Since the invention of the first telescope in the early 1600s, ground-based telescopes have been widely utilized in astronomical observation. At first, a simple refractive type was employed in a telescope design, but it has been replaced by a reflector type with the requirement of a large aperture for high resolution. In principle, a lens is inherent to chromatic aberration and fabrication of a large lens is impractical. In addition, support of a lens is permissible only at its edge, which makes it difficult to reduce the surface distortion induced by gravity. Currently the Yerkes Telescope is known to be the largest refractive telescope, having an objective lens 1 m in size [1], which is said to be the practical limit at that time. On the other hand, fabrication and aberration control of a mirror is achievable, and a mirror support can further relieve the mirror stress. As a result, the size of reflective telescopes has been increased, reaching 11.8 m in the Large Binocular Telescope (LBT) [2].

Meanwhile, a ground telescope utilizes an adaptive optics system to compensate for image distortion due to atmospheric turbulence [3, 4]. For example, the Keck Telescope employs 349 actuators on the external deformable mirror [5]. In the LBT, 700 actuators are arranged on the secondary mirror, which functions as a deformable mirror itself [6]. In those huge telescopes, bulky systems are required not only for turbulence compensation but also for precise spectrographic analysis. With modification of the optical path to a designated direction, a coudé mirror provides enough space for the external system. If a coudé path is identical to a telescope’s rotation axis, the complexity of the external system can be reduced. Unlike in a space telescope, in a ground telescope the mirror surface deformation induced by gravity depends on the telescope’s pointing direction. Self-weights and thermal variations of primary, secondary, and coudé mirrors and their mounts are critical to mirror surface distortions. Therefore, mirror supports should be dedicatedly designed, and many attempts have been reported to alleviate the surface distortion of a collimator [7–15]. Although a coudé mirror has a simple design compared to a primary or secondary mirror in a ground telescope, structural optimization of the coudé mirror design is also necessary to achieve high optical performance. To our knowledge, however, no coudé mirror optimization has been elaborately demonstrated in previous research.

In this research, we optimize the third mirror (M3) assembly installed on a 1-m class ground telescope. M3, which is the first coudé mirror, is going to be installed on the top of the primary mirror assembly, at 45° angle to the optical axis. To reduce mirror surface error (SFE), we optimize mirror support positions and the sizes of the mount. Here we minimize the SFE of M3 under gravity in both the vertical and horizontal directions, considering the altitude angle variation of the telescope. The mirror surface deformation is simulated using finite element analysis (FEA) and the coefficients of the Zernike polynomials are calculated in each individual design process. The sensitivities and tendencies of the Zernike terms for each design parameter are analyzed and reflected in the optimal M3 mount structure. The design criteria for the mirror mount mass, the structure of a tip-tilt stage, and the SFE due to temperature variation are also managed in this investigation. In chapter 2, the optimization process for the mirror-hole positions is demonstrated. In chapter 3, the optimization process of the mirror support structure and the results of the parametric study of the Zernike terms are introduced. In chapter 4, the thermally induced SFE is analyzed with respect to mirror bonding methods.

The configuration of the third mirror (M3) assembly is demonstrated in Fig. 1(a). It consists of the M3, its mount, and a tip-tilt plate. The M3 assembly is attached to the top of the primary mirror assembly; The red path indicates the coudé pass of collimated light. Zerodur (providing ultra-low thermal expansion, as shown in Table 1) is chosen for the M3 material. M3 has a flat elliptical surface, for which the major and minor axis lengths and thickness are 460, 330, and 50 mm respectively. The mirror size is determined by considering the size of the collimated beam. To reduce mirror distortion, the support needs to provide both axial and lateral symmetries [16, 17]. In the case of a circular flat mirror a 3-point support is typically used, and the number of points is increased by sustaining the symmetry to alleviate mirror stress. In our elliptical M3, the mirror supporting positions are managed as shown in Figs. 1(b) and 1(c). Hole 1 is positioned on the major axis, and the positions of holes 2 and 3 are symmetric about the major axis. Holes 2 and 3 are positioned at the same distance from the minor axis. The bonding area in Fig. 1(b) represents the fixed support as a boundary condition. It is set only on the mirror back surface, to reduce thermal deformation. This is discussed in detail in chapter 4.

TABLE 1. Material properties.

Material	Young’s Modulus (GPa)	Poisson’s Ratio	Density (kg/mm3)	Thermal Expansion Coefficients (K−1)
Zerodur	90.6	0.24	2.53 × 10−6	2.0 × 10−8
EC2216	0.69	0.43	1.32 × 10−6	1.02 × 10−4
Invar36	141	0.259	8.05 × 10−6	1.26 × 10−6

Figure 1. Third mirror assembly configuration: (a) Third mirror assembly with a primary mirror assembly, (b) isometric view,
(c) front view. Fixed support is expressed as red color.

The initial design points and design ranges are introduced in Table 2. The design ranges for hole depth and diameter are determined by considering the size of the studs attached to the holes. Figure 2 represents the design optimization process. First, a 3D mirror model is prepared using Ansys SpaceClaim (Ansys, PA, USA). Then the nodal deformations are determined for each design point using a commercial Ansys Mechanical program. We identify surface deformations under two different gravitational load directions, the z- and y-directions, in which the telescope is directed in the vertical (90° elevation) and horizontal (0° elevation) direction respectively. The nodal deformation values of the mirror surface are transferred to a Matlab opto-mechanical code, and surface distortion is represented by the coefficients of the Zernike polynomials. The response surface optimization of the Ansys Workbench is utilized to determine the points of minimal surface deformation. As expressed in Eq. (1), the objective function ( f ) returns the design point providing the lowest sum of the z-SFE (SFEz) and y-SFE (SFEy), after removing the piston and tilt terms. The weighting coefficients w^z and w^y of each term are both 0.5.

TABLE 2. Mirror design parameters (in units of mm).

Design Parameter	Hole 1 Position	Hole 2 Position	Hole 2–3 Distance	Hole Depth
Initial Design Point	90	90	120	30
Design Range	70–130	70–130	90–180	25–40
Optimal Design Point	115.8	91.7	158.4	40.0

Figure 2. The optimization process of the mirror hole design.

(1)$$f=w_z\times SF{E}_z+w_y\times SF{E}_y,w_z=w_y=0.5$$

In the response surface optimization, the Multi-Objective Genetic Algorithm is utilized based on a variant of the popular Non-dominated Sorted Genetic Algorithm-II. It supports multiple objectives and constraints and aims at finding the global optimum [18]. In our optimization process, it generates 1,000 samples initially and 1,000 samples per iteration, and converges after 3,588 evaluations. We finally check the convergence of the generated optimal point with removal of the piston and tilt terms using Matlab. The optimal design-parameter values for hole 1 and 2 positions, the distance between holes 2 and 3, and hole depth are 115.8, 91.7, 158.4, and 40.0 mm respectively. Here the design parameter step is determined as 0.1 mm by considering general manufacturing tolerance. The corresponding mirror surface deformations are demonstrated in Fig. 3. Figures 3(a) and 3(b) depict deformations of the initial design, and Figs. 3(c) and 3(d) depict the optimal design in the z- and y-direction, respectively. The peak-to-valley (PV) and average values in both directions are decreased by approximately 25% after optimization. As shown in Table 3, the surface errors after removing the piston (0^th) and tilt (1^st and 2^nd) terms in the z- and y-directions are 7.96 and 8.17 nm respectively. Those terms can be removed by mechanical control of the tip-tilt plate of the M3 mount. As shown in Fig. 4, lower-order Zernike terms, especially focus (4^th) and astigmatism-x (5^th) terms, are reduced by optimization. The higher order terms such as coma-y (8^th), spherical aberration (9^th), and 3-fold-y (11^th) are increased, but providing less sensitivities than the other terms do.

TABLE 3. Mirror surface errors (in units of nm).

Direction	z-direction	y-direction
Initial Design Point	10.41	10.60
Optimal Design Point	7.96	8.17

Figure 3. Mirror surface deformations: (a) and (b) initial design point deformations in the z- and y-direction respectively, (c) and (d) optimal design point deformations in the z- and y-direction respectively.

Figure 4. The amplitudes of Zernike terms for the initial and optimal design points.

After optimizing the mirror hole positions, we optimize an M3 mount structure to reduce the SFE. The mount structure is introduced in Fig. 5(a) and consists of the studs, mount, and tip-tilt plate. Here two design conditions are required for a coudé pass and optical tube assembly (OTA) control. First, the appropriate mount height is necessary to position the OTA’s rotational axis exactly at the M3 surface center. Second, the mount mass needs to be restricted by considering the OTA’s center of mass and assembly convenience. The design parameters are shown in Figs. 5(b) and 5(c), and the initial design points and design ranges are summarized in Table 4.

TABLE 4. Mirror mount design parameters (in units of mm).

Design Parameter	Stud Head Diameter	Bolt Contact Length	Stand Height	Plate Height	Plate Length
Initial Design Point	40	15	200	300	210
Design Range	27–50	8–20	150–250	230–320	180–240
Optimal Design Point	50	8	165	235	200
Design Constraints: Total Mass <15 kg

Figure 5. Configuration of the third mirror assembly: (a) The studs, mount, and tip-tilt plate shape, (b) stud design parameters, and (c) mount design parameters.

The stud design parameters consist of the head diameter, body diameter, and bolt contact length. The size of the bonding area is determined by the head diameter, and consequently its minimum diameter is determined such that the bond stress is less than the allowable stress. In our previous in-house test, the allowable bond stress between a glass mirror and Invar36 material is 4.13 MPa, with a safety factor of 1.5 [19]. With FEA simulation, the minimum head diameter is chosen as 27.0 mm. The maximum head diameter is determined by considering the thermally induced SFE. The stud bolting part is placed at its end to position a pivot point on the mirror centroid plane in y- direction. The minimum bolting length is chosen as 8 mm, to match the M8 bolt diameter utilized here. The distance between the M3 center and the M1 assembly top is approximately 300 mm. By considering the M1–M3 interface plate, the stand height ranges from 150 mm to 250 mm. Finally, the plate size range is determined to be enough to hold the stand.

Figure 6 shows the mount structure optimization process. Its strategy is the same as for the hole optimization process, and two design constraints are added for the total mount mass and height. After design optimization, the piston- and tilt-free SFEs are depicted in Fig. 7. Figures 7(a) and 7(b) demonstrate the initial SFEs in the z- and y-directions, while 7(c) and 7(d) demonstrate the optimized SFEs in the z- and y-directions respectively. The RMS errors and total mount mass are shown in Table 5. The SFEs decrease from 11.14 to 9.83 nm in the z-direction and from 11.95 to 10.97 nm in the y-direction after optimization. Figure 8 represents the SFE simulation results with respect to each design parameter. Figures 8(a), 8(b), and 8(c) represent the sensitivities and tendencies of SFEs with respect to the stud bolting length, stud head diameter, and stand height respectively. The SFE tendencies are similar in both directions for all parameters, and the stud length shows the largest sensitivity. The SFE is reduced with decrease in the stud length and with increase in the stud diameter and stand height. However, the stand height is determined as 165 mm, due to the mount mass constraint. Figure 9 demonstrates the amplitude change of each Zernike term by modifying the stud length and diameter.

TABLE 5. Mirror surface errors and total mount mass.

Design Point	RMS Error in z-direction (nm)	RMS Error in y-direction (nm)	Mass (kg)
Initial Design Point	11.14	11.95	18.63
Optimal Design Point	9.83	10.97	14.67

Figure 6. The optimization process for the M3 mount structure.

Figure 7. M3 surface errors (SFEs): (a) z-SFE in initial design, (b) y-SFE in initial design, (c) z-SFE in optimal design, and (d) y-SFE in optimal design.

Figure 8. M3 surface errors (SFEs): (a) SFE with respect to the stud bolting length, (b) SFE with respect to the stud diameter, and (c) SFE with respect to the stand height.

Figure 9. The coefficients of Zernike polynomials with respect to design parameters: (a)–(c) Astigmatism-x, coma-y, and 3-fold-y with respect to stud bolting length in the z-direction; (d)–(f) that in the y-direction; (g)–(i) Astigmatism-x, coma-y, and 3-fold-y with respect to stud head diameter in the z-direction; (j)–(l) that in the y-direction. The stud bolting length and stud head diameter are in units of mm.

The tendencies of the three most sensitive terms, astigmatism-x, coma-y, and 3-fold-y are analyzed for two design parameters: The stud bolting length and the stud head diameter. Figrues 9(a)–9(c) show the amplitudes of astigmatism-x, coma-y, and 3-fold-y with respect to the stud bolting length in the z-direction, and 9(d)–9(f) show that in the y-direction. The amplitudes of the astigmatism-x and coma-y terms are reduced with decrease in the stud length. In addition, the variation tendencies are identical in both directions. In the case of the 3-fold-y term, its amplitude increases when the stud length decreases, but the sensitivity is relatively low. Therefore, the RMS error rises with increase in stud length. Figures 9(g)–9(i) present the SFEs with respect to the stud head diameter in the z-direction, and Figs. 9(j)–9(l) represent that in the y-direction. Similar to the case of the stud bolting length, the tendencies of astigmatism-x and coma-y are opposite to that of 3-fold-y. The amplitudes of astigmatism-x and coma-y are reduced as the stud diameter increases.

In this chapter, we present the thermally induced SFEs of the two bonding schemes. Thermal deformation of the M3 surface is analyzed based on the optimized design in the previous section. The SFE budget for thermal variation is 0.6 nm/K. Figure 10(a) shows the final mount design, reflecting a 3-point tip-tilt structure. Between the two tip-tilt plates springs are arranged, to manually align plate distance. Figure 10(b) shows the generated mesh in Ansys Workbench, with total node number of 1,481,880. The four holes in the mount stands are added to reduce the mount mass. Their positions and sizes are determined not to degrade the gravitational SFEs of M3. The simulation results for thermal deformation with 5 K increase are demonstrated in Fig. 11. Considering the SFE budget for thermal variation, the SFE should be less than 3 nm for a 5 K increase. Figures 11(a) and 11(b) present the thermal SFEs for stud bonding type 1 and 2 respectively. Bonding type 1 indicates the bonding strategy in which the mirror back surface near the hole entrance and the hole’s inside wall are bonded with studs. Here the head and body of the stud are bonded to the mirror. Type 2 indicates that only the mirror back surface near the hole entrance is bonded to the stud head. As shown in Table 6, bonding type 1 provides an SFE of 8.34 nm RMS and type 2 provides 1.84 nm RMS for a 5 K variation. An increase in the bonding area produces more thermal stress, and the simulation results indicate that the mirror hole bonding with the stud body significantly affects the thermal distortion. As a result, bonding type 1 exceeds the budget for thermal SFE, and bonding type 2 is selected as our bonding scheme.

TABLE 6. Mirror surface thermal error.

Bonding Type	Thermal RMS Error with 5 K Variation (nm)
Type 1	8.34
Type 2	1.84

Figure 10. M3 assembly design: (a) 3D model, including a 3-point tip-tilt structure, (b) the mesh generated in Ansys Mechanical.

Figure 11. Thermally-induced M3 surface error (SFE): (a) bonding type 1 (stud back and side bonding), (b) bonding type 2 (stud back only).

In this research we have optimized the design of a coudé mirror assembly for a 1-m class ground telescope, including the mirror support positions and mount structure. All design parameters have been investigated using finite element analysis in Ansys Mechanical, and the coefficients of the Zernike polynomials have been calculated using Matlab. The optimized design parameters provided the minimal SFEs due to self-weight in both the vertical and horizontal gravitational directions. Not only gravity but also the mount mass, size, and thermal effects were reflected in the design criteria, for realization of proper beam transfer. By considering all of the design constraints, we demonstrated mirror SFEs of 9.83, 10.97, and 1.84 nm in the z-direction, y-direction, and for 5 K variation respectively. After manufacturing of M3 and its mount, the assembly process and performance test for SFE using an interferometer in the horizontal direction will be demonstrated in future work.

The configuration of the third mirror (M3) assembly is demonstrated in Fig. 1(a). It consists of the M3, its mount, and a tip-tilt plate. The M3 assembly is attached to the top of the primary mirror assembly; The red path indicates the coudé pass of collimated light. Zerodur (providing ultra-low thermal expansion, as shown in Table 1) is chosen for the M3 material. M3 has a flat elliptical surface, for which the major and minor axis lengths and thickness are 460, 330, and 50 mm respectively. The mirror size is determined by considering the size of the collimated beam. To reduce mirror distortion, the support needs to provide both axial and lateral symmetries [16, 17]. In the case of a circular flat mirror a 3-point support is typically used, and the number of points is increased by sustaining the symmetry to alleviate mirror stress. In our elliptical M3, the mirror supporting positions are managed as shown in Figs. 1(b) and 1(c). Hole 1 is positioned on the major axis, and the positions of holes 2 and 3 are symmetric about the major axis. Holes 2 and 3 are positioned at the same distance from the minor axis. The bonding area in Fig. 1(b) represents the fixed support as a boundary condition. It is set only on the mirror back surface, to reduce thermal deformation. This is discussed in detail in chapter 4.

TABLE 1. Material properties.

Material	Young’s Modulus (GPa)	Poisson’s Ratio	Density (kg/mm3)	Thermal Expansion Coefficients (K−1)
Zerodur	90.6	0.24	2.53 × 10−6	2.0 × 10−8
EC2216	0.69	0.43	1.32 × 10−6	1.02 × 10−4
Invar36	141	0.259	8.05 × 10−6	1.26 × 10−6

Figure 1. Third mirror assembly configuration: (a) Third mirror assembly with a primary mirror assembly, (b) isometric view,
(c) front view. Fixed support is expressed as red color.

The initial design points and design ranges are introduced in Table 2. The design ranges for hole depth and diameter are determined by considering the size of the studs attached to the holes. Figure 2 represents the design optimization process. First, a 3D mirror model is prepared using Ansys SpaceClaim (Ansys, PA, USA). Then the nodal deformations are determined for each design point using a commercial Ansys Mechanical program. We identify surface deformations under two different gravitational load directions, the z- and y-directions, in which the telescope is directed in the vertical (90° elevation) and horizontal (0° elevation) direction respectively. The nodal deformation values of the mirror surface are transferred to a Matlab opto-mechanical code, and surface distortion is represented by the coefficients of the Zernike polynomials. The response surface optimization of the Ansys Workbench is utilized to determine the points of minimal surface deformation. As expressed in Eq. (1), the objective function ( f ) returns the design point providing the lowest sum of the z-SFE (SFEz) and y-SFE (SFEy), after removing the piston and tilt terms. The weighting coefficients w^z and w^y of each term are both 0.5.

TABLE 2. Mirror design parameters (in units of mm).

Design Parameter	Hole 1 Position	Hole 2 Position	Hole 2–3 Distance	Hole Depth
Initial Design Point	90	90	120	30
Design Range	70–130	70–130	90–180	25–40
Optimal Design Point	115.8	91.7	158.4	40.0

Figure 2. The optimization process of the mirror hole design.

(1)$$f=w_z\times SF{E}_z+w_y\times SF{E}_y,w_z=w_y=0.5$$

In the response surface optimization, the Multi-Objective Genetic Algorithm is utilized based on a variant of the popular Non-dominated Sorted Genetic Algorithm-II. It supports multiple objectives and constraints and aims at finding the global optimum [18]. In our optimization process, it generates 1,000 samples initially and 1,000 samples per iteration, and converges after 3,588 evaluations. We finally check the convergence of the generated optimal point with removal of the piston and tilt terms using Matlab. The optimal design-parameter values for hole 1 and 2 positions, the distance between holes 2 and 3, and hole depth are 115.8, 91.7, 158.4, and 40.0 mm respectively. Here the design parameter step is determined as 0.1 mm by considering general manufacturing tolerance. The corresponding mirror surface deformations are demonstrated in Fig. 3. Figures 3(a) and 3(b) depict deformations of the initial design, and Figs. 3(c) and 3(d) depict the optimal design in the z- and y-direction, respectively. The peak-to-valley (PV) and average values in both directions are decreased by approximately 25% after optimization. As shown in Table 3, the surface errors after removing the piston (0^th) and tilt (1^st and 2^nd) terms in the z- and y-directions are 7.96 and 8.17 nm respectively. Those terms can be removed by mechanical control of the tip-tilt plate of the M3 mount. As shown in Fig. 4, lower-order Zernike terms, especially focus (4^th) and astigmatism-x (5^th) terms, are reduced by optimization. The higher order terms such as coma-y (8^th), spherical aberration (9^th), and 3-fold-y (11^th) are increased, but providing less sensitivities than the other terms do.

TABLE 3. Mirror surface errors (in units of nm).

Direction	z-direction	y-direction
Initial Design Point	10.41	10.60
Optimal Design Point	7.96	8.17

Figure 3. Mirror surface deformations: (a) and (b) initial design point deformations in the z- and y-direction respectively, (c) and (d) optimal design point deformations in the z- and y-direction respectively.

Figure 4. The amplitudes of Zernike terms for the initial and optimal design points.

This work was supported by the Defense Rapid Acquisition Technology Research Institute (DRATRI) - Grant funded by Defense Acquisition Program Administration (DAPA) (UC200012D).

Defense Acquisition Program Administration (DAPA) (UC200012D).