Simulation and analysis of deformable mirror driven by piezoelectric actuator array

1 Introduction

The deformable mirror (DM) is the critical component of adaptive optics (AO) systems, which achieve dynamic aberrations compensation through active surface deformation (Madec, 2012). In recent years, AO has been applied to optical communication (Wilks et al., 2002; Thompson et al., 2002; Toporovskiy et al., 2019; Weyrauch and Vorontsov, 2004; Li et al., 2014), laser beam shaping (Jochmann et al., 2013; Schroeder et al., 2010; Tran et al., 2014; Zheng et al., 2023), and microscopic imaging (Seyedahmadi and Vavvas, 2010; Zawadzki et al., 2005; Horsley et al., 2005; Chen et al., 2007), where it has shown significant potential. DM is used in AO systems as a wavefront corrector, and its various types are designed to address distinct application requirements (Man and Bifano, 2025; Toporovsky et al., 2023; Alaluf et al., 2018; Laslandes et al., 2015). For all this, Piezoelectric-driven DM remains a dominant choice, owing to their large stroke and high reliability (Toporovsky et al., 2022; Sun et al., 2018). The DMs driven by piezoelectric are primarily divided into bimorph mirror and actuator array mirror. A representative bimorph DM, noted for its effective corrections for low-order aberrations, structural simplicity, and low cost, is produced by Compagnie Industrielle des Lasers (CILAS) (Sinquin et al., 2008). However, the application of this DM is limited due to their low resonance frequency and unavoidable print-through effects (Rausch et al., 2015). Secondly, the DMs with piezoelectric actuator array are predominantly either unimorph piezoelectric sheets or piezoelectric stacks. The DMs driven by Stacked piezoelectric are the primary product of AOA Xinetics (AOX) (Wirth et al., 2013). Their mirrors have high resonant frequency, and large stroke, which can satisfy a wide range of requirements. Notably, the manufacturing process of this DM is complex and costly, and the replacement of a single actuator is a challenging task, which further augments the overall cost. In this regard, Vladimir Toporovsky et al. developed a modified stacked piezoelectric DM, which can achieve a reduction in the cost of replacing a single actuator (Toporovsky et al., 2021). However, the overall structure of this DM has become increasingly complex, and the manufacturing cost remains quite high. Consequently, the low-cost DM driven by unimorph piezoelectric actuator arrays have attracted considerable research interest (Ma et al., 2011; Ma et al., 2013).

Lei Tian et al. designed a DM driven by unimorph piezoelectric array with 19-actuators (Tian et al., 2017). This DM demonstrates significant application promise due to its large deformation capacity, low cost, and high resonant frequency. Guiming Zhong et al. additionally improved both the stroke and resonant frequency of this DM (Zhong et al., 2022; Han et al., 2023). However, configurations with a higher actuator counts (up to 100) were not conducted. The performance of the DM is typically characterized by actuator count, effective aperture, and mirror quality. Intuitively, under the same effective aperture, the actuator count of the DM is directly related to its correction capacity. While the DM with fewer than 100 actuators is sufficient for low-order aberrations correction. By determining the optimal count of actuators within this range, it is essential to balancing performance and cost (Lemaitre, 2014).

In this paper, typical actuator configurations with fewer than 100 units are discussed and analyzed in terms of correction performance and resonant frequency. The remainder of the paper is organized as follows. First, the finite element models of DMs driven by piezoelectric actuator arrays with 19, 37, and 61 units are respectively constructed. Subsequently, the actuator performance and low-order aberrations correction capability of each configuration are simulated and evaluated at their respective effective apertures. Finally, the fundamental resonant frequencies of the three DMs are determined. This work is valuable for designing such DMs in low-order aberrations correction at low cost.

2 The DM in adaptive optics

2.1 Overview of adaptive optics

The basic principle of AO is phase conjugation. In AO, the phase compensation is implemented as a linear correction technique that is both flexible and easy to realize in practice. An incoming optical field with a phase error is represented as (Equation 1).The symbol ϕ represents the phase aberrations of the optical field. The AO system generates a modulation that is phase-conjugate to E 1, as follows (Equation 2).These aberrations are compensated upon the superposition of E 1 and E 2. According to Equations 1, 2 the residual error is given by (Equation 3).The quality of static wavefront correction is quantified by Δ E.

E 1 = | E | e i ϕ (1)

E 2 = | E | e − i ϕ (2)

Δ E = E 1 − E 2 = | E | e i ϕ − e − i ϕ = 2 i | E | sin ⁡ ϕ (3)

While Δ E characterizes static correction accuracy, dynamic capability is critical for compensating atmospheric turbulence in astronomical imaging and other long-path applications. The corresponding atmospheric turbulence parameters are introduced below.

The theoretical foundation of AO is grounded in the characteristics of atmospheric turbulence. The spatial power spectrum model for refractive index fluctuations in turbulent media is established by the Russian scientist Kolmogorov, which is widely recognized as 2/3 law. It can be defined as (Equation 4).Where r is the distance between two points in space, and C n 2 indicates the refractive-index structure constant. The symbols l 0 and L 0 represent the inner and outer scales of turbulence respectively. The power spectral density of the refractive-index fluctuations in the inertial subrange is expressed as Equation 5.Where ρ is the spatial frequency.

D n r = C n 2 r 2 / 3 , l 0 ≪ r ≪ L 0 (4)

Φ n ρ = 0.033 C n 2 ρ − 11 / 3 (5)

A conventional AO system is composed of a wavefront sensor, a controller, and a wavefront corrector, as shown in Figure 1.

The wavefront aberrations are measured in real time by a wavefront sensor. The measured data are converted into control signals by the controller, and real-time correction is performed by the wavefront corrector. In AO systems, the wavefront aberrations are compensated by the wavefront corrector. To achieve a corrected wavefront near the diffraction-limit, the high-performance wavefront corrector is indispensable. The DM is used in AO systems as a wavefront corrector. Its ability to meet these dominant requirements is fundamentally determined by its structure and a set of key performance parameters.

2.2 Performance metrics of DMs

All conventional DMs compensate for the aberrations by using actuators to pull or push the facesheet to shape its surface. The DMs with actuator arrays are widely applicable. It is composed of the continuous facesheet, actuator arrays and substrate. In general, the piezoelectric actuator arrays are arranged in a specific layout on the rigid substrate. By driving the surface displacement through the actuator arrays, the DM compensates for incoming wavefront aberrations. The performance of the DM is typically evaluated in terms of effective aperture, surface quality, coupling, resonant frequency, stroke, and actuator count. The main performance parameters of the DM are described as follows.

The coupling coefficient is defined as the ratio of the displacement at the adjacent actuator positions to the displacement of the centrally driven actuator. The coupling value is governed by both the mirror material properties and the actuator array configuration, as follows Equation 6.Where E m is the Young’s modulus of the mirror, and h m is its thickness. The next parameters in question pertain to the DM’s actuators. The symbol E a denotes Young’s modulus, S a represents the cross-sectional area, d designates the pitch, and l a is the length. The ξ is a constant coefficient.

c = 1 / 1 + ξ E a S a d 4 l a E m h m 3 (6)

The resonant frequency of the DM is determined by its material properties and overall structure. A sufficiently high fundamental resonant frequency is required. This frequency must exceed the operating bandwidth to prevent structural resonance. It can be defined as Equation 7.Where k e is the equivalent stiffness of the mirror, and m e is its mass.

f n = 1 2 π k e m e (7)

The stroke of the DM is defined as the maximum mechanical displacement that an actuator can produce on the mirror surface. This parameter fundamentally determines the largest correctable wavefront amplitude. The dynamic range of aberrations correction is extended owing to a greater stroke. An empirical expression employed for the stroke is given by Equation 8.Where λ represents the working wavelength, λ k designates the number of waves, z denotes the propagation distance, and D is the clear aperture.

f s = 0.57 λ λ k 4 π z C n 2 D 5 / 3 . (8)

Complex mirror surface shapes for wavefront correction are produced by the combined action of multiple independent actuators. Within the same effective aperture, the higher the actuator density, the better the aberrations correction capability. The required count of actuators is represented as Equation 9.Where κ is the Kolmogorov constant, and S is the Strehl ratio.

N a = κ λ k 2 z C n 2 D 5 / 3 2.9 ⁡ ln 1 / S 6 / 5 (9)

2.3 Cost and alternatives

The high cost of conventional stacked piezoelectric DMs stems from multiple factors. Among these, only the actuator cost is briefly discussed in this paper.

Piezoelectric and electrostrictive materials are used as the stacked actuator materials. High-grade materials of the actuator are costly, and the process of producing qualified actuators is complex. As mentioned earlier, the overall cost has been further increased by the inability to individually replace failed actuators. Historically, the cost of a single actuator was as high as $3,000. The substantial cost of the actuator constitutes a major barrier to civilian, although it remains permissible within the scope of military projects.

Therefore, efforts were made to address this cost barrier, and some structural adjustments were explored. A more direct solution is to substitute the actuator with an inexpensive one. Commercially piezoelectric sheets are employed for this purpose, which are appropriate and readily available. Compared with stacked actuator, commercially available piezoelectric sheets are offered in a wide range of specifications at low cost, typically around $50 per unit (it is merely 1/60th of the cost of stacked actuator). The use of such off-the-shelf piezoelectric sheets therefore significantly reduces both the material expenditure and the overall complexity of the manufacturing process. This approach has been adopted by numerous researchers (Vdovin et al., 2005; Ma et al., 2011). The present paper also makes use of this method, which is a potentially low-cost design.

Overall, the cost of high-performance DMs driven by piezoelectric actuator arrays is substantial, which is largely imposed by the actuators. Research on low-cost DMs is therefore necessary.

3 DM configuration and finite element modeling

3.1 The DM’s design of structure

The structure is composed of a mirror, a piezoelectric actuator array with pillars, and a substrate in sequence from top to bottom, as shown in Figure 2.

The reflective layer is formed by the mirror facesheet, and the back of the facesheet is bonded to each actuator by slender pillars. The back of each actuator is glued to through-holes in the base, which are slightly smaller in size. The mirror is fabricated from monocrystalline silicon, with structural steel supporting pillars beneath it. The actuator is composed of a lead-zirconate-titanate piezoelectric (PZT) disc and an elastic membrane, with the quartz glass serving as the substrate material. The actuator bends due to the transverse inverse piezoelectric effect when a voltage is applied to the PZT disc. The displacement of the actuator is transferred to the mirror via the pillar located at the center of the actuator. Thus, wavefront aberrations is compensated with the active surface deformation of the mirror.

Overall, under the same effective aperture, the actuator of the DM is directly related to its correction capability, so it is important to conduct an accurate simulation of it. Low cost and readily available, the Commercial piezoelectric actuator is used as the modeling prototype in this paper, and it is composed of a PZT disc (piezoelectric layer) and an elastic membrane (passive elastic layer) made of brass, as shown in Figure 3.

The actuator’s equivalent Young’s modulus E e and Poisson’s ratio v e are given by Equations 10, 11 Where E p and E b are the Young’s modulus of the piezoelectric and brass layers respectively. While v p and v b are the Poisson’s ratios.

E e = c 1 E p + c 2 E b + c 1 E p c 2 E b v b − v p 2 c 1 E p 1 − v b 2 + c 2 E b 1 − v p 2 (10)

v e = c 1 E p 1 − v b 2 v p + c 2 E b 1 − v p 2 v b c 1 E p 1 − v s 2 + c 2 E b 1 − v p 2 (11)

The c 1 and c 2 are expressed as (Equation 12).Where h p, h b, and h are the thickness of the PZT layer, brass layer, and total thickness about the actuator. It indicates the ratio of each component to the total thickness. According to Equations 10, 11, the equivalent stiffness of the actuator D e can be obtained as (Equation 13).The equivalent elastic coefficient of the actuator k e is defined as (Equation 14).The torque M 0 produced by the PZT when a driving voltage is applied to it can be expressed as follows (Equation 15).where d 31 is the transverse piezoelectric coefficient and U is the applied voltage. According to Equations 13, 15, the radial deflection of the two-layer cross-section w 1 ( r ) and w 2 ( r ) can be given by Equations 16, 17 Where a, b, and r are the radii of the PZT layer, the brass layer, and the effective actuation area respectively. According to Equations 16, 17, the maximum displacement at the center of the actuator is written as (Equation 18).

c 1 = h p h , c 2 = h b h (12)

D e = E e h 3 12 1 − ν e 2 (13)

k e = 16 π D e b 2 (14)

M 0 = D e − d 31 U / h p h 2 + 2 h 1 E p h p + 1 E b h b D p + D b (15)

w 1 r = M 0 b 2 − a 2 a 2 − r 2 4 D e b 2 r ≤ a (16)

w 2 r = M 0 a 2 r 2 − b 2 − 2 b 2 ⁡ ln r b 4 D e b 2 a ≤ r ≤ b (17)

w max = w 1 b + w 2 a = M 0 a 2 ⁡ ln a b 2 D e (18)

Assume the actuators of the DM are arranged in a regular hexagonal array with a patch of d, and the mirror surface has a stiffness of D m. When a voltage is applied to the central actuator, the mirror elastic constraint limits the surface deflection to w 0, requiring a force f 0. Meanwhile, mechanical coupling induces deflections w 0 ′ and forces f 1 at the adjacent actuators. The following formula is obtained (Equation 19).where 6 f 1 represents the total force contributed by the six surrounding actuators. The facesheet can be treated as a thin plate with boundary constraints. And its deformation and the force acting on it can be depicted as follows (Equation 20).where α is a constant coefficient.

f 0 = 6 f 1 f 0 = D e ⋅ w max − w 0 f 1 = D e ⋅ w 0 ′ (19)

f 0 = α w 0 − w 0 ′ D m / d 2 (20)

The preceding analysis is grounded in the theory of plates and shells (BRODETSKY, 1941) and provides the basis for the subsequent simulation.

3.2 Material parameters and finite element model

The finite element models of DMs are constructed using Ansys multiphysics, which are driven by unimorph piezoelectric actuator arrays with 19, 37, and 61 units. An enumeration of the material parameters for the mirror, pillars, and piezoelectric actuators is provided in Table 1.

TABLE 1. Detailed parameters for each material.

The influence of the bonding layer present in actual fabrication is disregarded in the model. To conserve computational resources, the substrate is replaced by boundary constraints. The mirror was fixed with simply supported edges, and the actuator with fixed edges.

The actuator arrangement follows a regular hexagonal pattern, as show in Figure 4.

The center-to-center distance between adjacent actuators is defined as the actuator pitch, and all neighboring actuators are equally spaced. The actuator pitch of each DM measures 16 mm, 12 mm, and 8 mm, respectively. In all cases, each mirror surface has 0.35 mm thick and 76.2 mm (3 inches) diameter, with an effective aperture of 50 mm.

The piezoelectric layers of each actuator were modeled using SOLID226 (3-D 20-node coupled-field solid elements with piezoelectric capability), while the passive elastic layers were modeled with SOLID186 (3-D 20-node structural solid elements). The free-form meshing approach was selected due to the model’s regular structure and simplicity, with mesh size that can be controlled both globally and locally, as shown in Figure 5. A voltage of 40 V is applied to an actuator for each DM.

4 Performance analysis and discussion

4.1 Actuator characteristics

The results of the simulation for the 19-actuator DM are displayed in Figure 6.

The mirror deformations of the actuators with an effective aperture of 50 mm are 11.85 μm and 10.57 μm, respectively. The deformation of the mirror surface caused by each actuator displacement is within around 11 μm.

The simulation results for the 37-actuator DM are shown in Figure 7.

The mirror deformations of the actuators with an effective aperture of 50 mm are 6.278 μm, 6.258 μm, 5.376 μm, and 5.872 μm, respectively. The deformation of the mirror surface caused by each actuator displacement is about 6 μm.

The ensuing section will present the simulation results for the 61-actuator DM, as illustrated in Figure 8.

The mirror deformations of the actuators with an effective aperture of 50 mm are 3.539 μm, 3.537 μm, 3.536 μm, 3.513 μm, and 3.536 μm, respectively. The deformation of the mirror surface caused by each actuator displacement is approximately 3.5 μm.

It is important to distinguish between spatial correction capability and maximum correctable aberration amplitude. The present study focuses solely on the former, and the observed reduction in single-actuator stroke with increasing actuator count is an expected consequence of geometry scaling and does not imply a degradation in spatial correction resolution.

4.2 Aberration fitting performance

The relationship between the driving voltage and the deformation of the mirror surface is described by the DM’s influence function. In essence, it represents the deformation in mirror surface profile when a voltage is applied to a single actuator (Sheng-Yi et al., 2015; Silai et al., 2025). The influence function is generally approximated by the Gaussian function, defined as (Equation 21).Where I i x , y is the influence function of the i-th actuator, x i , y i is the position of the ith actuator, α is the Gaussian function exponent, d is the distance between adjacent actuators, and ω is the Coupling coefficient of the actuators.

I i x , y = exp ln ⁡ ω x − x i 2 + y − y i 2 d α (21)

The shape of the DM can be represented as a linear superposition of the influence functions of each actuator. Therefore, it is a weighted superposition of these influence functions, where the applied voltage corresponds to the weighting factor (Equation 22).Where W ′ x , y is the surface shape of the DM, n is the count of actuators, v i is the voltage applied to the i-th actuator, otherwise termed the weighting factor. The residual error (ERR) Δ W x , y is expressed as (Equation 23)

W ′ x , y = ∑ i = 1 n v i I i x , y (22)

Δ W x , y = W x , y − ∑ i = 1 n v i I i x , y (23)

For clarity, the coupling coefficients adopted in the present simulations are 0.25, 0.17, and 0.13 for the 19-, 37-, and 61-actuator configurations, respectively. These values were extracted from the computed influence functions according to the definition given in (Equation 6). A more detailed discussion of this parameter can be found in Ref. (Silai et al., 2025).

The subsequent wavefront reconstruction model is developed using Matlab. Random wavefronts are generated from the first 20 Zernike polynomials (tip/tilt excluded) with an initial root mean square (RMS) of 0.5 μm. DMs are fitted to 300 groups of random aberrations composed of the first 20 Zernike polynomials. The ERR and corrected PV are obtained, as shown in Figure 9.

The mean ERR values of DM of 19-, 37- and 61-actuators are 0.226, 0.046, and 0.029 μm, and their normalized ERR values are 43.2%, 9.2%, and 5.8%, respectively. The mean PV values of each DM are 2.156, 0.715, and 0.263 μm. Before and after correction wavefronts and PSF intensity distributions are randomly selected for each DM, as shown in Figure 10.

Intuitively, it is determined that the correction effects of each DM are better or worse within the first 20 Zernike modes. This determination is made by comparing the simulation results of aberrations and point spread functions before and after correction. In conclusion, the 19-actuator mirror is incapable of correcting the previous 20 Zernike polynomials, while the 37-actuator mirror possesses this capability, and the 61-actuator mirror is the most effective for this task.

The simulation results of the same group’s correction for wavefront aberrations by each DM are displayed in Figure 11.

It is imperative to reiterate the observation that the DM of 19- and 37-actuators exhibits a conspicuously low capacity for correction of the first 20 Zernike modes. This result is consistent with the conclusion previously reached.

The 19-actuator DM is fitted to 300 groups of random aberrations composed of the first 9 Zernike polynomials (tip/tilt excluded). Figures 12a and b presents the ERR and PV values.

The mean ERR value of this DM is 0.058 μm (normalized: 11.6%), and the mean PV is 0.580 μm. Before and after correction wavefronts and PSF intensity distributions were randomly selected, as shown in Figures 12c,d.

The RMS wavefront error per Zernike mode after correction for 300 groups is displayed in Figure 12e, and the comparison of Zernike coefficients before and after correction is shown in Figure 12f. A marked reduction in residual aberrations is observed after correction.

This is different from the previous correction results for the first 20 Zernike modes. The simulation results show that this 19-actuator DM is quite capable of correcting aberrations up to the first 9 Zernike modes.

The 37-actuator DM is fitted to 300 groups of random aberrations composed of the first 14 Zernike polynomials (tip/tilt excluded). As shown in Figures 13a,b, the mean ERR value of the DM is 0.011 μm, the normalized ERR is below 0.1λ, and the mean PV is 0.163 μm.

Before and after correction wavefronts and PSF intensity distributions are randomly selected, as illustrated in Figures 13c,d.

Figure 13e displays the RMS wavefront error per Zernike mode after correction for 300 groups, while Figure 13f illustrates the comparison of Zernike coefficients before and after correction.

Thus, effective correction is achieved for the first 14 Zernike modes by this 37-actuator DM.

4.3 Analysis of resonant frequency

The resonant frequency and modals are obtained through finite element simulation, as shown in Figures 14a–c.

The simulation results indicate that the fundamental resonant frequencies of each DM are 1.5 kHz, 2.7 kHz, and 5.5 kHz, respectively. As the actuator count is increased, the mirror diameter and thickness are held constant, and the spacing between actuators is correspondingly decreased. Consequently, the resonant frequencies of the DM are increased by a larger actuator count.

5 Conclusion

In this work, the models of DMs are constructed, which are driven by unimorph piezoelectric actuator arrays with 19, 37, and 61 units. The simulation and analysis results show that, at a voltage of 40 V, the wavefront deformation produced by each DM within an effective aperture of 50 mm can reach 11, 6, and 3.5 μm, respectively. As demonstrated in the simulation results, the 19-actuator DM exhibits moderate corrective efficacy for the first 9 Zernike polynomial aberrations, while the 37-actuator DM displays effective corrective performance for the first 14. Notably, the 61-actuator DM shows the capacity to attain ideal correction for the first 20 Zernike polynomial aberrations. In addition, the resonance frequencies of DMs are 1.5 kHz, 2.7 kHz, and 5.5 kHz, respectively. Finally, this work contributes to the design of a potentially low-cost DM for low-order aberrations correction.

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Footnotes

Edited by: Allen Yi, The Ohio State University, United States

Reviewed by: Huseyin Canbolat, Ankara Yıldırım Beyazıt University, Türkiye

Qingbing Chang, Northeast Forestry University, China