Optimization and Tolerance Scheme for a Mirror Mount Design Based on Optomechanical Performance

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1 Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010, pp Optimization and Tolerance Scheme for a Mirror Mount Design Based on Optomechanical Performance Hagyong Kihm and Yun-Woo Lee Center for Space Optics, Korea Research Institute of Standards and Science, Daejeon (Received 28 May 2010, in final form 13 August 2010) We present the design optimization and tolerance scheme for the fold mirror system in our satellite telescope. A new type of mirror and mount flexure is proposed. The mirror is light-weight and is designed with an asymmetric off-axis configuration. The mount is a combination of monolithic axial and lateral supports. The design parameters of the flexure mount are optimized for optomechanical performances by using a simulated annealing method. The sensitivities of the optomechanical performances to the design parameters are tabulated to obtain tolerance for the critical dimensions by using finite element analysis (FEA). Optical distortions are examined with Zernike polynomials for qualitative analysis and design feedback. This unprecedented mirror and mount design would be a promising candidate for an asymmetric or off-axis optical system. PACS numbers: Eq, v, Fw Keywords: Mirror mount, Optomechanics, Flexure, Tolerance, Space optics DOI: /jkps I. INTRODUCTION Optical systems require support structures that isolate the optical parts from mechanical and thermal loads. Mechanical loads are gravity, vibration, assembly or mounting errors, and fabrication residual stress. Thermal effects might distort optical surfaces and induce displacements due to a mismatch of CTE (coefficients of thermal expansion) or to a temperature gradient within the optical system. Therefore, the performance of optical mirrors, such as those employed in satellite telescopes, may be severely degraded by an inappropriate mounting configuration. The general design objective and philosophy of an optical mirror mount are introduced by Chin [1]. The concepts for a successful mounting should minimize optical distortions and provide a simple means of alignment. Also, the mount must be athermal. Achieving good thermal stability performance might be critical even in laboratory environment [2]. A kinematic mount is an ideal support constraining three orthogonal axes without redundancy. However, the point contact desirable for a kinematic support is not feasible in environmentally challenged systems. Instead, a semi-kinematic mount with a finite contact area is usually adopted to disperse local stresses. Flexure mounting may be regarded as a semi-kinematic design. A flexure is a monolithic structure providing elastic motions in a predefined way. The benefits of using flexures include hkihm@kriss.re.kr lack of the hysteresis and the friction effects inherent in semi-kinematic mounts. Also, maintenance is unnecessary and fabrication has become common practice with electrical discharge machining. A mirror mount flexure is not intended for linear or precise motions. Different from the flexure hinges used in actuator mechanism, a mirror mount flexure minimizes optical surface distortions and maintains optical alignment under operation or transport. Kinematic principles determine the location and the direction of a mounting flexure. The line of action of the flexure should pass through the mirror s center of gravity. Compliance should be provided to athermalize the mirror and mounting flexures. For example, radial compliance should be added in an axisymmetric mirror element. Tangential compliance is also required to prevent assembly stress from propagating toward the mirror surface [3]. Lateral mirror supports are usually responsible for radial and tangential compliance. Axial mirror supports, in the case of massive large aperture telescopes, reduce mirror surface distortion. Flexure mounts can be categorized according to the type of flexure element. Simple blade flexures are usually used as tangential edge supports for relatively small axisymmetric mirrors [3]. A bipod flexure, which is the most common support type, generally gives better results in terms of optical performances [4, 5]. Flexure hinges can also be categorized according to the crosssectional shape. There are corner-filleted flexure hinges and conic-section (circular, elliptical, parabolic, and hyperbolic) flexure hinges [6 8]. Corner-filleted flexures are more bending-compliant and induce lower stresses.

2 Optimization and Tolerance Scheme for a Mirror Mount Design Hagyong Kihm and Yun-Woo Lee Table 1. Design loads and requirements for the mirror assembly Design loads Optomechanical performances Safety factors for survival Material strength vibration 59 G (gravitation) assembly error 10 µm isothermal 20 C ± 12 fundamental frequency >210 Hz mirror distortion <15 nm mirror decenter <2 µm mirror tip/tilt <0.01 flexure yield >1.25 flexure buckling >10 mirror fracture >2 adhesive breakage >3 mirror fracture (Schott Zerodur) flexure yield (Invar36) adhesive (3M EC2216 B/A Gray) 10 MPa 210 MPa 6 MPa Less precision in rotational motion does not matter in mirror mount flexures. Most mirror mount flexures, be they blade or bipod, have corner-filleted cross sections for these reasons [3,9]. Optimal design of flexure hinges is possible for precision mechanisms based on theoretical closed-form solutions [10, 11] or finite element analysis (FEA) [12]. Empirical formulation and dimensionless graph analysis have also been reported [7,13]. Those research-efforts have provided initial design steps and even optimized simple cases for precision mechanisms. Mirror mount flexures can also benefit from those results being used as design guidelines. Optical performance, however, is a major criterion in mirror mounting flexures. For example, a theoretical derivation of optimum mount solution was made for a simple mirror disk [14]. The FEA has gained popularity in optical mirror mount design as the mirrors are light-weight and more complicated [15 17]. A comparison between a FEA and an interferometric measurement showed the two results proved to be in good agreement [18]. This paper presents a flexure design procedure with a simulated annealing (SA) algorithm [19] for optimal optomechanical performance. Also, we propose a new methodology to obtain tolerances for the critical dimensions by using a FEA. Section II. shows the overall configuration of a folding mirror system and design optimization. Section III. explains the performance results and tolerances based on a sensitivity table. Conclusions follow in Section IV. II. PARAMETRIC OPTIMIZATION USING SIMULATED ANNEALING The mirror and flexure assembly presented in this paper has a unique mounting scheme. The mount has three sets of bipod flexures on a single frame. The main bipod supports at the center of gravity, and the others prevent bending moments at the rear. Traditional mirror mounts have axial and/or lateral supports fabricated and assembled separately, but this mount is monolithic due to the mechanical interference with other mirror assemblies. With this concept, two different mirror sets are made in a space telescope system. The design and the tolerance processes developed in this paper are applied directly to those two cases. For optical systems for military or space missions, the design and tolerance processes have been veiled in the optomechanical engineer s expertise or have been determined by their experiences. Although this research seems to focus on a specific application, we tried to generalize the optomechanical design process with our example. Design loads and optomechanical requirements for the mirror system are summarized in Table 1. The vibration load comes from the launch environment and has three orthogonal directions. The assembly load is related to the fixture base flatness. Flexure mount is fixed with fasteners on flat bezel inserts, and their irregularities distort the flexure mount, resulting in optical distortion. The flatness tolerance of the bezel inserts is 10 µm. The satellite telescope suffers temperature variations in orbit, and a mismatch of CTE between the mirror, flexure, and adhesive should not affect the optical performance or the system s survival. The mirror surface distortion should be less than 15 nm for diffraction-limited system performance. Safety factors for system s survival differ depending on the parts material and the failure modes. Material strengths are taken from the vendors reports. The adhesive shear strength, however, was obtained by using an in-house coupon tests to reflect the real application. Figure 1 shows a schematic of the mirror assembly. The mirror s center of mass, which determines the angle and the position of bipod flexures, is marked for reference. Blade flexures provide radial compliance for thermal expansion mismatch, and bipod flexures have tangential compliance for the assembly load and the bending

3 -442- Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010 moments. The front flexure is positioned in the vicinity of mirror s center of mass while the other flexures support the rear. Critical dimensions of concern are denoted for parametric optimization. Their names and optimized dimensions are given in Table 2. The rotation centers of the bipod flexures coincide with the mirror s center of mass. The base frame of the flexure mount is triangleshaped and has tree holes for joining bolts with bezel inserts. We implemented a simulated annealing (SA) method for structural optimization [19]. SA is proven to be useful in global optimization and avoids being trapped at local minima [20]. The algorithm employs a random search, which accepts some increasing changes in the objective function f, as well as decreasing changes. Values increasing f are accepted with a probability ( p = exp δf ), (1) T where δf is the increase in f and T is referred to as the temperature. T plays a role similar to the temperature in a physical annealing process. To avoid getting trapped at local minima, the reduction rate should be slow. The objective of this study is to find a design meeting all the requirements shown in Table 1. We used MATLAB R for SA routine and optical analysis with Zernike polynomials [21]. MATLAB R gets FEA results from CATIA R and updates dimensional parameters after optomechanical analysis. We used four constraints for mechanical safety in each gravity direction. They are listed in Table 1 as safety factors for survival. In total, 13 constraints, including a frequency requirement, were used in the SA. Also, we tried to minimize optical distortions and stresses in the adhesive. However, we could not observe any significant improvement in optical distortions, which proved to be insensitive to dimensional variations of the flexures. We found from this result that optical performance was determined mainly by the mounting configuration rather than by the detailed dimensions of the flexures. This manifests the importance of proven design examples and optomechanical experiences. On the other hand, the stresses in the adhesive under vibrational loads were sensitive to the flexure dimensions. Figure 2 shows the parametric variations to minimize stresses in the adhesive pads. Over 90 iterations were conducted for convergence. Figure 3 shows the history of shear stress at the adhesive from Fig. 2. About a 16% improvement was observed. III. PARAMETRIC OPTOMECHANICAL PERFORMANCE EVALUATION Aside from design and optimization of the mirror and its mounting structures for the given requirement, tolerances for the critical dimensions are important. Tol- Fig. 1. (a) Schematic of the mirror assembly. (b) Section view showing the interface between the mirror and the flexure mount. (c) Independent design parameters for the flexure mount. Fig. 2. (Color online) Parametric variations from simulated annealing optimization. erance determines manufacturing methods and assembly procedures. Fabrication cost, time, and difficulty all depend on dimensional tolerance. Performance reproducibility also rely on tolerances. Mechanical designers usually decide the tolerance of each part from the viewpoint of manufacturability, assembly, and cost. In optomechanical systems, however, tolerance cannot be determined solely from mechanical viewpoints. We should consider optical surface distortion, tip-tilt, and a decenter induced by gravity, assembly load, and thermal load. Adhesive bonding, which is generally adopted for coupling an optical component with a metallic mount, might be a critical factor. Several authors have reported on optomechanical sensitivity and tolerances [22,23], but the results were confined to the lens assembly and the fabri-

4 Optimization and Tolerance Scheme for a Mirror Mount Design Hagyong Kihm and Yun-Woo Lee Table 2. Flexure design parameters and their values (mm). D 1 width D 2 Front radial flexure height D 3 thickness 2.04 D 4 Front tangential width 6.96 D 5 flexure thickness 4.17 D 6 height Rear radial flexure D 7 thickness 2.56 D 8 width Rear tangential flexure D 9 thickness 3.85 Fig. 3. Shear stress minimization with the parametric variations in Fig. 2. cation. In this paper, we propose to derive dimensional tolerances for a mirror mount from the sensitivities of the optomechanical performances, which, to our knowledge, is the time such an approach has been suggested. The tolerance is assumed to be inversely proportional to the sensitivity, which is defined as S ij = P i D j, (2) where S ij is the sensitivity of performance P i with respect to dimension D j. The tolerance of dimension D j is proportional to the allowable tolerance ratio T j as follows. T j min i P i δ i S ij, where δ i = Q i P i Q i. (3) Q i is the performance requirement, and δ i is the safety distance indicating how critical the dimension is. The tolerance is tightened when the safety distance has a small value. Table 3 shows the performance parameters and their values under environmental loads. The mirror s rigid body motion due to the assembly load is ignored as it can be adjusted during integration. Q i was calculated from the design load, safety factor, and material strength of each part. For example, Q 1 was obtained by dividing the material strength (210 MPa) by the gravity load (59 G) Fig. 4. (Color online) Mirror surface distortion under 1 G gravity in y-direction. and the safety factor (1.25). There are three orthogonal directions in gravity and three positions for the assembly load. In this paper, however, only maximum values of concern are presented. In the case of a 1-G gravity load, most performances give small safety distances, except for the mirror s tip/tilt. This means that the mechanical safety largely depends on the flexure design. The mirror s decenter under a 1-G gravity shows a small safety distance δ 12, which will be released in a non-gravity space environment. The CTE mismatch between the mirror, flexure, and adhesive results in a high shear stress at the adhesive, which is evident in the small distance δ 9. The Optical distortion of the mirror surface was evaluated by fitting the displacement with Zernike polynomials up to 49 terms [21]. Piston and tilt terms were used to calculate the decenter and the tip/tilt errors. The rest terms relate to the wavefront errors degrading the optical performances. Figure 4 shows the distortion map under 1-G gravity in the y-direction. Astigmatism is dominant with 23 nm, and the overall root-mean-square (rms) value is 4.1 nm, showing satisfactory performance. Figure 5 shows the distortion map under an assembly load (10 µm) at the rear insert position. The rms value is 5.2 nm, which also gives enough safety distance δ 17. Table 4 is the sensitivity S ij and derived tolerance of each dimension D j. The tolerance ratios T j were divided by the minimum value T 4 and are plotted in Figure 6

5 -444- Journal of the Korean Physical Society, Vol. 57, No. 3, September 2010 Table 3. Performance parameters and their values. i Performance parameter Design load Q i P i δ i 1 1 G gravity σ Von flexure (MPa) Assembly (10 µm) Isothermal (1 C) G gravity σ Von mirror (kpa) Assembly (10 µm) Isothermal (1 C) G gravity σ adhesive (kpa) Assembly (10 µm) Isothermal (1 C) Fundamental frequency Buckling factor G gravity Mirror decenter (µm) Isothermal (1 C) G gravity Mirror tip/tilt (deg) Isothermal (1 C) G gravity Mirror distortion (nm) Assembly (10 µm) Isothermal (1 C) Table 4. Sensitivities and derived tolerances. j S 1j S 2j S 3j T j tol. (mm) E E E E+3-8.9E E E E E E E E E E E E+3-1.0E E E+3 1.4E+3 3.1E E E E E+3-5.0E E Fig. 5. (Color online) Mirror surface distortion under an assembly load of 10 µm displacement at the rear insert. for qualitative comparison. One can then distribute the tolerances based on these parameters. Table 4 can also be used as a lookup table to interrogate performance degradation due to dimensional errors. The front radial flexure (D 1 3 ) has relatively large tolerances compared with other dimensions. Tangential flexures are proven to be critical for performance stability, so care must be taken in the manufacturing process. The last column shows the tolerance example for our case. IV. CONCLUSIONS We presented a new type of mirror/mount flexure design and proposed a new tolerance scheme for mirror mounting flexures based on FEA. The mirror is a pris- Fig. 6. Tolerance ratio T j for each dimension is plotted for comparison. matic off-axis folding mirror. The mount has three sets of monolithic flexures on a single frame, and each flexure is a combination of blade and bipod flexures. The optomechanical performance and the sensitivity method

6 Optimization and Tolerance Scheme for a Mirror Mount Design Hagyong Kihm and Yun-Woo Lee were proposed as new tolerance guidelines rather than relying on the arbitrary tolerances made by experienced engineers. We applied this procedure to two other sets of mirror assemblies for an infrared space telescope. Mission critical systems, like space and military optics, will benefit from our results. REFERENCES [1] O. Chin, Appl. Opt. 3, 895 (1964). [2] D. Wilson and K. D. Li Dessau, Opt. Photonics News 16, 40 (2005). [3] P. R. Yoder, Jr., P. Yoder, D. Vukobratovich and R. A. Paquin, Opto-Mechanical Systems Design, 3rd ed. (CRC Press, Boca Raton, FL, 2005). [4] T. Pamplona, Ch. Rossin, L. Martin, G. Moreaux, E. Prieto, P. Laurent, E. Grassi, J-L. Boit, L. Castinel, J. Garcia and B. Milliard, Proc. SPIE Int. Soc. Opt. Eng. 7018, (2008). [5] E. T. Kvamme and M. T. Sullivan, Proc. SPIE Int. Soc. Opt. Eng. 5528, 264 (2004). [6] Y. K. Yong, T-F. Lu and D. C. Handley, Precis. Eng. 32, 63 (2008). [7] Y. Tian, B. Shirinzadeh, D. Zhang and Y. Zhong, Precis. Eng. 34, 92 (2010). [8] N. Lobontiu, J. S. N. Paine, E. Garcia and M. Goldfarb, J. Mech. Des. 123, 346 (2001). [9] L. Furey, T. Dubos, D. Hansen and J. Samuels-Schwartz, Appl. Opt. 32, 1703 (1993). [10] Y. Wu and Z. Zhou, Rev. Sci. Instrum. 73, 3101 (2002). [11] Y. M. Tseytlin, Rev. Sci. Instrum. 73, 3363 (2002). [12] K-B. Choi and C. S. Han, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. 221, 385 (2007). [13] W. O. Schotborgh, F. G. M. Kokkeler, H. Tragter and F. J. A. M. van Houten, Precis. Eng. 29, 41 (2005). [14] G. Schwesinger, J. Opt. Soc. Am. 44, 417 (1954). [15] A. J. Malvick, Appl. Opt. 11, 575 (1972). [16] B. Mack, Appl. Opt. 19, 1000 (1980). [17] I. K. Moon, Y. W. Lee and Y-S. Kim, J. Korean Phys. Soc. 54, 1506 (2009). [18] D.-S. Wan, J. R. P. Angel and R. E. Parks, Appl. Opt. 28, 354 (1989). [19] S. Kirkpatrick, C. D. Gelatt, Jr. and M. P. Vecchi, Science 13, 671 (1983). [20] S. Kirkpatrick, J. Stat. Phys. 34, 975 (1984). [21] D. Malacara, Optical Shop Testing, 2nd ed. (Wiley- Interscience, New York, 1992). [22] S. Magarill, Proc. SPIE Int. Soc. Opt. Eng. 3786, 220 (1999). [23] C.-C. Cheng, Proc. SPIE Int. Soc. Opt. Eng. 6665, 66650H (2007).