Upgrading the GRAVITY fringe tracker for GRAVITY+: Tracking the white light fringe in the non-observable Optical Path Length state-space

Aims. As part of the ongoing GRAVITY+ upgrade of the Very Large Telescope Interferometer infrastructure, we aim to improve the performance of the GRAVITY Fringe-Tracker, and to enable its use by other instruments. Methods. We modify the group delay controller to consistently maintain tracking in the white light fringe, characterised by a minimum group delay. Additionally, we introduce a novel approach in which fringe-tracking is performed in the non-observable Optical Path Length state-space, using a covariance-weighted Kalman filter and an auto-regressive model of the disturbance. We outline this new state-space representation, and the formalism we use to propagate the state-vector and generate the control signal. While our approach is presented specifically in the context of GRAVITY/GRAVITY+, it can easily be adapted to other instruments or interferometric facilities. Results. We successfully demonstrate phase delay tracking within a single fringe, with any spurious phase jumps detected and corrected in less than 100 ms. We also report a significant performance improvement, as evidenced by a reduction of about 30 to 40% in phase residuals, and a much better behaviour under sub-optimal atmospheric conditions. Compared to what was observed in 2019, the median residuals have decreased from 150 nm to 100 nm on the Auxiliary Telescopes and from 250 nm to 150 nm on the Unit Telescopes. Conclusions. The improved phase-delay tracking combined with whit light fringe tracking means that from now-on, the GRAVITY Fringe-Tracker can be used by other instruments operating in different wavebands. The only limitation remains the need for an optical path dispersion adjustment.


Introduction
1 By combining the light from multiple telescopes, long-baseline 2 optical interferometry can achieve a much higher angular res-3 olution than is possible with a single-dish telescope.Even in 4 the upcoming era of telescopes exceeding 30 m, the Very Large 5 Telescope Interferometer (VLTI) will continue to outperform 6 single-dish observations in terms of angular resolution (for an 7 up-to-date overview, see Eisenhauer et al. 2023).This is par-8 ticularly useful for many applications, as shown by the broad 9 range of subjects to which it has been applied with groundbreak-10 ing results (GRAVITY Collaboration et al. 2018a,b,c, 2019a,b,c, 11 2020a,b,c,d,e,f, 2021, 2022).
12 One of the challenges that arise at this high angular resolu-13 tion is the need to manage disturbances, which can include atmo-14 spheric turbulence, telescope vibrations, and other noise sources, 15 and which remain the primary limitations to the precision of in-16 terferometric measurements.These disturbances lead to varia-These authors contributed equally.tions in the optical path length (OPL) of light travelling through each telescope to the beam combiner, resulting in an unstable interference pattern (unstable fringes).
To obtain stable interferometric measurements, these fringes need to be controlled in real time, and any OPL fluctuations need to be compensated for.In modern instruments, this is the role of a specific subsystem: the fringe tracker.The crucial role of fringe-tracking in optical interferometry has justified the extensive development efforts made over the past decades (Shao et al. 1988;Sorrente et al. 2001;Delplancke et al. 2006;Le Bouquin et al. 2008;Cassaing et al. 2008;Houairi et al. 2008;Colavita et al. 2010;Lozi et al. 2011;Menu et al. 2012;Choquet et al. 2014;Lacour et al. 2019).Various techniques have been proposed, based either on hardware or software solutions, but they all share a common objective: to measure and stabilise the fringes, thereby maximising the signal-to-noise ratio (S/N) of interferometric measurements.
For the GRAVITY instrument, the fringe-tracker was initially designed to perform control in the six-dimensional optical In the GRAVITY fringe tracker, the 24 outputs (4 outputs for 80 each of the six ABCD combiners) are dispersed over six wave-81 length channels in the K band and are recorded on a SAPHIRA 82 detector (Finger et al. 2014).We refer to Figure 1 in Paper 1 for an overview of the pixel arrangement on the detector real time display (RTD).As shown by Equations 1 to 7, for a given wavelength channel, the column vector (q 1 , . . ., q 24 ) T containing the 24 different intensity outputs and the column vector (F j , Γ j,k ) T containing the ten incoherent and coherent flux values are related by a multiplication by a single matrix: the P2VM (pixel to visibility matrix), P2VM • (q 1 , . . ., q 24 ) T = (F 1 , . . ., F 4 , Γ 1,2 , . . ., Γ 3,4 ) T . (8) The incoherent and coherent fluxes are computed in real time for each of the wavelength channels, which are treated independently.Similarly, when the polarisation is split, the two polarisations are treated independently.
In practice, the ABCD combiners are not perfectly balanced, and the phase offsets can be different from their fiducial values, which means that the coefficients linking the ABCD outputs to the complex amplitudes of the electric field in Equations 1 to 4 can vary.Therefore, the P2VM must be calibrated, which in GRAVITY is done during daytime using an internal source.This P2VM formalism was initially introduced by Tatulli et al. (2007) for the AMBER instrument (Petrov et al. 2007).It was then adapted to ABCD combiners by Lacour et al. (2008), and more details about its implementation can be found in these papers, along with Paper 1.

From fluxes to phase/group delays and closures
After the P2VM calculation, the resulting wavelength-dependent coherent fluxes Γ j,k,λ are still affected by dispersion (atmospheric dispersion, and dispersion in the fibre delay lines, FDDL), which at first order introduces a phase curvature of the form e iD(1−λ 0 /λ) 2 , where λ 0 = 2.2 µm.This is corrected for by using an empirical value for D that depends on the position of the star in the sky and on the position of the FDDLs.This correction was calibrated during the first year of on-sky observations.From this dispersion-corrected coherent flux measurement, the OPD φ j,k is calculated for baseline ( j, k) using These values are concatenated to form the OPD vector Φ, using the convention Φ = (φ 21 , φ 31 , φ 41 , φ 32 , φ 42 , φ 43 ) T .The closure phase θ PD j,k,l over the triangle linking telescopes T j , T k , and T l is calculated in units of length using where the incoherent flux is first averaged over 350 detector integration times (DITs, i.e., bout 385 ms at 909 Hz) to boost the signal-to-noise ratio, the phase closure being a notoriously noisy quantity.
The group delay ψ j,k and group closure θ GD j,k,l are obtained from the coherent flux, which is first corrected for any phase offset constant in wavelength by subtracting the phase delay, and then averaged over 150 DITs (about 165 ms at 909 Hz), again to boost signal-to-noise ratio, . (11) where ∆λ is the difference between the effective wavelength of 131 two pixel bins.From this, we also constructed another vector in 132 OPD space: Ψ = (ψ 21 , ψ 31 , ψ 41 , ψ 32 , ψ 42 , ψ 43 ) T .The closure of 133 the group delay can be written as 135 The last quantity of interest for the control algorithm is the The uncertainty on the phase can then be computed using a sim-143 ilar approach as in the appendix of Shao et al. (1988), In Equation 14, the variance of the real and imaginary parts of A first challenge that needs to be solved in the implementation 161 of a fringe tracker is the dichotomy between two distinct spaces.

162
The fringe tracker uses a set of four piezoelectric delay lines 163 (Pfuhl et al. 2014) to change the OPL of the four beams (one per 164 telescope).The feedback is provided by measurements of the 165 coherent fluxes (see Section 2), however, which are transformed into a set of six measurements of OPD φ (one per baseline) and group delay ψ.The OPL space itself is not observable, and this mismatch between the control space and the measurement space creates a paradox that needs to be solved by the system.
A second challenge arises from the need to regularly update the model that is used to represent and predict the disturbance.To do this, we chose to use a set of AR models, whose parameters are updated every few seconds via a model-fitting routine performed on a separate computer (see the hardware description in section 4.1).In order to perform the model fitting, this computer needs to be fed with measurements that only represent the disturbance, excluding the additional piezo command.The control must therefore work with a state model that explicitly separates the OPL into two components: a disturbance component L that represents both atmospheric and mechanical turbulence, and the OPL introduced by the piezo actuators X.Only the total OPD, which derives from the total OPL L + X, can be observed by the fringe tracker.In order to be able to update the disturbance model, however, the two components must be kept separated.
A third challenge comes from the fact that the main phase control loop is fed by phase measurements, which are only known to a modulo 2π (or λ 0 , in terms of OPD).Thus, this loop is blind to any potential fringe shift, and we therefore need the additional group delay loop.In Paper 1, the controller consisted of two completely independent loops running in parallel, with the potential of issuing conflicting commands.The algorithm developed at the time ensured that the integrator of the two controllers would not diverge too widely, but it did not strictly constrain the absolute group delay.The consequence was a tendency of the fringe tracker to jump between fringes during a single observation.The updated architecture detailed in this work provides a much more unified approach to controlling both the group and phase delays, thereby avoiding conflicting commands, mitigating phase jumps, and maintaining tracking of the white-light fringe.

State model
To solve these challenges, we adopted a state model in OPL space that explicitly separates the disturbance and the OPL introduced by the actuators.We used a set of two state vectors that both have the unit of length and that we denote X n and L n .For simplicity, these state vectors can be broken down into four independent vectors, corresponding to each telescope.For a given telescope T k , the vector of the last N values of OPL introduced by the piezo actuators (X n,k ) and by the disturbance (L n,k ) can be expressed by where x k (t j ) and l k (t j ) denote the OPL introduced by the piezo actuator and the disturbance on telescope T k at time t j , respectively.These vectors have a total length of 150, which matches the smoothing length of the group delay.The four telescopes can be concatenated to give our two final state vectors X n and L n of dimension 4 × 150 = 600, Each of the piezo control chains (which include the actuator response as well as the pure delay introduced by the processing) has a response function that we modelled as a fifth order, where the position of the piezo at time t n was derived from the command sent at the five last time steps, The c coefficients can be empirically measured by sending an impulse command (in Volts) through the control chain and measuring how the response evolves with time (Figure 1).
The command vectors U n,k = [u k (t n ), . . ., u k (t n−4 )] are then defined such that Again, the four telescopes can be concatenated to give U n of dimension 4 × 5 = 20, With these notations, the propagation of the piezo state vector X n takes the form where A X is a 600 × 600 block-diagonal matrix that just shifts the xs for each telescope, with and C is also a block-diagonal matrix, but of size 600×20, which contains the coefficients of the piezo responses, with To do this, we first note that for a four-telescope (six-baseline) interferometer such as GRAVITY, an OPD vector Φ = (φ 1,2 , φ 1,3 , φ 1,4 , φ 2,3 , φ 2,4 , φ 3,4 ) T and an OPL vector (l 1 , l 2 , l 3 , l 4 ) T are linked by the matrix equation where M is a 6 × 4 matrix, defined as Therefore, computing an OPL vector from the knowledge of the 249 OPD vector is under-constrained and yields multiple solutions.250 A solution entails using the Moore-Penrose pseudo-inverse of 251 M, For reference, the value of M + is 253 This particular choice ensures that the mean value of the OPLs 254 on each telescope is 0, which is beneficial for a fringe tracker.It 255 helps to avoid runaway situations in which all the actuators move 256 toward a large offset and saturate the system.

257
These equations can be extended to our OPL state-space vec-258 tor L n as defined in Equation 19, and a similarly defined OPD 259 vector Φ n = (Φ n,1,2 , . . ., Φ n,3,4 ) T using similar matrices in which 260 the 1s are replaced by identity matrices of size 150, With these matrices, the relations remain Our AR model fitting provides a set of six matrices A j,k that 263 can be used to propagate the individual Φ n, j,k .These can be com-264 bined into a block-diagonal matrix to propagate Φ n itself, Switching to OPL space, we have 266 which means that as long as our OPL vectors L n+1 contain values where the average on all telescopes is 0, we can consider that 269 M • M is the identity matrix.In other words, as long as we have Assuming temporarily that L n+1 fulfils the condition given by 272 Equation 39, we can left-multiply Equation 37 by M + to obtain A block matrix multiplication inside the brackets gives the final 274 form of our model in OPL state-space, with We can formulate two important remarks on the propagation 277 model given by Equation 42.Firstly, since Fig. 1.Open-loop response of the piezo control chain.This corresponds to the response of the piezo in micrometers per volt as a function of time (bottom axis) when submitted to an impulse command.Equivalently, this also gives the values of the coefficients of the fifth-order model used in Equation 20(top axis).Each colour represents a distinct actuator.
The process noise Q is derived from a set of 150 × 150 matrices Q j,k using a similar block construction, The matrices A j,k and Q j,k are recalculated every 5 seconds by an asynchronous machine (wgvkalm described in sec 4.1) based on OPD measurements.This computation makes use of the Python toolbox time-series analysis (Seabold & Perktold 2010).A detailed explanation of its implementation is provided in Appendix A.

Phase control
Without any feedback, that is, without any additional information about the state of the system, the internal model of the fringe tracker would simply continue to indefinitely propagate the state vector L using Equation 42 and to dispatch commands to the piezo actuators to respond to its own predictions.This behaviour would probably be acceptable over a few iterations, but given the uncertainties on the predictions of the disturbance, the internal state vector L n would quickly diverge from the real values.The internal model is aware of this, as Equation 45shows that in this scenario, the covariance matrix would slowly inflate due to the accumulating process noise Q.
To properly track the fringes over long periods, the fringe tracker therefore needs to integrate measurements into the loop.These measurements should not be taken at face value either, however, as they are also affected by varying levels of uncertainties.Integrating noisy measurements into the loop without careful consideration will create additional noise in the system, thereby reducing its overall performance.The main task of the Kalman controller in phase-delay loop is to determine how to best balance the information from the state model propagation and from the additional measurements performed on the system.In Paper 1, this was done using a fixed asymptotic Kalman gain, which could not react in real time to the quality of the model and measurements.This new version of the Kalman provides an optimal gain, calculated at each iteration to combine the two sources of information in the best possible way.
At each time step t n , a set of six measurements of the phase (or OPD) are obtained on each baseline, with an estimate of their for the context of this section, we simply denote Φ n = (φ j,k (t n )) 331 the column vector of size 6 containing the individual phase mea-332 surements (see Equation 9), and W n the associated 6 × 6 diagonal 333 covariance matrix, as defined from Equation 15 in Equation 16.

334
These measurements are related to our state vectors L n and 335 X n through the measurement equations, This means that although our state vectors have a length of 600 with H φ defined by 342 where v 1 = (1, 0, . . ., 0) T is the first basis vector with a length of 343 150.

344
Given our assumption that the piezo state-vector X n is per- The update state vector and its covariance matrix are then given 360 by 361 which indeed reduces to Pn regime and also to L n = Ln in the Pn W n regime.

363
The introduction of Φ CP in this equation ensures a zero simply recall here that Φ CP is constructed by setting three of the four closure phases θ PD j,k,l to the appropriate baselines, ignoring the closure phase on the triangle of the lowest signal-to-noise ratio, It is also important to note with this definition, two fundamentally different types of variations of Φ CP can occur: (1) Φ CP can change because of slowly varying closures phases in the measurements, in which case it simply tracks actual variations.
(2) Φ CP can switch between options given in Equation 54, which occurs when the triangle of the lowest signal-to-noise ratio changes.The second case corresponds to a change in the configuration, leading to an abrupt variation of Φ CP , and it is therefore banned while the science detector is integrating.This behaviour is controlled by an internal boolean parameter, called the mobility flag, which is "False" during science integrations.

Group-delay control
The initial version of the fringe tracker described in Paper 1 was designed for an instrument performing prolonged exposures at a consistent wavelength of 2.2 µm.As long as it stayed in the coherence envelope of the fringes, the fringe tracker had the freedom to transition from one fringe to the next.This is unacceptable if the GRAVITY fringe tracker is to be used to feed a different science instrument operating at a different wavelength, as this would blur the fringes on the science detector.In an effort to make the GRAVITY fringe tracker compatible with the L-band combiner MATISSE (Lopez et al. 2022), the group-delay controller was modified to consistently track the same fringe during a scientific exposure.This new MATISSE mode is called GRA4MAT (Woillez et al. in prep.), and is now routinely offered to the community.
For this to work, the group-delay control loop must determine in real time which fringe is being tracked by the phasecontrol loop, and it updates the state vector L n by integer multiple of λ 0 if required, depending on which fringe it wishes to track.These changes are completely transparent to the phasecontrol loop due to the modulo λ 0 in Equation (50).
To implement this behaviour, we used a formalism similar to the phase-delay loop.We first introduced the 6 × 600 measurement matrix H ψ to relate the group delay Ψ to the state vectors L n and X n , with and where 1 = (1, 1, . . ., 1) is a row vector with a length of 150 full of 1s.This is similar to the definition of H φ , but includes an averaging of the past 150 DITs.
We also introduced Ψ CP , an analogue of Φ CP , which is calculated using θ GD j,k,l instead of θ PD j,k,l in Equation 54, and an additional vector Ψ ZeroGD .This additional setpoint is used to select which specific fringe is set at a group delay of 0, to be tracked by the fringe tracker.Again, to avoid fringe jumps during a science integration, a mobility flag is used to freeze Ψ ZeroGD during integrations on the science detector.When the mobility flag allows changes, the Ψ Zero GD setpoint is estimated so that This condition is valid modulo λ 0 , which means that Equation 58does not exactly determine which fringe is tracked.To lift this uncertainty, two requirements must be added: R1: Vector Ψ ZeroGD must be as small as possible.
R2: Vector Ψ ZeroGD must not have a closure phase.
The first requirement is to ensure that the system tracks the white-light fringe.The second requirement is needed because the closure phase has already been nulled by the Psi CP term.
Therefore, the zero group-delay setpoint is estimated by the real-time computer as where M was introduced in Equation 29, and M † is a pseudoinverse weighted by the covariance matrix W n , that is, the matrix that solves the linear system Y = MX in the least-square sense, considering that W n from by Equation 16is the covariance matrix of X.This matrix M † can be calculated using The controller ensures that the group delay never deviates by more than λ 0 /2 from this setpoint.To do this, the controller calculates at each cycle an error in the form of Ψ− Ψ−Ψ CP −Ψ Zero GD .
It converts this group-delay error into OPL space using M † from Equation 60.When this error exceeds λ 0 /2 on any telescope, the controller subtracts λ 0 to all the 150 values corresponding to this telescope in the state vector L. Similarly, when an OPL value falls below −λ/2, the elements of L corresponding to this telescope are increased by λ 0 .These adjustments by integer multiples of λ 0 never disturb the phase controller due to the presence of a modulo λ 0 in Equation.52, and because the propagation model is stationary (more details in Appendix A).

Commands to the actuators
The command for the actuators is obtained from In this equation, U setpoint is used to modulate the position of the fringes in a process detailed in Paper 1 2 .This setpoint can now also be controlled externally by another machine, for example, to correct non-common OPD, as described in Section 4.1.
The term U search is used to search for the fringes when they are lost by one or more telescopes.This is made to ensure that the fringe tracker continues to track the fringes on the baselines where they remain visible, while searching on the others.
This is obtained to the best of our capabilities by taking 466 with C T k representing the integrated impulse response of the 467 piezo actuator at telescope k, where the c coefficients are the same as in Equation 20469 3.6.Summary of the new control loop 470 The architecture of the new control loop is illustrated in Figure 2. 471 In this updated architecture, the two control loops no longer run 472 in parallel, as was the case in Paper 1. To some extent, the phase 473 loop can now be seen as the main control loop and the group-474 delay control as an auxiliary loop that provides an additional 475 update of the state parameters to ensure proper tracking of the 476 group delay.The phase tracking is performed using a Kalman fil-477 ter that no longer relies on asymptotic gain, but uses a real-time 478 covariance-weighted combination of a model prediction (in OPL 479 space) and a measurement update (in OPD space).The control 480 algorithm consists of the following steps: 1.At iteration n, we start with an estimate of the state vector 482 Ln , propagated from the previous iteration. (66) 3. A measurement Φ of the OPDs is performed, which comes 496 with an estimate of its uncertainties, in the process described 497 in Section 2. The difference between this measurement and 498 the expected value Φn encodes some information on the state 499 of the disturbance.This information is combined with the a 500 priori knowledge of the state vector Ln to give an updated a 501 posterori estimate L n .This is a form of optimal data fusion 502 that uses the Kalman gain K n , defined in Equation 51503 2. Block diagram of the GRAVITY fringe-tracking controller within a single integration time, denoted n.The parameter states, illustrated in green, are central to the fringe tracker.They represent the disturbance conditions (L n ) and actuator positions (X n ).The blue segment indicates the group-delay feedback applied to the disturbance state, the red segment denotes the phase-delay feedback on L n , and the black segments represent the command send to the actuators.
The use of Φ CP arises from fact that the controller, which works in OPL, cannot account for a non-zero phase closure.At this step, the covariance P n on L n is propagated from Pn , using Equation 53.
4. The measured and expected group delays Ψ and Ψ are used to determine whether L n needs to be further modified by an increment of integer multiples of λ 0 .This is the group-delay tracking explained in Section 3.4.5.The vector L n contains our best estimate of the disturbance and is used to generate a command U n+1 in the process described by Equation 61.
6.The state vectors are propagated according to their respective model, The covariance matrix P n on L n is also propagated into Pn+1 using After this propagation, the next iteration can start.

Hardware implementation
In November 2022, a substantial upgrade of the GRAVITY fringe-tracker hardware was carried out to increase the computing power, as outlined by Abuter et al. (2016).As a result, the computing time was markedly reduced, opening up the possibility of implementing the updated and more complicated control algorithm.This new hardware setup is depicted in Figure 3.The upgraded real-time hardware of the controller, termed 528 wgvfft, is now based on a Linux workstation.The application 529 cycle is initiated by the arrival of detector data frames from the 530 ESO New General detector Control (NGC) workstation, facili-531 tated through an sFPDP communication link.This application is 532 executed on dedicated cores, with a portion of the RAM isolated 533 from the Linux kernel, ensuring efficient access by the reflective 534 memory (PCIE-5565PIORC) via direct memory access (DMA).535 The Kalman filter parameters (A L , Q, and F) are recalcu-536 lated every 5 seconds by another Linux workstation (wgvkalm).537 This workstation is also fitted with a PCIE reflective memory 538 card, which allows it to record the OPDs calculated by wgvfft 539 through the VLTI reflective memory network (RMN) ring.This 540 RMN ring is also used to transfer the Kalman filter parameters 541 back to wgvfft.Fig. 4. Observation of star HR 8799 using GRAVITY.The data are taken from a single DIT exposure on the spectrometer, spanning 100 seconds.The black curves denote the OPD, derived from the phase delay (Φ − Φ CP ).The blue curves illustrate observations from the group delay (Ψ − Ψ CP − Ψ ZeroGD ).Each panel represents a different baseline.At t 1 = 76.81 seconds, a noticeable fringe jump occurred on UT2.The effect of this jump across all baselines is depicted in the right panels.Impressively, the group delay identified and corrected the jump within 100 ms.The lower panel offers a magnified perspective of the baseline between UT2 and UT4, and the predictions from the controller state Φ and Ψ are overlaid.Notably, at t 2 = 76.9 s, all predictions shifted by 2200 nm, marking the detection of the jump.To shift the OPD by one λ, the phase delay took 3 DITs (3.3 ms), while the group delay took 150 DITs (165 ms), which is attributed to the smoothing length of the observable.
An optional external machine, also connected to the RMN  ).The atmospheric conditions were average, with a coherence time of 4 ms and a seeing of 0.8 .The planet was at a separation of 440 mas, and we observed using the off-axis dual-field mode of the fringe tracker.We used the roof mirror to split the field, so that all the flux from the star was injected in the fringe tracker fibre and all the flux of the planet in the science fibre.
Because the exoplanets are faint, these observations require extended integration times on the science channel, specifically, DITs of 100 s in this example.To optimise contrast, the fringe tracker was instructed to remain on the same fringe for the entire 100 s duration.Figure 4 shows that the fringe tracker was able to properly track the fringes on all six baselines.The residual OPD of this constant tracking is plotted in the upper left panel of Figure 4. HR 8799 is a bright star with a magnitude of 5.24 in K band and is no challenging target for the fringe tracker, which easily tracks these fringes at an interferometric S/N of ∼ 45.The power spectrum of the OPD residuals is shown in Figure 5 and in Appendix.B.
Over a 100-second integration period, the phase-delay controller consistently tracked constant OPD values, maintaining a standard deviation of 150 nm in the residual OPDs.Intriguingly, the group-delay residuals are notably higher, but they manifest at lower frequencies and vary over timescales of several seconds.The peak-to-peak amplitude of these variations remains below 1000 nm.As a result, the group-delay controller perceived no need for fringe adjustment and avoids any false triggers for a phase-jump detection.Consequently, the phase-delay fringe tracker operated smoothly and successfully maintained its tracking in the same phase.
Nonetheless, sporadic events can sometimes prompt the fringe tracker to transition abruptly from tracking one fringe to the next a full wavelength, λ 0 , apart.This phenomenon, unnoticed by the phase delay, is referred to as a fringe jump.A quintessential instance of this can be observed in Figure 4.This  This example perfectly illustrates the behaviour of the group-623 delay controller and shows that it required a span of 90 ms to de-624 tect the fringe jump.This interval represents the response time of 625 the group-delay control loop and is dictated by the group-delay 626 smoothing length (the 150 DITs).This setting represents a bal-627 ance between responsiveness and the potential for false positives 628 due to noise3 .As illustrated in the top right panels of Figure 4, 629 it is challenging to predict this noise because it does not appear 630 to be purely random.The value of 150 retained in the algorithm 631 has a mostly empirical basis.In Paper 1 we discussed the OPD residuals observed in 2019, 635 noting a significant dependence on the coherence time, τ 0 .Dur-636 ing unfavourable conditions (τ 0 < 3 mas), the values typically 637 exceeded 380 nm.Under average conditions (3 mas< τ 0 < 638 7 mas), these values ranged around 150 nm (ATs) or 250 nm 639 (UTs), and under optimal conditions (τ 0 > 7 mas), the UTs observed values down to 220 nm.We hypothesised that the inability of the UTs to reach residuals below 220 nm even in the best atmospheric conditions was due to vibrations.
Over the past four years, significant efforts have been made to mitigate these vibrations.Alongside the upgraded fringe tracker, these efforts have resulted in notable improvements.For the UTs, vibrations are predominant, exhibiting significant components at high frequencies, such as 20 nm vibrations at 300 Hz and 40 nm at 150 Hz.The control system strives to cancel these out, achieving partial success.However, it inadvertently reintroduces noise at other frequencies, resulting in no substantial improvement in the OPD residuals down to 50 Hz.This is evident in the cumulative spectrum, where the red and black curves exhibit similar amplitudes from 50 Hz upwards.Below 50 Hz, the fringe tracker is highly effective, cancelling even broad-frequency vibrations around 40 Hz.For the UTs, the most detrimental range is between 50 and 100 Hz, where a forest of spectral lines challenges the controller's ability to correct them, contributing to the observed ≈ 150 nm of residual.Potentially, a faster response time in the control loop could further reduce residuals within this range.
On the ATs, vibrations are not the dominant factor in the residuals, which are characterised by a white-noise pattern that aligns with the theoretically expected signal-to-noise ratio.For HR 7672, the dataset presented in Figure 5, the interferometric S/N is around 8, equating to a white-noise level of 60 nm.The residuals for different baselines range between 71 and 93 nm, which means that the system OPD residuals are close to the photon-and background-noise limitation.In this scenario, we observed minimal vibrations that did not degrade the interferometric S/N.This behaviour of the ATs is also substantiated statistically, as demonstrated by the dataset present in the ESO archive on datasets with a low signal-to-noise ratio (S/N below 3). Figure 6 shows the OPD residuals for all calibrators observed from June to August 2023.To expand the dataset, additional observations within the ExoGRAVITY large programme during this period were included.The upper panel of Figure 6 indicates that at lower fluxes for the ATs, the GRAVITY fringe tracker closely aligns with the theoretical phase error that arises from noise alone, indicating a reach towards the theoretical limit of the signal-to-noise ratio.However, when the signal-to-noise ratio surpasses 3 for both UTs and ATs, the performance begins to diverge from this limit, potentially due to piston noise from external factors such as the atmosphere.
In conclusion, the AT residuals at the 20%, 50%, and 80% percentiles are 95 nm, 120 nm, and 170 nm (1σ), respectively, and for the UTs, the corresponding values are 135 nm, 150 nm, and 185 nm.These values are derived from the histograms in the lower panels of Figure 6.The dependence on the coherence time for the performance is reduced.An optimal performance can now be achieved even at a low τ 0 of 2 ms.

Areas of improvement
At least three key areas require further attention to enhance the capabilities of fringe tracking: -Reducing the OPD residuals at high S/N 701 -Enhancing the signal-to-noise ratio for a given photon count 702 -Boosting the sensitivity by operating at slower speeds 703 OPD residuals.The critical question is whether OPD resid-704 uals can be further reduced at high signal-to-noise ratios.There 705 appears to be potential for improvement on the UTs, as their per-706 formance despite notable advances still lags that of the ATs.If 707 we ignore injection-related issues, the atmospheric piston impact 708 on an 8 m telescope should in theory be lower than on a 1.8 meter 709 telescope (Conan et al. 1995).Because the baselines in the as-710 trometric configuration used in our AT observations of HR 7672 711 are very similar to the UT baselines, the fringe tracker should 712 have achieved better results on the UTs and it should be possible 713 to reduce the OPD residuals below 80 nm on the UTs, with an 714 ultimate atmospheric limitation of 5 nm according to Courtney-715 Barrer et al. (2022).The discrepancy in performance might be 716 attributed to instrumental vibrations, which are particularly high 717 above 50 Hz (Figure 5).The control loop struggle to correct for 718 this although they seem to be properly captured by the model.719 Amplitude fluctuations induced by the performance of the adap-720 tive optics (AO), with lower Strehl ratios observed on the UTs, 721 are another major factor to take into account.A better control of 722 the vibrations, as well as a better AO correction, will certainly 723 help us to further decrease the OPD residuals on the UTs.

724
The signal-to-noise ratio Another question worthy of inter-725 est is whether the measured interferometric S/N matches the ex-726 pectations given the magnitude of our targets.The observations 727 of the bright target HR 8799 in July 2023 yielded an event rate 728 of n e − /s/UT = 10 7 photons per telescope per second, suggesting 729 a combined transmission for GRAVITY and VLTI of about 1%.730 The photometric S/N can be calculated as Here, the factor of 4 arises from the use of four telescopes, and γ 732 represents the efficiency of the recombination architecture, with 733 lower values being more desirable.At minimum, γ is inher-734 ently limited by the number of degrees of freedom in the sys-735 tem.Given the 16 degrees of freedom of our architecture (four 736 real fluxes and six complex coherent fluxes), an optimised beam 737 combiner could theoretically achieve a γ as low as 16.Assum-738 ing γ = 16, δt = 0.000854 and given the fact that the visibility is 739 close to one for this unresolved target, the interferometric S/N for 740 HR 8799 should be around 46, which indeed matches our mea-741 sured value of ∼ 45.This implies that the γ value for GRAVITY 742 is indeed close to 16, which is indicative of a well-designed com-743 biner.Further improvements will require innovative approaches 744 such as the hierarchical fringe tracker, which might also decrease 745 the γ value by reducing the system degrees of freedom (Petrov 746 et al. 2022).

747
The ultimate sensitivity.During the June commissioning 748 run, the exact sensitivity limit of the fringe tracker on the UTs 749 remained undetermined because the AO performance started to 750 degrade at magnitudes below 10 (see the UT data in Figure 6).751 The introduction of laser guide stars in GRAVITY+ is antici-752 pated to extend the AO limiting magnitude, potentially allow-753 ing us to ascertain the true sensitivity of the enhanced GRAV-754 ITY fringe tracker.When the signal-to-noise ratio limit at 1 kHz 755 is achieved, exploring a 100 Hz mode becomes a viable option 756 Therefore, the gain offered by this novel approach increases with the size of the array, thereby offering a compelling solution for future concepts such as the Planet Formation Imager (Monnier et al. 2018).
In the meantime, with this new version of the GRAVITY fringe tracker, ESO is now able to offer the interferometric community a facility fringe tracker that not only delivers a competitive performance, but also ensures seamless integration within the VLTI environment.This new fringe tracker is already in use with MATISSE (the GRA4MAT mode, described in Woillez et al. in prep.).Beyond this, it could also accommodate visitor instruments, becoming a useful tool for a much larger community.
Acknowledgements.The authors wish to express their gratitude to the referee, F. Cassaing for an excellent report, which contained both useful suggestions to improve the paper, corrections of important mistakes, and ideas for future work.This ensures the stationnarity of our model since 23 l=0 g l, j,k = 0.
The propagation matrices representing the AR model are then

136
uncertainty on the OPD estimate, σ j,k .This uncertainty can be 137 computed by first introducing the interferometric S/N, defined as 138 the ratio of the modulus of the coherent flux and the square root 139 of its variance.This variance is approximated as the half sum of 140 the variance of the real and imaginary parts of the coherent flux, 141 and the S/N is averaged over three DITs 1 , 142

145
the coherent flux are estimated for each frame from the back-146 ground, detector, and photon noises.Using these OPD noise es-147 timators, and under the assumption that each baseline noise is 148 uncorrelated, we also write the covariance matrix W on Φ, Γ is estimated from the variance on the individ-150 ual pixels q j , multiplied by the P2VM and P2VM T matrix, as 151 described in Section 3.4 of Paper 1. Compared to Paper 1, where 152 the σ j,k were computed coherently on five DITs, the new uncer-153 tainties are now computed on individual DITs and then averaged 154 over only three DITs.This change was motivated by the use of an 155 adaptive gain in the new control algorithm, which strongly ben-156 efits from a better estimate of the instantaneous signal-to-noise challenges of building a fringe tracker 160 our propagation model always guarantees that the condition 279 given by Equation 39 is fulfilled.Physically, this stems from the 280 fact that our propagation model really is the conversion of an 281 OPD model into OPL space, where we used the freedom given 282 by the under-constrained nature of the conversion to explicitly 283 guarantee that the average piston is always 0,through the defini-284 tion of M + in Equation 31.Secondly, the presence of L with a 285 hat in Equation 42 emphasises the fact that this is only a model-286 propagated estimate of L. There is no hat for X in Equation 23, 287 since we consider our piezo model to be perfect (or at least much 288 better than our atmospheric model).289 Equation 42 is linear, and the covariance matrix on L n is 290 propagated accordingly, using an additional 600 × 600 matrix 291 Q, which represents the process noise, 292 Pn+1 = A L • P n • A L T + Q.
345fectly known, the difference between the actual measurement 346 and the expected measurement Φn = H φ • ( Ln + X n ) is directly re-347 lated to the difference between the actual state of the disturbance 348 L n and its estimate Ln , 349Φ n − Φn = H φ • L n − Ln mod λ 0 ,(50)and ignoring the modulo, it is therefore tempting to simply update the state vector according to L n = Ln + H φ + • ( Φn − Φ).This 351 would reinject all the noise from the measurement into the con-352 trol loop, however, without any consideration for the respective 353 levels of confidence of the measurement and the prediction from 354 the model propagation.355 These levels of confidence are represented by the matrices Pn 356 and W n , which can therefore be used to calculate a Kalman gain 357 that properly weights the model prediction and the measurement 358 (Nowak 2019, section 4.3, for a proper derivation of this gain), 359 364 closure-phase on the measurement error.It is crucial to under-365 stand that because the model is in OPL space, it cannot account 366 for any non-zero closure phase.This closure-phase issue was al-367 ready present in the first version of the fringe tracker, and we 368 refer to Section 4.2 of Paper 1 for a detailed explanation.We

Fig. 3 .
Fig.3.Hardware architecture of the GRAVITY fringe tracker.The realtime controller operates within the wgvfft Linux workstation.Another Linux workstation, wgvkalm, is dedicated to calculating the Kalman filter parameters based on the observed OPDs.A Motorola CPU in the lgvttp workstation integrates the piston information with tip-tilt sensor data to accurately adjust the mirrors mounted on piezoelectric actuators in the tip-tilt piston (TTP).Additionally, an external workstation can provide the fringe tracker with piston setpoints and a mobility flag, which influences the controller behaviour.
543 ring, can be used by another instrument to control the fringe 544 tracker in real time.A workstation like this would have the possi-545 bility to send to wgvfft fringe-tracking setpoints (the U setpoint ).546 It can also be used to send the mobility flag introduced after 547 Equation 54. 548 The piston-correction values calculated by the wgvfft work-549 station are transmitted to another workstation named lgvttp us-550 ing the RMN ring network.The lgvttp workstation operates 551 with a Motorola CPU and is built around an mv6100 single-552 board computer, incorporating a VME bus for system commu-553 nication.There, they are combined with external measurements 554 of tip-tilt to form a comprehensive correction signal.This sig-555 nal in analog form is then sent out to control four active mirrors 556 mounted on piezoelectric tip-tilt platforms provided by Physik 557 Instrumente.558 The commissioning of the updated hardware was conducted 559 in late 2022 as part of the GRAVITY+ upgrade (Eisenhauer 560 2019; Gravity+ Collaboration et al. 2022) and was promptly 561 made available to the scientific community.We could enable 562 the white-light fringe tracking a few months later, allowing the 563 GRA4MAT mode of MATISSE (Woillez et al. in prep.).Simi-564 larly, the new Kalman filter state model, introduced in June 2023, 565 was immediately offered to the community upon commission-566 ing.
Performance of the group-delay control loop: Analysis of a fringe jump On 2 July 2023, we observed the exoplanet HR 8799 e during a scientific run of the ExoGRAVITY large programme (ESO LP 1104.C-0651

Fig. 5 .
Fig. 5. Spectra of the residual OPD (shown in black) and the reconstructed perturbation (H φ • L, depicted in red) for two observations.The upper panels present data on HR 8799 for one baseline from the UTs, corresponding to the data shown in Figure 4.The standard deviation of the OPD is 151 nm.The panel below shows a dataset from the ATs for HR 7672 with an OPD standard deviation of 86 nm.The cumulative spectrum for these two dataset are also shown.The data for all baselines are presented in Appendix B.
fringe jump manifested itself at t 1 = 76.81 seconds across all baselines related to UT2.The lower panel of the figure gives the phase delay (Φ − Φ CP ), represented by solid black curves.This phase delay has already accounted for the closure phase vector, as outlined in Equation (10).The dashed black curve displays the estimation Φ derived from the state parameters.Surrounding t 1 , the OPD fluctuates, but the Kalman filter overlooks the phase jump, and the λ 0 deviation in the OPD is not registered by Φ.This jump manifests itself in the group delay, represented by the solid blue curve in the figure.Owing to its smoothing over 150 DITs, the group delay gradually moves to a value close to −λ/2 at t 2 = 76.9 .At this point, all the UT2-related OPL state parameters of the phase control loop undergo a shift by λ 0 , induced by the group-delay controller.This is evidenced in the figure by the abrupt λ 0 shift of the phase-and group-delay state predictions ( Φ and Ψ).

Fig. 6 .
Fig. 6.OPD residuals.Top panel: OPD residuals plotted against the signal-to-noise ratio per baseline for all data collected from June to August 2023.The orange dots represent AT observations, and the blue dots denote UT observations.The dotted curve represents the theoretical limitation imposed by the measurement noise (λ/2πσ j,k ).The vertical dashed lines indicate the signal-to-noise ratio observed with the UTs of a star at a given star magnitude.It should be shifted by about 2.5 magnitude for the ATs.Bottom panels: Histogram of the same OPD residuals as in the upper panel for the ATs (left) and UTs (right).
Performance of the phase-delay control loop: OPD 633 residuals 634 Figure 5 displays the spectrum of the OPD residuals for two targets, HR 8799 and HR 7672, over a single representative baseline.The description of the dataset and spectrum for all baselines is presented in Appendix B.
GRAVITY was developed in a collaboration of the Max Planck Institute for Extraterrestrial Physics, LESIA of Paris Observatory, IPAG of Université Grenoble Alpes / CNRS, the Max Planck Institute for Astronomy, the University of Cologne, the Centro Multidisciplinar de Astrofisica from Lisbon and Porto, and the European Southern Observatory.This work used observations collected at the European Southern Observatory under ESO programme 1104.C-0651 and 109.238N.003.SL, JBL, and FM acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-21-CE31-0017 (project Ex-oVLTI).DD and RL have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement CoG -866070).PG acknowledges the financial support provided by FCT/Portugal through grants PTDC/FIS-AST/7002/2020 and UIDB/00099/2020.SL would like to warmly thank Sylvain Rousseau for spotting many typographical errors in the equations of Paper 1.
  I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150   I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 I 150 influence of the disturbance within a given number of future 461 DITs.This number of DITs, d, is made to be adjusted to the 462 mean response speed of the fringe tracker.Practically, it is d = 2 463 at 909 Hz, and d = 1 at lower frequencies.Hence, F is calculated 464 such that The term F • L n cancels out the disturbance calculated by the fringe tracker.More precisely, this term is used to negate 2 U setpoint is called U modulation in Paper 1. the From these state vectors, we construct the total OPL at each 490 telescope, which is simply Ln + X n , and then calculate the 491 expected values for the phase and group delay ( Φ and Ψn ) 492 using a set of measurement matrices H φ and H ψ , given in 493 Equations 49 and 57, respectively, and which convert our 494 OPL-based vectors into our OPD-based measurements, Our state vec-483 tor L contains the path length at each telescope over a given 484 number N = 150 previous iterations.This estimate, which 485 also comes with a covariance matrix Pn , represents our best 486 knowledge of the state parameters.We also have a state vec-487 tor X n that stores the position of the piezo actuators over the 488 last 150 iterations.489 2.