Research on a bimorph piezoelectric deformable mirror for adaptive optics in optical telescope

Hairen Wang

doi:10.1364/OE.25.008115

1. Introduction

Stars that can be observed by astronomers using a telescope are usually thousands to millions of light-years away. After star light travels such a long distance to our eyes, a ground-based astronomy telescope would cause serious wavefront errors, mainly because of the distorts due to atmosphere turbulence [1]. Adaptive optics (AO) is a means for real time compensation of the image degradation. The technique was first proposed by Babcock [2]. It consists of using a deformable mirror to correct the instantaneous wave-front distortions [3, 4]. The deformable mirror is usually supported by an array of active elements, which adjusts the mirror surface shape to compensate the light’s wavefront error caused by the atmospheric distortions [5–7]. The band-pass of atmospheric turbulence is more than10³Hz [8]. The effective bandwidth of this precision adaptive structure is typically limited by the controller’s signal processing speed and the resonance frequency of the deformable mirror assembly [9]. Recently, the aperture becomes larger, the cost of adaptive optics telescopes becomes much more expensive [10]. Thus, the system requires its actuators to have large stroke, high speed, high accuracy, high resonance frequency, and low cost. There are two kinds of deformable mirrors have been widely used in adaptive optics [11]. One kind of those deformable mirrors, known as conventional piezoelectric deformable mirrors (CPDM), is usually supported by an array of cylindrical piezoelectric actuators, coupled to a continuous mirrored face sheet [12–15]. Another kind, called bimorph piezoelectric deformable mirror (BPDM), is made of a glass sheet bonded to a piezoceramics layer with an appropriate electrode pattern on the back [16–20]. However, each of them has its advantages: the CPDM has a higher temporal response [21], while the BPDM has lower cost and larger stroke. A discrete-layout bimorph piezoelectric deformable mirror, which integrates the strengths of the CPDM and the BPDM, is developed to make the best of the advantages and minimize the disadvantages.

Several kinds of bending actuators had been analyzed in our previous works, the results have shown that the bending actuators, of a simple structure and at a low cost, have the large strokes and the high resonance frequencies [22–24]. As an application, here we propose a discrete-layout bimorph piezoelectric deformable mirror (DBPDM), which consists of a continuous mirrored face sheet and an array of one kind of these bending actuators—the bimorph beam piezoelectric bending actuator (see Fig. 1). The mirror face sheet connects to the mid-points of beams by push pads. The cross section of the push pad is a square with the side length being w_pad[see Fig. 2(a)and Fig. 2(c)]. The cross-section area of a push pad is A_pad = (w_pad)². As shown in Fig. 2(a), the structure of a simply supported bimorph beam piezoelectric bending actuator (BBPBA) consists of identical piezoelectric ceramic twin layers polarized in the thickness direction and a metallic layer in the middle. The metallic layer strengthens the bending stiffness of the BBPBA, and this configuration is useful to achieve a high resonance frequency of whole deformable mirror.

Fig. 1 The configuration of a discrete-layout bimorph piezoelectric deformable mirror.

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Fig. 2 (a) A simply-support BBPBA. (b) The cross section of the BBPBA. (c) The dome height of BBPBA and loading a concentrated force. (d)The cross section of the push pad.

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The geometric parameters of the actuator are shown in Fig. 2(a) and 2(b). The total thickness of the BBPBA is 2h, and the thickness of the metallic layer is 2c. The length and the width of the BBPBA are 2L (2L>>w_pad) and b, respectively. The top and the bottom electrodes are in parallel and have the same voltage across each one. The metallic middle layer is shared as the third electrode by the upper and lower piezoelectric layers. The BBPBA is simply supported at the boundary at x₁ = ± L, where any in-plane displacement is prohibited. When a load voltage V is applied at the BBPBA, one piezoelectric layer will be driven to extend and the other layer to contract, which induces the BBPBA into a state having a bending deformation. As shown in Fig. 2(d), the dome height d is the distance from the horizontal line to the mid-point of the beam plate during the deformation process, and it is also the deflection at x = 0. Distinct from the conventional cylindrical piezoelectric actuators (piezoceramics stacks), whose strokes are obtained by stretching out their length in the thickness direction, the BBPBA has a large stroke due to its bending deformation.

2. Realistic electro-mechanical model of discrete-layout bimorph piezoelectric deformable mirror (DBPDM)

Here, the piezoelectric model of the actuator in this paper is affected by the stress on the cross section of the push pad [see Fig. 2(a)and Fig. 2(d)].

F = \int_{A} T_{z} d S = {\bar{T}}_{z} A_{p a d} = w_{_{p a d}}^{2} {\bar{T}}_{z} .

where T_z and F denote the stress on the cross section and the effectively concentrated force on the cross section, respectively. When the deformable mirror is working, each actuator is acted upon by external forces from other active actuators, and F is the resultant of these external forces.

In the following, we consider the flexural deformation of the BBPBA in the x₃ direction by assuming that the BBPBA is slender, and 2L is much larger than the thickness 2h and the width b. We denote the deflection curve by u₃(x,t). According to the electrode configurations, the electric field in the BBPBA has the following components: E₁ = E₂ = 0, E₃ = −V/(h-c) at c<z<h, and E₃ = V/(h-c) at -h<z<-c. Using the usual one-dimensional stress approximation of beam [25, 26], we have the following stress components: σ₁ = σ₁(x,t), σ₂ = σ₃ = σ₄ = σ₅ = σ₆ = 0. From [25] and [27], we can obtain the axial stresses in the upper and lower layers

σ_{1}^{p} = - x_{3} s_{11}^{- 1} (\partial^{2} u_{3} / \partial x^{2}) - s_{11}^{- 1} d_{31} E_{3} + F x_{3} (x_{1} + L) / (2 I_{e}),

where I_e = 2I_p + b(h-c)(h + c)²/2 + nI_m is the effective moment of inertia of three-layer composite cross-section, and n = E^ms₁₁in which E^m is Young’s modulus of the metallic middle layer [27]. Here, the middle layer is assumed to be elastic, and its constitutive relation is stated as below:

σ_{1}^{m} = x_{3} E^{m} (\partial^{2} u_{3} / \partial x^{2}) + n F x_{3} (x_{1} + L) / (2 I_{e}) .

The bending moment is defined by the following integral over the cross section, which can be integrated explicitly with the expression forσ₁ in Eq. (2) and Eq. (3) [25,28]

M = \int x_{3} σ_{1} d x_{2} d x_{3} = - D (\partial^{2} u_{3} / \partial x^{2}) + (h + c) e_{31}^{p} V + Γ F (x_{1} + L) .

Where

e_{31}^{p} = b d_{31} s_{11}^{- 1}

, D = (2/3)[E^mc³ + (h³-c³)/s₁₁]b, and Γ = (nc³ + h³-c³)/(3I_e). D is the bending stiffness. The equation of equilibrium in a slender beam can be expressed by the following equation regarding the bending moment:

\partial^{2} M / \partial x_{1}^{2} = - D (\partial^{4} u_{3} / \partial x_{1}^{4}) = 0.

Because the BBPBA is simply supported at x₁ = ± L, we have

u_{3} (- L) = u_{3} (L) = 0, M (- L) = M (L) = 0.

A general solution to Eq. (5) can be written as

u_{3} = H_{1} + H_{2} x_{1} + H_{3} x_{1}^{2} + H_{4} x_{1}^{3} .

whereH₁-H₄ are constants to be determined by Eq. (6). Thus, we get the static displacement of the dome height

d_{s} = u_{3} (0) = H_{1} = - [2 e_{31}^{p} V (c + h) + L Γ F] L^{2} / (4 D) .

where d_s denotes the dome height at x₁ = 0, which is equal to the extensional displacement of the actuator. From Eq. (8), we can obtain

F = k_{s} d_{s} + k_{v} V,

where k_s = -2D/(L³Γ) and k_v = e^p₃₁(c + h)/(LΓ).

The BBPBAs at different sites drive the deformable mirror system. We treat the mirror as linear elastic structure impeding the motion of the BBPBA array. The number of BBPBAs in the mirror is N_act. The force-displacement relationship for the mirror structure in the BBPBA array is [15]

{F^{N}} = [K_{d m}^{M N}] {d_{s}^{N} - d_{0}} .

K_dm represents the N_act by N_act stiffness matrix of the mirror, while F and d are vectors with the size being N_act. The displacement d^N, is the displacement of the Nth BBPBA normal to the plane of the mirror face sheet, and F^N is the corresponding force at the BBPBA location. The superscripts denote the elements of the matrix and the vectors, and the repeated superscripts indicate the summation. d₀ is the average displacement of all the BBPBAs in the array. Since the mirror rests only on the BBPBA in the array, the reaction force caused by BBPBA displacements are non-zero relative to d₀. Compression occurred in BBPBAs with displacements above the average, while tension occurred in BBPBAs with displacements below the average. When all the BBPBAs had the same voltage and hence the same displacements, the reaction forces on the BBPBAs are zero.

The K_dm can be obtained via an FEA model, which only includes the mirror face sheet and push pads. This method has been discussed in detail in [15]. Using Eqs. (9) and (10), we obtain [29]

[K_{d m}^{M N}] {d_{}^{N} - d_{0}} = k_{s} {d^{N}} + k v {V^{N}} .

The average displacement for the BBPBA array is d₀ = (1/N_act)Σd^N. Thus, there is a linear system of N_act + 1 equations for the displacements. This system is solved numerically for the given BBPBA voltages. Finally, the normal displacement w(x,y) in the mirror face sheet is reconstructed from the influence functions f^N(x,y)using

w (x, y) = \sum_{N} f^{N} (x, y) d^{N} .

3. Numerical results and discussion

In the following numerical calculations, we take a deformable mirror of 21 elements as an example (see Fig. 3 and Fig. 4). The material of pads and face sheet is single crystal silicon with Young’s modulus E_si = 190 GPa, Poisson’s ratioν = 0.30, mass density is ρ = 2350 kg/m³, and the radius and final thickness of the face sheet are R = 15 mm and h₀ = 1 mm, respectively. We use PZT-5H [30] as our piezoelectric material of actuator, whose parameters are listed below:

Fig. 3 (a)The static dome height displacement d_s versus the electric field for different stresses of A_pad. (b) The . resonance frequency of the whole DBPDM is obtained with the mode analysis of whole finite element model.

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Fig. 4 A deformable mirror of 21 elements. (a)The locations of No.1, No.2 and No.3 BBPBAs. (b) FEA model of mirror face sheet and pads.

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(s₁₁,s₁₂,s₁₃,s₄₄) = (16.5,-4.78,-8.45,20.7,43.5) × 10⁻¹²m²/N,(d₃₁,d₁₅,d₃₃) = (−274,741,539) × 10⁻¹²C/N, ρ = 7500 kg/m³. We consider a beam flexural-mode piezoelectric actuator with 2L = 5 mm, Δh = h-c = 0.5mm, η = c/Δh = 1/2, b = 2mm, w_pad = 1mm, unless stated otherwise. Besides, the metallic layer is an aluminum (Al) alloy with Young’s modulus E_m = 70 GPa, Poisson’s ratioν = 0.33, and mass density is ρ = 2700 kg/m³.

The model predictions of d_s in response to the electric field are shown in Fig. 3(a) for different stresses. From the figure, d_s has a zero offset due to the external force. When ${\bar{T}}_{z}$ <0, the effectively concentrated force is a compression force, and the zero offset is a negative value. Conversely, the effectively concentrated force is a stretching force, and its zero offset is a positive value. From Eq. (8), it is observed that the zero offset is equal to L⁴FΓ/(4D) and proportional to F. From Fig. 3(a), the stroke of the actuator is larger than 10 microns at about |E₃| = 10MV/m, and it equates to the stroke of the deformable mirror [see Eqs. (10) and (11)]. According to the performance requirements of a deformable mirror with high stroke, its stroke must be at least 10 microns. Thus, the deformable mirror has a large stroke. Here, the finite element model of DBPDM is created with the commercial finite element software ANSYS. With the mode analysis of whole finite element model, we get its resonant frequency of 53.3kHz which is much larger than that of the BPDM. Without the metallic layer, the resonant frequency is 14.3 kHz, which is still very high. The high resonant frequency is caused by the introduction of a metallic layer into the BBPBAs.

With the realistic electro-mechanical model of DBPDM, consisting of Eqs. (1)-(12) and the FEA model of mirror face sheet and pads [see Fig. 4(b)], we can obtain the influence functions of all elements of the deformable mirror. For example, a vector of voltage {V^N}will be formed when only the No.1 BBPBA [1 in Fig. 4 (a)] is active with the applied voltage being 1 V, and 0 V are applied to other BBPBAs, respectively. With Eq. (11), we can obtain the displacement vector {d^N}, which is used as the input displacement loads to be applied to pads of the FEA model. The resulting displacement field in mirror face sheet defines the influence function f^N(x,y). Thus, we can get the influence function for the face sheet surface surrounding No.1 BBPBA [see Fig. 5(a)]. With the same way, we can get the influence functions of the other BBPBAs. Figure 5(b) shows the curves of the influence functions of No.1 BBPBA versus normalized radius for the different side lengths of the cross-section of the pad. It is observed that the top of the peak of the influence function widens with the increasing w_pad, and the peak foot of the influence function narrows as w_pad increases. Furthermore, the negative values of the influence function decrease with the increasing w_pad_. These phenomena also appear in the locations of No. 2 and No.3 BBPBAs with an increase of w_pad (see Fig. 6)_. When all sites' influence functions have been obtained, we can use the realistic electromechanical model to simulating wavefront correction, which will be addressed in future.

Fig. 5 (a) The 3D influence function for a unit voltage of the No.1 BBPBA. (b)The influence function versus normalized radius for different side lengths of pads at No.1 BBPBA.

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Fig. 6 (a) The influence function versus normalized radius for different side lengths of pads at No.2 BBPBA. (b) The influence function versus normalized radius for different side lengths of pads at No.3 BBPBA.

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4. Summary

In this paper, a discrete-layout bimorph piezoelectric deformable mirror (DBPDM) and its realistic electromechanical model have been proposed. As an example, a 21-elements DBPDM is studied. The achievement of a high resonance frequency of the whole DBPDM have been proved by the results of the mode analysis of whole finite element model. We have obtained the large stroke of the DBPDM through the analytical analysis of the behaviors of the actuator.

The numerical simulation is performed on the DBPDM using with the realistic electromechanical model. The results show that the size of the radius of the push pad has an impact on the influence function of the deformable mirror. In future, we can use the realistic electromechanical model to simulate wavefront correction errors in the study of the adaptive optics in optical telescope. Besides, the cost of the deformable mirror could be reduced due to the simple structure of the BBPBA. The results in this paper are useful for the design of the deformable mirror.

Funding

National Natural Science Foundation of China (Grant Nos. 10921063, 11190014, 11403109 and 11373073); Natural Science Foundation of Jiangsu Province (Grants No. BK20141042).

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Research on a bimorph piezoelectric deformable mirror for adaptive optics in optical telescope

Abstract

1. Introduction

2. Realistic electro-mechanical model of discrete-layout bimorph piezoelectric deformable mirror (DBPDM)

3. Numerical results and discussion

4. Summary

Funding

References and links

Cited By

Figures (6)

Equations (12)

Optics Express