1.

Introduction

The renowned Zernike polynomials, first proposed by Zernike,¹ are widely used to represent the optics aberrations in many optics systems. However, the research results have suggested that the Zernike polynomials work inefficiently in the dynamic range of the aberration corrections of the active optics and the adaptive optics.² The task of the active optics systems is to correct the optics aberrations in an effective way.³ This implies that a good method for correcting the aberrations is to remove the residual errors with minimum loads and the least computing resource. In our previous work,⁴ the free-vibration modes of an annular mirror (FVMAM) have been proposed to replace the annular Zernike polynomials. The results have shown that the mode shapes of the FVMAM resemble those of the annular Zernike polynomials, and there is almost a one-to-one match between each mode of the FVMAM and each mode of annular Zernike polynomials. Moreover, the qualitative results have shown that the FVMAM is more likely to perform more efficiently in the aberration corrections of the active optics and the adaptive optics. To verify this, we present a quantitative comparative study of the aberration corrections between the FVMAM and the annular Zernike polynomials.

2.

Mathematical Formulation

2.1.

Mathematical Formulation of the FVMAM

In our previous work,⁴ the FVMAM is stemmed from a thin annular plate. An annular mirror made of a glass annular plate is shown in Fig. 1. The thickness of the annular mirror is denoted by $2h$. The outer and the inner diameters of the annular mirror are denoted by $2b$ and $2a$, respectively. The edges of the outer surface $r=b$ and the inner surface $r=a$ are free. This means the following modes of the plate are derived from the free vibration. Since the detailed derivation of the FVMAM had been presented in the previous work,⁴ here the three-dimensional formulas of the FVMAM are given directly. An arbitrary wave-front error $U(r,\varphi )$ can be expanded with the infinite basis terms of the FVMAM:

Eq. (1)

$$U(r,\varphi )=\sum _{n=0}^{\infty }\sum _{m=1}^{\infty }{c}_{nm}{u}_{nm}(r)\cos (n\varphi ),$$

in which the $u_{nm}(r)\cos (n\varphi )$ denotes the orthonormal basis terms of the FVMAM, $m$ denotes the order, $n$ denotes the symmetry, and $c_{nm}$ denotes the coefficient of the modes of the FVMAM. Here

Eq. (2)

$$u_{mn}(r)=K_{nm}{\overline{u}}_{nm}(r),$$

where

Eq. (3)

$$K_{nm}=\sqrt{(b^2-a^2)/(\eta {\int }_a^b{\overline{w}}_{nm}(r)\mathrm{d}r)},\eta =\{\begin{array}{cc}2& n=0\\ 1& n>0\end{array},$$

Eq. (4)

$${\overline{u}}_{mn}(r)=H_{1,n}{J}_n({\lambda }_{m}r)+H_{2,n}{Y}_n({\lambda }_{m}r)+H_{3,n}{I}_n({\lambda }_{m}r)+H_{4,n}{K}_n({\lambda }_{m}r),$$

where $J_n({\lambda }_{m}r)$, $Y_n({\lambda }_{m}r)$, $I_n({\lambda }_{m}r)$, and $K_n({\lambda }_{m}r)$ are the $n$’th-order Bessel function of the first kind with argument ${\lambda }_{m}r$, the $n$’th-order Bessel function of the second kind with the argument ${\lambda }_{m}r$, the $n$’th-order modified Bessel function of the first kind with the argument ${\lambda }_{m}r$, and the $n$’th-order modified Bessel function of the second kind with the argument ${\lambda }_{m}r$, respectively. The four unknowns, $H_{1,n}$ to $H_{4,n}$, could be determined by the boundary conditions. Here, ${\lambda }_m$ is a function of the material parameters, geometric parameters, and natural frequencies of the annular mirror. (For more details see our previous work.⁴)

Fig. 1

The configuration of an annular plate.

2.2.

Aberration Compensation Algorithm

To be easy for a reader to understand, here the brief introduction of the aberration compensation algorithm has been presented. As an application example, the compensations of some low-order aberrations of the 2.5-m Wide-Field Survey Telescope (WFST) with the FVMAM will be presented. The WFST, which will use an advanced active optics control system, has a unique primary focus with a 7-lens corrector system. It provides not only the best image quality over a wide spectrum range from UV to infrared but also over a 3-deg wide field of view. Figure 2 shows the conceptual design model of WFST.⁵ The primary mirror of the WFST is supported by 54 pneumatic actuators (see Fig. 3), which are used to adjust the deformation shape of the reflective surface of the primary mirror to compensate the optical aberrations.

Fig. 2

The conceptual design model of WFST.

Fig. 3

The support system of primary mirror of WFST.

The task of the active optics systems is to correct the optics aberrations in an effective way, and a good method for correcting the aberrations is to remove the residual error with minimum loads and the least computing resource.³ Therefore, the aberration compensation can be served as a general optimization problem, which can be expressed as^5–7

Eq. (5)

$$\begin{gathered} \mathrm{\Omega }=\min \mathrm{\Omega }(\gamma )\xi =[{\xi }_1,{\xi }_2,\dots ,{\xi }_m] \\ \mathrm{s.t.}{\xi }_i^L\le {\xi }_i\le {\xi }_i^U(i=\mathrm{1,2},3,\dots ,m) \\ {\vartheta }_j^L\le {\vartheta }_j(\gamma )\le {\vartheta }_j^U(j=\mathrm{1,2},3,\dots ,n_1) \\ {h}_j^L\le h_j(\gamma )\le h_j^U(j=\mathrm{1,2},3,\dots ,n_2) \\ {w}_j^L\le w_j(\gamma )\le w_j^U(j=\mathrm{1,2},3,\dots ,n_3), \end{gathered}$$

in which $\mathrm{\Omega }$ is the objective function, and $\xi$ denotes the design variable. The terms $\vartheta$, $h$, and $w$ are the state variables. Here ${\xi }_i^U$, ${\vartheta }_j^U$, $h_j^U$, and $w_j^U$ are the upper limits of $\xi$, $\vartheta$, $h$, and $w$, respectively, and ${\xi }_i^L$, ${\vartheta }_j^L$, $h_j^L$, and $w_j^L$ are their lower limits. The terms $m$, $n_1$, $n_2$, and $n_3$ are the numbers of $\xi$, $\vartheta$, $h$, and $w$, respectively.⁷ Here the hybrid optimization algorithm^5,7 will be applied.

The damped least squares method, proposed by Levenberg and Marquardt, is also named Levenberg–Marquardt algorithm.⁸ The corrective forces ${\{f\}}_s$ of the primary mirror can be calculated with the damped least squares method:⁹

Eq. (6)

$${\{f\}}_s=-{({[\mathbf{C}]}_{r\times s}^{\mathrm{T}}{[\mathbf{C}]}_{r\times s}+\mathrm{p}{[\mathbf{I}]}_{s\times s})}^{-1}{[\mathbf{C}]}_{r\times s}^{\mathrm{T}}{\{\delta \}}_r,$$

where ${\{f\}}_s$, ${\{\delta \}}_r$, ${[\mathbf{C}]}_{r\times s}$, $[\mathbf{I}]$, and $p$ denote an $s$-dimensional vector of the corrective forces of the support points of the annular mirror, an $r$-dimensional vector of the displacements of the reflective surface nodes of the annular mirror, a compliance matrix, the unit matrix, and the damping factor, respectively. In this paper, $s=54$ [see Figs. 3 and 4(b)]. Here $p$ will serve as a design variable in the optimization algorithm. After the optimization, we can obtain the best value of $p$. In Eq. (6), the value of $p$ includes two components: the linear component and nonlinear component. To separate the linear component from $p$, a linear scale factor is introduced, as shown in the following equation:

Eq. (7)

$$\{f^S\}=\kappa \{f\}.$$

In this paper, both $\kappa$ and $p$ act as two design variables (the optimization factors) during the simulation of the aberration compensation of the active optics. The corrective forces vector $\{f^S\}$ is used as the corrective forces input to be loaded to the 54 supporting points of the primary mirror. The displacement vector $\{\mathrm{\Delta }\}$ of the reflective surface, produced by $\{f^S\}$, should be equal to $-\{\delta \}$ in an ideal state but not in practice. Thus, the residual displacement can be obtained as follows:

Eq. (8)

$$\{w_i\}=\{{\delta }_i\}+\{{\mathrm{\Delta }}_i\},$$

where $i$ denotes the $i$’th node of the reflective surface of the annular mirror. The paraboloidal shape surface meets the following equation¹⁰ provided that the paraboloidal shape of the primary mirror has on any residual displacements:

Eq. (9)

$$X^2+Y^2=4\mathrm{\Gamma }(Z+c),$$

in which $\mathrm{\Gamma }$ and $c$ are the focal length and the vertex $z$ coordinate, respectively. Provided that there are any residual displacements, the reflective surface of the new status meets the equation as follows:

Eq. (10)

$$X_1^2+X_1^2=4\mathrm{\Gamma }(Z_1+c),$$

where $c$, which is fixed, is equal to that of Eq. (9). For every node of a reflective surface, the direction cosines of the reflective surface normal can be written as

Eq. (11)

$$\begin{gathered} 2\cos {\phi }_1=-\frac{X_i}{\sqrt{\mathrm{\Gamma }(\mathrm{\Gamma }+Z_i+c)}}, \\ 2\cos {\phi }_2=-\frac{Y_i}{\sqrt{\mathrm{\Gamma }(\mathrm{\Gamma }+Z_i+c)}}, \\ 2\cos {\phi }_3=-\frac{2\mathrm{\Gamma }}{\sqrt{\mathrm{\Gamma }(\mathrm{\Gamma }+Z_i+c)}}. \end{gathered}$$

The displacement of the reflective surface node $i$ is denoted with $(u_i,v_i,w_i)$, and the distance to this best-fitted paraboloid is denoted with ${\sigma }_i$, we can obtain¹⁰

Eq. (12)

$$\begin{gathered} X-(X_i+u_i)=\pm {\sigma }_i\cos {\phi }_1, \\ Y-(Y_i+v_i)=\pm {\sigma }_i\cos {\phi }_2, \\ Z-(Z_i+w_i)=\pm {\sigma }_i\cos {\phi }_3. \end{gathered}$$

Fig. 4

The FEM of the annular primary mirror of the WFST. (a) The mesh of the FEM. (b) The 54 axial supports points of the primary mirror of the WFST.

We can get the geometrical RMS distance surface error by the following equation:¹⁰

Eq. (13)

$$RM{S}_g=\sqrt{\frac{1}{r}\sum _{i=1}^r{\sigma }_i^2}.$$

Further, we can acquire the effective residual surface error (the residual half path length error) as follows:¹⁰

Eq. (14)

$$\begin{gathered} RM{S}_e=\sqrt{\frac{1}{r}\sum _{i=1}^r{s}_i^2} \\ {s}_i={\sigma }_i\cos {\psi }_i, \end{gathered}$$

where $\psi$ is the angle between the surface normal and the reflective surface axis. According to Eqs. (13) and (14), here $RM{S}_{\mathrm{e}}$ is set as the objective function, $RM{S}_{\mathrm{g}}$ is set as the state variable, and $p$ and $\kappa$ are set as two design variables. Therefore, we can obtain

Eq. (15)

$$\begin{gathered} \mathrm{\Omega }=RM{S}_e=\sqrt{\frac{1}{r}\sum _{i=1}^r{s}_i^2} \\ [{\xi }_1,{\xi }_2,\cdots ,{\xi }_m]=[p,\kappa ] \\ \vartheta =RM{S}_g=\sqrt{\frac{1}{r}\sum _{i=1}^r{\sigma }_i^2}. \end{gathered}$$

In this paper, the hybrid optimization algorithm, proposed by us in previous works,^5,7 has been applied. The hybrid optimization algorithm exploits the complementary merits of both the zero- and first-order optimizations, with the former in global scale and the latter in small scale. Here the convergence criteria are the tolerance errors of $p$, $\kappa$, $RM{S}_{\mathrm{g}}$, and $RM{S}_{\mathrm{e}}$, which are constants of $1\times {10}^{-18}$, $1\times {10}^{-7}$, $1\times {10}^{-18}$, and $1\times {10}^{-18}$, respectively. In this paper, 55 is also set as the maximum of iteration in each optimization process, but the maximum is usually much less than 55 with the hybrid optimization algorithm providing a fast convergence with a high precision.

3.

Numerical Results and Discussion

To illustrate the effectiveness of the FVMAM, we present some numerical results in this section. In following the calculations, we use the compensations of some low-order aberrations of the primary mirror of 2.5-m WFST as the application examples. In all of the following calculations, we choose $2a=1000\mathrm{mm}$, $2b=2500\mathrm{mm}$, and $2h=120\mathrm{mm}$, unless stated otherwise. The material of the mirror is Cer-Vit material, whose mass density, Poisson’s ration, and Young’s modulus are $\rho =2530\mathrm{kg}{\mathrm{m}}^{-3}$, $\nu =0.24$, and $E_{\mathrm{m}}=91\mathrm{GPa}$, respectively.¹⁰ Figure 4(a) shows the finite element model (FEM) of the primary mirror of the WFST. Figure 4(b) shows the axial support design of the primary mirror, which has 54 supporting points (pneumatic actuators).

Because the astigmatism aberrations are the commonest aberrations of the active optics telescope, here we choose the correction simulations of four kinds of the astigmatism aberrations, including the $u_{21}(r)\cos 2\varphi$, $u_{21}(r)\sin 2\varphi$, $u_{31}(r)\cos 3\varphi$, and $u_{31}(r)\sin 3\varphi$, to illustrate the effectiveness of the FVMAM. These modes can be expressed as follows:

In this paper, the initial errors before the optimizing are 30 nm, unless stated otherwise. Figures 5(a), 8(a), 9(a), and 10(a) show the distribution of the initial errors of the four kinds of astigmatism aberrations. Figures 6, 7, 11, and 12 show the relationships between the objective function ${RMS}_{\mathrm{e}}$ and the design variables ($p$ and $\kappa$). Examining at Figs. 6, 7, 11, and 12, it is apparent that the objective function ${RMS}_{\mathrm{e}}$ is very sensitive to a very slight variation of the design variables ($p$ and $\kappa$). Figures 5(b), 8(b), 9(b), and 10(b) show the minimum residual errors of the four kinds of the astigmatism aberrations after one time of the optimizing, and the minimum residual errors of ${RMS}_{\mathrm{e}}=2.63$, 1.13, 3.71, and 5.22 nm have been stamped on Figs. 6, 7, 11, and 12, respectively. More detailed results of the aberration compensation by the FVMAM have been listed in Table 1. As can be seen from the data in Table 1, 91.23% of the initial errors of $u_{21}(r)\cos 2\varphi$, 96.23% of the initial errors of $u_{21}(r)\sin 2\varphi$, 87.63% of the initial errors of $u_{31}(r)\cos 3\varphi$, and 82.60% of the initial errors of $u_{31}(r)\sin 3\varphi$ could be removed after one time of the correction.

Fig. 5

The initial errors and the minimum residual errors. (a) The distribution of the initial errors of $u_{21}(r)\cos 2\varphi$. (b) The distribution of the minimum residual errors of $u_{21}(r)\cos 2\varphi$ after one time of correction.

Fig. 6

The relationship between the objective function ${RMS}_{\mathrm{e}}$ of $u_{21}(r)\cos 2\varphi$ and the design variables ($p$ and $\kappa$).

Fig. 7

The relationship between the objective function ${RMS}_{\mathrm{e}}$ of $u_{21}(r)\sin 2\varphi$ and the design variables ($p$ and $\kappa$).

Fig. 8

The initial errors and the minimum residual errors. (a) The distribution of the initial errors of $u_{21}(r)\sin 2\varphi$. (b) The distribution of the minimum residual errors of $u_{21}(r)\sin 2\varphi$ after one time of correction.

Fig. 9

The initial errors and the minimum residual errors. (a) The distribution of the initial errors of $u_{31}(r)\cos 3\varphi$. (b) The distribution of the minimum residual errors of $u_{31}(r)\cos 3\varphi$ after one time of correction.

Fig. 10

The initial errors and the minimum residual errors. (a) The distribution of the initial errors of $u_{31}(r)\sin 3\varphi$. (b) The distribution of the minimum residual errors of $u_{31}(r)\sin 3\varphi$ after one time of correction.

Fig. 11

The relationship between the objective function ${RMS}_{\mathrm{e}}$ of $u_{31}(r)\cos 3\varphi$ and the design variables ($p$ and $\kappa$).

Fig. 12

The relationship between the objective function ${RMS}_{\mathrm{e}}$ of $u_{31}(r)\sin 3\varphi$ and the design variables ($p$ and $\kappa$).

Table 1

The detail results of the aberration compensation by the FVMAM.

The mode shapes of the FVMAM resemble those of the annular Zernike polynomials, and there is almost a one-to-one match between each mode of the FVMAM and each mode of annular Zernike polynomials. Here Z5, Z6, Z9, and Z10, which denote the annular Zernike, correspond to $u_{21}(r)\cos 2\varphi$, $u_{21}(r)\sin 2\varphi$, $u_{31}(r)\cos 3\varphi$, and $u_{31}(r)\sin 3\varphi$, respectively (see our previous work⁴). Here Z5, Z6, Z9, and Z10 can be expressed as follows:

Table 2

The results of the aberration compensation by the annular Zernike polynomials.

Fig. 13

Comparison the FVMAM and annular Zernike polynomials on effectiveness.

4.

Summary

The goal of this paper is to present an effective way for the aberration compensation of the active optics. The FVMAM, derived from the elasticity theory, reflecting the natural properties of the physical phenomenon of the resonance, are applied in the diffraction theory of the optical aberrations. Moreover, a quantitative comparative study of the aberration corrections between the FVMAM and the annular Zernike polynomials has been presented. The main conclusion to be drawn from the results of the study is that FVMAM are more effective to correct the aberrations.