## Abstract

A cycloid-like variable curvature mirror (VCM) for zoom-imaging systems was investigated. An analytical-deformation solution to a thin-elastic plate with a cycloid-like thickness distribution and simply supported boundary condition under uniform pressure was found using a small parameter method. The finite-element analysis of the thin-elastic plate and designed VCM showed a good correlation with the analytical solution. The VCM was manufactured and polished to the initial shape with a root mean square (RMS) of 1/80λ. Finally, with air-pressure-based actuation testing under 0.07 MPa, the VCM deforms approximately 36.89 µm and maintains the RMS surface performance of 1/10λ, 1/40λ with and without spherical aberrations, respectively.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Variable curvature mirrors (VCMs) are advanced active optic components whose radius of curvature can be adjusted as needed by an external force to realize zoom imaging without moving elements and thereby replace a conventional mechanical zoom system [1–6], thermal lens compensation [7–10], and interferometric telescopes for pupil size stabilization upstream of the beam recombination [11–13]. The earliest VCM that can be searched dates back to 1972, which was designed and fabricated at the Hebrew University of Jerusalem using a solid-state laser [14]. In addition, the former Soviet Union [8,15,16], France [11–13], Germany [7,17], the US [18–20], Japan [21,22], and China [10,23–25] have also conducted extensive research on VCMs for different applications, forms, and actuation methods. Initially, VCMs were designed for use in solid-state lasers to improve the quality of the output beam. With such development, more and more applications have gradually been seen. During its development, various forms and actuation methods have been invented for various purposes and conditions. Most of them are of uniform thickness with a uniform actuation. With the gradual deepening of research, the benefits of a VCM with a variable thickness distribution have been discovered, and this type of VCM has been increasingly studied, such as by Kvantovaya Electron Co. [16], Lawrence Berkeley Laboratory [18], Sumitomo Electric Industries [22], and Laboratoire d’Astrophysique de Marseille-LAM, whose study on VCMs with such distributions has been considered as the most systematic cases [11–13]. In G. R. Lemaitre’s book *Astronomical Optics and Elasticity Theory*, several thickness distributions are presented which can achieve an ideal curvature radius variation during actuation, such as in exponential and cycloid-like types [26]. In our previous paper [27], we worked out and verified a simplified analytical deformation solution of an exponential thin elastic plate under a uniform load with a simply supported edge condition using a small-parameter method based on the *Theory of plates and shells* of Timoshenko [28]. As in [26], the cycloid-like thickness distribution, which is in the form of $h = {h_0}{\left( {1 - \frac{{{r^2}}}{{{a^2}}}} \right)^{1/3}}$, where *h* is the thickness varying with the radius, ${h_0}$ is the central thickness, *a* is the radius of the plate, *r* is the radius, is also a common form of a VCM. In this research, a systematic study of an elastic plate and a VCM with such form thickness was conducted. This VCM is expected to be used in zoom imaging systems in the future. Because the VCM under study has the thickness of a cycloid-like form, to analyze the deformation characteristics of cycloid-like VCMs, the first step is to obtain the analytical solution of the corresponding thin elastic plate, as in our previous study. In this study, we developed a simplified analytical solution of a cycloid-like elastic plate by applying the small-parameter method [29]. In addition, manufacturing, optical polishing, and air pressure procedures have also been conducted in later studies.

## 2. Analytical study

According to the theory of plates and shells [28], the curvature variation process can be physically modeled as a thin circular plate deformation caused by a uniform load with variable thickness and under a simply supported boundary condition, as shown in Fig. 1.

Whereas the thin circular plate is under a simply supported boundary condition with a uniform load, the stress condition and moment equilibrium of the plate are as shown in Fig. 2 [29].

The corresponding moment equilibrium equation is as follows:

where*r*represents the radial coordinate, $dr$ is the infinitesimal of

*r*, ${M_r}$ is the radial bending moment per unit length, ${\textrm{M}_t}$ is the tangential bending moment per unit length, and

*Q*is the shearing force per unit length.

The equations of ${M_r}$, ${M_t}$, and *Q* are

*w*is the vertical deflection, $D = \frac{{E{h^3}}}{{12({1 - {\mu^2}} )}}$ is the rigidity of the plate,

*h*is the thickness distribution, $\mu $ is the Poisson’s ratio, and

*q*is the load per unit area. Applying Eqs. (2)–(4) into Eq. (1), we obtain Eq. (5):

*h*is varied with

*r*, the rigidity of the plate is also changed. In this study, to make the analytical solution more suitable for a greater thickness distribution, the thickness distribution of the plate is expressed in the form of $h = {h_0}{\left( {1 - k\frac{{{r^2}}}{{{a^2}}}} \right)^{1/3}}$[26]. Different values of

*k*represent different thickness distribution modes. To solve the deflection

*w*of the plate, we firstly introduce the following substitutions:

*w*of the plate.

First, we set $\varepsilon = k$, and *y* is developed by the power series for $\varepsilon $ as in Eq. (11),

*x*, respectively. In addition, the right part of the equation, $h = {h_0}{({1 - kx} )^{1/3}}$, is developed by the power series in order of $\varepsilon $,

0) For ${y_0}$, the zero-order equation and the boundary conditions are

1) For ${y_1}$, the first-order equation and the boundary conditions are

2) For ${y_2}$, the second-order equation and the boundary conditions are

3) For ${y_3}$, the third-order equation and the boundary conditions are

4) For ${y_4}$, the fourth-order equation and the boundary conditions are

*y*, the influence decreases, and when it is higher than 4th order, the influence can be neglected as it is being too small. Therefore, we solve this only up to the 4th order. The overall deformation result is given by Eq. (11) after incorporating Eqs. (27), (24), (21), and (18).

Then, the central deflection (*x = *0), which is calculated in Eq. (11), can be developed as follows:

## 3. Finite element analysis (FEA)

To verify the accuracy of the analytical solution in analyzing the deformation of the elastic plate under different pressures, we built an elastic plate model by CATIA, conducted a finite element analysis (FEA) using MSC Patran/Nastran and compared the FEA results of the elastic plate model to the value calculated using Eq. (11). First, we conduct an FEA of the elastic thin plate model with a simply supported edge condition and pressure on the other side, as shown in Fig. 1, and extract its deformation values along the radial direction. We then compared it to the values calculated by using our analytical solution. In [27], we chose the verification parameters of the elastic plate according to the designed VCM actuated using a uniform air pressure with a closed cavity structure. Therefore, in this study, during the verification process, we use the parameters in [27] as a reference to verify the correctness of the solution, and we obtain a thickness distribution of $h = {h_0}{({1 - kx} )^{1/3}}$. Therefore, the diameter of an elastic plate is $2a = 134.5\; \textrm{mm}$, the central thickness is ${h_0} = 8.5\; \textrm{mm}$, and the material is ultralumin ($E = 72\textrm{GPa},\; \mu = 0.33$). Based on optimization, the parameters of the VCM are as follows: In addition, we choose $k = 0.9$ as the lower surface thickness distribution coefficient, and the thickness expression is $h = 8.5{({1 - 0.9x} )^{1/3}}$. The structure of the elastic plate model with several of the main parameters is shown in Fig. 3, and the VCM structure is shown in Fig. 4, with a collar thickness of $c = 0.62\; mm$.

## 4. Deformation comparison of FEA and analytical model

During the FEA process, we set the elastic thin plate under a simply supported edge condition and apply 0.1 MPa at its surface. We then obtain the deflection values, as shown in Table 1, and plot the deflection curve along its radius. A comparison between the FEA results of the elastic thin plate and the analytical solution is shown in Fig. 5. From Fig. 5, the difference between the two curves is extremely small, and the deformations show a good consistency. By subtracting the normalized deflection between the values of the two curves, we extract the difference between them, as shown in Fig. 6. The maximum difference is 0.36 µm at the central and edge parts, and the normalized residues value is almost 0.

Next, compare the central deflection values obtained through the FEA of the designed VCM and our analytical solution under a pressure of 0.01 to 0.1 MPa, as shown in Fig. 7. From Fig. 7, we can see that the overall central deflection values of the mirror are smaller than those of the thin elastic plate, which does not have a collar structure. In addition, we abstracted the deflection values of the FEA and analytical solution along the radius direction under 0.03, 0.06, and 0.09 MPa and plot them in Fig. 8.

From the comparison between the FEA and the analytical results shown in Figs. 7 and 8, we can see that the deflection curves obtained by these two methods are consistent, although the two values are not exactly the same. By adding a thin collar to the VCM structure, a deflection can be achieved by using an air pressure actuation. Meanwhile, the existence of a thin collar adds a semi-clamp effect to the mirror boundary condition instead of the simply supported edge condition [11]. This semi-clamp boundary condition reduces the deformation of the VCM compared to the elastic plate model, which has also been discussed in our previous study [27].

## 5. Air pressure actuation experiment

#### 5.1 Mirror manufacturing and actuation system building

The prototype mirror is manufactured using ultralumin LC4 with the parameters referred to in the FEA part, and there is a difference between the FEA and the analytical part in that the curvature radius of the mirror is 2807 mm. The overall manufacturing steps are as follows: *a)* obtain a blank using a CNC machining center, *b)* use single-point diamond turning to improve the accuracy of the mirror surface figure by better than λ/3, *c)* after the mechanical process, apply several thermal treatments to release the stress introduced by the mechanical process, *d)* improve the mirror’s surface by up to λ/50 through classical polishing, *e)* use ion-beam figuring (IBF) to improve the accuracy of the surface figure by up to λ/80. The mirror after applying IBF and the surface testing results are shown in Fig. 9.

For the actuation system, we have a pressure sensor, proportional valve, switch valve, micro pump, and integrated processor. The maximum pressure provided by the micropump is 0.1 MPa, and the control accuracy is 0.001 MPa. The air pressure actuation system structure is shown in Fig. 10, and the major equipment, pressure sensor, integrated processor, and micro pump are shown in Fig. 11.

#### 5.2 Analysis of actuation experiment and actuation results

After all of the apparatuses are assembled and connected together, the experimental testing system is as shown in Fig. 12. During the testing process, the surface performance of the VCM was measured using a ZYGO interferometer with a 6-inch aperture.

During the actuation process, we tested the performance of a mirror with a 120-mm diameter surface under pressure values of 0.02, 0.036, 0.053, and 0.070 MPa, and the accuracies of the corresponding surface figure are as shown in Fig. 13.

Through a decomposition of the performance of the mirror surface, the variation in the Zernike coefficients along with the pressure increase, particularly the Astigmatism 0/90, Astigmatism ±45, X coma, Y Coma, and Sphere 3 are as shown in Fig. 14. From Fig. 14, the coefficient values of Astigmatism 0/90, Astigmatism ±45, and Y Coma have nearly no change during the pressurization process. X Coma showed a slight increase. Only Sphere 3 increases sharply with an increase in the pressure value. Furthermore, the overall surface aberration was analyzed, as shown in Fig. 15. It is clear that the spherical aberration is also the main part after each actuation, when the others only change slightly during this process.

Through the analysis of the actuation testing results, the surface figure accuracy degrades to approximately λ/10 at 0.07 MPa, and the main aberration of this degradation is a spherical aberration, as values shown in Table 2. Based on this, the spherical aberration was removed from each test result. We found that although the surface performance slightly decreases with an increase in the pressure value, the surface performance of the VCM is still as high as λ/40 under a pressure of 0.07 MPa, as shown in Fig. 16. Therefore, a spherical aberration is the main aspect that needs to be removed in the following process of mirror research.

After the air pressure actuation, the deformation values were also analyzed. We found that the actuation deformation of the VCM under each air pressure value is larger than that of the VCM in Section 4. Based on the analysis, we found that this is because the tested VCM has an original curvature radius on its reflected surface of 2807 mm. By contrast, the analytical and FEA models are flat, and the other side obeys the thickness distribution. As a result, the thickness from the central part to the edge of the VCM is smaller than that of the flat surface model. It is then thinner than the overall model, except for the central thickness. Under the same air pressure value, the deformation is larger than that of the analytical model. Accordingly, a VCM model with an original curvature radius was built and analyzed using an FEA.

From the comparison, although the deformation results between the FEA model and the real fabrication model, which have the same theoretical structural parameters, are extremely close to each other, they are still different. The testing deformation of the mirror under 0.07 MPa is 36.89 µm, which is equal to that of a VCM model with a central thickness of 8.2-mm, as shown in Fig. 17. As is well known, the most influential reasons for a mirror deformation are the central thickness and the thickness of the collar, and thus there must be some operation that can influence these parameters during the manufacturing and polishing process. After examining the entire manufacturing process, we found that there are three processes that can reduce the thickness. First, the blank manufacturing process was not as accurate as the required parameter. Second, as before the classical polishing process, diamond turning was applied to the blank surface to improve the accuracy of the original surface. Third, the classical polishing process also contributes to its reduction, although it is extremely small. Because the mirror structure is not tested before it is sealed, its thickness cannot be verified. However, the thickness of its surface and collar could be two possible reasons.

## 6. Conclusion

In the first part of this paper, an analytical solution to a thin elastic plate with a cycloid-like thickness distribution with simply supported edge conditions and a uniform pressure was described by introducing an approximation of the small parameter method. The analytical solution was in a simple form for analysis. In the second part, the deformation of the analytical results of a thin elastic plate was compared to the FEA results of both the plate and the VCM. The analytical and FEA results for the thin elastic plate are almost the same. Therefore, the accuracy of the analytical solution is verified indirectly. In addition, the analytical results are consistent with the FEA results of the VCM, although the results are not exactly the same because of the difference in the real structures between the VCM and the elastic plate. Furthermore, in the following section, a VCM prototype manufactured and tested through the air pressure actuation is described. Deflection and surface performance values were collected and analyzed. The deflection and surface performance of a VCM under 0.07 MPa are 36.89 µm and 0.099 λ (with a spherical aberration), respectively. In this VCM, the main aberration that contributes to the surface performance deterioration is a spherical aberration. After removing the spherical aberration under each pressure, the surface performance of the VCM is still extremely good even at 0.07 MPa. Its value was approximately 0.024 λ. Therefore, the elimination of the spherical aberration of the VCM with a variable thickness distribution during the actuation process will be the focus of the following study. However, as shown in the experimental figures, the volume of the entire testing system in our lab is quite large. Considering its practicality in image systems, the devices must be further improved in terms of volume and integration to adapt to the system requirements.

## Funding

West Light Foundation of the Chinese Academy of Sciences (XAB2015A09); Research Fund for Young Star of Science and Technology in Shaanxi Province (2016KJXX-08); China Scholarship Council.

## Acknowledgments

The authors thank Prof. Emmanuel Hugot from Aix Marseille University, CNRS, Laboratoire d’Astrophysique de Marseille (LAM) for providing the useful advice.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The 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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