Research on elastic modes of circular deformable mirror for adaptive optics and active optics corrections

Hairen Wang; Mingzhu Zhang; Yingxi Zuo; Xianzhong Zheng

doi:10.1364/OE.27.000404

1. Introduction

The task of a primary mirror for active optics system or a deformable mirror for adaptive optics system is the removal of arbitrary aberrations by the elastic deformation of mirrors in the most effective way, which means the target is to minimize the residual error with minimum loads and little computational effort [1]. The well-known Zernike polynomials are widely used because of their immediate interpretation in terms of optical aberrations, but with somewhat low efficiency over the dynamic range [2]. Besides, some aberrations of Zernike polynomials are very difficult to correct, and these aberrations need very large loads even if they could be corrected, such as Z7, Z8 and so on. In an active optics correction system context or adaptive optics correction system, the choice of the so called “elastic modes” as the polynomials to represent the mechanical deformations is best justified by the derivation from mechanical properties [1–3]. The elastic modes, in which the mirrors deform most naturally and the deformations are significantly simpler to realize and require less loads, are easier to control. For thin meniscus primary mirrors the elastic modes which correct arbitrary wave-front aberrations in the most efficient way have been derived based on minimum-energy modes [1]. The deformable mirror of adaptive optics system, which is typically a circular plane mirror, is different from the thin meniscus mirrors which are typically spherical mirrors. Furthermore, the bimorph piezoelectric deformable mirror is made of a circular glass sheet bonded to a circular piezoceramics layer with an appropriate electrode pattern on the back [4–6]. Thus, the elastic modes derived by the thin meniscus mirrors are unsuited to deformable mirrors. In this paper, the elastic modes are derived by a bimorph piezoelectric deformable mirror (BPDM) based on an analytical method of linear piezoelectricity, and the three-dimensional formulas of elastic modes are presented. The elastic modes of BPDM can be extended to be applied in a more general deformable mirror whose mirror is only made of a circular glass sheet.

2. Derivation of elastic modes of BPDM

Figure 1 shows the configuration of a bimorph piezoelectric deformable mirror. From Fig. 1, the BPDM consists of a glass layer and a piezoelectric ceramic layer, which is polarized in the thickness direction. The thickness of the glass layer is h_s, and the thickness of the piezoelectric ceramic layer is h_p. The diameter of the circular bimorph is 2a. The distance between the median plane and the top surface of the BPDM is h_m1, and the distance between the median plane and the bottom is h_m2. Here two cases are considered: (a) The edge of the BPDM is considered free; (b) The BPDM is simply-supported at the outer boundary at r = a, where is no allowable to appear any in-plane displacement, as is shown in Fig. 1 (b).

Fig. 1 The configuration of a bimorph piezoelectric deformable mirror

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In the following, analytical solutions are derived from a cylindrical coordinate system (see Fig. 1 (a)). The z-direction is normal to the layers face, and r-direction and θ-direction are radial direction and circumferential direction, respectively. we consider the axis-symmetrically flexural deformation of the BPDM. Assuming, that is (h_s + h_p)/a<<1, and the deflection $w_{z} (r)$ of the BPDM can be expressed as follows:

ε_{r} = - z w_{z}_{, r r}, ε_{θ} = - \frac{z}{r} w_{z, r} - \frac{z}{r^{2}} w_{z, θ θ}, γ_{r θ} = - 2 {(\frac{1}{r} w_{z, θ})}_{, r} .

In the following,

E_{s}

and

ν_{s}

are the glass Young’s modules and Poisson’s ratio respectively ;

E_{p}

and

ν_{p}

are the piezoelectric ceramic’s Young’s modules and Poisson’s ratio respectively. The constitutive relations of glass layer can be written as follows:

σ_{r}^{s} = \frac{E_{s}}{1 - ν_{s}^{2}} (ε_{r} + ν_{s} ε_{θ}), σ_{θ}^{s} = \frac{E_{s}}{1 - ν_{s}^{2}} (ε_{θ} + ν_{s} ε_{r}), τ_{r θ}^{s} = \frac{E_{s}}{2 (1 + ν_{s})} γ_{r θ .}

Besides, we’ve also got the constitutive relations of the piezoelectric ceramic layer:

σ_{r}^{p} = \frac{E_{p}}{1 - ν_{p}^{2}} (ε_{r} + ν_{p} ε_{θ}), σ_{θ}^{p} = \frac{E_{p}}{1 - ν_{p}^{2}} (ε_{θ} + ν_{p} ε_{r}), τ_{r θ}^{p} = \frac{E_{p}}{2 (1 + ν_{p})} γ_{r θ .}

In order to simplify the derivation, the concept of median plane is introduced. Median plane, widely used in thin plate deformation theory, refers to a surface on which material deformation is absent when the plate itself bends. For a 2-layer bimorph in Fig. 1(a), h_m1 and h_m2 can be calculated through the simple expressions as follows [6]:

h_{m 1} = \frac{E_{s} h_{s}^{2} + E_{P} [{(h_{s} + h_{p})}^{2} - h_{s}^{2}]}{2 (E_{s} h_{s} + E_{P} h_{p})}, h_{m 2} = \frac{E_{p} h_{p}^{2} + E_{s} [{(h_{p} + h_{s})}^{2} - h_{p}^{2}]}{2 (E_{s} h_{s} + E_{P} h_{p})} .

By integrating principal stress over the thickness direction of the plates, we’ve obtained the bending moments per unit length. From constitutive relations of the layers, we get [5,7–10]

\begin{array}{l} M_{r} = \int_{- h_{m 2}}^{h_{m 1}} σ_{r} z d z = - D_{s} [w_{z, r r} + ν_{s} (\frac{1}{r^{2}} w_{z, θ θ} + \frac{1}{r} w_{z, r})] - D_{p} [w_{z, r r} + \\ ν_{p} (\frac{1}{r^{2}} w_{z, θ θ} + \frac{1}{r} w_{z, r})], \\ M_{θ} = \int_{- h_{m 2}}^{h_{m 1}} σ_{θ} z d z = - D_{s} [w_{z, θ θ} + ν_{s} w_{z, r r} + \frac{1}{r} w_{z, r})] - D_{p} [w_{z, θ θ} + ν_{p} w_{z, r r} + \frac{1}{r} w_{z, r})], \\ M_{θ r} = \int_{- h_{m 2}}^{h_{m 1}} τ_{r θ} z d z = - D_{s} (1 - ν_{s}) (_{\frac{1}{r} w_{z, θ}), r} - D_{p} (1 - ν_{p}) (_{\frac{1}{r} w_{z, θ}), r} . \end{array}

where

\begin{array}{l} D_{s} = \frac{E_{s} h_{s} [E_{s}^{2} h_{s}^{4} + 2 E_{s} E_{P} h_{s}^{3} h_{p} + E_{p}^{2} h_{p}^{2} (4 h_{s}^{2} + 6 h_{s} h_{p} + 3 h_{p}^{2})])}{12 {(E_{s} h_{s} + E_{p} h_{p})}^{2} (1 - ν_{s}^{2})} . \\ D_{p} = \frac{E_{p} h_{p} [E_{p}^{2} h_{p}^{4} + 2 E_{s} E_{P} h_{s}^{} h_{p}^{3} + E_{s}^{2} h_{s}^{2} (4 h_{p}^{2} + 6 h_{s} h_{p} + 3 h_{s}^{2})])}{12 {(E_{s} h_{s} + E_{p} h_{p})}^{2} (1 - ν_{p}^{2})} \end{array}

When E_s = E_p and ν_s = ν_p, the following derivation of the BPDM is reduced to that of a more general deformable mirror whose mirror is only made of a circular glass sheet. The transverse shear force per unit length is got through the following equations [11]:

Q_{r} = M_{r, r} + \frac{1}{r} M_{r θ, θ} + \frac{M_{r} - M_{θ}}{r}, Q_{θ} = M_{r θ, r} + \frac{1}{r} M_{θ, θ} + \frac{2 M_{r θ}}{r} .

Thus, the effective transverse shear force per unit length can be obtained

V_{r} = Q_{r} + \frac{\partial M_{r θ}}{r \partial θ}, V_{θ} = Q_{θ} + \frac{\partial M_{r θ}}{\partial r} .

The equation of motion in slim plate takes the following form:

\frac{Q_{r}}{r} + Q_{r, r} + \frac{Q_{θ, θ}}{r} = ξ {\ddot{w}}_{z} .

where $ξ = ρ_{s} h_{s} + ρ_{p} h_{p}$ . From the Eqs. (5),(7) we can obtain

- D \nabla^{2} \nabla^{2} w_{z} = ξ {\ddot{w}}_{z}

where $\nabla^{2} = \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2}}{\partial θ^{2}}$ and $D = D_{s} + D_{p}$ . For harmonic motions we use the complex notation [11,12]

w_{z} (r, θ, t) = Re {W_{z} (r, θ) \exp (i ω t)} .

Thus, Eq. (10) can be rewritten as

D \nabla^{2} \nabla^{2} W_{z} - ω^{2} ξ W_{z} = 0

The solutions of Eq. (12), which are characterized by a radial and an azimuthal part, are assumed to be of the type:

W_{z} (r, θ) = u_{n} (r) \cos (n θ) n=rotational symmetry

Substitution of (13)into (12) yields

D \nabla_{r}^{2} \nabla_{r}^{2} u_{n} (r) - ω^{2} m u_{n} (r) = 0

where

\nabla_{r}^{2} = \frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{n^{2}}{r^{2}}

. The general solution to Eq. (14) can be written as

u_{n} (r) = A_{1, n} J_{n} (λ r) + A_{2, n} Y_{n} (λ r) + A_{3, n} I_{n} (λ r) + A_{4, n} K_{n} (λ r) .

where

λ^{4} = ω^{2} ξ / D

, and

J_{n} (λ r)

,

Y_{n} (λ r)

,

I_{n} (λ r)

and

K_{n} (λ r)

are n-order Bessel function of the first and the second kind with argument

λ r

, n-order modified Bessel function of the first and the second kind with argument

λ r

respectively. Since

Y_{n} (α r)

and

K_{n} (α r)

at r = 0 are infinite, we obtain

A_{2, n} = A_{4, n} = 0

, and the Eq. (15) can be rewritten as

u_{n} (r) = A_{1, n} J_{n} (λ r) + A_{3, n} I_{n} (λ r) .

Here the dynamic boundary conditions of the two cases are presented. The first case: the edge of the BPDM is considered free[see Fig. 1 (a)], the dynamic boundary conditions are written as

M_{r} (a) = 0, V_{r} (a) = 0

The second case: the BPDM is simply supported at r = a [see Fig. 1 (b)], the dynamic boundary conditions are written as

u_{n} (a) = 0, M_{r} (a) = 0.

In either case, the boundary conditions lead to a set of 2 homogeneous equations with 2 unknowns where all the parameters depend on λ. In matrix form

(\begin{array}{l} b_{11} (λ) \\ b_{21} (λ) \end{array} \begin{array}{l} b_{12} (λ) \\ b_{21} (λ) \end{array}) (\begin{array}{l} A_{1, n} \\ A_{3, n} \end{array}) = (\begin{array}{l} 0 \\ 0 \end{array})

This is an eigenvalue problem in a nonstandard form. In order to have non-trivial solutions, the values of the determinant of the matrix (19) is equal to 0:

| \begin{array}{l} b_{11} (λ) \\ b_{21} (λ) \end{array} \begin{array}{l} b_{12} (λ) \\ b_{22} (λ) \end{array} | = 0

From the Eq. (20), we can get a series of values of λ. With substitution of each value of λ into the matrix(19), we can get 2 unknowns with an infinite number of solutions, and one particular solution can be obtained. One of the two unknowns can be written as a function of the other one. Thus,

A_{3, n}

can be divided by

A_{1, n}

. The particular solution

{\bar{u}}_{m n} (r)

, where

m

denotes the order of the mode depending on λ, can be obtained by setting

A_{1, n} =1

. The general normalization solution can be written as

u_{m n} (r) = K_{n m} {\bar{u}}_{n m} (r)

where

K_{n m} = \sqrt{\frac{a^{2}}{β \int_{0}^{a} {\bar{u}}_{n m} (r) d r}}, β = {\begin{cases} 2 n=0 \\ 1 n>0 \end{cases}

Thus, an arbitrary wave-front error

Γ (r, θ)

can be decomposed as:

Γ (r, θ) = \sum_{n = 0}^{\infty} \sum_{m = 1}^{\infty} c_{n m} u_{n m} (r) \cos (n θ) .

where n denotes the symmetry, m denotes the order, c_nm is the coefficient of the elastic modes.

3. Numerical results and discussion

In the following numerical calculations, we take a BPDM with an aperture of 165 mm as an example. The material of glass is the single crystal silicon with Young’s modulus E_s = 190 GPa, Poisson’s ratioν_s = 0.30, mass density is ρ_S = 2350 kg/m³, the thickness h_s = 1mm. We use PZT-5H [13] as the piezoelectric material of BPDM, whose parameters are listed below:

(c₁₁,c₁₂,c₁₃,c₃₃) = (12.6,7.95, 8.41,11.7) × 10¹⁰N/m²/N and ρ_p = 7500 kg/m³. From the above parameters of piezoelectric layer, we can get E_p = 55GPa and ν_p = 0.40. Here the thickness of piezoelectric layer is h_p = 2 mm.

Because of space constraints, only a small rotational symmetry number of $u_{m n} (r)$ and $u_{m n} (r, θ)$ are presented. Figure 2 shows the dependence of $u_{m n} (r)$ upon the radius r for m = 1. From the Eqs. (19), we know that $u_{m n} (r)$ is orthogonal with different rotational symmetry numbers n. For the same symmetry number n, $u_{m n} (r)$ is also orthogonal with different order numbers m. From Fig. 2 (a), we can see that: when n = 0, the value of curve decreases with increasing n; when n = 1, the value of curve climbs up and then decline with n increasing ; when n >1, the values of curves increases with n increasing; from n = 0 to n = 4, their outer edges are free vibrations. From Fig. 2(b), we can see that the edges of the BPDM are fixed and the peak values of the curves are closer to the edge of the deformable mirror with n increasing. The difference between Fig. 2(a) and Fig. 2(b) is attributed to their different boundary conditions. The 3D deformation function $u_{m n} (r, θ)$ of two different boundary conditions are shown in Fig. 3 and Fig. 4. From the derivation, $u_{m n} (r, θ)$ is also orthogonal.

Fig. 2 Deformation function $u_{m n} (r)$ versus radius r for m = 1. (a) The edge of the BPDM is considered free; (b) The edge of the BPDM is simply supported.

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Fig. 3 The 3D deformation function $u_{m n} (r, θ)$ when the edge of the BPDM is considered free. (a) The $u_{m n} (r, θ)$ of m = 1 and n = 0; (b) The $u_{m n} (r, θ)$ of m = 2 and n = 0; (c) The $u_{m n} (r, θ)$ of m = 1 and n = 1; (d) The $u_{m n} (r, θ)$ of m = 1 and n = 2; (e) The $u_{m n} (r, θ)$ of m = 1 and n = 3; (f) The $u_{m n} (r, θ)$ of m = 1and n = 4.

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Fig. 4 The 3D deformation function $u_{m n} (r, θ)$ when the edge of the BPDM is simply supported. (a) The $u_{m n} (r, θ)$ of m = 1 and n = 0; (b) The $u_{m n} (r, θ)$ of m = 2 and n = 0; (c) The $u_{m n} (r, θ)$ of m = 1 and n = 1; (d) The $u_{m n} (r, θ)$ of m = 1and n = 2; (e) The $u_{m n} (r, θ)$ of m = 1 and n = 3; (f) The $u_{m n} (r, θ)$ of m = 1and n = 4.

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Figure 3 shows the elastic modes of the free edge of the BPDM for different n and different m. Figure 3 (a), Fig. 3(b), Fig. 3(c), Fig. 3 (d), Fig. 3 (e), and Fig. 3 (f) are respectively similar to the defocusing aberrations of Zernike polynomials Z4, the spherical aberrations of Zernike polynomials Z11. the coma aberrations of Zernike polynomials Z7, the astigmatism aberrations of Zernike polynomials Z5, the astigmatism aberrations of Zernike polynomials Z9 and the astigmatism aberrations of Zernike polynomials Z14. Figure 4 shows the elastic modes of the simply-supported edge of the BPDM for different n and different m. Figure 4 (a), Fig. 4(b), Fig. 4(c), Fig. 4 (d), Fig. 4 (e) and Fig. 4 (f) are respectively similar to the defocusing aberrations of Zernike polynomials Z4, the spherical aberrations of Zernike polynomials Z11, the coma aberrations of Zernike polynomials Z7, the spherical aberrations of Zernike polynomials Z12, the spherical aberrations of Zernike polynomials Z18, the spherical aberrations of Zernike polynomials Z25.

Figure 5 (a) shows the comparison of Z4 and n = 0 of the elastic mode for two different dynamic boundary conditions. From the figure, curvature the curve of Z4 is agreeable with the curve of n = 0 of the elastic mode with the edge of the BPDM considered free. It is clear that the curve of Z4 with some whole motions could be agreeable with the curve of n = 0 of the elastic mode with the edge of the BPDM simply supported. Figure 5(b) shows the comparison of Z7 and n = 1 of the elastic mode for two different dynamic boundary conditions. We can see that the largest value of Z7 is nearer the outer edge, and that the values of Z7 and n = 1 of free edge are negative near the outer edge. The values of n = 0 of the edge simply supported are all larger than zero except for r = 0 and r = 1, whose values are equal to zero. Figure 5(c) shows the comparison of the Zernike polynomials(Z5,Z9,and Z14) and the deformation functions of the elastic modes(n = 2,n = 3 and n = 4) with the edge of the BPDM considered free. From the figure, the curvatures of Z5, Z9, and Z14 are respectively a little larger than those of n = 2, n = 3 and n = 4 with the edge of BPDM considered free. Figure 5(d) shows the comparison of the Zernike polynomials (Z12,Z18,and Z19) and the deformation functions of the elastic modes(n = 2,n = 3 and n = 4) with the edge of the BPDM simply supported. We can see that the variation trends of Z12,Z18,and Z19 are respectively agreeable with those of n = 2,n = 3 and n = 4 with the edge of the BPDM simply supported. The values of n = 2,n = 3 and n = 4 of the edge simply supported are all larger than zero except for r = 0 and r = 1, whose values are equal to zero, while the values of Z12,Z18,and Z19 are negative near the outer edge. From Fig. 3-Fig. 5, though Zernike polynomials are similar to the elastic modes, there are some differences between them. The Zernike polynomials are well agreeable with the elastic modes with the edge of a deformable mirror considered free. The elastic modes, in which the mirrors deform most naturally and the deformations require less loads, are stemmed from the minimum-energy modes of vibration of the mirror and easier to control [1]. Compared with the elastic modes, the Zernike modes are man-made, and some aberrations of Zernike modes are very difficult to correct, and these aberrations need very large loads even if they could be corrected, such as Z7, Z8 and so on. According to the principle of St.Venant, if a mirror is flexible enough to develop a given elastic-error mode, then the same error can be corrected by applying active forces of the same order of magnitude as the support forces [14]. The deformable mirror and the primary mirror both work on the similar principle. Thus, the St. Venant principle applies to the primary mirror and by analogy to others like the primary mirror, such as the deformable mirror, the second mirror for active optics, and so on. It means the elastic modes are more effective to decompose arbitrary wave-front errors of a deformable mirror.

Fig. 5 The comparisons of the radial functions of the Zernike polynomial and the deformation functions of the elastic modes.

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The values of λ and the corresponding natural frequency f of different n and different m, which are shown in Table 1, are dependent on boundary conditions, m, n, geometric parameters and material parameters of the BPDM. From the above analysis, the elastic modes are orthogonal with each other. Thus, deformation function of the BPDM, which are motivated to compensate the wave front errors, can be decomposed by the elastic modes.

Table 1. The values of λ and corresponding natural frequency f for different n and m.

View Table

The analytic solution method has been applied to analyze the performances of other piezoelectric structures [7,10,11,15–21], such as piezoelectric actuators, piezoelectric transformers and piezoelectric energy harvesters, etc. Furthermore, taking elastic modes with a free boundary condition as an example, we present a comparative study for the analytic analysis and the finite element analysis (FEA) (see Fig. 6 and Fig. 7). Figure 6 shows the FEA model of BPDM created by ANSYS. The mode shapes of the different elastic modes of the FEA are shown as Fig. 7. From the Fig. 3 and Fig. 7, the mode shapes of the analytic analysis of Fig. 3(a)-Fig. 3(f) are well agreeable with those of the FEA of Fig. 7(a)-Fig. 7(f), respectively. Furthermore, the resonant frequencies of the FEA of Fig. 7(a)-Fig. 7(f), which are separately 797Hz, 3342Hz, 1778Hz, 453Hz, 1053Hz and 1848Hz, are separately well agreeable with those of the analytic solutions of Fig. 3(a)-Fig. 3(f), too (see Table 1). We’ve found that the values of above frequencies are respectively closer to those of the analytic analysis with the more refined mesh of finite element analysis. It indicates that the results of FEA will be equal to those of the analytic solutions with the infinite fine mesh. Thus, the results of the analytic analysis and the FEA well agreeable with each other.

Fig. 6 The finite element model of BPDM.

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Fig. 7 The results of the elastic modes of finite element analysis when the edge of the BPDM is considered free. (a) The $u_{m n} (r, θ)$ of m = 1 and n = 0; (b) The $u_{m n} (r, θ)$ of m = 2 and n = 0; (c) The $u_{m n} (r, θ)$ of m = 1 and n = 1; (d) The $u_{m n} (r, θ)$ of m = 1 and n = 2; (e) The $u_{m n} (r, θ)$ of m = 1 and n = 3; (f) The $u_{m n} (r, θ)$ of m = 1 and n = 4.

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From Eqs. (1)-(23), it is known that the circular planes, which have different physical systems (including radii, thicknesses, and so on), have the same formulas of elastic modes. In other words, if the boundary conditions of the deformable mirrors of different physical systems are the same, they have the almost same mode shapes at the same n and m. Furthermore, taking the $u_{m n} (r, θ)$ of m = 1 and n = 3 in the free boundary condition as an example, presented is a comparative study of mode shapes for three different diameters 2a = 165 mm, 148.5 mm (the pupil size is 90% of the aperture of 165 mm), 140.25 mm (the pupil size is 85% of the aperture of 165mm). Figure 8(a), Fig. 8(b) and Fig. 8(c) respectively show the mode shapes of three different diameters including 2a = 165 mm, 148.5 mm, 140.25 mm. For the comparison, the mode shapes of Fig. 8 are displayed as symmetric with respect to the x-axis. Figure 8(a) is taken from Fig. 7(e). From the figures, we’ve found that the modes resemble each other in shape but not in resonance frequency. Besides, the comparisons of the deformation functions of the elastic modes for three different diameters are presented as shown in Fig. 9. The three curves of Fig. 9, which show the dependence of the deformation of z-direction on x axis after normalization, are taken from Fig. 8(a), Fig. 8(b), and Fig. 8(c), respectively. From the Fig. 9, it is observed that the three curves converge to a curve. We’ve found that only the resonance frequencies depend on the physical systems, and the different physical systems have little influence on the mode shapes after normalization. This implies that the elastic modes after normalization are also always orthogonal within a unit pupil. It is clear that there are no significant differences between the orthogonality within a unit pupil of the elastic modes and Zernike polynomials. Thus, if the pupil is set smaller than the mirror dimension, the elastic modes can also present a wavefront screen.

Fig. 8 The mode shapes for three different radii, 2a = 165, 148.5, 140.25 mm of the free boundary condition

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Fig. 9 The comparisons of the deformation functions of the elastic modes for three different radii, 2a = 165, 148.5, 140.25 mm of the free boundary condition

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4. Summary

The goal of this paper is to propose a useful decomposed aberration method for a deformable mirror. The orthogonal elastic modes of the deformable mirror, which could replace Zernike polynomials on decomposing the wave front errors, are proposed based on the theory of elasticity. As an example, some elastic modes of a BPDM with an aperture of 165 mm is presented. The useful mathematical formulas of elastic modes of a deformable mirror are obtained in two different kinds of dynamic boundary conditions, and the dependence of the elastic modes aberrations upon the orders and rotational symmetries is numerically evaluated. Moreover, a comparative study for elastic mode and Zernike polynomials is presented in the numerical analysis. The results have shown that the Zernike polynomials are well agreeable with the elastic modes for the edge of a deformable mirror considered free. The results have also implied that the elastic modes, in which the mirrors deform most naturally and the deformations require less loads, are more effective to decompose arbitrary wave-front errors of a deformable mirror.

Though the elastic modes are derived from a bimorph piezoelectric deformable mirror, their applications include but no limited a bimorph piezoelectric deformable mirror. When E_s = E_p and ν_s = ν_p, the derivation of the BPDM is reduced to that of a more general mirror which is only made of a circular glass sheet. It implies that the orthogonal elastic modes are easily extended to be applied in the circular flat mirrors for active optics. Moreover, since the curvature radii of the primary mirrors (the thin meniscus spherical mirrors) are usually larger (For example, the primary mirror of WFST [22–24]), the deformation shapes of the natural modes of the primary mirrors resemble those of the circular flat mirror. Thus, it is very likely that the elastic modes derived from the circular flat mirror, like the Zernike polynomials, could be used to represent the optical aberrations of the primary mirrors of the large optical telescopes. These works will be presented in the future.

Funding

National Natural Science Foundation of China (11873100, 11403109, 11773084); Natural Science Foundation of Jiangsu Province (BK20141042); and National Key Basic Research and Development Program (2018YFA0404702).

Acknowledgements

This work has been supported by the National Natural Science Foundation of China (Grant Nos. 11873100, 11403109 and 11773084), the Natural Science Foundation of Jiangsu Province (Grants No. BK20141042), the National Key Basic Research and Development Program (Grants No. 2018YFA0404702), the Youth Innovation Promotion Association, CAS, and the Ministry of Finance of China (MOF) and administrated by the Chinese Academy of Sciences (CAS).This work has also been supported by 2.5 m Wide Field Survey Telescope (WFST) project of China, the project has been started. The authors would like to thank Prof. Ji Yang and Prof. Ming Liang at Purple Mountain Observatory and Prof. Yuantai Hu and Prof. Hongping Hu at Huazhong University of Science and Technology for helpful discussions.

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symmetry	order	free conditions		simply-supported conditions
n	m	ë	f (Hz)	ë	f (Hz)
0	1	36.66	803	27.18	441
0	2	75.29	3388	66.16	2616
1	1	54.96	1805	45.32	1227
2	1	27.47	451	61.44	2256
3	1	41.97	1053	76.69	3515
4	1	55.71	1855	91.44	4998

Research on elastic modes of circular deformable mirror for adaptive optics and active optics corrections

Abstract

1. Introduction

2. Derivation of elastic modes of BPDM

3. Numerical results and discussion

4. Summary

Funding

Acknowledgements

References

Cited By

Figures (9)

Tables (1)

Equations (23)

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