Embedded parallel-actuated technology for deformable space segmented mirror

Deyi Dong; Donglin Xue; Junze Xiao; Junze Xiao; Chao Li; Yuxin Sun; Yu Zhang; Guanbo Qiao

doi:10.1364/OE.519616

1. Introduction

With the increasing demand for higher resolution and energy-harvesting ability of space telescopes, the aperture of the primary mirror of the optical system is growing larger. Due to limitations in manufacturing, transportation, safety, and other factors [1–3], 4-meter diameter is almost the limit of traditional monolithic mirror space systems [4]. To overcome this bottleneck, researchers have conducted research on new ultra-large aperture space optical systems and proposed many implementation solutions for ultra-large aperture space optical systems, such as membrane-based reflective imaging systems [5], deployable segmented imaging systems, optical interferometric synthetic aperture imaging systems [6] and diffractive imaging systems [7]. Among them, the deployable segmented imaging system has low technical implementation difficulty and high imaging quality compared with other forms [8], and therefore has widely concerned by researchers [9].

The primary mirror of a deployable segmented system is usually composed of segmented mirrors with the same curvature radius but different off-axis distances. Compared with traditional single-mirror optical systems, the error form of deployable segmented optical systems is more complex [10]. As shown in Fig. 1, according to the source of error, they can be divided into three categories: manufacturing error, alignment error, and environmental error. Alignment error occurs in the pose deviation of the sub-mirror, mainly including piston error, X/Y tilt error, eccentricity error and clock rotation error. These errors can be eliminated by a precise six-degree-of-freedom displacement adjustment mechanism, which is beyond the scope of this paper, so this part of the error is not considered in this study. Manufacturing error and environmental error occurs in the changes of the sub-mirrors surface shape and curvature radius, including the curvature consistency problem caused by the separate manufacturing and testing of each segmented mirror during the manufacturing process, and the additional deformation caused by transportation [11], gravity release/thermal stress, etc. It requires the sub-mirror’s own surface shape adjustment ability, which is the key problem that the deformable mirror needs to solve.

Fig. 1. The error source of the deployable segmented system.

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The JWST launched by NASA in 2021 is an important reference case for the deployable segmented system [12,13]. The primary mirror of JWST is composed of 18 nearly identical hexagonal sub-mirrors, each equipped with an active optical adjustment mechanism. A mechanical motor is installed at the center of the back of the sub-mirror, which is connected to the hexagon by six rods. The motor and rod mechanism can generate torque, realize the continuous adjustment of the curvature of each sub-mirror, and thus compensate for the curvature changes caused by the machining error and thermo-elastic deformation of the reflector. The actuation mode of the JWST sub-mirror belongs to the edge bending moment actuation mode, which is prone to produce large bending, can correct the curvature error, but has almost no active correction ability for other aberrations, so it is difficult to ensure the high-precision surface shape of the spliced whole mirror. Therefore, JWST can only be used in the infrared band. There are still many problems to be solved before achieving visible spectrum imaging applications.

For the future ultra-large aperture space telescope technology, to further break through the observation accuracy, it is necessary to achieve on-orbit nanometer-level high-precision surface shape control, while taking into account the curvature correction and aberration correction, therefore, we need to explore new active optics technology [14–16]. This paper starts from the control principle of the deformable mirror, and first explores two types of deformable mirror technologies that may be used for segmented sub-mirrors, including the vertical actuation active control and the embedded parallel-actuated active control technology, and based on the mathematical model of the influence function, the correction ability and characteristics of the two types of deformable mirrors for various Zernike aberrations are compared; then, the correction ability of the two for curvature and aberration is analyzed from the simulation perspective by finite element analyze. Finally, an embedded parallel-actuated deformable mirror with a diameter of 300mm for the space deployable segmented system is designed and manufactured, and the adjustment ability for typical aberrations is verified by experiment.

2. Principle of active control of deformable mirror

The on-orbit active control of deformable mirror belongs to the research field of active optics, which generally consists of three components: wavefront sensor, wavefront controller, and wavefront corrector. The wavefront sensor is used to measure the aberration of the incident wavefront, the wavefront controller converts the aberration into the corresponding control signal and sends it to the wavefront corrector, the wavefront corrector receives the signal and controls the actuation unit to output the correction force to the mirror surface to correct the wavefront aberration, therefore, the wavefront corrector is the core component of the active optical system [17].

For the wavefront corrector, according to the different ways of applying the actuation force, it can be divided into two forms: vertical actuation and parallel actuation, the actuation principle is shown in Fig. 2.

Fig. 2. Schematic diagram of different actuation modes. (a) Vertical actuation mode. (b) Parallel actuation mode.

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In the vertical actuation mode, a single actuator is mostly composed of discrete components of a motor and a mechanical structure. Many identical actuators are installed vertically on the bottom of the mirror, and the mirror surface is deformed by applying a vertical force. The actuator needs to design a rigid back plate separately to fix it, to ensure that the deformation force can strong enough to deform the mirror surface instead of the fixed back plate.

In the parallel actuation mode [18], the actuator is embedded in the support rib plate at the back of the mirror, and the mirror surface is deformed by the actuator’s deformation in the direction parallel to the mirror surface. Compared with the vertical actuation mode, this actuation mode does not require a rigid back plate, which can greatly reduce the envelope and weight of the adjustment mechanism.

2.1 Influence function

The influence of the actuator on the mirror surface is different for the two actuation modes. Generally, we define the effect of a single actuator applying a unit force on the mirror surface as the influence function $I_{n}(x,y)$ of the actuation unit. After measuring the influence function of each actuator, we can obtain the relationship between the correction target aberration $W(x,y)$ and the voltage $v_{n}$ that should be applied on each actuator. Assuming the number of actuators is $N$, the relationship can be written as:

(1)$$W(x,y)=\sum_{n=0}^{N}v_{v}I_n{(x,y)}$$

If the number of sampling points of the deformable mirror surface is $M$, the coordinates of the sampling points are $(x_{m},y_{m}),m=1,2,\ldots,M$, and the height value at the $m$ point is $W(x_{m},y_{m})$, then a matrix equation can be listed as Eq. (2), and the voltage matrix $V$ can be obtained by using the least squares method.

(2)$$\begin{pmatrix}W(x_1,y_1)\\W(x_2,y_2)\\\vdots\\W(x_M,y_M)\end{pmatrix}=\begin{pmatrix}I_1(x_1,y_1) & I_2(x_1,y_1) & \cdots & I_N(x_1,y_1)\\I_1(x_2,y_2) & I_2(x_2,y_2) & \cdots & I_N(x_2,y_2)\\\vdots & \vdots & & \vdots\\I_1(x_M,y_M) & I_2(x_M,y_M) & \cdots & I_N(x_M,y_M)\end{pmatrix}\begin{pmatrix}v_1\\v_2\\\vdots\\\nu_N\end{pmatrix}$$

Essentially, the control ability of the actuator on the mirror surface depends on the fitting ability of its influence function matrix to the target aberration. However, in the design and analysis stage, we cannot obtain the influence function by experiment, so it is very important to construct a mathematical model that can accurately represent the influence function.

The influence function of the deformable mirror with vertical actuation mode has been well studied [19,20], and can be approximated as a Gaussian type, whose expression is as follows:

(3)$$I(r)=exp[ln(\omega )\cdot (\frac{r}{d_{0}} )^{\alpha }]$$

where $\omega$ represents the coupling coefficient, which indicates the influence of the adjacent actuators at the center of this actuator, generally between $5{\% }$ and $15{\% }$; $d_{0}$ represents the distance between adjacent actuators; ${\alpha }$ is the Gaussian index, normally varies from 1.5 to 2.5; the values of the influence function parameters are determined by the structure and stiffness of the mirror and the actuator. Figure 3(a) shows the cross-sectional models of the Gaussian type influence functions under different coupling coefficients.

Fig. 3. Comparison of two types of influence functions. (a) Gaussian influence function. (b) Quadratic influence function.

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For the embedded parallel-actuated deformable mirror, there is less related work on the influence function research. To obtain the expression formula of the influence function of the parallel actuation, we derive the expression of the quadratic function type influence function based on the mechanical principle. The actuators embedded in the mirror reinforcement ribs expand in parallel, applying a bending moment force on the mirror surface. According to the two-dimensional bending moment equation Eq. (4), the strain induced by the bending moment force is a quadratic function [21].

(4)$$M(l)=\frac{ql^{2}}{2}$$

Therefore, the influence function at the center of action should also be quadratic. The expression of the quadratic function type influence function is constructed as follows:

(5)$$I(r)=[-(\frac{r}{b})^2+a]\cdot Circle(r_{0})$$

where $b$ is the actuation eigenfactor of the actuator, which describes the steepness of the action curve; $a$ is the response intensity, which describes the influence ability of the actuator on the surface shape; $Circle(r_{0})$ indicates the action range of the actuator. The values of the influence function parameters are determined by the structure and stiffness of the mirror and the actuator. Figure 3(b) shows the cross-sectional models of the quadratic function type influence functions under different actuation eigenfactors.

In fact, due to the constraint effect of adjacent cells at the boundary, the parallel-actuated influence function should also be smooth at the boundary. However, since this effect domain is generally small, in order to simplify the model, the correction term is no longer considered in this model. Only the shape characteristics of the influence function model in the central effect domain are considered.

2.2 Aberration correction capability comparison

Based on the error sources of the segmented mirror discussed in the introduction, we can identify the essential characteristics of the sub-mirrors. First, they should have good curvature adjustment ability to address the curvature consistency problem inherent in the spliced mirror. Second, they should have excellent aberration correction ability, especially for low-order and low-frequency aberrations, which are mainly induced by mechanical and thermal factors in the space optical system. Before exploring the curvature correction ability, we need to clarify the type and magnitude of aberrations caused by the variation of curvature radius. Assuming that the F-number of a single sub-mirror is $F^{{\# }}$, the curvature radius is $R$, and the sagittal height is $S$, when $S\ll R$, we can derive the sagittal height difference due to the change of curvature radius as:

(6)$$\delta S={-}\frac{1}{8(F^{{\#}})^{2}} \delta R$$

The wavefront aberration caused by the curvature radius deviation is:

(7)$$W=2\cdot \delta S={-}\frac{1}{4(F^{{\#}})^{2}} \delta R$$

It can be inferred that the wavefront aberration introduced by the change in the sub-mirror’s curvature radius is mainly defocus, which corresponds to the fourth term of the Zernike polynomial. So the curvature correction problem can be reduced to the sub-mirror’s ability to correct low-order aberrations.

Set the Zernike order as the independent variable to examine the correction ability of two types of deformable mirrors for each order of Zernike aberrations. In the large-aperture space active reflection mirror, the distance between each actuator is generally far, and the mutual influence is weaker. The coupling coefficient of the Gaussian-type influence function $\omega =5{\% }$, and the Gaussian index $\alpha =2$; the actuation eigenfactor of quadratic function-type influence function $b=0.29$ and the response intensity $a=2$. The number of actuators for both is set to 196 and is uniformly arranged.

Figure 4 illustrates the correction ability of two types of deformable mirrors for the first 50 Zernike aberrations. A comparison of the two influence functions reveals that the Gaussian-type influence function has superior control ability for high-order aberrations under the same number and density of actuators. Hence, in adaptive optics, vertically actuated deformable mirrors are commonly used to correct high-order aberrations induced by atmospheric turbulence; The quadratic function-type influence function is more adept at correcting low-order aberrations. For space optical reflection mirrors, the aberrations are mainly due to the mirror surface deformation caused by environmental factors such as force and heat, which result in low-order aberrations. Therefore, the embedded parallel actuator deformable mirror with a quadratic function-type influence function is more appropriate for active control in space optical systems.

Fig. 4. Comparison of the correction ability of two influence functions for the first 50 Zernike aberrations.

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3. Finite element analysis

For a space optical system, the sub-mirror should be as light as possible to minimize the challenges and costs of launch and transportation. However, it is difficult to quantify from the mathematical model of the influence function. Hence, a simulation model with physical parameters is needed. We designed two types of deformable mirrors with different actuation methods under the same aperture, using a hexagonal reflector with a diagonal distance of $300mm$ as an example. Then we used the finite element analysis method to further comparation, evaluating the active correction ability of the two actuation methods [22].

First, We create a finite element model. The vertical actuated mirror group has a size of 300mm $\times$ 300mm $\times$ 70mm and a design weight of 3.94kg, as shown in Fig. 5(a). The parallel actuated mirror group has a size of 300mm $\times$ 300mm $\times$ 30mm and a design weight of 1.95kg, as shown in Fig. 5(b). Next, we set the working conditions and apply a unit force to each actuator to solve the influence function. Then, we set three typical aberrations (astigmatism, spherical, and coma) with a one-wavelength perturbation to evaluate the aberration correction ability of the deformable mirror and set defocus and spherical with a one-wavelength perturbation to evaluate the curvature correction ability.

Fig. 5. (a) Finite element model of vertical actuated mirror. (b) Finite element model of parallel actuated mirror.

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Figure 6 displays the surface shape before and after correction. Table 1 and Table 2 present the correction results. The correction ability of vertically actuated and parallel actuated deformable mirrors were compared:

Fig. 6. Cloud map of correction ability for three typical aberrations.

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Table 1. Statistics of surface shape correction ability results

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Table 2. Statistics of curvature correction capability result

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For aberration correction ability, the parallel actuation method performs as well as the vertical actuation method for astigmatism and coma aberrations under the same disturbance input; however, it has a much better adjustment ability for spherical aberration; For curvature adjustment ability, whether it is a second-order defocus disturbance or a fourth-order spherical aberration disturbance, the active control capability of the embedded parallel-actuated method is better than that of the vertical actuated method; For weight and envelope, the parallelly actuated mode reduces both by more than $50{\% }$ compared to the vertically actuated mode. Therefore, the embedded parallel-actuated deformable mirror is more suitable for the sub-mirror of a segmented mirror system.

4. Experimental verification

4.1 Mirror structure design scheme

The reflector has a hexagonal shape with a diameter of 300mm. The back of the mirror body has a triangular lightweight design that enhances the structural rigidity and material utilization. A rectangular opening in the middle of the mirror body rib allows the installation of the actuator unit. The mirror body material is Si, and the mirror surface is a concave spherical surface with a curvature radius R of 2500mm. The surface density of the mirror body is 17.95kg/m$^{2}$. Figure 7 depicts the connection scheme between mirror body and flexible support. The material properties of the reflector mirror group are shown in Table 3.

Fig. 7. Connection scheme between mirror body and flexible support.

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Table 3. Material properties of the reflector mirror group

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We selected the PK4JMP2 model of the piezoelectric ceramic actuator from Thorlabs based on the required output force range of the experiment and the available installation space of the mirror body. The actuator has an outer dimension of 3.4 $\times$ 4.8 $\times$ 9.0 (mm) and a weight of 0.03g. To ensure the accurate installation position of each piezoelectric ceramic actuator, an actuator mounting bracket is designed. The bracket is attached to the opening of the mirror body rib by adhesive bonding. We installed 42 actuator units on the mirror body. Figure 8 shows the physical object of the installed mirror group.

Fig. 8. Physical picture of the experimental mirror group assembly.

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4.2 Results and discussion

The mirror surface deformation test optical path on a vibration isolation platform was established, as shown in Fig. 9. The main components included the active deformable mirror assembly (including the mirror, 42-unit piezoelectric actuator and 42-unit piezoelectric actuator connection structure, etc.), a PI adjustment platform, a 4D interferometer, an air-floating vibration isolation platform, and a 42-unit piezoelectric actuator controller. The 4D interferometer were monitored the surface shape of the active mirror in real time.

Fig. 9. Schematic diagram of the experimental optical path.

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As it is difficult to simulate the specific deformation of the mirror surface caused by environmental factors during on-orbit, we used a reverse validation approach to control the deformable mirror to produce specific aberrations instead of eliminating them to verify its aberration correction ability.

We measured the influence functions of each actuator by applying a unit voltage. Using the quadratic function type influence function derived from the theory as the basis function, the influence functions measured at different symmetrical positions are fitted, as shown in Fig. 10. The fitting effect is good, which verifies the correctness of the theoretical model.

Fig. 10. Top: The influence function test results of different symmetric positions. Bottom: Fitting the cross-section of influence function at the symmetrical position 1.

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Using Zernike polynomials, we generated defocus, coma, and spherical aberrations with a one-unit wavelength on a unit circle inscribed in a regular hexagon. We used them as target values and solved the matrix equation of the target image and the influence function to obtain the voltage values of each actuator. We controlled the surface shape by applying the voltage values to the corresponding actuators. Figure 11 shows the comparison between the target and generated aberrations.

Fig. 11. Comparison between target aberration and control generated aberration.

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Table 4 shows the reproduction ability of three typical aberrations. The aberration reproduction results show that each figure reproduces the typical features of each aberration as expected. The relative errors (|adjusted value-target value|/target value) of the PV and RMS values of each aberration before and after adjustment are within 10${\% }$, confirming that the embedded parallel-actuated form has good adjustment ability for the three typical aberrations.

Table 4. Reproduction ability of three typical aberrations

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As mentioned in Section 2 and 3, we can evaluate the advantages of embedded parallel-actuated deformable mirrors as spatially segmented sub-mirrors from two perspectives. First, the lightweight structure: embedded parallel-actuated deformable mirrors significantly reduce the surface mass density of the mirror compared to traditional support techniques for reflective mirrors, achieving a reduction by an order of magnitude (from the 100 kg range to the 10 kg range). In comparison to other active structures, such as vertically actuated mirrors, the total weight of mirror group can be reduced by 50${\% }$ while maintaining the same correction accuracy. Second, the ability to correct curvature and low-order aberrations: through an analysis of the influence functions for parallel and vertical actuated deformable mirrors, we mathematically demonstrate that parallel-actuated deformable mirrors exhibit superior correction capabilities for low-order aberrations. Under the same actuator arrangement, parallel-actuated deformable mirrors show a more pronounced advantage in correcting the first 15 Zernike polynomials.

5. Conclusion

This article focused on the research for high-performance active deployable system sub mirror technology.

Firstly, from the perspective of the mathematical model of the influence function of deformable mirrors, a comparative analysis was conducted on the adjustment ability of vertical actuation and embedded parallel-actuated deformable mirrors for three typical aberrations and curvature radius. The fitting effect of the influence function under different mathematical forms on various orders of Zernike polynomials was studied;

Secondly, a finite element model was established, and simulation results showed that the parallel-actuated deformable mirror not only has better ability to correct low order aberrations, but also has a lighter design. The advantages of embedded parallel-actuated deformable mirror in correcting low order aberrations and lightweight structures were determined;

Finally, an embedded parallel-actuated deformable mirror with a diameter of $300mm$ was designed and manufactured, and experimental tests on the correction ability of three typical aberrations were completed. The results showed that its correction ability for all three typical aberrations could reach over 90${\% }$.

Funding

National Natural Science Foundation of China (11873007, 62175234).

Acknowledgments

We gratefully thank Haixiang Hu and Xin Zhang at CIOMP for the discussion about the influence function for the different acutation forms and the help about the wavefront aberration caused by the curvature radius deviation.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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.

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Disturbance Type	Actuation Form	RMS(nm)			P-V(nm)
Disturbance Type	Actuation Form	Input Value	Corrected	Correction Ability	Input Value	Corrected	Correction Ability
Astigmatism (1 $λ$ )	Vertical	46.10	1.26	97.27 $%$	413.6	11.55	97.20 $%$
Astigmatism (1 $λ$ )	Parallel	46.10	2.18	95.27 $%$	413.6	21.80	94.73 $%$
Spherical (1 $λ$ )	Vertical	50.80	23.89	52.98 $%$	347.21	227.21	34.56 $%$
Spherical (1 $λ$ )	Parallel	50.80	10.27	79.78 $%$	347.21	119.30	65.64 $%$
Coma (1 $λ$ )	Vertical	48.75	8.74	82.08 $%$	532.78	118.49	77.76 $%$
Coma (1 $λ$ )	Parallel	48.75	5.65	88.41 $%$	532.78	54.66	89.74 $%$

Disturbance Type	Actuation Form	Curvature Radius Error (mm)	Correction Residuals (mm)	Correction Ability
Defocus (1 $λ$ )	Vertical	0.14	0.025	82.5 $%$
Defocus (1 $λ$ )	Parallel	0.14	0.0038	97.3 $%$
Spherical (1 $λ$ )	Vertical	0.038	0.0065	82.9 $%$
Spherical (1 $λ$ )	Parallel	0.038	0.0048	87.5 $%$

Material	Density $ρ$ ( $10^{- 6}$ kg/mm $^{3}$ )	Elastic Modulus E(GPa)	Linear Expansion Coefficient $α$ ( $10^{- 6}$ / $^{\circ}$ C)	Tensile Strength $σ$ b(MPa)	Poisson ratio $μ$	Components
Si	2.23	127	2.6	120	0.22	Mirror
Invar	8.1	141	2.4	470	0.25	Flexible support Backplate

Disturbance Type	RMS(nm)			P-V(nm)
Disturbance Type	Target Value	Reproducible Value	Reproduction Ability	Target Value	Reproducible Value	Reproduction Ability
Astigmatism (1 $λ$ )	46.1	42.5	92.18 $%$	413	387	93.7 $%$
Spherical (1 $λ$ )	50.8	48.2	94.9 $%$	347	327	94.4 $%$
Coma (1 $λ$ )	48.9	45.4	93.0 $%$	537	500	93.0 $%$

Disturbance Type	Actuation Form	RMS(nm)			P-V(nm)
Disturbance Type	Actuation Form	Input Value	Corrected	Correction Ability	Input Value	Corrected	Correction Ability
Astigmatism (1 $λ$ )	Vertical	46.10	1.26	97.27 $%$	413.6	11.55	97.20 $%$
Astigmatism (1 $λ$ )	Parallel	46.10	2.18	95.27 $%$	413.6	21.80	94.73 $%$
Spherical (1 $λ$ )	Vertical	50.80	23.89	52.98 $%$	347.21	227.21	34.56 $%$
Spherical (1 $λ$ )	Parallel	50.80	10.27	79.78 $%$	347.21	119.30	65.64 $%$
Coma (1 $λ$ )	Vertical	48.75	8.74	82.08 $%$	532.78	118.49	77.76 $%$
Coma (1 $λ$ )	Parallel	48.75	5.65	88.41 $%$	532.78	54.66	89.74 $%$

Embedded parallel-actuated technology for deformable space segmented mirror

Abstract

1. Introduction

2. Principle of active control of deformable mirror

2.1 Influence function

2.2 Aberration correction capability comparison

3. Finite element analysis

4. Experimental verification

4.1 Mirror structure design scheme

4.2 Results and discussion

5. Conclusion

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (4)

Equations (7)

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