Open Access
Add to BibSonomy
Abstract
In the present work, a discrete flexure for silicon carbide (SiC) deformable mirror (DM) with Lead-Magnesium-Niobate (PMN) stacked-actuators is designed using topology and size optimizations. In order to fulfill the designated surface strokes of the SiC mirror without adhesive failure, discrete flexures are adopted between the mirror faceplate and the actuators. As the same design process for the flexure of the final DM model, which has a 140 mm diameter mirror faceplate and 489 channels of piezoelectric PMN-30PT single crystal multilayer actuators, the topology and the sizes of the flexure for the 5×5 engineering DM model are optimized. The topology and the size optimizations are conducted by GTAM and DesignXplorer in ANSYS based on finite element method (FEM). The prototype of the designed model is built and the test results show appropriate mirror deformations comparing with the simulated results without adhesive failure.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Deformable mirror (DM) is an essential component in adaptive optics system, which can improve the image quality by correcting a distorted wavefront. The shape of the mirror faceplate of a DM is physically controlled by actuators and the distorted wavefront is compensated by the shape changes. In the research on adaptive optics by H. W. Babcock in 1953, the first DM was proposed [1]. Since then, numerous works on the development of DM have been conducted in various fields such as astronomy, microscopy, ophthalmic system, laser communication, high power laser (HPL) system, etc. [2–7]. In the HPL application fields, one of the critical points is that the thermal distortion may occur in the mirror faceplate of a DM. The analyses and resolutions on that issue have been conducted [8–13]. The resolution for the thermal distortion is using high reflective coating and using a passive or active cooling system. Also the selection of the material for the mirror faceplate is one of the key points. In terms of the material of the mirror faceplate, SiC shows superior properties such as thermal stability and thermal diffusivity. If thermal stability (κ/α, where κ is thermal conductivity and α is thermal expansion) is high, the amount of wavefront distortion is low [8]. The time for thermal relaxation is reduced if the thermal diffusivity (κ/(ρCp), where is $\rho$ is density and Cp is specific heat capacity) is high [12]. However, the stiffness of SiC is very high compared with other materials ULE and Zerodur commonly used for the mirror faceplate of a DM. Thus, it is harder to bend the mirror faceplate because of the high stiffness of SiC. It leads the stresses of the components of a DM such as mirror faceplate, actuator, and adhesive to be higher than those of the cases where ULE or Zerodur is used for the mirror faceplate. If SiC is adopted to the mirror faceplate in a DM, considering the adhesive stress is critical to design a DM. Defining the spacing of the actuators is related to the atmospheric condition (Fried parameter, r0) and the size of the telescope. Commercial DMs usually have 3.2 mm to 7 mm spacing. Considering adhesive stress becomes more important in DM design when the spacing of actuators is narrow.
In most DMs, pushers or adapters are adopted between a mirror faceplate and actuators. The pusher is glued to the top of an actuator and it pushes the mirror faceplate upwards. It makes the contact area with the mirror smaller and the influence function sharper. The adapter is glued to both a mirror faceplate and an actuator, and it holds the mirror faceplate firmly in the push-up and pull-down conditions. The pusher and adapter are in simple shapes and it is hard to find the research on the design of them. Ahn et al. introduced flexures between a mirror faceplate and actuators and it is shown that the flexures can reduce the adhesive stress effectively [14]. However, the shape of the flexure is a simple double blade type and the influence of the flexure shape changes is not reported.
In the present research, the optimal topology and the sizes of the flexure for a SiC DM are found by structural optimizations. Since the stiffness of SiC is very high, the flexure is designed to have appropriate axial and bending stiffness to reduce the adhesive stress. Also, the mirror surface stroke is maximized by optimization of the flexure design. To find the initial flexure design, topology optimization is performed with different objectives and constraints about the 5×5 channel engineering DM model, which is the simplified one of the final 489 channel DM. The initial flexure design is derived from the results of topology optimizations, and the major dimensions are defined as the design variables for size optimization. To perform the size optimization effectively, response surface optimization is applied. As a result of the response surface optimization, candidate design points are derived and they are verified by simulations. The prototype of the 5×5 channel engineering DM is built and the test results are compared with the simulation results. Section 2 describes the optimization process of the flexure. The specification and the operating conditions of the final DM and the engineering DM are explained and the influence of flexure is analyzed. Through topology optimization and size optimization, the optimal flexure design is derived and verified by simulations. In section 3, the mirror surface deformations are measured in operating conditions by a commercial interferometer, and also the results are compared with the simulation results. Section 4 concludes the present research.
2. Flexure design for SiC DM with PMN stacked-actuators
2.1 Specification and operating conditions of DM
The design specification of the DM is as follows. The mirror faceplate is chosen as chemical vapor deposition(CVD) SiC considering future high power laser applications. Comparing with conventional materials, SiC shows superior optomechanical properties such as high thermal stability, high thermal diffusivity, and high specific stiffness [13]. The diameter of the mirror is 140 mm and the thickness is 1 mm. The DM has 489 actuators with 5 mm spacing in both lateral directions. The actuator is made of PMN-30PT[Pb(Mg2/3Nb1/3)O3-30%PbTiO3] with PZT insulation caps at the top and the bottom. It is known that a PMN actuator has higher stroke and blocking force in the same volume comparing with the conventional PZT actuator [15]. A base plate is placed at the bottom to hold the actuators rigidly. The material of the base plate is Invar36 which has an extremely low thermal expansion coefficient (CTE). For gluing the mirror, the actuators and the base plate, epoxy adhesive (3M EC2216 B/A gray) is applied. The mechanical properties of the key materials used in the present work are listed in Table 1.
Table 1. Mechanical properties of key materials
The actuator is controlled in a voltage range of 0–200 V and the design free stroke of the actuator is ≥ 6 μm at 200 V. The initial offset voltage of the DM is 100 V to make the initial state of the mirror for push-up and pull-down. In other words, the moving range of the actuator is between + 3 μm (at the driving voltage 200 V) and – 3 μm (at the driving voltage 0 V) based on the offset voltage 100 V. One of the harshest operating conditions is the full stroke push-up case: a single actuator is driven at + 6 μm and others are driven at 0 μm. Another one is the full stroke pull-down case: a single actuator is driven at 0 μm and others are driven at + 6 μm. The two cases are symmetrical for displacement, but not for the adhesive stresses. In the push-up state of an actuator, the adhesives of the actuator are under compressive stresses and the adhesives of the neighbor actuators are under tensile stresses. If an actuator is in the pull-down state, the stress states are reversed. Thus, the two operating conditions should be considered together in flexure design.
The target mirror surface stroke is decided to be enough for successful wavefront corrections. The target of the flexure design in the present work is to make the mirror surface stroke Peak-to-Valley (PV) ≥ 2.0 μm at the full stroke push-up and push-down conditions. The 2 μm stroke range of the DM is decided to be able to cover the Korean weather condition at the telescope site.
The most crucial point of the flexure design is preventing failure of the adhesive regions during the operation of a DM. The applied epoxy adhesive would be detached or fractured if the normal or shear stress of the adhesive reaches to the critical value. In a DM, the critical values depend on what materials are glued together. The normal and shear failure stresses are measured by in-house tests according to ASTM D3528 (Type B specimen). The shapes of the test specimens are shown in Fig. 1 and the dimensions t1, t2, and L were designed following to ASTM D3528.
After each 10 specimens were made, the failure strengths are obtained by using a commercial test equipment MTS 810 and an extensometer. The B-values, which are usually used in space mission applications, are calculated from the measured results for normal and shear. The formula for the B-value is as follows.
where ${S_{Fn}}$ is the nth measured failure stress, N is the total number of tests, and $\sigma $ is the standard deviation.The B-values of the failure strengths, the safety factors and the allowable stresses are listed in Table 2.
Table 2. Failure and allowable stresses of EC2216 with respect to materials
2.2 Influence of flexures
Before starting specific design of a flexure, in this section, the influence of flexures is evaluated. The three example types of DMs are shown in Fig. 2 and 3: directly glued type, monolithic flexure type, and discrete flexure type. The DMs are simulated by ANSYS Mechanical and the results are compared. In the simulations, the engineering DM model which has 5 by 5 actuators, is adopted and the materials of the mirror faceplate, actuators, and base plate are the same as those of the final DM model. The thickness of the mirror faceplate is 1 mm and the diameter is 40 mm. The thickness of the base plate is 10 mm and the diameter is 47 mm. PZT and Invar36 are applied for the monolithic flexure and the discrete flexure, respectively.
Fig. 2. 5×5 channel engineering DM model of the directly glued type; (a) front view, (b) trimetric view.
Fig. 3. 5×5 channel engineering DM models of the monolithic and discrete flexure types; (a) front view of monolithic flexure type, (b) monolithic flexure (upside down) (c) front view of discrete flexure type, (d) discrete flexures.
Figure 4 shows the finite element model of the directly glued type DM and the displacement and boundary conditions for the three analysis cases. The first analysis case is the push-up condition of the center actuator where the stroke of the center actuator is + 6 μm and the strokes of others are + 3 μm, i.e. the center actuator is driven at 200 V and the other actuators are driven at 100 V. The second analysis case is the push-down condition. The stroke of the center actuator is 0 μm and the strokes of others are + 3 μm, i.e. the center actuator is driven at 0 V and other actuators are driven at 100 V. The last one is the inter-actuator case where the stroke of the center actuator is + 6 μm and that of the one neighbor actuator is 0 μm. The strokes of the rest actuators are + 3 μm. In this case, the center actuator, its one neighbor actuator and the rest actuators are driven at 200 V, 0 V, and 100 V, respectively. The simulation results of the directly glued type for the three analysis cases are shown in Fig. 5.
Fig. 4. Analysis model and applied boundary conditions for directly glued type DM; (a) finite element model, (b) boundary conditions for push-up case, (c) boundary conditions for pull-down case, (d) boundary conditions for inter-actuator case.
Fig. 5. Deformations of mirror surface in z direction of directly glued type DM; (a) single actuator stroke: push-up condition (b) single actuator stroke: pull-down condition, (c) inter-actuator stroke (push-up condition on the center actuator and pull-down condition on the next actuator).
Table 3 shows the maximum values of the normal and shear stresses of three adhesive regions in the push-up and pull-down conditions with the mirror surface strokes. In most cases, the normal and shear stresses are higher in the pull-down conditions. Also the normal stresses are higher than shear stresses in the pull-down condition. It is noticeable that the stresses are reduced by introducing flexures without serious mirror surface stroke decay. Also, the stresses are concentrated to the adhesive region between the flexure and the mirror in the monolithic flexure type, but those are appropriately distributed to the three adhesive regions in the discrete flexure type. As a result, introducing flexure occurs stress decreasing and the distribution of adhesive stresses can be controlled effectively by flexure designs even though the stroke of mirror surface decreases little amount.
Table 3. Simulation results of push-up and pull-down conditions with respect to flexure type
Regarding to the allowable stresses of the adhesive as shown in Table 2, the discrete flexure type is most relevant to achieve high mirror surface stroke and low adhesive stress. Also, the discrete flexure can be manufactured in more complex shape for controlling its compliance.
2.3 Structural optimization of a flexure
In section 2.2, the discrete flexure shows the advantages for achieving high mirror surface stroke and reducing adhesive stresses. To maximize the mirror surface stroke in the actuator control condition without adhesive failure, the structure of a flexure is optimized. Figure 6 shows the examples of adapter, pusher, and flexures for DM. The simple plate shape of an adapter is used to connect a mirror and an actuator, and there are no functions for reducing adhesive stresses about axial and bending load. Figure 6 (c) and (d) are the examples to have the mentioned functions.
Fig. 6. Examples of adapter, pusher, and flexure; (a) plate adapter, (b) pusher, (c) simple rod flexure, (d) double blade flexure.
In the flexure design, there are some points should be considered. The first one is the axial stiffness of the flexure. The flexure should be stiff enough to maximize the mirror surface stroke satisfying the adhesive stresses. The second one is the gluing areas between the flexure and the mirror and between the flexure and the actuator. The areas should be enough to avoid adhesive failure. To maintain the gaps during the enough curing time, the flexures are needed to be designed in considering gluing process and the gluing fixtures. To satisfy the aforementioned issues, in the present research, the optimal structure of a flexure for a DM is derived by topology optimization and size optimization as the process shown in Fig. 7.
As the first step of the structural optimization process, the optimal topology of a flexure is found within the limited design space. Then the optimal topology of the flexure becomes the initial design of the flexure for size optimization. Several major sizes of the flexure are defined as the design variables and then the design of experiments (DOE) is conducted. The response surface, which is formed by expected responses of a combination of design points, are derived by the results of DOE. Finally, the major sizes of the flexure are optimized by a generic optimization algorithm, which is a global search optimization method.
2.4 Topology optimization of a flexure
To maximize the mirror surface stroke in actuator operating conditions without adhesive failure, the topology of a flexure is optimized. As a simple explanation, the topology of a structure means the structural features such as the location, shape and number of holes in the structure. There are lots of ways to optimize the topology of a structure, but Solid Isotropic Material with Penalization (SIMP) is the most popular mathematical method. In the present work, GENESIS topology for ANSYS Mechanical (GTAM), which is also based on SIMP is used. The method is very straightforward since it keeps the mesh of the initial structure, but it controls the densities of the elements individually and the young's modulus of the material is interpolated by the penalization [16]. The formula for this process is represented in Eq. (2).
The second step is applying the load and boundary conditions. Symmetric boundary conditions are applied to the x-z plane and the y-z plane, and the circumference of the mirror is fixed. As described the pull-down condition of an actuator, minus force is applied to the bottom of the top isolation cap of the actuator.
The optimization result varies depending on the objective function and the constraints. Mass, volume, displacement, strain energy, stress, etc. can be set as the objective function and they also can be set as the constraints. To conduct optimizations efficiently, only one channel of the DM is modeled and proper boundary conditions for the actuator operating condition are applied. It is not efficient to use the full model for topology optimization since numerous structural analysis is required during the optimization. In order to reduce the computation time, a quarter model is modeled and analyzed.
Figure 8 represents the design space and the frozen regions of the quarter model for topology optimization.
Six different objective and constraint cases are applied and the results are compared. Table 4 shows the summary of the optimization cases.
Table 4. Objectives and constraints for topology optimizations
Figure 9 shows the optimization history of case 1. As can be seen, the objective is converging with the constraint satisfaction. The histories of other cases show similar convergences. For the four cases, the topology optimization results with the same mass fraction are compared. Figure 10 shows the topology optimization results of the four cases. The visualization of the results is set to represent only the material where the density is over 0.8 to help the comparison. It is shown that the lower half parts of the flexures look similar, but the upper half parts do not. The truss structure in the lower half part helps to reduce the mass and the axial stiffness. Cone shapes appear in the upper half parts of the flexures except case 3. The reason of the result is that there are no constraints on displacements in case 3. The top neck shape can be interpreted as a shape for reducing stress concentration, rather than it increases the mirror surface displacement.
In case 4, the neck shape becomes narrower with adding the displacement constraint. The common regions and the topology of the flexure from the results of the four cases are reinterpreted and remodeled for shape optimization. The discrete flexures are manufactured by CNC milling machining and wire electric discharge machining. Considering the manufacturing process and the sizes of the flexure, unnecessary complex shapes should be avoided in the remodeling process. Thus the truss structure of the lower half part is simplified by a rectangular hole in the remodeling process. The bottom pad gets thicker to be held easily by a gluing fixture in the gluing process. Also, the upper and lower parts of the rectangular box are remodeled as cone shapes based on the optimization results. The top gluing area is changed to a circular shape, and a reversed conic structure is adopted to avoid non-axisymmetric influence function and to distribute the adhesive stress more evenly. The initial design of the flexure, as a result, is derived as shown in Fig. 11.
Fig. 11. Remodeled flexure design based on the results of topology optimization; (a) front view, (b) trimetric view.
2.5 Size optimization of a flexure
The initial model is derived by topology optimization in the last section. To make it to a more detailed and specific model, size optimization about the major sized of the flexure is performed in this section. The major sizes of the initial design of the flexure are set as design variables and their upper and lower bounds set as shown in Fig. 12.
A brief description of the optimization process is as follows. First, a combination of the design variables is set as a design point and then the DM model is analyzed by finite element method (FEM). Next, a response surface is formed with a lot of simulation results about different design points. Then, by using an optimization algorithm, the optimal design point is derived. Since the response of the optimal design point is the predicted value, the design point is verified by conducting structural analysis.
The size optimization is performed via response surface optimization of DesignXplorer in ANSYS Mechanical. For the response surface optimization, DOE about the design variables should be performed first. The design points are generated by central composite design and the allowed values of the design variables are defined discretely by 0.1 mm considering manufacturability. A 5×5 channel engineering DM is modeled and the load and boundary conditions are applied. As the load condition, the full stroke pull-down case which is the harshest condition for adhesives is applied. In this condition, the strokes of all the actuators are driven at 6 μm except the center actuator. The bottom side of the base plate is fixed as the boundary condition. Thus the center flexure, actuator and adhesives are under tensile condition. The structural analyses of the generated design points are conducted by FEM, and then the response surface is generated by the simulation results. The specific values of the design variables and the simulation results of the design points are listed in Table 5. The responses are the maximum displacement of the mirror surface center, and the principle stresses of adhesives at two regions: between the mirror and the flexure and between the flexure and the actuator. Here, DV1, DV2, DV3, DV4 and DV5 indicate the pole diameter, the sidewall thickness, the square hole height, the square hole center height, and the bottom pad thickness, respectively.
Table 5. Simulation results of design points for DOE
The objectives and the constraints are as follows.
- - Objective 1: Maximize max. disp. of mirror surface in z direction
- - Objective 2: Minimize min. disp. of mirror surface in z direction
- - Constraint 1: Max. Principal stress of adhesive between mirror and flexure ≤ 4.13 MPa
- - Constraint 2: Max. Principal stress of adhesive between flexure and actuator ≤ 2.60 MPa
- - Constraint 3: Max. Principal stress of adhesive between actuator and base ≤ 2.60 MPa
Table 6 lists the predicted results of the candidate design points as the results of the size optimization. The candidate design points are simulated for verification and the results are listed together. As can be seen, the adhesive stress of the candidate point 1 in the region between the actuator and the flexure is over little much the constraint 2.6 MPa. Thus the candidate design points 2 and 3 are acceptable and the candidate design point 3 is a better choice.
Table 6. Predicted and verified results of candidate design points
3. Simulation and experimental results
3.1 Simulation results of engineering DM model
The candidate design point 3 is adopted and some of the sizes are slightly modified considering the dimensions and shapes of the actuator isolation cap and the shape of gluing fixture. Also, the base plate is slightly modified considering the actuator cables. The simulation results of the three actuators in maximum strokes + 6 μm are shown in Fig. 13. In the figure, the subscript means the id of the operating actuator.
Fig. 13. Simulated mirror strokes at No. 7, 8, and 13 actuators (driven at + 6 μm); (a) result of No. 7 actuator, (b) result of No. 8 actuator, (c) map of actuator positions, (d) result of No. 13 actuator.
To check the validity of the optimized flexure, simulations of DMs for a plate adapter and a double blade flexure, which are depicted in Fig. 6, are performed and the results are compared together. In the additional simulations, the diameter and the thickness of the plate adapter is 4 mm and 2 mm. For the double blade flexure, the diameter of the flexure, the thickness of the blade and the thickness of each horizontal pad are 4 mm, 1 mm and 2 mm, respectively. Table 7 represents the simulation results.
Table 7. Comparison of simulation results of plate adapter, double blade flexure and optimized discrete flexure
As can be seen, the optimized flexure shows a remarkable performance for reducing adhesive stresses. The adhesive stresses of the DMs with the plate adapter and the double blade flexure exceed the allowable stresses, which is listed in Table 2, but does not with the optimized flexure. In the pull-down condition with the optimized flexure, the maximum adhesive stress between the flexure and the mirror is decreased about 54% with 7.9% decay of the mirror surface stroke. Also, the mirror surface stroke is still over 2 μm.
3.2 Experimental results
The 5×5 channel engineering DM model as a prototype of the full channel DM is built and evaluated using a commercial Fizeau interferometer (ZYGO company). Figure 14 shows the experiment setup. The 3D shape of the mirror surface is measured by the interferometer after applying voltage in order to check the maximum mirror surface strokes. Since the boundary actuators serve to hold the mirror, only central nine actuators are tested. The pupil diameter of the interferometer is about 300 mm and the full resolution is 1024×1024 pixels. The size of the footprint of the DM are 271×271 pixels. The reference measured value of the mirror surface in the rest state is removed from the measured value of each actuation. The PV and rms values of the measured mirror surface in the rest state are 65 nm and 10.6 nm, respectively.
Figure 15 represents the measurement results of the mirror surface deformation, called an influence map. Here the subscript means the id of the operating actuator. Compared with the simulation results, it is shown that the shapes of the mirror surface are appropriate to their predicted shapes in section 3.2. Also, adhesive failure is not occurred.
Fig. 15. Measured mirror strokes and influence functions at central nine actuators positions when each actuator is driven at + 200 V and others 0 V.
The minimum, maximum, average and standard deviation of the measured free stokes of the manufactured actuators are 6.0 μm, 7.0 μm, 6.5 μm and 0.25 μm, respectively. For comparison of the measured results with the simulation results, the additional simulations for the actuator strokes 6.5, and 7 μm are performed, and the results are listed in Table 8 with the result of the actuator stoke 6 μm. Since the actuators are controlled by voltage and the actual strokes of the actuators are not the same, some mismatches on the PV values are occurred. However, the measured mirror surface strokes are between the simulation results of 6 μm and 7 μm conditions, and most of them are very close to the simulation result of 6.5 μm condition. For wavefront compensations, this can be resolved by a simple post-processing such as normalization of the influence functions.
Table 8. Simulation results with different actuator strokes for No. 7, 8, and 13 actuators
The cross sectional profiles of the measured influence functions are also compared with those of the simulation results. The profile for each actuator is captured at the cross section of the peak mirror stroke position. In order to remove the edge noises, the measured data only in the 95% mirror surface are used. Figure 16 shows the comparisons of the normalized profiles of No. 7, 8, and 13 actuators. As can be seen, there are some non-overlapping regions and differences of the sharpness, but the overall curvatures and the peak positions are much similar.
Fig. 16. Cross section comparisons of influence functions; (a) No. 7 actuator, (b) No. 8 actuator, (c) No. 13 actuator.
4. Conclusions
In the present work, the discrete flexure for SiC DM with PMN stacked-actuators is designed using structural optimizations. As a prototype of the final DM, which has 489 channels, the 5×5 channel engineering DM model is handled. By using topology optimization via GTAM in ANSYS, the optimal topology which maximizes the mirror surface stroke satisfying volume and stress constraints is found. Next, the major sizes of the initial flexure design, which is remodeled from the results of the topology optimizations, are set as the design variables and size optimization is performed. The response surface optimization of ANSYS DesignXplorer is used for the size optimization. Satisfying the upper and lower limits of design variables and the constraints of adhesive stresses, three candidate design points are derived, and one of them is applied to build the real engineering DM model. The built engineering DM model, which has 5×5 channels, is tested and compared with the simulation results. Considering the deviation of the actuator strokes at the control voltage + 200 V, it is shown that the influence function shapes and the PV values are appropriate without occurring adhesive failure. The flexure design influences to both the mirror surface stroke and the adhesive stresses, also the flexures in a DM influence each other. Thus, the proposed optimization process in the present work can be applied effectively for the flexure design of a DM.
Acknowledgments
The authors would like to thank Mr. Byoung-Hyok Jeon for assembling the prototype of engineering DM model and supporting tests.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. W. Babcock, “The possibility of compensating astronomical seeing,” Publ. Astron. Soc. Pac. 65(386), 229–236 (1953). [CrossRef]
2. P-Y. Madec, “Overview of deformable mirror technologies for adaptive optics and astronomy,” Proc. SPIE 8447, 844705–844718 (2012). [CrossRef]
3. O. Albert, L. Sherman, G. Mourou, T. B. Norris, and G. Vdovin, “Smart microscope: an adaptive optics learning system for aberration correction in multiphoton confocal microscopy,” Opt. Lett. 25(1), 52–54 (2000). [CrossRef]
4. L. Sherman, J. Y. Ye, O. Albert, and T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206(1), 65–71 (2002). [CrossRef]
5. E. J. Fernández, L. Vabre, B. Hermann, A. Unterhuber, B. Považay, and W. Drexler, “Adaptive optics with a magnetic deformable mirror: applications in the human eye,” Opt. Express 14(20), 8900–8917 (2006). [CrossRef]
6. S. Arnon and N. S. Kopeika, “Adaptive optical transmitter and receiver for space communication through thin clouds,” Appl. Opt. 36(9), 1987–1993 (1997). [CrossRef]
7. B. S. Vinevich, L. N. Evdokimovich, S. N. Smirnov, and A. G. Safronov, “Controlling the output radiation power of CO2 lasers by means of deformable mirrors,” J. Opt. Technol. 71(2), 65–70 (2004). [CrossRef]
8. E. J. Szetela and A. I. Chalfant, “Thermal distortion of mirrors,” Thermochim. Acta 26(1-3), 191–197 (1978). [CrossRef]
9. M. A. Ealey and J. A. Wellman, “Cooled ISOFLOW laser mirrors,” High Heat Flux Engineering , 1739, 374–382 (1993). [CrossRef]
10. G. Rabczuk and M. Sawczak, “Study on the possibilities of controlling the laser output beam properties by an intracavity deformable mirror,” Opto-Electron. Rev. 14(2), 136–142 (2006). [CrossRef]
11. A. V. Kudryashov and V. V. Samarkin, “Control of high power CO2 laser beam by adaptive optical elements,” Opt. Commun. 118(3-4), 317–322 (1995). [CrossRef]
12. R. H. Freeman and H. R. Garcia, “High-speed deformable mirror system,” Appl. Opt. 21(4), 589–595 (1982). [CrossRef]
13. K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “CVD SiC deformable mirror with monolithic cooling channels,” Opt. Express 26(8), 9724–9739 (2018). [CrossRef]
14. K. Ahn, H. G. Rhee, H. S. Yang, and H. Kihm, “Silicon carbide deformable mirror with 37 actuators for adaptive optics,” J. Korean Phys. Soc. 67(10), 1882–1888 (2015). [CrossRef]
15. H. C. Park, P. Adyatama, and H. Y. Lee, “A Single-crystal PMN-29PT Stack Actuator: Fabrication and Performance,” J. Korean Phys. Soc. 50(6), 545–550 (2013). [CrossRef]
16. M. P. Bendsoe and O. Sigmund, Topology optimization: theory, methods, and applications (Springer, 2013), Chap. 1.
17. T. Murata and H. Ishibuchi, “MOGA: Multi-objective genetic algorithms,” IEEE international conference on evolutionary computation1, 289–294 (1995).
18. ANSYS User’s Manual 2021 R1 (ANSYS Inc., 2021).
