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Hierarchical fringe tracking

Romain G Petrov, Fatmé Allouche, Massinissa Hadjara, Abdelkarim Boskri,

James Leftley, Amokrane Berdja, Stéphane Lagarde

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Romain G Petrov, Fatmé Allouche, Massinissa Hadjara, Abdelkarim Boskri, James Leftley, et al. Hierarchical fringe tracking. Optical and Infrared Interferometry and Imaging IX, SPIE, Jun 2024, Yokohama, France. pp.89, 10.1117/12.3018417. hal-05123126

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Hierarchical Fringe Tracking

Romain G. Petrov*<sup>a</sup>, Fatmé Allouche<sup>a,f</sup>, Massinissa Hadjara<sup>b,c</sup>, Abdelkarim Boskri<sup>d</sup>, James
Leftley<sup>a</sup>, Amokrane Berdja<sup>e</sup>, and Stéphane Lagarde<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>LPHEA Laboratory, Oukaimeden Observatory, Cadi Ayyad University/FSSM, BP 239, Marrakesh, Morocco; 
<sup>e</sup>Centro de Astronomía (CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta 1270300, Chile; 
<sup>f</sup>European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile.

ABSTRACT

Hierarchical Fringe Trackers (HFT) maximize the sensitivity and accuracy of fringe tracking. Their performances are independent from the number of apertures. They cophase pairs of telescopes, then pairs of pairs and so on. We report the key results of mathematical analysis, design, manufacturing, optical tests and simulated performances of 4 telescopes HFT chips for the VLTI and 6 to 8 telescopes HFT chips for CHARA or a VLTI extension. An end-to-end simulation with realistic input piston and flux, based on the experimental characterization of the signals on the test bench, validates the servo loop and state machine architectures and supports the performance analysis, confirming the expected performance gain of about 3 magnitudes (with a limiting magnitude $K>16$ on the VLTI with UTs) and the fact that the performances do not decrease with the number of apertures. The performance gain is based on the combination of the HFT architecture with a very broad band HFT covering the 1.1 to 2.2 µm domain with 3 to 5 HFT chips working in broad sub bands in J, H and K. Analysis of fringe jumps and losses at the sensitivity limit show that an HFT manages then more efficiently than the standard pairwise architecture. The impact of HFT characteristics on AGN science programs for optical interferometers is illustrated, showing that this architecture is the key for fascinating applications including direct distance measurements of AGNs accurate enough to contribute to the Hubble tension problem.

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

# 1. INTRODUCTION

The hierarchical fringe tracker (HFT) is intended to maximize the limiting sensitivity of optical long baseline interferometers, such as the VLTI, CHARA or future interferometers with a larger number of apertures, with performances that does not decrease with the number of apertures. This is a key characteristic for homodyne interferometers as well as for heterodyne or intensity interferometers. 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 main goals of this paper. Based on experimental data measured on prototype HFT chips, we have simulated the

<footnote>*Romain.PETROV@univ-cotedazur.fr. Tel : +33 4 89 15 03 50.</footnote>
limiting performances, on the VLTI, of an HFT working in the J, H and K bands thanks to a feeding system that actively corrects chromatic effects and allows to use broad band integrated optics HFT chips (see figure 3).

The basic component of an HFT chip, as shown in Figs. 1 & 2, is a Two Beams Spatial Filter (TBSF) derived from the standard ABCD beam combiner. Two the outputs of the ABCD chip are injected into an X coupler that produces a "constructive" C+ output containing up to ~80% of the total incoming flux when the input beams are co-phased and a "nulling" C- output that should be ~0 at co-phasing. Figure 1 illustrates this concept and shows the key equations of a TBSF, its four outputs A, B, C+ and C- and of its phase delay estimator. Our preferred phase delay estimator is:

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

where the $(...+\pi)$ modulo$(2\pi)-\pi$ operation is used to recenter between $\pm\pi$.

This estimator is independent from the injected flux ratio. This estimator needs that a fraction of the C+ output is directly send to the detector. This decreases the flux injected in the level n+1 but maintains the fact that this level receives more flux than the level 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 (the $F(a_1, a_2, \Delta\varphi, \Delta\varphi_{cons})\psi \neq 0$ when $\Delta\varphi \neq 0$ and $a_1 \neq a_2$) and this generates fringe jumps at higher flux).

<figure><img src="image_3.png" /><figcaption>Figure 1. Principle and key equations of a Two Beam Spatial Filter. The four outputs A, B, C+ and C- are combined to produce an exact phase delay estimator.</figcaption></figure>

The use of this phase delay estimator requires diverting a fraction of the flux C+ directly to the detector as illustrated in figure 2. The largest that fraction the better the cophasing at level n but the lower the flux injected in the level n+1. The simulation [1] shows that any number between 0.25 and 0.33 gives the optimum compromise for the global sensitivity limit set by minimizing the fringe jumps and the tracking residuals at all levels.
## 2. NEW GENERATION FRINGE TRACKER

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2. 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 that is fed into the level n, a fraction of the flux is derived directly to an output. 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 [2].</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 and of about 40% when we have to use a Laser Guide Star. 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) 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). We propose a device that can obtain these two effects with a minimum transmission loss (and in particular without using dichroic plates). This is sketched in figure 3. For each VLTI beam, we place a dispersive element (P) in a focal plane which allows to have a dispersed pupil but non dispersed focal images. The dispersed pupil is placed on a MEMS mirror (0.025 µm to 0.05 µm for each individual mirror). The global orientation of the mirrors in a band allows to select it and feed it in an Integrated Optics HFT chip, one per photometric band. Individual single channel pupils on the MEMS mirror segments are (i) tip-tilted to correct transverse chromatic dispersion and optimize injection and (ii) translated in a piston correcting the chromatic OPD differences within sub bands. Selected mirrors can pick spectral windows around emission lines redshifted anywhere between 1 and 2.2 µm to feed it to an external instrument or to an internal module specifically optimized for line differential interferometry with resolution ~1500 (±500) [3]. This design yields conservative transmission estimates of 2% in J, 3% in H and 4% in K based on the measured VLTI transmission [4], 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.

### Expected sensitivity gain

We expect a sensitivity gain with regard to the current GRAVITY Fringe Tracker from the improved transmission other the 1% current value of the GRAVITY fringe tracker [2][3] and from the broader spectral bands. With the Strehl ratios expected in the GRAVITY+ AO context [5], this yields a gain of a factor ~9 in flux.

The HFT architecture provides another sensitivity gain. In the pair wise ABCD GRAVITY Fringe Tracker, the flux of each telescope is divided by 3 and we have a phase delay estimate for each of the 6 baselines. These 6 phase delay estimates can be combined to retrieve the 3 piston differences between the 4 telescopes. In practice we show [3] that this combination can be applied only when the phase delay estimators remain linear, i.e. the phase delay error per baseline is lower than 1 radian. This is a first reason to argue that the limit of the pairwise FT is set by the flux necessary to yield this 1 radian error with one third (i.e. $(n_T-1)^{-1}$ for $n_T$ telescopes) of the flux of each telescope. Moreover, when the science channels are not at the same wavelength than the tracking channels we have to avoid or to recover very rapidly fringe jumps [5]. Fringe jumps become probable when the phase delay error reaches ~1 radian (lambda/6 corresponds to ~2 fringe jumps in 100 fringe tracker frames). A fringe jump on any baseline will automatically produce a fringe jump in
any linear combination of phase delays. In the HFT we directly measure the phase delay that is applied to each delay line and the fringe jump probability is set by the 1 radian phase delay error corresponding to the full flux collected by the first two apertures. Hence the HFT specific gain corresponds to nT-1. Another way to say this is to repeat that the performance of a standard pairwise FT decreases as nT-1 with the number nT of telescopes. Hence our prediction is a gain of a factor 27 in flux on the VLTI or magnitudes over the 12.5 magnitude for the GRAVITY FT compatible with observation in the J band. As shown in the next chapters this is in good agreement with our simulation results.

<figure><img src="image_6.png" /><figcaption>Figure 3. Possible concept of a feeding system allowing to distribute the light among various TBSF with an active correction of chromatic effects both in OPD and in image position to allow using very broad band chips at wavelength as short as the J band. A dispersive element in a pupil plane sends a dispersed pupil on a MEMS mirror. Each segment of the MEMS, corresponding to roughly a spectral channel of resolution 500, can be moved in tip-tilt and in piston. We use tilts to select the relevant spectral bands. Then, on focal plane we have a non-dispersed image fed into a single mode fiber.</figcaption></figure>

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Find $P_0$ and $a$ that maximize $F(P_0, a)$

$$F(P_0, a) = \sum_{i=1}^{n_\lambda} \sum_{j=1}^{n_G} \cos(2\pi(P_0 + aj)\sigma_i - \varphi_{ij}) \text{sinc}\left(\frac{P\delta\lambda_i}{\lambda_i^2}\right)$$
$$P = P_0 + aj$$

The group delay piston after $n_G$ frames is $P = P_0 + an_G$

<caption>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 (see figure 1) used for group delay estimation.</caption>
### 3. SIMULATION OF A BROAD BAND J,H,K<sub>S</sub> HFT PERFORMANCE

We run many simulations of a full, hence simplified, fringe tracker algorithm, featuring variations of the sampling frequency (200 Hz), the time to estimate the phase delay (5 to 30 ms), the number of phase delay estimates used for group delay measurement, the number of spectral channels (either one in J, one in H, one in K short or two in J, two in H and one in K short and various design parameters for the HFT chips (the $\Delta\phi_{cons}$, the fraction of C+ transmitted to the level n+1). The simulation is described in another paper at the same conference [1]. The tracking is based on a correction of the piston corresponding to each measured phase delay shift (defined with a $k\bar{\lambda}$ ambiguity) combined with group delay correction of the absolute piston when a fringe jump is detected. The group delay estimator is based on the combination of the phase delay estimators in the different spectral channels. It is described in the figure 4. We look for the absolute piston that best fits all phase delays. To avoid the effect of unpredictable $2\pi$ jumps when the phase delay approaches $\pm\pi$, we minimize the geometrical distance between in the complex plane between the phase delay $e^{i\Delta\phi(\lambda)}$ 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).

<figure><img src="image_8.png" /><figcaption>Figure 5. RMS of the residual piston in $\mu$m as a function of magnitude for several phase delay computation time and number of phase delays used for group delay. The vertical lines near each magnitude indicate the lock factor. The residual RMS is computed assuming that the fringe tracker closes the shutters of the science instrument as soon as the group delay detects a fringe jump and as long as it has not closed the loop again.</figcaption></figure>

Then we produce statistics of fringe delay tracking performances based on two parameters: the rms of the residual piston (in $\mu$m) and the lockup factor i.e. the fraction of time of effective tracking. The residual piston rms is computed for the tracking period as we are experiencing a “close shutter” function that feeds the science instrument only when the group delay estimator and phase delay snr estimators indicates that we are tracking. The figure 5 illustrates this type of statistics that we ran for several parameters, here the time $\tau$ used to estimate the phase delay and the time $n_G\tau$ used for group delay estimation as a function of the target magnitude in K. Here we see that at magnitude K=16 we have combinations of $\tau$ and $n_G$ yielding a good tracking with a residual better than 100 nm good enough for high quality J band observations. At
K=16.5 we have some fringe jumps rapidly recovered by the group delay tracking and at K=17 tracking becomes problematic for the J band with a rms residual larger than 250 nm and frequent fringe losses that still recoverable by the group delay tracking with some delay.

<figure><img src="image_9.png" /></figure>

Figure 6: tracking simulations on the VLTI in good seeing conditions (best 20%). We overplot the atmospheric piston (fin gray curve) and the tracking residual (in µm) in green for the baseline 1-2, red for the baseline 3-4 and purple for the baseline 12-34 (difference between the average pistons on apertures 1-2 and 3-4). The lower plot is for the tracking residual on the pair 12-34. The open shuter time and rms are computed assuming that the shuters of the science instrument are closed when the fringes are not phase delay tracked as explained in the text.
The figure 7 illustrate this situation. Each plot corresponds to one of the dots in figure 5 as indicated in the label. At K=16 we have a good tracking solution with 20 ms time to estimate the phase delay (coherent integration of 4 frames of 5 ms). The group delay is checked every 400 ms but is not used in that perfect tracking case. It is important to note that the rms residual on baselines 12-34 (level 2) is significantly lower than this on the 1-2 and 3-4 level 1 baselines, with respectively 75 nm and 100 nm. At τ=10 ms, with the same atmospheric piston we have some fringe jumps, rapidly detected and corrected by the group delay tracking, yielding an acceptable open shutter residual on the baseline 12-34 that is tracked during 93% of the time. The fringe jumps on baselines 1-2 and 3-4 also impact baseline 12-34. It might be possible to mitigate this by an improved state machine. At K=16.5, the fringe jumps or losses are still recovered by the group delay tracking but this takes more time, although the closed shutter tracking residual remains acceptable at 124 nm with a closed loop situation in the 80-90% range. At K=17, the group delay tracker with τnG>600 ms remains able to recover the fringes in a few seconds (in most cases) but the cumulated fringe losses make all baselines other than 1-2 and 3-4 improper for observations even if the lock factor is close to 50% and full fringe losses, implying starting a fringe search loop occur only typically every 3-4 mn.

We have also simulated a 6 telescopes combination (as in figure 2 left) to further illustrate that we have a system with performances independent from the number of apertures, when the first levels are well cophased, and to show that the HFT architecture is not limited to 2^nd^ telescope numbers. The results are illustrated in figure 7, with a perfect tracking at K=16 (assuming 6 telescopes with the collecting power of an UT) and an acceptable tracking at K=16.5. in that last case we have forced a fringe jump on one baseline to illustrate the group delay recovery. In the good tracking case, the residual rms is the best on baseline 123-456, while 1-2 and 12-3 as well as 4-5 and 45-6 have similar performances. Note that baselines ij-k receive less flux than a ij-kl baseline in the 4 telescope case.

<figure><img src="image_10.png" /><figcaption>Figure 7: tracking simulations for 6 "UTs" in the conditions of figure 6. For the 1m telescopes of CHARA, assuming AO giving the same Strehl ratios, the corresponding magnitudes would be 11.5 and 12.</figcaption></figure>

In all cases we have assumed that the source is unresolved on all baselines. This would be the case for an off-axis tracking star at these kinds of magnitudes. It would also be the case for type 1 AGNs that would remain very substantially unresolved for any baseline smaller than 1 km. For resolved targets, the magnitudes given above have to be replaced by this corresponding to the average coherent flux in all implied baselines.

Our initial motivation to push the fringe tracking sensitivity on the VLTI to its maximum value was the need to resolve a large number of BLRs in type 1 AGNs that are small enough to allow the combination of spectro-astrometry and Reverberation mapping and resolved enough to allow differential visibility measurements. The combination of the angular size from spectro-astrometry with the linear size from reverberation mapping yield a direct distance measurement that can contribute to confirm and quantify the Hubble tension between expansion rates estimates from the late and early universe. The differential visibility measurements are critical to remove degeneracies between geometrical and kinematical parameters to make the model fitting of differential visibility and phase more accurate in the estimation of the mass of the central super massive black hole as well as for unbiased distance estimate [6]. Figure 8, which was already displayed in [7] shows the AGN whose BLRs can be resolved by spectro-astrometry and by differential visibility with different limiting sensitivities of fringe tracking. The limiting K magnitudes of K=12.8 for the GRAVITY FT and 15.2 for a broad band HFT were from analytical estimates given in [3] that included a model of the anisopistonic error. Here we show that these sensitivity estimates were quite conservative and the gain in number of resolved AGN offered by an HFT will be even better. In figure 9, we show the time lags of the observable objects from figure 8. A time lag of 400
light days is a strong practical limitation for a RM program of less than 4-5 years, as covering at least 3-4 time lags is necessary for an accurate RM measure. We note the very strong increase in the overlap between RM and Differential interferometry offered by a FT sensitivity extended to 15.2.

## 4. IMPACT ON J BAND OBSERVATIONS OF QUASARS

<figure><img src="image_11.png" /><figcaption>Figure 8: Potential J band observations of QSOs Broad Line regions (resolution 1000, two hours of observations) at the VLTI with the UTs with the GRAVITY Fringe Tracker GFT) after completion of the GRAVITY+ upgrade and with the HFT. Upper left: GFT on-axis; upper right: GFT off-axis; bottom left: HFT on axis; bottom right HFT off-axis. The blue dots indicate a spectro-astrometric resolution of the BLR with a SNR>10 on the photocenter displacement. The orange dots indicate targets allowing visibility measurements of the inclination and position angle of the BLR. There are no QSO BLRs that can be imaged in the J band at the VLTI.</figcaption></figure>

## 5. CONCLUSION

We have described an updated HFT chip design with optimized phase delay and group delay estimators and we have simulated the fringe tracking loop with a simplified state machine. Our simulation does not include yet vibrations. We have only estimated that all significant vibrations have frequencies lower than 100 Hz, allowing a fringe sensing sampling at 200 Hz. We have not implemented any Kalman filter like optimization. These two points are the next step of our work. However, we have displayed evidence that the noise propagation between the different levels of an HFT can be controlled up to magnitudes that indeed exceed our initial analytical estimates and confirmed that an HFT can have a high limiting sensitivity that does not decrease with the number of apertures, at least up to the 6-8 apertures level. Analyzing the case of an aperture number in the 9-16 range (4 levels HFT) and in the 17-32 range (5 levels HFT) still has to be implemented in our simulator but the fact that the tracking performances increase from level 1 to 2 and from level 2 to 3 when the first levels are tracking correctly is a strong indication that an HFT can work for a large number of apertures. The intrinsic bootstrapping capacity of an HFT strongly depends from the architecture of the array and this also is a topic for further investigation.
<figure><img src="image_12.png" /><figcaption>Figure 9: time lag histogram of the observable targets defined in figure 8. In orange, the targets observable with the current FT sensitivity of K=10.5. In red, the targets observable in J with a fringe tracking sensitivity similar to this of the GRAVITY FT in a GRAVITY+ context. In purple the targets accessible with a very conservative estimate of the HFT limiting magnitude.</figcaption></figure>

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] F. Allouche, M. Hadjara, A. Berdja, A. Boskri, S. Lagarde, R. G. Petrov, “End-to-end simulations of a Hierarchical fringe Tracker”, Proc. SPIE 13095, Optical and Infrared Interferometry and Imaging IX (18 June 2024).

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

[3] 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

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

[5] 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.

[6] 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.

[7] 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).