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Detecting radius of curvature (ROC) mismatch among primary mirror (PM) segments

Zhirong Lu, Guohao Ju, Xiaoquan Bai, Chunyue Zhang, Fengyi Jiang, Boqian Xu, Jihong Dong, and Shuyan Xu

Open Access Open Access

Abstract

A segmented primary mirror (PM) is an efficient solution to the problems of a monolithic PM manufacture, testing, transportation, and launch. However, the problem of the radius of curvature (ROC) matching among PM segments will arise, which if not solved will seriously degrade the final imaging quality of the system. Accurately detecting ROC mismatch among PM segments from the wavefront map is of crucial importance for efficiently correcting this kind of manufacturing error, while currently there are few related studies. Based on the inherent relation between the PM segment’s ROC error and corresponding sub-aperture defocus aberration, this paper proposes that the ROC mismatch can be accurately estimated from the sub-aperture defocus aberration. Secondary mirror (SM) lateral misalignments will influence the accuracy of estimating ROC mismatch. A strategy is also proposed to reduce the influence of SM lateral misalignments. Detailed simulations are performed to demonstrate the effectiveness of the proposed method for detecting ROC mismatch among PM segments. This paper paves a road for detecting ROC mismatch using image-based wavefront sensing methods.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Segmented PM structure has become an important trend in the development of super-large-aperture telescopes in the future. A traditional large-aperture telescope with monolithic PM has great difficulties in mirror manufacturing and testing, mechanical support, and lightweight design, as well as the transportation and launch of the whole telescope body. The segmented PM is an effective way to solve these problems, which has been adopted by James Webb Space Telescope (JWST) [1–3].

After a segmented PM space telescope is deployed on orbit. It needs to go through several complicated steps to gradually complete the alignment and phasing of different PM segments and achieves imaging performance close to the diffraction limit [4,5]. The ROC mismatch is one of the problems that need to be solved in the alignment and phasing process [5–7], and it is because each individually produced PM segment will have a different ROC error. This error will induce additional wavefront aberrations, which is one of the key problems affecting the final imaging performance of the segmented PM telescope [8,9]. Aberrations induced by ROC mismatch among PM segments cannot be compensated by simply adjusting a system element [10,11]. Therefore, this error should be accurately detected and corrected before the telescopes can be applied to astronomical observation.

There exist several problems which need to be solved when detecting ROC mismatch among PM segments. The first problem is to select a suitable wavefront sensing method that is not only suitable for on-orbit applications but also can obtain the required wavefront aberrations. The second problem is the method for detecting ROC mismatch among PM segments. The last problem is the method for eliminating the influence of SM lateral misalignments on detecting ROC mismatch among PM segments (because SM lateral misalignments are the most common type of misalignments for space telescopes and aberration analysis’ result in Section 4 also shows that this problem needs to be solved). However, there are few related studies concerning the aberration properties of ROC mismatch among PM segments at present [11,12], and nearly no research is conducted on decoupling the influence of ROC mismatch among PM segments and SM lateral misalignments on the wavefront distribution. Although JWST can detect ROC mismatch among PM segments, public research has not disclosed its technical details.

In this paper, we establish a framework for detecting ROC mismatch among PM segments on-orbit. First, we introduce the image-based phase retrieval algorithm. Then, we analyze the effects of the PM segment’s ROC error and SM lateral misalignments on the wavefront aberrations by using Nodal aberration theory (NAT), respectively. According to the above aberration analysis results, we propose the method for detecting ROC mismatch among PM segments (including the method for decoupling the sub-aperture wavefront aberrations induced by SM lateral misalignments and PM segment’s ROC error). Finally, the effectiveness of the detection method is fully verified by simulation.

This paper is organized as follows. Section 2 describes the basic principle of an image-based phase retrieval algorithm. Section 3 discusses NAT and the influence of the PM segment’s ROC error and SM lateral misalignments on wavefront aberrations. Based on Section 3, Section 4 proposes a method to detect ROC mismatch among PM segments. The method for decoupling the sub-aperture wavefront aberrations induced by SM lateral misalignments and the PM segment’s ROC error is also proposed in Section 4. In Section 5, the effectiveness of the method proposed in Section 4 is verified by simulation. This paper is concluded in Section 6.

2. Image-based iterative phase retrieval algorithm

In this section, we will describe the basic principle of the image-based iterative phase retrieval algorithm. The phase retrieval algorithm is one of the wavefront sensing methods, which is used to reconstruct the wavefront aberration of the optical system. Compared with the traditional wavefront sensing methods such as interferometer and SH sensor, the image-based iterative phase retrieval algorithm only needs to collect a few Point Spread Function (PSF) images in the focal plane. Then the wavefront aberration can be reconstructed by the phase retrieval algorithm. Therefore, this method hardly needs to rely on additional hardware and a complex optical path. Simple structure and high precision make it more suitable for on-orbit applications [13,14]. The most famous application of the image-based phase retrieval algorithm is wavefront sensing of the Hubble Space Telescope (HST) [15]. JWST, as the successor of HST, also adopted this method in its wavefront sensing and control process [16]. This paper will use the image-based iterative phase retrieval algorithm to reconstruct the wavefront aberration of the optical system. Its basic principles are described below.

For general imaging optical systems, there is the following relationship between generalized pupil function $P({\varepsilon ,\eta } )$, system impulse response $\tilde{h}({x,y} )$, and PSF [17]:

$$\begin{array}{c} P({\varepsilon ,\eta } )= p({\varepsilon ,\eta } ){\textrm{e}^{\textrm{i}\varphi ({\varepsilon ,\eta } )}}\mathrel{\mathop{{\buildrel \longrightarrow \over \longleftarrow}}\limits_{{{\cal F}^{ - 1}}}^{\cal F}}\tilde{h}({x,y} )= |{\tilde{h}({x,y} )} |{\textrm{e}^{\textrm{i}\varPhi ({\varepsilon ,\eta } )}},\\ PSF({x,y} )= \tilde{h}({x,y} ){{\tilde{h}}^\ast }({x,y} )= {|{\tilde{h}({x,y} )} |^2}, \end{array}$$
where ${\cal F}$ denotes Fourier Transform (FT), ${{\cal F}^{ - 1}}$ denotes inverse Fourier transform (IFT), $p({\varepsilon ,\eta } )$ is pupil function (i.e., a 0-1 function representing the shape of the pupil), $\varphi ({\varepsilon ,\eta } )$ is the phase of wavefront aberrations. Based on this relationship, Gerchberg and Saxton first proposed the iterative phase retrieval algorithm in 1972, also known as the GS algorithm (GSA) [18]. The principle of GSA is shown in Fig. 1. Take the random phase ${\varphi _0}({\varepsilon ,\eta } )$ as the initial estimation of wavefront aberrations. The modulus of the pupil function $|{f({\varepsilon ,\eta } )} |$ and the square root of PSF$|{F({x,y} )} |$ are known constraints of the pupil plane and image plane, respectively. $g({\varepsilon ,\eta } )$ and $G({x,y} )$ are estimates of generalized pupil function and system impulse response, respectively. Iterations are carried out between $g({\varepsilon ,\eta } )$ and $G({x,y} )$ by FT and IFT. The constraint condition of the corresponding plane needs to be applied after each transformation. The termination condition of the iteration is to obtain the solution that satisfies the convergence condition. $\varphi ({\varepsilon ,\eta } )$ is the reconstructed wavefront aberration.

 figure: Fig. 1.

Fig. 1. The Gerchberg-Saxton Algorithm (GSA).

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The phase retrieval problem can be regarded as the process of solving a multivariate high-dimensional ill-conditioned equation set in the complex number domain. Therefore, the performance of GSA is poor because few constraints. The modified GSA is often used in practice such as JWST [16]. The performance of the algorithm is enhanced by introducing multiple PSF images into the algorithm. The GSA uses multiple PSF images in this paper as shown in Fig. 2. PSF1 ∼ PSF5 are the PSF collected at five different defocus positions (the amount of defocus is known). The algorithm starts from the defocus position corresponding to PSF1. After defocusing compensation, the result phase ${\varphi _1}({\varepsilon ,\eta } )$ will be used as the initial value of iterations at the corresponding defocus position of PSF2. Similar procedures are repeated until phase retrieval is completed at all positions. The weighted average $\bar{\varphi }({\varepsilon ,\eta } )$ will be calculated as the initial value of the next iteration. The algorithm will also continue until the convergence condition is satisfied.

 figure: Fig. 2.

Fig. 2. The modified GSA using multiple PSF images.

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3. Influence of ROC error and lateral misalignments on wavefront aberrations

This section discusses Nodal Aberration Theory (NAT) and the influence of the PM segment’s ROC error and SM lateral misalignments on wavefront aberrations. The content in this section is the basis for detecting ROC mismatch among PM segments and decoupling aberrations induced by the PM segment’s ROC error and SM lateral misalignments in Section 4.

3.1 Basic theory of Nodal aberration theory (NAT)

Nodal Aberration Theory (NAT) is proposed by Shack based on Buchroeder’s research. NAT is a tool for analyzing wavefront aberrations of misaligned on-axis optical systems [19,20]. The wavefront aberrations of misaligned on-axis optical systems can be expressed as [21]:

$$\begin{array}{c} {W_{on \cdot axis}} = \sum\limits_j {\sum\limits_{p = 0}^\infty {\sum\limits_{n = 0}^\infty {\sum\limits_{m = 0}^\infty {{W_{klm \cdot j}}{{[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} )\cdot ({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} )} ]}^p}{{({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )}^n}{{[{({{\boldsymbol H} - {{\boldsymbol \sigma }_j}} )\cdot {\boldsymbol \rho }} ]}^m}} } } } ,\\ k = 2p + m,\quad l = 2n + m, \end{array}$$
where ${W_{klm \cdot j}}$ is the specific type of aberration coefficient of surface j, ${\boldsymbol H}$ is the normalized field vector, ${\boldsymbol \rho }$ is the normalized pupil vector, ${{\boldsymbol \sigma }_j}$ is the aberration field decenter vector of surface j. The value of (k + l) or (p + n + m) reflects the order of wavefront aberrations: k + l = 4 (or p + n + m = 2) corresponds to third-order aberrations; k + l = 6 (or p + n + m = 3) corresponds to fifth-order aberrations.

NAT can be extended to a misaligned off-axis optical system [22,23]. The pupil-offset off-axis optical system can be regarded as a part of an on-axis optical system. To analyze the wavefront aberrations of a pupil-offset off-axis optical system, the pupil decenter vector ${\boldsymbol s}$ is induced. Let normalized effective field vector ${{\boldsymbol H}_{{\boldsymbol A} \cdot j}} = {\boldsymbol H} - {{\boldsymbol \sigma }_j}$, the wavefront aberrations of a misaligned pupil-offset off-axis optical system can be expressed as [24]:

$$\begin{array}{c} {W_{off \cdot axis}} = \sum\limits_j {\sum\limits_{p = 0}^\infty {\sum\limits_{n = 0}^\infty {\sum\limits_{m = 0}^\infty {{W_{klm \cdot j}}{{({{{\boldsymbol H}_{{\boldsymbol A} \cdot j}} \cdot {{\boldsymbol H}_{{\boldsymbol A} \cdot j}}} )}^p}{{[{({{\boldsymbol \rho } + {\boldsymbol s}} )\cdot ({{\boldsymbol \rho } + {\boldsymbol s}} )} ]}^n}{{[{{{\boldsymbol H}_{{\boldsymbol A} \cdot j}} \cdot ({{\boldsymbol \rho } + {\boldsymbol s}} )} ]}^m}} } } } ,\\ k = 2p + m,\quad l = 2n + m. \end{array}$$

Equation (2) and Eq. (3) are the theoretical basis for analyzing the influence of the PM segment’s ROC error and SM lateral misalignments on wavefront aberrations.

3.2 Influence of PM segment’s ROC error on sub-aperture wavefront aberrations

For an on-axis optical system with segmented PM, each PM segment and the rest of the system (except other PM segments) can be regarded as a pupil-offset off-axis subsystem of the on-axis parent system. PM segment’s ROC error will influence the corresponding sub-aperture wavefront aberrations. According to Eq. (3), in each off-axis subsystem, the expression of sub-aperture wavefront aberrations induced by surface j can be obtained as follows:

$$\begin{aligned} {W_{off \cdot axis \cdot j}} &= \left\{ \begin{array}{l} {W_{020 \cdot j}} + 4{W_{040 \cdot j}}({{\boldsymbol s} \cdot {\boldsymbol s}} )+ \\ {W_{220\textrm{M} \cdot j}}({{{\boldsymbol H}_{{\boldsymbol A} \cdot j}} \cdot {{\boldsymbol H}_{{\boldsymbol A} \cdot j}}} )+ 2{W_{131 \cdot j}}({{{\boldsymbol H}_{{\boldsymbol A} \cdot \textrm{PMs}}} \cdot {\boldsymbol s}} )\end{array} \right\}({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )\quad \cdots \quad Defocus\\& + \left\{ {2{W_{040 \cdot j}}{{\boldsymbol s}^2} + \frac{1}{2}{W_{222 \cdot j}}{\boldsymbol H}_{{\boldsymbol A} \cdot j}^2 + {W_{131 \cdot j}}{{\boldsymbol H}_{{\boldsymbol A} \cdot j}}{\boldsymbol s}} \right\} \cdot {{\boldsymbol \rho }^2}\quad \cdots \quad Ast\\& + \{{4{W_{040 \cdot j}}{\boldsymbol s} + {W_{131 \cdot j}}{{\boldsymbol H}_{{\boldsymbol A} \cdot j}}} \}\cdot {\boldsymbol \rho }({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )\quad \cdots \quad Coma\\& + \{{{W_{040 \cdot j}}} \}{({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )^2}\quad \cdots \quad SA3, \end{aligned}$$
where ${W_{220\textrm{M} \cdot j}} = {W_{020 \cdot j}} + {{{W_{222 \cdot j}}} / 2}$.

If we use subscript PS to represent PM segments and replace the subscript j in Eq. (4). Then, Eq. (4) becomes the expression of sub-aperture wavefront aberrations induced by PM segments. PM segment’s ROC error induces wavefront aberration coefficients’ variation, $\Delta {W_{klm \cdot \textrm{PS}}}$. Therefore, the PM segment’s ROC error will induce additional sub-aperture aberrations as follows: $\Delta {W_{020 \cdot \textrm{PS}}}$, $\Delta {W_{040 \cdot \textrm{PS}}}$, $\Delta {W_{220\textrm{M} \cdot \textrm{PS}}}$ and $\Delta {W_{131 \cdot \textrm{PS}}}$ induce sub-aperture defocu; $\Delta {W_{040 \cdot \textrm{PS}}}$, $\Delta {W_{222 \cdot \textrm{PS}}}$ and $\Delta {W_{131 \cdot \textrm{PS}}}$ induce sub-aperture third-order astigmatism; $\Delta {W_{040 \cdot \textrm{PS}}}$ and $\Delta {W_{131 \cdot \textrm{PS}}}$ induce sub-aperture third-order coma; $\Delta {W_{040 \cdot \textrm{PS}}}$ induces sub-aperture third-order spherical aberration. When we use Zernike fringe polynomials to describe wavefront aberrations, the variation of sub-aperture aberration coefficients, $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$∼ $\Delta {C_{9 \cdot \textrm{sub} \cdot \textrm{PS}}}$, will be induced by the PM segment’s ROC error (subscript sub represents sub-aperture, ${C_{4 \cdot \textrm{sub}}}$ corresponds sub-aperture defocus, ${C_{5 \cdot \textrm{sub}}}$ and ${C_{6 \cdot \textrm{sub}}}$ correspond sub-aperture third-order astigmatism, ${C_{7 \cdot \textrm{sub}}}$ and ${C_{7 \cdot \textrm{sub}}}$ correspond sub-aperture third-order coma, and ${C_{9 \cdot \textrm{sub}}}$ corresponds sub-aperture third-order spherical aberration).

3.3 Influence of SM lateral misalignments on full-aperture and sub-aperture wavefront aberrations

The SM lateral misalignments include four types: SM decenters along the X and Y axis and SM tip-tilts around the X and Y axis. The influence of SM lateral misalignments on wavefront aberrations needs to be discussed in the on-axis parent system and off-axis subsystem respectively. According to Eq. (2), the expression of full-aperture wavefront aberrations induced by surface j can be obtained as follows:

$$\begin{aligned} {W_{on \cdot axis \cdot j}} &= \{{{W_{020 \cdot j}} + {W_{220\textrm{M} \cdot j}}({{{\boldsymbol H}_{{\boldsymbol A} \cdot j}} \cdot {{\boldsymbol H}_{{\boldsymbol A} \cdot j}}} )} \}({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )\quad \cdots \quad Defocus\\& + \frac{1}{2}\{{{W_{222 \cdot j}}{\boldsymbol H}_{{\boldsymbol A} \cdot j}^2} \}\cdot {{\boldsymbol \rho }^2}\quad \cdots \quad Ast\\& + \{{{W_{131 \cdot j}}{{\boldsymbol H}_{{\boldsymbol A} \cdot j}}} \}\cdot {\boldsymbol \rho }({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )\quad \cdots \quad Coma\\& + \{{{W_{040 \cdot j}}} \}{({{\boldsymbol \rho } \cdot {\boldsymbol \rho }} )^2}\quad \cdots \quad SA3, \end{aligned}$$
where ${W_{220\textrm{M} \cdot j}} = {W_{020 \cdot j}} + {{{W_{222 \cdot j}}} / 2}$

If we replace the subscript j in Eq. (5) with subscript SM. Equation (5) becomes the expression of full-aperture wavefront aberrations induced by SM. The SM lateral misalignments mainly induce the variations of the aberration field decenter vector, $\Delta {{\boldsymbol \sigma }_{\textrm{SM}}}$, i.e. the variations of the normalized effective field vector, $\Delta {{\boldsymbol H}_{{\boldsymbol A} \cdot \textrm{SM}}}$. Therefore, The SM lateral misalignments will induce additional full-aperture aberrations, mainly including full-aperture defocus, full-aperture third-order astigmatism, and full-aperture third-order coma. When we use Zernike fringe polynomials to describe wavefront aberrations, the variation of full-aperture aberration coefficients, $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$∼$\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$, will be induced by SM lateral misalignments (subscript full represents full-aperture, ${C_{4 \cdot \textrm{full}}}$ corresponds full-aperture defocus, ${C_{5 \cdot \textrm{full}}}$ and ${C_{6 \cdot \textrm{full}}}$ correspond full-aperture third-order astigmatism, ${C_{7 \cdot \textrm{full}}}$ and ${C_{8 \cdot \textrm{full}}}$ correspond sub-aperture third-order coma).

If we replace subscript j in Eq. (4) with subscript SM. Equation (4) becomes the expression of sub-aperture wavefront aberrations induced by SM lateral misalignments. It can be seen that SM lateral misalignments will induce additional sub-aperture aberrations, mainly including sub-aperture defocus, sub-aperture third-order astigmatism, and sub-aperture third-order coma. When we use Zernike fringe polynomials to describe wavefront aberrations, the variation of sub-aperture aberration coefficients, $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}$∼$\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}$, will be induced by SM lateral misalignments.

Importantly, it can be seen that both the PM segment’s ROC error and SM lateral misalignments will induce sub-aperture defocus.

4. Method of detecting ROC mismatch among PM segments

4.1 Method for detecting ROC mismatch among PM segments

Based on Section 3.2, when the field of view (FOV) is small and the ROC error of PM segments is within a certain range, $\Delta {W_{020 \cdot \textrm{PS}}}$ is major (so that $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ is major) [25]. For each PM segment, we suppose that the functional relationship between $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ and PM segment’s ROC error can be expressed as follows:

$${R_{\textrm{e} \cdot \textrm{PS}}} = f({\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}} ),$$
where ${R_{\textrm{e} \cdot \textrm{PS}}}$ represents the PM segment’s ROC error. If we can obtain the $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ of each PM segment, then the PM segment’s ROC error can be calculated by Eq. (6). When the magnitude of ROC error is comparatively small, we can simplify the function between ${R_{\textrm{e} \cdot \textrm{PS}}}$ and $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$, and consider that ${R_{\textrm{e} \cdot \textrm{PS}}}$ is proportionate to $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$.

Here we propose a method for detecting ROC mismatch among PM segments. First, the wavefront aberrations of the optical system with ROC mismatch among PM segments are reconstructed by using the phase retrieval algorithm described in Section 2. Second, the defocus coefficient $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ of each sub-aperture wavefront is fitted. Finally, PM segments’ ROC errors are calculated by Eq. (6). The ROC mismatch among PM segments can be obtained.

4.2 Method for decoupling aberrations induced by PM segment ROC error and SM lateral misalignments

Based on Section 3.3 and compared Eq. (4) and Eq. (5), it can be seen that the variations of full-aperture aberration $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ induced by SM lateral misalignments can convert into the variations of sub-aperture defocus $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}$. Therefore, It will be coupled with the sub-aperture defocus aberration induced by the PM segment’s ROC error ($\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$) which affects the detection of ROC mismatch among PM segments. Therefore, if we can calculate and remove $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ from wavefront distribution, the sub-aperture defocus aberration induced by SM lateral misalignments and the PM segment’s ROC error can be decoupled.

The full-aperture wavefront aberrations of an optical system with segmented PM are discontinuous due to ROC mismatch among PM segments and phasing error. Therefore, we cannot obtain the full-aperture aberrations induced by SM lateral misalignments by direct fitting. Here we will consider using sub-aperture aberrations of PM segments to calculate the full-aperture aberrations induced by SM lateral misalignments.

For a PM segment with number sn, we suppose that the functional relationships between the PM segment’s ROC error (${R_{\textrm{e} \cdot \textrm{PS}}}$) and sub-aperture aberrations induced by ${R_{\textrm{e} \cdot \textrm{PS}}}$($\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$, $\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{PS}}}$, $\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{PS}}}$) can be expressed as follows:

$${\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{PS}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{PS}}}} \end{array}} \right]_{sn \cdot \textrm{PS}}} = {\left[ {\begin{array}{{c}} {{f_{4 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )}\\ {{f_{7 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )}\\ {{f_{8 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )} \end{array}} \right]_{sn}}.$$

For the same PM segment as above, we also suppose that there are functional relationships between full-aperture aberrations induced by SM lateral misalignments and sub-aperture aberrations converted by the former. Such as the sub-aperture aberrations $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}$, $\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}$ converted by full-aperture aberrations $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$ can be expressed as follows:

$${\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 4\textrm{fS}}} = {\left[ {\begin{array}{{c}} {{f_{4 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{7 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{8 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )} \end{array}} \right]_{sn}},$$
where subscript 4fS represents aberrations converted by $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$. In the same way, we can write the following expression about full-aperture aberrations $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ and sub-aperture aberrations $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}$, $\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}$:
$${\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 7\textrm{fS}}} = {\left[ {\begin{array}{{c}} {{f_{4 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{7 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{8 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )} \end{array}} \right]_{sn}},$$
$${\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 8\textrm{fS}}} = {\left[ {\begin{array}{{c}} {{f_{4 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{7 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{8 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )} \end{array}} \right]_{sn}}.$$

The sub-aperture aberrations of the PM segment with number sn induced by SM lateral misalignments are the sum of Eq. (8) ∼ Eq. (10):

$$\begin{aligned} {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot \textrm{SM}}} &= {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 4\textrm{fS}}} + {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 7\textrm{fS}}} + {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot 8\textrm{fS}}}\\& = {\left[ {\begin{array}{{c}} {{f_{4 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{4 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{4 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{7 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{7 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{7 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{8 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{8 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{8 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )} \end{array}} \right]_{sn}} \end{aligned}$$

Therefore, for the PM segment with number sn, the total variations of sub-aperture aberrations induced by the PM segment’s ROC error and SM lateral misalignments are the sum of Eq. (7) and Eq. (11):

$$\begin{aligned} {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub}}}}\\ {\Delta {C_{7 \cdot \textrm{sub}}}}\\ {\Delta {C_{8 \cdot \textrm{sub}}}} \end{array}} \right]_{sn}} &= {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{PS}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{PS}}}} \end{array}} \right]_{sn \cdot \textrm{PS}}} + {\left[ {\begin{array}{{c}} {\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{7 \cdot \textrm{sub} \cdot \textrm{SM}}}}\\ {\Delta {C_{8 \cdot \textrm{sub} \cdot \textrm{SM}}}} \end{array}} \right]_{sn \cdot \textrm{SM}}}\\& = {\left[ {\begin{array}{{c}} {{f_{4 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )+ {f_{4 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{4 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{4 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{7 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )+ {f_{7 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{7 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{7 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )}\\ {{f_{8 \cdot \textrm{PS}}}({{R_{\textrm{e} \cdot \textrm{PS}}}} )+ {f_{8 \cdot 4\textrm{fS}}}({\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{8 \cdot 7\textrm{fS}}}({\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}} )+ {f_{8 \cdot 8\textrm{fS}}}({\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}} )} \end{array}} \right]_{sn}}. \end{aligned}$$

It can be seen that the number of unknown parameters (${R_{\textrm{e} \cdot \textrm{PS}}}$, $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$) is greater than the number of equations in Eq. (12). Fortunately, for any PM segments, we can all write an expression similar to Eq. (12). These expressions can construct an overdetermined equation set, and the unknown parameters can be solved by the least square method.

The framework for detecting ROC mismatch among PM segments is shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Framework for detecting ROC mismatch among PM segments.

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5. Simulation of detecting ROC mismatch among PM segments

This section verifies the effectiveness of the methods proposed in Section 4 by simulation. The simulation considers two cases: one is only ROC mismatch among PM segments exists (to verify the effectiveness of the method for detecting the ROC mismatch among PM segments proposed in Section 4.2); The other is both ROC mismatch among PM segments and SM lateral misalignments exist (to verify the effectiveness of the method for decoupling sub-aperture defocus aberration induced by SM lateral misalignments and PM segment’s ROC error proposed in Section 4.3).

In addition, we consider that the actual wavefront sensing is usually affected by noise, i.e., there is noise when using CCD to collect PSFs. Therefore, for the simulation in the above two cases, we further consider the case of noise existing. To simulate noise, we model each PSF to have Gaussian CCD readout noise with a standard deviation of 23 e- and a dark current of 0.2 e-/s over a 1 s integration time. The photon noise which is dependent on intensity follows a Poisson distribution. The peak signal-to-noise ratio (PSNR) is defined as:

$$PSNR = 20\textrm{lo}{\textrm{g}_{10}}\left( {\frac{{{S_{peak}}}}{{\sqrt {{S_{peak}} + \sigma_{read}^2 + \sigma_{dark}^2} }}} \right)$$
where ${S_{peak}}$ is the peak value of noise-free PSF, $\sigma _{read}^2$ and $\sigma _{dark}^2$ are the variances associated with the readout noise and the dark current noise at each pixel, respectively. The peak of PSF is set to 10,000 photons, which is restricted to full-well electron numbers. The final PSNR is approximately equal to 40 dB. The simulation process is shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. The process of simulation.

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5.1 Establishment of the simulation optical system

JWST’s design is a three-mirror anastigmat (TMA) telescope whose PM is composed of 18 hexagonal mirror segments [26]. In this section, a similar optical system is established in Code V with reference to the design parameters of JWST [27]. Some of the parameters of JWST and the simulation optical system are shown in Table 1. The optical layout of the simulation optical system is shown in Fig. 5(a). The segmented PM is shown in Fig. 5(b), where 18 segments are divided into three categories A, B, C, and numbered.

 figure: Fig. 5.

Fig. 5. (a) Optical layout of the simulation optical system. (b) The segmented PM.

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Table 1. Parameters of JWST and the simulation optical system

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First, we determine the relationship between PM segments’ ROC error and the variation of sub-aperture aberrations. PM segment C1 is taken as an example. The pupil-offset off-axis subsystem is shown in Fig. 6(a). As shown in Table 2, the sub-aperture wavefront aberrations of segment C1 can be easily read in Code V by introducing known ROC errors of PM segments. It can be seen that the variation of sub-aperture defocus aberration is the greatest, which is consistent with the conclusion in Section 4.1. As shown in Fig. 6(b), obviously, there is a linear relationship between the PM segment’s ROC error and sub-aperture defocus aberration. For each PM segment, the above relationship can be determined in the same way.

 figure: Fig. 6.

Fig. 6. (a) Pupil-offset off-axis subsystem corresponding to PM segment C1. (b) The relationship between the PM segment’s ROC error and sub-aperture defocus aberration of PM segment C1. It can be seen that there is a linear relationship between the PM segment’s ROC error and sub-aperture defocus aberration.

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Table 2. Variation of sub-aperture aberrations due to ROC error of PM segment C1

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Then, we determine the relationships between full-aperture aberrations induced by SM lateral misalignments and sub-aperture aberrations converted by the former. A set of random SM lateral misalignments will induce additional full-aperture wavefront aberrations $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$. The influence of $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ on sub-aperture wavefront aberrations of segment C1 is shown in Table 3. The relationship between full-aperture and sub-aperture wavefront aberrations is also linear. The influence of $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ on sub-aperture wavefront aberrations of the PM segment can be determined in the same way. Then we can obtain Eq. (12) for PM segment C1. In this paper, we use the sub-aperture wavefront aberrations of PM segment A1, A4, B1, B4, C2, and C5 to construct overdetermined equations set to solve $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$.

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Table 3. The influence of ${C_{8\,full}}$ on sub-aperture wavefront aberrations of segment C1

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5.2 Simulations for the case where only ROC mismatch among PM segments exists

A set of random PM segments’ ROC errors (the absolute range of ROC error is 0.01∼0.02 mm; positive and negative signs are also random) are inputted into the simulation optical system to simulate ROC mismatch among PM segments. Five PSF images are collected at five different defocus positions of ±6 mm, ± 3 mm, and 0 mm respectively. The simulation will be carried out in a noise-free case and a noisy case with 40 dB PSNR, respectively.

Phase retrieval results are shown in Fig. 7. The root mean square errors (RMSE) of the recovered wavefront aberrations are 0.00066 λ (Noise = 0 dB) and 0.00570 λ (Noise = 40 dB) respectively (λ = 2000nm). The calculated results of PM segments’ ROC error are shown in Table 4. The average RMSE of the PM segment’s ROC errors is 0.00054 mm (Noise = 0 dB) and 0.00259 mm (Noise = 40 dB) respectively.

 figure: Fig. 7.

Fig. 7. Phase retrieval results when only ROC mismatch among PM segments exist. (a) The result is when there is no noise in PSFs. (b) The result is when there is 40 dB noise in PSFs. It can be seen that the phase retrieval algorithm can reconstruct the wavefront accurately regardless of whether there is noise or not.

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Table 4. Result of calculated ROC error of PM segments for the first perturbation condition

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To verify the stability of the proposed method, 100 groups of Monte Carlo tests are carried out and the results are presented in Fig. 8. The average RMSE of recovered wavefront aberration is 0.00091 λ (Noise = 0 dB) and 0.00596 λ (Noise = 40 dB), respectively (λ = 2000nm). The average RMSE of PM segments’ ROC errors is 0.00058 mm (Noise = 0 dB) and 0.00221 mm (Noise = 40 dB).

 figure: Fig. 8.

Fig. 8. Monte Carlo simulations when only ROC mismatch among PM segments exists. The green line is the required ROC error detection accuracy. (a) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is no noise (100 groups of Monte Carlo tests). (b) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is 40 dB noise (100 groups of Monte Carlo tests). It can be seen that the method proposed in this paper can work well regardless of whether there is noise or not, PM segments’ ROC errors can be solved accurately, i.e. the information of ROC mismatch among PM segments can be obtained accurately. When there is 40 dB noise, although the accuracy of wavefront and ROC error decreases, the ROC error is still much lower than the green line.

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It can be proved that the method for detecting ROC mismatch among PM segments by using sub-aperture defocus aberration $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ is feasible. The detection accuracy is affected by the noise of the PSF images. RMSE of recovered wavefront aberration for the case where PSNR = 40 dB increases compared to the noise-free case, which leads to the decrease in the accuracy of calculated PM segment ROC error. However, the decrease of accuracy is acceptable because it is still lower than the upper limit of allowable accuracy (0.01 mm) [7].

5.3 Simulation for the case where both ROC mismatch among PM segments and SM lateral misalignments exist

Based on there is the ROC mismatch among PM segments, a group of random SM lateral misalignments are also induced. The range of SM decenters along the X and Y axis is [ -0.05 mm, 0.05 mm], range of SM tip-tilts around the X and Y axis is [-0.005°, 0.005°]. Five PSF images are collected at five different defocus distances of ±6 mm, ± 3 mm, and 0 mm, respectively. The simulation will be carried out in a noise-free case and a noisy case with a PSNR of 40 dB, respectively.

The phase retrieval results are shown in Fig. 9. The RMSE of the recovered wavefront aberrations are 0.00105 λ (Noise = 0 dB) and 0.00598 λ (Noise = 40 dB), respectively. Full-aperture wavefront aberrations $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ induced by SM lateral misalignments are calculated. The results are shown in Table 5. Then, we remove $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ from the full-aperture wavefront aberrations and calculate the ROC error of PM segments according to the method proposed in this paper. The results are shown in Table 6. The average RMSE of PM segments’ ROC errors is 0.00042 mm (Noise = 0 dB) and 0.00210 mm (Noise = 40 dB), respectively.

 figure: Fig. 9.

Fig. 9. The phase retrieval result when both ROC mismatch among PM segments and SM lateral misalignments exist. (a) The result is when there is no noise in PSFs. (b) The result is when there is 40 dB noise in PSFs. It can be seen that the phase retrieval algorithm can reconstruct the wavefront accurately regardless of whether there is noise or not.

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Table 5. Full-aperture wavefront aberrations induced by SM lateral misalignments

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Table 6. Result of calculated ROC error of PM segments for the second perturbation condition

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To further verify the effectiveness of the proposed method, 100 groups of Monte Carlo tests are carried out and the results are presented in Fig. 10. The average RMSE of recovered wavefront aberration is 0.00122 λ (Noise = 0 dB) and 0.00595 λ (Noise = 40 dB), respectively. The average RMSE of PM segments’ ROC errors is 0.00086 mm (Noise = 0 dB) and 0.00244 mm (Noise = 40 dB), respectively.

 figure: Fig. 10.

Fig. 10. Monte Carlo simulations when both ROC mismatch among PM segments and SM lateral misalignments exist. (a) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is no noise (100 groups of Monte Carlo tests). (b) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is 40 dB noise (100 groups of Monte Carlo tests). The red triangle represents the result without using the decoupling method proposed in this paper. The blue cross represents the result of using the decoupling method. The green line is the required ROC error detection accuracy. It can be seen that the decoupling method has no effect on the accuracy of the wavefront, but without using the decoupling method, PM segments’ ROC errors cannot be solved accurately, i.e., the information of ROC mismatch among PM segments cannot be obtained accurately.

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It can be seen that SM lateral misalignments do not affect the accuracy of recovered wavefront aberration, but it will lead to the failure of detecting ROC mismatch among PM segments. Without using the decoupling method, PM segments’ ROC errors cannot be solved accurately, i.e. the information of ROC mismatch among PM segments cannot be obtained accurately. Decoupling the sub-aperture wavefront aberrations induced by ROC error of PM segments and SM lateral misalignments is very important for accurately detecting the ROC mismatch among PM segments. Simulation results demonstrate the effectiveness of the proposed method.

6. Conclusion

Accurately detecting ROC mismatch among PM segments from the wavefront map is of crucial importance for efficiently correcting this kind of manufacturing error, while currently there are few related studies. In this paper, we propose a method for detecting ROC mismatch among PM segments. Based on NAT, the influence of the PM segment’s ROC error and SM lateral misalignments on wavefront aberrations is analyzed. Combined with the data obtained from the simulation optical system, the following conclusions are drawn.

  • (1) The PM segment’s ROC error mainly induces sub-aperture defocus aberration $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$, the relationship between error and aberration is linear.
  • (2) The $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ induced by SM lateral misalignments will generate sub-aperture defocus aberration $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{SM}}}$, which will be coupled with the $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$ induced by PM segment’s ROC error.

According to the above conclusions, we proposed a method that is suitable for detecting ROC mismatch among PM segments on orbit as follows:

  • (1) Wavefront aberration is recovered by using the image-based phase retrieval algorithm.
  • (2) Using sub-aperture wavefront aberrations, the full-aperture aberration $\Delta {C_{4 \cdot \textrm{full} \cdot \textrm{SM}}}$, $\Delta {C_{7 \cdot \textrm{full} \cdot \textrm{SM}}}$ and $\Delta {C_{8 \cdot \textrm{full} \cdot \textrm{SM}}}$ induced by SM lateral misalignments is calculated and removed (i.e., decoupling the aberrations induced by PM segment’s ROC error and SM lateral misalignments).
  • (3) The PM segments’ ROC errors are calculated according to the sub-aperture defocus aberration $\Delta {C_{4 \cdot \textrm{sub} \cdot \textrm{PS}}}$, and ROC mismatch among PM segments can be obtained.

Finally, the simulation results under various working conditions show that the method proposed in this paper can accurately detect the ROC mismatch among PM segments. The method is effective. The research done in this paper provides a feasible scheme for detecting ROC mismatch among PM segments.

Funding

National Natural Science Foundation of China (61905241, 62205334).

Disclosures

The authors declare that they have no conflicts of interest related to this article.

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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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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Figures (10)
Fig. 1.
Fig. 1. The Gerchberg-Saxton Algorithm (GSA).

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Fig. 2.
Fig. 2. The modified GSA using multiple PSF images.

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Fig. 3.
Fig. 3. Framework for detecting ROC mismatch among PM segments.

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Fig. 4.
Fig. 4. The process of simulation.

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Fig. 5.
Fig. 5. (a) Optical layout of the simulation optical system. (b) The segmented PM.

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Fig. 6.
Fig. 6. (a) Pupil-offset off-axis subsystem corresponding to PM segment C1. (b) The relationship between the PM segment’s ROC error and sub-aperture defocus aberration of PM segment C1. It can be seen that there is a linear relationship between the PM segment’s ROC error and sub-aperture defocus aberration.

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Fig. 7.
Fig. 7. Phase retrieval results when only ROC mismatch among PM segments exist. (a) The result is when there is no noise in PSFs. (b) The result is when there is 40 dB noise in PSFs. It can be seen that the phase retrieval algorithm can reconstruct the wavefront accurately regardless of whether there is noise or not.

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Fig. 8.
Fig. 8. Monte Carlo simulations when only ROC mismatch among PM segments exists. The green line is the required ROC error detection accuracy. (a) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is no noise (100 groups of Monte Carlo tests). (b) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is 40 dB noise (100 groups of Monte Carlo tests). It can be seen that the method proposed in this paper can work well regardless of whether there is noise or not, PM segments’ ROC errors can be solved accurately, i.e. the information of ROC mismatch among PM segments can be obtained accurately. When there is 40 dB noise, although the accuracy of wavefront and ROC error decreases, the ROC error is still much lower than the green line.

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Fig. 9.
Fig. 9. The phase retrieval result when both ROC mismatch among PM segments and SM lateral misalignments exist. (a) The result is when there is no noise in PSFs. (b) The result is when there is 40 dB noise in PSFs. It can be seen that the phase retrieval algorithm can reconstruct the wavefront accurately regardless of whether there is noise or not.

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Fig. 10.
Fig. 10. Monte Carlo simulations when both ROC mismatch among PM segments and SM lateral misalignments exist. (a) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is no noise (100 groups of Monte Carlo tests). (b) RMSE distribution of wavefront aberrations and ROC errors (calculate by wavefront aberrations) when there is 40 dB noise (100 groups of Monte Carlo tests). The red triangle represents the result without using the decoupling method proposed in this paper. The blue cross represents the result of using the decoupling method. The green line is the required ROC error detection accuracy. It can be seen that the decoupling method has no effect on the accuracy of the wavefront, but without using the decoupling method, PM segments’ ROC errors cannot be solved accurately, i.e., the information of ROC mismatch among PM segments cannot be obtained accurately.

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Tables (6)

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Table 1. Parameters of JWST and the simulation optical system

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Table 2. Variation of sub-aperture aberrations due to ROC error of PM segment C1

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Table 3. The influence of C 8 f u l l on sub-aperture wavefront aberrations of segment C1

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Table 4. Result of calculated ROC error of PM segments for the first perturbation condition

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Table 5. Full-aperture wavefront aberrations induced by SM lateral misalignments

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Table 6. Result of calculated ROC error of PM segments for the second perturbation condition

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Equations (13)

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P ( ε , η ) = p ( ε , η ) e i φ ( ε , η ) ⟵ ⟶ F − 1 F h ~ ( x , y ) = | h ~ ( x , y ) | e i Φ ( ε , η ) , P S F ( x , y ) = h ~ ( x , y ) h ~ ∗ ( x , y ) = | h ~ ( x , y ) | 2 ,
W o n ⋅ a x i s = ∑ j ∑ p = 0 ∞ ∑ n = 0 ∞ ∑ m = 0 ∞ W k l m ⋅ j [ ( H − σ j ) ⋅ ( H − σ j ) ] p ( ρ ⋅ ρ ) n [ ( H − σ j ) ⋅ ρ ] m , k = 2 p + m , l = 2 n + m ,
W o f f ⋅ a x i s = ∑ j ∑ p = 0 ∞ ∑ n = 0 ∞ ∑ m = 0 ∞ W k l m ⋅ j ( H A ⋅ j ⋅ H A ⋅ j ) p [ ( ρ + s ) ⋅ ( ρ + s ) ] n [ H A ⋅ j ⋅ ( ρ + s ) ] m , k = 2 p + m , l = 2 n + m .
W o f f ⋅ a x i s ⋅ j = { W 020 ⋅ j + 4 W 040 ⋅ j ( s ⋅ s ) + W 220 M ⋅ j ( H A ⋅ j ⋅ H A ⋅ j ) + 2 W 131 ⋅ j ( H A ⋅ PMs ⋅ s ) } ( ρ ⋅ ρ ) ⋯ D e f o c u s + { 2 W 040 ⋅ j s 2 + 1 2 W 222 ⋅ j H A ⋅ j 2 + W 131 ⋅ j H A ⋅ j s } ⋅ ρ 2 ⋯ A s t + { 4 W 040 ⋅ j s + W 131 ⋅ j H A ⋅ j } ⋅ ρ ( ρ ⋅ ρ ) ⋯ C o m a + { W 040 ⋅ j } ( ρ ⋅ ρ ) 2 ⋯ S A 3 ,
W o n ⋅ a x i s ⋅ j = { W 020 ⋅ j + W 220 M ⋅ j ( H A ⋅ j ⋅ H A ⋅ j ) } ( ρ ⋅ ρ ) ⋯ D e f o c u s + 1 2 { W 222 ⋅ j H A ⋅ j 2 } ⋅ ρ 2 ⋯ A s t + { W 131 ⋅ j H A ⋅ j } ⋅ ρ ( ρ ⋅ ρ ) ⋯ C o m a + { W 040 ⋅ j } ( ρ ⋅ ρ ) 2 ⋯ S A 3 ,
R e ⋅ PS = f ( Δ C 4 ⋅ sub ⋅ PS ) ,
[ Δ C 4 ⋅ sub ⋅ PS Δ C 7 ⋅ sub ⋅ PS Δ C 8 ⋅ sub ⋅ PS ] s n ⋅ PS = [ f 4 ⋅ PS ( R e ⋅ PS ) f 7 ⋅ PS ( R e ⋅ PS ) f 8 ⋅ PS ( R e ⋅ PS ) ] s n .
[ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 4 fS = [ f 4 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) f 7 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) f 8 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) ] s n ,
[ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 7 fS = [ f 4 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) f 7 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) f 8 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) ] s n ,
[ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 8 fS = [ f 4 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 7 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 8 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) ] s n .
[ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ SM = [ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 4 fS + [ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 7 fS + [ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ 8 fS = [ f 4 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 4 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 4 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 7 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 7 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 7 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 8 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 8 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 8 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) ] s n
[ Δ C 4 ⋅ sub Δ C 7 ⋅ sub Δ C 8 ⋅ sub ] s n = [ Δ C 4 ⋅ sub ⋅ PS Δ C 7 ⋅ sub ⋅ PS Δ C 8 ⋅ sub ⋅ PS ] s n ⋅ PS + [ Δ C 4 ⋅ sub ⋅ SM Δ C 7 ⋅ sub ⋅ SM Δ C 8 ⋅ sub ⋅ SM ] s n ⋅ SM = [ f 4 ⋅ PS ( R e ⋅ PS ) + f 4 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 4 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 4 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 7 ⋅ PS ( R e ⋅ PS ) + f 7 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 7 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 7 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) f 8 ⋅ PS ( R e ⋅ PS ) + f 8 ⋅ 4 fS ( Δ C 4 ⋅ full ⋅ SM ) + f 8 ⋅ 7 fS ( Δ C 7 ⋅ full ⋅ SM ) + f 8 ⋅ 8 fS ( Δ C 8 ⋅ full ⋅ SM ) ] s n .
P S N R = 20 lo g 10 ( S p e a k S p e a k + σ r e a d 2 + σ d a r k 2 )
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