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Applications of the elastic modes of a circular plate in wavefront correction of the adaptive optics and the active optics

Hairen Wang, Mingzhu Zhang, Jingjing Gao, Yue Lan, Yingxi Zuo, and Xianzhong Zheng

Open Access Open Access

Abstract

The elastic modes of a general circular thin plate (EMCTP), reflecting the natural deformation of the resonance, are applied to the diffraction theory of the optical aberrations in this paper. Our work has shown that the mode shapes of the EMCTP resemble those of the Zernike polynomials. As an application example, the compensations of some low order aberrations of the 2.5 m-wide field survey telescope (WFST) have been performed with the EMCTP. Moreover, a quantitative comparative study of the active optics corrections for the EMCTP and the Zernike polynomials is presented in the numerical analysis. The quantitative analysis results have demonstrated that the efficiency of the EMCTP is superior to the standard Zernike polynomials as well as the annular Zernike polynomials.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optical systems have their own optical aberrations due to their designs and manufacturing errors. The well-known Zernike polynomials, proposed by Frits Zernike [1], whose orthogonal functions are defined on the unit circle, are widely used to represent the optics aberrations. The optical aberrations can usually be corrected by active optics or adaptive optics. The task of the adaptive optics or the active optics is to adjust the deformable mirror to compensate the optical aberrations in the most effective way [2]. However, studies have shown that Zernike polynomials perform somewhat less efficiency over the dynamic range of the aberration corrections of the active optics systems [3]. Moreover, some aberrations represented by the Zernike polynomials are not easy to compensate. The so-called minimum-energy modes, proposed by Noethe, are a more effective way to represent optical aberrations [2]. However, the minimum-energy modes are applicable only to the thin meniscus spherical mirror of the active optics system. Some researchers also had proposed other polynomial bases (e.g., Bessel circular functions, gradient orthogonal base and so on) [4–6]. However, those polynomial bases, which are either in a purely mathematical sense or targeting the particular mirrors systems. In our previous work [7], the elastic modes of a bimorph piezoelectric deformable mirror (EMBPDM) have been proposed. However, the EBPDM, which is aimed mostly at the aberrations representations of a bimorph piezoelectric deformable mirror, cannot be directly applicable to other optical systems (e.g. the active optics system of the WFST). Therefore, here furtherly proposed is the elastic modes of a general circular thin plate (EMCTP), which are more easily extended to be used instead of the standard Zernike polynomials in optics systems. The elastic modes, in which the mirrors deform most naturally and the deformations require less loads, are derived from theory of elasticity, reflecting the intrinsic physical properties of the vibration of the mirror and obeying the minimum energy principle. Here the EMCTP has been applied to the simulations of the aberrations compensations of the active optics systems, and a quantitative comparative study of the active optics corrections between the EMCTP and the standard Zernike polynomials is presented.

2. Theoretical derivation of the EMCTP

Figure 1 shows the configuration of a thin circular plate. The thickness of the thin circular plate is denoted by h, and the diameter of the thin circular plate is denoted by 2a. The edge of the thin circular plate is considered free.

 figure: Fig. 1.

Fig. 1. The configuration of a circular thin plate

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The detailed derivation of the EMCTP has been presented in Appendix A. From Appendix A, it is observed that an arbitrary wave-front error $w(\rho ,\varphi )$ can be decomposed as [2]:

$$w(\rho ,\varphi ) = \sum\limits_{n = 0}^\infty {\sum\limits_{m = 1}^\infty {{c_{nm}}{w_{nm}}(\rho )\cos (n\varphi ).} }$$
in which the ${w_{nm}}(\rho )\cos (n\varphi )$, a set of the orthonormal functions in the unitary circle, is the function basis of the EMCTP. here the elastic modes, like Zernike polynomials, also form a completely orthogonal set, and this is based on its analytical solutions. Here n denotes the symmetry, m denotes the order, and cnm denotes the coefficient of the elastic modes. The formulation of ${w_{nm}}(\rho )$ can be expressed as [8]
$${w_{nm}}(\rho ) = {K_{nm}}{\bar{w}_{nm}}(\rho )$$
where
$${K_{nm}} = \sqrt {{a^2}/\beta \int\limits_0^a {{{\bar{w}}_{nm}}(\rho )d\rho } } ,\textrm{ }\beta \textrm{ = }\left\{ \begin{array}{l} 2\textrm{ n = 0}\\ 1\textrm{ n > 0} \end{array} \right.,{\bar{w}_{nm}} = {A_{1,n}}{J_n}({\chi _m}\rho ) + {A_{3,n}}{I_n}({\chi _m}\rho ).$$
${J_n}\textrm{(}{\chi _m}\rho \textrm{)}$ denotes the n-order Bessel function of the first with argument $\chi \rho $, and ${I_n}\textrm{(}{\chi _m}\rho \textrm{)}$ denotes the n-order modified Bessel function of the first with the argument $\chi \rho $. ${A_{1,n}}$ and ${A_{3,n}}$ can be determined by the conditions in which the boundary conditions, n, m and ${\chi _m}$ have been given. ${\chi _m}$ is a function of the natural frequencies, the material parameters, and geometric parameters of the thin plate. (More details see Appendix A). In general, elastic modes are equated with vibrational modes. However, here the elastic modes emphasis the elastic deformation of the modes, while vibrational modes emphasis the vibrations of the modes. In this paper, we are more interested in the elastic deformation of the modes than in resonance frequencies.

3. Physical characteristics of the EMCTP

To demonstrate the physical characteristics of the EMCTP, some numerical calculations are carried out. Here we have taken a circular thin plate with a diameter of 100 mm as an example. In this section, 2a=100 mm, h=2 mm, unless stated otherwise. The material of the circular thin plate is the Cer-Vit material with Young’s modulus E=91 GPa, Poisson’s ratioν=0.24, and mass density ρz=2530 kg/m3.

The mode shapes of the free edge of the EMCTP for different n and m have been shown in Fig. 2. The mode shape of w01, as shown in Fig. 2(a), resembles that of Z4. The mode shape of w11cosφ, as shows in Fig. 2(b), resembles that of Z8. The mode shape of w21cos2φ, as shown in Fig. 2(c), resembles that of Z6. The mode shape of w31cos3φ, as shown in Fig. 2(d), resembles that of Z10. The mode shape of w41cos4φ, as shown in Fig. 2(e), resembles that of Z14.The mode shape of w51cos5φ, as shown in Fig. 2(f), resembles that of Z20. Because of the space constraints, only some physical characteristics of wnmcosnφ are presented. The results have shown that the mode shapes of the EMCTP resemble those of the standard Zernike polynomials.

 figure: Fig. 2.

Fig. 2. The mode shapes of the EMCTP are from the analytic solutions. (a) the mode shape of w01, whose resonant frequency is 2013Hz; (b) the mode shape of w11cosφ, whose resonant frequency is 4631 Hz; (c) the mode shape of w21cos2φ, whose resonant frequency is 1258 Hz; (d) the mode shape of w31cos3φ, whose resonant frequency is 2907 Hz; (e) the mode shape of w41cos4φ, whose resonant frequency is 5087 Hz; (f) the mode shape of w41cos4φ, whose resonant frequency is 7785 Hz.

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The analytic solutions of the mode shapes of the EMCTP have been demonstrated by the finite element method (The details please see Appendix B).

Besides, the quantitative comparative studies of the mode shapes of the EMCTP and the standard Zernike polynomials are presented in Fig. 3. The two curves per graph in Fig. 3, which show the dependence of the deformation of z-direction on x-axis (the definition of the direction is coincident with Fig. 17 in Appendix B) after normalization, are taken from the corresponding analytic results (see Fig. 2), and the corresponding mode shapes of the standard Zernike polynomials, respectively. From the figures, it is observed that there are some differences between the curvatures of the curves of the EMCTP and the standard Zernike polynomials. The resonant modes are stemmed from the free vibration modes, where the mirrors deform most naturally and the deformations require fewer loads. According to the principle of the Saint-Venant's principle, 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 the magnitude as the supporting forces [9]. In other words, the elastic deformation of modes, which can be produced in vibrations, also can be produced with the actuators of the mirror supporting. Compared with the resonant modes, the standard Zernike polynomials are man-made (not physical, in a purely mathematical sense), and some aberrations represented by the standard Zernike polynomials are not easy to compensate. It implies that the EMCTP are more effective to correct aberrations in adptive optics or active optics. The analytic solutions of the RMCTP modes are Bessel polynomials. In mathematics, the Bessel polynomials are an orthogonal sequence of the polynomials. The orthogonality had been proved by its inventor [10] and other researchers [11,12]. Besides, the elastic modes of a general circular plate belong to resonance modes, whose orthogonality has been universally acknowledged in mechanics [8,13]. Thus, the resonant modes after normalization, like the standard Zernike polynomials, are also always orthogonal within a unit pupil.

 figure: Fig. 3.

Fig. 3. Comparative studies of the mode shapes of the EMCTP and the standard Zernike polynomials. (a) a comparative study of the mode shapes of w01 and Z4; (b) a comparative study of the mode shapes of w11cosφ and Z8; (c) a comparative study of the mode shapes of w21cos2φ and Z6; (d) a comparative study of the mode shapes of w31cos3φ and Z10; (e) a comparative study of the mode shapes of w41cos4φ and Z14;(f) a comparative study of the mode shapes of w51cos5φ and Z20.

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4. Application of the EMCTP

The orthogonal resonant modes, derived from a general circular thin plate, is expected to be more easily extended to be applied to the optical systems where the Zernike polynomials are currently used to represent the optical aberrations. As an application example, here the EMCTP is used to simulate the compensations of some low order aberrations of the active optics system of the 2.5 m-wide field survey telescope (WFST). The conceptual design model of the WFST is shown in Fig. 4(a). Based on an advanced primary focus system, the novel optical design will enable the telescope to precisely survey a wide swathe of sky at wide wave bands. The focal plane is equipped with a 0.9 gigapixel mosaic CCD camera, allowing the entire northern hemisphere to be surveyed every three nights. When completed in 2022, the WFST will be the most advanced of its type in the northern hemisphere, gathering valuable observatory data to monitor astronomical events efficiently.

 figure: Fig. 4.

Fig. 4. (a)The conceptual design model of the WFST;(b) The support system of the primary mirror

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The mirror axial support system comprises of 54 pneumatic linear actuators, which are used to adjust the deformation shape of the reflective surface of the primary mirror to compensate the optical aberrations [see Fig. 4(b) and Fig. 5(b)].

 figure: Fig. 5.

Fig. 5. The finite element model (FEM) of the primary mirror of the WFST. (a) The mesh of the FEM; (b) The 54 axial supports points of the primary mirror of the WFST.

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Figure 5 shows the finite element model of primary mirror, which is used in the simulation of the corrections of some low order aberrations of the active optics system of the WFST. The primary mirror thickness is h=120.00 mm, the inner diameter ϕ1=1000.00 mm, and the outer diameter ϕ2 = 2500.00 mm. From Fig. 4, the central obscuration is produced by the camera system in primary focus, and the diameter of the central obscuration is a little less than the inner diameter of primary mirror. In the FEM, Cer-Vit material is chosen to be the material of the primary mirror, and its density, Young’s modulus, and Poisson ratio are respectively 2530 kgm-3, 91GPa, and 0.24. The primary mirror’s axial support design is 54 supporting points [see Fig. 5(b)]. During the process of the simulations of the active optics corrections, the FEM model is statically determined by supporting on three mirror points①, ② and ③, which are in the inner ring and form an equilateral triangle. At each of the three points we only fix two degrees (φ, z) in the cylindrical coordinate system. For this reason, the support is statically determinate. The remaining 51 supporting points are loaded by ${\{ {f^S}\} _r}$(see Eq. (7)). The equilibrium of the remaining 51 points, whose forces and moments are in balance, have been considered. The values of r1, r2 and r3 are respectively 619.23 mm, 874.67 mm and1125.00 mm, and the values of φ1, φ2 and φ3 are respectively π/12, π/18 and π/24 [14,15].

4.1 A. The Optical Aberration Compensation Algorithm (OACA)

Here, the aberration compensations are simulated the optical aberration compensation algorithm (OACA) [16].The task of a primary mirror for active optics system or a deformable mirror for adaptive optics system is the removal of the arbitrary aberrations by the elastic deformation of the mirrors in the most effective way, which means the target is to minimize the residual error with minimum loads and least computational effort [2,7]. Thus, the optical aberrations compensation usually can be treated as a general optimization problem.

$$\begin{array}{l} \prod = \min \prod (\gamma ),\textrm{ }\gamma \textrm{ = [}{\gamma _1},{\gamma _2},{\gamma _3}, \cdot{\cdot} \cdot ,{\gamma _m}\textrm{]}\\ \textrm{s}\textrm{.t}\textrm{.}\\ \gamma _i^L \le {\gamma _i} \le \gamma _i^U(i = 1,2,3, \cdot{\cdot} \cdot ,m)\\ g_j^L \le {g_j} \le g_j^U(j = 1,2,3, \cdot{\cdot} \cdot ,{n_1})\\ h_j^L \le {h_j} \le h_j^U(j = 1,2,3, \cdot{\cdot} \cdot ,{n_2})\\ w_j^L \le {w_j} \le w_j^U(j = 1,2,3, \cdot{\cdot} \cdot ,{n_3}) \end{array}$$
where Π denotes the objective function, γ is the design variable, and g, h, and w denote state variables. $\gamma _i^U$, $g_j^U$, $\textrm{ }h_j^U$ and $\textrm{ }w_j^U$ denote respectively the upper limits of γ, g, h and w, while $\gamma _i^L$, $g_j^L$, $h_j^L$ and $\textrm{ }w_j^L$ denote respectively their lower limits. m, n1, n2 and n3 denote respectively the numbers of γ, g, h and w .

The vector of the corrective forces ${\{ f\} _s}$ of the primary mirror can be obtained with the least square method [17]:

$${\{ f\} _s} ={-} {([{\mathbf C}]_{r \times s}^\textrm{T}{[{\mathbf C}]_{r \times s}})^{ - 1}}[{\mathbf C}]_{r \times s}^\textrm{T}{\{ \delta \} _r}$$
where ${\{ \delta \} _r}$, an r-dimensional vector, denotes the deformations shapes of the reflective surface of the primary mirror; ${\{ f\} _s}$, an s- dimensional vector, denotes the corrective forces of the actuators of the supporting points of the primary mirror, where s=54 (see Fig. 4(b) and Fig. 5(b)); ${[{\mathbf C}]_{r \times s}}$ denotes the compliance matrix between ${\{ f\} _s}$ and ${\{ \delta \} _r}$. However, ${\{ f\} _s}$, given by Eq. (5), is usually not the most appropriate corrective forces for the aberration compensations [18]. Thus, the damp least square method is introduced to obtain the best corrective forces. According to the damp least square method, Eq. (5) can be rewritten as:
$${\{ {f^\eta }\} _s} ={-} {([{\bf C}]_{r \times s}^\textrm{T}{[{\bf C}]_{r \times s}} + p{[{\bf I}]_{s \times s}})^{ - 1}}[{\bf C}]_{r \times s}^\textrm{T}{\{ \delta \} _r},$$
where p denotes a damping factor, $[{\mathbf I}]$ denotes a unit matrix. Here ${\{ {f^\eta }\} _s}$ given by Eq. (6), is used to replace ${\{ f\} _s}$. Here p serves as a design variable in the optimization algorithm. After optimization, we can obtain the best value of p. According to the damp least square method, p must be a positive value. In Eq. (6), the value of p includes two components: a linear component and a nonlinear component. In order to separate the linear component from p, a linear scale factor is introduced, as shown in the following equation
$$\{ {f^S}\} = \kappa \{ {f^\eta }\} ,$$
where $\kappa $ denotes the linear scale factor. $\kappa $ and p act as the two design variables (optimization factors) during the simulating of the aberration corrections of the active optics system. With Eq. (7), we can obtain the corrective force vector $\{ {f^S}\} $, which is used as the corrective force input to be applied to support points of the finite element model in the simulations. The resulting displacement field in mirror face defines the deformation shapes vector $\{ \Delta \} $, which should be equal to $- \{ \delta \} $ in an ideal state but not in practice. Thus, the residual displacements can be obtained:
$$\{ {w_i}{\} = \{ }{\delta _i}{\} + \{ }{\Delta _i}{\} ,}$$
where i denotes the i-th point of the reflective surface of the primary mirror.

For a paraboloidal shape of primary mirror without the residual displacements, the reflective surface meets the following equation [19]:

$$x_{}^2 + {y^2} = 4F(z + c),$$
where F and c denote respectively the focal length and the vertex z coordinate. With the residual displacements, the surface points deviate from the previous positions. The reflective surface of the new status is as follows:
$$x_1^2 + y_1^2 = 4F({z_1} + c),$$
where the focal distance F, which is fixed, is equal to that of Eq. (9). For every point of the reflective surface, the direction cosines of the surface normal can be expressed as:$2\cos {\theta _1} ={-} {x_i}/\sqrt {F(F + {\textrm{z}_i} + c)} ,2\cos {\theta _2} ={-} {y_i}/\sqrt {F(F + {z_i} + c)} ,\cos {\theta _3} ={-} F/\sqrt {F(F + {z_i} + c)} .$ The displacement of the i-th point of the reflective surface is denoted by (ui, vi, wi), and the distance to the best fitted paraboloid is indicated as σi, we can get [19]:
$$x - ({x_i} + {u_i}) ={\pm} {\sigma _i}\cos {\theta _1},\textrm{ }y - ({y_i} + {v_i}) ={\pm} {\sigma _i}\cos {\theta _2},z - ({z_i} + {w_i}) ={\pm} {\sigma _i}\cos {\theta _3}.$$

The geometrical RMS distance surface error can be expressed as the following equation [19]

$$RM{S_g} = \sqrt {(\sum\limits_{i = 1}^r {\sigma _i^2)/r} } .$$

On the basis of Eq. (12), we can obtain the effective surface error (the residual half path length error) as [19]:

$$RM{S_e} = \sqrt {\frac{1}{r}\sum\limits_{i = 1}^r {s_i^2} } ,\textrm{ }{s_i} = {\sigma _i}\cos {\psi _i},$$
where ψ denotes the angle between the reflective surface axis direction and the surface normal direction. According to Eq. (4), here RMSe serves as the objective function, RMSg serves as the state variable, and p and κ act as the design variables. Therefore, we can obtain
$$\Pi = RM{S_e} = \sqrt {(\sum\limits_{i = 1}^r {s_i^2)/r} } ,\quad \textrm{[}{\gamma _1},{\gamma _2}]\textrm{ = [}p,\kappa ],\quad g = RM{S_g} = \sqrt {(\sum\limits_{i = 1}^r {\sigma _i^2)/r} } .$$

4.2 B. Optimization results

Because the defocusing aberrations and the astigmatism aberrations are the commonest aberrations of the active optics telescope, here we choose the compensation simulations of ${w_{\textrm{01}}}(\rho )\cos (\textrm{0})$, ${w_{\textrm{21}}}(\rho )\cos (\textrm{2}\varphi )$, ${w_{\textrm{21}}}(\rho )\sin (\textrm{2}\varphi )$, ${w_{31}}(\rho )\cos (3\varphi )$, and ${w_{\textrm{31}}}(\rho )\sin (\textrm{3}\varphi )$, whose expressions are written as following:

$${w_{01}}\cos (0) = {K_{01}}[{A_{1,0}}{J_0}({\chi _1}\rho ) + {A_{3,0}}{I_0}({\chi _1}\rho )]\cos (0).$$
$${w_{21}}\cos (2\varphi ) = {K_{21}}[{A_{1,2}}{J_2}({\chi _1}\rho ) + {A_{3,2}}{I_2}({\chi _1}\rho )]\cos (2\varphi ).$$
$${w_{21}}\sin (2\varphi ) = {K_{21}}[{A_{1,2}}{J_2}({\chi _1}\rho ) + {A_{3,2}}{I_2}({\chi _1}\rho )]\sin (2\varphi ).$$
$${w_{31}}\cos (3\varphi ) = {K_{31}}[{A_{1,3}}{J_3}({\chi _1}\rho ) + {A_{3,3}}{I_3}({\chi _1}\rho )]\cos (3\varphi ).$$
$${w_{31}}\sin (3\varphi ) = {K_{31}}[{A_{1,3}}{J_3}({\chi _1}\rho ) + {A_{3,3}}{I_3}({\chi _1}\rho )]\sin (3\varphi ).$$

In the following, the five aberrations, represented by the EMCTP modes, are compensated with the optical aberrations compensation algorithm (OACA). The simulations are based on the hypothesis that the start errors are 30nm, which is about 1/10 of λ=320 nm.

Figure 6(b), Fig. 7(b), Fig. 8(b), Fig. 9(b), and Fig. 10(b) have respectively shown the mode shapes of ${w_{\textrm{01}}}(\rho )\cos (\textrm{0})$, ${w_{\textrm{21}}}(\rho )\cos (\textrm{2}\varphi )$, ${w_{\textrm{21}}}(\rho )\sin (\textrm{2}\varphi )$, ${w_{\textrm{31}}}(\rho )\cos (\textrm{3}\varphi )$, and ${w_{\textrm{31}}}(\rho )\sin (\textrm{3}\varphi )$ of the primary mirror with the start errors of 30 nm. Figure 6(a), Fig. 7(a), Fig. 8(a), Fig. 9(a), and Fig. 10(a) show the dependence of the residual errors RMSe upon p and κ for the ${w_{\textrm{01}}}(\rho )\cos (\textrm{0})$, ${w_{\textrm{21}}}(\rho )\cos (\textrm{2}\varphi )$, ${w_{\textrm{21}}}(\rho )\sin (\textrm{2}\varphi )$, ${w_{\textrm{31}}}(\rho )\cos (\textrm{3}\varphi )$, and ${w_{\textrm{31}}}(\rho )\sin (\textrm{3}\varphi )$ respectively. From the five figures, we can observe that the min RMSe of the above five mentioned modes are respectively 4.20 nm, 1.96 nm, 1.26 nm, 3.91 nm and 4.53 nm, which have been stamped in the above figures. Figure 6(c), Fig. 7(c), Fig. 8(c), Fig. 9(c), and Fig. 10(c)show the residual displacement errors RMSe of the five kinds of the aberrations after one time of the correction with the OACA. From the figures, RMSe = 0.013 λmin (λmin=320) of ${w_{01}}$, RMSe = 0.006λmin (λmin=320) of ${w_{21}}\textrm{cos}2\varphi$, RMSe = 0.004λmin (λmin=320) of ${w_{21}}\sin 2\varphi$, RMSe = 0.012 λmin (λmin=320) of ${w_{31}}\textrm{cos}3\varphi$ and RMSe = 0.014 λmin (λmin=320) of ${w_{31}}\sin 3\varphi$ are obtained after one time of the correction with the OACA. The results have shown that the five kinds of the aberrations could be compensated effectively by the OACA. The above results of the aberration corrections are summarized in Table 1.

 figure: Fig. 6.

Fig. 6. The aberration corrections of w01. (a) The dependence of the residual errors RMSe of w01 on p and κ ;(b) The aberration of Z4 with start errors 30 nm before correction;(c) The residual errors RMSe of w01 after one time of the correction with the OACA is 4.20 nm.

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 figure: Fig. 7.

Fig. 7. The aberration corrections of w21cos2φ. (a) The dependence of the residual errors RMSe of w21cos2φ on p and κ; (b) The aberration of w21cos2φ with start errors 30 nm before correction; (c) The residual errors RMSe of w21cos2φ after one time of the correction with the OACA is 1.96 nm.

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 figure: Fig. 8.

Fig. 8. The aberration corrections of w21sin2φ. (a) The dependence of the residual errors RMSe of w21sin2φ on p and κ; (b) The aberration of w21sin2φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w21sin2φ after one time of the correction with the OACA is 1.23 nm.

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 figure: Fig. 9.

Fig. 9. The aberration corrections of w31cos3φ. (a) The dependence of the residual errors RMSe of w31cos3φ on p and κ ; (b) The aberration of w31cos3φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w31cos3φ after one time of the correction with the OACA is 3.91 nm.

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 figure: Fig. 10.

Fig. 10. The aberration corrections of w31sin3φ. (a) The dependence of the residual errors RMSe of w31sin3φ on p and κ; (b) The aberration of w31sin3φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w31sin3φ after one time of the correction with the OACA is 4.53 nm.

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Tables Icon

Table 1. The results of the aberration corrections with the EMCTP.

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Besides, a comparative study of the EMCTP and the standard Zernike polynomials is presented. Other being equal (FEA model, etc), Z4, Z5, Z6, Z9 and Z10 are used separately to validate ${w_{01}}$, ${w_{21}}\textrm{cos}2\varphi$, ${w_{21}}\sin 2\varphi$, ${w_{31}}\cos 3\varphi$ and ${w_{31}}\sin 3\varphi$ quantitatively.

The expressions of the standard Zernike polynomials Z4, Z5, Z6, Z9 and Z10 are shown as following:

$$\begin{array}{l} Z4 = \sqrt 3 (2{\rho ^2} - 1),\textrm{ }Z5 = \sqrt 6 {\rho ^2}\sin (2\varphi ),\textrm{ }Z6 = \sqrt 6 {\rho ^2}\cos (2\varphi ),\\ Z9 = \sqrt 8 (3{\rho ^3} - 2\rho )\sin (3\varphi ),\textrm{ }Z10 = \sqrt 8 (3{\rho ^3} - 2\rho )\cos (3\varphi ). \end{array}$$

The results have shown that the residual errors, after a compensation by the EMCTP, are much better than those of the circular Zernike polynomials (see Table 1 and Table 2). The results have suggested that compared with the standard Zernike polynomials, the EMCTP can cut the most of the limited values of the residual half path length errors obtained by over half. Moreover, the values of the elements of the ${\{ {f^S}\} _s}$ of the EMCTP are smaller than those of the standard Zernike polynomials (see Table 1 and Table 2). This implies that there are two advantages of using EMCTP: (a) the smaller magnitude of the corrective forces required, which meants that the use of the actuators with a smaller range and better resolution/repeatability could be allowed, and a reduction of the random force errors that limit the quality of the correction can be achieved; (b) reduced print-through of the high-spatial-frequency bumps on the mirror surface, which spreads light widely in the image. Here, the comparing the EMCTP and the standard Zernike polynomials on effectiveness is also presented (see Fig. 11). The effectiveness metric of each mode, which is defined as dividing the RMS surface change by the corresponding RMS of the actuator forces, refers to the approach for comparing correction modes which was proposed by Dr. Jerry Nelson for TMT [16]. In this paper, the effectiveness of w21cos2φ, which is the highest, serves as the criterion. Thus, the effectiveness of w21cos2φ is set as 100%, and those of other modes and Zernike polynomials the ratios of their values (nm/N) to the value (nm/N) of w21cos2φ. From Fig. 11(a), we can observe that the effectiveness of the modes of the EMCTP is much larger compared with that of the standard Zernike polynomials. The EMCTP, where the mirrors deform most naturally and the deformations require fewer loads, is stemmed from the minimum-energy modes of the vibration of the mirror and easier to control. That's the primary reason the EMCTP is much more effective to compensate the aberrations by comparison with the standard Zernike polynomials.

 figure: Fig. 11.

Fig. 11. (a) Comparison between the EMCTP and the Standard Zernike polynomials on effectiveness; (b) Comparison between the EMCTP and the annular Zernike polynomials on effectiveness.

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Moreover, a comparative study of the EMCTP and the annular Zernike polynomials is also presented (see Fig. 11(b) and Table 3). The expressions of the annular Zernike polynomials Z4, Z5, Z6, Z9 and Z10 are shown as following [16,20]:

$$\begin{array}{l} Z4 = \sqrt 3 ( - 2{\rho ^2} + {\varsigma ^2} + 1)/({\varsigma ^2} - 1),\textrm{ }Z5 = \sqrt 6 {r^2}\sin 2\varphi /\sqrt {{\varsigma ^4} + {\varsigma ^2} + 1} ,\\ Z6 = \sqrt 6 {r^2}\cos 2\varphi /\sqrt {{\varsigma ^4} + {\varsigma ^2} + 1} ,\textrm{ }Z9 = 2\sqrt 2 {r^3}\sin 3\varphi /\sqrt {{\varsigma ^6} + {\varsigma ^4} + {\varsigma ^2} + 1} ,\textrm{ }\\ Z10 = 2\sqrt 2 {r^3}\cos 3\varphi /\sqrt {{\varsigma ^6} + {\varsigma ^4} + {\varsigma ^2} + 1} . \end{array}$$

Tables Icon

Table 2. The results of the aberration corrections with the standard Zernike polynomials.

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From Fig. 11(b), we can see that the effectiveness of the modes of the EMCTP is also superior to that of the annular Zernike polynomials. From Table 2 and Table 3, the Remaining RMSs (nm) of Z5, Z6 and Z10 of the annular Zernike polynomials, which are respectively a little better than those of Z5, Z6 and Z9 of the standard Zernike polynomials, are larger than those of the corresponding modes of the EMCTP.

Tables Icon

Table 3. The results of the aberration compensation by the annular Zernike polynomials [16].

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5. Conclusion

In summary, the resonant modes of the circular thin plates (EMCTP) have been proposed to represent the aberrations of adaptive (active) optics system. The physical properties of a circular plate on resonant modes have been studied with an analytic dynamics model of the thin plate theory. The results have shown that the mode shapes resemble those of the standard Zernike polynomials, and there is almost a one-to-one match between each resonant mode and each standard Zernike polynomials. Besides, the results obtained by the analytic method have been demonstrated by the finite element method. The study has shown that the results of the analytic method are in agreement with those of FEM.The simulations of the aberration corrections of the active optics system with the EMCTP also have been presented. Serving as an example, the compensations of some low order aberrations of the 2.5 m-wide field survey telescope (WFST) have been performed with the EMCTP, the standard Zernike polynomials and the annular Zernike polynomials. The quantitative analysis results have shown that the EMCTP can not only be used instead of both the standard Zernike polynomials and the annular Zernike polynomials, but also more effective to compensate the aberrations compared to the two kinds of the Zernike polynomials.

Appendix. A detailed theoretical derivation of the EMCTP

The EMCTP are derived from a cylindrical coordinate system, which is shown in Fig. 1(a). The z-direction is normal to the circular thin plate surface, and ρ-direction and φ-direction are radial direction and circumferential direction, respectively. Assume that is h/a<<1, and we can obtain the strains with the deflection ${u_z}(\rho )$ of the circular thin plate:${\varepsilon _\rho } ={-} z{u_z}_{,\rho \rho }$, ${\varepsilon _\varphi } ={-} z{u_{z,\rho }}/\rho - z{u_{z,\varphi \varphi }}/{\rho ^2}$ and ${\varepsilon _{\rho \varphi }} ={-} 2{({u_{z,\varphi }}/\rho )_{,\rho }}$ . From the equations of the strains, we can obtain the stresses :${\tau _\rho } = E({\varepsilon _\rho } + \nu {\varepsilon _\varphi })/(1 - {\nu ^\textrm{2}})$, ${\tau _\varphi } = E({\varepsilon _\varphi } + \nu {\varepsilon _\rho })/(1 - {\nu ^\textrm{2}})$ and ${\tau _{\rho \varphi }} = E{\varepsilon _{\rho \varphi }}/[2(1 + \nu )]$, where $E$ and $\nu$ are the Young’s modules and Poisson’s ratio of the thin plate, respectively. By integrating principal stresses over the thickness direction of the circular thin plate, the bending moments per unit length can be obtained [7]

$$\begin{array}{l} {M_\rho } = \int_{ - h/2}^{h/2} {{\tau _\rho }zdz} ={-} D[{u_{z,\rho \rho }} + \nu ({u_{z,\varphi \varphi }}/{\rho ^2} + {u_{z,\rho }}/\rho )],\\ {M_\varphi } = \int_{ - h/2}^{h/2} {{\tau _\varphi }zdz} ={-} D[{u_{z,\varphi \varphi }} + \nu {u_{z,\rho \rho }} + {u_{z,\rho }}/\rho )],\textrm{ }\\ {M_{\rho \varphi }} = \int_{ - h/2}^{h/2} {{\tau _{\rho \varphi }}zdz} ={-} D(1 - \nu ){({u_{z,\varphi }}/\rho )_{,\rho }},\textrm{ }D = 3{h^3}E/[12(1 - {\nu ^2})]. \end{array}$$

The transverse shear forces per unit length are obtained through the equation [13]: ${Q_\rho } = {M_{\rho ,\rho }} + {M_{\rho \varphi ,\varphi }}/\rho + ({M_\rho } - {M_\varphi })/\rho ,\quad {Q_\varphi }\textrm{ = }{M_{\rho \varphi ,\rho }} + {M_{\varphi ,\varphi }}/\rho + 2{M_{\rho \varphi }}/\rho \textrm{.}$ Thus, the effectively transverse shear forces per unit length can be obtained [21]:${V_\rho } = {Q_\rho } + \rho {M_{\rho \varphi ,\varphi }}$ and ${V_\varphi } = {Q_\varphi } + {M_{\rho \varphi ,\rho }}$ .The equation of motion in slim plate takes the following form:${Q_\rho }/\rho + {Q_{\rho ,\rho }}\textrm{ + }{Q_{\varphi ,\varphi }}/\rho = \xi {\partial ^2}{u_z}/\partial {t^2}.$ where $\xi = {\rho _z}h$. The equation of motion could be simplified as [8] :$- D{\nabla ^2}{\nabla ^2}{u_z} = \xi {\partial ^2}{u_z}/\partial {t^2}$, where ${\nabla ^\textrm{2}}\textrm{ = }{\partial ^\textrm{2}}/\partial {\rho ^2} + (\partial /\partial \rho )/\rho + ({\partial ^2}/\partial {\varphi ^2})/{\rho ^2}$. For harmonic motions we use the complex notation:${u_z}(\rho ,\varphi ,t) = Re \{ {U_z}(\rho ,\varphi )exp (i\omega t)\} .$ Thus, the simplified equation of motion can be rewritten as [8]$D{\nabla ^2}{\nabla ^2}{U_z} - {\omega ^2}\xi {U_z} = 0$, whose solutions characterized by a radial and an azimuthal part are assumed to be of the type [7]:

$${U_z}(\rho ,\varphi ) = {w_n}(\rho )\cos (n\varphi )\textrm{ n = rotational symmetry}$$

Substituting Eq. (23)into the $D{\nabla ^2}{\nabla ^2}{U_z} - {\omega ^2}m{U_z} = 0$, we can obtain [7]:

$$D\nabla _\rho ^2\nabla _\rho ^2{w_n}(\rho ) - {\omega ^2}\xi {w_n}(\rho ) = 0$$
where $\nabla _\rho ^2 = {\partial ^2}/\partial {\rho ^2} + (\frac{\partial }{{\partial \rho }})/\rho - {n^2}/{\rho ^2}$ . The general solution to equation(24) is written as [7]
$${w_n}(\rho ) = {A_{1,n}}{J_n}(\chi \rho ) + {A_{2,n}}{Y_n}(\chi \rho ) + {A_{3,n}}{I_n}(\chi \rho ) + {A_{4,n}}{K_n}(\chi \rho ).$$
where ${\chi ^\textrm{4}}\textrm{ = }{\omega ^\textrm{2}}\xi /D$, and ${J_n}\textrm{(}\chi \rho \textrm{)}$, ${Y_n}\textrm{(}\chi \rho \textrm{)}$, ${I_n}\textrm{(}\chi \rho \textrm{)}$ and ${K_n}\textrm{(}\chi \rho \textrm{)}$ are n-order Bessel function of the first and the second kind with argument $\chi \rho $, n-order modified Bessel function of the first and the second kind with argument $\chi \rho $ respectively. Since ${Y_n}\textrm{(}\chi \rho \textrm{)}$ and ${K_n}\textrm{(}\chi \rho \textrm{)}$ at ρ=0 are infinite, we obtain ${A_{2,n}} = {A_{4,n}} = 0$, and the equation can be rewritten as [7]
$${w_n}(\rho ) = {A_{1,n}}{J_n}(\chi \rho ) + {A_{3,n}}{I_n}(\chi \rho ).$$

In this paper, the dynamic boundary conditions of the free vibration are written as:${M_\rho }(a) = 0$ and ${V_\rho }(a) = 0.$ The boundary conditions lead to a set of two homogeneous equations with two unknowns where all the parameters depend on λ. In matrix form [17]

$$\left( \begin{array}{l} {b_{11}}(\chi )\\ {b_{21}}(\chi ) \end{array} \right.\textrm{ }\left. \begin{array}{l} {b_{12}}(\chi )\\ {b_{21}}(\chi ) \end{array} \right)\left( \begin{array}{l} {A_{1,n}}\\ {A_{3,n}} \end{array} \right) = \left( \begin{array}{l} 0\\ 0 \end{array} \right)$$

This is an eigenvalue problem in a nonstandard form. In order to have non-trivial solutions, the values of the determinant of the matrix(27) are equal to 0 [17]:

$$\left|\begin{array}{l} {b_{11}}(\chi )\textrm{ }\\ {b_{21}}(\chi ) \end{array} \right.\left. \begin{array}{l} {b_{12}}(\chi )\\ {b_{22}}(\chi ) \end{array} \right|= 0$$

For each value of n, a series of values of χ can be obtained by the Eq. (28). and the orders of χ corresponds to the modes’ orders m. With substitution of each value of λ into the matrix(27), we can get 2 unknowns with an infinite number of the 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 ${A_{\textrm{1},n}}$. The particular solution ${\bar{u}_{mn}}(\rho )$, where m denotes the order of the mode depending on λ, can be obtained by setting ${A_{\textrm{1},n}}\textrm{ = 1}$. The general normalization solution can be written as [8]

$${w_{mn}}(\rho ) = {K_{nm}}{\bar{w}_{nm}}(\rho )$$
where
$${K_{nm}} = \sqrt {{a^2}/(\beta \int\limits_0^a {{{\bar{w}}_{nm}}(\rho )d\rho } )} ,\textrm{ }\beta \textrm{ = }\left\{ \begin{array}{l} 2\textrm{ n = 0}\\ 1\textrm{ n > 0} \end{array} \right.,{\bar{w}_{nm}} = {A_{1,n}}{J_n}({\chi _m}\rho ) + {A_{3,n}}{I_n}({\chi _m}\rho ).$$
where ${\chi _m}$, means the orders of χ corresponds to the modes’ orders m. Thus, an arbitrary wave-front error $w(\rho ,\varphi )$ can be decomposed as [2]:
$$w(\rho ,\varphi ) = \sum\limits_{n = 0}^\infty {\sum\limits_{m = 1}^\infty {{c_{nm}}{w_{nm}}(\rho )\cos (n\varphi ).} }$$
where n denotes the symmetry, m denotes the order, and cnm is the coefficient of the elastic modes.

Appendix. B A detailed proof of the analytic solutions of EMCTP with FEM

For prove the analytic analysis of the EMCTP, the finite element method (FEM) analysis of the EMCTP is carried out (see Fig. 12 and Fig. 13). The finite element model of a circular thin plate, which has the identical physical materials and identical geometrical parameters, is shown as Fig. 11. Figure 13 shows the results of the mode shape of the EMCTP of FEM. From the Fig. 2 and Fig. 4, the mode shapes of the analytic solutions of Figs. 2(a)–2(f) are respectively match with those of the FEM results of Figs. 13(a)–13(f). The values of resonant frequencies f of wnmcosnφ of both analytic method and FEM, which are listed in Table 1, are dependent on boundary conditions, m, n, geometric parameters and material parameters of the circular thin plate. The analytic value of the resonant frequency f can be obtained from the equation ${\chi ^\textrm{4}}\textrm{ = }{\omega ^\textrm{2}}\xi /D$, when the eigenvalue χ is known. Here $\omega \textrm{ = 2}\pi f$.The FEM value of the resonant frequency f can be extracted directly from the FEM analysis results. From Table 4, we can see that the resonant frequencies of the FEM are also respectively conformed to those of the analytic analysis of the EMCTP. Furthermore, the quantitative comparative studies of the analytic and the FEM solutions of the mode shapes of the EMCTP is presented in Fig. 14. The two curves of per graph in Fig. 14, which shows the dependence of the deformation of z-direction on x axis after normalization, are taken from the corresponding analytic results (see Fig. 2) and the corresponding FEM results (see Fig. 13), respectively. From Fig. 14, we can see that the curve of each mode shape of FEM is exactly the same as that of the analytic method, and two curves converge to a curve. Thus, the analytic solutions of the mode shapes of the EMCTP are in good agreement with the FEM solutions.

 figure: Fig. 12.

Fig. 12. The finite element model of a circular thin plate.

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 figure: Fig. 13.

Fig. 13. The mode shapes of the EMCTP from the finite element method solutions. (a) the mode shape of w01, whose resonant frequency is 1937Hz; (b) the mode shape of w11cosφ, whose resonant frequency is 4474 Hz; (c) the mode shape of w21cos2φ, whose resonant frequency is 1249 Hz; (d) the mode shape of w31cos3φ, whose resonant frequency is 2871 Hz; (e) the mode shape of w41cos4φ, whose resonant frequency is 4997 Hz; (f) the mode shape of w41cos4φ, whose resonant frequency is 7611 Hz.

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 figure: Fig. 14.

Fig. 14. Comparative studies of the finite element method solutions and the analytic solutions of mode shapes of the EMCTP. (a) w01; (b) w11cosφ; (c) w21cos2φ; (d) w31cos3φ; (e) w41cos4φ;(f) w51cos5φ.

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Table 4. The values of the resonant frequencies f of wnmcosnφ of both analytic method and FEM.

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Funding

Key Research Program of the Chinese Academy of Sciences (ZDBS-LY-SLH019); National Natural Science Foundation of China (11873100, 11403109, 11773084); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019315).

Acknowledgments

This work has been supported by the Key Research Program of the Chinese Academy of Sciences, Grant No.ZDBS-LY-SLH019, and the National Natural Science Foundation of China (Grant Nos. 11873100, 11403109 and 11773084), and the Youth Innovation Promotion Association CAS (Grant No. 2019315). This work has also been supported by 2.5 m Wide Field Survey Telescope (WFST) project of China, which has been started. The authors would like to thank Dr. Qian and Prof. Cheng at Purple Mountain Observatory for providing the conceptual design model of WFST.

Disclosures

The authors declare no conflicts of interest.

References

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3. N. Hubin and L. Noethe, “Active optics, adaptive optics, and laser guide stars,” Science 262(5138), 1390–1394 (1993). [CrossRef]  

4. J. P. Trevino, J. E. Gómez-Correa, D. R. Iskander, and S. Chávez-Cerda, “Zernike vs. Bessel circular functions in visual optics,” Ophthal Physl Opt. 33(4), 394–402 (2013). [CrossRef]  

5. M. Segel, S. Gladysz, and K. Stein, “Optimal modal compensation in gradient-based wavefront sensorless adaptive optics,” in Laser Communication and Propagation through the Atmosphere and Oceans VIII, (International Society for Optics and Photonics, 2019), 111330V.

6. M. Hadipour, M. Tahtali, and A. J. Lambert, “Vibrating membrane mirror concept for adaptive optics,” in Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation II, (International Society for Optics and Photonics, 2016), 99123O.

7. H. Wang, M. Zhang, Y. Zuo, and X. Zheng, “Research on elastic modes of circular deformable mirror for adaptive optics and active optics corrections,” Opt. Express 27(2), 404–415 (2019). [CrossRef]  

8. W. Soedel and M. S. Qatu, “Vibrations of shells and plates,” (Acoustical Society of America, 2005).

9. R. Wilson, F. Franza, and L. Noethe, “Active Optics: I. A System for Optimizing the Optical Quality and Reducing the Costs of Large Telescopes,” J. Mod. Opt. 34(4), 485–509 (1987). [CrossRef]  

10. H. L. Krall and O. Frink, “A new class of orthogonal polynomials: the Bessel polynomials,” Trans. Amer. Math. Soc. 65(1), 100 (1949). [CrossRef]  

11. S. Roman, The umbral calculus (Springer, 2005).

12. E. Grosswald, “Moments and orthogonality on the unit circle,” in Bessel Polynomials (Springer, 1978), pp. 25–33.

13. J. N. Reddy, Theory and analysis of elastic plates and shells (CRC press, 2006).

14. H. Wang, J. Cheng, Z. Lou, Y. Qian, X. Zheng, Y. Zuo, and J. Yang, “Multi-variable H-β optimization approach for the lateral support design of a wide field survey telescope,” Appl. Opt. 55(31), 8763–8769 (2016). [CrossRef]  

15. H. Wang, Z. Lou, Y. Qian, X. Zheng, and Y. Zuo, “Hybrid optimization methodology of variable densities mesh model for the axial supporting design of wide-field survey telescope,” Opt. Eng. 55(3), 035105 (2016). [CrossRef]  

16. H. Wang, D. Yao, Y. Zuo, X. Zheng, and J. Yang, “Study on the application of the free-vibration modes of an annular mirror in the active optics system,” J. Astron. Telesc. Instrum. Syst. 6(1), 019002 (2020). [CrossRef]  

17. R. R. Stoll, Linear algebra and matrix theory (Courier Corporation, 2013).

18. Q. Hu and Q. Yao, The design of astronomical telescope (Tsinghua University press, Beijing, 2013).

19. J. Cheng, The principles of astronomical telescope design (Springer, 2010), Vol. 360.

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Figures (14)
Fig. 1.
Fig. 1. The configuration of a circular thin plate

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Fig. 2.
Fig. 2. The mode shapes of the EMCTP are from the analytic solutions. (a) the mode shape of w01, whose resonant frequency is 2013Hz; (b) the mode shape of w11cosφ, whose resonant frequency is 4631 Hz; (c) the mode shape of w21cos2φ, whose resonant frequency is 1258 Hz; (d) the mode shape of w31cos3φ, whose resonant frequency is 2907 Hz; (e) the mode shape of w41cos4φ, whose resonant frequency is 5087 Hz; (f) the mode shape of w41cos4φ, whose resonant frequency is 7785 Hz.

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Fig. 3.
Fig. 3. Comparative studies of the mode shapes of the EMCTP and the standard Zernike polynomials. (a) a comparative study of the mode shapes of w01 and Z4; (b) a comparative study of the mode shapes of w11cosφ and Z8; (c) a comparative study of the mode shapes of w21cos2φ and Z6; (d) a comparative study of the mode shapes of w31cos3φ and Z10; (e) a comparative study of the mode shapes of w41cos4φ and Z14;(f) a comparative study of the mode shapes of w51cos5φ and Z20.

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Fig. 4.
Fig. 4. (a)The conceptual design model of the WFST;(b) The support system of the primary mirror

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Fig. 5.
Fig. 5. The finite element model (FEM) of the primary mirror of the WFST. (a) The mesh of the FEM; (b) The 54 axial supports points of the primary mirror of the WFST.

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Fig. 6.
Fig. 6. The aberration corrections of w01. (a) The dependence of the residual errors RMSe of w01 on p and κ ;(b) The aberration of Z4 with start errors 30 nm before correction;(c) The residual errors RMSe of w01 after one time of the correction with the OACA is 4.20 nm.

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Fig. 7.
Fig. 7. The aberration corrections of w21cos2φ. (a) The dependence of the residual errors RMSe of w21cos2φ on p and κ; (b) The aberration of w21cos2φ with start errors 30 nm before correction; (c) The residual errors RMSe of w21cos2φ after one time of the correction with the OACA is 1.96 nm.

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Fig. 8.
Fig. 8. The aberration corrections of w21sin2φ. (a) The dependence of the residual errors RMSe of w21sin2φ on p and κ; (b) The aberration of w21sin2φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w21sin2φ after one time of the correction with the OACA is 1.23 nm.

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Fig. 9.
Fig. 9. The aberration corrections of w31cos3φ. (a) The dependence of the residual errors RMSe of w31cos3φ on p and κ ; (b) The aberration of w31cos3φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w31cos3φ after one time of the correction with the OACA is 3.91 nm.

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Fig. 10.
Fig. 10. The aberration corrections of w31sin3φ. (a) The dependence of the residual errors RMSe of w31sin3φ on p and κ; (b) The aberration of w31sin3φ with Start errors 30 nm before correction; (c) The residual errors RMSe of w31sin3φ after one time of the correction with the OACA is 4.53 nm.

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Fig. 11.
Fig. 11. (a) Comparison between the EMCTP and the Standard Zernike polynomials on effectiveness; (b) Comparison between the EMCTP and the annular Zernike polynomials on effectiveness.

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Fig. 12.
Fig. 12. The finite element model of a circular thin plate.

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Fig. 13.
Fig. 13. The mode shapes of the EMCTP from the finite element method solutions. (a) the mode shape of w01, whose resonant frequency is 1937Hz; (b) the mode shape of w11cosφ, whose resonant frequency is 4474 Hz; (c) the mode shape of w21cos2φ, whose resonant frequency is 1249 Hz; (d) the mode shape of w31cos3φ, whose resonant frequency is 2871 Hz; (e) the mode shape of w41cos4φ, whose resonant frequency is 4997 Hz; (f) the mode shape of w41cos4φ, whose resonant frequency is 7611 Hz.

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Fig. 14.
Fig. 14. Comparative studies of the finite element method solutions and the analytic solutions of mode shapes of the EMCTP. (a) w01; (b) w11cosφ; (c) w21cos2φ; (d) w31cos3φ; (e) w41cos4φ;(f) w51cos5φ.

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Tables (4)

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Table 1. The results of the aberration corrections with the EMCTP.

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Table 2. The results of the aberration corrections with the standard Zernike polynomials.

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Table 3. The results of the aberration compensation by the annular Zernike polynomials [16].

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Table 4. The values of the resonant frequencies f of wnmcosnφ of both analytic method and FEM.

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Equations (31)

Equations on this page are rendered with MathJax. Learn more.

w ( ρ , φ ) = ∑ n = 0 ∞ ∑ m = 1 ∞ c n m w n m ( ρ ) cos ⁡ ( n φ ) .
w n m ( ρ ) = K n m w ¯ n m ( ρ )
K n m = a 2 / β ∫ 0 a w ¯ n m ( ρ ) d ρ ,   β  =  { 2  n = 0 1  n > 0 , w ¯ n m = A 1 , n J n ( χ m ρ ) + A 3 , n I n ( χ m ρ ) .
∏ = min ∏ ( γ ) ,   γ  = [ γ 1 , γ 2 , γ 3 , ⋅ ⋅ ⋅ , γ m ] s .t . γ i L ≤ γ i ≤ γ i U ( i = 1 , 2 , 3 , ⋅ ⋅ ⋅ , m ) g j L ≤ g j ≤ g j U ( j = 1 , 2 , 3 , ⋅ ⋅ ⋅ , n 1 ) h j L ≤ h j ≤ h j U ( j = 1 , 2 , 3 , ⋅ ⋅ ⋅ , n 2 ) w j L ≤ w j ≤ w j U ( j = 1 , 2 , 3 , ⋅ ⋅ ⋅ , n 3 )
{ f } s = − ( [ C ] r × s T [ C ] r × s ) − 1 [ C ] r × s T { δ } r
{ f η } s = − ( [ C ] r × s T [ C ] r × s + p [ I ] s × s ) − 1 [ C ] r × s T { δ } r ,
{ f S } = κ { f η } ,
{ w i } = { δ i } + { Δ i } ,
x 2 + y 2 = 4 F ( z + c ) ,
x 1 2 + y 1 2 = 4 F ( z 1 + c ) ,
x − ( x i + u i ) = ± σ i cos ⁡ θ 1 ,   y − ( y i + v i ) = ± σ i cos ⁡ θ 2 , z − ( z i + w i ) = ± σ i cos ⁡ θ 3 .
R M S g = ( ∑ i = 1 r σ i 2 ) / r .
R M S e = 1 r ∑ i = 1 r s i 2 ,   s i = σ i cos ⁡ ψ i ,
Π = R M S e = ( ∑ i = 1 r s i 2 ) / r , [ γ 1 , γ 2 ]  = [ p , κ ] , g = R M S g = ( ∑ i = 1 r σ i 2 ) / r .
w 01 cos ⁡ ( 0 ) = K 01 [ A 1 , 0 J 0 ( χ 1 ρ ) + A 3 , 0 I 0 ( χ 1 ρ ) ] cos ⁡ ( 0 ) .
w 21 cos ⁡ ( 2 φ ) = K 21 [ A 1 , 2 J 2 ( χ 1 ρ ) + A 3 , 2 I 2 ( χ 1 ρ ) ] cos ⁡ ( 2 φ ) .
w 21 sin ⁡ ( 2 φ ) = K 21 [ A 1 , 2 J 2 ( χ 1 ρ ) + A 3 , 2 I 2 ( χ 1 ρ ) ] sin ⁡ ( 2 φ ) .
w 31 cos ⁡ ( 3 φ ) = K 31 [ A 1 , 3 J 3 ( χ 1 ρ ) + A 3 , 3 I 3 ( χ 1 ρ ) ] cos ⁡ ( 3 φ ) .
w 31 sin ⁡ ( 3 φ ) = K 31 [ A 1 , 3 J 3 ( χ 1 ρ ) + A 3 , 3 I 3 ( χ 1 ρ ) ] sin ⁡ ( 3 φ ) .
Z 4 = 3 ( 2 ρ 2 − 1 ) ,   Z 5 = 6 ρ 2 sin ⁡ ( 2 φ ) ,   Z 6 = 6 ρ 2 cos ⁡ ( 2 φ ) , Z 9 = 8 ( 3 ρ 3 − 2 ρ ) sin ⁡ ( 3 φ ) ,   Z 10 = 8 ( 3 ρ 3 − 2 ρ ) cos ⁡ ( 3 φ ) .
Z 4 = 3 ( − 2 ρ 2 + ς 2 + 1 ) / ( ς 2 − 1 ) ,   Z 5 = 6 r 2 sin ⁡ 2 φ / ς 4 + ς 2 + 1 , Z 6 = 6 r 2 cos ⁡ 2 φ / ς 4 + ς 2 + 1 ,   Z 9 = 2 2 r 3 sin ⁡ 3 φ / ς 6 + ς 4 + ς 2 + 1 ,   Z 10 = 2 2 r 3 cos ⁡ 3 φ / ς 6 + ς 4 + ς 2 + 1 .
M ρ = ∫ − h / 2 h / 2 τ ρ z d z = − D [ u z , ρ ρ + ν ( u z , φ φ / ρ 2 + u z , ρ / ρ ) ] , M φ = ∫ − h / 2 h / 2 τ φ z d z = − D [ u z , φ φ + ν u z , ρ ρ + u z , ρ / ρ ) ] ,   M ρ φ = ∫ − h / 2 h / 2 τ ρ φ z d z = − D ( 1 − ν ) ( u z , φ / ρ ) , ρ ,   D = 3 h 3 E / [ 12 ( 1 − ν 2 ) ] .
U z ( ρ , φ ) = w n ( ρ ) cos ⁡ ( n φ )  n = rotational symmetry
D ∇ ρ 2 ∇ ρ 2 w n ( ρ ) − ω 2 ξ w n ( ρ ) = 0
w n ( ρ ) = A 1 , n J n ( χ ρ ) + A 2 , n Y n ( χ ρ ) + A 3 , n I n ( χ ρ ) + A 4 , n K n ( χ ρ ) .
w n ( ρ ) = A 1 , n J n ( χ ρ ) + A 3 , n I n ( χ ρ ) .
( b 11 ( χ ) b 21 ( χ )   b 12 ( χ ) b 21 ( χ ) ) ( A 1 , n A 3 , n ) = ( 0 0 )
| b 11 ( χ )   b 21 ( χ ) b 12 ( χ ) b 22 ( χ ) | = 0
w m n ( ρ ) = K n m w ¯ n m ( ρ )
K n m = a 2 / ( β ∫ 0 a w ¯ n m ( ρ ) d ρ ) ,   β  =  { 2  n = 0 1  n > 0 , w ¯ n m = A 1 , n J n ( χ m ρ ) + A 3 , n I n ( χ m ρ ) .
w ( ρ , φ ) = ∑ n = 0 ∞ ∑ m = 1 ∞ c n m w n m ( ρ ) cos ⁡ ( n φ ) .
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