Active Alignment of Large-Aperture Space Telescopes for Optimal Ellipticity Performance.

Ellipticity performance of space telescopes is important for exploration of dark matter. However, traditional on-orbit active optical alignment of space telescopes often takes "minimum wavefront error across the field of view" as the correction goal, and the ellipticity performance after correcting the wave aberration is not optimal. This paper proposes an active optical alignment strategy to achieve optimal ellipticity performance. Based on the framework of nodal aberration theory (NAT), the aberration field distribution corresponding to the optimal full field-of-view ellipticity is determined using global optimization. The degrees of freedom (DOFs) of the secondary mirror and the folded flat mirror are taken as the compensation DOFs to achieve the optimal ellipticity performance. Some valuable insights into aberration field characteristics corresponding to optimal ellipticity performance are presented. This work lays a basis for the correction of ellipticity for complicated optical systems.

Fast High-Resolution Phase Diversity Wavefront Sensing with L-BFGS Algorithm.

The presence of manufacture error in large mirrors introduces high-order aberrations, which can severely influence the intensity distribution of point spread function. Therefore, high-resolution phase diversity wavefront sensing is usually needed. However, high-resolution phase diversity wavefront sensing is restricted with the problem of low efficiency and stagnation. This paper proposes a fast high-resolution phase diversity method with limited memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm, which can accurately detect aberrations in the presence of high-order aberrations. An analytical gradient of the objective function for phase-diversity is integrated into the framework of the L-BFGS nonlinear optimization algorithm. L-BFGS algorithm is specifically suitable for high-resolution wavefront sensing where a large phase matrix is optimized. The performance of phase diversity with L-BFGS is compared to other iterative method through simulations and a real experiment. This work contributes to fast high-resolution image-based wavefront sensing with a high robustness.

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

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.

Jitter-Robust Phase Retrieval Wavefront Sensing Algorithms.

Phase retrieval wavefront sensing methods are now of importance for imaging quality maintenance of space telescopes. However, their accuracy is susceptible to line-of-sight jitter due to the micro-vibration of the platform, which changes the intensity distribution of the image. The effect of the jitter shows some stochastic properties and it is hard to present an analytic solution to this problem. This paper establishes a framework for jitter-robust image-based wavefront sensing algorithm, which utilizes two-dimensional Gaussian convolution to describe the effect of jitter on an image. On this basis, two classes of jitter-robust phase retrieval algorithms are proposed, which can be categorized into iterative-transform algorithms and parametric algorithms, respectively. Further discussions are presented for the cases where the magnitude of jitter is unknown to us. Detailed simulations and a real experiment are performed to demonstrate the effectiveness and practicality of the proposed approaches. This work improves the accuracy and practicality of the phase retrieval wavefront sensing methods in the space condition with non-ignorable micro-vibration.

Channeled spectropolarimeter with arbitrary retarder orientation settings.

A channeled spectropolarimeter can simultaneously obtain intensity, spectral, and polarization information. In the traditional model, the retarders must be oriented at specific angles. However, misalignments of the retarders are inevitable during assembly, and the status of the retarders is sensitive to environmental perturbations, which affects the performance of the channeled spectropolarimeter. In this study, a general channeled spectropolarimeter model was derived, in which the retarder orientations can be arbitrary and unknown. Meanwhile, the system is unaffected by environmental perturbation because it can self-calibrate to avoid fluctuations in the retarder orientations and phase retardations. The effectiveness and robustness of the model were verified through simulations and experiments.

Active alignment of space astronomical telescopes by matching arbitrary multi-field stellar image features.

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.

Deep learning wavefront sensing for fine phasing of segmented mirrors.

Segmented primary mirror provides many crucial important advantages for the construction of extra-large space telescopes. The imaging quality of this class of telescope is susceptible to phasing error between primary mirror segments. Deep learning has been widely applied in the field of optical imaging and wavefront sensing, including phasing segmented mirrors. Compared to other image-based phasing techniques, such as phase retrieval and phase diversity, deep learning has the advantage of high efficiency and free of stagnation problem. However, at present deep learning methods are mainly applied to coarse phasing and used to estimate piston error between segments. In this paper, deep Bi-GRU neural work is introduced to fine phasing of segmented mirrors, which not only has a much simpler structure than CNN or LSTM network, but also can effectively solve the gradient vanishing problem in training due to long term dependencies. By incorporating phasing errors (piston and tip-tilt errors), some low-order aberrations as well as other practical considerations, Bi-GRU neural work can effectively be used for fine phasing of segmented mirrors. Simulations and real experiments are used to demonstrate the accuracy and effectiveness of the proposed methods.

Cross-iteration multi-step optimization strategy for three-dimensional intensity position correction in phase diverse phase retrieval.

Parameters mismatching between the real optical system and phase retrieval model undermines wavefront reconstruction accuracy. The three-dimensional intensity position is corrected in phase retrieval, which is traditionally separated from lateral position correction and axial position correction. In this paper, we propose a three-dimensional intensity position correction method for phase diverse phase retrieval with the cross-iteration nonlinear optimization strategy. The intensity position is optimized via the coarse optimization method at first, then the intensity position is cross-optimized in the iterative wavefront reconstruction process with the exact optimization method. The analytic gradients about the three-dimensional intensity position are derived. The cross-iteration optimization strategy avoids the interference between the incomplete position correction and wavefront reconstruction during the iterative process. The accuracy and robustness of the proposed method are verified both numerically and experimentally. The proposed method achieves robust and accurate intensity position correction and wavefront reconstruction, which is available for wavefront measurement and phase imaging.

Aberration compensation strategy for the radius of curvature error of the primary mirror in off-axis three-mirror anastigmatic telescopes.

This paper discusses compensation strategies for the aberration fields caused by the error in the radius of curvature (ROC) of the primary mirror (PM) in pupil-offset off-axis three-mirror anastigmatic (TMA) astronomical telescopes. Based on the nodal aberration theory framework, the specific astigmatic and coma aberrations of the off-axis three-mirror system in the presence of the ROC error of the PM are derived. It is demonstrated that some field-dependent aberration components can be induced by ROC error in the off-axis TMA telescopes, apart from the dominating field-constant aberration terms. To reduce the influence of the ROC error on the aberration fields, we propose two aberration compensation strategies: adjusting the position of the PM and introducing axial misalignment of the secondary mirror (SM). Through theoretical analysis and simulations, we conclude that the compensation strategy of changing the axial position of the PM can make the aberration distribution close to the nominal state; the compensation strategy of axially adjusting the SM can make the aberration distribution meet the observation requirements, which is more suitable for space applications. We also discuss compensating the effect of the ROC error using lateral misalignments.

Non-propagation fast phase diverse phase retrieval for wavefront measurement with generalized FFT-based basis function.

Phase retrieval is an attractive optical testing method with a simple experimental arrangement. The sampling grids wave propagation computation based on the FFT operations is usually involved in each iterative process for the classical phase retrieval model. In this paper, a novel non-propagation optimization phase retrieval technique with the FFT-based basis function is proposed to accelerate wavefront measurement. The sampling grids wave diffraction propagation computation is converted to matrix-vector products that have small dimensions to reduce the computational burden. The diffraction basis function based on generalized numerical orthogonal polynomial and two-step Fresnel propagation is deduced, which is suitable for the generally shaped pupil. This paper provides a universal non-propagation framework to accelerate phase retrieval which is applicable to the arbitrarily shaped wavefront measurement.

Aberration fields of pupil-offset off-axis two-mirror astronomical telescopes induced by ROC error.

This paper presents a systematic and deep discussion on the aberration field characteristics of pupil-offset off-axis two-mirror astronomical telescopes induced by the radius of curvature (ROC) error based on the framework of the nodal aberration theory (NAT). The expressions of the third-order aberrations in off-axis two-mirror astronomical telescopes with ROC error are derived first. Then the astigmatic and coma aberration fields are discussed, and it is shown in a field constant astigmatism and coma will be induced by ROC error. The aberration compensation between axial misalignments and ROC error are further discussed, and it is shown that the net astigmatic and coma aberration field induced by ROC error can well be compensated by axial misalignments. Importantly, it is also demonstrated that the focal plane shift induced by ROC error can also be compensated at the same time. Also, this paper briefly analyzes the aberration field characteristics when there is the error of conic constant in optical system. Some other discussions are also presented concerning the ROC inconsistency in astronomical telescopes with a segmented primary mirror. This work will lead to a deep understanding of the influence of ROC error in pupil-offset off-axis astronomical telescopes.

Aberrational interactions between axial and lateral misalignments in pupil-offset off-axis two-mirror astronomical telescopes.

Due to the absence of rotational symmetry, the effects of axial and lateral misalignments couple tightly together, which leads to special aberration field characteristics. This paper will present an in-depth and systematic discussion on the interactions between the effects of axial and lateral misalignments in pupil-offset off-axis two-mirror astronomical telescopes. The aberration function of this class of telescopes in the presence of axial and lateral misalignments is derived. The specific expressions of two dominant non-rotationally symmetric aberrations, i.e., astigmatism and coma, are obtained and the aberration field characteristics are discussed. Importantly, it is shown that under certain conditions, a node will arise in the field of view for these two kinds of aberrations. Then the aberrational compensation mechanisms between axial and lateral misalignments are quantitatively explicated, and it is shown that the non-rotationally symmetric aberrations induced by axial misalignments can well be compensated by lateral misalignments. However, we also find that in this process, the defocus aberration induced by these two kinds of misalignments will accumulate (rather than cancel out). Therefore, in practice, it is better to separate these two kinds of misalignment. Finally, we propose a simple method to decouple axial misalignments from lateral misalignments with wavefront measurement at one field position. Most of this work can be extended to other kind of pupil-offset off-axis astronomical telescopes, such as off-axis three-mirror anastigmatic telescopes.

Object-independent image-based wavefront sensing approach using phase diversity images and deep learning.

This paper proposes an image-based wavefront sensing approach using deep learning, which is applicable to both point source and any extended scenes at the same time, while the training process is performed without any simulated or real extended scenes. Rather than directly recovering phase information from image plane intensities, we first extract a special feature in the frequency domain that is independent of the original objects but only determined by phase aberrations (a pair of phase diversity images is needed in this process). Then the deep long short-term memory (LSTM) network (a variant of recurrent neural network) is introduced to establish the accurate non-linear mapping between the extracted feature image and phase aberrations. Simulations and an experiment are performed to demonstrate the effectiveness and accuracy of the proposed approach. Some other discussions are further presented for demonstrating the superior non-linear fitting capacity of deep LSTM compared to Resnet 18 (a variant of convolutional neural network) specifically for the problem encountered in this paper. The effect of the incoherency of light on the accuracy of the recovered wavefront phase is also quantitatively discussed. This work will contribute to the application of deep learning to image-based wavefront sensing and high-resolution image reconstruction.

Cross-iteration deconvolution strategy for differential optical transfer function (dOTF) wavefront sensing.

Differential optical transfer function (dOTF) is a promising analytic image-based wavefront sensing approach, which is simple in both hardware implementation and mathematical operation. However, there is one deep-rooted problem inherent in this approach, i.e., the essential trade-off between the signal ratio and resolution due to the effect of convolution. In this Letter, a cross-iteration deconvolution strategy is proposed to solve this problem with two different dOTFs, based on the understanding of an underlying prior knowledge when pupil blockage is used to introduce pupil modification. This Letter contributes to the development of a deterministic, efficient, and precise image-based wavefront sensing technique.

Aberration fields of off-axis astronomical telescopes induced by rotational misalignments.

Due to the absence of rotational symmetry, off-axis astronomical telescopes with off-set pupil become subject to rotational misalignments. Rotational misalignments of large off-axis mirrors with reference to their geometric center can greatly degrade the imaging quality. This paper presents an in-depth discussion on the net aberration fields of off-axis astronomical telescopes induced by rotational misalignments. Aberration function of off-axis telescopes with rotational misalignments is derived based on the framework of nodal aberration theory. Expressions of several important aberrations are obtained under some approximations. Then the specific field characteristics of these aberrations are presented and explicated. Meanwhile, we demonstrate that rotational misalignments can be converted to a kind of surface decenters; on the other hand, the effects of rotational misalignments have their special features which are different from the effects of general surface decenters. Besides, some other insightful discussions are further presented. This work is also applicable to the rotational misalignments of the off-axis segments of primary mirror in segmented mirror astronomical telescopes.

Efficient solution to the stagnation problem of the particle swarm optimization algorithm for phase diversity.

The phase diversity (PD) technique needs optimization algorithms to minimize the error metric and find the global minimum. Particle swarm optimization (PSO) is very suitable for PD due to its simple structure, fast convergence, and global searching ability. However, the traditional PSO algorithm for PD still suffers from the stagnation problem (premature convergence), which can result in a wrong solution. In this paper, the stagnation problem of the traditional PSO algorithm for PD is illustrated first. Then, an explicit strategy is proposed to solve this problem, based on an in-depth understanding of the inherent optimization mechanism of the PSO algorithm. Specifically, a criterion is proposed to detect premature convergence; then a redistributing mechanism is proposed to prevent premature convergence. To improve the efficiency of this redistributing mechanism, randomized Halton sequences are further introduced to ensure the uniform distribution and randomness of the redistributed particles in the search space. Simulation results show that this strategy can effectively solve the stagnation problem of the PSO algorithm for PD, especially for large-scale and high-dimension wavefront sensing and noisy conditions. This work is further verified by an experiment. This work can improve the robustness and performance of PD wavefront sensing.

Nonrotationally symmetric aberrations of off-axis two-mirror astronomical telescopes induced by axial misalignments.

In unobscured off-axis astronomical telescopes with an offset pupil, the effects of axial misalignments are very different from those in on-axis ones. Specifically, a series of nonrotationally symmetric aberrations with characteristic field dependence will be induced by axial misalignments. This paper takes off-axis two-mirror astronomical telescopes as an example to discuss the field characteristics of several important nonrotationally symmetric aberrations (including astigmatism, coma, and trefoil aberration) induced by axial misalignments in off-axis astronomical telescopes. The expressions of these aberrations are derived under some approximations. The accuracy of the proposed expressions is demonstrated. The specific field characteristics of these aberrations are presented and explicated. It is shown that the effects of axial misalignments bear strong similarities to the effects of the lateral misalignments in the symmetry plane of the off-axis system. On the other hand, the inherent relationships between astigmatism and coma induced by axial misalignments are further revealed, which are different from those induced by lateral misalignments. This fact presents the possibility of separating the effects of axial misalignments and lateral misalignments. Most of this work can be extended to other off-axis astronomical telescopes with more freedom.

Feature-based phase retrieval wavefront sensing approach using machine learning.

A feature-based phase retrieval wavefront sensing approach using machine learning is proposed in contrast to the conventional intensity-based approaches. Specifically, the Tchebichef moments which are orthogonal in the discrete domain of the image coordinate space are introduced to represent the features of the point spread functions (PSFs) at the in-focus and defocus image planes. The back-propagation artificial neural network, which is one of most wide applied machine learning tool, is utilized to establish the nonlinear mapping between the Tchebichef moment features and the corresponding aberration coefficients of the optical system. The Tchebichef moments can effectively characterize the intensity distribution of the PSFs. Once well trained, the neural network can directly output the aberration coefficients of the optical system to a good precision with these image features serving as the input. Adequate experiments are implemented to demonstrate the effectiveness and accuracy of proposed approach. This work presents a feasible and easy-implemented way to improve the efficiency and robustness of the phase retrieval wavefront sensing.

Field diversity phase retrieval method for wavefront sensing in monolithic mirror space telescopes.

To guarantee the uniqueness of the solution for the wavefront phase, a series of intensity images with known phase diversities is usually needed in the current phase retrieval wavefront sensing methods. However, to obtain these intensity images with deliberately added diversity phases, some additional instruments (e.g., beam splitters) or operations (e.g., adjustment of the focus) are usually needed, which can pose a challenge for wavefront sensing in space telescopes. This paper proposes a new concept for retrieving the wavefront phase of monolithic mirror space telescopes with perturbations, where the intensity measurements with phase diversities are directly obtained from different field positions of one image, without the need for any additional instruments or operations. To realize this new concept, we present a modified phase diversity method to account for the unknown phase diversities between these intensity measurements based on an in-depth understanding of the net aberration fields induced by misalignments and figure errors. Relevant simulations for different cases are performed to demonstrate the feasibility and accuracy of the proposed method. Since in this method the phase diversities between different intensity measurements are mainly induced by the diversities in the field position, we call it the field diversity phase retrieval method. This work can present great facility for wavefront sensing in monolithic mirror space telescopes.

Elimination of the field-dependent aberrations of the JWST-like space telescopes in the multi-field fine-phasing process.

This paper investigates the alignment strategies for eliminating the field-dependent aberrations of the class of large three-mirror anastigmatic (TMA) space telescopes with a segmented primary mirror (PM) like the James Webb Space Telescope (JWST) in the multi-field fine-phasing process based on the framework of nodal aberration theory. During the single-field (on-axis field) fine-phasing process, the individual segment tip, tilt, and piston errors, as well as the de-space of the secondary mirror, are well corrected, and the PM is also adjusted to compensate for those aberrations induced by the misalignments of other mirrors at the center of the science field of view. However, interrogating off-axis field points can reveal the presence of large wavefront errors due to mirror misalignments. Eliminating these field-dependent aberrations is the main goal of the multi-field fine-phasing process. This paper first presents an analytic study on an established alignment strategy used for eliminating the field-dependent aberrations. While it is demonstrated that this alignment strategy has the ability to reduce the field dependency of the wavefront errors, it will, however, also be revealed that this strategy still exhibits some problems, and its alignment efficiency is low. Then, a new alignment strategy with higher alignment efficiency is further proposed. Detailed simulations with a TMA telescope that has similar parameters with the JWST are performed to illustrate the efficiency and rationality of the proposed strategy. This work can not only contribute to an in-depth understanding of the multi-field fine-phasing process, but also present a possibility to improve the efficiency of this process.