Phasing piston error in segmented telescopes

Junlun Jiang; Weirui Zhao

doi:10.1364/OE.24.019123

1. Introduction

As the demand for space exploration increasing, telescope's resolution has become higher and higher. In order to get a high resolution, the telescope primary mirror’s aperture must be enlarged. This makes optical processing and testing, and transportation or launch very difficult. 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 this problem. Currently, most of large telescopes adopt segmented primary mirror for the high resolution. The primary mirror diameter of the European Extremely Large Telescope (E-ELT) [1] is 42m, and made of 984 segments. The 30m diameter primary mirror of the Thirty Meter Telescope (TMT) [2] is made of 492 segments. The 6.5m diameter primary mirror of the James Webb Space Telescope (JWST) [3] is composed of 18 segments. The advanced large aperture space telescope (ATLAST) [4] is a concept for a next generation flagship astrophysics mission to study the universe. Its primary mirror of the segmented 9.2-meter aperture has 36 hexagonal 1.1315m glass mirrors.

The relative position of the segments, such as tip-tilt and piston error, will be affected by manufacturing error, external disturbances, thermal deformations, misalignments induced by launch, segment deployment errors, and spacecraft jitter. So with the telescope of segmented and deployable primary mirror design, problem of co-phasing all individual mirror segments arises. In order to achieve a spatial resolution comparable to that of a monolithic mirror, the optical path difference (OPD) between the segments should be reduced to λ/40 RMS. Many cophasing methods have been studied. Modified Shack-Hartmann sensing [5,6] and curvature sensing [7,8] are used on segmented telescope, and both have shown great success. The capture range of the broadband Shack-Hartmann algorithm (BSH) is ± 10λ, 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 dispersed fringe sensor (DFS) [9] 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. The two-dimensional dispersed fringe sensor [10] was proposed based on the DFS. A broadband source and a dispersing prism are used in the sensor. The broadband spectum interference pattern formed by the adjacent segments is dispersed according to the different wavelength. The dispersion direction is perpendicular to the baseline. The piston error can be obtained by analyzing the intensity distribution on these two directions. This method’s capture range is 200μm, measurement error is less than 20nm. 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 [11]. The simulation of different modulation functions(MTFs) for various piston error has been carried out and show that the best fitted function is the Gaussian function. MTF can be obtained by performing a Fourier transform for the point spread function (PSF) recorded on the image plane, and then the piston error can be retrieved by the Gaussian function. This method's capture range is close to the coherence length of the input light, but its accuracy is lower at the extremes of the capture range.

In this paper, we propose a novel method 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. A mask with a sparse subpupil configuration is set on the exit-pupil plane to sample the wave reflected by the segmented primary mirror. The relationship between the piston error and the heights of the MTF’s surrounding peaks is derived according to the Fourier optics principle. Simulation has been done to verify the relationship is correct. Therefore, the piston error can be detected by using this relationship after measuring the MTF’s surrounding peaks. Finally, experiment has been carried out on the ACAT [12] to validate the feasibility of this method. This method is adaptable to any segmented and deployable primary mirror telescope.

2. Proposition

Our proposition is based on Fourier optics. PSF can be obtained by imaging a point source with a segmented primary mirror telescope. A mask with a sparse subpupil configuration is set on the exit-pupil plane of the segmented telescope to research the relationship of the image quality and the piston error between the segments. For simplicity, we use a two-segment telescope with the subpupil configuration depicted in Fig. 1. At present, the hexagonal segments have a widespread adoption, such as JWST. So the hexagonal segment model [11] is usually used to analyze the problems of phasing error.

Fig. 1 Telescope segments(hexagons) and sparse subpupil(circles) configuration.

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In Fig. 1, a mask with two subpupils is set on the exit-pupil plane. The light, reflected by the segments, is sampled by each subpupil and focused on a CCD where the PSF can be recorded. Then, a 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 coordinates of the subpupil center are $(B / 2, 0), (- B / 2, 0)$ respectively, and the diameter of the subpupil is D. We can define the complex amplitude $G (x, y)$ of the wave in the subpupil plane $(x, y)$ as

G (x, y) = A (x, y) [c i r c (\frac{x - \frac{B}{2}, y}{D / 2}) * e^{i φ_{1}} + c i r c (\frac{x + \frac{B}{2}, y}{D / 2}) * e^{i φ_{2}}],

where

φ_{1}

and

φ_{2}

are phases of the waves reflected by the corresponding segments.

A (x, y)

represents the amplitude of

G (x, y)

, and assume

A (x, y) = {\begin{cases} 1 in the subpupils \\ 0 out of the subpupils \end{cases} .

The phase difference between two segments is

Δ φ

and shown as

Δ φ = φ_{2} - φ_{1} = \frac{2 π}{λ} p,

where

p

is the piston error,

λ

is the observation wavelength.

The PSF is defined on the image plane (u, v) as

\begin{array}{l} P S F_{m} (u, v, λ) = {| F T [G (x, y)] |}^{2} = 2 \frac{\frac{D^{2}}{2} J_{1}^{2} (π D \sqrt{u^{2} + v^{2}})}{u^{2} + v^{2}} [1 + \cos (\frac{2 π}{λ} p - 2 π u B)] \\ = F_{d i f f r a c t i o n} \cdot F_{int e r f e r e n c e}, \end{array}

where FT represents the Fourier transform operation,

J_{1}

is first order Bessel function,

u = x / (λ f)

,

v = y / (λ f)

,

f

is the focal length of the imaging lens,

F_{d i f f r a c t i o n} = 2 \frac{\frac{D^{2}}{2} J_{1}^{2} (π D \sqrt{u^{2} + v^{2}})}{u^{2} + v^{2}}

is diffraction factor,

F_{int e r f e r e n c e} = [1 + \cos (\frac{2 π}{λ} p - 2 π u B)]

is interference factor.

From Eq. (4), we can see that the intensity distribution on image plane contains the diffraction factor $F_{d i f f r a c t i o n}$ and the interference factor $F_{int e r f e r e n c e}$ . The diffraction factor is simple superposition formed by two single subpupil diffraction, and the interference factor is coherent superposition formed by the two subwaves sampled by the two subpupils respectively. The piston error is included in the interference factor. The intensity distribution on the image plane is changed with the piston error. According to the interference principle, the interference factor will disappear when the piston error exceeds the coherence length of the light used in the optical imaging system. At this time, coherent imaging of two subpupil becomes two single-subpupil diffraction imaging. The intensity distribution on image plane is the simple superposition of the two subpupil diffraction. Thus, the capture range of piston error is limited by the coherence length of the input light.

L = \frac{L_{c}}{2} = \frac{λ_{0}^{2}}{2 Δ λ},

where

L

is the capture range of piston error,

λ_{0}

is the central wavelength,

Δ λ

is the spectrum bandwidth of the input light or spectral filter,

L_{c}

is the coherence length and

L_{c} = λ_{0}^{2} / Δ λ

[13], the factor 1/2 in Eq. (5) comes from the input light reflection on the segmented primary mirror, giving an OPD equal to twice the step between segments.

Bringing $u$ and $v$ into Eq. (4), the image intensity distribution on the image plane can be expressed as

P S F_{m} (x, y, λ) = 2 \frac{\frac{D^{2}}{2} f^{2} λ^{2} J_{1}^{2} (\frac{π D}{λ f} \sqrt{x^{2} + y^{2}})}{x^{2} + y^{2}} [1 + \cos (\frac{2 π p}{λ} - \frac{2 π B}{λ f} x)] .

Taking the 2D Fourier transform of $P S F_{m} (x, y, λ)$ gives the complex OTF.

\begin{matrix} O T F (f_{x}, f_{y}) = F T [P S F_{m} (x, y, λ)] \\ = 2 O T F_{s u b} (f_{x}, f_{y}) + O T F_{s u b} (f_{x} + \frac{B}{λ f}, f_{y}) e^{- i \frac{2 π}{λ} p} + O T F_{s u b} (f_{x} - \frac{B}{λ f}, f_{y}) e^{i \frac{2 π}{λ} p}, \end{matrix}

where

(f_{x}, f_{y})

is the spatial frequency in the x and y direction respectively,

O T F_{s u b} (f_{x}, f_{y})

is the OTF of single aperture diffraction system.

O T F_{s u b} (f_{x}, f_{y}) = {\begin{cases} \frac{2}{π} [arc \cos (\frac{λ f}{D} ρ) - (\frac{λ f}{D} ρ) \sqrt{1 - {(\frac{λ f}{D} ρ)}^{2}}] ρ \leq \frac{D}{λ f} \\ 0 ρ > \frac{D}{λ f} \end{cases},

where

ρ = \sqrt{f_{x}^{2} + f_{y}^{2}}

.

According to the Eq. (7) and Eq. (8), the $O T F (f_{x}, f_{y})$ consists of three parts: one central part and two side lobes. These three parts distribute in three different regions, and have their own different center spatial frequency. The center spatial frequency of these parts is $(0, 0)$ , $(B / λ f, 0)$ , and $(- B / λ f, 0)$ respectively. Under the certain $λ$ and $f$ , the size of the each region is decided by $D$ , the distance between the side lobes or between the side lobe and central part is decided by $B$ . So, $B$ should be big enough and $D$ should be small enough to make the side lobe terms separate from the central part. Thus, $M T F (f_{x}, f_{y})$ can be obtained by doing modulus operation for the three parts of Eq. (7) respectively and shown as

\begin{matrix} M T F (f_{x}, f_{y}) = | O T F (f_{x}, f_{y}) | \\ = | 2 O T F_{s u b} (f_{x}, f_{y}) | + | O T F_{s u b} (f_{x} + \frac{B}{λ f}, f_{y}) e^{- i \frac{2 π}{λ} p} | + | O T F_{s u b} (f_{x} - \frac{B}{λ f}, f_{y}) e^{i \frac{2 π}{λ} p} | \\ = 2 M T F_{s u b} (f_{x}, f_{y}) + M T F_{s u b} (f_{x} + \frac{B}{λ f}, f_{y}) + M T F_{s u b} (f_{x} - \frac{B}{λ f}, f_{y}), \end{matrix}

where

M T F_{s u b} (f_{x}, f_{y})

is the modulus of the

O T F_{s u b} (f_{x}, f_{y})

_.

M T F_{s u b} (f_{x}, f_{y}) = | O T F_{s u b} (f_{x}, f_{y}) | .

According to Eq. (7) and Eq. (9), we can see that $M T F (f_{x}, f_{y})$ has three peaks at the center spatial frequencies of the three parts. We name them central peak and surrounding peak respectively.

When the input light is not monochromatic light, and is centered at $λ_{0}$ with the bandwidth $Δ λ$ . The PSF is defined as

P S F_{c} (x, y, λ) = 2 \int_{λ_{0} - \frac{Δ λ}{2}}^{λ_{0} + \frac{Δ λ}{2}} P S F_{m} (x, y, λ) S (λ) d λ,

where

S (λ)

is PSF weight of different wavelengths, assuming

S (λ) = 1

.

The integral of Eq. (11) is difficult to compute, it can be expressed as differential summation approximately and shown as Eq. (12). The integral range, bandwidth $Δ λ$ , is divided into n intervals equally.

\begin{matrix} P S F_{c} (x, y, λ) = \sum_{t = 1}^{n} [\frac{P S F_{m} (x, y, λ_{t}) Δ λ}{n}] \\ = \frac{Δ λ}{n} \sum_{t = 1}^{n} {2 \frac{(\frac{D^{2}}{2}) f^{2} λ_{t}^{2} J_{1}^{2} (\frac{π D}{λ_{t} f} \sqrt{x^{2} + y^{2}})}{x^{2} + y^{2}} [1 + \cos \frac{2 π}{λ_{t}} (p - \frac{B}{f} x)]} . \end{matrix}

The bigger the n is, the smaller the error introduced by this approximation will be.

Taking the 2D Fourier transform of Eq. (12) gives the complex OTF in broad spectrum as

\begin{matrix} O T F_{a r r a y} (f_{x}, f_{y}) = F T [P S F_{c} (x, y, λ)] \\ = \frac{Δ λ}{n} \sum_{t = 1}^{n} [2 O T F_{s u b} (f_{x}, f_{y}) + O T F_{s u b} (f_{x} + \frac{B}{λ_{t} f}, f_{y}) e^{- i \frac{2 π}{λ_{t}} p} + O T F_{s u b} (f_{x} - \frac{B}{λ_{t} f}, f_{y}) e^{i \frac{2 π}{λ_{t}} p}] . \end{matrix}

From Eq. (13), we can see that the OTF contains three peaks: one central peak and two surrounding peaks. The spatial frequency coordinate of the surrounding peaks is

(B / λ_{t} f, 0)

and

(- B / λ_{t} f, 0)

respectively. For different wavelength, the coordinates are different. But the coordinate difference introduced by different wavelength is very small, so the surrounding peaks position can be approximately considered constant as

(B / λ_{0} f, 0)

and

(- B / λ_{0} f, 0)

. And also, the piston error only affects the OTF’s surrounding peaks (OTF_ph), and doesn’t affect the central peak of OTF. Then one of the OTF’s side lobe can be expressed as

O T F_{p h} (f_{x}, f_{y}) = \frac{Δ λ}{n} O T F_{s u b} (f_{x} + \frac{B}{λ_{0} f}, f_{y}) [\sum_{t = 1}^{n} e^{- i \frac{2 π}{λ_{t}} p}] .

The

M T F_{p h} (f_{x}, f_{y})

is the amplitude part of the

O T F_{p h} (f_{x}, f_{y})

and given by

\begin{matrix} M T F_{p h} (f_{x}, f_{y}) = | O T F_{p h} (f_{x}, f_{y}) | \\ = \frac{Δ λ}{n} M T F_{s u b} (f_{x}, f_{y}) \sqrt{n + [\sum_{j = 1}^{n - 1} \sum_{t = j}^{n - 1} \cos (\frac{2 π}{λ_{t}} p + \frac{2 π}{λ_{t + 1}} p)]} . \end{matrix}

Thus, the piston error between the segments can be obtained by measuring the heights of the

M T F_{p h} (f_{x}, f_{y})

at different

(f_{x}, f_{y})

. We measure its peak height at the position

(B / λ_{0} f, 0)

or

(- B / λ_{0} f, 0)

and normalize it, and name it

M T F_{n p h}

. It can be expressed as

M T F_{n p h} (\frac{B}{λ_{0} f}, 0) = M T F_{n p h} (- \frac{B}{λ_{0} f}, 0) = \frac{1}{n} \sqrt{n + [\sum_{j = 1}^{n - 1} \sum_{t = j}^{n - 1} \cos (\frac{2 π}{λ_{t}} p + \frac{2 π}{λ_{t + 1}} p)]} .

In order to make the differential summation more close to the integral, we make the n = 100. When the input light’s central wavelength is 632.8nm and bandwidth is 1nm, we can get the relation curve between the $M T F_{n p h}$ and $p$ shown as Fig. 2 according to the Eq. (16).

Fig. 2 MTF_nph-p curve of the Eq. (16).

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Figure 2 shows that with the piston error increasing, the $M T F_{n p h}$ decreases. When the piston error increases to 200μm, the half of the coherent length $L_{c}$ , the $M T F_{n p h}$ decreases nearly to zero, and will increase with the expansion of the spectrum bandwidth. That means the maximum capture range depends on the coherent length of the light used in the telescope.

L = \frac{L_{c}}{2} = \frac{λ_{0}^{2}}{2 Δ λ} = \frac{{(632.8 n m)}^{2}}{2 \times 1 n m} \approx 200 μ m .

From Fig. 2 we can see that a little change of the $M T F_{n p h}$ corresponds to a large change of the piston error when close to phasing. Since the tip-tilt error can reduce the $M T F_{n p h}$ , before measuring the piston error with Eq. (16), the tip-tilt error should be corrected as better as possible.

3. Simulation

In order to validate the mathematical derivation for the Eq. (16), the relationship of the $M T F_{n p h}$ and $p$ , extensive numerical simulations have been performed. We simulate two sparse circular subpupils shown as Fig. 1. One subpupil is the reference pupil, which samples the light reflected by the reference segment. The other is the measured pupil, which samples the light reflected by an adjustable segment. Piston error is introduced on the measured pupil. The diameter of the circular sub-pupil is a and two subpupils center distance is b. The values of a and b should ensure to make MTF's side lobe terms separate from its central part. We set a spectral filter, its central wavelength is 632.8nm and its bandwidth is 1nm.. The program calculates the corresponding images on the image plane for different $p$ . When $p = 0$ , we can obtain the PSF shown as Fig. 3.

Fig. 3 The PSF of optical system (a = 0.5m, b = 1.73m).

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Then, we can get the MTF shown as Fig. 4 by Fourier transform of the PSF shown in Fig. 3. The heights of the surrounding peaks become maximum when $p = 0$ .

Fig. 4 The MTF corresponding to the Fig. 3.

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When $p = 200 λ$ , we get the PSF shown as Fig. 5. And get the MTF shown as Fig. 6 by Fourier transform of the PSF shown in Fig. 5. We can see that the heights of the surrounding peaks of MTF decrease.

Fig. 5 The PSF of optical system (a = 0.5m, b = 1.73m).

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Fig. 6 The MTF corresponding to the Fig. 5.

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By introducing piston error on the measured pupil with 0.1- $λ$ steps, we can get a set of data about the heights of the surrounding peaks $M T F_{n p h}$ . The curve about the relation between the $M T F_{n p h}$ and the $p$ can be plotted (see Fig. 7). We find that this curve is same as the curve derived by the Eq. (16). Thus, the relationship derivation of the $M T F_{n p h}$ and $p$ , the Eq. (16), is correct.

Fig. 7 MTF_nph as a function of the $p$ between two segments, Circles express individual simulation data in 10 $λ$ steps, solid line is the curve of the Eq. (16).

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Using Eq. (16) to calculate the piston error in a realistic segmented telescope directly is complex. We try to find a kind of function similar to the Eq. (16) and could be used to measure the piston error easily. So we fit the Eq. (16) with Gaussian, linear, parabola, cubic polynomial, quartic polynomial, and piecewise quartic polynomial functions. The best fitted one is the piecewise quartic polynomial function. We fit Fig. 2, the curve of the Eq. (16), with a two-piece quartic polynomial function curve shown as Fig. 8.

Fig. 8 The two-piece quartic polynomial function curve for Eq. (16).

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The two-piece quartic polynomial function is

M T F_{n p h} = {\begin{cases} A_{1} p^{4} + B_{1} p^{3} + C_{1} p^{2} + D_{1} p + E_{1} 0 \leq p < L_{s} \\ A_{2} p^{4} + B_{2} p^{3} + C_{2} p^{2} + D_{2} p + E_{2} L_{s} \leq p \leq L \end{cases},

where

A_{1}, B_{1}, C_{1}, D_{1}, E_{1}, A_{2}, B_{2}, C_{2}, D_{2}, E_{2}

are fitting parameters,

L

expresses the capture range of the piston error detection,

L_{s}

corresponds to a piston error and

0 < L_{s} < L

,

A_{1} = 4.776 \times 10^{- 10}, B_{1} = 5.629 \times 10^{- 10}, C_{1} = - 4.07 \times 10^{- 5}, D_{1} = 4.336 \times 10^{- 8}, E_{1} = 1, A_{2} = 2.585 \times 10^{- 11},

B_{2} = 1.298 \times 10^{- 7}, C_{2} = - 5.523 \times 10^{- 5}, D_{2} = 7.212 \times 10^{- 4}, E_{2} = 0.987, L_{s} = 34.75 μ m, L = 200 μ m .

These parameters will be different for the different segmented telescope imaging system and spectrum bandwidth of the source used in the system.

The difference between the fitting curve of the two-piece quartic polynomial function and the curve plotted by Eq. (16) is shown as Table 1. In Table 1, the settled piston error expresses the piston error introduced on the measured pupil, the calculated $M T F_{n p h}$ and the fitting $M T F_{n p h}$ is obtained by the Eq. (16) and the Eq. (17) respectively.

Table 1. Difference Between the Calculated MTF_nph and the Fitting MTF_nph

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Taking the fitting $M T F_{n p h}$ into the Eq. (17), we can obtain the corresponding $p$ , and name it measured piston error. The difference between the measured piston error and the settled piston error is shown as Table 2 and Fig. 9. By the error analysis, the difference is 1.8nm RMS.

Table 2. Difference Between the Measured Piston Error and the Settled Piston Error

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Fig. 9 The difference between the measured piston error and the settled piston error.

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We simulated 3161 sets of data, and due to the length of the article, only a part of data is shown in Table 1 and Table 2.

Thus, the piston error between the segments can be retrived by using the piecewise quartic polynomial function. But the relationship of the MTF_nph-p must be calibrated in advance.

4. Experiment

Validation experiment has been implemented for this method on the ACAT [12]. ACAT was set up to validate some methods for cophasing errors detection. The method of two-dimensional dispersed fringe sensor [10] is used to calculate the piston error between the adjacent segments. The tip-tilt errors are obtained by calculating the differences between the every two centroid positions of the images which were formed by the segments on the focal plane. The phasing errors can be corrected by several sensitive piezoelectric transducer (PZT) actuators in three degrees of freedom (piston, tip, and tilt) in a range of 15mm with a resolution of 0.3nm according to the phasing errors. The process of the cophasing errors detection and correction can be carried out under the computer closed-loop control in nanoscale.

ACAT consists of a source module (SM), beam splitting module (BSM), piston error detection module (PEDM), tip-tilt error detection module (TEDM), sensitive micro-displacement actuator module (SMAM), segmented mirror module (SMM), FISBA spherical interferometer, laser plane interferometer (LPI), and Computer Control System (CCS). Its optical layout is shown as Fig. 10 [12]. All the modules of the ACAT are installed on a bed with upper and lower layers. Segmented mirror and actuators coupled with the segments are placed on the lower layer, the other modules are placed on the upper layer.

Fig. 10 Optical layout of the ACAT.

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The SM consists of a continuous spectrum halogen light source with fiber bundle output, a group of lenses, and a 5μm pinhole (as a point object). The wave from the pinhole is collimated to provide a point object at infinity. After the BSM, the plane wave is converged and then reaches the spherical segmented mirror with a spherical curvature radius of 1500mm and a diameter of about 330mm. The segmented mirror is divided into three segments. One of them (Seg. 1) is fixed as a reference one. The others (Seg. 2 and Seg. 3) can be adjusted in three degrees of freedom in a range of 15mm with a resolution of 0.3nm according to the phasing errors using the PZT actuators. The each adjustable segment is coupled with the three actuators distributing in 120°interval on a circle. Waves reflected from the segmented mirror are recollimated and then incident to the PEDM and the TEDM after the BSM again. Piston and tip-tilt error between Seg. 1 and each adjustable segment are detected, respectively. According to the cophasing error, correction signals are calculated and output to the driver of the actuators by computer. Thus the position of each adjustable segment is corrected relative to Seg. 1, respectively. After several iterations, the phasing error will go steadily to a minimum. After phasing, the moveable plane reflector will be driven by a precision motor and inserted into the optical path to introduce the FISBA interferometer, which can evaluate the performance of the piston and tip-tilt error detection-correction closed-loop. The process of phasing error detecting and correcting can be monitored in real time by the LPI.

To validate the feasibility of the method proposed in this paper by experiment on the A CAT, we set a mask with three sparse circular subpupils, shown as Fig. 11, on the exit-pupil plane of the ACAT (the “Mask” shown in Fig. 10) to sample the sub-waves reflected from the three segments. Then the segmented plane wave with the piston error, normally incident on this mask and then image at the focal plane of the lens3. Thus the Fourier, i.e. far field regime is satisfied. The intensity distribution (PSF) formed by the sub-waves is recorded on a CCD. Making a Fourier transform for this PSF to get $M T F_{n p h}$ , then the piston error between the segments can be obtained by Eq. (17). Then on ACAT, this piston error will be used as a feedback signal to control the PZT actuators to correct the step between the segments.

Fig. 11 The mask with three sparse circular subpupils.

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As shown in Fig. 11, the diameter of the mask is 28mm.The subpupils' diameter and their position should ensure to avoid overlap of the MTF side lobes which formed by each pair of subwaves sampled by corresponding subpupils. Here, the diameter of the subpupil is 7.4mm, the center of the three subpupils distribute on a circle, the diameter of this circle is 15.9mm, and the three subpupils distribute in 120° interval on this circle.

The spectrum bandwidth of the ACAT’s source is 300nm, and central wavelength $λ_{0}$ is 550nm. The coherence length is 1μm. Firstly, on ACAT, initial piston errors of less than 200μm are introduced manually, and start close-loop control to phase the segments. Secondly, the mask with three sparse circular subpupils is set on the exit-pupil plane of the ACAT. Then, we calibrate the relation of the MTF_nph-p. For simplity, we calibrate the relation for Seg.3. We introduce piston error to Seg.3 from 0 to 0.5μm with a step of 0.005 $λ_{0}$ , and record the corresponding PSF and then calculate the $M T F_{n p h}$ . The relation of the MTF_nph-p is fitted with a two-piece quartic polynomial functions and shown as Eq. (18).

M T F_{n p h} = {\begin{cases} 15.8859 p^{4} + 0.5265 p^{3} - 7.4497 p^{2} + 0.0012 p + 1 0 \leq p < 0.1953 (μ m) \\ 15.0458 p^{4} + 4.4962 p^{3} - 10.8357 p^{2} + 0.9790 p + 0.9086 0.1953 \leq p \leq 0.5 (μ m) \end{cases} .

After calibration, Seg.2 is phased with the reference one, and initial piston error is introduced on Seg.3. The PSF, deformed by the piston error of the seg.3, is recorded by the CCD shown as Fig. 12.

Fig. 12 The PSF deformed by the piston error of the Seg.3.

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Mathematically, a Fourier transform is performed over the PSF to obtain the OTF and then the MTF can be gotten by modulus operation (shown as Fig. 13).

Fig. 13 The MTF corresponding to the Fig. 12.

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In Fig. 13, we can see that the MTF has six surrounding peaks and one central peak. Four of the six surrounding peaks are influenced by the piston error of the Seg.3.

Then using the Eq. (18) instead of the two-dimensional dispersed fringe sensor to measure the piston error. The $M T F_{n p h}$ is 0.5202. Taking the $M T F_{n p h}$ into the Eq. (18), we can retrieve the piston error 0.2815μm.

At last, the close-loop control of ACAT is started. After phasing, FISBA interferometer is used to evaluate the phasing result. The measurement result is shown in Fig. 14, PV = 0.58λ, RMS = 0.07λ, the piston error between Seg.1 and Seg.3 is 0.03λ, typically tip-tilt error is 0.049″. According to the Fig. 14, we can see that every segment figure error and the optical system error contribute more to the PV and RMS values of the whole aperture, instead of the tip-tilt and piston errors.

Fig. 14 Result evaluated by FISBA interferometer after closed-loop correction. The individual segment figure error is about PV = 0.24λ, RMS = 0.03λ.

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We repeated the process of the co-phasing error detection and correction for Seg. 3, and evaluation with FISBA for 10 times, and achieved typical figure values of PV = 0.53λ, RMS = 0.07λ, accuracy of detection and correction of the piston error is 0.026λ RMS, and 0.049″for the tip-tilt error. Thus, the method proposed in this paper is used to measure the piston error successfully, and its feasibility is validated also.

5. Conclusion

In this paper, we put forward a novel method to detect the piston error in segmented telescopes. A mask with a sparse sub-pupil configuration is set on the exit-pupil plane to sample the wave from the segments. The relation between the piston error and the heights of MTF’s surrounding peaks (MTF_nph) is derived by analyzing the intensity distribution on the image plane according to the Fourier optics principle. The piston error can be obtained by using this relation after measuring the MTF_nph. This method's capture range is the input light’s coherence length, the accuracy is better than 0.026λ RMS. The hardware requirements of this method are very small, just needs to attach a mask with a sparse sub-pupil configuration on the exit-pupil plane. Thus, with this method, cophasing is no longer be divided into coarse and fine regimes which involving separate dedicated hardware solutions. The simulation and experiments have been carried out to prove the validity and feasibility of this method. To show the method’s potential, it can be used to realize piston error parallel detection for a multi-segment primary mirror telescope with a high accuracy in a large capture range, just need to set a mask with a sparse multi-subpupil configuration on the exit-pupil plane of the telescope or rotate the mask to cophase the entire segmented mirror. The design principle for the sparse multi-subpupil configuration is to avoid overlap of the MTF side lobes which formed by each pair of subwaves sampled by corresponding subpupils. 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.

Acknowledgments

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

References and links

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Settled piston error(μm)	Calculated MTF_nph	Fitting MTF_nph	Difference
0	1	1	-5.15e-08
18.3512	0.9835	0.9835	1.48e-08
37.3352	0.9327	0.9327	-7.74e-05
56.3192	0.8509	0.8511	1.83e-04
75.3032	0.7430	0.7429	-1.42e-4
94.2872	0.6157	0.6156	-1.58e-04
113.2712	0.4767	0.4768	9.22e-05
125.9272	0.3815	0.3817	1.98e-04
144.9112	0.2410	0.2410	3.95e-05
163.8952	0.1100	0.1097	-2.41e-04

Settled piston error(μm)	Measured piston error(μm)	Difference(μm)
0	4.8425e-04	4.84e-04
18.3512	18.3512	8.27e-06
37.3352	37.3358	6.43e-04
56.3192	56.3188	-4.16e-04
75.3032	75.3035	3.11e-04
94.2872	94.2879	6.84e-04
113.2712	113.2714	1.98e-04
125.9272	125.9258	-0.0014
144.9112	144.9125	0.0013
163.8952	163.8940	-0.0012

Settled piston error(μm)	Calculated MTF_nph	Fitting MTF_nph	Difference
0	1	1	-5.15e-08
18.3512	0.9835	0.9835	1.48e-08
37.3352	0.9327	0.9327	-7.74e-05
56.3192	0.8509	0.8511	1.83e-04
75.3032	0.7430	0.7429	-1.42e-4
94.2872	0.6157	0.6156	-1.58e-04
113.2712	0.4767	0.4768	9.22e-05
125.9272	0.3815	0.3817	1.98e-04
144.9112	0.2410	0.2410	3.95e-05
163.8952	0.1100	0.1097	-2.41e-04

Settled piston error(μm)	Measured piston error(μm)	Difference(μm)
0	4.8425e-04	4.84e-04
18.3512	18.3512	8.27e-06
37.3352	37.3358	6.43e-04
56.3192	56.3188	-4.16e-04
75.3032	75.3035	3.11e-04
94.2872	94.2879	6.84e-04
113.2712	113.2714	1.98e-04
125.9272	125.9258	-0.0014
144.9112	144.9125	0.0013
163.8952	163.8940	-0.0012

Phasing piston error in segmented telescopes

Abstract

1. Introduction

2. Proposition

3. Simulation

4. Experiment

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (14)

Tables (2)

Equations (19)

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