Large segmented sparse aperture collimation by curvature sensing

Qichang An; Xiaoxia Wu; Xudong Lin; Ming Ming; Jianli Wang; Tao Chen; Jingxu Zhang; Hongwen Li; Lu Chen; Jing Tang; Rui Wang; Rui Wang; Hongchao Zhao

doi:10.1364/OE.413599

1. Introduction

Many future large telescopes have been designed with mirrors that are segmented into several smaller units. Examples include the 10-meter Keck telescope with 36 segmented mirrors [1], the Chinese 12-meter optical/infrared telescope (LOT) [2], the Thirty Meter Telescope (TMT) with 492 M1 Units, and the European Extremely Large Telescope (E-ELT) with 798 M1 Units [3,4]. Shark–Hartman sensors are used to phase the segments during observation. However, when in the mirror lab, a large flat or collimator is needed to align the system [5]. For large scale telescopes, it is not feasible to process alignment using an equivalent flat or collimator.

Sub-aperture stitching will benefit from lower requirements for the measurement instruments and realize metrology at considerably lower costs. Sub-aperture stitching was first proposed by Stuhlinger [6]. Its basic principle is to obtain full-aperture surface data by overlapping smaller apertures and combining them under certain principles. The most significant advantage of the sub-aperture stitching algorithm lies in its expandability. However, with increasing size, the number of sub-apertures will increase significantly. In addition, multiple measurements are often required for sub-aperture stitching, only one of which is used in the final shoot, meaning a waste of time and manual labor [7]. The sparse aperture measurement method was proposed by Kim in 1983 [8]. Its main idea is to build a small aperture standard flat array to replace the collimating optical path of a single large flat and realize giant optical elements or the systems’ wavefront detection. In the early phase of system integration detection, the low effectiveness of sub-aperture detection seriously delays the system assembly and adjustment; however, in the cross-check process of observation, a large number of sub-apertures will cause information redundancy.

Thus, many great jobs followed the work of Kim. In 2010, Stokes divided the nine sparse apertures into three groups and took the cut-off frequency as the optimization objective, obtaining a group of 9-aperture layout schemes [9]. In 2016, after analyzing the relationship between the position of discrete sub-aperture and the spatial resolution, Salvaggio proposed a six-sparse-aperture scheme, which achieved the highest spatial resolution with the same number of detection apertures [10]. In 2011, Yan used the sparse aperture method to detect optical elements. In a certain space frequency range (the first 36 Zernike polynomial in Noll), he achieved the same accuracy as the full aperture detection [11]. In 2016, Xu analyzed the detection error of the 0.2-meter aperture planar reflector of a high-energy laser weapon system according to the unique mirror error distribution produced by mirror “ring-polish” processing. Using the sparse aperture with uniform diagonal distribution, Xu verified that the sparse aperture can still be perfectly competent on the premise of a super-smooth surface [12]. In 2017, Wu compared the results of sparse aperture detection and sub-aperture stitching with the overlapping area for 0.2 mm × 0.3 mm bar planar mirrors and proved the equivalence of the two methods at medium frequency (with axial symmetry less than three and circumferential symmetry less than 4). Furthermore, Wu pointed out that the sparse aperture can reduce the second-order error of the sub-aperture introduced in detection [13].

In 1991, the National Ignition Laboratory (NIF) proposed characterizing a specular surface with a power spectral density function (PSD). In 2010, Parks proposed the characterization of optical systems in the full frequency domain using structural functions [14]. However, PSD and structure functions are two-dimensional indexes, and there are some ambiguities as to quantitative criteria. Point spread function (PSF) is equivalent to the impulse response function of the optical system. Its shape can represent the diffraction limit of the system, atmospheric disturbance, pupil shape, and error of the system itself. However, as a three-dimensional evaluation method, PSF cannot form a single value index, which is easy to compose or decompose. Based on the index of the full width, half-height, and 80% energy concentration of the point spread function with the shape of the point spread function, the system cannot be characterized correctly when the shape of the PSF deviates from a Gaussian shape.

In this study, we try to develop a sparse aperture testing method based on curvature sensing, for the large, segmented telescope. Moreover, we use PSSn as the evaluation of the changes in telescope performance during telescope collimation. Furthermore, we consider a curvature-sensing-based algorithm designed to solve the restricted wavefront sensing problems of determining both the collimation of the telescope and the mirror figure of each segment (as we shall show) to control the Warping Harness [15].

The article is organized as follows: the first part is concerned with the accuracy of single aperture testing and the extent to which that accuracy influences the estimation of the mirror figure from the curvature signal. The combination of curvature sensing and sparse aperture detection is analyzed and verified by simulation and experiment. Finally, all errors will be evaluated using the thirty-meter telescope performance matrices, normalized point source sensitivity (PSSn), and the final alignment error of the large, segmented telescope is analyzed. With these sparse apertures, the figure of the full telescope aperture is reconstructed.

2. Materials and methods

2.1 Theoretical background and methodology

In this section, the curvature sensing model and error is analyzed based on Fourier optics theory, and the applicability and robustness of curvature sensing in sparse aperture testing are studied.

Curvature sensing is a popular method for correcting atmospheric turbulence in adaptive optics. Moreover, there are many advantages to its application in active telescope alignment with sparse aperture, such as its simple structure, robustness to aperture obstruction, and broad dynamic range.

The basic principle of segmented mirror alignment using a sparse aperture is shown in Fig. 1. The flat mirror, which collimates the light beam back, is approximately 1/4 of the full aperture size. The flat will be moved, accomplishing testing of the full large, segmented mirror. In the single aperture, the figure will be measured before the flat moves to a nearby location. To correct relative tilt and focus, one segment is theoretically sufficient, but additional flats can reduce the random error introduced by the environment and instrument.

Fig. 1. Basic principle of segmented mirror alignment by sparse aperture. (a) Schematic diagram of sparse aperture detection. (b) Sparse aperture shifting deviation diagram.

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For the James Webb Space Telescope (JWST) at intermediate focus, a laser beam is emitted in the direction of the secondary mirror, and the primary mirror and a collimated beam is formed. The collimated beam is reflected through the discrete auto collimating flats (ACFs) and returned to the scientific terminal through the telescope. Based on the scientific instruments’ images, the secondary mirror alignment, and field of view tilt adjustment, the system wavefront detection is realized.

Different from the traditional auto-collimation test in which all elements reflect twice, the light source is located in the middle image plane, and the components between the focal plane and the middle image plane only receive the light from the secondary mirror; thus, it is called “pass and a half test.” The testing profile of this work is similar to that of JWST, and it also contains three collimation sparse apertures.

Here, the principle of curvature sensing is analyzed by a wavefront with a single spatial component, which is represented by Eq. (1). The amplification was set as a unit. Here, $\lambda$ is the wavelength, A is the amplitude of the single-phase spatial frequency component, ${\textbf u}$ is the spatial coordinate in the pupil, and ${\textbf f}$ is the spatial frequency domain coordinate. $\varphi$ is the initial phase.

(1)$$W({{\textbf u},0} )= \textrm{exp} \left[ {j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]$$

The curvature of this wavefront is shown in Eq. (2). It is assumed that the term in the exponent is sufficiently small.

(2)$${\nabla ^2}W({{\textbf u},0} )\approx \frac{\partial }{{\partial {{\textbf u}^2}}}\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } ) ={-} \frac{{2\pi A}}{\lambda }{({2\pi {\textbf f}} )^2}\sin ({2\pi {\textbf {uf}} + \varphi } )$$

The accuracy of sparse aperture testing by curvature sensing is influenced by tipping in a single testing step, shifting effects of sparse aperture, and mirror seeing.

The light intensity distribution at extra-focus is shown in Eq. (3).

(3)$$\begin{aligned} {W_ + } &= {|{W({{\textbf u},{z_ + }} )} |^2}\\ &= \textrm{exp} \left[ {{e^{j\pi {{|{\textbf f} |}^2}{z_ + }\lambda }}j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\textrm{exp} \left[ { - {e^{ - j\pi {{|{\textbf f} |}^2}{z_ + }\lambda }}j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &= \textrm{exp} \left[ {({{e^{j\pi {{|{\textbf f} |}^2}{z_ + }\lambda }} - {e^{ - j\pi {{|{\textbf f} |}^2}{z_ + }\lambda }}} )j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &= \textrm{exp} \left[ { - \frac{{4\pi A}}{\lambda }\sin ({\pi {{|{\textbf f} |}^2}{z_ + }\lambda } )\sin ({2\pi {\textbf {uf}} + \varphi } )} \right] \end{aligned}$$

The light intensity distribution at intra-focus is shown in Eq. (4).

(4)$$\begin{aligned} {W_ - } &= {|{W({{\textbf u},{z_ - }} )} |^2}\\ &= \textrm{exp} \left[ {{e^{j\pi {{|{\textbf f} |}^2}{z_ - }\lambda }}j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\textrm{exp} \left[ { - {e^{ - j\pi {{|{\textbf f} |}^2}{z_ - }\lambda }}j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &= \textrm{exp} \left[ {({{e^{j\pi {{|{\textbf f} |}^2}{z_ - }\lambda }} - {e^{ - j\pi {{|{\textbf f} |}^2}{z_ - }\lambda }}} )j\frac{{2\pi A}}{\lambda }\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &= \textrm{exp} \left[ { - \frac{{4\pi A}}{\lambda }\sin ({\pi {{|{\textbf f} |}^2}{z_ - }\lambda } )\sin ({2\pi {\textbf {uf}} + \varphi } )} \right] \end{aligned}$$

The curvature information can be extracted by the difference in the light intensity distribution.

(5)$$\begin{aligned} {W_ + } - {W_ - } &= {|{W({{\textbf u},{z_\textrm{ + }}} )} |^2} - {|{W({{\textbf u},{z_ - }} )} |^2}\\ &= \textrm{exp} \left[ { - \frac{{4\pi A}}{\lambda }\sin ({\pi {{|{\textbf f} |}^2}({z_\textrm{ + }})\lambda } )\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &- \textrm{exp} \left[ { - \frac{{4\pi A}}{\lambda }\sin ({\pi {{|{\textbf f} |}^2}{z_ - }\lambda } )\sin ({2\pi {\textbf {uf}} + \varphi } )} \right]\\ &\approx{-} 2\frac{{4\pi A}}{\lambda }\pi {|{\textbf f} |^2}(\Delta z)\lambda \sin ({2\pi {\textbf {uf}} + \varphi } )\end{aligned}$$

Referring to the Fourier optics theory, the light translates the free space for a distance of z, as shown by Eq. (5). Here, $\lambda$ is the wavelength, A is the amplitude of the single-phase spatial frequency component, ${\textbf u}$ is the spatial coordinate in the pupil, ${\textbf f}$ is the spatial frequency domain coordinate, and ${z_{+/- }}$ is the defocusing amount. Visibly, ${W_ + } - {W_ - }$ is proportional to the wavefront curvature, which can be estimated by the difference in intensity. Here we note ${z_\textrm{ + }} - {z_ - } = 2\Delta z$.

The Poisson equation of the Zernike mode with non-zero curvature is simulated, and the Poisson equation based on an iterative Fourier transform is realized.

The light intensity distribution at intra and extra focus for the trefoil-dominated wavefront is shown in Figs. 2(a) and (b). Based on the distribution, the reconstructed wavefront is shown in Figs. 2(d) and (f).

Fig. 2. Wavefront curvature sensing solution of (a) light distribution of intra focus, and (b) light distribution of extra focus. (c) Light intensity difference. (d) Original wavefront phase. (f) Wavefront phase retrieval result. (e) Wavefront phase retrieval residual error.

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Here, we use Fourier theory to model and analyze the accuracy of curvature sensing. The accuracy of the trefoil in Fringe Zernike polynomials is more than 15%, while the accuracy was almost 2% after two iterations.

For the stitching step, there are in total three kinds of errors that will be taken into consideration: the shift error between the two sub-apertures under stitching, the piston and tip/tilt error between them, and the turbulence (mainly coming from air turbulence in the lab, vibration, noise in detectors, etc.). Thus, the stitching error can be expressed as follows:

(6)$${\Delta _{sti}}({\textbf u} )= P + {\textbf {Tu}} + n$$

Here P is the coefficient for piston error, ${\textbf T}$ is the coefficient matrix for tip/tilt error, and n is the turbulence in Eq. (7).

(7)$$\begin{aligned} W({{\textbf u},0} ) &= \textrm{exp} \left[ {j\frac{{2\pi A\sin ({2\pi {\textbf {uf}} + \varphi } )\textrm{ + }{\Delta _{sti}}({\textbf u} )}}{\lambda }} \right]\\ &\textrm{ = }\textrm{exp} \left[ {j\frac{{2\pi A\sin ({2\pi {\textbf {uf}} + \varphi } )\textrm{ + }P + {\textbf {Tu}} + n}}{\lambda }} \right] \end{aligned}$$

The curvature can be expressed by Eq. (8). Therefore, the tipping error of the sparse aperture can be ignored owing to the curvature sensing.

(8)$$\begin{aligned} {\nabla ^2}W({{\textbf u},0} ) &\approx \frac{\partial }{{\partial {{\textbf u}^2}}}\frac{{2\pi A\sin ({2\pi {\textbf {uf}} + \varphi } )\textrm{ + }P + {\textbf {Tu}} + n}}{\lambda }\\ & ={-} \frac{{2\pi A}}{\lambda }{({2\pi {\textbf {uf}}} )^2}\sin ({2\pi {\textbf {uf}} + \varphi } )\end{aligned}$$

The mark coordination is measured under input sub-aperture coordinates. These two systems are rotated by $Z^{\prime}$ the axis normal line of the mirror Z’, and the light axis shifted by $\delta {\textbf u}$, as shown in Eq. (9):

(9)$$\delta W({{\textbf u},0} )= \frac{\partial }{{\partial {\textbf u}}}W({{\textbf u},0} )\delta {\textbf u}$$

The curvature of the wavefront with a shifting error in the aperture location is shown in Eq. (10):

(10)$$\begin{aligned} {\nabla ^2}\delta W({{\textbf u},0} ) &\approx \frac{\partial }{{\partial {{\textbf u}^3}}}\frac{{2\pi A\sin ({2\pi {\textbf {uf}} + \varphi } )}}{\lambda }\\ & ={-} \frac{{2\pi A}}{\lambda }{({2\pi {\textbf f}} )^3}\cos ({2\pi {\textbf {uf}} + \varphi } )\end{aligned}$$

Considering that the deviation is a high-order small quantity, it can be ignored in the testing process. This method can reduce the balance of the error generated by the misalignment of the sparse aperture center and its ideal location.

2.2 Analysis of normalized point source sensitivity in sparse testing

Although in the previous section we analyzed the applicability and robustness of curvature sensing in discrete aperture measurement, the numerical characterization of its detection effect, especially the single-value numerical characterization, was not described in detail.

This section discusses the characterization of discrete aperture detection results from the perspective of frequency-domain characterization.

We gradually establish an evaluation method for sparse testing based on the PSSn. Seo proposed PSSn in 2009 [16], which uses the average of the point spread function in the imaging area.

The optical transfer function is a two-dimensional index, which inevitably has some limitations in evaluating a continuous process and as an optimization index. The error evaluation and prediction of large-aperture telescopes can be realized comprehensively, quickly, and conveniently by combining a single value evaluation index. By comparing with the traditional single value index, the PSSn is selected as the evaluation index, and it can be expressed by the optical transfer function (OTF) yielding:

(11)$$PSSn = \frac{{\int {{{|{OT{F_e}({\vec{\theta }} )} |}^2}{{|{OT{F_{t + a}}({\vec{\theta }} )} |}^2}} }}{{\int {{{|{OT{F_{t + a}}({\vec{\theta }} )} |}^2}} }}$$

where $OT{F_e}$ is the optical transfer function of the error telescope, and $OT{F_{t + a}}$ denotes the optical transfer function of the ideal telescope exposed to the atmosphere.

3. Results

Previously, the basic model and error influence of curvature sensing were analyzed. In this section, we will start with the numerical simulation of single-aperture curvature sensing and the analysis of multi-aperture curvature sensing to verify the feasibility of this method. Meanwhile, taking PSSn as an evaluation index, the process of system collimation based on curvature sensing is revealed. An improvement in the optical imaging quality of the system is shown.

3.1 Single-aperture detection analysis

Curvature sensing will be used to assess the figure in a single aperture. Additionally, we can use this testing to optimize the tilting of the segmented mirror if necessary. The iteration step is chosen as 0.3 as shown in Fig. 3. The aberration is well corrected, but the residual error is mainly from the tilt. The tilt-introduced boresight will be corrected by a light LED bar around the pupil just as with JWST [17].

Fig. 3. Closed-loop performance of split-field wavefront curvature sensing. (a) Original wavefront phase. (b) Wavefront phase retrieval result. (c) Wavefront phase retrieval residual error. (d) Relation between convergence rate and step size.

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The light path is shown in Fig. 4. The curvature sensing can determine the different phases of collimation in a single aperture.

Fig. 4. Experimental light path diagram.

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The light intensity distribution at intra and extra focus for the wavefront under different misalignment states is shown in Fig. 5. Based on the distribution, the reconstructed wavefront is shown in Fig. 6.

Fig. 5. Light intensity distribution under different misalignment states. (a) Intra focus light distribution before correction. (b) Extra focus light distribution before correction. (c) Difference of the light distribution before correction. (d) Intra focus light distribution after correction. (e) Extra focus light distribution after correction. (f) Difference of the light distribution after correction.

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Fig. 6. Optimization of the highly segmented mirror figure. (a) Mirror figure before correction. (b) Mirror figure after correction.

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A comparison of the Zernike polynomials before and after correction is shown in Fig. 7. Because each element of the segmented deformable mirror moves, a higher order (Z15-Z25) is selected when the aberration is applied.

Fig. 7. Comparing the Zernike polynomials before and after correction.

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We analyze and calculate these designs and determine by comparing the variations in the Multitude transform function (MTF).

3.2 Sparse aperture detection analysis

The total number and distribution of the sparse aperture is discussed in this section. Here, we optimize the size and location of the sparse apertures under the principle that all the target spatial frequencies must be covered. The MTF is used to monitor the optimization process. In addition to this sparse testing, the figure of the primary mirror can be extracted from the sparse testing result. It is aimed at specifying the aberration introduced by the second and tertiary mirrors if necessary. The process is shown in Fig. 8.

Fig. 8. Process of sparse aperture testing.

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The calibration can be obtained at different zenith angles, and the aberration introduced by gravity and print will be minimized. First, we use the translation device to move the plane mirror to the baseline and use a light source located at the focus. The collimating beam is formed by the primary and secondary mirrors and the planar mirror. We can then obtain the surface figure of the telescope using the corresponding curvature sensing algorithm based on the light distribution of intra and extra focus. Taking three sparse apertures as examples, the simulation result of sparse aperture testing is shown in Figs. 9 and 10. PSSn has some reliability and relatively strong characterization ability for a wide spatial frequency. The experimental results are consistent with the variation in the full frequency. As shown in Fig. 11, we know that the PSSn changes with the misalignment of the telescope (presented by the low order aberration).

Fig. 9. Simulation results of sparse aperture testing. (a) Curvature sensing results. (b) Residual error.

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Fig. 10. PSF correction by wavefront curvature sensing. (a) PSF before correction. (b) PSF after correction.

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Fig. 11. Variation of PSSn during correction.

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4. Discussion

This method can also be applied to a movable, segmented flat, such as LAMOST. The segmented mirror does not suffer from the dramatically increasing cost of fabrication. However, it calls for hardware and strategy to keep the mirrors in a nearly perfect face. For LAMOST M1, the Schmidt corrector plate manufacturer adopts both thin mirror active optics and active optics technology. The relative position of each sub mirror is fed back by the edge sensor placed between the sub-mirrors.

The imaging quality decreases significantly with the increasing extent of system misalignment. Therefore, the large telescope needs to be carefully aligned. Collimation relies mainly on low-order information. It is not necessary to detect the full aperture of a large, segmented mirror. Sparse aperture testing is sufficient to estimate a large telescope's misalignment and complete the final alignment.

5. Conclusions

This paper focuses on analyzing the detection accuracy of a low-order surface and discusses the accuracy of sparse aperture detection using curvature sensing. By exploring the combined application of curvature sensing and sparse aperture testing, we model the high-efficiency detection of figure profiles in large, segmented telescopes. In addition, we solve the contradiction between the increasingly urgent need for a large flat or collimator and the high cost of this inspection equipment.

Funding

National Key Research and Development Program of China (2017YFE0102900); National Natural Science Foundation of China (11703026, 11803034, 62005279); Jilin Scientific and Technological Development Program (20180201059GX); Norman Bethune Medical Engineering and Equipment Center (BQEGCZX2019042); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020221).

Acknowledgments

This research was carried out at the Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Science. The authors are very grateful to Dr. Wang for providing valuable suggestions and help.

Disclosures

The authors declare no conflict of interest, and the final version of the manuscript has been reviewed and approved for publication by all authors. All authors agree with this submission. The work has not been published or submitted for publication elsewhere, either completely or in part, or in another form or language; no materials are reproduced from another source.

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Large segmented sparse aperture collimation by curvature sensing

Abstract

1. Introduction

2. Materials and methods

2.1 Theoretical background and methodology

2.2 Analysis of normalized point source sensitivity in sparse testing

3. Results

3.1 Single-aperture detection analysis

3.2 Sparse aperture detection analysis

4. Discussion

5. Conclusions

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (11)

Equations (11)

Optics Express