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Abstract
In order to reduce the cost of the piezo actuator array deformable mirror (DM), a piezoelectric DM driven by unimorph actuator arrays on multi-spatial layers is proposed. The actuator density can be multiplied by increasing the spatial layers of the actuator arrays. A low-cost DM prototype with 19 unimorph actuators located on three spatial layers is developed. The unimorph actuator can generate a wavefront deformation up to 11 µm at an operating voltage of 50 V. The DM can reconstruct typical low-order Zernike polynomial shapes accurately. The mirror can be flattened to 0.058 µm in RMS. Furthermore, a focal spot close to Airy spot is obtained in the far field after the aberrations of the adaptive optics testing system being corrected.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Deformable mirror (DM) is the critical component of adaptive optics (AO) system [1,2]. It can compensate the aberrations in the optical path by changing the mirror shape in real time, so that the optical system can reach the diffraction limit. Among various kinds of DMs [3–8], piezoelectric DMs have the advantages of simple structure, fast response and high reliability [9,10]. They have been widely used in the areas of astronomical imaging [11,12], retinal imaging [13], optical communication [14], and laser beam shaping [15]. The piezoelectric DMs can be usually divided into two categories. One is piezoelectric actuator array DM which comprises a continuous facesheet actuated by the underlying piezoelectric actuators [16,17]. It has a good correction capability for high-order aberrations. The other one is unimorph or bimorph DM which is formed by one or multiple piezoelectric layers bonded to the mirror layer [18–20]. The transverse piezoelectric effect is used to produce the bending deformation of the mirror. Unimorph or bimorph DMs are usually used in the correction of low-order aberrations.
The traditional piezo actuator array DM is driven by piezoelectric stack actuators. The high price limits its applications [21,22]. In order to reduce the cost, G. Vdovin et al. [23] proposed a DM driven by piezo tube actuators, but the price is still high. E. H. Yang et al. [24] and X. H. Xu et al. [25] proposed a piezoelectric MEMS DM driven by unimorph microactuators. MEMS manufacturing method can miniaturize the device and has the potential of mass production. However its complex manufacturing process restricts the realization of high-performance mirror membrane. So far, it has not entered the practical stage. In order to reduce the processing difficulty, G. M. Zhong et al. [26] proposed a DM driven by commercial piezoelectric unimorph actuators. The actuator is made of a copper disc and a piezoelectric disc. That DM has the advantages of simple fabrication, low cost and large deformation. For the regular design of the DM, the pupil size of the DM increases with the actuator number. In order to improve the actuator density, it is necessary to reduce the actuator size, resulting in a reduction in the deformability of the actuator, which is not conducive to the application.
To solve this problem, a new structure of unimorph actuator array DM is proposed. The mirror of the proposed DM is driven by piezo unimorph actuator arrays which are located on multi-spatial layers. The actuator density of the DM can be multiplied by increasing the number of the actuator arrays without reducing the actuator size. A low-cost DM prototype with 19 unimorph actuators located on three layers is fabricated and characterized. It is proved that the proposed DM has a good correction performance and has application prospects in low-cost AO systems.
2. Structure and principle
The structure diagram of the piezoelectric DM driven by unimorph actuators located on the same layer is shown in Fig. 1(a). The DM consists of a mirror, a piezoelectric unimorph actuator array, a substrate, and the connecting pillars. The piezoelectric unimorph actuator is formed by gluing a piezoelectric disc on an elastic membrane. The piezoelectric disc can be placed on the bottom or top of the elastic membrane. Here, the piezoelectric disc is on the bottom for the convenience of the electrical connection. It produces bending deformation due to the transverse inverse piezoelectric effect when a voltage is applied. The displacement of the actuator is transferred to the mirror through a connecting pillar at the center of the actuator, forming a local mirror deformation of a Gaussian-like function. When all of the piezoelectric actuators work together, the DM mirror can produce a deformation conjugate to the distorted wavefront for aberration compensation. The diameter of the through hole on the substrate is larger than that of the connecting pillar, so the connecting pillar can move freely in the through hole without hitting the side wall. For this structure, it needs to reduce the actuator size for increase of the actuator density, resulting in a reduction in the deformability of the actuator. The structure diagram of the piezoelectric DM driven by unimorph actuators located on multi-spatial layers is shown in Fig. 1(b). The actuators on different space layers are connected to the mirror through the connecting pillars with different lengths. The connecting pillars of the lower layer actuator arrays pass through the substrates of the upper layer actuator arrays. Based on this structural design, the actuator density of the DM can be multiplied by increasing the number of the actuator arrays without changing the actuator size. On the other hand, the size of the actuator can be increased under the premise of keeping the density of the actuator constant, as shown in Fig. 1, which is beneficial for increasing the actuator deformability.
Fig. 1. Schematic diagram of piezoelectric DM driven by unimorph actuators located on the same layer (a) and on multi- spatial layers (b).
Since the connecting pillars of the lower layer actuator arrays need to pass through the substrates of the upper layer actuator arrays, the number of spatial layers of the actuator array is limited. For the piezo actuator array DM, the actuators are conventionally arranged in a square arrangement or a triangle arrangement. The arrangement of multi-layer actuator arrays is shown in Fig. 2. The actuators on different space layers are mutually crosswise arranged. For square arrangement, two layers or four layers can be used. For triangle arrangement, three layers or four layers can be used. The spatial order of the layers can be interchangeable. Compared to the actuator spacing l of single-layer DM, the equivalent actuator spacing (the spacing of adjacent actuators regardless of the spatial layer) l’ of the multi-layer DM is reduced, which is equal to $\sqrt 2 {\kern 1pt} {\kern 1pt} /2$l for two layers, $\sqrt 3 {\kern 1pt} {\kern 1pt} /3$l for three layers and 1/2 l for four layers. This means that the actuator spacing can be reduced by increasing the number of layers effectively, thus the actuator density of the DM is increased.
Fig. 2. The arrangement of multi-layer actuator arrays. (a) Single layer of square arrangement, (b) single layer of triangle arrangement, (c) two layers of square arrangement, (d) three layers of triangle arrangement, (e) four layers of square arrangement, (f) four layers of triangle arrangement
The deflection of the actuator is transferred to the mirror through the connecting pillar at the center of the actuator. The pillars need to be rigid enough to transfer the displacement of the actuator to the mirror. At the same time, they have a certain degree of flexibility, which can produce slight bending to smooth the mirror deformation during the working process. The impact of the length of the connecting pillars is investigated using finite element method. Here, a three-layer piezoelectric unimorph actuator array DM is used. The connecting pillar adopts structural steel with a diameter of 0.5 mm. The pillars in different space layers have different lengths. The lengths of the pillars on the bottom layer, middle layer and top layer are 24 mm, 17 mm and 10 mm, respectively. A voltage of 100 V is applied to an actuator on each space layer. The simulation results are shown in Fig. 3. The deformation of the connecting pillars is mainly in the vertical direction. The deformations of the connecting pillars on different layers (from bottom to top) are 1.6 µm, 1.1 µm and 0.6 µm, respectively. The strains are 66 ppm, 64 ppm and 60 ppm, which are very close. Deformation of the connecting pillars results in a loss of actuator deformation. The mirror deformations of the actuators on different layers (from bottom to top) are 9.7 µm, 10.4 µm and 10.6 µm, respectively. A ten percent deformation loss is not serious for a larger deformation capacity. The diameter and length of the pillars can be appropriately increased to reduce the loss, but the flexibility will be reduced.
Fig. 3. Deformation simulation of the actuators on the top layer (a), the middle layer (b) and the bottom layer (c). The figures from left to right are the schematic diagram of voltage application, the DM structural deformation and the mirror deformation. The unit of the deformation is µm.
3. DM prototype
Figure 4 shows the prototype of the proposed DM driven by 19 piezo unimorph actuators located on three layers. The structure diagram is shown in Fig. 1(b). There are 7, 6 and 6 actuators on the bottom layer, the middle layer, and the top layer, respectively. The actuator is made of a copper disc and a piezoelectric disc. The diameters are 20 mm and 15 mm, and the thicknesses are 0.15 mm and 0.2 mm, respectively. The price of the unimorph actuator is only one-tenth or less expensive than piezoelectric stack actuator. The actuator spacing of single-layer actuator array is 21 mm. The equivalent actuator spacing of the three-layer DM is 12.1 mm. The effective aperture of the DM is 35 mm. The substrates are made of aluminum alloy with a thickness of 5 mm. There are two kinds of through-hole arrays on the substrates. The large through holes are used for mounting the actuators. The small through holes are used to allow connecting posts to pass through. A commercial polished silicon wafer with a thickness of 400 µm and a diameter of 3 inches is used as the mirror. The mirror is coated with a protective silver film. The preparation process is briefly described as follows. Firstly, the unimorph actuators were bonded on the substrates using epoxy, which are coaxial with the large through holes. Then, the connecting pillars with different lengths were glued to the mirror with the help of positioning mold. The three-layer piezoelectric unimorph actuator arrays were subsequently glued on the connecting pillars carefully. It is worth noting that multiple actuator arrays do not increase the difficulty of fabrication with the help of the positioning mold. Finally, the DM was electrically connected and packaged.
Fig. 4. Prototype of the DM. (a) fabricated DM, (b) piezoelectric unimorph actuators, and (c) side view of the DM before packaging.
4. Characterization
4.1 Test system
In order to verify the performance of the fabricated DM, an AO test system based on Shack-Hartmann wavefront sensor was built, as shown in Fig. 5. A single-mode fiber-coupled semiconductor laser with a wavelength of 635 nm was used. The laser beam firstly passes through the beam splitting prism BS1, and then is collimated by a lens L1 with a focal length of 400 mm. The collimated light reaches the mirror of the DM, and then returns through the lens L1 in the same path. The beam reaches the beam splitting prism BS1 again. Half of the beam is reflected in the direction of 90° and passes through a lens L2 with a focal length of 40 mm. The lens L1 and lens L2 form a 10-fold beam shrinking system. Finally, the beam passes through a second beam splitting prism BS2, where 50% of the beam is received by a wavefront sensor (Thorlabs WFS150-7AR). A 23 × 23 microlens array is used for measurement, and 65 Zernike polynomials are used for wavefront aberration fitting. The wavefront information is measured by the wavefront sensor in real time and used as the feedback of the control system to generate the target wavefront using the DM [27]. In the experiment, the voltage range from 0 V to 100 V was used for testing the fabricated DM. The mirror deformation when a bias voltage of 50 V is applied to all the actuators is set as a reference, thus the equivalent voltage range is from -50 V to 50 V. The other 50% of the beam is focused by the lens L3 with a focal length of 200 mm. The focal spot in the far-field is measured by a CCD camera to evaluate the correction performance.
4.2 Actuator performance
The wavefront deformation response (influence function) of each actuator was obtained by subtracting the initial mirror surface from the measured wavefront of the DM under a driving voltage of 50 V. The wavefront deformations of all actuators in the effective aperture are shown in Fig. 6. The actuator produces a Gaussian-like local deformation of the mirror. The measured deformations are basically consistent with the simulation results. The DM has a large stroke at a low operating voltage. The deformation cross-sections of the typical actuators are shown in Fig. 7. The peak-to-valley (PV) values of the generated deformations of actuator 1, 2, 8, and 9 in the effective aperture are 11.5 µm, 11.8 µm, 7.9 µm and 11.3 µm, respectively. The PV value of actuator 8 is smaller than others since it is at the corner and farther from the center of the mirror. The inter-actuator coupling coefficient, which shows how much the movement of one actuator displaces its neighbors, is also very important for DM. If the coupling coefficient is too large, the ability of correcting high-order aberrations will be reduced. If the coupling coefficient is too small, the local fluctuation of the actuator will be formed, which is not conducive to correcting low-order aberrations. The central deformation (a point) of actuator 1 is 11.5 µm. The deformation at the adjacent actuators (b point) is about 2.6 µm. The calculated coupling coefficient is about 23%. According to previous research results [28], this value is appropriate for wavefront correction.
Fig. 6. The wavefront deformation of the all actuator in the effective aperture at a voltage of 50 V, from left to right are the actuator layout diagram, experimentally measured and simulated wavefront deformations.
4.3 Correction capability
Reconstruction of Zernike polynomial shapes was performed to evaluate the correction capability of DM. In this work, the first 14 Zernike polynomial shapes (except the tip/tilt terms) were reconstructed using the fabricated DM. The results are shown in Fig. 8. The experimentally generated shapes fit the theoretical shapes quite well. The root-mean-square (RMS) and the residual error (ERR, the RMS value of the difference between the measured shape and the ideal shape) of each actual reconstructed Zernike polynomial aberration were calculated. The residual error is much smaller than the RMS values. Furthermore, the PV values and the normalized residual error (the ratio of ERR to RMS) of each Zernike polynomial shape were caluculated and ploted in Fig. 9. The reconstruction amplitude decreases gradually with the increase of aberration order. The normalized residual error increases with the increasing of Zernike polynomial term. The PV amplitudes of the reconstructed astigmatisms (Z3 and Z5) are about 14.7 µm with a normalized residual error of about 1.3%. The PV amplitude of the reconstructed defocused aberration (Z4) is about 12.1 µm with a normalized residual error of about 2.5%. The PV amplitudes of the reconstructed trefoil aberrations (Z6 and Z9) are about 8.1 µm with a normalized residual wavefront error of about 6.3%. The PV amplitudes of the reconstructed coma aberrations (Z7 and Z8) are about 5.9 µm with a normalized residual error of about 8.6%. The experimental correction residual error is less than 10% for the first 9 Zernike polynomial shapes, indicating that the fabricated DM has good reconstruction capability. This DM also has good correction capability for the 10th to 14th Zernike aberrations, though the DM has only 19 actuators. The correction capability of the DM can be further improved by increasing the number of actuators.
Fig. 9. PV values and normalized residual errors of the reproduced Zernike polynomial shapes from the 3rd to 14th mode.
4.4 Mirror quality
Due to the bonding stress during the fabrication process and the inconsistency of the thermal expansion coefficients of the materials, the fabricated mirror surface will be deformed inevitably. The initial wavefront deformation of the mirror was measured and flattened by itself. The results are shown in Fig. 10(a). The initial deformation is mainly comprised by low-order aberrations and can be corrected by the DM in a closed-loop control. The PV value of the initial deformation is 6.034 µm, and the corresponding RMS value is 1.305 µm. After correction the PV value of the deformation is substantially decreased to 0.591 µm with a corresponding RMS value of 0.058 µm (only about 1/11 wavelength). The equivalent voltage used for flattening is from -30 V to 30 V. Since the operating voltage range of the actuator with a 200 µm thick piezoelectric layer can be 0-200 V (equivalent to ±100 V), only 30% of the voltage range is used to correct the initial aberrations of the mirror surface, which indicates that there is enough voltage for AO applications. Since there only 19 actuators were used for correction, the correction capability is limited and the high-order aberrations cannot be corrected. The comparison of Zernike coefficients before and after flattening is shown in Fig. 10(b). The remaining aberrations after flattening are high order aberrations. The residual wavefront error can be further reduced by increasing the number of actuators. The far-field focal spots before and after flattening were measured by a CCD camera. The results are shown in Fig. 10(c). The focal spot before correction is distorted seriously due to the aberrations caused by the DM and the optical system. Since the intensity of the focal spot before correction was very weak, the exposure time required was much longer than that after correction. After flattening, the focal spot is close to Airy spot, with some low stray spots at the surround of the main spot. This means that the DM has a good mirror quality after correction.
Fig. 10. Mirror flattening. (a) Initial and flattened mirror surface (unit: µm), (b) Zernike polynomial decomposition of the wavefront aberrations before and after correction, and (c) the far-field focal spot before and after correction.
5. Conclusions
In this work, we demonstrated a new structure of piezoelectric DM that driven by unimorph actuator arrays on multi-spatial layers. Compared with the traditional DM where all the actuators are distributed on the same layer, the proposed DM can improve the actuator density by increasing the spatial layers of actuators while keeping the overall size of the DM unchanged. The space layouts of the actuators were designed for square arrangement and triangle arrangement. A DM prototype with 19 actuators located on three layers was fabricated and characterized. The test results show that the wavefront deformation generated by single actuator can reach 11 µm at a low voltage of 50 V. It can accurately reconstruct the first 9 Zernike polynomial shapes with normalized residual error less than 10%. The mirror can be flattened to 0.058 µm in RMS. After the mirror aberrations as well as the system aberrations were corrected, the far-field focal spot close to Airy spot was obtained. This proposed DM has the characteristics of large stroke, low operating voltage, low cost and good correction performance. The DM can be scaled up easily for low-cost AO applications even for astronomy application.
Funding
Ningbo Science and Technology Innovation 2025 (2021Z068); Zhejiang Province Public Welfare Technology Application Research Project (LGG22E050002).
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.
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