## Abstract

We propose a method to simultaneously detect the multi-piston between segments. The relation between the piston of any two segments and amplitude of the modulation transfer function (MTF) sidelobes (MTF* _{nph}*) is derived. The piston can be retrieved by this relation after measuring the MTF

*. This method's capture range is the operating light’s coherence length, the accuracy is 0.026λ (λ = 633 nm) RMS. The MTF model of a mask with sparse multi-subaperture configuration is established. The arrangement rules, to avoid the sidelobes overlapping, are obtained. The mask with a sparse 18 sub-aperture configuration is designed, which makes the MTF sidelobes distribution non-redundant.*

_{nph}© 2017 Optical Society of America

## 1. Introduction

In order to observe more distant and fainter object with a better resolution, larger primary mirror telescopes are needed to improve the diffraction limit and increase the collected light energy. This leads to problems with monolithic primary mirror manufacture and testing, transportation and launch. At present, it is hard to build a monolithic primary mirror with 10 meters diameter or more. A segmented and deployable primary mirror was adopted to address these issues. Currently, most of large telescopes adopt segmented primary mirror. Keck telescope [1,2] with a 10 m diameter and 36 segments, Giant Magellan Telescope (GMT) [3,4] with a 24.5 m diameter and 7 segments in Las Campanas, El Gran Telescopio de Canarias (GTC) [5] with a 10.4 m diameter and 36 segments at La Palma Island, the European Extremely Large Telescope (E-ELT) [6] with a 42 m diameter and 984 segments in Paranal, the Thirty Meter Telescope (TMT) [7] with a 30 m diameter and 492 segments at Mauna Kea in Hawaii, the James Webb Space Telescope (JWST) [8] with a 6.5 m diameter and 18 segments deployable primary mirror, and the advanced technology large aperture space telescope (ATLAST) [9] with a 9.2 m diameter and 36 segments deployable primary mirror, among of them the Keck and GTC are already in operation, the others are planned to be in operation sometime during the next decade. However, segmented mirror introduces another new problem, that of cophasing the segments closely match the optimum mirror shape, in order to be diffraction-limited over whole aperture. For the earth-based segmented primary telescope with a diameter over 10 m, it is very difficult to realize diffraction-limited imaging because of the atmospheric turbulence although with an adaptive optical correction. Therefore, a space-based segmented telescope has been developed. After launch and deployment, the segments must be cophased to achieve a spatial resolution comparable to that of a monolithic mirror also. The performance of this kind of space-based telescope may be compromised by some factors such as manufacturing and alignment errors, on orbit thermal deformations, segment deployment errors, launch induced misalignments, and spacecraft jitter [10].

Many cophasing methods have been studied. Modified Shack-Hartmann sensing [11,12] and curvature sensing [13,14] are used on segmented telescope, and both have shown great success. The capture range of the broadband Shack-Hartmann algorithm (BSH) is ± 10 λ, and the accuracy is λ/3 RMS. The capture range of the narrow Shack-Hartmann algorithm (NSH) is ± λ/4, and the accuracy is λ/140 RMS. The capture range of the curvature sensing is λ/8, and the accuracy is λ/80 RMS. These methods are generalizations of traditional wavefront sensing techniques, utilizing physical optics to model the effects of diffraction from a discontinuous surface. The capture range of the Pyramid sensor [15] is ± 10 λ, and the accuracy is 10 nm RMS. The dispersed fringe sensor (DFS) [16] was proposed for coarse phasing of the Next Generation Space Telescope (NGST). The capture range of the DFS is ± 100 μm, and the accuracy is 100 nm. Dispersed Hartman Sensors (DHS) [17] was researched and used in the JWST for coarse phasing. The two-dimensional dispersed fringe sensor [18] was proposed based on the DFS. And this method’s capture range is 200 μm, measurement error is less than 20 nm. In 2015, intersegment piston sensor based on the coherence measurement of a star image was put forward to realize a coarse phasing in segmented telescope [19]. Its capture range is close to the coherence length of the operating light, but its accuracy is lower at the external position of the capture range.

But current cophasing methods allow phasing process to be divided into coarse and fine regimes which involve separate dedicated hardware solutions. We have been trying to address this issue. In 2016, we proposed an novel method [20] to detect the piston error based on analyzing the intensity distribution on the image plane of a segmented telescope, which is of a large capture range and high accuracy. With this method, cophasing is no longer be divided into coarse and fine regimes. For simplicity, we used a two-segment telescope model and set a mask with a sparse clear sub-aperture configuration in conjugate plane of the segmented primary mirror. We derived the functional relationship between the piston error and the amplitudes of MTF’s sidelobes. The amplitudes can be obtained by analyzing the point spread function (PSF) recorded in the image plane with Fourier transform. And then, the piston error can be retrieved in terms of the functional relationship. Its capture range is the coherence length of the operating light used in the detection. Experiments have been carried out on the Active Cophasing and Aligning Testbed (ACAT) [21] to validate the feasibility of this method. The result stated that the accuracy of detecting the piston error is 0.026 λ RMS (λ = 633 nm). This method is adaptable to any segmented and deployable primary mirror telescope. The hardware requirements of this method are very small, just a mask with a sparse clear sub-aperture configuration is needed to attach in the conjugate plane of the segmented primary mirror.

In this paper, we propose a method to simultaneously detect the multi-piston errors of segments. A mask with a sparse circular clear multi-subaperture configuration is set in the exit pupil plane of a segmented telescope to fragment the pupil. According to the Fourier optics principle, the intensity distribution in the image plane is analyzed, and MTF distribution is obtained. The key to realize a simultaneous multi-piston measurement of our method is to avoid overlap of the MTF sidelobes which formed by each pair of subwaves sampled by the corresponding pair of the sub-apertuers. We research and derive the MTF model of a mask with a sparse circular clear multi-subaperture configuration. By analyzing this model, the rules for an non-redundant arrangement of the MTF sidelobes are obtained. Taking the 18-segment mirror as an example, we design a mask with a sparse 18 circular clear sub-apertures configuration to realize the MTF sidelobes non-redundant distribution according to the rules. Thus, the amplitudes of the sidelobes can be simultaneously obtained. And then the piston errors of the whole aperture can be retrieved simultaneously by the relationship between the piston error of any two segments and amplitudes of MTF sidelobes we derived in the literature [20]. This method can be used in a multi-segment primary mirror telescope, such as JWST, to phase the piston simultaneously over whole aperture. For a large number of segments system, options to use our method to cophase an entire mirror include rotating the mask or utilizing multiple aperture masks.

In this paper, the piston error is measured at the wavefront rather than on the segments’ surfaces. And the piston error is the differential piston between segments and not the absolute one. For a reflective segmented primary mirror, the distance between segments’ surfaces will produce a wavefront error (or optical path difference) which equals to twice as this distance.

## 2. MTF sidelobes non-redundant distribution

#### 2.1 Piston measurement principle

In the literature [20], we derive the functional relationship between the piston error of any two segments and the amplitudes of the MTF sidelobes, and propose to retrieve the piston error by using this relationship. Simulation and experiments have proven that this method can measure the piston error with a large capture range and high accuracy. A detailed explanation of this method can be found in the literature [20]. Since this method is the basis of our implementation of multi-piston measurement, we present a concise summary of the method below.

For the purpose of building the functional relations, we use a two-segment telescope system and set a mask with two sparse circular clear sub-apertures in the conjugate plane of the segmented primary mirror to sample the wavefront from the segments. We define $\left(x,y\right)$to be the coordinates in the image plane, and $\left({x}_{0},{y}_{0}\right)$to be the coordinates in the conjugate plane. The two sparse sub-apertures are set in the ${x}_{0}$ direction, and have the same diameter $D$. The center distance between the sub-apertures is $B$, and $B>D$.

The input coherent light (central wavelength ${\lambda}_{0}$, spectral bandwidth $\Delta \lambda $), reflected by the segments, is sampled by each sub-aperture and focused on a CCD where the PSF is recorded. This image is equivalent to the square modulus of the Fourier transform of the sub-aperture configuration. Here, the PSF (the intensity distribution in the image plane) is the cosine interference fringe modulated by the sub-aperture diffraction, and the piston error is included in the interference term. Then, an inverse Fourier transform is performed for the PSF to obtain the optical transfer function (OTF) which is composed of the MTF and phase transfer function (PTF). The MTF is the amplitude part of the OTF and given by [20]

From Eq. (1) we can see that, $MTF\left({f}_{x},{f}_{y}\right)$formed by the two sub-apertures consists of three parts. One part, $2MT{F}_{sub}\left({f}_{x},{f}_{y}\right)$, is MTF central peak. The other two parts, $MT{F}_{sub}\left({f}_{x}+B/\lambda f,{f}_{y}\right)$and$MT{F}_{sub}\left({f}_{x}-B/\lambda f,{f}_{y}\right)$, are MTF sidelobes. The two sidelobes have the same amplitude and symmetricly distribute on both sides of the central peak in the baseline of this pair of sub-apertures. The center frequency difference between the sidelobes is related to the $B$. The amplitudes of the sidelobes are related to the coherence degree of the PSF and the coherent degree is related to the OPD or piston error. When the piston error is null, the amplitude of the MTF sidelobe is maximum; as the piston error increases, the amplitude of the MTF sidelobes become smaller and smaller, until they disapper completely. The MTF central peak height is always the same regardless of the coherence degree, its height is the maximum amplitude of the sidelobe multiplied by the number of sub-aperture. According to the Fourier optics principle, we derived functional relationship between the piston error and the amplitude of MTF sidelobe and given by

For a reflective primary mirror, the piston capture range is equal to the half of the coherent length of the light used in detection. If the central wavelength of the light is 633 nm, spectral bandwidth is 1 nm, the piston error capture range will be 200 μm.

Experiments have been carried out on the ACAT to validate the feasibility of this method [20]. The result stated that the accuracy of the piston error is 0.026 λ RMS (λ = 633 nm).

Thus, we just place a mask with a sparse multi-subaperture configuration in the conjugate plane of the segmented primary mirror, and do the Fourier transform for the PSF recorded by the CCD to obtain the amplitudes of the MTF sidelobes. Then the multi-piston errors between segments can be simultaneously retrieved by Eq. (3) with a high accuracy in a large capture range. The key is to avoid overlap of the MTF sidelobes formed by each pair of sub-wavefronts sampled by the corresponding pair of sub-apertures, which makes us simultaneously obtain the amplitudes of every pair of sidelobes and trace the corresponding every pair of sub-apertures, and then every pair of segments. Thus, the piston errors of entire segmented primary mirror can be retrieved by Eq. (3). Therefore, the MTF model of a mask with a sparse clear multi-subaperture configuration and the rules for an non-redundant arrangement of the MTF sidelobes must be researched.

#### 2.2 MTF model

For a segmented telescope with N segments, we set a mask in the telescope exit pupil plane to partition the pupil into N identical circular sub-apertures with the same diameter D. Each sub-aperture corresponds to one segment, and samples the wavefront reflected by the segment. Figure 1 takes an 18-segment mirror as an example to show the corresponding position relation between the every sub-aperture and segment in the exit pupil plane. The gray area is opaque, the blue areas express circular clear sub-apertures.

For the purpose to establish the MTF model, a diffraction limited imaging system without optical aberrations is assumed. The pupil function of the mask can be regarded as the convolution of a single sub-aperture transmittance and a two-dimension $\delta $function, and given by

We take 4 circular clear sub-apertures as an example shown in Fig. 2. From Fig. 2 (a) we can see that, there are six baselines L1, L2, L3, L4, L5 and L6, and six pairs of sub-aperture A-B, A-C, B-C, D-A, D-C, and D-B ($N\left(N-1\right)/2\text{=}\text{4}\times \left(\text{4-1}\right)/\text{2}\text{=6}$). In Fig. 2 (b),there are one MTF central peak and 12 MTF sidelobes ($N\left(N-1\right)\text{=4}\times \left(\text{4-1}\right)\text{=12}$). Four sub-MTFs (N = 4) overlap at the position where the center spatial frequency is zero to form this central peak. Every pair of sub-apertures produces a pair of MTF sidelobes. Six pairs of sub-apertures produce six pairs of sidelobes or 12 sidelobes, A-B vs 1-1', A-C vs 2-2', B-C vs 3-3′, D-A vs 4-4', D-C vs 5-5′, D-B vs 6-6'. The sidelobes symmetricly distribute on both sides of the central peak in the baseline of this pair of sub-apertures, the center spatial frequency difference between this pair of sidelobes is decided by the center distance of this pair of the sub-apertures.

From Eq. (8) we can see that, whether the sidelobes overlap depend on the diameter and positions of these sparse sub-apertures. And Eq. (8) is the basis of the position arrangement. If all of sidelobes do not overlap, their amplitudes could be obtained at same time, hence the piston errors between the every segments of entire mirror will be retrieved simultaneously by using Eq. (3).

#### 2.3 Arrangement rule

The number of the segment will increase with the enlargement of the segmented telescope primary mirror aperture, which leads to the increase in the sub-aperture quantity of the mask. This will inevitably make multiple sub-apertures distribute in the same baseline, and some baselines parallel to each other or have same baseline vector, as well relative positon of the sidelobes closer. According to Eq. (8), it can conclude that these could make some sidelobes completely or partly overlapping in the spatial frequency domain, which leads to the redundant distribution of the MTF sidelobes. The complete overlap of multiple MTF sidelobes will make a misjudgment of sidelobe amplitude, which will lead to a mistake in piston error measurement with Eq. (3). If the partial overlap of the sidelobes makes the sidelobe amplitude change, the measurement error will be introduced in piston detection with Eq. (3) also. Here, we respectively analyze these redundant distribution of the sidelobes as the following.

If the central distance between any pair of sub-apertures with the same shape and size, arranged on the same baseline, is equal to each other the redundant distribution of the MTF side-lobes will occur. The example of 4 sub-apertures is shown as Fig. 3. According to Eq. (8), if the sidelobes distribute non-redundantly, four sub-apertures should produce 12 MTF sidelobes, but Fig. 3 just shows 8 MTF sidelobes. This is because that the center distance between the sub-aperture A and B is equal to the center distance of the sub-aperture C and D, which makes the center positions of the pair of sidelobes, produced by the sub-aperture A and B, same as the center positions of the pair of sidelobes, produced by the sub-aperture C and D. And since these sub-apertures have same shape and same size, the spatial frequency area of the sidelobe is equal to each other. Consequently, these two pairs of the sidelobes are completely overlapping. As shown in Fig. 3, the sidelobes 2 and 2′are produced by the sub-apertures A and B, and C and D together, and their amplitudes are doubled. For the same reason, the sidelobes 4 and 4′are produced by the sub-aperture A and C, and B and D together.

If the center distance of the two sub-apertures on a baseline is equal to the center distance of the two sub-apertures arranged on the paralleled baseline, the redundant distribution of the MTF side-lobes will occur. The example of 4 sub-apertures is shown as Fig. 4. L1 is the baseline of the pair of sub-apertures A and B. L2 is the baseline of the pair of sub-apertures C and D. The L1 and L2 are parallel to each other, that means they have the same baseline vector. Hence, although these two pairs of the sub-apertures are located in the different baselines, their corresponding two pairs of the MTF sidelobes are overlapping because of the same baseline vector, same center distance of the pairs of sub-apertures, same sub-aperture shape and same size. As shown in Fig. 4, the sidelobes 4 and 4′are produced by the sub-apertures A and B, and C and D together, and their amplitudes are doubled.

If the center frequency difference of the sidelobes is smaller, partial overlap of the sidelobes will occur. Figure 5 shows that two pairs of sub-apertures locate in two different baselines. As can be seen from Fig. 5, the three sub-apertures A, B and C produce six MTF sidelobes ($N\left(N-1\right)=3\times 2=6$). Three pairs of sub-apertures A and B, B and C, A and C respectively produce the corresponding pair of side lobes 3 and 3′, 2 and 2′, 4 and 4′. Because of the small angle between the baseline L1 and L2, the sidelobes 2 and 3, and 2′and 3′partly overlap. But this overlap do not change the amplitude of the sidelobes. Therefore, the amplitudes can still be substituted into Eq. (3) to realize the piston retrieve. However, when the sidelobes overlap further, it will result in a change in the sidelobe amplitude, the measurement error will be introduced in piston detection with Eq. (3).

We simulate the process of two sidelobes overlap to research the influence of the overlap on sidelobe amplitude, and is shown as Fig. 6. The green line and blue line express two sidelobes respectively, and the red line expresses the synthesis sidelobe after the two lobes overlapping. b is the center frequency difference of the two sidelobes, R_{f} is the radius of the area occupied by the sidelobe in spatial frequency domain.

From the simulation process shown in Fig. 6, we can see that the overlap does not affect the amplitude of the each sidelobe as long as the relation $b\ge {R}_{f}$is satisfied. R_{f} is related to the sub-aperture’s diameter D, and given by

Usually, one reference segment should be designated for the phasing segmented mirror. We call the sub-aperture on the mask corresponding to the reference segment as the reference sub-aperture, the other sub-apertures on the mask corresponding to the other segments as measured sub-aperture. According to the above theory and simulation analysis, the arrangement rules are shown as below.

- I. The reference sub-aperture and measured sub-apertures are located on the same baseline. And the measured sub-apertures are located in the different baseline vector as possible as we can.
- II. For the sub-apertures which locate on the same baseline or in the same baseline vector, the center distance of any two sub-apertures should not be equal to the center distance between the reference sub-aperture and the measured sub-aperture.
- III. When a sidelobe partly overlaps with the pair of sidelobes produced by the reference and measured sub-apertures, the center frequency difference of the two overlap lobes must be larger than the radius of the frequency circle corresponding to the each sidelobe's.

## 3. MTF sidelobes non-redundant distribution for the 18-segment mirror

We take the 18-segment mirror, similar to JWST, as an example, and realize the MTF sidelobes non-redundant distribution by following Eq. (8) and the arrangement rules. Figure 7 shows the mask with a sparse 18 circular clear sub-apertures configuration. This mask is set in the conjugate plane of the 18-segment mirror during virtual piston error measurement.

The grey area is opaque, and the blue circle expresses a clear sub-aperture. The b sub-aperture is the reference one, the others are the measured sub-apertures. Nine baselines are selected such as a-b-c-d, h-b-i-j, e-b-f-g, b-m-n, b-p-q, b-k, b-l, b-s, b-r shown in Fig. 7. All of the 18 sub-apertures are located in these nine baselines. Each baseline includes the reference sub-aperture b in order to measure the piston errors between the reference segment and the other segments. Therefore, all of the MTF sidelobes respectively produced by the reference sub-aperture and all of the measured sub-apertures distribute on these nine baselines. We just need to ensure these sidelobes not to overlap. Meanwhile the diameter of the sub-apertures should be increased as much as possible to get better signal-to-noise ratio.

For a convenient statement, we describe the mask in its conjugate plane, i.e., on the segmented primary mirror. The coordinate origin is set at the center of the 18-segment mirror. The diameter of the segmented mirror is 8.7 m. The sub-aperture’s diameter is 0.3 m. The center coordinates of the sub-apertures are shown as below (unit: m).a (0, 4.07), b (0, 1.73), c (0,-1.73), d (0, −2.84), e (−0.96, 2.28), f (2.02, 0.56), g (3.52, −0.305), h (0.96, 2.28), i (−2.02, 0.56), j (−3.52, −0.305), k (−2.5, 1.73), l (3.42, 1.13), m (−0.8, −0.86), n (−1.47, −3.11), p (0.8, −0.86), q (1.47, −3.11), r (3, −1.5), s (−3, −1.5).

Figure 8 is the MTF distribution corresponding to the mask shown in Fig. 7. To display clearly, we use asterisk to indicate sidelobe. The nine baselines and the sidelobes in these nine baselines are marked in different colors. The other sidelobes are marked with dark asterisk.

According to Eq. (8), if the MTF sidelobes distribution is non-redundant, the quantity of the sidelobes should be equal to $N\left(N-1\right)$, N is the quantity of the sub-apertures. For example, the baseline such as a-b-c-d, h-b-i-j, and e-b-f-g, four sub-apertures are located on each of them. Twelve sidelobes should be produced on each of these three baselines. From Fig. 8 we can see that there are 12 sidelobes distributing on each of these three baselines. This means that the MTF sidelobes distribution in these three baselines are non-redundant. And so on, on the other six baselines, b-m-n, b-p-q, b-k, b-l, b-s, and b-r, the MTF sidelobes distribution are non-redundant also. The total 56 MTF sidelobes distribute in these nine baselines. Consequently, the MTF sidelobes non-redundant distribution is realized. Meanwhile, the validity of Eq. (8) and the feasibility of the arrangement rules are proved.

Further more, we can calculate the size of the mask according to the telescope optical imaging system parameters and set it in the exit pupil plane of an 18-segment primary mirror telescope. By performing the Fourier transform analysis for the PSF recorded on the CCD, the non-redundant distribution of the MTF sidelobes shown in Fig. 8 can be obtained. According to the selected baselines vectors and the designed positions of the sub-apertures we can confirm the positions of the sidelobes respectively produced by the reference sub-aperture and all of the measured sub-apertures. Thus, the amplitudes of the sidelobes in the nine baselines can be simultaneously obtained. To substitute these amplitudes respectively into Eq. (3), the piston errors of the whole aperture can be retrieved simultaneously.

## 4. Conclusion

In this paper, we put forward a method to simultaneously detect the multi-piston errors between the segments. Firstly, the relation between the piston error of any two segments and the amplitude of MTF sidelobes (MTF* _{nph}*) is derived by analyzing the intensity distribution on the image plane according to the Fourier optics principle. The piston error can be retrieved by using this relation after measuring the MTF

*. This method's capture range is the operating light’s coherence length, the accuracy is 0.026 λ (λ = 633 nm) RMS. Secondly, the MTF model of a mask with a sparse circular clear multi-subaperture configuration is researched and established. And then the arrangement rules, to avoid the MTF sidelobes respectively produced by the reference sub-aperture and all of the measured sub-apertures overlapping, are discussed and obtained according to the MTF model. Finally, the 18-segment mirror, similar to JWST, is taken as an example, the mask with a sparse 18 circular clear sub-apertures configuration is designed by following the MTF model and the arrangement rules, which makes the MTF sidelobes distribution non-redundant. Thus, the MTF*

_{nph}*of the sidelobes can be simultaneously obtained, and used to retrieve the piston errors of these 18 segments by using the relation of the piston error and the MTF*

_{nph}*. The hardware requirements of our method are very small, we just need to attach a mask with a sparse multi-subaperture configuration in the exit pupil plane of the segmented telescope, or rotate the mask particular for a larger quantity of segments, to cophase the entire segmented mirror. And this mask should ensure the MTF sidelobes non-redundant distribution. Thus, with this method, multi-piston measurement can be implemented simultaneously, and piston detection is no longer be divided into coarse and fine regimes which involving separate dedicated hardware solutions. This method can be adapted to any segmented and deployable primary mirror telescope, whatever the shape of the segmented mirror and the number of the segments is.*

_{nph}## Acknowledgments

This work was supported by the Beijing Key Lab for Precision Optoelectronics Measurement Instrument and Technology.

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