Context. Extending the potential of multi-aperture telescopes toward the higher magnitudes necessary for extragalactic science needs an off-axis reference source to cophase the interferometric array. Aims. The off-axis reference star and the science source must be within the isopistonic angle, and evaluating this parameter is crucial for estimating the potential of long-baseline interferometers for faint sources, differential astrometry, and phase-reference imaging. Methods. We derived an analytical method of deducing the isopistonic angle from the standard atmospheric optical parameters, and used a full description of the atmospheric turbulence to validate numerical integrations. Results. We used the analytical expression for the isopistonic angle for predictions concerning the VLT, Keck, and LBT interferometers, as well as for an array of small apertures in the Antarctica site Dome C, indicating that is might be a unique site on earth for interferometric observations of faint sources.

The limiting magnitude is a key issue for optical interferometry. Pairwise fringe trackers based on the integrated optics concepts used for example in GRAVITY seem limited to about K=10.5 with the 8m Unit Telescopes of the VLTI, and there is a general "common sense" statement that the efficiency of fringe tracking, and hence the sensitivity of optical interferometry, must decrease as the number of apertures increases, at least in the near infrared where we are still limited by detector readout noise. Here we present a Hierarchical Fringe Tracking (HFT) concept with sensitivity at least equal to this of a two apertures fringe trackers. HFT is based of the combination of the apertures in pairs, then in pairs of pairs then in pairs of groups… The key HFT module is a device that behaves like a spatial filter for two telescopes (2TSF) and transmits all or most of the flux of a cophased pair in a single mode beam. We give an example of such an achromatic 2TSF, based on very broadband dispersed fringes analyzed by grids, and show that it allows piston measures from very broadband fringes with only 3 to 5 pixels per fringe tracker. We show the results of numerical simulation indicating that our device is a good achromatic spatial filter and allowing a first evaluation of its coupling efficiency, which is similar to this of a single mode fiber on a single aperture. Our very preliminary results indicate that HFT has a good chance to be a serious candidate for the most sensitive fringe tracking with the VLTI and also interferometers with much larger number of apertures. On the VLTI the first rough estimate of the magnitude gain with regard to the GRAVITY internal FT is between 2.5 and 3.5 magnitudes in K, with a decisive impact on the VLTI science program for AGNs, Young stars and planet forming disks.

The Planet Formation Imager (PFI) is a project for a very large optical interferometer intended to obtain images of the planet formation process at scales as small as the Hill sphere of giant exoplanets. Its main science instruments will work in the thermal infrared but it will be cophased in the near infrared, where it requires also some capacity for scientific imaging. PFI imaging and resolution specifications imply an array of 12 to 20 apertures and baselines up to a few kilometers cophased at near infrared coherent magnitudes as large as 10. This paper discusses various cophasing architectures and the corresponding minimum diameter of individual apertures, which is the dominant element of PFI cost estimates. From a global analysis of the possible combinations of pairwise fringe sensors, we show that conventional approaches used in current interferometers imply the use of prohibitively large telescopes and we indicate the innovative strategies that would allow building PFI with affordable apertures smaller than 2 m in diameter. The approach with the best potential appears to be Hierarchical Fringe Tracking based on "two beams spatial filters" that cophase pairs of neighboring telescopes with all the efficiency of a two telescopes fringe tracker and transmit most of the flux as if it was produced by an unique single mode aperture to cophase pairs of pairs and then pairs of groups of apertures. We consider also the adaptation to PFI of more conventional approaches such as a combination of GRAVITY like fringe trackers or single or multiple chains of 2T fringe trackers.

The full scientific potential of the VLTI with its second generation instruments MATISSE and GRAVITY require fringe tracking up to magnitudes K>14 with the UTs and K>10 with the ATs. The GRAVITY fringe tracker (FT) will be limited to K~10.5 with UTs and K~7.5 with ATs, for fundamental conceptual reasons: the flux of each telescope is distributed among 3 cophasing pairs and then among 5 spectral channels for coherencing. To overcome this limit we propose a new FT concept, called Hierarchical Fringe Tracker (HFT) that cophase pairs of apertures with all the flux from two apertures and only one spectral channel. When the pair is cophased, most of the flux is transmitted as if it was produced by an unique single mode beam and then used to cophase pairs of pairs and then pairs of groups. At the deeper level, the flux is used in an optimized dispersed fringe device for coherencing. On the VLTI such a system allows a gain of about 3 magnitudes over the GRAVITY FT. On interferometers with more apertures such as CHARA (6 telescopes) or a future Planet Formation Imager (12 to 20 telescopes), the HFT would be even more decisive, as its performance does not decrease with the number of apertures. It would allow building a PFI reaching a coherent magnitude H~10 with 16 apertures with diameters smaller than 2 m. We present the HFT concept, the first steps of its feasibility demonstration from computer simulations and the optical design of a 4 telescopes HFT prototype.

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