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# End-to-end simulation of hierarchical fringe tracking

Fatmé Allouche, Hadjara Massinissa, Amokrane Berdja, Abdelkarim Boskri, James Leftley, Stéphane Lagarde, Romain G Petrov

► To cite this version:

Fatmé Allouche, Hadjara Massinissa, Amokrane Berdja, Abdelkarim Boskri, James Leftley, et al. "End-to-end simulation of hierarchical fringe tracking. Optical and Infrared Interferometry and Imaging IX, SPIE, Jun 2024, Yokohama, France. pp.90, 10.1117/12.3020393. hal-05123152

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https://hal.science/hal-05123152v1

Submitted on 20 Jun 2025

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# End-to-end simulation of Hierarchical Fringe Tracking

Fatmé Allouche<sup>a,f*</sup>, Massinissa Hadjara<sup>b,c</sup>, Amokrane Berdja<sup>d</sup>, Abdelkarim Boskri<sup>e</sup>, James Leftley<sup>a</sup>,
Stéphane Lagarde<sup>a</sup> and Romain G. Petrov<sup>a†</sup>

<sup>a</sup>Université Côte d'Azur, Observatoire de la Côte d'Azur, CNRS, Laboratoire Lagrange, Nice, France;
<sup>b</sup>Nanjing Institute of Astronomical Optics & Technology, National Astronomical Observatories, Chinese Academy of Sciences, Nanjing 210042, China; 
<sup>c</sup>Chinese High Angular Resolution Southern Astronomical Laboratory (CHARSAL) / Astro-Photonics Laboratory/Space and Planetary, Exploration Laboratory (SPEL), Dept. of Electrical Engineering, FCFM, Universidad de Chile, Av. Tupper 2007, Santiago, Chile; 
<sup>d</sup>Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile; 
<sup>e</sup>LPHEA Laboratory, Oukaimeden Observatory, Cadi Ayyad University/FSSM, BP 239, Marrakesh, Morocco; 
<sup>f</sup>European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile.

## ABSTRACT

We discuss a new generation fringe tracker (FT) that implements a Hierarchical Fringe Tracker (HFT) architecture with a very broad band near infrared spectral coverage from 1.1 to 2.2 µm in the J, H and K bands. The goal is to approach the absolute maximum fringe tracking sensitivity in optical long baseline interferometry, first on the VLTI, and to show that an HFT has performances independent from the number of apertures, a key characteristic for larger interferometers from CHARA to the VLTI with more UTs or combining all UTs and ATs to future very large interferometers. This paper describes the development in progress of an end-to-end simulator of such a system based on our first laboratory tests of prototype HFT. This simulator already allowed us to define a new optimization for the integrated optics HFT chips, to discuss a set of operating parameters for our new generation fringe tracker and to confirm that it is applicable to an indefinite number of apertures and should approach or even exceed a limiting sensitivity on the VLTI of K~16, which is a gain of at least 3 magnitudes over the expected performance of the current GRAVITY FT in the context of the ongoing GRAVITY+ VLTI upgrade.

**Keywords:** Astrophysics, Instrumentation, Optical and Infrared Interferometry, Fringe Tracking.

## 1. INTRODUCTION

A major axis of development of new astrophysical applications of optical interferometry is the observation of AGNs with differential interferometry. The Broad Line Region (BLR) of AGNs is dynamically tied to the central super massive black hole (SMBH). Fully resolving a significant number of BLRs would require an interferometer with kilometric baselines able to reach a K band magnitude of the order of at least 14. However, it has been shown [1] [2] that the application of differential interferometry to unresolved BLRs can constrain their angular size and a set of geometrical parameters allowing a model dependent but quite reliable estimate of the mass of the SMBH. This has been achieved with success by GRAVITY on the VLTI [3]. Extending this achievement to Quasars up to redshift larger than z~2, to study the coevolution between SMBH and host galaxies at “cosmic noon” is one of the main motivations of the GRAVITY+ upgrade of the VLTI sensitivity [5]. Combining the angular size obtained from spectro-astrometry (SA) using phase differential interferometry (PDI) and the linear size estimated by Reverberation Mapping (RM) yields a direct distance estimate in a unprecedented range of cosmic distances with a potentially decisive impact on the Hubble tension problem that is currently one of the major questions in fundamental (astro)physics [4]. However, making SMBH mass and distance

<footnote>* Fatme.Allouche@eso.org.</footnote>

<footnote>† Romain.PETROV@univ-cotedazur.fr.</footnote>
estimates accurate enough to be decisive requires more geometrical information [2]. We have shown that this can be obtained by amplitude differential interferometry (ADI) the VLTI in the J band if the sensitivity of on-axis and off-axis fringe tracking can be increased above K~15 [12][13]. The ongoing GRAVITY+ upgrade of the VLTI [5] will allow Adaptive Optics correction on almost any target on the sky using either a natural guide star (NGS), yielding a Strehl Ratio in K ~0.6, or a combination of laser guide star (LGS) and NGS, yielding a Strehl ratio in K ~0.4. The two other features of GRAVITY+ are a substantial active reduction of vibrations in the UT Coude train [14] and the possibility to fringe track off-axis with a separation between the science source and the guide star of at least 30 arcseconds [15]. However, the most optimistic estimates of the limiting sensitivity of fringe tracking with the GRAVITY Fringe Tracker (GFT) in the GRAVITY+ context give a maximum magnitude of K~13 for the tracking source, which limits both the number of AGNs that allow fringe tracking on-axis and the sky coverage for tracking off-axis. The GFT is a pair wise fringe tracker operating in the K band. Pair wise fringe trackers divide the flux of each of the nT apertures among the nT-1 baselines implying this aperture. This decreases the limiting sensitivity of the FT with the number of baselines and constitutes a first objection to the extension of the VLTI to more UTs, or to a combination between UTs and ATs. It limits the development of large interferometers with more apertures. These considerations triggered our ambition to develop a fringe tracker than can track on targets fainter than K~15 on the VLTI in the GRAVITY+ context and with performances that do not decrease with the number of apertures. Such a device, called the Hierarchical Fringe Tracker (HFT) is described in another paper in these proceedings [7] and in a previous SPIE conference [13]. Here we describe the ongoing development of an end-to-end simulation of an HFT for 4 and 6 to 8 telescopes based on integrated optics chips.

<figure><img src="image_3.png" /><figcaption>Figure 1: Testing a first 4T HFT chip on the optical bench. The chip is described in [13] and sketched in the upper left corner. It featured photometric derivations to assess fully the different components. The optical test bench features a super continuum source, a collimator and a mask producing six coherent beams, a first set of piezo actuators to tune and perturbate the OPD between beams, a second set of piezo actuators with an absolute position capacitive gauge to scan and correct the OPD, a fiber injection optics feeding fibers that a V-groove aligns on the inputs of the HFT chip displayed in the lower left corner. The output of the HFT chip is imaged on an infrared camera through a low dispersion prism yielding the signals in several spectral channels in the H band. We record these output signals as a function of the OPD introduced by the input piezo actuators that allow to scan the fringes. The HFT chip is about 30 mm long, much smaller and hence in principle more transmissive, than the corresponding ABCD GFT like chip.</figcaption></figure>
## 2. HFT CONCEPT AND BASIC EQUATIONS

Unlike pairwise systems like the GFT that divide the flux of each telescope among many baselines or “all-in-one” systems like AMBER and MATISSE that use a large number of pixels (and hence collect a large readout noise) to analyze interferograms containing all spatial frequencies, an HFT uses all the flux from a pair of apertures to cophase them and transmits all or most of the cophased flux to a next hierarchical level that cophases pairs of pairs of telescopes and so on. A HFT first combines two beams and uses all the incoming flux to cophase them and sends the co-phased beam to the next level that co-phases pairs of pairs of apertures and so on. This is done without any opto-mechanical actuation and results from the design of the HFT chip. If the level n+1 receives more flux than the level n when the pairs are cophased at level n, then the performances of the system (for unresolved sources) are set only by the cophasing of the first level pairs, i.e. is given by the cophasing limit of a pair of telescopes, whatever the total number of apertures. This assumes that the propagation of tracking errors is controlled and checking this point is one of the goals of our simulation.

The starting point of our simulator was the test on an optical bench of our first HFT integrated optics prototype chip [13] that is described in figure 1. The results are described in figure 2. They showed that the basic equations of an HFT chip, as described in figure 3, are correct, that the second level indeed receives more flux than the first level and that we have an usable although not perfectly linear phase delay estimator both for the first and second level. The second level allows to estimate the OPD between the mean optical path of beams 1 and 2 and the mean optical path of beams 3 and 4 whatever the cophasing status of pairs 1-2 and 3-4, with of course a loss of signal level when beams i-j are anti cophased.

<figure><img src="image_4.png" /><figcaption>Figure 2: measured signals and phase delay for the first level (left, here produced when we scan the OPD between beams 1 and 2) and the second level (right: showing the signals as a function of the OPD between beams 1+2 and 3+4). The lines 1,2 and 4 show the outputs A, B, and C, the line 3 is for A-B and the line 5 is for the phase delay estimator arg(A-B, C-A-B). Each color represents a spectral channel (respectively 1.517, 1.551, 1.586, 1.620, 1.689 and 1.724 µm). We see that there is a unique estimate for the phase delay between ±λ/2 but that the phase delay estimator is not perfectly linear.</figcaption></figure>

The figure 3 describes the evolution and gives the key equations of the basic component of an HFT that is the two beams spatial filter (TBSF). It is derived from the standard ABCD beam combiner where two of its outputs are merged by an X coupler to produce a pair of constructive and nulling outputs C+ and C-. An additional achromatic phase delay $\Delta\varphi_{cons}$ can be used to maximize the flux in C+ or ease the control of the phase transmitted by that single mode channel. The four outputs A, B, C+ and C- are combined to produce an exact phase delay estimator ($\Delta\varphi = \frac{2\pi P}{\lambda}$ modulo(2$\pi$) yielding the piston measure P used for phase delay tracking. This phase delay estimator is independent from the ratio of coherent flux injected in the TBSF, as indicated by the equations in figure 3.

$$ \Delta\varphi = \left[ \arg[2(B - A); C_+ + C_- - B - A] - \frac{\pi}{4} + \pi \right] \text{modulo}(2\pi) - \pi $$

where the $(...+\pi)\text{modulo}(2\pi)-\pi$ operation is used to recenter $\Delta\varphi$ between $\pm\pi$ and to ensure $P = \frac{\Delta\varphi}{2\pi}$.

This estimator needs that a fraction of the C+ output is directly measured on the detector, as indicated in figure 5. This decreases the flux injected in the level n+1 but maintains the fact that this level receives more flux than the level n if this fraction is small enough. There are estimators of the phase delay based only on A, B and C-, allowing to inject all the cophased flux in the level n+1 but they are sensitive to flux ratio fluctuations when the input beams are not cophased ($F(a_1, a_2, \Delta\varphi, \Delta\varphi_{cons}) \neq 0$ when $\Delta\varphi \neq 0$ and $a_1 \neq a_2$). One of the goals of our simulator is to compare the performances
of these estimators based either on C+ and C. or on C. only and to find the optimum share between the fraction of C+ sent to the next hierarchical level and this measured directly on the detector.

<figure><img src="image_5.png" /><figcaption>Figure 3. Principle and key equations of a Two Beam Spatial Filter (TBSF). Two outputs of a standard ABCD beam combiner are merged by an X coupler to produce a pair of constructive and nulling outputs C+ and C. An additional achromatic phase delay $\Delta\varphi_{cons}$ can be used to maximize the flux in C+ or ease the control of the phase transmitted by that single mode channel. The four outputs A, B, C+ and C- are combined to produce an exact phase delay estimator ($\Delta\varphi = 2\pi P/\lambda$) yielding the piston measure P used for phase delay tracking. This phase delay estimator is independent from the ratio of coherent flux injected in the TBSF.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4. Group Delay estimator from the phase delays in different spectral channels. Maximizing F ($P_0, a$) yields the piston $P = P_0 + an_G$ that best fits the $n_\lambda$ (number of spectral channels) and $n_G$ (number of frames) phase delay measures used for group delay estimation.</figcaption></figure>

The phase delay estimator yields a piston estimate with a $k\lambda$ ambiguity. This is removed by the group delay estimator described in figure 4 that is based on the combination of the phase delay estimators in different spectral bands. We look for the absolute piston that best fits all phase delays. To avoid the effect of unpredictable $2\pi$ jumps when the noisy phase
delay approaches $\pm\pi$, we minimize the geometrical distance in the complex plane between the phase delay $e^{i\Delta\varphi(\lambda)}$ and the piston $e^{i2\pi P/\lambda}$ phasors. As the group delay is estimated on a large $n_G$ number of frames, we estimate a linear piston drift over that period (first order behavior when the phase delay tracking is lost for a short time or after a fringe jump). The optimum number of spectral bands is another expected result from the simulation.

Figure 5 shows how two beams spatial filters can be combined in an HFT chip for 4 or 6 telescopes.

<figure><img src="image_7.png" /><figcaption>Figure 5. Design of an HFT chip for 4 telescopes (right) and one possible 6 telescopes setup (left). To have access to C+ from the level n-1 a fraction of the flux is derived directly to an output to the detector (in the figure that's 1/3). The fraction of derived flux is an optimization parameter of the system. The actual 4T chip is about 3cm long, much shorter (and hence more transmissive) than the equivalent pair-wise ABCD chip [8].</figcaption></figure>

### 3. THE SIMULATOR

<figure><img src="image_8.png" /><figcaption>Figure 6: A simplified servo loop for a 2 beams tracker. The input piston and flux yield the A, B, C+ and C- signal recorded on the detector that are averaged over time and eventually over spectral channels as indicated in the text. These signals are used to retrieve phase and group delays according to the estimators in figures 3 and 4. These phase and group delays are combined to drive the piezo actuator that corrects the OPD variations. The quality of the tracking is defined by the lock factor and by the rms of the tracked piston $P_c(t)$.</figcaption></figure>

We analyze the limiting sensitivity of a new generation fringe tracker in the context of the ongoing AO upgrade on the UTs that should provide for any target a Strehl ratio in K of about 60% when a bright enough natural guide star is available (NGS mode) and of about 40% when we have to use a Laser Guide Star (LGS mode). To maximize the limiting sensitivity on the VLTI in that context we add two more features to the HFT architecture. The first is a very broad band operation with several HFT chips optimized for large spectral band in J, H and K. We have considered two options: (i) three chips covering respectively the J band (1.1-1.4 µm), the H band (1.5-1.8 µm) and the “non thermal” part of the K band (2-2.2 µm).
µm) and (ii) 5 chips covering 1.1-1.25 µm, 1.25-1.4 µm, 1.5-1.65 µm, 1.65-1.8 µm and 2-2.2 µm. The first option reduces the number of spectral channels and hence the readout noise but has a very limited coherence length in J and hence is less efficient for the group delay recovery of fringe losses. Our provisional choice used here is the 5 channels option (ii). Our quite large spectral channels in J and H implies a correction of chromatic dispersion effects both in the image plane (for an optimum injection in the fibers) and in the OPD direction (for a maximum broad band fringe contrast). In [7] we propose a device that can obtain these two effects with a minimum transmission loss (and in particular without using dichroic plates). This design yields conservative transmission estimates of 2% in J, 3% in H and 4% in K based on the measured VLTI transmission [10], on the number of optical elements and on the conservative estimate of our chip transmission (50%). These values have been combined to the different Strehl ratio values estimated for the different spectral bands using the Marechal approximation.

<figure><img src="image_9.png" /><figcaption>Figure 7: the global flow chart of an HFT simulation. Each TBSF generates a piston correction, combining phase delay and group delay that is used to correct one input OP Difference between two beams in a way that does not change the mean OP of the two beams. The signals are set by the input OP and fluxes and the source characteristics (magnitude and complex visibility). The correction are applied with a gain (that can be zero) set by the “state machine” from the measures recorded in a recent past (at least the interval used to compute the group delay). The improvement of this state machine and register are one of the main development axes of the simulator.</figcaption></figure>

Figure 6 displays the sketch of the servo loop for a pair of beams. The input signals are a simulated atmospheric piston, in agreement with the piston structure function yield by the standard seeing parameters $r_0$, coherence time $\tau_0$, wind speed and direction $V$, outer scale of turbulence $L_0$ and interferometric array setup i.e. baselines and telescope diameter [16]. We also simulate an injection flux fluctuation from the preliminary simulations of the GRAVITY+ AO performance [6] where we set the expected mean Strehl ratio and the typical rms fast fluctuations. The input signal are sampled at 2 KHz, a frequency in principle quite higher than any atmospheric or instrument fluctuations. Then, the signals on the detector, which are computed accordingly to the equations in figure 3, are averaged over the Detector Integration Time (DIT). Then we compute a phase delay estimated over a frame time $\tau_f$. If $\tau_f$ contains several DITs we can either average, for each wavelength, the coherent flux estimated for each DIT or the phase delay estimated from each DIT. The first option, called
coherent integration, assumes either small or predictable phase changes over $\tau_f$. We compute a broad band phase delay, from an average of all signals over all spectral channels and a phase delay per spectral band that can be eventually averaged over lambda. The broad band phase delay has a higher SNR and yields more accurate tracking, but it has a very small coherence length and is therefore very sensitive to tracking losses. The best way to use these two types of phase delay estimators is still under investigation. Here we report the results given by an average over lambda of the phase delay and we use the dispersion of phase delay over wavelength with regard to the rms of the broad band phase delay as one indicator of tracking status. The phase delays per spectral channel and over several $\tau_f$ are used to estimate the group delay as indicated in figure 4. This group delay computation is heavy, but it is repeated only every $N_G\tau_f$, which is typically several times 100 ms. The phase delay correction is applied every $\tau_f$ with currently a fixed gain (that is optimized for each seeing condition). Eventually, the correction signal will be produced at the 2 KHz sampling frequency to include the response of the piston correction actuator. Every $N_G\tau_f$ we check the group delay and correct it, by an integer number of mean wavelengths $\lambda$, if it's found outside of a $\pm\lambda/2$ interval. When we detect that we "are not on the right fringe" either from the comparison of the broad band and narrow band phase delays or from the group delay value we "close the shutters of the science instrument" i.e. we exclude the corresponding interval of time from the computation of the residual tracking rms as well as of the lock factor time. This feature is intended to allow very long exposure times in a science channel that is not recorded at the mean wavelength of the fringe tracking and is hence sensitive to fringe jumps [11]. The limiting magnitude and performances of this mode are still being evaluated but we display some preliminary results below.

Figure 7 displays the global flowchart of our simulator, that includes the 2 beams module of figure 6 for each pair of beams. Each TBSF corrects only one OPD difference ($\Delta P_{ij} = P_j - P_i$) in a way that does not change the average piston ($P_j + P_i)/2$ that is by applying $P_j = P_j - \Delta P_{ij}/2$ and $P_i = P_i + \Delta P_{ij}/2$. This avoids the propagation of tracking noise as long as the ij pair is tracked. For the 2nd level tracking that corrects $\Delta P_{ijkl} = (P_l + P_k)/2 - (P_j + P_i)/2$ we apply a correction on all 4 pistons: $P_k = P_k - \frac{\Delta P_{ijkl}}{2}$ and $P_l = P_l - \frac{\Delta P_{ijkl}}{2}$ and $P_i = P_i + \frac{\Delta P_{ijkl}}{2}$ and $P_j = P_j + \frac{\Delta P_{ijkl}}{2}$.

The $C_+$ output of each TBSF transmits a coherent flux that contains a parasite phase term (in addition to the phase corresponding to $(P_j + P_i)/2$). This parasite phase term, that is zero when the TBSF input beams are cophased, can be computed with some noise from the A, B, $C_+$ and $C_-$ measures and removed from the $\Delta P$ correction. Our simulation indicates that this would globally decrease the performance as the noise on the parasite phase estimate is larger than this parasite phase itself when we are close to the sensitivity limit. Our current results are without correction of this parasite phase.

The flow chart in figure 7 can be straightforwardly extended to 8 apertures. It can also be used for 3 apertures if instead of combining at level 2 two pairs 1-2 and 3-4 we combine a pair 1-2 with the signal directly from a telescope 3. We have used such a setup to simulate the 6 telescopes combination given in figure 5.

Currently we use a very simplified state machine. The phase delay correction at the level 1 is systematically applied if $C_+ + C_- - A - B$ exceeds a threshold as $(C_+ = C_- = A = B)$ indicates that one of the beams has no flux. This makes the system partially resilient to flux losses: if we lose beam i in the pair ij the lower levels track only using beam j. The group delay correction is applied if two group delay estimators are coherently out of the $\pm\lambda/2$ range. A group delay correction at level n inhibits all correction at level n+1 involving this pair of beams. Eventually we will develop a more sophisticated state machine using all the information recorded and updated over the last $N_G\tau_f$ time interval to validate the phase delay and group delay correction and adjust the gain. Ultimately, we will implement a Kalman or another optimum filtering together with a realistic simulation of non-atmospheric piston fluctuations, from vibrations in the Coude train from example.

# 4. RESULTS

So far, the results of our simulation are:

* An improved design of our HFT chips represented in figure 5. These new chips have been ordered and will be tested on the optical bench later in 2024 or early 2025.

* The validation of our phase delay estimators.
* The validation and the optimization of our group delay estimator given in figure 4. The weighting with a sinc function sensitive to the coherence length triggered a major improvement in fringe jumps correction.

* The choice of $\Delta\varphi_{cons} = 0$. The value that optimizes the $C_+$ intensity at cophasing is $\Delta\varphi_{cons} = \pi/4$ and the value that minimizes the parasite noise propagation is $\Delta\varphi_{cons} = \pi/8$ but $\Delta\varphi_{cons} = 0$ is very close to maximize the stability of the phase delay estimator and has the advantage to simplify the manufacturing of the chip.

* The choice of the $C_+C_-$ phase delay estimator described here instead of estimators using only $C_-$ (and hence maximizing the flux transmitted to the next level) that produce more fringe jumps or require a preliminary calibration phase.

* An estimate of the noise propagation between levels: when the levels n are tracking, the residual tracking rms on the downstream levels is systematically lower.

<figure><img src="image_10.png" /><figcaption>Figure 8: 4T tracking simulation, with VLTI UT, in median (best 50%) seeing conditions. At K=15.5, we have good tracking (with t=20 ms, that might be unrealistic with regard to instrument piston perturbations) or acceptable tracking (with t=10 ms). At K=16, the tracking is either poor, with a lot of fringe losses that the group delay tracking recovers, or fails, with a tracking lost and out of range of group delay correction. In good (best 20%) seeing conditions, the same behavior is observed at 0.5 to 1 magnitudes fainter as shown in [7].</figcaption></figure>

* The resilience with regard to flux losses, even near the limiting sensitivity: when a telescope is lost, only the baselines involving this telescope are lost. For example, if we lose beam 1, the pair 1-2 is not tracked, but the 2^nd^ level tracks on beam 2 and the baselines 2-3 and 2-4 (and of course 3-4) are tracked.

* Fringe jumps are propagated to downstream levels: if the pair i-j tracks to a wrong value, then the mean piston $(P_j + P_i)/2$ is changed and this impacts all baselines implying beams i and j.

* A preliminary investigation of the bootstrapping capability of our very broadband HFT. Independently from the HFT architecture, we have a good wavelength bootstrapping because of our very large broad band, as many targets would be resolved only in some spectral bands for a given baseline. The HFT architecture itself has some
built-in bootstrapping features. The first levels would be systematically implemented in the shortest baselines.
The next levels, on longer baselines would see lower visibility but more flux, the second partially compensating
the first. Moreover, for the pair 12-34 for example, the average equivalent visibility seen by this TBSF is the
average (V<sub>13</sub> + V<sub>14</sub> + V<sub>23</sub> + V<sub>34</sub>)/4. Even if one of these visibilities is near zero, the average will not necessarily
be ~zero and the pair and all the involved baselines will be tracked. The optimization of this bootstrapping
depends on the interferometer baseline distribution. We still have to fully investigate that aspect but we have
shown that an object resolved of the largest baseline (UT1-UT4 for example) can be tracked, with of course a
loss of sensitivity with regard to an unresolved target.

<figure><img src="image_11.png" /><figcaption>Figure 9. 6T tracking simulation with VLTI UT in good (best 20%) seeing conditions. Note the forced fringe jump (beam 3 step 40000) and the fast recovery on all baselines close, in spite of the fact that we are close to the limiting sensitivity. We see that when level 1 is tracking, levels 2 and 3 are also (almost always) tracking with a tracking rms smaller that level 1 (because they receive more flux). However, a tracking error in a level propagates to the downstream levels. The recovery time increases with the tracking level (the recovery at level n+1 starts when it has been achieved on level n). Therefor the sensitivity limit is indeed set by the good tracking limit at level 1.</figcaption></figure>
* A tool to compare the performances as a function of the design parameters discussed here as well as a function of operation parameters such as the sampling frequency, the frame time used for the phase delay estimator and the number of phase delay frames used for group delay estimation that have to be tuned as a function of the injected piston characteristics. More details on this optimization as a function of seeing can be found in [7].

* A test of the efficiency of our weighted broad band group delay estimator. Actually even at the sensitivity limit the group delay correction is usually sufficient to recover the fringes without need to activate a fringe search loop.

* A general confirmation of the feasibility of an HFT architecture, of its level of performances with a sensitivity limit between K=16 and K=16.5 in good seeing conditions [7] and between K=15 and K=15.5 in poor seeing condition.

* A general confirmation that the performances, set by “good tracking on level 1 with all the flux available from two telescopes” are independent from the number of apertures.

<figure><img src="image_12.png" /><figcaption>Figure 10: Test of the impact of the C+ fraction diverted for direct measurements from 15% (top left), 25% (top right) and 33% (bottom). Decreasing that fraction increases the fringe jump or loss probability at level n-1 (with a threshold around 20%) and the tracking rms at level n-1 while it slightly improves the rms at level n. Our provisional conclusion is that 25% is close to an optimum but any value between 25% and 33% seem to give very similar performances.</figcaption></figure>

## 5. CONCLUSION

We are developing and end-to-end simulator of a broad band J-H-K fringe tracker based on a Hierarchical Fringe Tracker architecture. We use a realistic simulation of the atmospheric piston, a pragmatic simulation of injected flux (instantaneous Strehl ratio fluctuations) that needs to be updated the results of the commissioning of the GRAVITY+ AO (2nd semester 2024). We still have to implement a simulation of telescope vibrations.

Our simulator allowed us to improve our chip design and to finalize our phase and group delay estimators. The performance estimates have been based on the flux expected from the VLTI/UTs as a function of magnitude using the known VLTI transmission, a conservative estimate of the HFT transmission and the GRAVITY+ AO specifications.

We conclude that the HFT has indeed performances that are independent from the number of apertures if the first level are tracking properly with a sensitivity set by a 2T fringe tracker using all the flux delivered by two apertures. Our system
has a built up “redundancy” as the tracking for a given baseline combines all information relevant to that baseline and has
a bootstrapping capability that has to be optimized by matching the HFT design to the interferometric array configuration.

We confirm the potential gain in sensitivity that should allow fringe tracking on the VLTI on targets with coherent
magnitude K between 15 and 16.5 according to seeing conditions. However this still needs confirmation when we
implement realistic vibration simulations as well as an improved state machine with Kalman filters.

ACKNOWLEDGEMENTS

This work has been supported by the French Agence Nationale de la Recherche (ANR) through the grant "AGN MELBa" ANR-21-CE31-0011 and by the Institut des Sciences de l'Univers (INSU-CNRS). The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation program under Grant Agreement 101004719 (ORP).

REFERENCES

[1] R. G. Petrov et al, “AMBER: a near infrared focal instrument for the VLTI”, Comptes rendus de l'Académie des sciences - Série IV - Physique - Astrophysique, Volume 2, Numéro 1, pp. 67-97, arXiv:astro-ph/0507398

[2] S. Rakshit, et al., “Differential interferometry of QSO broad-line regions - I. Improving the reverberation mapping model fits and black hole mass estimates”, MNRAS 447, 2015

[3] Gravity collaboration et al, “Spatially resolved rotation of the broad-line region of a quasar at sub-parsec scale”, Nature 563, 2018

[4] Jian-Min Wang A parallax distance to 3C 273 through spectroastrometry and reverberation mapping”, Nature Astronomy, 4, 2020

[5] GRAVITY+ Collaboration, R. Abuter et al, “The GRAVITY+ Project: towards all-sky, faint-science, high-contrast near-infrared interferometry at the VLTI”, The Messenger, 189, 2022, The GRAVITY+ Project: towards all-sky, faint-science, high-contrast near-infrared interferometry at the VLTI

[6] J. Woillez et al, “The VLTI in the GRAVITY+ era”, Proc. SPIE 13095, Optical and Infrared Interferometry and Imaging IX (18 June 2024).

[7] R. G. Petrov, F. Allouche, M. Hadjara, A. Berdja, A. Boskri, S. Lagarde “Hierarchical Fringe Tracker”, Proc. SPIE 13095, Optical and Infrared Interferometry and Imaging IX (18 June 2024).

[8] S. Lacour, R. Dembet, R. Abuter et al, “The GRAVITY fringe tracker”, A&A, 624, A99, 2019.

[9] A. Boskri, R. G. Petrov, T. El Halkouj et al, “Potential and sky coverage for off-axis fringe tracking in optical long baseline interferometry”, MNRAS, 506, 2021

[10] J. Woillez, J. Allonso et al, “The 2nd generation VLTI path to performance”, Proc. SPIE 9907, id. 990706 9 pp. 2016.

[11] J. Woillez, R. G. Petrov, R. Abuter et al, «GRAVITY for MATISSE -- Improving the MATISSE performance with the GRAVITY fringe tracker », to appear in A&A, May 2024, arXiv:2405.20730.

[12] R. G. Petrov, J. Leftley, M. Hadjara et al, «Combining Differential Interferometry and Reverberation Mapping for AGN in the J band », MASCA workshop, Nice, France, October 2023.

[13] R. G. Petrov, F. Allouche et al, « Hierarchical Fringe Tracking, sky coverage and AGNs at the VLTI », Proc. SPIE 12183, id. 121830I 14 pp. (2022).

[14] R. Laugier et al, «VLTI Unit Telescope Coude train vibration control upgrade for GRAVITY+», Pro. SPIE 13095, Optical and Infrared Interferometry and Imaging IX (18 June 2024).

[15] GRAVITY+ Collaboration, et al, «First light for GRAVITY Wide. Large separation fringe tracking for the Very Large Telescope Interferometer », A&A 665, 2022

[16] A. Berdja et al, «Quasi-Random Frequency Sampling for Optical Turbulence Simulations », to be published in Optics Continuum, 2024, arXiv:2404.02978