# SPICA-FT: The new fringe tracker of the CHARA array

Cyril Pannetier  Laboratoire Lagrange, Université Côte d’Azur, Observatoire de
la Côte d’Azur, CNRS, Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex
4, France  Philippe Berio  Laboratoire Lagrange, Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Boulevard de l’Observatoire, CS 34229,
06304 Nice Cedex 4, France  Denis Mourard  Laboratoire Lagrange, Université
Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Boulevard de
l’Observatoire, CS 34229, 06304 Nice Cedex 4, France  Sylvain Rousseau
Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France  Fatme
Allouche  Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la
Côte d’Azur, CNRS, Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex 4,
France  Julien Dejonghe  Laboratoire Lagrange, Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Boulevard de l’Observatoire, CS 34229,
06304 Nice Cedex 4, France  Christophe Bailet  Laboratoire Lagrange,
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Boulevard de
l’Observatoire, CS 34229, 06304 Nice Cedex 4, France  Daniel Lecron
Laboratoire Lagrange, Université Côte d’Azur, Observatoire de la Côte d’Azur,
CNRS, Boulevard de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France
Frédéric Cassaing  ONERA, DOTA, Paris Saclay University, France  Jean-Baptiste
Le Bouquin  Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Karine
Perraut  Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  John D.
Monnier  University of Michigan, Ann Arbor, MI 48109, USA  Narsireddy Anugu
CHARA Array, Georgia State University, Atlanta, GA 30302, USA  Theo ten
Brummelaar  CHARA Array, Georgia State University, Atlanta, GA 30302, USA

###### Abstract

SPICA-FT is part of the CHARA/SPICA instrument which combines a visible 6T
fibered instrument (SPICA-VIS) with a H-band 6T fringe sensor. SPICA-FT is a
pairwise ABCD integrated optics combiner. The chip is installed in the MIRC-X
instrument. The MIRC-X spectrograph could be fed either by the classical 6T
fibered combiner or by the SPICA-FT integrated optics combiner. SPICA-FT also
integrates a dedicated fringe tracking software, called the opd-controller
communicating with the main delay line through a dedicated channel. We present
the design of the integrated optics chip, its implementation in MIRC-X and the
software architecture of the group-delay and phase-delay control loops. The
final integrated optics chip and the software have been fully characterized in
the laboratory. First on-sky tests of the integrated optics combiner began in
2020. We continue the on-sky tests of the whole system (combiner + software)
in Spring and Summer 2022. We present the main results, and we deduce the
preliminary performance of SPICA-FT.

###### keywords:

Interferometry, Fringe tracker

##  1 Introduction

Unlike VLTI with GRAVITY-FT[1] , the fringe tracker of the 4-beam spectro-
interferometer GRAVITY[2] , the CHARA array was not equipped with such an
instrument. On VLTI, the need for a fringe-tracker was in particular justified
by the scientific objective of GRAVITY to precisely measure the dynamical
processes around the central black hole of the Milky Way, Sgr A*[3] . In the
context of the SPICA instrument[4] , we developed the fringe tracker SPICA-FT.
It operates simultaneously with SPICA in order to push its sensitivity. Our
goal is to stabilize the fringes over 100–200 ms with an accuracy of the order
of 100 nm. Even if SPICA-FT was developed for SPICA, it could be used to
stabilize fringes for other beam combiners of CHARA. It could be considered as
the CHARA fringe tracker.

SPICA-FT is inspired from GRAVITY-FT. It recombines up to 6 telescopes in H
band with a pairwise ABCD encoding in an integrated optics (IO) chip. The IO
chip is installed in the MIRC-X instrument[5] already in operation at CHARA
and should provide a higher sensitivity[6] than the all-in-one optical setup.
Such integrated optics have first been proposed for astronomical purpose in
1999[7] . The chip output is placed at the entrance of the MIRC-X
spectrograph. SPICA-FT could be operated at several spectral resolution, the
lowest spectral resolution (R=22) giving the best sensitivity.

In addition to the optical module, we developed a dedicated fringe tracker
software. It is fully integrated in the MIRC-X control software. It is also
inspired from the GRAVITY fringe tracker software. It allows Group Delay or
Phase Delay tracking. The algorithm follows the formalism presented in Ref. 1.
Its architecture has been defined in such a way that it could be used with
several beam combiners: SPICA-FT of course but also with the standard MIRC-X
combiner or with the MYSTIC combiner[8] .

A first version of the SPICA-FT chip has been installed and tested on sky in
2020. We obtained fringes with 5 telescopes (one telescope was not available
at the time). Then we worked on the optimization of the main functions of the
IO Chip and on the development of the fringe tracker software. In May 2022, we
tested the final version of the software on sky with the standard MIRC-X
combiner. The final IO chip will be installed during Summer of 2022.

In this paper, we present the IO combiner and its performance in section 2,
the fringe tracker software in section 3 and the results obtained on sky in
May 2022 in section 4.

##  2 Beam Combiner

###  2.1 Design and Manufacturing of the IO chip

In October 2017, we consulted at least five different companies for the
fabrication of an integrated optics chip able to combine, following the ABCD
scheme, 6 beams over the H band. It is a pairwise scheme (Figure 1) leading to
15 baselines and $15\times 4=60$ outputs. The initial scheme was based on the
design proposed by Pierre Labeye in his PhD manuscript [9]. We have defined
the following specifications:

  * •

operating wavelength range: 1.5 to 1.8 µm; 6 inputs, 15 baselines with ABCD
encoding,

  * •

single mode waveguides over the H band with a Numerical Aperture (at 1/e²)
between 0.13 and 0.15,

  * •

Phase shifts 90°+-10° over the spectral band,

  * •

Throughput larger than 60% over the H band,

  * •

Contrast level larger than 95% in polarized light,

  * •

Flux balancing between two entrance waveguides with a tolerance of 15%, Flux
cross-talk below 0.5%.

Figure 1: Design (VLC-OCA) of the SPICA-FT IO beam combiner. Each beam is
divided in 5 equal parts thanks to a smart succession of 60/40 (3), 50/50 (2),
and 66/33 (4) couplers. The 15 possible pairs are combined inside a ABCD
circuit (1), sampling the fringes with fixed phase shifts of $0$, $\pi/2$,
$\pi$, and $3\pi/2$. This permits the instantaneous measurements of the phase-
delay. Optical paths are equalized to avoid any bias in closure phase
measurements.

In December 2017, we started a feasibility study with the VLC Photonics
company on the technology of Planar Lighwave Circuit, with 1.8% of difference
of refractive index between the core guiding structure and the cladding layers
(technology of doped silica). The detailed study was then started in December
2018 and we received our first chips in May 2019. Unfortunately many of these
chips were corrupted by large defaults during the manufacturing process (large
bubbles and poor polishing of the entrance and exit faces). After analysis, we
identified additional unbalances in the 66/33 and in the 50/50 splitter
functions. However in terms of transmission (55%) and in terms of phase shift
quality the best chip was very close to the specifications.

A second iteration for the manufacturing process was done after a series of
meetings with the company and the foundry to improve the cleaning process
before the fabrication of our large chips (82x35$mm$) whereas the foundry was
mainly manufacturing a large number of small chips where defaults just lead to
rejection of a few percentage of the fabrication. In our case of fabrication
of a small number of chips (considering the cost and the size of a wafer) the
probability of pollution should be reduced as much as possible. With this in
mind, the foundry fabricated a second series of chips where the splitter
functions have been redesigned. This second series was of much better quality
but we ended with still a poor flux balancing because of a residual default in
the 66/33 splitter function. Moreover our characterization demonstrated that
the design made by the company was not exempt of internal closure phase. This
second generation was tested on sky in January 2020. After new exchanges with
the VLC Photonics company, we obtained, with the third generation received in
October 2021, excellent results with low internal closure phases, good
balancing, and overall a good match with our specifications.

The beams coming from the telescope are injected into the IO beam combiner
thanks to optical fibers. We selected polarization maintaining single mode
fibers (HB1000 6/125). The fibers were assembled by Leukos company into a
silicon V-groove array. Slow polarization axes are guaranteed by Leukos to be
perpendicular to the main axis of the front face of the V-groove to within
$\pm 3\deg$. The fibers were then length-matched ($\pm$ 0.5 mm) and
connectorized (FC/PC). The V-groove is then glued to the entrance of the IO
beam combiner. During the gluing process, we fed the fibers with a source and
we measured the flux at the IO beam combiner outputs to check the matching
between the fibre core and IO waveguides. When the matching is maximized, the
glue is UV polymerized to “freeze” the adjustment.

At the output of the IO beam combiner, we glued a micro lens array (MLA). It
has been specified in order to feed the MIRCx spectrograph with beams with the
correct f-number. This fused silica MLA was manufactured by Advanced
Microoptic Systems GmbH company. It is a 1D-array of refractive positive
microlenses with a pitch of 120 µm, a lens diameter of 120 µm, a lens clear
aperture of 100 µm and a curvature radius of 130 µm. The MLA thickness is 0.3
mm and it is anti-reflection coated on both sides.
Figure 2 shows a picture of the whole V-groove/chip/MLA inside its mount.

Figure 2: Picture of the whole V-groove/chip/MLA. The V-groove is on the left
of the IO beam combiner and the MLA is on the right.

###  2.2 Performance

We used laboratory testbenchs to check the main characteristics of all
manufactured IO chips. The global transmission has been measured on a
dedicated testbench in Grenoble while we developed a 6T fringe tracker
testbench in Nice in order to characterize the interferometric properties of
the IO chips.

First, we measured the global throughput of all the manufactured assemblies.
We found an average throughput at 1.62 µm higher than 55% which is close to
our specifications.

To characterize the interferometric properties of the IO chip, we used the
P2VM formalism[10]. The P2VM is measured following the procedure described in
Lapeyrere et al 2014[11]. The P2VM characterizes the relations of the
photometry, the coherence, and the phase between the 6 inputs and the 60
outputs of the IO beam-combiner. The $\kappa\\_$matrix contains the
contribution of each beam to each fringe pattern. It could be extracted from
the P2VM. In a perfect case, a fifth of the flux of each beam contributes to
each fringe pattern. With our best chip, we found a contribution of $20\pm
3\%$ instead of $20\%$ and we measured an averaged cross-talk of $-0.1\pm
0.1\%$. The corresponding $\kappa\\_$matrix is presented in Table 1.

Table 1: $\kappa\\_$matrix: contribution (in %) of each beam to each fringe pattern | Beam 1 | Beam 2 | Beam 3 | Beam 4 | Beam 5 | Beam 6
---|---|---|---|---|---|---
Base 12 | 19.3 | 22.7 | 0.0 | 0.2 | 0.1 | 0.1
Base 13 | 22.7 | -0.2 | 17.4 | 0.1 | 0.1 | 0.1
Base 23 | -0.1 | 22.7 | 23.1 | -0.3 | 0.0 | 0.0
Base 24 | 0.0 | 20.3 | -0.3 | 18.8 | -0.3 | -0.1
Base 14 | 17.6 | -0.1 | -0.2 | 18.0 | -0.3 | -0.3
Base 15 | 18.2 | 0.0 | -0.2 | -0.1 | 16.3 | -0.3
Base 25 | -0.1 | 16.0 | -0.1 | 0.0 | 16.7 | -0.1
Base 16 | 22.5 | -0.1 | -0.2 | -0.3 | -0.1 | 21.1
Base 26 | -0.1 | 19.9 | -0.1 | -0.4 | -0.2 | 19.2
Base 36 | 0.0 | -0.1 | 18.4 | -0.1 | -0.3 | 16.5
Base 34 | 0.0 | 0.0 | 19.2 | 16.9 | -0.1 | -0.2
Base 35 | 0.0 | -0.1 | 23.2 | 0.0 | 21.3 | -0.2
Base 45 | 0.0 | 0.0 | -0.1 | 25.1 | 23.1 | -0.1
Base 46 | 0.0 | 0.0 | 0.0 | 22.2 | -0.1 | 27.6
Base 56 | 0.0 | 0.0 | 0.0 | 0.0 | 23.7 | 16.6

The P2VM provides also information on the instrumental visibility and on the
phase shifts between the ABCD outputs. We found an instrumental visibility
(averaged over 15 baselines and over 27 spectral channels between 1.45 and
1.65 µm) of $74\pm 13\%$. It should be noted that these visibility
measurements were done in natural light and without any compensation of the
birefringence of the fibers. This last effect is corrected when the component
is installed inside MIRCx. For all baselines and all the wavelengths, the ABCD
outputs should be in quadrature (i.e. phase differences around $90\deg$). We
found phases (averaged over 15 baselines and over 27 spectral channels between
1.45 and 1.65 µm) of output B, C and D with respect to output A equal to
$84.1\pm 16.0\deg$, $181.3\pm 2.7\deg$ and $264.4\pm 16.1\deg$ respectively.
As an example, Figure 3 shows the instrumental contrast and the phase shifts
of the BCD outputs versus the A output for the Baseline 26.

Figure 3: Instrumental Contrast and Phase Shifts for the baseline 16.

###  2.3 Installation in MIRCX

The SPICA-FT IO beam combiner is installed at the entrance of the MIRCx
spectrograph. In figure 4, the MIRCx detector is represented on the left, the
standard MIRCx beam combiner is on the bottom right. The spectrograph is
located between these two modules (dispersive optics wheel and collimator). A
removable folding mirror could be used to feed the spectrograph with the
SPICA-FT beam combiner (pink module in figure 4). The optical fibers of the IO
beam combiner could be connected in the MIRCx injection modules (represented
on top of figure 4) instead of the optical fibers of the standard MIRCx beam
combiner. In the future, we plan to develop a new injection module which will
allow to keep the optical fibers of both combiners connected. This will reduce
the risks when handling the fibers.

Figure 4: 3D view of the MIRCx focal instrument with the SPICA-FT IO beam
combiner (in pink).

##  3 Fringe Tracker Software

###  3.1 Software Architecture

The SPICA-FT software follows the CHARA compliant architecture, based on
client/server architecture and GNOME ToolKit (GTK) Graphical User Interfaces
(GUIs). It runs on the same machine (Linux Xubuntu operating system) than the
MIRCx control command Software. The credone_control_server (MIRCx software)
reads frames from the camera and writes them to a shared memory. The
mircx_server reads the frames from this shared memory and performs the real-
time image processing. It writes the photometry of each beam, the correlated
flux of each baseline and the associated errors into another shared memory.

The SPICA-FT software is fed by this last shared memory. For each frame, the
first process, called phase_sensor, reads the shared memory and computes the
Phase Delay (PD), the Group Delay (GD), the PD SNR and the closure phases.
Then these computed quantities are sent to the second process, called
opd_controller, which handles the fringe tracker state machine and computes
the commands to be sent to the CHARA delay lines. The opd_controller follows
the algorithm described in next subsection.

###  3.2 Algorithm

The algorithm of SPICA-FT is mainly an adaptation of GRAVITY’s algorithm
presented in Lacour et al 2019[1]. It is fed with the 15 complex coherent flux
$\Gamma_{ij}$ of baselines $ij$ and the 6 photometries $F_{i}$ of telescopes
$i$ estimated by the fringe sensor. It comes either from the standard MIRC-X
combiner, the pairwise IO combiner of SPICA-FT, or MYSTIC combiner. These
coherent flux enable the computation of several observables necessary for the
generation of the control commands $u_{i}$ sent to the delay lines: the 15
phase-delays $\mathbf{\Phi}=\left(\Phi_{ij}\right)$ and group-delays
$\mathbf{\Psi}=\left(\Psi_{ij}\right)$, the 20 closure phases
$\mathbf{\Theta}=\left(\Theta_{ijk}\right)$ and an estimation of the variance
of the phase-delay estimator ($Var\left(\Phi_{ij}\right)$), derived from the
$\mathrm{SNR}\left(\Gamma_{ij}\right)$, for $(i,j,k)\in[\\![1,6]\\!]^{3}$ and
$i<j<k$.

###  3.3 Computation of the observables

Figure 5: Block-diagram of the SPICA-FT TRACK state. For clarity, we don’t
show the closure phase computation that feed the reference vectors
$\Phi_{ref}$ and $\Psi_{ref}$.

As shown on the bottom part of the block-diagram in figure 5, PD and GD
estimators are computed with different temporal frequencies. The phase-delays
at time $n$ are computed using only the last coherent flux of all spectral
channels,

$\Phi^{n}_{ij}=\arg{\left(\sum_{\lambda}^{N_{\lambda}}\Gamma^{\prime
n}_{ij,\lambda}\right)},$ (1)

artificially reconstructing a white fringe from all spectral channels of the
spectrograph. On the other hand, the group-delays are computed using the
averaged coherent flux of each spectral channel on the $N_{gd}$ last frames
corrected from the instantaneous phase-delays

$\Gamma^{\prime\prime
n}_{ij,\lambda}=\left\langle\Gamma^{\prime}_{ij,\lambda}\exp{\left(-i\Phi_{ij}\right)}\right\rangle_{N_{gd}}$
(2)

and computing the phase difference of these averaged phasors between all
neighbouring spectral channels:

$\Psi^{n}_{ij}=\arg\left(\sum_{\lambda=1}^{N_{\lambda}-1}\Gamma^{\prime\prime
n}_{ij,\lambda+1}\Gamma^{\prime\prime n*}_{ij,\lambda}\right).$ (3)

$\mathbf{\Psi^{n}_{ij}}$ is not exactly the group-delays but only the phase
differences proportional to the physical group-delays $\mathbf{GD}$ with the
relation

$\mathbf{GD}=\dfrac{\bar{\lambda}R}{2\pi}\cdot\mathbf{\Psi},$ (4)

when expressed in microns, where $R$ and $\bar{\lambda}$ account for the
spectral resolution and the mean wavelength of the instrument. From this, we
see that the physical group-delays are defined modulo $\bar{\lambda}R$, i.e. a
horizon $R$ times wider than the phase-delays which are defined, in microns,
modulo $\bar{\lambda}$. The number of frames used for computing the group-
delays must both increase enough the low SNR of the instantaneous group-delay
estimator and remain shorter than the temporal scale of the drift of the
fringe packet it compensates for. Acccording to Lawson et al 2000 [12], the GD
estimator is expected to be 20 times noisier than PD estimator, suggesting an
optimal averaging length $N_{gd}=20$ which corresponds to 80 ms at the fastest
frame rate.

The closure phases of $\mathbf{\Phi}$ and $\mathbf{\Psi}$, expected to be
independent from atmosphere, are computed on a longer temporal scale
(typically $N_{cp}=300~{}frames$) to maximise their SNR and because they are
expected to vary more slowly than $\mathbf{GD}$ and $\mathbf{PD}$ over time:

$\Theta^{PD}_{ijk}=\arg\left(\left\langle\sum_{\lambda=1}^{N_{\lambda}}\Gamma^{\prime}_{ij,\lambda}\sum_{\lambda=1}^{N_{\lambda}}\Gamma^{\prime}_{jk,\lambda}\sum_{\lambda=1}^{N_{\lambda}}\Gamma^{\prime*}_{ik,\lambda}\right\rangle_{N_{cp}}\right)$
(5)

and

$\Theta^{GD}_{ijk}=\arg\left(\left\langle\Gamma^{(3)}_{ij}\Gamma^{(3)}_{jk}\Gamma^{(3)*}_{ik}\right\rangle_{N_{cp}}\right)$
(6)

where

$\Gamma^{(3)}_{ij}=\sum_{\lambda=1}^{N_{\lambda}-1}\Gamma^{\prime\prime}_{ij,\lambda+1}\Gamma^{\prime\prime*}_{ij,\lambda}.$
(7)

An additional estimator at the heart of the SPICA-FT controller is the PD
variance estimator. It is necessary for the controller in order to perform
weighted least-square minimization of the PD and GD estimators and for its
state-machine. So far we tried two computations for the PD variance, the first
one following the definition of Equation 14 of Lacour et al 2019[1] and the
second following the expression:

$\sigma^{2}_{\phi}=\dfrac{\displaystyle\sum_{\lambda}\mathrm{Var}\left({\Re\left\\{\Gamma_{\lambda}\right\\}}\right)+\mathrm{Var}\left(\Im\left\\{\Gamma_{\lambda}\right\\}\right)}{2\displaystyle\sum_{\lambda}\left\langle\left|\Gamma_{\lambda}\right|^{2}\right\rangle_{5DIT}}$
(8)

which is more robust to PD jitters and thus seems better suited to fringe
detection as we will see in Section. 4.

###  3.4 Computation of the commands

The SPICA-FT fringe-tracking algorithm is based on the switch between two main
states, TRACK and RELOCK. The latter is triggered each time flux is lost on
one of the available telescopes. At the beginning of each observation, a
specific third state, named SCAN, is triggered with the role of scanning the
OPDs of all baselines to set the SNR thresholds parameters and find the
fringes. This state-machine is schematized in figure 6.

During the permanent regime, TRACK and RELOCK controls are generated at each
time $n$ such that the commands sent to the delay lines are

$\mathbf{u}^{n}=\mathbf{u}^{n}_{TRACK}+\mathbf{u}^{n}_{RELOCK}$ (9)

where both $\mathbf{u}^{n}_{TRACK}$ and $\mathbf{u}^{n}_{RELOCK}$ are null
when respectively no telescopes and all telescopes are cophased and both are
non-null when only a subset of telescopes are cophased.

Figure 6: State-machine of SPICA-FT.

####  3.4.1 The TRACK state

The control logic of the TRACK state is drawn as the upper block-diagram of
figure 5. Since it is an adaptation to six telescopes of the GRAVITY fringe-
tracking algorithm already explained in details in Lacour et al 2019[1], we
focus here only on the implementations that differ from the article or that
are of particular interest.

The definition of the reference vectors $\Phi_{ref}$ and $\Psi_{ref}$ is not
straightforward. The role of these vectors is to handle the closure phases due
to the instrument or to the astrophysical target. The closure phases make the
measured OPDs inconsistent with pure atmospheric pistons. If we don’t subtract
it a priori, they will thus spread over all OPDs when processing the least-
square minimisation at the next step and the final projection into piston-
space will be incorrect. For six telescopes, there are 20 closure phases of
which 10 are independent. The correction is done by choosing 10 independent
closure phases $\Theta_{ijk}$ and, for each one, subtracting it to the PD and
GD estimators of one of its baselines. A simple way to choose 10 independent
closure phases is to take all those which involve the same telescope. The
reference vectors play thus this role of subtraction on the suitable
baselines, using the slowly evolving estimated closure phases. To be robust to
cases where some telescopes are not available, making impossible the
measurement of the closure phases it belongs to, we defined six reference
vectors

$\displaystyle\mathbf{\Phi_{ref}}=\begin{pmatrix}0\\\ 0\\\ 0\\\ 0\\\ 0\\\
\Theta_{123}\\\ \Theta_{124}\\\ \Theta_{125}\\\ \Theta_{126}\\\
\Theta_{134}\\\ \Theta_{135}\\\ \Theta_{136}\\\ \Theta_{145}\\\
\Theta_{146}\\\ \Theta_{156}\\\ \end{pmatrix}$
$\displaystyle,\begin{pmatrix}0\\\ -\Theta_{123}\\\ -\Theta_{124}\\\
-\Theta_{125}\\\ -\Theta_{126}\\\ 0\\\ 0\\\ 0\\\ 0\\\ \Theta_{234}\\\
\Theta_{235}\\\ \Theta_{236}\\\ \Theta_{245}\\\ \Theta_{246}\\\
\Theta_{256}\\\ \end{pmatrix},\begin{pmatrix}\Theta_{123}\\\ 0\\\
-\Theta_{134}\\\ -\Theta_{135}\\\ -\Theta_{136}\\\ 0\\\ -\Theta_{234}\\\
-\Theta_{235}\\\ -\Theta_{236}\\\ 0\\\ 0\\\ 0\\\ \Theta_{345}\\\
\Theta_{346}\\\ \Theta_{356}\\\ \end{pmatrix},\begin{pmatrix}\Theta_{124}\\\
\Theta_{134}\\\ 0\\\ -\Theta_{145}\\\ -\Theta_{146}\\\ \Theta_{234}\\\ 0\\\
-\Theta_{245}\\\ -\Theta_{246}\\\ 0\\\ -\Theta_{345}\\\ -\Theta_{346}\\\ 0\\\
0\\\ \Theta_{456}\\\ \end{pmatrix},\begin{pmatrix}\Theta_{125}\\\
\Theta_{135}\\\ \Theta_{145}\\\ 0\\\ -\Theta_{156}\\\ \Theta_{235}\\\
\Theta_{245}\\\ 0\\\ -\Theta_{256}\\\ \Theta_{345}\\\ 0\\\ -\Theta_{356}\\\
0\\\ -\Theta_{456}\\\ 0\\\ \end{pmatrix},\begin{pmatrix}\Theta_{126}\\\
\Theta_{135}\\\ \Theta_{146}\\\ \Theta_{156}\\\ 0\\\ \Theta_{236}\\\
\Theta_{245}\\\ \Theta_{256}\\\ 0\\\ \Theta_{346}\\\ \Theta_{356}\\\ 0\\\
\Theta_{456}\\\ 0\\\ 0\\\ \end{pmatrix}$ (10)

suitable to the situations where respectively each of the telescope 1, 2, 3,
4, 5 and 6 is cophased with the array. The GD reference vector
$\mathbf{\Psi_{ref}}$ follows the same definition.

The PD and GD errors $\epsilon_{PD}$ and $\epsilon_{GD}$, compatible with
piston-space projection, are finally obtained from the difference between the
$\mathbf{\Phi}$ and $\mathbf{\Psi}$ estimators and their respective reference
vectors $\mathbf{\Phi_{ref}}$ and $\mathbf{\Psi_{ref}}$.

From the values of PD variances of all baselines provided by the first stage
of the fringe-tracker, we compute the matrices $\mathbb{I}^{15}_{PD}$ and
$\mathbb{I}^{15}_{GD}$ which play the roles of:

  * •

estimating, by a weighted least-square minimization, 15 new
$\epsilon^{\prime}_{PD}$ and $\epsilon^{\prime}_{GD}$ from the 15 PD and GD
errors computed by the first stage of the fringe-tracker: the baselines whose
SNR is lower than $\mathrm{SNR}_{GD}$ are not taken into account whereas the
others are basically weighted by their SNR. $SNR_{GD}$ is set by the operator
or determined by the state SCAN at the beginning of the observation.

  * •

filtrating the PD commands on pistons for which the variance is estimated
lower than a given threshold, meaning this command would have no physical
origin.

Then the GD errors go through an extra step that nulls the errors whose
absolute value is lower than a given threshold, currently fixed to $\lambda$,
to prevent the two controls PD (that, because of its wrapping over $2\pi$,
corrects only absolute errors lower than $\lambda/2$) and GD to send twice the
same command to the delay lines. These errors are then projected into piston-
space thanks to the pseudo-inverse of the matrix of projection from piston to
OPDs:

$\mathbf{M}=\setcounter{MaxMatrixCols}{15}\begin{pmatrix}1&1&1&1&1&0&0&0&0&0&0&0&0&0&0\\\
-1&0&0&0&0&1&1&1&1&0&0&0&0&0&0\\\ 0&-1&0&0&0&-1&0&0&0&1&1&1&0&0&0\\\
0&0&-1&0&0&0&-1&0&0&-1&0&0&1&1&0\\\ 0&0&0&-1&0&0&0&-1&0&0&-1&0&-1&0&1\\\
0&0&0&0&-1&0&0&0&-1&0&0&-1&0&-1&-1\end{pmatrix}^{T}$ (11)

to get the pistons errors

$\mathbf{\epsilon}_{p,PD}=\mathbf{M}^{\dagger}\cdot\mathbf{\epsilon}^{\prime}_{PD}$
(12)

and

$\mathbf{\epsilon}_{p,GD}=\mathbf{M}^{\dagger}\cdot\mathbf{\epsilon}^{\prime\prime}_{GD}$
(13)

Finally, at each time $n$, the controllers PD and GD are integrated following
the classical PI controller equations

$\mathbf{u}_{PD}^{n}=\mathbf{u}_{PD}^{n-1}+K_{PD}\cdot\mathbf{\epsilon}_{p,PD}^{n}$
(14)

and

$\mathbf{u}_{GD}^{n}=\mathbf{u}_{GD}^{n-1}+K_{GD}\cdot\mathbf{\epsilon}_{p,GD}^{n}$
(15)

before the GD control go again through an extra step that converts the piston
commands into an integer number of the mean wavelength $\bar{\lambda}$ of the
fringe-sensor to prevent from blurring the fringes.

Thus, the total command from the TRACK state is:

$\mathbf{u}^{n}_{TRACK}=\mathbf{u}^{n}_{GD}+\mathbf{u}^{n}_{PD}$ (16)

####  3.4.2 The RELOCK state

The fringe-tracker enters the RELOCK state when the SNR on all baselines
involving a given telescope are below the threshold $\mathrm{SNR}_{GD}$,
leading to the reduction of the rank of the matrix $\mathbb{I}^{15}_{GD}$ by
as much unities as lost telescopes, and remains below the threshold during at
least one second. An additional condition for triggering the RELOCK state is
that flux is present on the lost telescopes, to avoid looking for unavailable
telescopes. This state is of a major importance on SPICA-FT which is a fiber-
fed spectro-interferometer. Injection losses can occurs often, all the more so
as there are six fibers to feed with six different adaptive optic systems with
their own weaknesses.

It is worth mentioning that during the RELOCK state, the GD and PD control
loops keep generating $\mathbf{u}_{TRACK}$ commands for cophasing as much
telescopes as possible. When it starts, the RELOCK state launches a sawtooth
function that will be used to generate additional commands on the telescopes.
The RELOCK command sent to all telescopes is computed as follows:

$\mathbf{u}^{n}_{RELOCK}=\mathbb{K}^{6,n}\cdot\mathbf{v}^{T}\cdot u^{n}_{saw}$
(17)

where the multiplication of the vector velocities $\mathbf{v}^{T}$ by the
matrix

$\mathbb{K}^{6,n}=\mathbf{I}^{6}-\mathbb{I}^{6,n}_{GD}$ (18)

with
$\mathbb{I}^{6,n}_{GD}=\mathbf{M}\cdot\mathbb{I}^{15,n}_{GD}\cdot\mathbf{M}^{T}$,
works such that:

  * •

the components of the vector $\mathbf{v}$ of lost telescopes remains
unchanged,

  * •

the components of the vector $\mathbf{v}$ corresponding to telescopes
belonging to a same cophased group are added up together.

Doing so, the cophased telescopes are not disturbed by the RELOCK command
while the lost telescopes are scanning. However, this implementation works
only with specific velocity vectors $\mathbf{v}$ non redundant in a sense that
for any subset of cophased telescopes, the addition of its components never
equals the computed velocity of another subset (one or more) of the
complementary telescopes. This is achieved with the vector

$\mathbf{v}=\begin{pmatrix}-8.25&-7.25&-4.25&1.75&3.75&8.75\end{pmatrix}^{T}$
(19)

except for one very specific situation: when at the same time the telescopes
1, 3 and 4 are cophased together while the telescopes 2, 5 and 6 are also
cophased together. If this occurs, we only need to change the sign of the
velocity of one of the two groups to keep searching.

The sawtooth function triggered at the beginning of RELOCK state is designed
to scan around zero, alternating in both directions with a little overlap at
each change, while gradually moving away faster and faster. Its behaviour is
visible on figure 5 of Lacour et al 2019[1].

####  3.4.3 The SCAN state

The SCAN state is specific to the beginning of an observation. After the delay
lines are set up to the positions where the coherence of the telescopes are
expected according to the coordinates of the star and of the stellar
interferometer, they receive new commands

$\mathbf{u}_{SCAN}=\mathbf{v_{0}}\cdot u_{saw},$ (20)

where $u_{saw}$ is the same sawtooth function than for RELOCK state and

$\mathbf{v_{0}}=\begin{pmatrix}0&1&2&3&4&5\end{pmatrix}^{T}$ (21)

is a velocity vector chosen to scan all telescopes except the reference one
(here the first one) around the start positions while recording the minimal
SNR, corresponding to the absence of fringes, and maximal SNR, corresponding
to the center of the fringe packet. The positions where the null GD is found
between the reference telescope and the five others is also recorded in order
to send the telescopes to these positions at the end of the SCAN state.

For each baseline, the records of the minimal and maximal SNRs enable to set
the PD and GD thresholds $\mathrm{SNR}_{PD}$ and $\mathrm{SNR}_{GD}$, two
parameters of high importance for the RELOCK and TRACK states. These
thresholds can differ from one baseline to another since the visibilities
depend on the baselength and the injection conditions can be very different
depending on the telescope.

##  4 Results on Sky

###  4.1 Phase-tracking performance

On figure 7, we show a $\sim 38.5$ seconds sequence of phase-tracking with
four telescopes (E1, S1, S2, W1) that occurred during our last commissioning
run in May 2022. The tracked star is HD186155, a star of magnitude
$m_{H}\simeq 4.2$ and diameter $\o_{H}\simeq 0.58$ mas. The visibility of the
star is between 0.4 and 0.6 at 1.6 µm for all involved baselines but the
shortest one S1S2 which barely resolves it. During this sequence, SPICA-FT
reached PD residuals below 200 nm rms on the baselines of the triangle E1S2W1
and 250 nm rms on the other baselines. The GD estimator remains around 0 with
standard-deviation of about 400 nm for the triangle E1S2W1 and 800 nm on the
three other baselines. The PD estimator of non-tracked baselines are all
showing PD estimator rms between 420 and 460 nm.

Figure 7: Phase-tracking with E1S1S2W1 on the star HD186155, magnitude 4 and
diameter $\o_{H}\simeq 0.58$ mas leading to $\left|V_{H}\right|\simeq 0.4$ on
the longest baseline E1S1.

###  4.2 Limitations and expected improvements

####  4.2.1 Fringe jumps

Figure 8: Zoom on the phase-tracking sequence with E1S1S2W1.

Although the phase-delay residuals were low, we experienced during the fringe-
tracking more fringe jumps that we expect for. This is illustrated on figure 8
with a zoom on the phase-tracking sequence already shown in figure 7. Their
frequency compared to the dispersion evolution time scale is too high to be
explained by the normal functioning of the GD loop. Fringe jumps can occur
when the OPD moves of more than a wavelength, either very quickly compared to
the PD loop latency or during a loss of flux that prevents the PD loop to see
the shift. But the observing conditions were sufficiently good to put aside
this explanation, at least considering such a high frequency of fringe jump
events.

We thus looked more precisely at the behaviour of the GD reference vector,
appearing at the bottom of figure 8, which give the targets on all baselines
for the GD control loop and we saw that it is far noisier than the PD
reference vector. As a consequence, the implementation that we chose, putting
the component of the reference vector to zero as soon as its SNR is too low,
implies a lot of resets to zero which could explain such an instability for
the GD control loop. The plan now is to test a phase-tracking either without
GD reference vector or with a different implementation of it.

We believe though that a smarter implementation of the GD reference vector
will lead to a substantial reduction of the fringe jumps frequency.

####  4.2.2 SNR estimation and fringe detection

Another source of improvement of SPICA-FT will be on the estimation and usage
of the SNR of the measurements. We already defined in section 3.3 the two ways
of computing the SNR of the PD estimator. figure 9 shows the difference of
behaviour of these two estimators during a sequence when MIRC-X is group-
tracking only on the telescopes E1, W1 and W2. We plot on these figures the
curves concerning also the telescope S1 for comparison with the tracked
baselines. The fact that MIRC-X is only group-tracking implies that the
fringes are still moving a lot. But we were monitoring during the sequence
recording that the fringes were always present. The integration time was 20
ms, longer than the atmospheric coherence time at this moment ($\sim 10$ ms).
The estimator relying on the coherent integration often reaches very high
values, which is a signal for the presence of fringes. However, it also falls
down very often to values corresponding to the absence of fringes. The error
bars in the bar diagram show that the rms value of this estimator is very
high, making it poorly reliable to detect jittering fringes. Shortening the
integration time to 8 ms didn’t show improvement in this regard. On the other
side, the estimator summing up the modules of coherent flux shows lower values
in presence of fringes but also a lower standard deviation, which as a whole
seem to make it more suitable for detecting jittering fringes. For this
reason, we will keep this estimator for the moment.

(a) Squared SNR of phase-delay estimator averaged over 40 frames, using
Equation 14 of Lacour et al 2019[1].

(b) Squared SNR of phase-delay estimator averaged over 40 frames, using
Equation 8.

Figure 9: Comparison between the estimation of squared SNR relying on the
integration of coherent fluxes over 5 DIT and the one relying on the
incoherent addition of coherent flux modules tried during the commissioning of
May 2022. During the computation, MIRC-X was group-tracking the telescopes E1,
W1 and W2.

On figure 9(b), we also see that the estimation of the SNR on the non-tracked
baselines, where there is no fringe, can be rather different from a baseline
to another (0.47 on E1E2 and 0.82 on S1E2). This suggests that defining one
threshold per baseline will improve the state-machine of SPICA-FT.

##  5 Conclusion

We report on the first on-sky tests of the SPICA-FT fringe tracker installed
at the CHARA array. During an observing run in May 2022, we stabilized the
fringes of 4 telescopes with an OPD residuals around 200nm during several
minutes on several stars with magnitude between 4 and 4.5 in H band.

We identify several limitations in the fringe tracker software that could be
easily overcome in the near future. We would like to reduce substantially the
fringe jumps and reach OPD residuals of 100 nm with 6 telescopes. In addition
to the software improvements, we plan to install our IO beam combiner at the
entrance of the MIRCx spectrograph. Before this installation, we will develop
a new fiber injection module which will allow to switch easily between the
MIRCx and SPICA-FT fibers.

We think SPICA-FT (IO beam combiner + Software) will be ready by the end of
2022. In the next 6 months, we will derive its full performance in terms of
sensitivity and OPD residuals.

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