Alignment algorithm of nonsymmetric off-axis reflective astronomical telescopes based on the modified third-order nodal aberration theory

Jinxin Wang; Jinxin Wang; Xu He; Jing Luo; Xiaohui Zhang; Tianxiao Xu; Tianxiao Xu

doi:10.1364/OE.453005

1. Introduction

The typical characteristics of nonsymmetric off-axis reflective optical systems discussed in this paper is that there is an offset aperture stop on primary mirror (PM), and tilts and decenters are introduced in optical elements during the design process. The design of such optical systems effectively avoid the problem of aperture obscuration caused by the secondary mirror (SM) and its supporting structure, and greatly increases the effective field of view (FOV) of space astronomical telescopes [1–3]. While maintaining the same optical performance as on-axis astronomical telescopes, as the aperture and FOV of off-axis astronomical telescopes increase, its optical performance has been further improved, such as higher energy-collection efficiency, better diffraction characteristics, higher dynamic range, so on [4]. In addition, the nonsymmetric off-axis reflective astronomical telescopes also have obvious advantages over on-axis telescopes in terms of ellipticity performance [5], which is essential for detecting dark matter and dark energy through weak gravitational lensing measurement. High-precision computer-aided alignment (CAA) algorithm is an important link to realize the key performances of the nonsymmetric off-axis astronomical telescopes. However, for the nonsymmetric off-axis reflective telescopes, the coupling of the degrees of freedom of misalignment of each optical surface is stronger than that of on-axis telescopes, and the variation of aberration fields caused by misalignments is more complicated. This makes it difficult for traditional numerical CAA algorithms [6–8] to decouple the mapping relationship between the misalignments and wave aberrations. Analytical alignment algorithms based on nodal aberration theory (NAT) are dedicated to analyzing the characteristics of aberration fields of misaligned optical systems, and deducing the complex functional relationship between misalignments and aberrations, which is of great significance for the alignment of the nonsymmetric off-axis reflective telescopes.

NAT was first discovered by Shack [9] and then developed from third order to fifth order by Thompson [10–13]. It is a powerful tool for guiding optical alignment of astronomical telescopes. Schmid et al. [14] studied the aberration field characteristics of misaligned Ritchey Chretien astronomical telescopes based on NAT, and showed that ensuring perfect performance on axis alone is insufficient to achieve the alignment of large telescopes. Thompson et al. [15] demonstrated how to use NAT to determine the alignment strategies for three-mirror anastigmatic telescopes. These works provide valuable insights and theoretical guidance for the alignment of astronomical telescopes.

On the basis of the above studies, some researchers began to study the quantitative calculation method of misalignments. Generally, there are mainly two types of misalignments in the fine alignment process of astronomical telescopes, namely axial misalignments [16] and lateral misalignments [17]. The axial misalignments are also called longitudinal misalignments, which refers to the dislocation of optical surfaces along the axial direction. The Lateral misalignments are also called transverse misalignments, which refers to the decenter and tip–tilt of optical surfaces in the lateral direction. Sebag et al. [18] used NAT to study the quantitative calculation method of lateral misalignments, and made an alignment plan for Large Synoptic Survey Telescope (LSST). Gu et al. [19] took an on-axis three-mirror telescope as the research object, and proposed a NAT-based method to calculate lateral misalignments. Zhang et al. [20] calculated the lateral misalignments of an off-axis two-mirror telescope and an off-axis three-mirror telescope based on NAT. However, most of these works still lack the effective support and verification of physical experiments. In addition, these studies ignore the effect of axial misalignments on the aberration fields of optical systems when constructing calculation models.

For on-axis optical systems, axial misalignments mainly affect rotationally symmetric aberrations of systems (e.g. defocus and spherical aberration), and lateral misalignments mainly affect the non-rotationally symmetric aberrations of systems (e.g. astigmatism and coma) [21]. Since these two types of misalignments usually have no coupling effects on the aberration fields of on-axis optical systems, these two types of misalignments can often be dealt with separately during alignment process, which makes the alignment of on-axis telescopes relatively simple and straightforward. However, this is not the case for off-axis telescopes. In off-axis telescopes, axial and lateral misalignments exhibit strong coupling due to the absence of symmetry. The astigmatic and coma aberration fields of the misaligned optical systems are not only effected by lateral misalignments, but also closely related to axial misalignments [16]. Therefore, when constructing solution models of off-axis optical systems, if net aberration fields induced by axial misalignments are not derived and calculated together, it is impossible to accurately solve misalignments. As a result, when there are axial misalignments in off-axis telescopes, the traditional NAT-based misalignment algorithm is invalid, which makes it difficult to apply to actual CAA experiments. In order to align off-axis telescopes that contain both axial and lateral misalignments in practical applications, some new work should be done in this paper.

2. Wave aberration expression of off-axis optical systems with designed tilts and decenters in the presence of both axial and lateral misalignments

In this section, the wave aberration expression of the misaligned off-axis optical systems with designed tilts and decenters is derived, and third-order astigmatic, third-order coma, and third-order spherical net aberration fields induced by axial and lateral misalignments are separated from the total aberration fields.

The wave aberration expression in vector form of the misaligned rotationally symmetric optical systems is an important theoretical basis for deriving the wave aberration expression of the misaligned off-axis optical systems. It can be expressed as [10]

(1)$${W_{on - axis}} = \sum\limits_j {\sum\limits_p^\infty {\sum\limits_n^\infty {\sum\limits_m^\infty {{{({{W_{klm}}} )}_j}} } } } {({{{\vec{H}}_{Aj}} \cdot {{\vec{H}}_{Aj}}} )^p}{({\vec{\rho } \cdot \vec{\rho }} )^n}{({{{\vec{H}}_{Aj}} \cdot \vec{\rho }} )^m},$$

with

(2)$${\vec{H}_{Aj}} = \vec{H} - {\vec{\sigma }_j},$$

where j is the optical surface number, $k = 2p + m$, $l = 2n + m$, and ${W_{klm}}$ represents the aberration coefficients of different types of aberrations. $\vec{\rho }$ represents the normalized pupil vector. ${\vec{H}_{Aj}}$ and $\vec{H}$ respectively represent the normalized effective field vectors before and after the aberration field center shift, as shown in Fig. 1(a), $\vec{\sigma }$ denotes the decenter vector of the center position of the aberration field, which is linearly related to the tilts and decenters of optical surfaces [22].

Fig. 1. Schematic diagram of (a) effective field vectors and aberration field decenter vectors of the misaligned optical system and (b) the pupil vector coordinate transformation of the off-axis system relative to its on-axis parent system.

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It is worth noting that when tilts and decenters are introduced in the design process of optical systems, the aberration field decenter vector $\vec{\sigma }$ consists of two parts and can be expressed as

(3)$$\vec{\sigma } = {\vec{\sigma }^\# } + \vec{\sigma }^{\prime},$$

where ${\vec{\sigma }^\# }$ is caused by designed tilts and decenters, and $\vec{\sigma }^{\prime}$ is caused by lateral misalignments.

For an off-axis optical system with an offset pupil, it can be obtained by decentering the pupil of a parent on-axis system while the other design parameters (e.g. radius, conic constants) of the system remain unchanged. In other words, the pupil of the off-axis system is compressed and offset relative to the pupil of the parent on-axis system. The mathematical relationship between the pupil of the off-axis system and the pupil of the parent on-axis system can be expressed as [20,23]

(4)$$\left. {\begin{array}{c} {\vec{\rho } = M\vec{\rho }^{\prime} + \vec{h}}\\ {M = \frac{r}{R}} \end{array}} \right\},$$

where $\vec{\rho }^{\prime}$ and $\vec{\rho }$ respectively represent the normalized off-axis pupil vector and on-axis pupil vector, as shown in Fig. 1(b). M denotes the pupil scaling factor. r and R respectively denote the pupil radius of the off-axis system and the on-axis system. $\vec{h}$ represents the normalized position change vector of the off-axis pupil center relative to the on-axis pupil center.

Then substituting Eq. (4) into Eq. (1), the wave aberration expression of the misaligned off-axis optical systems with designed tilts and decenters can be expressed as

(5)$$\begin{aligned} &{W_{off - axis}}\\ &= {\sum\limits_j {\sum\limits_p^\infty {\sum\limits_n^\infty {\sum\limits_m^\infty {{{({{W_{klm}}} )}_j}{{({{{\vec{H}}_{Aj}} \cdot {{\vec{H}}_{Aj}}} )}^p}[{({M\vec{\rho }^{\prime} + \vec{h}} )\cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ]} } } } ^n}{[{{{\vec{H}}_{Aj}} \cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ]^m}, \end{aligned}$$

The introduction of ${\vec{H}_{Aj}}$ in Eq. (5) can describe the effect of lateral misalignments on the wave aberrations of off-axis optical systems, but the effect of axial misalignments is not reflected in the Eq. (5). Considering that axial misalignments will cause the aberration coefficient ${W_{klm}}$ to change, Eq. (5) can be modified as

(6)$$\begin{aligned} &{W_{off - axis}}\\ &= {\sum\limits_j {\sum\limits_p^\infty {\sum\limits_n^\infty {\sum\limits_m^\infty {{{({{W_{klm}} + \Delta W_{klm}^d} )}_j}{{({{{\vec{H}}_{Aj}} \cdot {{\vec{H}}_{Aj}}} )}^p}[{({M\vec{\rho }^{\prime} + \vec{h}} )\cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ]} } } } ^n}{[{{{\vec{H}}_{Aj}} \cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ]^m}, \end{aligned}$$

where $\Delta W_{klm}^d$ denotes net change of ${W_{klm}}$ due to the axial misalignments of optical surfaces.

In practical engineering applications, the Fringe Zernike polynomials (hereinafter referred to as Zernike polynomials) is usually used to fit the exit pupil wavefront errors (WFE) of an optical system [24]. Therefore, in order to facilitate the subsequent establishment of the misalignment calculation model, it is necessary to clearly express the dependence of FOVs and misalignments for different aberration types expressed by Zernike polynomials. To this end, combined with [25–27], Eq. (6) is modified as

(7)$${W_{off - axis}} = {\sum\limits_i {\left[ {\sum\limits_k^\infty {\sum\limits_l^\infty {\sum\limits_m^\infty {\sum\limits_a^\infty {\vec{E}_{klm}^{(a )}({{H_x},{H_y}} )\cdot \vec{R}_{klm}^{(a )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )} } } } } \right]} _i}{Z_i}({\rho ,\varphi } ),$$

with

(8)$${\sum\limits_i {\left[ {\sum\limits_k^\infty {\sum\limits_l^\infty {\sum\limits_m^\infty {\sum\limits_a^\infty {\vec{E}_{klm}^{(a )}({{H_x},{H_y}} )\cdot \vec{R}_{klm}^{(a )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )} } } } } \right]} _i} = \sum\limits_i {{C_i}({{H_x},{H_y}} )} ,$$

where ${H_x}$ and ${H_y}$ represent the x component and y component of $\vec{H}$, respectively. ${\sigma _x}$ and ${\sigma _y}$ respectively represent the x component and y component of $\vec{\sigma }$. $\rho$ and $\varphi$ represent the components of $\vec{\rho }$, and a represents the power of $\vec{\sigma }$. $\vec{E}_{klm}^{(a )}({{H_x},{H_y}} )$ is a vector used to describe the FOVs dependence of different aberration types, and $\vec{R}_{klm}^{(a )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )$ is a vector used to describe the misalignments dependence of different aberration types. ${Z_i}({\rho ,\varphi } )$ denotes the i-th term of the Zernike polynomial, and ${C_i}({{H_x},{H_y}} )$ is the corresponding Zernike coefficient. On this basis, Eq. (8) can be further expanded as

(9)$$\sum\limits_i {\sum\limits_k^\infty {\sum\limits_l^\infty {\sum\limits_m^\infty {\left\{ \begin{array}{l} \vec{E}_{klm}^{(0 )}({{H_x},{H_y}} )\cdot \vec{R}_{klm}^{(0 )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )\\ + \vec{E}_{klm}^{(1 )}({{H_x},{H_y}} )\cdot \vec{R}_{klm}^{(1 )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )\\ + \sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )}({{H_x},{H_y}} )\cdot \vec{R}_{klm}^{(a )}({{\sigma_x},{\sigma_y},\Delta W_{klm}^d} )} \end{array} \right\} = } } } } \sum\limits_i {{C_i}({{H_x},{H_y}} )} ,$$

where $\sum\limits_i {\sum\limits_k^\infty {\sum\limits_l^\infty {\sum\limits_m^\infty {\vec{E}_{klm}^{(0 )}} } } } \cdot \vec{R}_{klm}^{(0 )}$ represents the intrinsic residual error of the nominal optical system, which is independent of misalignments but related to with designed tilts and decenters. The remaining polynomial represents the changes in wave aberrations before and after the system is aligned, that is, the net aberrations induced by misalignments. When there are no misalignments in the optical system, the remaining polynomial is zero. Furthermore, $\sum\limits_i {\sum\limits_k^\infty {\sum\limits_l^\infty {\sum\limits_m^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}({a > 2} )} } } }$ does not exist when only third-order astigmatism, third-order coma and third-order spherical aberration are considered. The specific expressions of $\vec{E}_{klm}^{(0 )} \cdot \vec{R}_{klm}^{(0 )}$, $\vec{E}_{klm}^{(1 )} \cdot \vec{R}_{klm}^{(1 )}$, and $\sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}}$ for different Zernike coefficients are show in Table 1, and the derivation process of these expressions is given in Appendix A.

Table 1. The expressions of $\vec{E}_{klm}^{(0 )} \cdot \vec{R}_{klm}^{(0 )}$, $\vec{E}_{klm}^{(1 )} \cdot \vec{R}_{klm}^{(1 )}$, and $\sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}}$ for different Zernike coefficients.

View Table | View all tables in this article

The expressions of $\vec{A}_{klm}^\#$, ${\vec{A}^{\prime}_{klm}}$, $\vec{A}_{klm}^d$, $\vec{B}_{klm}^\#$, ${\vec{B}^{\prime}_{klm}}$, and $\vec{B}_{klm}^d$ in Table 1 are

(10)$$\left. {\begin{array}{c} {{{\vec{A}}^\# }_{klm} = \sum\limits_j {{{({{W_{klm}}} )}_j}{{\vec{\sigma }}^\# }_j} }\\ {{{\vec{A}^{\prime}}_{klm}} = \sum\limits_j {{{({{W_{klm}}} )}_j}{{\vec{\sigma }^{\prime}}_j}} }\\ {\vec{A}_{klm}^d = \sum\limits_j {{{({\Delta W_{klm}^d} )}_j}({\vec{\sigma }_j^\# + {{\vec{\sigma }^{\prime}}_j}} )} }\\ {\vec{B}_{klm}^\# = {{\sum\limits_j {{{({{W_{klm}}} )}_j}({\vec{\sigma }_j^\# } )} }^2}}\\ {{{\vec{B}^{\prime}}_{klm}} = \sum\limits_j {{{({{W_{klm}}} )}_j}[{{{({{{\vec{\sigma }^{\prime}}_j}} )}^2} + 2\vec{\sigma }_j^\# {{\vec{\sigma }^{\prime}}_j}} ]} }\\ {\vec{B}_{klm}^d = \sum\limits_j {{{({\Delta W_{klm}^d} )}_j}[{{{({\vec{\sigma }_j^\# } )}^2} + {{({{{\vec{\sigma }^{\prime}}_j}} )}^2} + 2\vec{\sigma }_j^\# {{\vec{\sigma }^{\prime}}_j}} ]} } \end{array}} \right\}.$$

In Table 1, ${\vec{A}^\# }_{klm,x}$ and ${\vec{A}^\# }_{klm,y}$ are the x component and y component of ${\vec{A}^\# }_{klm}$. ${\vec{A}^{\prime}_{klm,x}}$ and ${\vec{A}^{\prime}_{klm,y}}$ are the x component and y component of ${\vec{A}^{\prime}_{klm}}$. $\vec{A}_{klm,x}^d$ and $\vec{A}_{klm,y}^d$ are the x component and y component of $\vec{A}_{klm}^d$. $\vec{B}_{klm,x}^\#$ and $\vec{B}_{klm,y}^\#$ are the x component and y component of $\vec{B}_{klm}^\#$. ${\vec{B}^{\prime}_{klm,x}}$ and ${\vec{B}^{\prime}_{klm,y}}$ are the x component and y component of ${\vec{B}^{\prime}_{klm}}$. $\vec{B}_{klm,x}^d$ and $\vec{B}_{klm,y}^d$ are the x component and y component of $\vec{B}_{klm}^d$. The x component of $\vec{h}$ is usually regarded as zero because of the physical meaning of the meridian and sagittal directions of $\vec{h}$ is the same in the pupil-offset off-axis systems.

It can be seen from Table 1 that the net Zernike astigmatism (C5/6) induced by misalignments contains three components, i.e., a field-constant component, field-linear component, and field-quadratic component. The expressions of field-constant astigmatism component contain scalars related to lateral and axial misalignments, i.e., ${\vec{A}^{\prime}_{131,x}} + \vec{A}_{131,x}^d, {\vec{A}^{\prime}_{131,y}} + \vec{A}_{131,y}^d, {\vec{B}^{\prime}_{222,x}} + \vec{B}_{222,x}^d, {\vec{B}^{\prime}_{222,y}} + \vec{B}_{222,y}^d,$ and $\sum\limits_j {\Delta W_{040j}^d}$. The expressions of field-linear astigmatism component contain scalars related to lateral and axial misalignments, i.e. ${\vec{A}^{\prime}_{222,x}} + \vec{A}_{222,x}^d, {\vec{A}^{\prime}_{222,y}} + \vec{A}_{222,y}^d,$ and $\sum\limits_j {\Delta W_{131j}^d}$. The expressions of field-quadratic astigmatism component contain only a scalar related to axial misalignments, i.e., $\sum\limits_j {\Delta W_{222j}^d}$. In addition, it can be seen that the net Zernike coma (C7/8) induced by misalignments contains two components, i.e., the field-constant component and field-linear component. Among them, the expressions of field-constant coma component contain scalars related to lateral and axial misalignments, i.e., ${\vec{A}^{\prime}_{131,x}} + \vec{A}_{131,x}^d, {\vec{A}^{\prime}_{131,y}} + \vec{A}_{131,y}^d,$ and $\sum\limits_j {\Delta W_{040j}^d} ,$ and the expressions of field-linear coma component contain only a scalar related to axial misalignments, i.e., $\sum\limits_j {\Delta W_{131j}^d}$.

These results indicate that in misaligned pupil-offset off-axis optical systems, axial and lateral misalignments do not generate new aberration types, but they change the field dependence. The net astigmatism fields and net coma fields induced by misalignments have not only the contribution of lateral misalignments but also the contribution of axial misalignments. In other words, lateral misalignments and axial misalignments are coupled, and the wider the FOV of optical systems, the stronger the coupling effect due to the influence of field-linear astigmatism, field- quadratic astigmatism and field-linear coma. Therefore, when both axial and lateral misalignments exist and the FOV is wide, it is impossible to accurately calculate misalignments without simultaneously deriving and calculating the net aberration fields induced by axial misalignments when constructing the solution models of the off-axis systems.

3. Misalignment calculation model

In this section, the net aberrations induced by misalignments is used to compute misalignments. Therefore, the item that is unrelated to misalignments in Table 1 can be ignored. In this case, Eq. (11) can be obtained by sorting out Table 1.

(11)$$\scalebox{0.8}{$\left[{\begin{array}{@{}c@{}} {\Delta {C_5}}\\ {\Delta {C_6}}\\ {\Delta {C_7}}\\ {\Delta {C_8}}\\ {\Delta {C_9}} \end{array}} \right] = \left[ {\begin{array}{@{}ccccccccc@{}} 0&{{M^2}{h_y}}&{ - {M^2}{H_x}}&{{M^2}{H_y}}&{ - 2{M^2}h_y^2}&{ - {M^2}{h_y}{H_y}}&{\frac{{{M^2}}}{2}({H_x^2 - H_y^2} )}&{\frac{{{M^2}}}{2}}&0\\ { - {M^2}{h_y}}&0&{ - {M^2}{H_y}}&{ - {M^2}{H_x}}&0&{{M^2}{h_y}{H_x}}&{{M^2}{H_x}{H_y}}&0&{\frac{{{M^2}}}{2}}\\ { - \frac{{{M^3}}}{3}}&0&0&0&0&{\frac{{{M^3}}}{3}{H_x}}&0&0&0\\ 0&{ - \frac{{{M^3}}}{3}}&0&0&{\frac{{4{M^3}}}{3}{h_y}}&{\frac{{{M^3}}}{3}{H_y}}&0&0&0\\ 0&0&0&0&{\frac{{{M^4}}}{6}}&0&0&0&0 \end{array}} \right]\left[ {\begin{array}{@{}c@{}} {{{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d}\\ {{{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d}\\ {{{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d}\\ {{{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d}\\ {\sum\limits_j {\Delta W_{040}^d} }\\ {\sum\limits_j {\Delta W_{131}^d} }\\ {\sum\limits_j {\Delta W_{222}^d} }\\ {\vec{B}^{\prime}_{222.x} + \vec{B}_{222,x}^d}\\ {\vec{B}^{\prime}_{222,y} + \vec{B}_{222,y}^d} \end{array}} \right].$}$$

For the convenience of description, Eq. (11) can be expressed as

(12)$$\Delta C = H\Delta X,$$

where $\Delta C$, $H$ and $\Delta X$ are called Zernike coefficient matrix, FOV matrix, and misalignment matrix, respectively. When the number of selected FOV points is greater than or equal to two, the nine elements in the misalignment matrix can be solved according to the overdetermined equations formed by Eq. (12). In principle, when the number of misalignment dimensions is less than or equal to nine, the nine elements in the misalignment matrix can be used to further quantitatively calculate misalignments. However, the functional relationship between the nine elements in the misalignment matrix and misalignments is complicated, and the number of misalignment dimensions is greater than nine for the off-axis three-mirror telescope to be aligned in this paper. Therefore, it is difficult to calculate misalignments through analytical method only by using these nine elements. In order to solve this problem, on the basis of Eq. (12), a method of fitting the partial derivative coefficient matrix of misalignments is proposed. Based on the Taylor Function Multivariate theorem, the i-th (i = 1-9) element in the misalignment matrix can be expressed as

(13)$${X_i}({N + \Delta M} )= \sum\limits_{p = 0}^m {\frac{1}{{p!}}} {\left( {\sum\limits_{q = 1}^n {\Delta {m_q}\frac{\partial }{{\partial {x_q}}}} } \right)^p}{X_i}(N )+ {R_{N,m}}({\Delta M} ),$$

where $N + \Delta M = ({{x_1},{x_2}, \cdots ,{x_n}} )$ represents any condition of n misalignment degrees of freedom of an optical system, and $N = ({{x_{10}},{x_{20}}, \cdots ,{x_{n0}}} )$ represents the initial condition of the n misalignment degrees of freedom. $\Delta M = ({\Delta {m_1},\Delta {m_2}, \cdots \Delta {m_n}} )$ represents the misalignments of n degrees of freedom, and ${R_{N,m}}({\Delta M} )$ is the remainder term. Considering that misalignments of optical surfaces in CAA is usually small, the linear approximation of Eq. (13) and the first-order (i.e., order p = 1) expansion are performed to obtain

(14)$$\begin{aligned} {X_i}({N + \Delta M}) &= {X_i}(N )+ \left( {\sum\limits_{q = 1}^n {\Delta {m_q}\frac{\partial }{{\partial {x_q}}}} } \right){X_i}(N )+ {R_{N,m}}({\Delta M}) \\ &\approx {X_i}(N )+ \sum\limits_{q = 1}^n {\Delta {m_q}\frac{{\partial {X_i}(N )}}{{\partial {x_q}}}} . \end{aligned}$$

Equation (14) can be expressed in matrix form, and it is obtained that

(15)$$\Delta X = \left[ {\begin{array}{c} {{X_1}({N + \Delta M} )}\\ \vdots \\ {{X_i}({N + \Delta M} )}\\ \vdots \end{array}} \right] - \left[ {\begin{array}{c} {{X_1}(N )}\\ \vdots \\ {{X_i}(N )}\\ \vdots \end{array}} \right] = \left[ {\begin{array}{cccc} {\frac{{\partial {X_1}(N )}}{{\partial {x_1}}}}&{\frac{{\partial {X_1}(N )}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {X_1}(N )}}{{\partial {x_n}}}}\\ \vdots & \vdots & \vdots & \vdots \\ {\frac{{\partial {X_i}(N )}}{{\partial {x_1}}}}&{\frac{{\partial {X_i}(N )}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {X_i}(N )}}{{\partial {x_n}}}}\\ \vdots & \vdots & \vdots & \vdots \end{array}} \right]\left[ {\begin{array}{c} {\Delta {m_1}}\\ {\Delta {m_2}}\\ \vdots \\ {\Delta {m_n}} \end{array}} \right].$$

Based on the principle of multiple linear regression, the fitting method of the partial derivative coefficient matrix in Eq. (15) is as follows: firstly, the misalignment threshold range of each misalignment dimensions should be determined according to the initial alignment accuracy of an actual optical system. Within the given misalignment threshold ranges, V sets of misalignments are randomly generated according to a standard uniform distribution, and these misalignments are respectively introduced into the optical design software to represent V misalignment samples. For different misalignment samples, the Zernike coefficients can be obtained from the optical design software. After that, according to Eq. (12), the sample values of V misalignment matrixes under V misalignment samples can be calculated, and the obtained sample data has the following structure.

(16)$$\left. {\begin{array}{c} {\Delta {X_i}{{(N )}_1} = \frac{{\partial {X_i}(N )}}{{\partial {x_1}}}\Delta {m_{11}} + \frac{{\partial {X_i}(N )}}{{\partial {x_2}}}\Delta {m_{12}} + \cdots + \frac{{\partial {X_i}(N )}}{{\partial {x_n}}}\Delta {m_{1n}}}\\ {\Delta {X_i}{{(N )}_2} = \frac{{\partial {X_i}(N )}}{{\partial {x_1}}}\Delta {m_{21}} + \frac{{\partial {X_i}(N )}}{{\partial {x_2}}}\Delta {m_{22}} + \cdots + \frac{{\partial {X_i}(N )}}{{\partial {x_n}}}\Delta {m_{2n}}}\\ \vdots \\ {\Delta {X_i}{{(N )}_V} = \frac{{\partial {X_i}(N )}}{{\partial {x_1}}}\Delta {m_{V1}} + \frac{{\partial {X_i}(N )}}{{\partial {x_2}}}\Delta {m_{V2}} + \cdots + \frac{{\partial {X_i}(N )}}{{\partial {x_n}}}\Delta {m_{Vn}}} \end{array}} \right\},$$

where $\Delta {m_{V1}},\Delta {m_{V2}}, \cdots ,\Delta {m_{Vn}}$ are n misalignments randomly generated in the V-th misalignment sample.

Let

(17)$$\left. {\begin{array}{c} {{\alpha_i} = \left( {\begin{array}{c} {\Delta {X_i}{{(N )}_1}}\\ {\Delta {X_i}{{(N )}_2}}\\ \vdots \\ {\Delta {X_i}{{(N )}_V}} \end{array}} \right)} \;\;\;{{\beta_i} = \left( {\begin{array}{cccc} {\Delta {m_{11}}}&{\Delta {m_{12}}}& \cdots &{\Delta {m_{1n}}}\\ {\Delta {m_{21}}}&{\Delta {m_{22}}}& \cdots &{\Delta {m_{2n}}}\\ \vdots & \vdots & \vdots & \vdots \\ {\Delta {m_{V1}}}&{\Delta {m_{V2}}}& \cdots &{\Delta {m_{Vn}}} \end{array}} \right)}\\ {\chi_i} = \left( {\begin{array}{c} {\frac{{\partial {X_i}(N )}}{{\partial {x_1}}}}\\ {\frac{{\partial {X_i}(N )}}{{\partial {x_2}}}}\\ \vdots \\ {\frac{{\partial {X_i}(N )}}{{\partial {x_n}}}} \end{array}} \right) \end{array}} \right\}.$$

At this time, Eq. (16) can be expressed as

(18)$${\alpha _i} = {\beta _i}{\chi _i}.$$

Finally, using least squares algorithm, the i-th row vector in the partial derivative coefficient matrix can be fitted.

(19)$${\chi _i} = {({{\beta_i}^T{\beta_i}} )^{ - 1}}{\beta _i}^T{\alpha _i},$$

where $T$ denotes the matrix transpose operation, and the superscript $- 1$ denotes the matrix inversion operation.

Substituting the fitted partial derivative coefficient matrix into Eq. (12), a misalignment calculation model of off-axis optical systems with designed tilts and decenters based on third-order NAT can be obtained.

(20)$$\Delta C = H\left[ {\begin{array}{cccc} {\frac{{\partial {X_1}(N )}}{{\partial {x_1}}}}&{\frac{{\partial {X_1}(N )}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {X_1}(N )}}{{\partial {x_n}}}}\\ \vdots & \vdots & \vdots & \vdots \\ {\frac{{\partial {X_i}(N )}}{{\partial {x_1}}}}&{\frac{{\partial {X_i}(N )}}{{\partial {x_2}}}}& \cdots &{\frac{{\partial {X_i}(N )}}{{\partial {x_n}}}}\\ \vdots & \vdots & \vdots & \vdots \end{array}} \right]\left[ {\begin{array}{c} {\Delta {m_1}}\\ {\Delta {m_2}}\\ \vdots \\ {\Delta {m_n}} \end{array}} \right].$$

4. Design parameters of an wide-field off-axis three-mirror telescope

In order to verify the correctness and practicability of the misalignment calculation model, a real wide-field off-axis three-mirror telescope was used for alignment experiments. The aperture stop of the telescope is located on PM, and both SM and third mirror (TM) include a decenter designed along the Y-direction and a tilt designed around the X axis. The design parameters (e.g. radius, conic constants) of the off-axis three-mirror telescope are the same as a theoretical rotationally symmetric parent system, but only certain off-axis sections of the parent system are fabricated and used. The specific optical design parameters of the off-axis three-mirror telescope are shown in Table 2 and Table 3, and the optical layout is shown in Fig. 2. In Fig. 2, it can be seen that the pupil scaling factor M is $250/710$, and the normalized pupil eccentricity ${h_y}$ is $- 460/710$.

Fig. 2. Schematic diagram of optical layout of the off-axis three-mirror telescope.

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Table 2. Constructional parameters of the off-axis three-mirror telescope.

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Table 3. Primary parameters of the off-axis three-mirror telescope.

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The full field displays (FFDs) of RMS wavefront error (WFE), Fringe Zernike astigmatism (C5/6), Fringe Zernike coma (C7/8), and Fringe Zernike spherical aberration (C9) in the nominal telescope are shown in Fig. 3.

Fig. 3. FFDs of the nominal off-axis three-mirror telescope. (a) RMS WFE, (b) Fringe Zernike astigmatism (C5/6), (c) Fringe Zernike coma (C7/C8), (d) Fringe Zernike spherical aberration (C9).

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In order to ensure the accuracy and robustness of the algorithm, the FOV points used in the alignment process should cover the effective FOV of the telescope as much as possible. The red spots in Fig. 3(a) represent nine typical FOV points to be used when calculating misalignments. The coordinates of these nine FOV points are F0 (0°, −0.5°), F1 (−0.5°, −0.5°), F2 (−0.5°, 0°), F3 (0°, 0°), F4 (0.5°, 0°), F5 (0.5°, −0.5°), F6 (0.5°, −1°), F7 (0°, −1°), and F8 (−0.5°, −1°).

5. Simulation

5.1 Simulation alignment process and results

In this section, the dynamic data connection function of mathematical simulation and optical design software will be used to perform Monte Carlo simulation alignment experiments. To simulate the real misaligned telescope, the randomly generated misalignments and the actually measured figure errors of each mirror are introduced into the optical design model. The wavefront fitting function of optical design software is adopted to obtain the Zernike coefficients. After the Zernike coefficients are obtained, the misaligned telescope can be aligned through the constructed solution model. The alignment process of the misaligned telescope is an iterative optimization process. After each alignment, the RMS WFE will be calculated, and the correction result will be compared with the previous alignment result. If it is better than the last alignment result, the iterative process will continue, if it is worse than the last result, the iteration will be ended.The specific simulation alignment process is shown in Fig. 4.

Fig. 4. The specific simulation alignment process.

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In the actual alignment process, in order to reduce the misalignment dimensions to be aligned, an optical surface in optical system is usually fixed as a datum. Considering that in the off-axis three-mirror telescope the volume and mass of the PM are relatively large, which leads to difficulties in high-precision adjustment. Therefore, the PM is used as the alignment reference, and the SM and TM are adjusted to correct the wave aberrations of the telescope.

In addition, prior to CAA, the telescope needed to go through an initial alignment step. The current initial alignment methods are mostly based on the principle of the mechanical characteristic transfer of optical components [28,29], and the initial alignment of systems are completed by the theodolite and laser tracker. For the telescope in section 4, the linear residual error of about 0.5-1mm and the angular residual error of about 0.05-0.1° are introduced after the initial alignment by the existing methods. Therefore, in simulation alignment experiments, it is assumed that SM and TM have the misalignment threshold ranges shown in Table 4. The XDE, YDE, ADE, and BDE in Table 4 are the lateral misalignments of optical surfaces. XDE and YDE are the vertex decenters of optical surfaces in the x-z and y-z planes, respectively. ADE and BDE are the tip–tilt of optical surfaces in the x-z and y-z planes, respectively. ZDE are the axial misalignments, which represents the vertex decenters of optical surfaces along the z direction. According to the misalignment threshold ranges in Table 4, the partial derivative coefficient matrix of misalignments fitted by using the method mentioned in Section 3 is presented in Appendix B.

Table 4. The misalignment threshold range of the SM and TM.

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Within the given misalignment ranges, 500 sets of misalignments were randomly generated according to the uniform distribution, representing 500 random misalignment conditions. In addition, in the actual alignment environment, there are some factors that affect the measurement accuracy of the Zernike coefficients. In order to simulate this situation, the noise model shown in Eq. (21) is used to simulate the wavefront measurement errors in the actual alignment process.

(21)$$e = {\omega _{normrnd}}({\mu = 0,\sigma } )+ {C_{figure}},$$

where ${\omega _{normrnd}}({\mu = 0,\sigma } )$ is a random number that obeys a normal distribution with a mean value of $\mu$ and a standard deviation of $\sigma$, which can simulate random noises caused by uncertain factors such as airflow disturbance, temperature, vibration, etc. Under the laboratory environment conditions that will be shown in the next section, the fluctuation value of the multiple measurement results of Zernike coefficients at a single FOV point over a period of time is used to evaluate the measurement uncertainty of the Zernike coefficients. The standard deviation $\sigma$ of multiple Zernike coefficients measurement value is taken as the measurement errors caused by uncertain factors, and the results are shown in Table 5.

Table 5. Standard deviation of the measured Zernike coefficients ($\lambda = 632.8$nm)

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${C_{figure}}$ represents the residual figure errors of each optical surface after processing. An interferometer and compensators were used to measure the figure errors of the PM, SM, and TM in the actual optical system, and the measurement results were added to the optical design model. The FFDs of RMS wavefront error (WFE), Fringe Zernike astigmatism (C5/6), Fringe Zernike coma (C7/8), and Fringe Zernike spherical aberration (C9) are obtained after the figure errors are added into the nominal optical system, as shown in Fig. 5. After extracting the Zernike coefficients, the misalignments can be calculated according to the overdetermined equations formed by Eq. (20).

Fig. 5. FFDs of the nominal off-axis three-mirror telescope with figure errors. (a) RMS WFE, (b) Fringe Zernike astigmatism (C5/6), (c) Fringe Zernike coma (C7/C8), (d) Fringe Zernike spherical aberration (C9).

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After several iterations of alignment, the convergence results of the misalignments in each dimension are shown in Fig. 6. The abscissa in Fig. 6 represents the number of experiments for simulated alignment, and the ordinate represents the value of misalignments. The pink scatter plot represents the value of the simulated misalignments introduced by the optical system, and the blue scatter plot represents the value of the residual misalignments after the optical system is aligned. Furthermore, in order to evaluate the convergence accuracy of misalignments in a more accurate way, the following formula is introduced to calculate the root-mean-square deviation (RMSD) of the residual misalignments.

(22)$$RMSD = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {{{[{M{{(i)}_{error}}} ]}^2}} }$$

where $M{(i)_{error}}$ represents the residual misalignments,. n = 500. The calculation results are shown in Table 6.

Fig. 6. When there are figure errors and random noises, the convergence results of the SM and TM in each dimension.

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Table 6. The RMSD of residual misalignments.

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As can be seen from Fig. 6, the linear misalignments of the SM and TM basically converge to within ± 0.05mm, and the angular misalignments basically converge to within ±0.01°. As is shown in Table 6, the RMSD of linear and angular residual misalignments is maintained at the order of ${10^{ - 2}}$ and ${10^{ - 3}}$, respectively. Compared with the results without considering the wavefront measurement errors, the RMSD of residual misalignments in each dimension has basically increased by one order of magnitude. This indicates that when the residual misalignments converge to a certain range, the net aberrations induced by the misalignments will be coupled with the net aberrations induced by the figure errors and random noises, which makes it difficult to further calculate the correct misalignments.

Figure 7 shows the convergence results of the average RMS WFE of the full FOV, which are calculated with 12 × 12 equally spaced FOV points in 1° × 1°. The red scatter plot represents the average RMS WFE before the system is aligned. The blue point represents the average RMS WFE after the system is aligned. It can be seen from Fig. 7 that the average RMS WFE are all corrected to below 0.06 waves for different misalignment conditions when the figure errors and random noises are not considered. The average RMS WFE are all corrected to around 0.1 waves when considering figure errors and random noises. The simulation results show that the proposed algorithm is feasible for the correction of the off-axis three-mirror telescope with both axial and lateral misalignments.

Fig. 7. Average RMS WFE before and after alignment of the optical system (a) Wavefront measurement errors are not present. (b) Wavefront measurement errors are present.

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5.2 Comparison experiment

In this section, a comparison experiment is carried out with the traditional NAT method. The traditional NAT method for solving misalignments tends to solving it through a completely analytical method. The misalignment solution steps of the traditional NAT method are as follows:

(1) Derive the net wave aberration expressions (expressed by Zernike polynomials) induced by the lateral misalignments, and construct a system of equations to solve aberration field decenter vectors $\vec{\sigma }$.
(2) Through Buchroeder's [22,30] paraxial ray tracing method, derive analytical expressions between the lateral misalignments and aberration field decenter vectors, so as to construct a system of linear equations for solving the lateral misalignments (the aberration field decenter vectors and the lateral misalignments are linearly related).
(3) When actually computing the misalignment value, first of all, obtain the aberration field decenter vectors of each misaligned mirror according to the system of equations constructed in the first step, and then use the solved aberration field decenter vectors to compute corresponding lateral misalignments according to the equations derived in the second step.

References [18–20] all adopt the above method in telescopes alignment, and only the lateral misalignments are considered. However, in this paper, after obtaining net wave aberration expressions induced by axial and lateral misalignments, the method described in the second step wasn’t chosen to compute the misalignments, instead, a method of fitting misalignment partial derivative coefficient matrix is proposed. The purpose of this is to be able to solve axial and lateral misalignments at the same time. In order to illustrate the necessity of considering the net aberrations induced by axial misalignments, combined with [19,20], the traditional method is used to construct a corresponding lateral misalignment calculation model(LMCM), and a simulation alignment experiment is carried out for the misaligned off-axis three-mirror telescope. It is worth noting that for the telescope with designed tilts and decenters, after calculating aberration field decenter vectors $\vec{\sigma }$ using LMCM, ${\vec{\sigma }^\# }$ induced by designed tilts and decenters should be subtracted. In the experiment, only the lateral misalignments can be calculated in the LMCM, therefore, only the calculated lateral misalignments can be used for alignment. When the wavefront measurement errors are not considered, the simulation results are shown in Table 7 and Fig. 8.

Fig. 8. Average RMS WFE before and after alignment of the optical system.

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Table 7. The RMSD of residual lateral misalignments

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Compared Table 6 and Table 7, it can be found that when the axial misalignments are not introduced, the lateral misalignments convergence accuracy of LMCM is higher than that of the algorithm proposed in this paper. However, after introducing the axial misalignments, the LNCM became ineffective. As can be seen from Fig. 8, the RMS WFE of the telescope does not converge.

6. Experiment

In order to further validate the practicability of the constructed calculation model, an actual CAA experiment of the above-mentioned off-axis three-mirror telescope is performed in this section. The alignment experiment site of the off-axis three-mirror telescope is shown in Fig. 9.

Fig. 9. The experimental site of the off-axis three-mirror telescope.

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In order to obtain the exit pupil wave aberrations of the system, the flat reflecting mirror, off-axis three-mirror optical system, and the interferometer in Fig. 9 form a self-collimation optical path. The entire experimental device is located ten meters underground and is placed on an air-bearing platform to suppress the influence of vibration on the wavefront detection of the optical system. The PM is placed on a fixed bracket as the alignment reference of the entire system, and a mask is used to accurately control its actual aperture size. The SM and the TM are placed on two high-precision hexapods respectively, which can be accurately translated and rotated by controlling the hexapods. The azimuthal and pitching angles of the flat reflecting mirror can be precisely controlled by a theodolite and an adjustment frame connected to the flat reflecting mirror, so as to realize the wavefront measurement of the telescope at different FOV points. It should be noted that, in order to facilitate the placement of the experimental device, the symmetry plane of the off-axis three-mirror system in Fig. 9 is in the horizontal direction, which is different from optical simulation software such as CODE V (the symmetry plane of the system is usually in the vertical direction in CODE V). Therefore, some coordinate transformations between the hexapods coordinate system and the CODE V coordinate system are required in the actual alignment to convert the calculated misalignments to the actual adjustment value of the hexapods, for our theoretical calculations are based on the CODE V coordinate system. After the initial alignment of the telescope is completed, the constructed calculation model can be used for CAA.

In the CAA, the fringe Zernike coefficients of nine typical FOV points in Fig. 3(a) need to be measured by interferometer first. For the wavefront measurement of each FOV point, it is necessary to use a theodolite (there is a theodolite located behind the flat reflecting mirror, which is not shown in Fig. 9) to accurately monitor and adjust the azimuthal and pitching angles of the flat reflecting mirror. In order to minimize the influence of factors such as airflow disturbance, vibration and temperature on the measurement results, in each alignment iteration process, the fringe Zernike coefficients of each FOV point will be measured repeatedly for thousands of times with the interferometer and averaged. The wavefront plots of nine typical FOV points before CAA are shown in Fig. 10.

Fig. 10. The wavefront plots of 9 typical FOV points before CAA.

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After five alignment iterations, the wavefront plots of nine typical FOV points are shown in Fig. 11 and Fig. 12. Figure 13 shows the convergence of the fringe Zernike coefficients and RMS WFE during five alignment iterations. The experimental results in Fig. 10-Fig. 13 show that the wave aberrations of the telescope gradually decreases with the increase of the number of iterations. When the number of iterations is less than three, the wave aberrations converge faster because the influence of the misalignments on the wave aberrations is dominant at this time. However, when the number of iterations is greater than three, because the influence of the misalignments on the wave aberrations is covered by the figure errors and random noises, it is difficult for the wave aberrations to converge further and basically reach a stable state. By using the NAT-based CAA algorithm constructed in this paper, after 3-5 alignment iterations, the average RMS WFE of the system within the 1° × 1° square FOV is corrected from 1.9$\lambda$ to 0.12 $\lambda$($\lambda = 632.8$nm).

Fig. 11. The wavefront plots of nine typical FOV points after two alignment iterations.

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Fig. 12. The wavefront plots of nine typical FOV points after five alignment iterations.

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Fig. 13. The convergence of the RMS WFE and fringe Zernike coefficients with the increase of the number of alignment iterations.

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7. Conclusion

This paper mainly studies the third-order NAT misalignment calculation algorithm for nonsymmetric off-axis reflective telescopes that have both axial and lateral misalignments. For such misaligned telescopes, the traditional misalignment calculation algorithm based on NAT is difficult to be applied because it cannot solve the axial and lateral misalignment simultaneously. In order to solve this problem, the work done is summarized as follows. Firstly, the expressions of fringe Zernike wave aberrations for misaligned off-axis optical systems with designed decenters and tilts are derived when there are both axial and lateral misalignments. The third-order net astigmatism (C5/C6), third-order net coma (C7/C8) and third-order net spherical aberration (C9) induced by axial and lateral misalignments are separated from the total aberration fields of the misaligned systems. Then, based on the derived aberration equations and using the proposed method of fitting the partial derivative coefficient matrix of misalignments, a third-order NAT misalignment calculation model that can simultaneously solve the axial and lateral misalignments is constructed. Finally, the CAA of a nonsymmetric off-axis three-mirror telescope with designed decenters and tilts is performed by using the constructed calculation model. The correctness and practicability of the calculation model are comprehensively verified by simulations and experiments.

Appendix A: derivation process of $\vec{E}_{klm}^{(0 )} \cdot \vec{R}_{klm}^{(0 )}$, $\vec{E}_{klm}^{(1 )} \cdot \vec{R}_{klm}^{(1 )}$, and $\sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}}$

When only the third-order aberrations are considered, the expressions of third-order astigmatism, third-order coma, and third-order spherical aberration can be obtained by expanding Eq. (6), as shown in Eq. (23)-Eq. (25). The expansion process can be referred to [10].

(23)$$W_{off - axis}^{Astig} = \frac{1}{2}\sum\limits_j {[{({{W_{222j}} + \Delta W_{222j}^d} )\vec{H}_{Aj}^2 \cdot {{({M\vec{\rho }^{\prime} + \vec{h}} )}^2}} ]} ,$$

(24)$$W_{off - axis}^{Coma} = \sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} [{{{\vec{H}}_{Aj}} \cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ][{({M\vec{\rho }^{\prime} + \vec{h}} )\cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ],$$

(25)$$W_{off - axis}^{Sph} = \sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} {[{({M\vec{\rho }^{\prime} + \vec{h}} )\cdot ({M\vec{\rho }^{\prime} + \vec{h}} )} ]^2}.$$

According to the vector multiplication proposed by Shack [9–10], Eq. (23)-Eq. (25) can be expanded as

(26)$$W_{off - axis}^{Astig} = \frac{1}{2}\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} )} ({{M^2}\vec{H}_{Aj}^2 \cdot {{\vec{\rho }}^2} + 2M\vec{H}_{Aj}^2 \cdot \vec{h}\vec{\rho } + \vec{H}_{Aj}^2 \cdot {{\vec{h}}^2}} ),$$

(27)$$W_{off - axis}^{Coma} = \sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} \left[ \begin{array}{l} {M^3}({{{\vec{H}}_{Aj}} \cdot \vec{\rho }} )({\vec{\rho } \cdot \vec{\rho }} )+ {M^2}{{\vec{H}}_{Aj}}\vec{h} \cdot {{\vec{\rho }}^2} + 2M({\vec{h} \cdot \vec{h}} )({{{\vec{H}}_{Aj}} \cdot \vec{\rho }} )\\ + M{{\vec{h}}^2} \cdot {{\vec{H}}_{Aj}}\vec{\rho } + 2{M^2}({{{\vec{H}}_{Aj}} \cdot \vec{h}} )({\vec{\rho } \cdot \vec{\rho }} )+ ({\vec{h} \cdot \vec{h}} )({{{\vec{H}}_{Aj}} \cdot \vec{h}} )\end{array} \right],$$

(28)$$W_{off - axis}^{Sph} = \sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} \left[ \begin{array}{l} {M^4}{({\vec{\rho } \cdot \vec{\rho }} )^2} + 4{M^3}({\vec{h} \cdot \vec{\rho }} )({\vec{\rho } \cdot \vec{\rho }} )+ 2{M^2}{{\vec{h}}^2} \cdot {{\vec{\rho }}^2}\\ + 4{M^2}({\vec{h} \cdot \vec{h}} )({\vec{\rho } \cdot \vec{\rho }} )+ 4M({\vec{h} \cdot \vec{h}} )({\vec{h} \cdot \vec{\rho }} )+ {({\vec{h} \cdot \vec{h}} )^2} \end{array} \right].$$

Substitute ${\vec{H}_{Aj}} = \vec{H} - \vec{\sigma }_j^\# - {\vec{\sigma }^{\prime}_j}$ into the above formulas, and ignore the piston, tilt and defocus terms, Eq. (26)-Eq. (28) can be further expanded as

(29)$$W_{off - axis}^{Astig} = \frac{{{M^2}}}{2}\left[ \begin{array}{l} \sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){{\vec{H}}^2} - 2\vec{H}\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} )({\vec{\sigma }_j^\# + {{\vec{\sigma }^{\prime}}_j}} )} } \\ + \sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){{({\vec{\sigma }_j^\# } )}^2}} + \sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){{({{{\vec{\sigma }^{\prime}}_j}} )}^2}} \\ + 2\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} )\vec{\sigma }_j^\# {{\vec{\sigma }^{\prime}}_j}} \end{array} \right] \cdot {\vec{\rho }^2},$$

(30)$$\begin{aligned} W_{off - axis}^{Coma} &= \left[ {{M^3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} \vec{H} \cdot \vec{\rho } - {M^3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} ({\vec{\sigma }_j^\# + {{\vec{\sigma }^{\prime}}_j}} )\cdot \vec{\rho }} \right]({\vec{\rho } \cdot \vec{\rho }} )\\ &+ \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} \vec{H}\vec{h} - {M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} ({\vec{\sigma }_j^\# + {{\vec{\sigma }^{\prime}}_j}} )\vec{h}} \right] \cdot {{\vec{\rho }}^2}, \end{aligned}$$

(31)$$W_{off - axis}^{Sph} = \sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} [{{M^4}{{({\vec{\rho } \cdot \vec{\rho }} )}^2} + 4{M^3}({\vec{h} \cdot \vec{\rho }} )({\vec{\rho } \cdot \vec{\rho }} )+ 2{M^2}{{\vec{h}}^2} \cdot {{\vec{\rho }}^2}} ].$$

{\textrm{Let}}\quad\quad \left\{ {\begin{array}{c} {{{\vec{A}}^\# }_{klm} = \sum\limits_j {{{({{W_{klm}}} )}_j}{{\vec{\sigma }}^\# }_j} }\\ {{{\vec{A}^{\prime}}_{klm}} = \sum\limits_j {{{({{W_{klm}}} )}_j}{{\vec{\sigma }^{\prime}}_j}} }\\ {\vec{A}_{klm}^d = \sum\limits_j {{{({\Delta W_{klm}^d} )}_j}({\vec{\sigma }_j^\# + {{\vec{\sigma }^{\prime}}_j}} )} }\\ {\vec{B}_{klm}^\# = {{\sum\limits_j {{{({{W_{klm}}} )}_j}({\vec{\sigma }_j^\# } )} }^2}}\\ {{{\vec{B}^{\prime}}_{klm}} = \sum\limits_j {{{({{W_{klm}}} )}_j}[{{{({{{\vec{\sigma }^{\prime}}_j}} )}^2} + 2\vec{\sigma }_j^\# {{\vec{\sigma }^{\prime}}_j}} ]} }\\ {\vec{B}_{klm}^d = \sum\limits_j {{{({\Delta W_{klm}^d} )}_j}[{{{({\vec{\sigma }_j^\# } )}^2} + {{({{{\vec{\sigma }^{\prime}}_j}} )}^2} + 2\vec{\sigma }_j^\# {{\vec{\sigma }^{\prime}}_j}} ]} } \end{array}} \right.,

then Eq. (29) and Eq. (30) can be simplified as

(32)$$W_{off - axis}^{Astig} = \frac{{{M^2}}}{2}\left[ {\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){{\vec{H}}^2} - 2\vec{H}({\vec{A}_{222}^\# + {{\vec{A}^{\prime}}_{222}} + \vec{A}_{222}^d} )+ \vec{B}_{222}^\# + {{\vec{B}^{\prime}}_{222}} + \vec{B}_{222}^d} } \right] \cdot {\vec{\rho }^2},$$

(33)$$\begin{aligned} W_{off - axis}^{Coma} &= \left[ {{M^3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} \vec{H} \cdot \vec{\rho } - {M^3}({\vec{A}_{131}^\# + {{\vec{A}^{\prime}}_{131}} + \vec{A}_{131}^d} )\cdot \vec{\rho }} \right]({\vec{\rho } \cdot \vec{\rho }} )\\ &+ \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} \vec{H}\vec{h} - {M^2}({\vec{A}_{131}^\# + {{\vec{A}^{\prime}}_{131}} + \vec{A}_{131}^d} )\vec{h}} \right] \cdot {{\vec{\rho }}^2}. \end{aligned}$$

Combined with [27], Eq. (31)-Eq. (33) can be expanded into scalar form, and get

(34)$$\begin{aligned} W_{off - axis}^{Sph} &= \frac{{{M^4}}}{6}\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} ({6{{|\rho |}^4} - 6{{|\rho |}^2} + 1} )- 2{M^2}h_y^2\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} {|\rho |^2}\cos 2\varphi \\ &+ \frac{{4{M^3}{h_y}}}{3}\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} [{({3{{|\rho |}^3} - 2|\rho |} )\sin \varphi } ], \end{aligned}$$

(35)$$\begin{aligned} W_{off - axis}^{Astig} &= \frac{{{M^2}}}{2}\left[ \begin{array}{l} \sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} )({H_x^2 - H_y^2} )- 2{H_x}({\vec{A}_{222,x}^\# + {{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d} )} \\ + 2{H_y}({\vec{A}_{222,y}^\# + {{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d} )+ \vec{B}_{222,x}^\# + {{\vec{B}^{\prime}}_{222,x}} + \vec{B}_{222,x}^d \end{array} \right]({{{|\rho |}^2}\cos 2\varphi } )\\ &+ \frac{{{M^2}}}{2}\left[ \begin{array}{l} 2\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){H_x}{H_y}} - 2{H_y}({\vec{A}_{222,x}^\# + {{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d} )\\ - 2{H_x}({\vec{A}_{222,y}^\# + {{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d} )+ \vec{B}_{222,y}^\# + {{\vec{B}^{\prime}}_{222,y}} + \vec{B}_{222,y}^d \end{array} \right]({{{|\rho |}^2}\sin 2\varphi } ), \end{aligned}$$

(36)$$\begin{aligned} W_{off - axis}^{coma} &= \left[ {\frac{{{M^3}}}{3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {H_x} - ({\vec{A}_{131,x}^\# + {{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d} )} \right]({3{{|\rho |}^3} - 2|\rho |} )\cos \varphi \\ &+ \left[ {\frac{{{M^3}}}{3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {H_y} - ({\vec{A}_{131,y}^\# + {{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d} )} \right]({3{{|\rho |}^3} - 2|\rho |} )\sin \varphi \\ &- \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {h_y}{H_y} - {h_y}({\vec{A}_{131,y}^\# + {{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d} )} \right]({{{|\rho |}^2}\cos 2\varphi } )\\ &+ \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {h_y}{H_x} - {h_y}({\vec{A}_{131,x}^\# + {{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d} )} \right]({{{|\rho |}^2}\sin 2\varphi } ), \end{aligned}$$

For the optical systems discussed in this paper, the x component of $\vec{h}$ is zero, so in this expansion, let ${\vec{h}_x}$=0. At this point, according to Eq. (34)-Eq. (36), it can be obtained that

(37)$$\begin{aligned} {C_5} &= \frac{{{M^2}}}{2}\left[ \begin{array}{l} \sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} )({H_x^2 - H_y^2} )- 2{H_x}({\vec{A}_{222,x}^\# + {{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d} )} \\ + 2{H_y}({\vec{A}_{222,y}^\# + {{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d} )+ \vec{B}_{222,x}^\# + {{\vec{B}^{\prime}}_{222,x}} + \vec{B}_{222,x}^d \end{array} \right]\\ &- 2{M^2}h_y^2\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} - \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {h_y}{H_y} - {h_y}({\vec{A}_{131,y}^\# + {{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d} )} \right], \end{aligned}$$

(38)$$\begin{aligned} {C_6} &= \frac{{{M^2}}}{2}\left[ \begin{array}{l} 2\sum\limits_j {({{W_{222j}} + \Delta W_{222j}^d} ){H_x}{H_y}} - 2{H_y}({\vec{A}_{222,x}^\# + {{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d} )\\ - 2{H_x}({\vec{A}_{222,y}^\# + {{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d} )+ \vec{B}_{222,y}^\# + {{\vec{B}^{\prime}}_{222,y}} + \vec{B}_{222,y}^d \end{array} \right]\\ &+ \left[ {{M^2}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {h_y}{H_x} - {h_y}({\vec{A}_{131,x}^\# + {{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d} )} \right], \end{aligned}$$

(39)$${C_7} = \left[ {\frac{{{M^3}}}{3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {H_x} - ({\vec{A}_{131,x}^\# + {{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d} )} \right],$$

(40)$${C_8} = \left[ {\frac{{{M^3}}}{3}\sum\limits_j {({{W_{131j}} + \Delta W_{131j}^d} )} {H_y} - ({\vec{A}_{131,y}^\# + {{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d} )} \right] + \frac{{4{M^3}{h_y}}}{3}\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} ,$$

(41)$${C_9} = \frac{{{M^4}}}{6}\sum\limits_j {({{W_{040j}} + \Delta W_{040j}^d} )} .$$

Table 1 can be obtained by sorting out Eq. (37)-Eq. (41) and separating the intrinsic design residual aberration terms of the nominal optical system from the aberration terms caused by misalignments.

Appendix B: partial derivative coefficient matrix of the off-axis three-mirror telescope

When Eq. (11) is used to fit the partial derivative coefficient matrix, it is found that when the number of randomly generated misalignment samples V is greater than 100, the fitted partial derivative coefficient matrix will not change significantly. In this paper, let V = 200, and the fitted partial derivative coefficient matrix is

\scalebox{0.7}{$M = \left[ {\arraycolsep2.7pt\begin{array}{@{}cccccccccc@{}} { - 19.67}&{9.5 \times {{10}^{ - 11}}}&{ - 3.1 \times {{10}^{ - 9}}}&{16.37}&{ - 9.2 \times {{10}^{ - 9}}}&{ - 2.52}&{1.7 \times {{10}^{ - 9}}}&{ - 1.9 \times {{10}^{ - 9}}}&{2.19}&{2.7 \times {{10}^{ - 9}}}\\ { - 4.2 \times {{10}^{ - 3}}}&{ - 21.47}&{ - 18.16}&{ - 1.38}&{ - 0.43}&{ - 1.9 \times {{10}^{ - 4}}}&{ - 2.56}&{ - 2.25}&{3.9 \times {{10}^{ - 4}}}&{0.32}\\ { - 1.39}&{3.1 \times {{10}^{ - 8}}}&{4.7 \times {{10}^{ - 8}}}&{6.71}&{ - 2.3 \times {{10}^{ - 8}}}&{0.30}&{ - 1.8 \times {{10}^{ - 7}}}&{5.3 \times {{10}^{ - 8}}}&{ - 3.55}&{ - 7.1 \times {{10}^{ - 9}}}\\ { - 1.3 \times {{10}^{ - 3}}}&{ - 1.51}&{ - 7.01}&{ - 0.73}&{0.12}&{1.2 \times {{10}^{ - 4}}}&{0.34}&{3.63}&{ - 9.2 \times {{10}^{ - 5}}}&{ - 0.15}\\ { - 3.9 \times {{10}^{ - 3}}}&{ - 1.34}&{ - 0.74}&{ - 0.03}&{4.14}&{ - 2.6 \times {{10}^{ - 4}}}&{ - 0.04}&{ - 0.05}&{ - 2.0 \times {{10}^{ - 4}}}&{ - 0.19}\\ {4.5 \times {{10}^{ - 4}}}&{ - 9.0 \times {{10}^{ - 3}}}&{ - 0.04}&{ - 1.15}&{ - 0.35}&{ - 6.2 \times {{10}^{ - 5}}}&{ - 0.01}&{ - 0.07}&{ - 1.8 \times {{10}^{ - 3}}}&{0.02}\\ { - 3.8 \times {{10}^{ - 3}}}&{0.07}&{0.09}&{1.47}&{ - 0.33}&{4.3 \times {{10}^{ - 6}}}&{0.03}&{0.04}&{2.3 \times {{10}^{ - 4}}}&{0.61}\\ {3.5 \times {{10}^{ - 3}}}&{5.03}&{1.79}&{ - 0.26}&{ - 10.41}&{0.02}&{ - 0.89}&{ - 2.20}&{0.13}&{0.53}\\ { - 0.88}&{ - 2.9 \times {{10}^{ - 8}}}&{ - 4.7 \times {{10}^{ - 8}}}&{ - 0.67}&{1.6 \times {{10}^{ - 8}}}&{1.07}&{1.8 \times {{10}^{ - 8}}}&{ - 4.9 \times {{10}^{ - 8}}}&{ - 2.34}&{8.5 \times {{10}^{ - 9}}} \end{array}} \right].$}

Funding

National Natural Science Foundation of China (12003033, 61875190).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Zernike coefficient	$\vec{E}_{klm}^{(0 )} \cdot \vec{R}_{klm}^{(0 )}$	$\vec{E}_{klm}^{(1 )} \cdot \vec{R}_{klm}^{(1 )}$	$\sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}}$
${C_5}$	${\left[ {\begin{array}{{20}{c}} { - 2{M^2}h_y^2}\\ { - {M^2}{h_y}{H_y}}\\ {{M^2}{h_y}}\\ { - {M^2}{H_x}}\\ {{M^2}{H_y}}\\ {\frac{{{M^2}}}{2}({H_x^2 - H_y^2} )}\\ {\frac{{{M^2}}}{2}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {\sum\limits_j {{W_{040j}}} }\\ {\sum\limits_j {{W_{131j}}} }\\ {\vec{A}_{131,y}^\# }\\ {\vec{A}_{222,x}^\# }\\ {\vec{A}_{222,y}^\# }\\ {\sum\limits_j {{W_{222j}}} }\\ {\vec{B}_{222,x}^\# } \end{array}} \right]$	${\left[ {\begin{array}{{20}{c}} {{M^2}{h_y}}\\ { - {M^2}{H_x}}\\ {{M^2}{H_y}}\\ { - 2{M^2}h_y^2}\\ { - {M^2}{h_y}{H_y}}\\ {\frac{{{M^2}}}{2}({H_x^2 - H_y^2} )} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {{{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d}\\ {{{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d}\\ {{{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d}\\ {\sum\limits_j {\Delta W_{040j}^d} }\\ {\sum\limits_j {\Delta W_{131j}^d} }\\ {\sum\limits_j {\Delta W_{222j}^d} } \end{array}} \right]$	$\frac{{{M^2}}}{2}{\vec{B}^{\prime}_{222,x}} + \vec{B}_{222,x}^d$
${C_6}$	${\left[ {\begin{array}{{20}{c}} {{M^2}{h_y}{H_x}}\\ { - {M^2}{h_y}}\\ { - {M^2}{H_y}}\\ { - {M^2}{H_x}}\\ {{M^2}{H_x}{H_y}}\\ {{M^2}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {\sum\limits_j {{W_{131j}}} }\\ {\vec{A}_{131,x}^\# }\\ {\vec{A}_{222,x}^\# }\\ {\vec{A}_{222,y}^\# }\\ {\sum\limits_j {{W_{222j}}} }\\ {\vec{B}_{222,y}^\# } \end{array}} \right]$	${\left[ {\begin{array}{{20}{c}} { - {M^2}{h_y}}\\ { - {M^2}{H_y}}\\ { - {M^2}{H_x}}\\ {{M^2}{h_y}{H_x}}\\ {{M^2}{H_x}{H_y}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {{{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d}\\ {{{\vec{A}^{\prime}}_{222,x}} + \vec{A}_{222,x}^d}\\ {{{\vec{A}^{\prime}}_{222,y}} + \vec{A}_{222,y}^d}\\ {\sum\limits_j {\Delta W_{131j}^d} }\\ {\sum\limits_j {\Delta W_{222j}^d} } \end{array}} \right]$	$\frac{{{M^2}}}{2}\vec{B}^{\prime}_{222,y} + \vec{B}_{222,y}^d$
${C_7}$	${\left[ {\begin{array}{{20}{c}} {\frac{{{M^3}}}{3}{H_x}}\\ { - \frac{{{M^3}}}{3}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {\sum\limits_j {{W_{131j}}} }\\ {\vec{A}_{131,x}^\# } \end{array}} \right]$	${\left[ {\begin{array}{{20}{c}} { - \frac{{{M^3}}}{3}}\\ {\frac{{{M^3}}}{3}{H_x}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {{{\vec{A}^{\prime}}_{131,x}} + \vec{A}_{131,x}^d}\\ {\sum\limits_j {\Delta W_{131j}^d} } \end{array}} \right]$	0
${C_8}$	${\left[ {\begin{array}{{20}{c}} {\frac{{4{M^3}}}{3}{h_y}}\\ {\frac{{{M^3}}}{3}{H_y}}\\ { - \frac{{{M^3}}}{3}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {\sum\limits_j {{W_{040j}}} }\\ {\sum\limits_j {{W_{131j}}} }\\ {\vec{A}_{131,y}^\# } \end{array}} \right]$	${\left[ {\begin{array}{{20}{c}} { - \frac{{{M^3}}}{3}}\\ {\frac{{4{M^3}}}{3}{h_y}}\\ {\frac{{{M^3}}}{3}{H_y}} \end{array}} \right]^{\rm T}}\left[ {\begin{array}{{20}{c}} {{{\vec{A}^{\prime}}_{131,y}} + \vec{A}_{131,y}^d}\\ {\sum\limits_j {\Delta W_{040j}^d} }\\ {\sum\limits_j {\Delta W_{131j}^d} } \end{array}} \right]$	0
${C_9}$	$\frac{{{M^4}}}{6}\sum\limits_j {{W_{040j}}}$	$\frac{{{M^4}}}{6}\sum\limits_j {\Delta W_{040j}^d}$	0

Surface	Radius (mm)	Thickness (mm)	Conic Constant	Decenter X (mm)	Decenter Y (mm)	Tilt About X (deg)	Tilt About Y (deg)
Object	Infinity	Infinity	0	–	–	–	–
Stop	Infinity	0	0	–	−460	–	–
PM	−3600.41	−1551.77	−0.921	0	0	0	0
SM	−910.903	1558.771	−4.828	0	−8.241	1	0
TM	−1219.413	−1533.43	−0.292	0	−24.122	1.752	0
Image	Infinity	0	–	–	–	–	–

Specification	Value
Entrance pupil	PM
Entrance pupil diameter	500mm
Effective focal length	6000mm
Effective FOV	1° × 1°
Central FOV	(0°, −0.5°)

	XDE	YDE	ZDE	ADE	BDE
SM	±0.8mm	±0.8mm	±0.8mm	±0.1°	±0.1°
TM	±0.5mm	±0.5mm	±0.5mm	±0.1°	±0.1°

Wavefront measurement errors are not present
	XDE	YDE	ZDE	ADE	BDE
SM	$1.012 \times {10^{ - 3}}$	$9.167 \times {10^{ - 4}}$	$1.710 \times {10^{ - 3}}$	$2.063 \times {10^{ - 4}}$	$8.910 \times {10^{ - 5}}$
TM	$9.332 \times {10^{ - 4}}$	$2.614 \times {10^{ - 3}}$	$9.958 \times {10^{ - 4}}$	$9.015 \times {10^{ - 5}}$	$1.007 \times {10^{ - 4}}$
Wavefront measurement errors are present
SM	$2.108 \times {10^{ - 2}}$	$2.032 \times {10^{ - 2}}$	$3.607 \times {10^{ - 2}}$	$6.272 \times {10^{ - 3}}$	$4.654 \times {10^{ - 3}}$
TM	$2.117 \times {10^{ - 2}}$	$3.843 \times {10^{ - 2}}$	$3.419 \times {10^{ - 2}}$	$7.351 \times {10^{ - 3}}$	$7.139 \times {10^{ - 3}}$

Alignment algorithm of nonsymmetric off-axis reflective astronomical telescopes based on the modified third-order nodal aberration theory

Abstract

1. Introduction

2. Wave aberration expression of off-axis optical systems with designed tilts and decenters in the presence of both axial and lateral misalignments

3. Misalignment calculation model

4. Design parameters of an wide-field off-axis three-mirror telescope

5. Simulation

5.1 Simulation alignment process and results

5.2 Comparison experiment

6. Experiment

7. Conclusion

Appendix A: derivation process of $\vec{E}_{klm}^{(0 )} \cdot \vec{R}_{klm}^{(0 )}$, $\vec{E}_{klm}^{(1 )} \cdot \vec{R}_{klm}^{(1 )}$, and $\sum\limits_{a = 2}^\infty {\vec{E}_{klm}^{(a )} \cdot \vec{R}_{klm}^{(a )}}$

Appendix B: partial derivative coefficient matrix of the off-axis three-mirror telescope

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (13)

Tables (7)

Equations (43)

Optics Express

Axial misalignments are not present
	XDE	YDE	ADE	BDE
SM	$2.431 \times {10^{ - 5}}$	$7.221 \times {10^{ - 5}}$	$8.203 \times {10^{ - 6}}$	$9.001 \times {10^{ - 6}}$
TM	$6.439 \times {10^{ - 4}}$	$6.023 \times {10^{ - 4}}$	$6.431 \times {10^{ - 4}}$	$4.493 \times {10^{ - 5}}$
Axial misalignments are present
SM	$7.765 \times {10^{ - 1}}$	$1.148$	$1.134$	$3.441 \times {10^{ - 1}}$
TM	$2.302$	$9.671 \times {10^{ - 1}}$	$1.368$	$6.733 \times {10^{ - 1}}$