Adaptive piston correction of sparse aperture systems with stochastic parallel gradient descent algorithm

Zongliang Xie; Haotong Ma; Xiaojun He; Bo Qi; Ge Ren; Li Dong; Yufeng Tan

doi:10.1364/OE.26.009541

1. Introduction

Optical sparse aperture systems have grown out of the quest for the next generation of large aperture telescopes [1]. There are two implementation forms, segmented telescopes such as the James Webb Space Telescope (JWST) [2] and Extremely Large Telescope (ELT) [3], and telescope arrays such as Large Binocular Telescope (LBT) [4] and STAR-9 [5]. To achieve a high resolution equivalent to that of a monolithic mirror, the sub-apertures must be phased within a fraction of the wavelength. The crucial point is correcting the piston errors, which can’t be directly detected by traditional wavefront sensors like Shack-Hartmann.

Many approaches of piston sensing with point sources or star images have been studied. Such methods commonly can be divided into two types: (1) intensity techniques where the measures are done directly from the intensity distribution or fringes, such as the modified Shack-Hartmann sensing [6,7], the dispersed fringes sensors [8,9], and the interferometry with masks and diffracting components [10]; and (2) modulation transfer function (MTF) techniques, capable of extracting piston errors from surrounding peaks of MTFs, such as the chromatic phase diversity [11], the intersegment piston sensor [12], and the calibration-retrieve sensing [13,14]. Though being of large capture ranges and high accuracy, these cophasing methods are limited to the size of the object. Before observing an extended object, we can first phase the sparse aperture systems with a point source, usually a star, while it would be difficult for us to correct the systems using the methods mentioned above when pistons again inevitably appear due to turbulence or thermal deformations during the observation of the extended object. Phase diversity, an image-based wavefront sensing method [15,16], allows the piston reconstruction whatever the size of the object observed. In spite of successfully phasing STAR-9 [5], it can only work within a narrow spectral band, causing 2π ambiguity.

In this paper, we introduce a close-loop control for piston correction through image quality optimization using stochastic parallel gradient descent (SPGD) algorithm. SPGD algorithm has been used in wavefront sensorless adaptive optics (AO) [17], while here it is used to phase a multi-aperture imaging system. Inspired by image-based AO [18], the image quality is considered as a function with respect to control signals of piston actuators. When the phasing close-loop starts, the pistons are required being less than the coherence length so that the intensity interference occurs and the image metric is relevant to the piston. Only in this start condition, can SPGD algorithm use piston perturbations to search the right gradient direction for optimization. During the phasing procedure, SPGD algorithm adaptively optimizes the compensating pistons to maximize the image quality. With the convergence achieved, the sparse aperture system is regarded as being phased.

The proposed technique for piston correction has following advantages. First, by removing the requirement of any additional optical components, it simplifies the configuration and is compatible with all existing multi-aperture systems. Secondly, compared with other phasing techniques using point sources, it is applicable to both point objects and extended scenes. And thirdly, unlike phase diversity, it doesn’t suffer from ambiguity. Also, there is a limitation that the reported technique requires the high signal to noise ratio. At low-light regime, it is easy for the metric variation caused by piston perturbations to be submerged by that caused by the noise, thus making it difficult for SPGD algorithm to find the right gradient direction for optimization and phase the system. However, the technique could be widely used in high flux applications, such as solar astronomy or active imaging.

This paper is structured as follows. In section 2, the optical model of sparse aperture imaging and how SPGD algorithm corrects the pistons are described. Section 3 gives a simulation analysis. Section 4 presents the experimental results with laser and white light point sources, as well as an extended object. Finally, we conclude this paper in section 5.

2. Principle of piston correction

2.1 Optical model

Based on the theory of Fizeau interferometry, the synthetic image detected on the CCD can be expressed as:

I (x, y) = o (x, y) \otimes h (x, y) + n (x, y)

where

I (x, y)

is the captured synthetic image,

o (x, y)

is the ideal geometrical image,

h (x, y)

represents the point spread function (PSF) of the sparse aperture system,

n (x, y)

is the additive noise,

\otimes

denotes a convolution, and

(x, y)

are the coordinates of image plane.

In the incoherent and monochromatic case, the PSF at the wavelength $λ$ can be expressed as

h (x, y, λ) = {| FT (P (u, v, λ)) |}^{2}

where

P (u, v, λ)

is the generalized pupil function,

(u, v)

are the pupil plane coordinates and

FT (\cdot)

represents Fourier transform.

For multi-aperture systems, the alignment errors seriously affect the imaging performance, including tip-tilt and pistons. Tip-tilt errors can be sensed by traditional wavefront sensors but pistons can’t. The purpose of this paper is to correct the pistons. So here we assume that all sub-apertures are equipped with AO and the phase distortions except pistons are corrected. In principle, pistons can be involved in the expression of $P (u, v, λ)$ , shown as follows

P (u, v, λ) = p (u - u_{1}, v - v_{1}) + \sum_{n = 2}^{N} p (u - u_{n}, v - v_{n}) \exp (\frac{2 π i}{λ} O P D_{n})

where N is the total number of sub-apertures,

p (u, v)

is the binary pupil function of an elementary,

(u_{n}, v_{n})

is the center position of the nth sub-aperture, and

O P D_{n}

is the relative piston error between the nth sub-aperture and the first one taken as the reference.

For the broadband case where the wavelengths range from $λ_{1}$ to $λ_{M}$ , the PSF should be an integral modeled as

h (x, y) = \int_{λ_{1}}^{λ_{M}} h (x, y, λ) s (λ) d λ

where

s (λ)

is the weight of different wavelengths.

Up to now, we have built the sparse aperture imaging model associated with the pistons. For conventional single-aperture imaging system, Muller and Buffington defined several image sharpness metrics and proved that these quality functions are maximized only when the phase distortions are eliminated. Inspired by their work, in this paper we also consider the phasing issue as an optimization problem. The basis of this correction method is that the chosen sharpness metric must be a function with respect to the pistons, which reaches the extrema only in the absence of pistons. There are many definition functions proposed to quantify the image quality. Here we discuss four candidates listed in Table 1.

Table 1. Different candidate image sharpness metrics

View Table

Using the optical model derived above, we explore the variation of the candidate sharpness metrics with respect to pistons for two apertures. The $λ_{1}$ and $λ_{M}$ of Eq. (4) are chosen as 500 nm and 700 nm, respectively. The discrete wavelength interval is 20 nm and $s (λ)$ is assumed as 1. The relation curves for the 4 definition functions between the PSF image metrics and pistons are plotted as Fig. 1, from which we can see that all the 4 definition metrics are functions associated with pistons, which can reach maxima when the pistons are zero. So in theory, it is available to correct the pistons by maximizing the image definition functions. It can be also seen from Fig. 1 that J₂ and J₃ generate many local extremas, which will influence the optimization procedure while the curves yielded by J₁ and J₃ show much better monotonicity. Though the curve for J₃ has a steeper variation than that for J₁, the J₃ calculation with a Fourier transform takes much more time than the J₁ calculation. In the following simulation and experiment, the simple but useful definition function J₁ is used as the image metric.

Fig. 1 The relation curves between the PSF sharpness metrics and pistons using 4 candidate definition functions.

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2.2 Piston correction with SPGD algorithm

In this paper, we use SPGD algorithm to search the correcting amounts of the piston actuators for maximization of the image sharpness metric. SPGD algorithm, capable of avoiding local extremas with fast convergence speed [19], has been widely used as a control algorithm for AO. Now here it is used to phase sparse aperture systems.

In any sparse aperture system, it is necessary to design some forms of actuation for each sub-aperture to allow for adjustments of optical path lengths so that the pistons could be compensated as follows

O P D'_{n} = O P D_{n} - m_{n}

where

O P D'_{n}

is the residual piston error and m_n is the compensating optical path length handled by piston actuator for the nth sub-aperture. The whole phase compensation, introduced by the piston actuators, can be combined linearly as follows:

w (u, v, t) = \sum_{n = 1}^{N} m_{n} (t) S_{n} (u, v)

where

S_{n} (u, v)

is the phase response functions of the nth sub-aperture. Specifically, in sparse aperture system, the response function of piston actuator for the nth sub-aperture can be expressed as

S_{n} (u, v) = \frac{2 π}{λ} p (u - u_{n}, v - v_{n})

Then according to Vorontsov’s derivation [20], piston perturbations are generated to estimate the gradient and the control signal of the nth sub-aperture piston actuator is updated at each iteration k by the following equation

m_{n}^{(k + 1)} = m_{n}^{(k)} + γ δ J^{(k)} δ m_{n}^{(k)}

where γ is the gain update,

δ J

is the image metric variation resulting from small random perturbations

{δ m_{1}, δ m_{2}, ..., δ m_{N - 1}}

loaded on all the piston actuators except for the reference one. Specifically, the metric variation is obtained by using both positive and negative piston perturbations, following the expression below

δ J = J (m + δ m) - J (m - δ m)

where

m

is the control vector and

δ m

is the perturbation vector for the N-1 piston actuators. As shown in Fig. 2, how SPGD algorithm works in each step can be briefly described as follows.

Fig. 2 Flowchart of SPGD phasing algorithm.

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First, statistically independent random perturbations $δ m = {δ m_{1}, δ m_{2}, ..., δ m_{N - 1}}$ on piston actuators are generated, where all $δ m_{n}$ are small values with zero mean and equal variances, $〈 δ m_{n} 〉 = 0$ , $〈 δ u_{n} δ u_{m} 〉 = σ^{2} δ_{n m}$ , where $δ_{n m}$ is the Kronecker symbol.

Next, we output the control signal with positive perturbations to piston actuators and calculate the sharpness metric function based on the intensity captured by CCD, denoted as $J_{+} = J {m_{1} + δ m_{1}, m_{2} + δ m_{2}, ..., m_{N - 1} + δ m_{N - 1}}$ .

Then, we output the control signal with negative perturbations to piston actuators and calculate the sharpness metric function based on the intensity captured by CCD, denoted as $J_{-} = J {m_{1} - δ m_{1}, m_{2} - δ m_{2}, ..., m_{N - 1} - δ m_{N - 1}}$ .

Finally, the control signals are updated by $m = m + γ δ m (J_{+} - J_{-})$ , where $γ$ is the update gain, $γ > 0$ and $γ > 0$ according to the procedure of maximization and minimization, respectively. In this paper, $γ > 0$ is chosen to maximize the image quality.

The iteration stops until the sharpness metric achieves its maximum value. The increasing process of the metric function means the pistons are being reduced gradually, and the final convergence is an indication that the system is phased well.

3. Numerical simulation

We first demonstrate the effectiveness of the proposed phasing technique by means of simulation. One advantage of this method is handling extended objects while most current methods can only work with point sources. To prove this attractive capability, the simulation is performed with an extended target, specifically the USAF 1951 resolution test chart, shown as Fig. 3(a). Then a sparse aperture system consisting of 4 sub-apertures with diameters of 100 mm is simulated shown in Fig. 3(b). The equivalent focus length is 1900 mm and the pixel size of the detector is 1.67 μm, which closely match our experimental testbed. The imaging broadband ranges from 500 nm to 700 nm, which is approximately simulated by a differential summation with 20 nm sampling ratio between adjacent wavelengths (11 bands). The weight of different wavelengths $s (λ)$ is assumed to be 1. Then a realization of piston errors are generated for the system as shown in Fig. 3(c), where the left lower sub-aperture is taken as the reference, and the relative pistons of the other three are 200, 300 and 400 nm, respectively.

Fig. 3 Simulation setup. (a) The USAF 1951 resolution test chart used as an extended object. (b) The four aperture configuration. (c) The loaded piston errors [200, 300, 400] nm.

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The close-loop control then starts to correct the piston errors. To improve the realism of the simulation, 1% Gaussian noises (variance $σ^{2}$ = 1% of that of signal values) are added for each frame of the intensity images recorded by CCD. In the close-loop, SPGD algorithm, performed as a piston corrector, adaptively maximizes the sharpness metric by controlling the piston actuators. The evolution of the normalized metric with SPGD algorithm proceeding is shown as Figs. 4(a), from which it can be found that the metric approximately reaches its maximum value after about 30 iterations. As a result, the residual pistons of the three sub-apertures of interest after compensation are 12 nm, 4 nm and 16 nm, respectively, as presented in Fig. 4(b). The high phasing accuracy demonstrates that SPGD algorithm has almost achieved the global optimization with fast convergence speed. Fig. 4(c) and 4(d) exhibit the images without and with close-loop, respectively. From their comparison, it is easy to recognize that the sharpness and contrast of the corrected image is obviously enhanced. To give a more explicit improvement in image quality, the line traces of the vertical bars labeled with blue and red dotted lines in Figs. 4(c) and 4(d) are plotted as an example in Fig. 4(e). The exhibition from the plots that the differences between the peaks and valleys for the corrected image become larger than those of the uncorrected one, further demonstrating the improvement in terms of sharpness and contrast.

Fig. 4 Results of simulation. (a) The evolution of the sharpness metric as SPGD algorithm proceeds. (b) The residual piston errors [12, 4, 16] nm after close-loop control. The images without (c) and with (d) piston correction. (e) The trace lines of the same sets of bars of (c) and (d) marked with blue and red dotted lines, respectively.

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Then we analyze the influence of the update gain of SPGD phasing algorithm. To reduce the disturbance of other factors, the noise isn’t added here. Then based on the same settings, we perform 8 phasing controls using the update gains of 400, 600, 800, 1000, 1200, 1400, 1600, and 1800, respectively. The phasing evolutions generated by SPGD algorithm are shown as Fig. 5. It can be seen that the update gain mainly influences the convergence speed and stability. In general, the metric converges faster with the update gain increasing. But the update gain seems to have a threshold value, corresponding to the fastest convergence speed. As the update gain increases beyond the threshold value, the convergence speed decreases and even the stability declines with serious fluctuation appearing. This simulation could provide a useful guide for the choice of the update gain in further applications.

Fig. 5 The evolutions of the sharpness metrics generated by SPGD algorithm using different update gains.

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4. Experimental results

The close-loop phasing algorithm has been applied to our binocular telescope testbed, of which the configuration is shown as Fig. 6. The testbed consists of two Maksutov-Cassegrain telescopes (Meade’s product, ETX-125), arranged horizontally in a center-to-center distance of 180 mm, each with a diameter of 127 mm. The equivalent focal length is 1900 mm and the pixel pitch of the detector is 1.67 μm. A pyramidal mirror mounted on a coupled motorized translation stage, shown at the lower right of Fig. 6, is served as the piston actuator. There are two translation stages for high and low precision control respectively. It might be difficult for the close-loop control to realize phasing when the interference doesn’t happen. In that case, the intensity is formed simply by summation so that the metric is irrelevant to the piston perturbations, leading to difficulty of SPGD algorithm in searching the right gradient direction. Thus SPGD algorithm mainly commands the high-precision one (PI’s product, E-727) with a spatial resolution of 1 nm for fine phasing in the status where the interference occurs. In the experimental environment, the commercial telescope suffers weak aberrations and the tip-tilt errors have been corrected well. In such a condition, the piston is the dominant distortion. The proposed close-loop control is used to successfully phase the testbed with laser and white point sources, as well as with an extended object. Here detailed experimental results are presented.

Fig. 6 The optical configuration of the binocular telescope testbed. The piston actuator (the lower right), is composed of a pyramidal mirror and a coupled motorized translation stage. The high-precision one (PI’s product, E-727) can achieve a spatial resolution of 1 nm.

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4.1 Laser source

To start with, a fiber laser is used as the source to phase the binocular telescope using SPGD algorithm. Although there exists 2π ambiguity for the monochromatic case, we can test the effectiveness within a certain period. Besides as an initial demonstration, this set of experiment is also of significance for active sparse aperture imaging [21]. Figure 7(a) shows the phasing evolution of the close-loop control, in which the image metric converges to its maximum value after about 80 iterations. The images without and with close-loop control are shown as Figs. 7(b) and 7(c), respectively. The absence of other piston sensors in our lab limits us to quantitatively measure the phasing accuracy. However, two aspects can be also summarized to indicate that the system is phased well. On the one hand, the intensity power is increased without amplifying the source energy or prolonging the exposure time. The only explanation for the intensity improvement is because of piston correction. The intensity is finally converged, denoting that the piston has been corrected. On the other hand, from the comparison between the uncorrected and corrected images, it can be seen that the corrected one exhibits better fringes with two side lobes distributed symmetrically and presents improved contrast, indicating correction of piston errors.

Fig. 7 Experimental phasing results with a laser source. (a) The evolution of the sharpness metric as SPGD algorithm proceeds. The images without (b) and with (c) piston correction.

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4.2 White-light source

Next, we demonstrate the phasing performance with a white-light source. During the close-loop procedure, SPGD algorithm controller continuously updates the movement of the high-precision translation stage to compensate the piston for maximizing the metric function, generating an evolution curve plotted as Fig. 8(a). It can be found that after about 25 iterations, SPGD algorithm converges to its maximum value. However, since the white-light source is not as stable as the laser, the intensity fluctuation after convergence seems more serious. Averaging more images can reduce such fluctuation at expense of time while here three images are averaged for one metric calculation. The images in open-loop and close-loop are presented in Figs. 8(b) and 8(c), respectively. Similarly, the increased converged power indicates the piston correction. Nevertheless, due to the weak coherence, the interference fringes having lower contrast can’t be observed directly from the intensity distribution. Luckily, the spatial frequencies of the fringes can be identified in the modulation transfer function (MTF). We perform the Fourier transforms on Figs. 8(b) and 8(c), and then obtain their spectrums shown as Figs. 9(a) and 9(b), respectively, which are also the MTFs in the case of point sources. The peaks of MTF side lobes will increase with the piston decreasing from the theoretical derivation of the literature [13]. The spectrum of the uncorrected image in Fig. 9(a) exhibits side lobes faintly while that of the corrected image in Fig. 9(b) presents them obviously with their energy much increased. Specifically, the logarithmic side lobe peak with piston correction is 1.769 while that without piston correction is only 0.701. The increased side lobe peak and final convergence indicate that the pistons have been corrected.

Fig. 8 Experimental phasing results with a white-light source. (a) The evolution of the sharpness metric as SPGD algorithm proceeds. The images without (b) and with (c) piston correction.

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Fig. 9 The spectrums of the images without (a) and with (b) piston correction (log scale).

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4.3 Extended object

Finally, we test the attractive advantage of our proposed method, specifically phasing realization with extended objects. A binary grating is used here as an extended scene to phase the testbed. Fig. 10(a) and 10(b) present the imaging results before and after close-loop control, respectively. The corresponding phasing evolution is shown as Fig. 10(c), exhibiting that the sharpness metric is converged after 30 iterations. Similar to the previous analysis, piston correction is indicated from two points. First, the intensity power of the corrected image is improved thanks to piston reduction, and the final convergence indirectly demonstrates that the testbed has been phased well. Second, compared with the uncorrected image, the corrected one shows enhanced sharpness and contrast, which can be found especially from the larger peak-valley differences shown by the line traces of Fig. 10(d). That the experimental improvement of the image definition and contrast, as well as the fluctuating variation of the line traces, coincides with the simulation, is an indication that we achieve fine phasing.

Fig. 10 Experimental phasing results with an extended object. The images without (a) and with (b) piston correction. (c) The evolution of the sharpness metric as SPGD algorithm proceeds. (d) The trace lines of (a) and (b) marked with blue and red dotted lines, respectively.

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The proposed SPGD close-loop algorithm succeeds in phasing the binocular telescope testbed, and in turn the fine experimental results also demonstrate its high optimization capability. SPGD algorithm works for the laser source, white light source, and extended object, indicating that its optimization is robust to different conditions. Also, in each condition, SPGD algorithm always generates a fast convergence speed. The two points both verify that SPGD algorithm has high optimization capability.

5. Conclusion

In conclusion, an adaptive piston correction method using SPGD algorithm has been proposed and demonstrated for sparse aperture systems. The piston errors can be corrected by iteratively searching for the maximum value of the image sharpness metric. SPGD algorithm controlling the compensating optical path lengths is performed here as a piston corrector. The simulation with four apertures and experiment based on our binocular telescope are conducted to prove the effectiveness of the technique. It is shown from the results that our phasing close-loop control using SPGD algorithm can’t only work with the laser and white-light sources, but also with the challenging extended objects. The technique proposed in this paper is promising owing to the needlessness of any additional optics and applicability to both point sources and extended objects, which we believe will have wide applications in segmented telescopes and telescope arrays.

Funding

National Natural Science Foundation of China (NSFC) (61205144, 61775239); Youth Innovation Promotion Association, CAS.

References and links

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Definition functions	Description
$J_{1} = \sum_{x} \sum_{y} I {(x, y)}^{2}$	The basic gray level variance
$J_{2} = I (x_{0}, y_{0})$	The intensity of the central point (x₀, y₀)
$J_{3} = \sum_{x} \sum_{y} {{[I (x, y) - I (x + 1, y + 1)]}^{2} + {[I (x + 1, y) - I (x, y + 1)]}^{2}}^{1 / 2}$	The Roberts gradient sharpness function
$J_{4} = {\sum_{f_{x}} \sum_{f_{y}} (f_{x}^{2} + f_{y}^{2}) \| FT [I (x, y)] \|}^{2}$	The mean square radius of the image spectrum, (f_x, f_y) are frequency coordinates

Adaptive piston correction of sparse aperture systems with stochastic parallel gradient descent algorithm

Abstract

1. Introduction

2. Principle of piston correction

2.1 Optical model

2.2 Piston correction with SPGD algorithm

3. Numerical simulation

4. Experimental results

4.1 Laser source

4.2 White-light source

4.3 Extended object

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Tables (1)

Equations (9)

Optics Express