. 14. The experiment setup mainly consists of a HSMA, an interferometer, a compensator assembly, an adjustment mount and a test frame. Figures 15(a1) and (a2) show the measured surface shapes and interference fringes of the secondary mirror, while corresponding Zernike coefficients are exhibited in Fig. 15(a3). It can be seen that the surface distortion of the secondary mirror within the 10th order is mainly the oblique trefoil aberration with the PV value of 0.136µm and RMS values of 13.289 nm, respectively. Figure 15(a4) shows the variation of Zernike coefficients of the surface shape at the zenith angle of $0^\circ $, taking the zenith angle of $90^\circ $ as the reference. The Zernike coefficient within the 10th order is almost unchanged. The effect of the slight change of surface shape on the performance of the telescope could be ignored. It can be concluded that the proposed HSMA is able to maintain high-precision surface shape of the secondary mirror at different attitudes.
Fig. 14. Experiment setup for the gravity-induced distortion measurement of the HSMA at the zenith angle $\theta $ of 0$^\circ $.
Fig. 15. Experimental results of the surface shape of the secondary mirror at the zenith angle $\theta $ of 0$^\circ $. (a1) is the measured surface shape of the secondary mirror with PV value of 0.136µm and RMS value of 13.289 nm, (a2) is the interference fringe patterns of (a1). (a3) is the corresponding Zernike decomposition coefficients. And (a4) is the variations of Zernike coefficients for the zenith angle $\theta $ of 0$^\circ $, taking the Zernike coefficients of the $\theta $ of 90$^\circ $ (Fig. 14(b2)) as the reference.
In order to measure the positioning accuracy of the secondary mirror at different attitudes, a verification experiment setup by using three dial indicators was built. As shown in Fig. 16(a1), dial indicators A and B are installed at the triangular backplate of the HSMA to measure the displacement of the secondary mirror in z-axis, while dial indicator C is used to measure the displacement in y-axis. In the experiment, the measured data of dial indicators A, B and C are first recorded as reference values at the zenith angle θ of 0°. Then the HSMA is rotated to point horizontally and displacement of the secondary mirror related to the backplate is measured by the three dial indicators. During the rotation of the HSMA from 0° to 90°, all the indicators of the three dial indicators remain almost no change and the displacements of the secondary mirror are smaller than 0.1µm in y-axis and z-axis. It should be noted that in the experiment, the measurement processes are repeated more than five times and the measurement results show that the HSMA has good repeatability and stability.
Fig. 16. Experiment setup for positioning accuracy measurement of the secondary mirror at different attitudes from 0$^\circ $ to 90$^\circ $. (a) shows the schematic diagram of the experiment setup. (b) shows the practical experiment setup.
6. Conclusion
In order to maintain high-precision surface shape of the secondary mirror over a wide range of temperature, a hybrid ball-hinged secondary mirror assembly (HSMA), mainly consisting of a secondary mirror, a triangular backplate and a hybrid ball hinge unit, is proposed in this paper. The HSMA could isolate the secondary mirror from external thermal disturbance by using a central positioning ball hinge, three sets of axial positioning ball hinges and three sets of axial preload spring units. The finite element model of the HSMA is established and the temperature-induced distortion characteristics of the secondary mirror are investigated in simulation. Simulation results show that the temperature-induced surface shape distortion is mainly the defocus aberration when ambient temperatures are 70°${C}$ and -30°${C}\; $($\Delta T$ of ±50°${C}$ from the initial temperature). As the defocus term of the surface shape distortion can be eliminated by axial alignment of the secondary mirror during the telescope operation, the residual distortion without defocus term is furtherly considered. The calculated residual distortions are both mainly oblique trefoil aberration with the maximum PV value of 16.31 nm and RMS value of 3.005 nm for the ambient temperature from -30°${C}$ to 70°${C}$. The change of secondary mirror surface shape distortions over a wide range of temperature are minimal and negligible. In the experiment, a HSMA fabricated in our lab at the room temperature of 25°$\mathrm{{C}}$ is adopted and a test optical path based on a temperature control box is built. The experimental ambient temperature variation range is set ranging from 5°${C}$ to 45°${C}$. Experiment results well match the simulation and show that the surface shape of the secondary mirror almost remains unchanged when the ambient temperature varies ±20°$\mathrm{{C}}$ from the initial temperature. Furthermore, since the secondary mirror is also subject to the influence of gravity at different attitudes, the capabilities of the HSMA to maintain high-precision surface shape and accurate alignment of the secondary mirror at different attitudes are investigated. Both simulation and experimental results show that the gravity-induced distortion is well depressed at different attitudes from 0$^\circ $ to 90$^\circ $. It can be concluded that the proposed HSMA has good isolation capability against complex external environment. It is able to maintain high-precision surface shape of the secondary mirror over a wide range of temperature and at different attitudes from 0$^\circ $ to 90$^\circ $.
Funding
Tsinghua Initiative Scientific Research Program; Tsinghua University Education Foundation; National Natural Science Foundation of China (61775112).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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