The Opto-Mechanical–Thermal Coupling Analysis and Verification of an All-Aluminum Freeform Imaging Telescope

Abstract

A freeform imaging telescope (FIT) can achieve a large field of view, high resolution, light weight, and small volume at the same time. Single-point diamond turning (SPDT) is usually used to fabricate FITs, which is made entirely of aluminum alloy. Compared with a traditional telescope, whose reflector is made of glass and whose structure is aluminum, the coefficient of thermal expansion (CTE) of the structure and reflector of which is non-conforming, the CTE of the structure and reflector in an all-aluminum FIT is identical. Therefore, it was expected to theoretically have athermalization properties. In this paper, an all-aluminum off-axis three-mirror FIT was verified. The opto-mechanical–thermal coupling analysis of the FIT at −20 °C was carried out, including data processing and coordinate transformation. The reflector node deformation data of the global coordinates obtained from the finite-element analysis were converted into XY polynomial coefficients of the local coordinate system in ZEMAX. The results showed that the modulation transfer function (MTF) of the FIT at −20 °C~+40 °C still reached the diffraction limit. Moreover, the MTF of the FIT at −20 °C was 0.291 through a thermal environmental test, which was almost the same as the MTF at 22 °C. These results showed that the all-aluminum FIT could achieve athermalization properties.

1. Introduction

At present, off-axis three-mirror anti-astigmatism optical systems are widely used in the field of space remote sensing and detection [1]. With the increasing demand for space technology with a large field of view and high resolution, the diameter of off-axis three-mirror systems will increase further, which will result in an increase in assembly and manufacturing costs. Therefore, the optical surfaces of off-axis three-mirror systems are designed as freeform surfaces [2,3], which can be arbitrarily configured according to actual needs. Thus, a large field of view can be realized with a compact structure [4].

Nowadays, the single-point diamond turning (SPDT) of aluminum alloy is the most common method for fabricating freeform surface mirrors [5,6]. Aluminum alloy is the most commonly used metal structural material used in space, as it has the advantages of low density, low cost, and good machinability. Hence, it can effectively reduce the weight of opto-mechanical structures and the manufacturing costs of optical telescopes [7,8]. However, compared with other commonly used structural materials (such as TC4, invar, and AL-SIC composite materials), aluminum alloys are even less rigid and have a larger coefficient of thermal expansion (CTE), which is not suitable for the extreme environments in space. When the temperature changes, the aluminum alloy material produces large deformations, which will change the reflector’s shape, spacing, and position, and the optical performance of the freeform imaging telescope (FIT) will be also affected. However, for FITs with all-aluminum materials used for the opto-mechanical parts, the CTE of the reflectors and the mechanical structure are identical. So, in theory, such all-aluminum FITs will not suffer thermal defocusing. Therefore, in order to verify the athermalization characteristics of all-aluminum FITs, it is necessary to conduct an opto-mechanical–thermal coupling analysis and experiment. At present, the opto-mechanical–thermal coupling analysis of all-aluminum telescopes is mainly based on coaxial optical systems [9,10], and there are relatively few reports on the opto-mechanical–thermal coupling analysis of all-aluminum off-axis three-mirror FITs.

In this paper, an all-aluminum FIT was designed. Its environmental temperature adaptability was studied through opto-mechanical–thermal coupling analysis. Through coordinate transformation, the reflector node deformation data of the global coordinates obtained from the finite-element analysis were converted into XY polynomial coefficients of the local coordinate system. The latter were imported into ZEMAX to analyze the modulation transfer function (MTF) of the optical system. The MTFs of the all-aluminum FIT at different temperatures were measured by thermal environmental testing, and the results were fairly consistent with the opto-mechanical–thermal coupling analysis.

2. Design of All-Aluminum Freeform Imaging Telescope

2.1. Optical Design

The specifications of the off-axis three-mirror FIT are summarized in

Table 1. Specifications of the FIT.

F#	Field of View	Entrance Pupil Diameter (mm)	Wavelength (nm)	Pixel Size (μm2)
4	4° × 4°	40	400–700	5.5 × 5.5

. After multiple optimizations, the final optical design was achieved as shown in Figure 1, which not only satisfied the imaging requirements, but also guaranteed the feasibility of the structural design.

The MTFs of the obtained optical system under different fields of view are shown in Figure 2a. The MTF values of all the fields were greater than 0.7 at 90 lp/mm, which could approach the diffraction limit as shown in Figure 2a. The RMS spot diameters of all the fields were smaller than 1 μm (see Figure 2b), which was much smaller than the pixel size of 5.5 × 5.5 μm², thus it could satisfy the needs of optical remote sensing imaging.

To guarantee an as-built performance of MTF > 0.5 at 90 lp/mm of the optical system, the tolerances were analyzed in ZEMAX. Since the most sensitive tolerance is the tip/tilt error of M1, M1 was selected as the assembling reference. For the other two mirrors, the tip/tilt tolerance was ±0.017°, the clocking tolerance was ±0.03°, the x/y decenter tolerance was ±25 μm, and the vertex distance tolerance was ±50 μm. The tip/tilt and position of the image plane was taken as the compensator. The surface error of the three freeform mirrors was randomly modeled by Zernike polynomials, and the effect was also simulated in ZEMAX. Considering the manufacturing capability, the surface error was assigned as RMS 0.03λ at 632.8 nm. Overall, the surface errors caused an MTF degradation of 0.12 at 90 lp/mm.

2.2. Structural Design

The 3D model of the all-aluminum FIT contained three freeform mirrors (M1~M3), the support frame, two cover plates, and the external hood (see Figure 3). The support frame was machined from a whole piece of aluminum alloy to guarantee the accuracy of the three mirrors’ relative positions. The three mirrors were optical–mechanical integrated structures, which were directly fixed on the support frame with screws. The cover plates were also fixed on the support frame by screws, which made the FIT’s structure closed and avoided the influence of external stray light. The size of the FIT was 210 × 200 × 116 mm. The material of the mirrors and the support structure was AL6061-T6, whose material properties are shown in

Table 2. Material properties of AL6061-T6.

Density (kg/m3)	Thermal Expansion Coefficient (K−1)	Elastic Modulus (N/m2)	Poisson’s Ratio	Thermal Conductivity (W/(m·K))
2710	2.4 × 10−5	6.9 × 1010	0.33	154

. AL6061-T6 has good turning performance and can realize a freeform profile by using single-point diamond turning (SPDT). It is also a lightweight structural material, which is commonly used in the aerospace field [11,12]. After the optical and structural design, the weight of the FIT was estimated as 2.1 kg according to the volume and density of AL6061-T6 using the 3D modeling software.

3. Opto-Mechanical–Thermal Coupling Analysis

3.1. Principles of Opto-Mechanical–Thermal Coupling Analysis

Opto-mechanical–thermal coupling analysis is a technology that integrates multidisciplinary knowledge and comprehensive analysis [13], mainly including optics, mechanics, heat, etc. The data transfer and coordinate transformation are the key factors in the coupled analysis [14]. For a freeform surface, many researchers commonly convert the node deformation data obtained from the mechanical and thermal analysis into the coefficients of Zernike polynomials [15,16]. The process includes unifying the node deformation data, normalizing the data, converting the polar coordinates, and obtaining the coefficients of the Zernike polynomial by matrix calculation. Generally, the order of the polynomial is determined according to the accuracy of the analysis. However, the basic function of the Zernike polynomial is continuous and orthogonal in the unit circle region, and its application is limited.

Except for Zernike polynomials, there are many other mathematical expressions including aspheric surface, XY polynomial, non-uniform rational B-spline [17], and so on. The freeform surface expressed by the XY polynomial is obtained by adding the power terms of each order of $x$ and $y$ to the quadratic surface base, and its expression is as follows:

$$z=\frac{\mathrm{c}\left(x^2+y^2\right)}{1+{[1-(1+\mathrm{k})c^2\left(x^2+y^2\right)]}^{1/2}}+{\displaystyle \sum _{\mathrm{m}=0}^p{\displaystyle \sum _{n=0}^p{C}_{(m,n)}}}{x}^m{y}^{n}1\le \mathrm{m}+n\le p$$

(1)

For designers, an XY polynomial has a relatively high degree of freedom and a strong aberration correction ability. Since the optical system was symmetric about the YOZ plane, the freeform mirror could be characterized using XY polynomials without the coefficients of odd items, which could simplify the complexity of the surface representation. In addition, it could match to the form of the computer-numerical-control (CNC) machining of the freeform surfaces, which is the most widely used form of freeform surfaces. Therefore, this paper used an XY polynomial to describe the freeform surface. The theory of opto-mechanical–thermal coupling analysis is as follows: Firstly, the temperature field distribution inside the telescope was obtained through thermal analysis. Secondly, the node data of the three mirrors and focus surface obtained from the static analysis was calculated with the temperature field distribution data. Thirdly, the XY polynomial coefficients of three mirrors were fitted, which needed correct node data and coordinate transformation. Finally, the MTF of the FIT at this temperature field was analyzed by importing the XY polynomial coefficients into ZEMAX. The flow chart of the opto-mechanical–thermal coupling analysis is shown in Figure 4.

3.2. Data Correction

There will be discretization errors when using the finite-element method for static structure analysis. The errors mainly result from the mesh division of the model [18] and are mainly restricted by the number of elements and nodes. Since the calculation accuracy and calculation resources need to both be considered in the finite-element method, the mesh cannot be infinitely approached to the actual surface shape. Thus, discretization errors will occur. Moreover, when extracting the node coordinates, the decimal places are rounded in order to conform to the finite word length of the computer, which will cause the loss of node coordinate accuracy.

The node data obtained in the opto-mechanical–thermal coupling analysis mainly includes the coordinate values ($x_0$, $y_0$, $z_0$) and the deformation values ($x_d$, $y_d$, $z_d$) of the mirror nodes. The discretization errors will affect both the coordinate values and the deformation values, but the coordinate values are generally of the order of ${10}^{-3}$ m, and the highest order of the deformation value data is generally ${10}^{-6}$ m. So, the coordinate value data has a greater impact when fitting and calculating the XY polynomial coefficients. Therefore, in order to guarantee more accurate fitting polynomial coefficients, it is necessary to correct the node data of the coordinate values. It can be seen from the XY polynomial Equation (1) that the value along the optical axis is $\text{(}{z}_0+z_d\text{)}$, and the method used in this paper was to correct the data along the optical axis. Firstly, the new optical axis direction coordinate $z_0^{\prime }$ was calculated using the $x_0$, $y_0$ and polynomial coefficients of the initial design, as shown below:

$$z_0^{\prime }=\frac{\mathrm{c}\text{(}{x}_0^2+y_0^2\text{)}}{1+{[1-(1+\mathrm{k})c^2\left(x_0^2+y_0^2\right)]}^{1/2}}+{\displaystyle \sum _{\mathrm{m}=0}^p{\displaystyle \sum _{n=0}^p{C}_{(m,n)}}}{x}_0^m{y}_0^n$$

(2)

The $z_0$ correction process in a two-dimensional coordinate system is shown as Figure 5. $z=f\left(x,y\right)$ is the initial freeform surface, the red circle is the extracted node ($x_0$, $z_0$), the green pentagram is the corrected node ($x_0^{\prime }$, $z_0^{\prime }$), and $z_0^{\prime }$ is the value calculated by the XY polynomial. Taking M1 as an example, the node coordinate distribution before and after the correction is shown in Figure 6a,b, respectively. Since $z_0^{\prime }$ reduces the discretization errors by polynomial calculation, the corrected node coordinate distribution is closer to the initial freeform surface.

Finally, the optical axis direction data is $(z_0^{\prime }+z_d)$ when fitting the new polynomial coefficient. Compared with the extracted $z_0$, $(z_0^{\prime }+z_d)$ has no discrete error and loss of data information, which will lead to a smaller coefficient error.

3.3. Coordinate Transformation

The node coordinates and deformation data extracted from the finite-element software belonged to the global coordinate system, while the XY polynomial coefficients imported into ZEMAX belonged to the local coordinate system of the three mirrors (M1~M3) in the optical configuration. Therefore, the data in the global coordinate system needed to be converted to the local coordinate system, because the XY polynomial coefficients of a single mirror were calculated in the local coordinate system.

The node A of M1 in the global coordinate system O-XYZ and the local coordinate system $O^{\prime }-X^{\prime }{Y}^{\prime }{Z}^{\prime }$ are $A\left(x_0,y_0,z_0\right)$ and $A^{\prime }\left(x_0^{\prime },y_0^{\prime },z_0^{\prime }\right)$, respectively. The schematic diagram is show in Figure 7.

The node A transformation process includes the translation and rotation of coordinates, and the transformational relationships are shown as Equations (3) and (4). $A_{0pan}(x_{0pan},y_{0pan},z_{0pan})$ is the coordinate of node A after translation.

$$\left[\begin{array}{c}{x}_{0pan}\\ y_{0pan}\\ z_{0pan}\\ 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& dx\\ 0& 1& 0& dy\\ 0& 0& 1& dz\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\\ 1\end{array}\right]$$

(3)

$$\left[\begin{array}{c}{x}_0^{\prime }\\ y_0^{\prime }\\ z_0^{\prime }\\ 1\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos \theta & \sin \theta & 0\\ 0& -\sin \theta & \cos \theta & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \alpha & 0& -\sin \alpha & 0\\ 0& 1& 0& 0\\ \sin \alpha & 0& \cos \alpha & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}\cos \beta & \sin \beta & 0& 0\\ -\sin \beta & \cos \beta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}_{0pan}\\ y_{0pan}\\ z_{0pan}\\ 1\end{array}\right]$$

(4)

3.4. Opto-Mechanical–Thermal Coupling Analysis

The required ambient temperature range for the all-aluminum freeform imaging telescope was −55 °C~+70 °C. In this paper, the process of the opto-mechanical–thermal coupling analysis was mainly focused on −20 °C, and only M1 was taken as an example.

Firstly, the 3D model was simplified. The features such as fillets, chamfers, and pin holes that did not affect the results were all removed. Secondly, the thermal simulation model in the steady-state thermal module was established using finite-element software. This process included meshing the FIT, setting the ambient temperature to −20 °C, imposing fixed constraints on eight bottom holes, and obtaining the three directional (x, y, z) temperature deformation of the nodes on the surface of the three mirrors. Figure 8a,b represents the finite-element model of the all-aluminum freeform telescope and the thermal deformation of M1 in the global coordinate system, respectively.

Thirdly, the node thermal deformation data in the global coordinate system were converted to the local coordinate system, and the surface profile of M1 was obtained, as shown in Figure 9a. The rigid displacement of the surface shape was removed, and the RMS of M1 was 0.44 λ, as shown in Figure 9b, where the component of the surface was mainly power. The coefficients of the XY polynomial included the surface profile and its deformation for the three freeform mirrors, as shown in

Table 3. The coefficients of XY polynomial for the three freeform mirrors.

Term	M1	M2	M3	Term	M1	M2	M3	Term	M1	M2	M3
X1Y0	−3.46 × 10−5	−8.14 × 10−5	−1.53 × 10−3	X1Y3	−6.34 × 10−11	3.15 × 10−11	9.04 × 10−6	X2Y4	−1.86 × 10−12	−7.60 × 10−12	−7.45 × 10−4
X0Y1	1.70 × 10−4	1.46 × 10−4	−1.72 × 10−3	X0Y4	2.51 × 10−9	−1.22 × 10−7	−1.02 × 10−2	X1Y5	1.29 × 10−13	5.60 × 10−14	−8.28 × 10−6
X2Y0	−1.99 × 10−4	−7.69 × 10−4	−8.37 × 10−2	X5Y0	1.01 × 10−11	5.39 × 10−14	7.87 × 10−6	X0Y6	−2.29 × 10−13	−9.92 × 10−12	−3.58 × 10−4
X1Y1	−3.5 × 10−8	1.84 × 10−8	1.09 × 10−5	X4Y1	2.70 × 10−11	1.58 × 10−9	1.00 × 10−3	X7Y0	5.62 × 10−15	1.23 × 10−16	1.87 × 10−5
X0Y2	−2.80 × 10−4	−2.11 × 10−3	−3.17 × 10−1	X3Y2	−2.12 × 10−11	4.12 × 10−13	−4.65 × 10−5	X6Y1	−1.34 × 10−14	−2.05 × 10−12	−7.75 × 10−4
X3Y0	−4.11 × 10−9	−1.44 × 10−11	−3.90 × 10−5	X2Y3	1.40 × 10−11	1.83 × 10−9	1.88 × 10−4	X5Y2	−5.11 × 10−15	−1.15 × 10−16	−1.07 × 10−5
X2Y1	−1.79 × 10−6	2.73 × 10−7	7.67 × 10−3	X1Y4	−3.12 × 10−11	−2.86 × 10−13	−5.29 × 10−5	X4Y3	1.12 × 10−14	2.25 × 10−12	7.54 × 10−4
X1Y2	1.26 × 10−8	−2.30 × 10−9	4.06 × 10−5	X0Y5	3.95 × 10−12	5.58 × 10−10	−2.41 × 10−4	X3Y4	−2.48 × 10−14	−2.00 × 10−15	−7.52 × 10−5
X0Y3	−2.21 × 10−6	−8.57 × 10−6	−2.20 × 10−2	X6Y0	−3.29 × 10−13	3.54 × 10−13	−1.34 × 10−4	X2Y5	1.79 × 10−14	2.29 × 10−12	3.42 × 10−4
X4Y0	8.39 × 10−9	4.85 × 10−8	−2.67 × 10−3	X5Y1	−1.92 × 10−14	−1.12 × 10−13	6.05 × 10−6	X1Y6	−1.64 × 10−14	9.19 × 10−16	−4.55 × 10−5
X3Y1	−3.23 × 10−12	5.80 × 10−11	−6.95 × 10−6	X4Y2	5.80 × 10−13	9.22 × 10−12	−3.64 × 10−4	X0Y7	2.78 × 10−15	−3.78 × 10−13	−1.27 × 10−4
X2Y2	1.03 × 10−8	−3.73 × 10−8	−1.07 × 10−2	X3Y3	1.90 × 10−13	9.70 × 10−14	−1.13 × 10−5

. The residual error included the model conversion error, finite-element meshing error, etc.

Then, the XY polynomial coefficients were substituted into the optical design software for simulation. The geometric diffuse spot distribution and MTF curve of the optical system are shown in Figure 10a,b, respectively. The imaging quality was still close to the diffraction limit, which proved that the athermalization effect was excellent.

Due to the ambient temperature requirements of the all-aluminum FIT, the optical performance at multiple characteristic temperatures was analyzed by opto-mechanical–thermal coupling.

The analysis results are shown in

Table 4. The MTF of multiple characteristic temperatures.

Temperature	−55°C	0°C
MTF
Temperature	20°C	70°C
MTF

. It was found that the all-aluminum FIT had good athermalization characteristics.

4. Thermal and Optical Testing

We accomplished the manufacture and assembly of the FIT by using SPDT and computer-aided assembly (CAA), respectively. The wavefront aberration of multiple fields of view were used to guide the adjustment of the position and orientation of the mirrors during the assembly process. The three freeform mirrors and the prototype of the FIT are shown in Figure 11a,b, respectively.

The thermo-optical performance of the all-aluminum freeform telescope was tested and verified. The equipment used in the test included a high- and low-temperature tank of 1.5 L, a collimator (1.6 m focal length and 200 mm diameter), a light source (visible light emitted by an integrating sphere), and a target (55 μm line width). The test setup is shown in Figure 12.

The all-aluminum FIT was placed in the high- and low-temperature test tank. The entrance of the telescope was aligned with the window of the test tank. The collimator and the light source were placed outside the tank. The optical axis of the collimator and the telescope were aligned well. The telescope’s MTFs were measured at 20 °C and −20 °C during the tests. The target images taken at 20 °C and −20 °C are shown in Figure 13a,b, respectively. The inset of Figure 13a shows the actual target. The grayscale response value of the dark stripes and the adjacent bright stripes are shown in the picture. The MTF of a telescope can be calculated by Equation (5), and the results were 0.298 and 0.291 at 20 °C and 0.291 at −20 °C, respectively. The above result verified the athermalization characteristics of the all-aluminum FIT.

$$MTF=\frac{D{N}_{\max }-D{N}_{\min }}{D{N}_{\max }+D{N}_{\min }}\times \frac{\pi }{4}$$

(5)

5. Conclusions

This paper introduced the optical design, structural design, and, in particular, the opto-mechanical–thermal coupling analysis of an all-aluminum FIT. The data conversion and coordinate transformation between the finite-element node data and ZEMAX optical parameters and the athermalization characteristics of the telescope were analyzed. The process of opto-mechanical–thermal coupling analysis was carried out at −20 °C, with emphasis on the method of node data correction. The analysis showed that the temperature had little effect on the optical performance, and the MTF of the telescope could reach the diffraction limit at −55~+70 °C. Finally, a thermo-optical test was carried out on the telescope. It was verified that the MTF of the telescope at 20 °C and −20 °C were basically the same, indicating the athermalization properties of the all-aluminum FIT.