Issue 
A&A
Volume 567, July 2014



Article Number  A87  
Number of page(s)  8  
Section  Astronomical instrumentation  
DOI  https://doi.org/10.1051/00046361/201321486  
Published online  16 July 2014 
Cophasing of a diluted aperture synthesis instrument for direct imaging
II. Experimental demonstration in the photoncounting regime with a temporal hypertelescope
^{1}
Équipe photonique −
XLIM (CNRS UMR 7252 ), University of Limoges,
123 avenue Albert Thomas,
87060
Limoges Cedex,
France
email:
laurent.bouyeron@unilim.fr
^{2}
CODECHAMP, 23190
Champagnat,
France
^{3}
LEUKOS SAS, 87069
Limoges Cedex,
France
Received:
15
March
2013
Accepted:
7
April
2014
Context. Amongst the new techniques currently developed for highresolution and highdynamics imaging, the hypertelescope architecture is very promising for direct imaging of objects such as exoplanets. The performance of this instrument strongly depends on the cophasing process accuracy. In a previous highflux experimental study with an eighttelescope array, we successfully implemented a cophasing system based on the joint use of a genetic algorithm and a subaperture piston phase diversity using the object itself as a source for metrology.
Aims. To fit the astronomical context, we investigate the impact of photon noise on the cophasing performance operating our laboratory prototype at low flux. This study provides experimental results on the sensitivity and the dynamics that could be reached for real astrophysical observations.
Methods. Simulations were carried out to optimize the critical parameters to be applied in the cophasing system running in the photoncounting regime. We used these parameters experimentally to acquire images with our temporal hypertelescope test bench for different photon flux levels. A data reduction method allows highly contrasted images to be extracted.
Results. The optical path differences have been servocontrolled over one hour with an accuracy of 22.0 nm and 15.7 nm for 200 and 500 photons/frame, respectively. The data reduction greatly improves the signaltonoise ratio and allows us to experimentally obtain highly contrasted images. The related normalized point spread function is characterized by a 1.1 × 10^{4} and 5.4 × 10^{5} intensity standard deviation over the dark field (for 15 000 snapshots with 200 and 500 photons/frame, respectively).
Conclusions. This laboratory experiment demonstrates the potential of our hypertelescope concept, which could be directly transposed to a spacebased telescope array. Assuming eight telescopes with a 30 cm diameter, the Iband limiting magnitude of the main star would be 7.3, allowing imaging of a companion with a 17.3 mag.
Key words: instrumentation: high angular resolution / instrumentation: interferometers / techniques: interferometric
© ESO, 2014
1. Introduction
The hypertelescope concept (Labeyrie 1996) is a promising concept for the next generation of high angular resolution instruments. The main feature of this multiaperture interferometer is to provide direct images in the milliarcsecond range at optical wavelengths. For this purpose, the hypertelescope allows isolation of the light of each resolved element of the object on one pixel of the image. The related potential astrophysical targets can be extended sources and binary systems with high contrasts. For example, in the case of an exoplanetary system, the photons coming from the faint companion are always imaged separately from the main bright source. Thanks to this property, this instrument shows a higher sensitivity than a classical interferometer (Lawson 1997; Ten Brummelaar et al. 2005; Petrov et al. 2007) sampling the spatial spectrum of the object. However, the expected performance strongly depends on the accuracy of the cophasing system.
Fig. 1 THT testbench theoretical PSF on logarithmic (left) and linear (right) scale. In both cases, the intensity is normalized. The telescopearray configuration is optimized for highcontrast imaging. DR is the dynamic range of the PSF, CLF is the clean field of view and DF is the dark field. 
In 2007, we proposed an alternative hypertelescope architecture (Reynaud & Delage 2007) called the temporal hypertelescope (THT). The main advantage of this instrument is its versatility. It can be easily reconfigured to fit the observed object geometry, and is well suited for spatial and terrestrial instruments. A THT test bench was implemented at the XLIM laboratory (Bouyeron et, al. 2010) thanks to the support of CNES and Thales Alenia Space. This instrument consists of eight telescopes linked to an integratedoptics eighttoone interferometer through optical fibres. In a previous step (Bouyeron et al. 2012), we developed a cophasing method based on a joint use of a genetic algorithm (GA, Brady 1985) and the subaperture piston phase diversity technique (SAPPD, Bolcar & Fienup 2005, 2009). This method relies on the use of the current image and an aberrated one in order to infer the instrument phase without any a priori knowledge on the observed object. This phase shift is used as an error signal and is sent to the feedback loop dedicated to the cophasing of the instrument. We experimentally obtained a λ/ 400 (4 nm) stabilization of the optical path difference over one hour, allowing the acquisition of highcontrast images (1:10 000). However, this first experiment was realized using sources much brighter than available astronomical objects. Consequently, the impact of photon noise on the cophasing system was not significant.
In this article, we address this last point and report on experimental cophasing results obtained in the photoncounting regime (PCR) with the THT test bench. Dynamic and point spread function (PSF) measurements are achieved using a double output interferometer. The first one operates at low flux and is used to stabilize the optical paths of the instrument. The second one displays the image in a high flux regime and is used as a reference measurement to check the cophasing accuracy. The experimental setup is described in Sect. 2. The optimization of the phaseshift evaluation (SAPPD parameters) involved in the cophasing process in the photoncounting regime is presented in Sect. 3 through simulations. Section 4 presents the experimental tests of the cophasing process and image acquisitions obtained with the THT instrument operating in the photoncounting regime. Finally, using our experimental data as a starting point, the extrapolations of these results to a real spacebased instrument are presented in Sect. 5.
2. General description of the THT test bench
A THT is a multiaperture interferometer providing a direct image of the observed object. The instrument is based on an experimental configuration able to maintain the phase relationship between the optical fields coming from the telescopes and to be mixed in accordance with the golden rule of imaging interferometry (Traub 1986). In the THT (Reynaud & Delage 2007), these phases are temporally modulated, taking the input pupil configuration into account. The image is temporally acquired point by point through a raster scan process. At a given wavelength λ, the image I is described over the field of view by $\begin{array}{ccc}\mathit{I}\mathrm{=}\mathrm{PSF}\mathrm{\otimes}\mathit{O}& & \end{array}$where PSF denotes the instrument PSF, O the object angular intensity distribution, and ⊗ the convolution operator. The corresponding image spatial spectrum $\begin{array}{c}{\mathrm{\u02dc}}_{}\\ \mathit{I}\end{array}$ is equal to $\begin{array}{ccc}\begin{array}{c}\mathrm{\u02dc}\\ \mathit{I}\end{array}\mathrm{=}\mathrm{\mathcal{F}}\mathrm{\left(}\mathit{I}\mathrm{\right)}\mathrm{=}\mathrm{OTF}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ \mathit{O}\end{array}& & \end{array}$where OTF is the instrument optical transfer function, $\begin{array}{c}{\mathrm{\u02dc}}_{}\\ \mathit{O}\end{array}$ the object spatial spectrum, and ℱ(I) the Fourier transform of the image I.
The PSF can be optimized for highcontrast imaging (see Fig. 1). This ability is characterized by the PSF dynamic range (DR), which is defined as the ratio between the PSF main peak intensity I_{max} and the maximum intensity level I_{DF} over the PSF dark field (DF: the PSF area where residual side lobes are located): $\begin{array}{ccc}\mathrm{DR}\mathrm{=}\frac{{\mathit{I}}_{\mathrm{max}}}{{\mathit{I}}_{\mathit{DF}}}\mathrm{\xb7}& & \end{array}$The object angular size must not exceed the array clean field of view (CLF) to avoid any aliasing effect. The clean field of view is defined as $\begin{array}{ccc}\mathrm{CLF}\mathrm{=}\frac{\mathit{\lambda}}{{\mathit{B}}_{\mathrm{min}}}\mathit{,}& & \end{array}$where B_{min} is the smallest array baseline. The dark field corresponds to the CLF area where a faint object could be detected.
Our main goal is to provide images with both high angular resolution and high contrast using a limited number of telescopes (fewer than ten). For this purpose, in a previous study (Armand et al. 2009), the telescope spatial distribution in the THT input pupil and the relative photon flux collected by each subaperture were optimized to maximize the dynamic range when imaging a highcontrast linear object, such as a starplanet system.
The THT test bench can be divided into four main parts (see Fig. 3).

The binarystar simulator: The star simulatorconsists of two singlemode fibres fed by independentdistributedfeedback (DFB) lasers at 1.55 μm located in the focal plane of a lens. These sources are linearly polarized. The intensity ratio between the two pointlike sources can be adjusted to set the intensity distribution of the object under test. Their angular separation is equal to 25 μrad. In the framework of this experimental study, we only use quasimonochromatic light.

The telescope array: The telescope array consists of eight lenses acting as telescopes. The input pupil mapping is reported in Fig. 2. This configuration is equivalent to a redundant linear array when observing a binary star (i.e. 1D object along a vertical direction). By applying the apodization technique to maximize the dynamic range, the flux levels can be adjusted by an iris in front of each subaperture to fit the theoretical optimized configuration. The wavefront portions collected by each lens are then injected into singlemode polarization maintaining fibres. This leads to a spatial filtering (Lardière et al. 2007) and a control of the polarization of the optical fields along the beam propagation. The clean field of view is CLF = 62 μrad. The optimization was done for the following dark field area DF = [−31; −15.5 ] ∪ [ 15.5;31 ] μrad.

The eightarm interferometer: Each arm consists of a fibre delay line and a fibre optical path modulator. The first device enables the balancing of the optical path lengths between the interferometer arms. The second device enables the temporal modulation of optical path required to achieve the image acquisition with the THT instrument. The optical path difference OPD_{kl}, generated between the arms k and l, is driven by the golden rule of imaging interferometry that requires a linear relationship between the telescope baseline and the related optical path difference: $\begin{array}{ccc}{\mathrm{OPD}}_{\mathit{kl}}\mathrm{=}\mathit{\lambda}\frac{{\mathit{B}}_{\mathit{kl}}}{{\mathit{B}}_{\mathrm{min}}}\left(\frac{\mathit{t}}{{\mathit{t}}_{\mathrm{0}}}\mathrm{}\frac{\mathrm{1}}{\mathrm{2}}\right)\mathit{,}& & \end{array}$where B_{kl} = (k − l)B_{min} is the baseline between telescopes k and l, and t_{0} is the scan time of the entire clean field of view. The temporal modulation of optical paths allows acquisition of a single image (called the shortexposure image) every t_{0} = 100 ms. Finally, beams collected by each subaperture are mixed in an eighttoone optical combiner.

The signal acquisition and processing: At the eighttoone optical combiner output (see Fig. 4), the light is split into two channels by using a 90/10 Y optical junction creating high flux – low flux channels.
Fig. 2 Theoretical (white disks) and experimental (black disks) telescope array configuration. Owing to the experimental constraints, the configuration of the telescope array is not linear. However, the projection of the telescope baseline along the vertical axis (direction of the 1D object) is fully identical to the theoretical redundant configuration. These two configurations are equivalent to a 1D vertical array when observing a 1D vertical object. 
Fig. 3 Temporal hypertelescope testbench scheme. This setup is divided into four main parts: (a) the unbalanced binarystar simulator, (b) the telescope array, (c) the eightarm interferometer, and (d) the signal acquisition chain. 
Fig. 4 Signalacquisition chain. Two signals are simultaneously acquired. The first one (top curve) is a shortexposure image obtained in the photoncounting regime (PCR) by using a hybrid detector, the second one is the image simultaneously detected with a high flux level 
The first channel uses an InGaAs photodiode to directly detect the infrared high flux level. It is not used in the servo loop of the cophasing system. In the second channel, the light passes through an adjustable fibre attenuator combined with a hybrid detector (Roussev et al. 2004), which reduces the 1.55 μm light flux to the photoncounting regime and mimicks a real astronomical observation. The InGaAs detector at this infrared wavelength has a high darkcount level and a low detection efficiency. To overcome this problem, we used a hybrid detector developed in our laboratory (Ceus et al. 2012). This detector is based on a sumfrequency generation process used for its inherently noiseless properties (Louisell et al. 1961). In a nonlinear optical waveguide (PPLN crystal) pumped by a 1.064 μm YAG laser, the 1.55 μm infrared light is converted into a 633 nm visible signal (see Fig. 5). This way, in the photoncounting channel, the signal can be acquired by a silicon photoncounting photodiode with a very low dark count (2.5 photons/frame).
Two channels are simultaneously recorded by the same data acquisition board. The first channel is a shortexposure highflux image used as a reference measurement. The second channel records singlephoton events captured by the photoncounting detector. Knowing the phasemodulation applied on the interferometer arms, the arrival time of each photon can be linked to an angular position in the image. Figure 4 illustrates such an acquisition.
3. Cophasing in the photoncounting regime
3.1. The principle of cophasing process
The cophasing system, previously implemented on the THT test bench, is based on a joint use of the subaperture piston phase diversity technique (SAPPD, Bolcar & Fienup 2005, 2009), and a genetic algorithm (GA, Brady 1985).
The phase diversity technique allows us to compute a phase criterion sensitive to the phase aberrations of the multiaperture instrument, but independent of the unknown geometry of the astronomical target. This process involves acquiring two different images of the same object. The first one, called the standard image I_{0}, is the current image of the object for which the phase piston errors must be cancelled. The second one, called the diversity image I_{d}, is obtained by applying known aberration errors to the instrument.
On the THT test bench, the diversity function is generated using optical path modulators located on each interferometer arm. These variations between the optical paths are directly related to the piston phase. The piston diversity range (PDR) is the span over which the values of piston phase diversity are randomly chosen.
The two images are used to compute a phase criterion χ^{ref} that does not depend on the target intensity distribution and that is a relevant signature of the actual piston errors. Kendrick et al. (1994) have proposed four different metrics to get the χ^{ref} criterion
$\begin{array}{ccc}& & {\mathit{M}}_{\mathrm{1}}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{=}\frac{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}\\ & & {\mathit{M}}_{\mathrm{2}}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{=}\frac{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{+}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}\\ & & {\mathit{M}}_{\mathrm{3}}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{=}\frac{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{+}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{+}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}\\ & & {\mathit{M}}_{\mathrm{4}}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{=}\frac{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}{\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{0}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathrm{+}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}\mathit{.}\begin{array}{c}\mathrm{\u02dc}\\ {\mathit{I}}_{\mathrm{d}}^{\mathrm{\ast}}\end{array}\mathrm{\left(}{\mathit{\nu}}_{\mathit{i}}\mathrm{\right)}}\mathit{,}\end{array}$where $\begin{array}{c}{\mathrm{\u02dc}}_{}\\ {\mathit{I}}_{\mathrm{0}}\end{array}$ and $\begin{array}{c}{\mathrm{\u02dc}}_{}\\ {\mathit{I}}_{\mathrm{d}}\end{array}$ are the standard and diversity image Fourier spectra, respectively. Here, ν_{i} = i.B_{min}/λ is one among the eight spatial frequencies sampled by the redundant telescope array, i varies from 0 to 7, and z^{∗} denotes the complex conjugate of z. All these metrics are independent of the object because the object spectrum is simultaneously involved in the numerator and the denominator of each formula.
The whole phase criterion is computed with $\begin{array}{ccc}{\mathit{\chi}}_{\mathit{j}}^{\mathrm{ref}}\mathrm{=}\sum _{{\mathit{\nu}}_{\mathit{i}}}{\mathit{M}}_{\mathit{j}}\mathrm{(}{\mathit{\nu}}_{\mathit{i}}\mathrm{)}\mathit{,}& & \end{array}$where j is the label of the metric. The value of the phase criterion χ^{ref} has a nonlinear relationship to piston errors in the instrument. The genetic algorithm is then used to solve this noninvertible problem.
Fig. 5 Principle of the upconversion by sum frequency generation. The beams coming from the pump source and the astronomical target are injected into a PPLNwaveguide. The infrared signal is converted into a visible signal that can be detected using a silicon photoncounting detector. 
Fig. 6 Simulations of the dynamic range evolution DR_{s}, computed from the shortexposure images as a function of the number of cycles in the photoncounting regime for various pairs in the metricphase diversity range. The number of photons per frame is close to 1000. The cophasing algorithm works with the data from the channel operated in the photoncounting system. DR_{s} evolution is measured on high flux level data. In this framework, we assume that the THT is ideal (i.e. no more instrumental defects) and that only the photon noise corrupts the shortexposure image. The metrics M_{2} and M_{3} are more efficient than the M1 and M4 ones and require a large PDR [ − π;π ]. 
This technique has been used to design and optimize a phased antenna array (AresPena et al. 1999; Marcano & Durán 2000). The genetic algorithm principle stems from the Darwin evolution theory, where a population of individuals is confronted with its natural environment. The mostwell adapted individuals with respect to their environment have greater chances of reproducing themselves and so transmitting their genotype. New individuals are obtained by crossing the genotype of two parents (i.e. selected individuals of the previous generation). The genotype of this new individual can finally diverge from its parents by a mutation effect: each gene can acquire a modified value that is slightly different from its parents. In our case, the optical path lengths linked to the interferometer arms are the only free parameters defining the genotype of an individual. In our instrument, a cophasing cycle follows the sequence:

image ${\mathit{I}}_{\mathrm{0}}^{\mathrm{ref}}$ acquisition,

random generation of piston phase diversity,

acquisition of diversity image ${\mathit{I}}_{\mathrm{d}}^{\mathrm{ref}}$,

computation of the reference phase criterion χ^{ref},

sending χ^{ref} and piston diversity values into the genetic algorithm,

iterative evaluation of the piston errors with the genetic algorithm,

adjustment of the optical path length in each interferometer arm.
Finally, a new cycle of cophasing begins again.
3.2. Selection of the most efficient metric
In a previous study using a high flux level (Bouyeron et al. 2012), we numerically tested the cophasing efficiency of each metric for different phase diversity ranges ([ − π/ 10; + π/ 10 ], [ − π/ 4; + π/ 4 ], [ − π/ 2; + π/ 2 ], [−π; + π ]). The metric’s efficiency was evaluated through the dynamic range evolution (DR_{s}) of the shortexposure image as a function of the number of cophasing cycles. We observed that M_{1} and M_{4} were more efficient when applied with a narrow phase diversity range ([−π/ 10; + π/ 10 ]), whereas M_{2} and M_{3} required a wide one ([−π; + π ]). Thus, to operate with high flux levels, each metric gives a similar performance when selecting their more suitable phase diversity range. Conversely, in the low flux regime, the photon noise affects the selection of the metric. This noise brings about additional random fluctuations of the intensity when shortexposure images are acquired. Figure 6 shows simulation results and illustrates the cophasing behaviour for each metric at the low flux level. We demonstrate that when photon noise is significant, a wide phase diversity range gives the best results regardless of the metrics. Since M_{2} and M_{3} are intrinsically more efficient with wide phase diversity ranges, these two metrics lead to better performance in the photoncounting regime.
4. Data processing and experimental results
4.1. Experimental cophasing tests of the temporal hypertelescope operating in the photoncounting regime
The purpose of this experimental study is to validate our cophasing method dedicated to stabilize THT optical path lengths in the photoncounting regime. As a first step, the instrument target is a pointlike source. According to the simulation results, we use the M_{2} [ − π;π ] metric for performing experimental tests. The cophasing algorithm is fed by the data acquired on the photoncounting regime channel. To assess the efficiency of the cophasing process, a shortexposure image is simultaneously acquired on the high flux level channel for each cophasing cycle. The dynamic range evolution is extracted from these high flux data. Figure 7 shows the dynamic range evolution for two photonflux levels (200 and 500 photons/frame). The averages of dynamic range obtained taking all shortexposure images into account are 1000 and 1700, respectively.
Fig. 7 Experimental dynamic range (DR_{s}) evolution versus cophasing cycle number obtained from the shortexposure images acquired with the THT testbench operating in the photoncounting regime. The observed object is a pointlike source. Two photon flux are investigated (200 and 500 photons/frame). The cophasing algorithm works with the photoncounting regime data. The DR_{s} is measured with the high flux level data. The mean values of DR_{s} are equal to 1000 for 200 photons/frame and 1700 for 500. 
Using the relationship between the dynamic range and the phasing error numerically computed through a Monte Carlo statistical approach (see Fig. 8), it is possible to plot the evolution of the residual error of the optical path difference (standard deviation of the OPD between the interferometric arms) in the instrument (see Fig. 9). The mean values of the OPD error are close to 22.0 ± 4.0 nm for 200 photons/frame and 15.7 ± 1.0 nm for 500. These results demonstrate the feasibility of stabilizing the instrument’s optical path lengths over a significant duration (here 25 min) with only a few hundred photons/frame detected at the output of the THT. Obviously, the cophasing quality is lower than this one, which was previously obtained in high flux level (i.e. 4 nm, see Bouyeron et al. 2012), but as we see below, it is sufficient for acquiring highcontrast longexposure images.
Fig. 8 Evolution of the average of the dynamic range value of shortexposure images as a function of the standard deviation of the phase piston (or OPD) between interferometer arms. This was numerically obtained through a Monte Carlo statistical approach. Each point is evaluated over 10 000 computed PSF. This curve is computed using the actual testbench characteristics (apodization coefficients, array configuration, working wavelength, etc.). The only instrumental defect taken into account in this simulation is the piston phase error. For example, the DR_{s} mean value of 1000 corresponds to a 22 nm OPD standard deviation. 
Fig. 9 Residual errors of the optical path difference experimentally observed during the acquisition of the THT pointspread function (here 25 min). These curves were obtained by using the relationship between the dynamic range and the RMS value of the OPD error (see Fig. 8). The mean values of the OPD standard deviation are close to 22 nm for 200 photons/frame and 15.7 nm for 500. 
4.2. Longexposure image acquisition in the photoncounting regime
4.2.1. The different steps of the data reduction
The raw data acquired in the photoncounting regime cannot be used directly to get sharp images of the observed object. Data reduction is mandatory for obtaining highcontrast images. Figure 10 presents the different steps in this data processing. First, the photon frames (Fig. 10a) are converted from quantized data to analogue data by using a binning process. Each frame is converted in an N_{p} pixel image by adding together the photonevents detected over each sample (Fig. 10b). The object position in the image is a free parameter and can shift during the acquisition of successive frames. Therefore, in a second step, each image is recentred in the middle of the frame (Figs. 10c and 10d).
In a third step, the images are stacked to produce the longexposure analogue image. Comparing the two longexposure images simultaneously obtained with low and high flux levels, we observe a background shift between these two images (Fig. 10e). This gap is due to the photoncounting detector dark counts. This contribution is randomly superimposed on the photon frames and results in a homogeneous intensity background over longexposure images. Finally, once we know the photoncounting detector’s dark count rate, this bias can be removed and the two longexposure images (resulting from the photoncounting regime and high flux level data) match perfectly (Fig. 10f).
Fig. 10 Data reduction process: a) photoncounting regime frame; b) binned data; c) image before recentring and d) image after recentring; e) longexposure images (photoncounting regime data and highflux data) before dark count debiasing; f) longexposure images after dark count debiasing. 
One can observe a difference between the theoretical instrument PSF and experimental longexposure image (see Fig. 11 left). This difference is due to the limited accuracy of the cophasing process. Some photons are consequently incoherently superimposed on the coherent image (both on the photoncounting regime and high flux level channels). This phenomenon results in homogeneous noise on the whole image and can be removed by subtracting the mean value of this background (see Fig. 11 right). The origin of this bias is mainly due to the photon noise limiting the cophasing accuracy and to the limited resolution of the piezoelectric optical path modulator. The intensity bias to be subtracted is a function of the photonflux level and is a reproducible process that can be evaluated by acquiring a calibration PSF.
Fig. 11 Substraction of the incoherent light. Due to the limited accuracy of the cophasing process, a homogeneous background appears on the whole of the longexposure image. The bias amplitude on a longexposure image (left) is a function of photonflux level and of number of shortexposure images. Its subtraction leads to an image close to the ideal PSF (right). 
Assuming that this background bias is mainly due to the photon count statistics, the noise behaviour on the dark field area can be analysed as follows. During a shortexposure image collecting N_{ph} photons per frame (see Fig. 12), the mean value of the background N_{sb} for each snapshot image can be approximately inferred from the snapshot dynamic range DR_{s}, $\begin{array}{ccc}{\mathit{N}}_{\mathrm{sb}}\mathrm{\approx}\frac{{\mathit{N}}_{\mathrm{ph}}}{\mathrm{D}{\mathrm{R}}_{\mathrm{s}}\mathit{.}{\mathit{N}}_{\mathrm{pp}}}\mathit{,}& & \end{array}$where N_{pp} corresponds to the number of pixels of the PSF peak. The full width at half maximum of this peak is about onesixth of the clean field of view (CLF). Over an N_{f}frame integration, we stack N_{ib} = N_{f}.N_{sb} photon events on each pixel of the dark field. This number approximately corresponds to the background offset 1/DR of the normalized longexposure image and can be removed. Thus, the residual RMS fluctuation observed in each pixel of the dark field of the normalized longexposure image is equal to $\begin{array}{ccc}\mathrm{RM}{\mathrm{S}}_{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{DR}\mathit{.}\sqrt{{\mathit{N}}_{\mathrm{ib}}}}\mathrm{\xb7}& & \end{array}$
Fig. 12 Definition of the parameters required to estimate the intensity fluctuations in the dark field of the longexposure image. 
Fig. 13 Experimental images of a single star and a binary star obtained with the THT test bench in the photoncounting regime. Two photon flux are investigated (top: 500 and bottom: 200 photons/frame). The number of pixels is equal to 50 across the clean field of view. These images are obtained by stacking 30 000 and 20 000 shortexposure images for 200 and 500 ph/fr respectively. The intensity ratio between the two components of the unbalanced binary star is set to 1:1000 and their angular separation is 25 μrad. The dynamic range (DR) of each longexposure PSF measured before the background bias subtraction are close to 450 and 730 for 200 and 500 ph/fr, respectively. 
4.2.2. Image acquisition of an unbalanced binary star system
In this section, we report some experimental results obtained with the THT breadboard in the photoncounting regime. The cophasing algorithm uses the M_{2} [−π; + π ] metric. Two photonflux levels have been investigated: 200 and 500 photons/frame. For each one, two longexposure images have been acquired in the photoncounting regime: the instrument PSF, using a pointlike source, and the image of an unbalanced binary system. In this second case, the companion was 1000 (±10%) times weaker than the star. Figure 13 shows the comparison between a calibration PSF and the image of the binary star for the two photonflux levels. These longexposure images are obtained by stacking 30 000 shortexposure images for 200 ph/fr and 20 000 for 500 ph/fr. In both cases, the angular position and amplitude ratio between the main star and the companion are well measured (25 μrad and factor 1000). The standard deviations of the intensity fluctuations measured on the PSF dark field are equal to 1.1 × 10^{4} 200 ph/fr and 5.4 × 10^{5} for 500 ph/fr. If we only consider the effect of the photon noise, the calculated RMS fluctuations are equal to 8.3 × 10^{5} for 200 ph/fr and 5.1 × 10^{5} for 500 ph/fr. These values are close to the measured intensity fluctuations and prove that the cophasing process is mainly limited by the photon noise. The corresponding signaltonoise ratio are 9000 and 18 000. The faint companion is at least ten times brighter than the noise observed experimentally. The signaltonoise ratio value observed for 500 photons/frame reaches the limitation previously recorded with our cophasing process using high flux data (Bouyeron et al. 2012). This limitation is intrinsic to our test bench and is mainly due to the misalignment of the neutral axis of maintaining polarization fibre at each connection.
These experimental results demonstrate the imaging ability of the THT to properly operate in the photoncounting regime and, when using an appropriate data reduction process, the ability of this instrument to perform as well as in the highflux regime.
5. Conclusion and discussion
Our previous study experimentally demonstrated the effectiveness of our system to cophase an aperture synthesis instrument dedicated to direct imaging with a bright source. In this paper, we extend this performance even with the very weak photonflux level in the range of the astrophysical sources. We optimized the algorithm parameters through numerical investigations (metric/PDR). By applying these parameters to our experimental setup, a stabilization of the OPDs in the range of 20 nm using only 200 photons per frame has been demonstrated experimentally. Using an averaging process over about 20 000 acquisitions and a data reduction method usable in the photoncounting regime, we produced highcontrast images of a (1:1000) unbalanced binary object. This measurement has been achieved with a signaltonoise ratio equal to 10. This new experimental step demonstrates that the concept of hypertelescope is fully relevant for astrophysical applications.
For illustrative purposes, we extrapolated the results presented here to a largescale and spacebased instrument with the same architecture as our breadboard (linear array of eight telescopes in a redundant configuration). This case is close to our current setup with very low wavefront distortion before the telescope array and no variation in the baselines of the array as a function of time. A groundbased configuration would need to take the additional perturbations of the atmosphere and the geometric evolution of the baselines over the observation into account. We did not focus on the angular resolution specification that can be managed with the telescopearray configuration and the scale of the baselines. Therefore, the assumptions used in our extrapolation were as follows.

The acquisition time of a single frame (shortexposure image) is set to 0.1 s.

The operating wavelength is 0.9 μm to directly benefit from both silicon photoncounting detectors and optical fibres. The reference spectral illuminance in I band is 8.3 × 10^{13} W cm^{2}μm^{1}. The mean energy per photon in this band is 2.2 × 10^{19}J.

Global photometric losses are equal to 22 dB (coupling: 9 dB; polarization: 3 dB, apodization: 4 dB, insertion: 5 dB, detector quantum efficiency: 1 dB).

With 200 photons per frame, the instrument is able to provide longexposure images with a 1:10 000 contrast.

The optical bandwidth is chosen equal to 100 nm. This assumption is made to allow the computation of the instrument theoretical performance. However, it has not been tested yet on our test bench. This will be one of the next steps in our investigations.
This way, we can estimate the limiting magnitude of the instrument when observing an unbalanced binary star. This computation is done for various mirror diameters (20 to 50 cm) to be in agreement with reasonable tradeoff for a space mission. Results are shown in Table 1. Here, M_{s} is the limiting magnitude of the main star, and M_{c} is the magnitude of its faint companion (Δm = 10). These results show that, even for relatively small mirror diameters, such an instrument can obtain highly resolved and highcontrast images of stellar systems (Ollivier et al. 2009).
Limiting magnitude for an unbalanced binary star observation.
Acknowledgments
This works is supported by Thales Alenia Space and the Centre National d’Études Spatiales (CNES). We are grateful to A. Dexet for his suggestions regarding all the specific mechanical components.
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All Tables
All Figures
Fig. 1 THT testbench theoretical PSF on logarithmic (left) and linear (right) scale. In both cases, the intensity is normalized. The telescopearray configuration is optimized for highcontrast imaging. DR is the dynamic range of the PSF, CLF is the clean field of view and DF is the dark field. 

In the text 
Fig. 2 Theoretical (white disks) and experimental (black disks) telescope array configuration. Owing to the experimental constraints, the configuration of the telescope array is not linear. However, the projection of the telescope baseline along the vertical axis (direction of the 1D object) is fully identical to the theoretical redundant configuration. These two configurations are equivalent to a 1D vertical array when observing a 1D vertical object. 

In the text 
Fig. 3 Temporal hypertelescope testbench scheme. This setup is divided into four main parts: (a) the unbalanced binarystar simulator, (b) the telescope array, (c) the eightarm interferometer, and (d) the signal acquisition chain. 

In the text 
Fig. 4 Signalacquisition chain. Two signals are simultaneously acquired. The first one (top curve) is a shortexposure image obtained in the photoncounting regime (PCR) by using a hybrid detector, the second one is the image simultaneously detected with a high flux level 

In the text 
Fig. 5 Principle of the upconversion by sum frequency generation. The beams coming from the pump source and the astronomical target are injected into a PPLNwaveguide. The infrared signal is converted into a visible signal that can be detected using a silicon photoncounting detector. 

In the text 
Fig. 6 Simulations of the dynamic range evolution DR_{s}, computed from the shortexposure images as a function of the number of cycles in the photoncounting regime for various pairs in the metricphase diversity range. The number of photons per frame is close to 1000. The cophasing algorithm works with the data from the channel operated in the photoncounting system. DR_{s} evolution is measured on high flux level data. In this framework, we assume that the THT is ideal (i.e. no more instrumental defects) and that only the photon noise corrupts the shortexposure image. The metrics M_{2} and M_{3} are more efficient than the M1 and M4 ones and require a large PDR [ − π;π ]. 

In the text 
Fig. 7 Experimental dynamic range (DR_{s}) evolution versus cophasing cycle number obtained from the shortexposure images acquired with the THT testbench operating in the photoncounting regime. The observed object is a pointlike source. Two photon flux are investigated (200 and 500 photons/frame). The cophasing algorithm works with the photoncounting regime data. The DR_{s} is measured with the high flux level data. The mean values of DR_{s} are equal to 1000 for 200 photons/frame and 1700 for 500. 

In the text 
Fig. 8 Evolution of the average of the dynamic range value of shortexposure images as a function of the standard deviation of the phase piston (or OPD) between interferometer arms. This was numerically obtained through a Monte Carlo statistical approach. Each point is evaluated over 10 000 computed PSF. This curve is computed using the actual testbench characteristics (apodization coefficients, array configuration, working wavelength, etc.). The only instrumental defect taken into account in this simulation is the piston phase error. For example, the DR_{s} mean value of 1000 corresponds to a 22 nm OPD standard deviation. 

In the text 
Fig. 9 Residual errors of the optical path difference experimentally observed during the acquisition of the THT pointspread function (here 25 min). These curves were obtained by using the relationship between the dynamic range and the RMS value of the OPD error (see Fig. 8). The mean values of the OPD standard deviation are close to 22 nm for 200 photons/frame and 15.7 nm for 500. 

In the text 
Fig. 10 Data reduction process: a) photoncounting regime frame; b) binned data; c) image before recentring and d) image after recentring; e) longexposure images (photoncounting regime data and highflux data) before dark count debiasing; f) longexposure images after dark count debiasing. 

In the text 
Fig. 11 Substraction of the incoherent light. Due to the limited accuracy of the cophasing process, a homogeneous background appears on the whole of the longexposure image. The bias amplitude on a longexposure image (left) is a function of photonflux level and of number of shortexposure images. Its subtraction leads to an image close to the ideal PSF (right). 

In the text 
Fig. 12 Definition of the parameters required to estimate the intensity fluctuations in the dark field of the longexposure image. 

In the text 
Fig. 13 Experimental images of a single star and a binary star obtained with the THT test bench in the photoncounting regime. Two photon flux are investigated (top: 500 and bottom: 200 photons/frame). The number of pixels is equal to 50 across the clean field of view. These images are obtained by stacking 30 000 and 20 000 shortexposure images for 200 and 500 ph/fr respectively. The intensity ratio between the two components of the unbalanced binary star is set to 1:1000 and their angular separation is 25 μrad. The dynamic range (DR) of each longexposure PSF measured before the background bias subtraction are close to 450 and 730 for 200 and 500 ph/fr, respectively. 

In the text 
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