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Abstract
The high-frequency aberrations (HIFAs) that are corresponding to the actuator array have been reported to exist on the initial surface shape of many deformable mirrors (DMs), such as the bimorph DM, the unimorph DM, the monomorph DM, and the membrane DM. This actuator-corresponding high-frequency aberration (AC-HIFA) could not be corrected effectively by the DM and would limit the correction ability of the DM. In this paper, we presented the AC-HIFA in a stacked array piezoelectric (PZT) DM, which may result in ghost damage that is dangerous in the high power laser system. More importantly, we investigated a solution through simulation and experiment that by using a mirror plate and a long thin post array, which were machined integrally from a piece of BK7 glass, the AC-HIFA could be eliminated completely. In addition, the structural parameters’ influences on the AC-HIFA were investigated in the simulation, which could help other researchers to determine appropriate parameters of the mirror and the posts and to make a fine surface shape in their own DMs.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The DM is widely used to correct the distorted wavefront in many applications, such as the high-power laser systems [1–3], the microscopy [4,5] and the astronomical telescopes [6–8]. However, restricted by the structure and the mechanism, the DM with limited actuators is only used to correct the low order wavefront aberrations and has limitations to correct the HIFAs effectively [9–11]. The HIFAs may arise from the initial surface shape of the DM and be left in the corrected wavefront or intensity [12,13]. Therefore, during the practical manufacturing process, it is essential to ensure a fine initial surface shape, for which various methods have been developed, such as the fine machining techniques [14], the double-sided coating techniques [15], and the constant temperature/humidity control [16]. Random static HIFAs in the initial surface shape are reduced greatly by the above-mentioned techniques. However, some AC-HIFAs still exist in the initial surface shape (or the residual surface shape) of the DMs, including the membrane DM [12,17], the monomorph DM [13], the unimorph DM [18], and the bimorph DM [19].
The AC-HIFAs exhibit a convexity array in the initial surface shape that matches the actuator array at the backside of the mirror. Some researches have been done to depress or eliminate the AC-HIFAs. In a boundary actuation DM, forces and/or torques are applied on the mirror rim to produce distortions of the mirror interior. Since the forces are applied far from the optical surface, the AC-HIFA could be avoided [20]. In an ultralightweight DM, a thin substrate of optical surface is coated with continuous active layers. Any stiff backing structure for the mirror surface is eliminated and microfabrication technologies are exploited to provide tight integration of the active materials into the mirror structure, to avoid AC-HIFA [21]. The dynamic AC-HIFA, which appears during the correction process, could be eliminated by realizing the strict perpendicular relation between the actuator and the mirror plate [22].
The residual stress that arises from the adhesive curing [23,24] has been identified as an important reason for the AC-HIFA [17,18,25]. When the mirror is glued to an actuator array or a continuous actuator disc together with an electrode array, the discontinuous distribution of the adhesive results in the inhomogeneous residual stress across the mirror, which leads to the inhomogeneous distortions and a convexity array in the initial surface shape. Some methods are brought forward to reduce or eliminate the adhesive curing induced AC-HIFAs in the initial surface shape, including the re-grinding and re-polishing of the assembled coated mirror [17], adopting a thick mirror [25] or the adhesive with ultralow residual stress. Nevertheless, these methods are difficult to be applied to the stacked array PZT DM for the high power laser system. As to the method of re-grinding and re-polishing the assembled coated mirror, the mirror has to be recoated after it is ground and polished, using the low-temperature coating technique to protect the PZT and the adhesive from the thermal damage. However, the breakdown threshold of the low-temperature coating film is relatively low, which would limit the application of the DM. The adhesives should be non-absorbent to the incident laser radiation to maintain a high strength in the high power laser system. However, the radiation resistance of the adhesive with ultralow residual stress has not been experimentally ensured yet. Also, the thick mirror is not a good solution. Although a thick mirror may enhance the surface fidelity and protect the surface from harmful stress, nevertheless, the effective stroke of the DM would be reduced distinctly by the thick mirror for a given drive on the actuator, which will greatly limit the application of a DM. Fortunately, a mirror with a post array structure has been presented to be effective in eliminating the adhesive curing induced AC-HIFA of the PZT DM [26]. However, the detailed investigation has not been presented.
In this paper, the post solution to eliminate the adhesive curing induced AC-HIFA on the initial surface shape of a stacked array PZT DM is investigated systematically. This paper is organized as follows. In section 2, the AC-HIFA of the stacked array PZT DM is presented. In section 3, a simulation model is built to analyze the parameter influences of a conventional mirror on the AC-HIFA. Simulation results show that by adopting a thick mirror or a low-stress adhesive, the AC-HIFA could only be depressed to some extent. In section 4, an improved DM model, using a mirror with a post array, is presented to overcome the influence of the adhesive and achieve the fine initial surface shape without any AC-HIFA. Simulational results show that by adopting a relatively long and thin post array, the AC-HIFA could be essentially eliminated from the initial surface shape of the DM. In section 5, an experiment is conducted, in which no AC-HIFA is observed in the initial surface shape of the DM or in the intensity distribution of a laser beam.
2. AC-HIFA in a stacked array PZT DM
Owing to the high breakdown threshold, the large aperture and the large stroke, the stacked array PZT DMs are commonly used in the high power laser systems [1–3,25]. It is important to ensure the flatness of the initial surface shape, as the HIFAs in the initial surface shape could not be corrected effectively by the actuators and might result in the unexpected shift of the focal point when the laser is in operation, which is particularly dangerous. Figure 1 shows a lab-manufactured 116-actuator DM, which consists of a conventional mirror plate, a hexagonally distributed PZT array (P885.91, Physik Instrumente GmbH, maximum travel range of 38 μm, operating voltage range of −20 V to 120 V.), a flexure plate array, and a base (the DM is referred to as the conventional DM hereinafter). The mirror is bonded to the flexure plate array with the adhesive (Hysol EA 9330). The structural parameters are listed in Table 1. The mirror surface shape is considerably flat before bonding, without AC-HIFA on it.
Fig. 1 A lab-manufactured 116-actuator DM with a conventional mirror plate. (a) The photo. (b) The structure sketch. HG is the mirror thickness.
Table 1. Parameters of the conventional DM in the experiment
Figure 2 shows the interferometer (wavelength 632.8nm, 6〞VERIFIRE XPZ of Zygo Corp.) measurement results of the conventional DM after the adhesive is completely cured. During the measurement, the DM is not controlled. It could be seen that AC-HIFA appears on the initial surface shape as a result of the residual stress of the cured adhesive, with a convexity array matching the PZT array. The peak to valley (PV) value of the initial surface shape is 731.75nm, which is considerably large.
Fig. 2 Interferometer measurement results of the conventional DM. PV = 731.75nm. (a) Two-dimensional color temperature diagram. (b) Three-dimensional color temperature diagram. (c) Interference fringe diagram.
In order to investigate the influence of the AC-HIFA on the incident light intensity, a laser radiation experiment is made as shown in Fig. 3. The divergent beam from a fiber-coupled laser (1053 nm wavelength, 100 mW, 400 μm core diameter, 0.22 NA) is collimated by a lens (Lens1) and is transformed to a parallel beam. The parallel beam is incident on the tested surface and is reflected in the opposite direction. Two lenses (Lens1 and Lens2, with the focal length of F1 = 900mm and F2 = 60mm, respectively) are used to form a telescope system. The reflected beam is condensed by the telescope and is captured by a CCD camera (1004 × 1004 pixels, sensor size 7.4 mm × 7.4 mm, piA1000-48gm of Basler, Inc.) When a lenslet array (array size 10 mm × 10 mm, lenslet pitch 300 μm, focal length 8.7 mm) is used in front of the CCD camera to investigate the point array intensity, the focal length of Lens2 is changed to 100 mm to expand the beam size and to increase the point number so that the distribution of the point intensity could be shown accurately.
Fig. 3 Sketch of a laser radiation experiment. The incident beam is represented by the pink color. The reflected beam is represented by the blue color.
A calibration experiment is conducted to investigate the system influence on the CCD image, in which an optical flat (flatness of λ/20, λ = 632.8nm) is tested [Fig. 4]. Figure 4(a) shows that when there is no lenslet array, the light intensity of the CCD image is relatively homogeneous. The vague fringes and circles arise from the inhomogeneous intensity of the fiber-coupled laser and the defects of the optical elements, which is inherent to the optical system and could not be removed. In Fig. 4(b), when the lenslet array is applied (i.e. the CCD becomes a Shack-Hartmann CCD), the point array is regular and matches with the lenslet array. It indicates that the fringes from the fiber-coupled laser do not introduce notable wavefront aberrations and would not influence the experiment results in the following.
Fig. 4 Calibration result by an optical flat. (a) CCD image without the lenslet array. (b) CCD image with the lenslet array.
Figure 5 shows the measurement results of the conventional DM in the laser radiation experiment. In Fig. 5(a), a dark-spot array that matches the PZT array appears, which is the image of the AC-HIFA in the initial surface shape that results from the adhesive curing. It illustrates that the AC-HIFA in the initial surface shape has an intensity modulation on the laser beam. In Fig. 5(b), when the lenslet array is applied, the point array has an inhomogeneous intensity distribution with a dark-hole array, which corresponds to the convexity array in the initial surface shape. It could be explained in the view of geometrical optics. The reflected beam from the conventional DM is not a parallel beam, which could be divided into two parts. The reflected beams from the convexity array of the DM surface form a divergent beam array. The other part of the reflected beam remains parallel. As the image distance of the divergent beam is further than that of the parallel beam, the point intensity of the divergent beam array is lower than that of the parallel beam on the sensor surface (namely the image plane of the parallel beam), which results in the dark-hole array in Fig. 5(b). Figure 5(c) is the same as Fig. 5(b), in which the red circles are used to help to identify a hexagonally distributed dark-hole array.
Fig. 5 CCD images with the conventional DM. (a) Without the lenslet array. (b) With the lenslet array. (c) With the lenslet array. The red circles are used to identify a hexagonally distributed dark-hole array from Fig. 5(b).
It could be concluded that the residual stress of the cured adhesive results in AC-HIFA on the initial surface shape of the conventional DM, which has an intensity modulation on the incident laser beam. In order to eliminate the intensity modulation, the structure of the DM should be optimized to ensure that the initial surface shape is not affected by the residual stress of the cured adhesive.
A finite element model is built in ANSYS to investigate the influence of the structural parameters on the initial surface shape and to seek for an optimized DM structure. Before the investigation, the model of the conventional DM [Fig. 1, Table 1] is built to verify the validity of the simulation. The whole model is built with a 10-node tetrahedral structure solid element. The material parameters are listed in Table 2. The degree of freedom (DOF) of the base bottom is set to zero. The surface shape of the mirror is plane before bonding. A residual stress (denoted by S) of 10MPa is applied to the adhesive to represent the influence of the adhesive curing. Under these conditions, the initial surface shape of the conventional DM is obtained, as shown in Fig. 6.
Table 2. Material parameters in finite element simulation
Fig. 6 Simulational result of the lab-manufactured conventional DM in Fig. 1. (HG = 1mm, S = 10MPa). S represents the residual stress of the cured adhesive. (a) Two-dimensional initial surface shape. (b) Three-dimensional initial surface shape. (c) Residual surface shape.
Figure 6(a) is the simulational two-dimensional initial surface shape with a convexity array that matches the PZT array at the backside of the mirror. The PV value of Fig. 6(a) is 729.65nm, which is very close to the measurement results from the interferometer (731.75nm in Fig. 2). By contrasting Fig. 2(a) and Fig. 6(a), it could be concluded that the simulation result coincides well with the interferometer result. The same conclusion could be drawn by contrasting the three-dimensional initial surface shape of the interferometer result [Fig. 2(b)] and the simulation result [Fig. 6(b)]. Figure 6(c) is the residual surface shape when the target surface shape is an absolute plane. Under the control of the algorithm, the actuators are supplied with different voltages and provide a pulling or pushing to the mirror, which realizes the deformation. Once the residual error is smaller than a target, the correction process ends. It can be seen that the residual surface shape has the same convexity array as the initial surface shape, with the PV value slightly decreasing from 729.65nm to 711.80nm. Figure 6(c) illustrates that the AC-HIFA in the initial surface shape, which results from the adhesive curing, could not be effectively corrected by the actuators of the DM. Therefore, it is important to achieve a fine initial surface shape without any AC-HIFA.
3. Influence of the parameters on the AC-HIFA in a conventional DM
In a conventional DM, the mirror is a cuboid or a cylinder, which is bonded to the actuators by the adhesive. The residual stress of the cured adhesive is applied directly to the back surface of the mirror, which may result in a convexity array on the front surface, as shown in section 2. In this section, the influence of the parameters on the AC-HIFA of a conventional DM will be investigated. The purpose is to verify whether the AC-HIFA could be eliminated by simply changing the parameters, namely the mirror thickness (denoted by HG) and the residual stress (denoted by S) of the adhesive. In the simulations below, the finite element model is the same as that used in section 2, except for the variables under investigation.
3.1 Mirror thickness
In Fig. 7 and Fig. 8(a), when the mirror thickness increases from 1mm to 5mm with an increment of 1mm, the AC-HIFA exists in the initial surface shape and the PV value decreases from 729.65nm to 245.10nm. As the maximum stroke is limited by the mirror thickness, the mirror thickness larger than 5mm is not in consideration. Figure 8(b) and 8(c) show that the initial surface shapes mainly consist of several Zernike polynomials, which result in the AC-HIFA. By adopting a thicker mirror [Fig. 8(c)], the Zernike coefficients could be decreased. However, a uniform coefficient distribution among the Zernike polynomials could not be achieved. Figure 9 shows the residual surface shape of the Fig. 7. It could be seen that the AC-HIFA could be decreased by the correction; however, the convexity array still remains in the residual surface shape. The simulation results indicate that a thicker mirror could only decrease the AC-HIFA in a conventional DM, but is unable to eliminate them.
Fig. 7 Initial surface shape with a varying mirror thickness (HG) in a conventional DM. (a) HG = 1mm. (b) HG = 2mm. (c) HG = 3mm. (d) HG = 4mm. (e) HG = 5mm. (S = 10MPa).
Fig. 8 (a) PV values of the initial and the residual surface shape with varying mirror thickness in a conventional DM. (b) Zernike coefficient of the initial surface shape for HG = 1mm. (c) Zernike coefficient of the initial surface shape for HG = 5mm.
Fig. 9 Residual surface shape with a varying mirror thickness (HG) in a conventional DM. (a) HG = 1mm. (b) HG = 2mm. (c) HG = 3mm. (d) HG = 4mm. (e) HG = 5mm. (S = 10MPa).
3.2 Residual stress
In Fig. 10 and Fig. 11(a), when the residual stress increases from 2MPa to 10MPa with an increment of 2MPa, the AC-HIFA appears in the initial surface shape and the PV value increases linearly from 145.93nm to 729.65nm. Figure 11 and Fig. 12 show that the initial surface shape mainly consists of several Zernike polynomials and the AC-HIFA could not be corrected effectively by the actuators. After correction, the depressed AC-HIFA will remain in the residual surface shape of the DM [Fig. 12]. The simulation results indicate that the AC-HIFA in a conventional DM could not be eliminated by simply adopting a low-stress adhesive.
Fig. 10 Initial surface shape with varying stress (S) in a conventional DM. (a) S = 2MPa. (b) S = 4MPa. (c) S = 6MPa. (d) S = 8MPa. (e) S = 10MPa. (HG = 1mm).
Fig. 11 (a) PV values of the initial and the residual surface shape with varying stress in a conventional DM. (b) Zernike coefficient of the initial surface shape for S = 2MPa. (c) Zernike coefficient of the initial surface shape for S = 10MPa.
Fig. 12 Residual surface shape with varying stress (S) in a conventional DM. (a) S = 2MPa. (b) S = 4MPa. (c) S = 6MPa. (d) S = 8MPa. (e) S = 10MPa. (HG = 1mm).
From the discussions above, it could be concluded that the AC-HIFA caused by the adhesive curing in a conventional DM could not be eliminated by simply changing the values of the mirror parameters (e.g. the mirror thickness and the adhesive stress). In order to eliminate the unwanted AC-HIFA, the DM structure should be improved.
4. Model of an improved DM without the AC-HIFA
It has been discussed in section 3 that the AC-HIFA arises from the direct contact of the adhesive and the back of the mirror plate. If the designed structures could separate the adhesive from the back of the mirror plate, the issue may be solved. In consideration of this point, an improved mirror, using a mirror plate with a post array, might be the solution. The post array is part of the mirror plate, while they are machined integrally from a piece of BK7 glass [Fig. 13, Fig. 14]. Figure 13 shows the schematic diagram of the adhesive influence on different mirror structures. In Fig. 13(a), the adhesive is directly applied to the back of the mirror plate. Distortion appears at the back of the mirror and is transferred to the front surface. As a result, a large convexity appears in the initial surface shape when the adhesive is completely cured, as has been shown in section 1. In Fig. 13(b), the mirror has short and wide posts on the back. These posts could separate the adhesive from the back surface of the mirror. However, limited by the length of the posts, the influence of the adhesive curing could only be restricted partly and small convexities still emerge in the initial surface shape. In Fig. 13(c), the mirror has long and thin posts. The influence of the adhesive could be completely restricted within the post and no distortion appears in the front surface. Therefore, the initial surface shape will not be distorted by the adhesive curing.
Fig. 13 Schematic diagram of the adhesive influence on different mirror structures. (a) Conventional mirror without a post. (b) Mirror with a short and wide post. (c) Mirror with a long and thin post.
Fig. 14 (a) Structure sketch of the improved DM with a post array. (b) Distribution of the post array.
An improved DM model, which consists of a mirror with a post array, a flexure array, a PZT array, and a base, is built in ANSYS to verify the discussion above and to investigate the influence of the structural parameters on the initial surface shape [Fig. 14]. Except for the post array (made of BK7 glass) and the mirror thickness, the other parts of the improved DM are the same as they are in the conventional DM. The investigated parameters and their symbols are listed in Table 3.
Table 3. Parameters and symbols in the simulation
4.1 Post length
In the simulation, the parameters are set as follows: HG = 1mm, DH = 3mm, S = 10MPa, while the post length is set increasing from 1mm to 3mm with an increment of 0.5mm. It could be seen from Fig. 15 that when the post is shorter than 2mm, the AC-HIFA emerges in the initial surface shape [Fig. 15(a) and 15(b)]. When the post gets longer than 2mm, the AC-HIFA would gradually disappear [Fig. 15(c)] and achieve an initial surface shape with random aberration [Fig. 15(d), 15(e)]. By comparing Fig. 15(a) with Fig. 6(a), it could be seen that the post is effective in reducing the PV value of the initial surface shape, even with a small length of 1mm. The PV value of the initial surface shape is inversely proportional to the post length [Fig. 16(a)]. When the post length is set 3mm, a balanced coefficient distribution among the Zernike polynomials [Fig. 16(c)] is achieved for the initial surface shape [Fig. 15(e)], which is quite different from the distribution for the post length of 1mm [Fig. 16(b)]. Figure 17 shows the residual surface shape from the initial surface shape shown in Fig. 15. In Fig. 17, the residual surface shape is similar to the initial surface shape, which indicates that the distortion resulting from adhesive curing mainly consists of HIFAs that could not be effectively corrected by the actuators. It could be concluded that the post length has a great influence on the initial surface shape. By adopting the long posts, the AC-HIFA could be eliminated from the initial surface shape and the residual surface shape.
Fig. 15 Initial surface shape with varying post length HH. (a) HH = 1mm. (b) HH = 1.5mm. (c) HH = 2mm. (d) HH = 2.5mm. (e) HH = 3mm. (HG = 1mm, DH = 3mm, S = 10MPa).
Fig. 16 (a) PV values of the initial and the residual surface shape with varying post length. (b) Zernike coefficient of the initial surface shape for HH = 1mm. (c) Zernike coefficient of the initial surface shape for HH = 3mm.
Fig. 17 Residual surface shape with varying post length HH. (a) HH = 1mm. (b) HH = 1.5mm. (c) HH = 2mm. (d) HH = 2.5mm. (e) HH = 3mm. (HG = 1mm, DH = 3mm, S = 10MPa).
4.2 Post diameter
In the simulation, the parameters are set as follows: HG = 1mm, HH = 2mm, S = 10MPa, while the post diameter is set increasing from 2mm to 4mm with an increment of 0.5mm. It could be seen from Fig. 18 that when the post diameter is smaller than 3mm, random HIFAs appear in the initial surface shape [Fig. 18(a), 18(b)]. When the post diameter gets larger than 3mm, AC-HIFAs starts to appear in the initial surface shape and become distinct when the post diameter is as large as 4mm [Fig. 18(e)]. The PV value achieves a minimum when the DH is set 3mm [Fig. 19(a)]. For the DH of 2mm, the random initial surface shape [Fig. 18(a)] consists of the Zernike polynomials with similar amplitudes [Fig. 19(b)]. By contrast, for the DH of 4mm, the regular initial surface shape [Fig. 18(e)] mainly consists of several Zernike polynomials. Figure 20 shows the residual surface shape from the initial surface shape shown in Fig. 18. By comparing Fig. 18 with Fig. 20, it could be seen that the AC-HIFAs could not be effectively corrected by the actuators and remain in the residual surface shape. It could be concluded that the post diameter has a great influence on the initial surface shape. By adopting the posts with a small diameter, the AC-HIFA could be eliminated from the initial surface shape and the residual surface shape.
Fig. 18 Initial surface shape with varying post diameter DH. (a) DH = 2mm. (b) DH = 2.5mm. (c) DH = 3mm. (d) DH = 3.5mm. (e) DH = 4mm. (HG = 1mm, HH = 2mm, S = 10MPa).
Fig. 19 (a) PV values of the initial and the residual surface shape with varying post diameter. (b) Zernike coefficient of the initial surface shape for DH = 2mm. (c) Zernike coefficient of the initial surface shape for DH = 4mm.
Fig. 20 Residual surface shape with varying post diameter DH. (a) DH = 2mm. (b) DH = 2.5mm. (c) DH = 3mm. (d) DH = 3.5mm. (e) DH = 4mm. (HG = 1mm, HH = 2mm, S = 10MPa).
4.3 Mirror thickness
In the simulation, the parameters are set as follows: HH = 2mm, DH = 2mm, S = 10MPa, the mirror thickness is set increasing from 1mm to 5mm with an increment of 1mm. It could be seen from Fig. 21 and Fig. 22(a) that the PV value of the initial surface shape is inversely proportional to the mirror thickness in the improved DM. The initial surface shape consists of random HIFAs when the mirror thickness is set 1mm [Fig. 21(a)]. As the mirror gets thicker, the relative proportion of the Zernike polynomials change accordingly [Fig. 22(c)] and results in a different random surface shape [Fig. 21(e)]. Figure 23 shows the residual surface shape from the initial surface shape shown in Fig. 21. It could be seen that all the residual surface shapes in Fig. 23 only consist of random HIFAs. The PV value is in inverse proportion to the mirror thickness. It could be concluded that a relatively thick mirror is preferable in the improved DM, which would reduce the PV value of the surface shape enormously. Meanwhile, it should be noted that the thick mirror could reduce the maximum stroke of the DM. Thus, there is a trade-off between the surface fidelity and the maximum stroke while determining the value of the mirror thickness.
Fig. 21 Initial surface shape with varying mirror thickness HG in the improved DM. (a) HG = 1mm. (b) HG = 2mm. (c) HG = 3mm. (d) HG = 4mm. (e) HG = 5mm. (HH = 2mm, DH = 2mm, S = 10MPa).
Fig. 22 (a) PV values of the initial and the residual surface shape with varying mirror thickness in an improved DM. (b) Zernike coefficient of the initial surface shape for HG = 1mm. (c) Zernike coefficient of the initial surface shape for HG = 5mm.
Fig. 23 Residual surface shape with varying mirror thickness HG in the improved DM. (a) HG = 1mm. (b) HG = 2mm. (c) HG = 3mm. (d) HG = 4mm. (e) HG = 5mm. (HH = 2mm, DH = 2mm, S = 10MPa).
4.4 Residual stress
In the simulation, the parameters are set as follows: HG = 1mm, HH = 3mm, DH = 3mm, while the residual stress of the cured adhesive is set increasing from 10MPa to 50MPa with an increment of 10MPa. It can be seen from Fig. 24 and Fig. 25 that the initial surface shape is composed of random HIFAs and the PV value is linear with the residual stress. Note that the residual stress could only decide the PV value of the initial surface shape. The distribution of the initial surface shape is decided by the structural parameters (e.g. the mirror thickness, the post length and the post diameter). In order to achieve the residual surface shape with small PV value [Fig. 26], it is recommended that adhesive with low residual stress be used to reduce the HIFAs both in the initial surface shape and the residual surface shape.
Fig. 24 Initial surface shape with varying stress S in the improved DM. (a) S = 10MPa. (b) S = 20MPa. (c) S = 30MPa. (d) S = 40MPa. (e) S = 50MPa. (HG = 1mm, HH = 3mm, DH = 3mm).
Fig. 25 (a) PV values of the initial and the residual surface shape with varying residual stress in an improved DM. (b) Zernike coefficient of the initial surface shape for S = 10MPa. (c) Zernike coefficient of the initial surface shape for S = 50MPa.
Fig. 26 Residual surface shape with varying stress S in the improved DM. (a) S = 10MPa. (b) S = 20MPa. (c) S = 30MPa. (d) S = 40MPa. (e) S = 50MPa. (HG = 1mm, HH = 3mm, DH = 3mm).
4.5. Simulational result of the improved DM
From the discussions above, it is clear that in order to make the initial surface shape resistant to the adhesive curing, a thick mirror, long and thin posts, and a low-stress adhesive is preferred in the design. The optimum parameters are listed in Table 4. Other structures, including the flexure plate, the PZT, and the base are the same as that in the conventional DM (listed in Table 1). The simulation results are shown in Fig. 27. Different from the conventional DM, the initial surface shape of the improved DM only consists of random HIFAs without any AC-HIFA. The PV value is 6.23nm and 4.19nm for the initial surface shape and the residual surface shape respectively, which would have a negligible influence on the incident beam. By comparing Fig. 27 with Fig. 6, it could be seen that the post array is effective in eliminating the AC-HIFA from the initial surface shape. By changing the structural parameters in the simulation, the AC-HIFA could be eliminated in other DMs as well.
Table 4. Final parameters of the improved DM
Fig. 27 Simulation result of the improved DM. (a) Two-dimensional initial surface shape. (b) Three-dimensional initial surface shape. (c) Residual surface shape.
Note that the best values for HH and S in section 4.1 and 4.4 are used in Table. 4. However, the best value for DH and HG are not. The best value for DH in section 4.2 is 2 mm, while a value of 3 mm is used in Table 4. It is because a thin post is difficult to machine and easy to result in a fracture in practice. A 3-mm-diameter post is easy to machine and could eliminate the AC-HIFA, therefore, it is used in the prototype of the improved DM. In addition, the best value for HG in section 4.3 is 5 mm, while a value of 2.5 mm is used in Table 4. It is because when a PZT stretches to shape the mirror surface, the mirror will provide a force back on the PZT. A thicker mirror would generate a larger force and a smaller deformation of the surface. A simulation is conducted in Fig. 28. The mirror thickness is 5 mm and 2.5 mm in Fig. 28(a) and Fig. 28(b), respectively. A voltage of 20V is applied to the central PZT (maximum travel range of 38 μm, operating voltage range of −20 V to 120 V) of the two DMs respectively. The PV value of the deformed surface shape is 1.35μm and 3.64μm for the 5 mm and 2.5 mm thickness, respectively. Therefore, the value for HG is set to 2.5 mm to obtain a larger deformation in the improved DM.
Fig. 28 Mirror surface shape under a voltage of 20 V applied to the central PZT. (a) HG = 5mm. PV = 1.35μm. (b) HG = 2.5mm, PV = 3.64μm. (HH = 3mm, DH = 3mm)
5. Experiment of the improved DM
In the experiment, an improved DM, using a mirror with a post array, is manufactured in our lab [Fig. 29], whose structure is the same as the simulation model described in section 4.5. Note that the mirror with a post array is machined from a piece of BK7 glass [Fig. 29(a)]. The adhesive is applied to bond the post array of the mirror to the flexure array of the base. Therefore, the adhesive is separated from the back of the mirror plate by the post array.
Fig. 29 The lab-made 116-actuator improved DM with a post array. (a) Before assembly. (b) After assembly.
In Fig. 30, the initial surface shape of the improved DM is measured by a Zygo interferometer. The color temperature diagrams show a random shape, in which no AC-HIFA appears as in the conventional DM [Fig. 2]. The PV value of the improved DM is 207.55nm, which is much smaller than that of the conventional DM (731.75nm). The interferometer result verifies that the influence of the adhesive curing is eliminated by the post array and no AC-HIFA appears on the initial surface shape of the improved DM.
Fig. 30 Interferometer measurement result of the improved DM. PV = 207.55nm. (a) Two-dimensional color temperature diagram. (b) Three-dimensional color temperature diagram. (c) Interference fringe diagram.
In addition, a laser radiation experiment with the setup illustrated in Fig. 3 is made to test the influence of the initial surface shape of the improved DM on the laser beam. Experiment results show that, when there is no lenslet array, the light intensity of the CCD image is relatively uniform. Only some inherent fringes and circles are observed [Fig. 31(a)]. When the lenslet array is applied, the point array is regular and matches with the lenslet array [Fig. 31(b)]. It indicates that the improved DM does not introduce AC-HIFA to the incident wavefront and has negligible modulation on the intensity of the laser beam, which is similar to the result of the optical flat [Fig. 4] and completely different from the result of the conventional DM [Fig. 5].
6. Conclusion
In conclusion, the AC-HIFA is reported to exist in the initial surface shapes of a stacked array PZT DM and remains in the intensity of a laser beam in this paper. The mechanism of the AC-HIFA is analyzed. A finite element model is built to investigate the influence of the structural parameters on the AC-HIFA in a conventional DM, in which the adhesive is applied directly to the back of the mirror to realize the bonding. Simulation results show that by adopting a thick mirror or a low-stress adhesive, the AC-HIFA could only be reduced, but not be eliminated. An improved DM model, using an integral mirror with a post array, is recommended to be the solution. Simulational results indicate that the post array is effective in controlling the AC-HIFA. By adopting a long and thin post array, the AC-HIFA could be completely eliminated from the surface shape of the DM. In the experiment, a Zygo interferometer is used to measure the initial surface shape of the improved DM. The measurement results show that no AC-HIFA exists in the initial surface shape of the improved DM. In the laser radiation experiment, the improved DM has negligible modulation on the intensity of the laser beam. The experiment results indicate that the post array is effective in eliminating the AC-HIFA in a stacked array PZT DM.
Funding
National Natural Science Foundation of China (NSFC) (61775112).
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