## Abstract

Space-based optical astronomical telescopes are susceptible to mirror misalignments due to space disturbance in mechanics and temperature. Therefore, it is of great importance to actively align the telescope in orbit to continuously maintain imaging quality. Traditional active alignment methods usually need additional delicate wavefront sensors and complicated operations (such as instrument calibration and pointing adjustment). This paper proposes a novel active alignment approach by matching the geometrical features of several stellar images at arbitrary multiple field positions. Based on nodal aberration theory and Fourier optics, the relationship between stellar image intensity distribution and misalignments of the system can be modeled for an arbitrary field position. On this basis, an objective function is established by matching the geometrical features of the collected multi-field stellar images and modeled multi-field stellar images, and misalignments can then be solved through nonlinear optimization. Detailed simulations and a real experiment are performed to demonstrate the effectiveness and practicality of the proposed approach. This approach eliminates the need for delicate wavefront sensors and pointing adjustment, which greatly facilitates the maintainance of imaging quality.

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

## 1. Introduction

Astronomical telescopes represent an important class of astronomical observation instruments. They play an important role in the field of astronomy, especially those space-borne ones, which can achieve sharper images than ground-based telescopes due to completely free of the atmospheric turbulence and absorption effects [1,2].

It is important to note that the imaging performance of space-based large-aperture astronomical telescopes is susceptible to space perturbations, which can gradually lead to mirror misalignments during long-term on-orbit observation and operation. These misalignments originating from various thermal/mechanical stresses on the optics will introduce wavefront aberrations and degrade the imaging quality. Therefore, in order to maintain the imaging quality of the optical system, it is critical to periodicity actively align the system and correct those aberration induced by misalignments [3–5].

A typical class of misalignment calculation method can be called numerical method, including sensitive table method (STM) [6,7], artificial neural networks method (ANN) [8], merit function (MF) regression method [9–11] and differential wavefront sampling (DWS) [12,13]. STM mainly uses the linear relationship between Zernike aberration coefficients and different misalignment parameters to calculate misalignments; ANN is a mathematical fitting tool to establish the nonlinear mapping relation between misalignments and Zernike coefficients; MF regression method deals with current and target aberration values to obtain misalignments; DWS method pays attention to the coupling effects of inter-elements. These methods are widely used in alignment or active alignment of astronomical telescopes.

Analytic aberration theories have also been deeply investigated and widely applied in the field of optical alignment. Nodal aberration theory (NAT), discovered by Shack and developed by Thompson, is one of the analytic tools in studying the aberration behavior of on-axis optical systems which contain misaligned, or intentionally tilted and/or decentered components [14–19]. NAT has been applied in the alignment of on-axis two-mirror and three-mirror telescope systems [20–22]. In recent years, NAT has been expanded to freeform surfaces [23–27] and pupil-offset off-axis systems [28–32]. The effects of axial [28], lateral [29,30] and rotational misalignments [31] on the aberration fields of the pupil-offset off-axis telescope systems had been analyzed respectively. Besides, there are some other analytic theories discussing misalignment-induced aberrations [33–35], and many aspects of these works were developed independent of the NAT. These works contain strong relevance to the theory of misalignment-induced aberrations and their image shape patterns.

However, these methods usually depend heavily on wavefront sensing for several certain field points. Wavefront sensing methods for space telescopes mainly include Shack-Hartmann wavefront sensor method [36], curvature sensing [37], phase retrieval (PR) method [38,39] and phase diversity (PD) method [40,41]. Shack-Hartmann sensor and curvature sensing usually need additional optical elements and need to be calibrated to meet the accuracy requirements, while on-orbit calibration further increases the complexity of the system [42,43]. In addition, while PR and PD can use stellar images at arbitrary field points to recover the wavefront phase, the field positions needed for wavefront sensing are usually fixed (because the sensitivity table is fixed and it is related to field position). In other words, if there is no suitable star located at these pre-set field positions, the pointing of the telescope should be adjusted, which also increases wavefront sensing complexity and greatly decreases the alignment efficiency. Therefore, it is necessary to develop a novel alignment approach that are not restricted by wavefront sensing for fixed field positions.

This paper proposes a novel active alignment approach by matching the geometrical features of several stellar images at arbitrary multiple field positions. Based on nodal aberration theory and Fourier optics, the relationship between stellar image intensity distribution and misalignments of the system can be modeled for an arbitrary field position. After feature extraction, the relations between features of stellar image and misalignments can also be established for an arbitrary field position. On this basis, an objective function is established by matching the geometrical features of the collected multi-field stellar images and modeled multi-field stellar images. Misalignments can then be solved through nonlinear optimization. Detailed simulations and a real experiment are performed to demonstrate the effectiveness and practicality of the proposed approach.

This paper is organized as follows. Section 2 introduce some basic theories of nodal aberration theory. Section 3 analytically establishes the relations between stellar image features and system misalignments for an arbitrary field position. Section 4 proposes the approach of solving misalignments by matching multi-field stellar image features through nonlinear optimization. Detailed simulations and a real experiment are presented in Section 5 and Section 6, respectively. This paper is concluded in Section 7.

## 2. Third-order wave aberration function in the presence of misalignments based on the nodal aberration theory

The sum of the total aberrations of the system can be attributed to all individual surface contributions. Wave aberration function of astronomical telescope can be expressed as [44]

*j*, $\vec{H}$ denotes a normalized vector for the field height in the field height, $\vec{\rho }$ is the normalized pupil vector describing the position in the pupil, $\vec{s}$ represents the location of the off-axis pupil relative to the on-axis pupil which is also normalized by semi-diameter of the off-axis aperture. Importantly, for on-axis system, $\vec{s} = 0$.

According to nodal aberration theory, the aberration field center associated with each individual surface will shift in presence of lateral misalignments, and the normalized field height $\vec{H}$in Eq. (1) should be replaced with a new normalized field vector, ${\vec{H}_{Aj}}$, which can be described as ${\vec{H}_{Aj}} = \vec{H} - {\vec{\sigma }_j}$. Here ${\vec{\sigma }_j}$ represents the position of the shifted aberration field center for *j*th individual surface, which is directly related to the lateral misalignment parameters of the optical system [45]. In this case, the wave aberration function can be rewritten as

Equation (3) shows that the wave aberration terms induced by misalignments represent a linear distribution in the full FOV (all the aberration terms containing ${\vec{\sigma }_j}$ are field-linear or field-constant).

## 3. Deterministic relationship between the features of the stellar image and misalignments for an arbitrary field position

According to nodal aberration theory, we can model the relationship between aberration field and misalignments. On this basis, we can further model the relationship between stellar image intensity distribution at certain field position and misalignments based on Fourier optics principle. After feature extraction, the relations between features of stellar image at certain field position and misalignments can also be established.

#### 3.1 Analytic relationship between the stellar image and misalignments for an arbitrary field position

According to Eq. (3), the 4^{th}-9^{th} fringe Zernike coefficients (defocus, 0° astigmatism, 45° astigmatism, 0° coma, 90° coma and spherical aberration) corresponding to the third-order aberrations at special field point can be expressed as

*j*th fringe Zernike coefficient of the misaligned optical system at the field point $\vec{H}\textrm{ = }{({{h_x},{h_y}} )^T}$, ${\mathbf v}$ is the vector composed by misalignment parameters, $C_j^0({\vec{H}} )$represents the

*j*th fringe Zernike coefficient in the nominal state, and$\Delta {C_j}({\vec{H},{\mathbf v}} )$ represents the net change of aberration coefficient at the field point $\vec{H}$ induced by misalignments.

Since $\Delta {C_j}({\vec{H},{\mathbf v}} )$ represents a linear distribution in the full FOV, it can be further expressed as

The wavefront aberration at special field point $\vec{H} = ({{h_x},{h_y}} )$ can be expressed as a linear combination of fringe Zernike polynomials,

*j*th fringe Zernike polynomial, and

*j*is generally taken from 4 to 9 for on-orbit alignment. Here ${\mathbf W}$ and ${{\mathbf Z}_j}$ are matrices which represent the two-dimensional distribution of wave aberration and Zernike coefficient on the pupil surface respectively.

According to Fourier optics, the stellar spot image (point spread function) obtained from telescope in the presence of misalignments can be expressed as

*i*is an imaginary unit, $F{T^{ - 1}}$ is the inverse Fourier transform, and${\mathbf A}$is a binary two-dimensional matrix, and the value of the matrix element is 1 inside the normalized aperture and the rest is 0.

Given a set of misalignment parameters, we can easily model the intensity of a stellar image for an arbitrary field position according to Eqs. (4)–(7) and Table 1.

#### 3.2 Feature extraction of stellar images using Tchebichef moments

In this paper, the discrete orthogonal Tchebichef moments are introduced to extract or represent the features of the stellar spot image. The Tchebichef moments of the stellar spot image can be calculated by

*x*and

*y*are the pixel index numbers of stellar spot gray image in two dimensions, ${\mathbf I}$ is the image data matrix of stellar defocus spot, ${{\mathbf t}_p}(x )$ and ${{\mathbf t}_q}(y )$ are two one-dimensional vectors, which represent the specific values of Tchebichef moments in two dimensions of gray image (each pixel index number corresponds to a value). The specific expression can be expressed as

## 4. Active alignment of the astronomical telescope by matching arbitrary multi-field stellar image features through nonlinear optimization

Since there is a relationship between the features of a stellar image at certain field position and misalignments, the features of stellar images at several arbitrary positions within field of view can be used to solve misalignments of the system.

We suppose that the number of stellar images used to solve misalignments is *s*, the stellar spot images collected from the telescope are ${{\mathbf I}_1}$, ${{\mathbf I}_2}$,…, ${{\mathbf I}_s}$and the corresponding field coordinates are ${\vec{H}_1}$, ${\vec{H}_2}$,…, ${\vec{H}_s}$. On one hand, low-order Tchebichef moments can be used to directly extract the image feature vectors of *s* stellar spot images collected from the telescope, which can be expressed as in the form of a feature vector:

*s*stellar images of the corresponding

*s*field positions can be modeled according to Eqs. (4)-(11). The feature vectors corresponding to these

*s*modeled spot images can also be calculated using Tchebichef moments, which can be expressed as another feature vector:

The process of objective function establishment is illustrated in Fig. 1.

Procedures of active alignment of astronomical telescopes is presented below (also illustrated in Fig. 2):

- (1) Collect multi-field stellar images from the system. Note that the field position of the available star is usually random. However, the presented alignment approach can be applicable to arbitrary multi-field stellar images (the field position of each stellar image should be known, and it is better that these stars are distributed throughout the field of view). Besides, it is better that the stellar images are defocused and therefore more information can be used.
- (2) Analyze the imaging quality and determine if it has degenerated. If the imaging quality satisfies the requirements for astronomical observation, then quit the active alignment process.
- (3) Solving the misalignments of system using the approach presented above if the imaging quality has degenerated. In the process, particle swarm optimization (PSO) is selected as the optimization tool. If the objective function is small enough after optimization (optimization criterion is satisfied), then get the value of misalignments.
- (4) Align the system by applying the inverse of the solved misalignments to the system, and then repeat the above procedures.

Importantly, for astronomical telescopes, the number of the adjustable degrees of freedom is usually less than the number of misalignments. For example, for a TMA telescope, typically only the secondary mirror (SM) can be adjusted because the aberrations induced by misalignments of primary mirror (PM) and tertiary mirror (TM) can be effectively compensated by adjusting SM. Besides, it is expensive to actively adjust the position of PM and TM. In this case, the vector ${\mathbf v}$ in Eq. (14) and Fig. 2 no longer represents misalignment parameters, but represents adjustable degree of freedoms.

The minimum number of field positions used in this approach is three. The main reason is that the field-linear component is the dominant field-dependent aberrations and those higher-order components induced by misalignments can be neglected. Three field points are needed to determine the magnitude and orientation of the field-linear components.

## 5. Simulations

In this section, detailed alignment simulations will be carried out to demonstrate the practical feasibility and accuracy of the proposed approach under certain misalignment conditions and noise levels. The optical system for the Supernova/Acceleration Probe (SNAP) Mission will be used in the simulation. The optical prescription of SNAP is presented in Table 2 [46] and the optical layout of the preliminary SNAP telescope is shown in Fig. 3. Note that the position of each mirror is defined with reference to its previous mirror (not a global coordinate system).

#### 5.1 Single simulation used to show the intermediate processes

In this section, a single simulation will be performed to illustrate the intermediate procedures of the proposed approach, which is listed below:

- (1) Introduction of a set of misalignments. The PM is selected as coordinate reference, and the randomly introduced misalignments of SM and TM are listed in Table 3, where
*XDE*and*YDE*are the mirror vertex decenters in the x-z and y-z plane, respectively,*BDE*and*ADE*are the mirror tip-tilts in the x-z and y-z plane, respectively, and ZDE represents the error in spacing between mirrors. Then they are introduced to the optical simulation software to simulate an initial misalignment state. - (2) Acquisition of several defocus stellar images at randomly selected multiple field positions (the field position of each stellar image should be known, and it is better that these stars are distributed throughout the field of view). The deliberately introduced defocus distance is 2 mm. The four defocused stellar images are shown in Fig. 4.
- (3) Addition of noise. Here we mainly consider photon noise, readout noise and dark current noise. Signal-to-noise (
*SNR*) ratio of each pixel is expressed as$$SNR = 20{\log _{10}}(\frac{{{S_{(u,v)}}}}{{\sqrt {{S_{(u,v)}} + \sigma _{read}^2 + \sigma _{dark}^2} }}),$$where ${S_{(u,v)}}$is the peak pixel value of the noise-free image, $\sigma _{read}^2$and $\sigma _{dark}^2$ are the variances associated with the readout noise and the dark current noise, respectively. The peak SNR is used to evaluate the added noise level in one stellar image, and the peak SNR of the defocused stellar images is 30 dB [47]. These defocused stellar images with noise are shown in Fig. 5. - (4) Determine the adjustable degrees of freedom used to compensate for system misalignments. In this simulation, the five degrees of freedom of SM ($XD{E_{SM}}$, $YD{E_{SM,}}$, $AD{E_{SM}}$, $BD{E_{SM}}$ and $ZD{E_{SM}}$) and 3 degrees of freedom of folding mirror ($AD{E_{FM}}$, $BD{E_{FM}}$ and $ZD{E_{FM}}$) are selected as the adjustable degrees of freedom. Here the SM is used to compensate for the non-rotationally symmetric aberrations of the TM, and the folding mirror is used correct the inclination of image plane. The vector ${\mathbf v}$ in Eq. (14) and Fig. 2 now represents these adjustable degrees of freedom.
- (5) Solving the value of these adjustable degrees of freedom (contained in vector ${\mathbf v}$) with the proposed approach, which is used to compensate for the system misalignments.

After matching the multi-field stellar image features through optimization, the feature values of the collected stellar images (column A) and modeled stellar images (column B) are presented in Table 4 (before calculating image features, the sum of the image intensity is normalized to 1). We can recognize that these two sets of feature values are consistent with each other.

The calculation results of these adjustable degrees of freedom are presented in Table 5, which are the equivalent misalignments of the SM and FM that can introduce the same amounts of misalignment-induced aberrations. The negative values of the Table 5 are the delta adjustment to each degree of freedom. Subtracting those values in Table 5 from the corresponding values in Table 3 is the final position of each degree of freedom. Full field displays (FFDs) for different aberration types (astigmatism C5/C6, coma C7/C8, and medial focal surface C4) and different alignment states (nominal state, misaligned state and aligned state) are shown in Fig. 6. We can see that different aberrations are nearly corrected to the nominal state, and the average RMS is reduced from 0.389λ to 0.033λ, which demonstrate the effectiveness of the proposed active alignment approach.

#### 5.2 Monte Carlo simulations

In this subsection, Monte-Carlo simulations will be performed to demonstrate the effectiveness of the proposed method. Two cases with different noise levels are considered. Specifically, one is the noise-free case and no adjustment error is considered; the peak SNR in the other case is 30 dB and the error in adjustment is further considered. The adjustment error is defined as 1/100 of the adjustment value plus a random value in certain range (within ±1µm for decenters and ±1″for tip-tilts). In each case, 100 pairs of misalignments are randomly generated following a uniform distribution within the specified perturbation ranges shown in Table 6. For each perturbation state, the simulation process presented above will be performed. The average RMS WFE of 11×11 field points will be used to evaluate the image quality of SNAP telescope. The average RMS WFEs for 100 different sets of misalignment states are shown in Fig. 7. The average RMS WFEs after active alignment using the proposed approach for the two cases are shown in Fig. 8 (for the second case, we conduct a second alignment iteration).

In the noise-free cases, the average RMS WFE of the optical system after active alignment is very close to the nominal state, as shown in Fig. 8(a). Even in the noisy case with a peak SNR of 30 dB, the average RMS WFE after alignment is still smaller than 0.07λ, as shown in Fig. 8(b). These results demonstrate the effectiveness and practicality of the proposed method, while the noise can actually decrease the accuracy of the proposed method to some extent.

#### 5.3 Influence of noise on the accuracy of the proposed approach

The existence of image noise reduces the alignment accuracy of the optical system. In this subsection, the relationship between alignment accuracy of the proposed approach and noise level will be further discussed. For each noise level (evaluated using peak SNR), 100 sets of random misalignments following a uniform distribution are generated and introduced to the optical simulation software. For each misalignment state, defocused stellar images from different field points are collected. Different levels of noise are added to the stellar images. Then the proposed approach is utilized to align the system. The root mean square deviation (RMSD) between the RMS WFE of the alignment state and nominal state of the 100 sets of misalignments is calculated, which can be expressed by

The simulation results for different noise levels are shown in Fig. 10. It demonstrates that the alignment accuracy of the proposed approach is actually related to the noise level. As the peak pixel signal-to-noise increases, the alignment accuracy of the system gradually improves. When the peak signal-to-noise ratio is higher than 35 dB, the influence of noise on the alignment accuracy of the proposed approach can be ignored.

Note that the value of 0.07 waves in Fig. 9 has different meanings with the value of 0.07 waves for Fig. 8(b). The value of 0.07 waves is the tolerance for Fig. 8(b), while the value of 0.07 waves in Fig. 9 is the RMSD, which evaluates the residual error of active alignment (i.e., deviation between the wavefront error of nominal state and actively aligned state).

## 6. Experiment

In this section, a two-mirror telescope will be used to further validate the effectiveness of the proposed approach. The entrance pupil diameter of the off-axis two-mirror telescope is 60 mm, the effective focal length of the system is 839.9 mm and the magnitude for pupil offset is 90 mm. The experimental procedures are listed below:

- (1) Align the two-mirror telescope close to the nominal state.
An interferometer (PhaseCam 6000) is used to align the two-mirror telescope. The sketch and physical map of the self-collimation optical path used to align the two-mirror telescope is shown in Fig. 10. A PI hexapod is used to accurately adjust the position and pose of the SM. The wavefront errors at four marginal field positions ((-0.3°, -0.3°), (0.3°, -0.3°), (-0.3°, 0.3°), (0.3°, 0.3°)) of the aligned initial system are measured using interferometer, which are shown in Fig. 13(a).

- (2) Introduce a set of misalignments to the system.
A set of SM misalignments is randomly generated and introduced to the system using hexapod. Then wavefront error at four marginal field positions signified above are measured using interferometer, which are shown in Fig. 13(b). We can see that a large field-constant astigmatism is introduced to the system.

- (3) Collect multi-field defocused point spread functions (PSFs)
In order to generate stellar image in this experiment, we use the light generated by the collimator to simulate the light emitted by the star. Sketch and physical map of the experimental setup for generating and collecting multi-field PSF images are presented in Fig. 11. Four defocused PSFs corresponding to the above 4 field positions are collected, which are shown in Fig. 12. Meanwhile, a theodolite is used to transfer coordinate and determine the pointing of the collimator for a certain field.

- (4) Solve the adjustment of the SM using the proposed approach and apply it to the system.

After matching the multi-field PSF features through optimization, the feature values of the collected defocused PSFs and modeled PSFs are presented in Table 7. We can recognize that these two sets of feature values are consistent with each other. After solving the adjustment of the SM and applying it to the system, multi-field wavefront aberration measurements are shown in Fig. 13(c). We can see that the system is nearly corrected to the initial state, which demonstrates the effectiveness of the proposed approach.

## 7. Conclusion

This paper proposes a novel active alignment approach by matching the geometrical features of several stellar images at arbitrary field positions through nonlinear optimization. Based on the framework of nodal aberration theory and Fourier optics, the relation between stellar images for an arbitrary field point and misalignment parameters can be modeled. Tchebichef moments which are discrete orthogonal are introduced to represent stellar image features. By comparing Tchebichef moment features of the modeled PSFs between collected stellar images at several arbitrary field points, an objective function is established. Using particle swarm optimization, we can obtain the amount of mirror position adjustment for active alignment of space telescopes. Compared with traditional alignment methods which usually need wavefront sensing for certain field positions, this approach has the following advantages:

- (1) First of all, based on the insights about aberration field characteristics provided by nodal aberration theory, several PSF images at arbitrary field positions are handled at one time to character the wavefront aberrations across the whole field and calculate misalignments. On the other hand, PR or PD can only process the PSF images for one certain field at one time, they do not utilize the inherent relations between PSFs at different field of view.
- (2) Secondly, compared to PR or PD which directly use intensity distribution of PSF images, we first effectively compress the image data using discrete orthogonal Tchebichef moments, which provides some advantages in feature matching (for example, the computational load is smaller, and we need not to consider the error in image registration between the collected image and the modeled image using Fourier optics.
- (3) Finally, in some traditional alignment methods, wavefront sensing at certain field points is needed, because the sensitivity table is fixed and it is related to field position. In this case, if no suitable star is located at corresponding field positions, we should adjust the pointing of the telescope. In our approach, wavefront errors at arbitrary filed position can be used align the system for nodal aberration theory is inherent in this approach.

In the paper, to guarantee the accuracy of the proposed approach, we use 16 Tchebichef moments to describe each PSF. However, it is hard to reveal some intuitive insights into the relations between these image features and misalignments (or aberrations). In fact, there are some references discussing the analytic relations between image features and aberrations [48–50]. They utilize the shape of PSF and does not require Fourier transform in its estimation. Therefore, they can be easily implemented with rapid aberration estimation.

## Funding

National Natural Science Foundation of China (61905241, 61705220); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019219).

## Acknowledgments

We thank Synopsys for providing the educational license of CODE V.

## 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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