# Evaluating and Enhancing Candidate Clocking Systems for CHIME/FRB VLBI
Outriggers

Savannah Cary Department of Astronomy, Wellesley College, 106 Central Street,
Wellesley, MA 02481, USA  MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139, USA  Juan Mena-Parra MIT Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, 77 Massachusetts
Ave, Cambridge, MA 02139, USA  Calvin Leung MIT Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology, 77
Massachusetts Ave, Cambridge, MA 02139, USA  Department of Physics,
Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA
02139, USA  Kiyoshi Masui MIT Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, 77 Massachusetts Ave,
Cambridge, MA 02139, USA  Department of Physics, Massachusetts Institute of
Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA  J.F. Kaczmarek
Dominion Radio Astrophysical Observatory, Herzberg Research Centre for
Astronomy and Astrophysics, National Research Council Canada, PO Box 248,
Penticton, BC V2A 6J9, Canada  Tomas Cassanelli David A. Dunlap Department of
Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada  Dunlap Institute for Astronomy & Astrophysics,
University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada

###### Abstract

As the Canadian Hydrogen Intensity Mapping Experiment (CHIME) has become the
leading instrument for detecting Fast Radio Bursts (FRBs), CHIME/FRB
Outriggers will use very-long-baseline interferometry (VLBI) to localize FRBs
with milliarcsecond precision. The CHIME site uses a passive hydrogen maser
frequency standard in order to minimize localization errors due to clock
delay. However, not all outrigger stations will have access to a maser. This
report presents techniques used to evaluate clocks for use at outrigger sites
without a maser. More importantly, the resulting algorithm provides
calibration methods for clocks that do not initially meet the stability
requirements for VLBI, thus allowing CHIME/FRB Outriggers to remain true to
the goal of having milliarcsecond precision.

Radio astronomy (1338), Radio transient sources (2008), Radio pulsars (1353),
Astronomical instrumentation(799), Very long baseline interferometry (1769)

††journal: Research Notes of the AAS††software: AllanTools (Wallin et al.,
2018)

##  1 Introduction

To localize Fast Radio Bursts (FRBs, millisecond-long radio bursts originating
from distant galaxies, Lorimer et al., 2007), the CHIME/FRB Outriggers project
is using VLBI with telescopes situated across North America. VLBI localizes
its sources by recording the signals’ arrival times between two or more
telescopes and measuring the propagation delay. Imperfections in a VLBI
station’s internal clock leads to undesired contributions to the measured
delay; this report focuses on mitigating this timing offset (referred to as
“clock delay”). Outrigger stations will rely on well-localized, bright sources
suitable for phase calibration on large baselines (referred to as
calibrators), alongside a local frequency reference (i.e. clock) that is
stable over timescales for which there are no calibrators in the field of view
(FOV).

Our goal is to correct for unwanted delay due to clock errors, ultimately
creating a system where timing errors do not exceed 200 ps on timescales of at
least 1000 s (Mena-Parra et al., 2021 in prep.). We therefore have created a
data analysis pipeline to determine a clock’s performance against our
specification for CHIME/FRB Outriggers. Our pipeline handles measured and
simulated clock data.

In this note we present our methods for evaluating candidate clocking systems
for VLBI. Using measured data from experimental tests, we then analyze the
performance of the EndRun Technologies Meridian II US-Rb rubidium oscillator
and determine if this clock can be used at outrigger sites.

##  2 Evaluating Clock Performance

###  2.1 Providing a timestream

Testing clock performance requires a timestream that represents the system’s
timing errors. This algorithm utilizes real and mock timestreams, where the
mock timestreams are simulated based on Allan deviations provided by the clock
manufacturer. Because the instability of most clocks can be modeled as a sum
of power-law type noises (Thompson et al., 2017), it is assumed that the Allan
variance follows a linear combination of the following power-law type noises:
white phase modulation $(\sigma_{y}\propto\tau^{-1})$, white frequency
modulation $(\sigma_{y}\propto\tau^{-1/2})$, flicker frequency modulation
$(\sigma_{y}\propto\tau^{0})$, and random walk frequency modulation noise
$(\sigma_{y}\propto\tau^{1/2})$.

###  2.2 Interpolation Techniques

Calibrators are used to infer clock timing offsets. For our analysis, a
calibrator is a reliable delay measurement. We can run the recorded or
simulated measurement of clock delays through an interpolation algorithm that
simulates the presence and absence of calibrators in the FOV; the timestream
is masked for periods corresponding to when there is no observable calibrator.
The chosen clock must be stable on the timescales spanning between calibrator
observations.

To evaluate performance based on varying separation times between calibrators,
we interpolate between observations of the mock calibrators to estimate the
masked clock delay and compare to true delay. Two mock calibrators are placed
at a random section of the timestream with a separation time
$\tau_{\mathrm{{sep}}}$ between the end of the first transit and beginning of
the second. This is motivated by the fact that CHIME and the Outriggers are
stationary instruments, relying on Earth’s rotation to survey the sky.
Calibrators are observable for 540 s, which corresponds to the transit time
through the CHIME FOV at zenith. Data from these 540 s are “recorded” with a
chosen simulated integration period of $\tau_{\mathrm{int}}$, providing 540
s/$\tau_{\mathrm{int}}$ data points collected per calibrator. All clock delay
measurements within this span of $\tau_{\mathrm{int}}$ are averaged and
considered to be the recorded delays associated with the integration period.
The timestamp associated with each sample is taken to be the middle-time of
each integration. These data points, 540 s$/\tau_{\mathrm{int}}$ for each
calibrator, the basis for inferring the measurements of timing offsets of the
timestream.

Because calibrators are not infinitely bright, we expect uncertainty with each
measurement of clock delay. Varying degrees of calibration errors,
$\sigma_{\mathrm{cal}}$, are added to each data point via a normal
distribution to make the simulation as realistic as possible. The calibration
error is based off varying the calibrator’s signal-to-noise ratio (SNR). Other
noise contributions are the noise generated when injecting the clock signal
into the telescope’s correlator, $\sigma_{\mathrm{transfer}}$, and can be
calculated from the timestream itself. For simulated timestreams, this noise
is added.

In Fig. 1, the top left plot is an example of this masking method for
simulating two calibrators. We interpolate between calibrations using the best
fit line obtained from the collected calibration points.

For this clock, a linear fit provided the lowest RMS uncertainty. However, our
methods can handle other forms of fitting functions, such as cubic, quadratic,
and general cross validation smoothing splines (Craven & Wahba, 1978),
dependent on what is best for a given clock’s performance.

Each delay is weighted based on the inverse variance $1/\sigma^{2}$; $\sigma$
is determined from the calibration error and transfer noise and equals
$(\sigma_{\mathrm{cal}}^{2}+\sigma_{\mathrm{transfer}}^{2})^{1/2}$. The best
fit function is what would be used to account for clock delay when phasing the
array.

The process is repeated for several random sections of the timestream. The top
right plot of Fig. 1 has the resulting residuals for a single
$\tau_{\mathrm{sep}}$. Simulating the worst-case scenario, the RMS is
calculated using the maximum residual per iteration between the fitting
function and the timestream. The lower plot of Fig. 1 represents having run
multiple random interpolations across the timestream, per desired
$\tau_{\mathrm{sep}}$.

Figure 1: Simulated calibration procedure using measured timing offsets for a
rubidium clock. The top left graph is a single iteration of the process
described in Section 2.2 for a timestream with noise added according to an SNR
of 20. Two calibrators are simulated to be in the FOV for 540 s (shown in the
shaded area) with a separation time $\tau_{\mathrm{sep}}$ 1000 s. The green
diamonds are the unmasked data, with the horizontal error bars being the
integration period per calibrator. The vertical error bars represent the
calibration error and clock transfer noise
$(\sigma_{\mathrm{cal}}^{2}+\sigma_{\mathrm{transfer}}^{2})^{1/2}$. The upper
right plots the residuals from interpolating 10 out of 100 iterations for a
$\tau_{\mathrm{sep}}$ = 1000 s. The corresponding RMS error for this specific
instance is marked by the blue square in the bottom panel. Shown also is how
well we expect the clock to perform given different values of noise given via
SNRs of 10, 15, 20, and infinite. The horizontal dashed line represents the
desired threshold for accounting for clock delay: 200 ps. Plotted is also the
result of the calibration if no interpolation is applied, labeled “Free Run”.
By implementing this algorithm, at infinite SNR and 1000 s, the clocking
errors are reduced by over a factor of $\sim$3.

##  3 results from the rubidium oscillator

The results in Fig. 1 are from the rubidium oscillator, a clock now proved to
be more cost effective compared to a maser. Without use of our algorithm, at
infinite SNR the clock meets stability requirements for $\tau_{\mathrm{sep}}$
< 200 s; however, after applying our interpolation methods, the clock meets
our requirements at times $\geq$ 1000 s for SNR >10.

In conclusion, these methods make it possible to evaluate clocks for stations
without a maser for CHIME/FRB Outriggers. These methods also make it possible
to extend the stability of sub-optimal high-precision timing standards,
allowing for a more cost-effective solution to accounting for clock errors. In
the future, this analysis can be implemented where frequency standards are not
originally stable enough on their own, enabling the CHIME/FRB Outriggers VLBI
network to meet its strict timing and localization requirements.

## References

  * Craven & Wahba (1978) Craven, P., & Wahba, G. 1978, Numerische Mathematik, 31, 377, doi: 10.1007/BF01404567
  * Lorimer et al. (2007) Lorimer, D. R., Bailes, M., McLaughlin, M. A., Narkevic, D. J., & Crawford, F. 2007, Science, 318, 777, doi: 10.1126/science.1147532
  * Mena-Parra et al. (2021 in prep.) Mena-Parra, J., Leung, C., Cary, S., & Masui, K. 2021 in prep.
  * Thompson et al. (2017) Thompson, A. R., Moran, J. M., & Swenson Jr, G. W. 2017, Interferometry and Synthesis in Radio Astronomy, 3rd edn. (Springer, Cham)
  * Wallin et al. (2018) Wallin, A. E. E., Price, D. C., Carson, C. G., & Meynadier, F. 2018, allantools: Allan deviation calculation.  http://ascl.net/1804.021


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# Progress of the CHARA/SPICA project

Pannetier C  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, France  ONERA/DOTA, Université Paris Saclay, 92322
Châtillon, France  Mourard D  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, France  Berio P  Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France  Cassaing F
ONERA/DOTA, Université Paris Saclay, 92322 Châtillon, France  Allouche F
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, France  Anugu N  Steward Observatory, Department of Astronomy,
University of Arizona, Tucson, USA  University of Michigan, Ann Arbor, MI
48109, US  School of Physics and Astronomy, University of Exeter, Exeter,
Stocker Road, EX4 4QL, UK  Bailet C  Université Côte d’Azur, Observatoire de
la Côte d’Azur, CNRS, Laboratoire Lagrange, France  ten Brummelaar T  The
CHARA Array, Mount Wilson Observatory, Mount Wilson, CA 91023  Dejonghe J
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, France  Gies D  The CHARA Array, Mount Wilson Observatory, Mount
Wilson, CA 91023  Jocou L  Institut de Planetologie et d’Astrophysique de
Grenoble, Grenoble 38058, France  Kraus S  School of Physics and Astronomy,
University of Exeter, Exeter, Stocker Road, EX4 4QL, UK  Lacour S  LESIA,
Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Univ. Paris
Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
Lagarde S  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, France  Le Bouquin J.B  Institut de Planetologie et
d’Astrophysique de Grenoble, Grenoble 38058, France  Lecron D  Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France
Monnier J  University of Michigan, Ann Arbor, MI 48109, US  Nardetto N
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, France  Patru F  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, France  Perraut K  Institut de
Planetologie et d’Astrophysique de Grenoble, Grenoble 38058, France  Petrov R
Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire
Lagrange, France  Rousseau S  Université Côte d’Azur, Observatoire de la Côte
d’Azur, CNRS, Laboratoire Lagrange, France  Stee P  Université Côte d’Azur,
Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, France  Sturmann J
The CHARA Array, Mount Wilson Observatory, Mount Wilson, CA 91023  Sturmann L
The CHARA Array, Mount Wilson Observatory, Mount Wilson, CA 91023

###### Abstract

CHARA/SPICA (Stellar Parameters and Images with a Cophased Array) is currently
being developed at Observatoire de la Côte d’Azur. It will be installed at the
visible focus of the CHARA Array by the end of 2021. It has been designed to
perform a large survey of fundamental stellar parameters with, in the possible
cases, a detailed imaging of the surface or environment of stars. To reach the
required precision and sensitivity, CHARA/SPICA combines a low spectral
resolution mode $R=140$ in the visible and single-mode fibers fed by the AO
stages of CHARA. This setup generates additional needs before the
interferometric combination: the compensation of atmospheric refraction and
longitudinal dispersion, and the fringe stabilization. In this paper, we
present the main features of the 6-telescopes fibered visible beam combiner
(SPICA-VIS) together with the first laboratory and on-sky results of the
fringe tracker (SPICA-FT). We describe also the new fringe-tracker simulator
developed in parallel to SPICA-FT.

###### keywords:

long baseline interferometry, fringe-tracking, CHARA

##  1 Introduction

###  1.1 Scientific rationale

Measuring the angular diameter of stars is critical for constraining the
stellar and planet fundamental parameters[1, 2, 3]. In addition to the high
interest for exoplanet characterisation since the first discovery in 1995[4],
stellar physics has seen a recent reawakening with the discovery of
oscillating processes in more than a thousands of stars with WIRE[5], MOST[6],
CoRoT[7] and Kepler[8] and measuring their fundamental properties (diameter,
temperature, mass, age) has become critical in many domains of astronomy.
Indirectly, the stellar angular diameters are also used to derive the distance
of eclipsing binaries in Large Magellanic Cloud[9] and Small Magellanic
Cloud[10] through the so-called Surface-Brightness Color Relations (SBCR). It
exists currently in the literature many relations derived with different
subsets of the JMMC Measured Stellar Diameters Catalog (JMDC), a catalog[11]
that gathers all the star diameters directly measured. When using these
relations for deriving diameters of stars of magnitude V=6, the uncertainty
lies between 2% for V-K=3 and 9% both for early-type (V-K=0) and late-type
(V-K=5) stars[12]. With its capability of resolving such distant objects,
stellar interferometry has long been used for calibrating these relations.
However, due to sensitivity and accuracy limitations, among the 1500 star
diameters of the JMDC, only 11% are known with an accuracy better than 1%,
which partly explains the SBCR dispersion. The three other reasons for this
dispersion are the fact that the stellar diameters of the JMDC come from
heterogeneous measurement techniques, from different star selection
criteria[13], and probably because of circumstellar material (for e.g. wind),
binarity or rotation that are not considered when analysing the data.

Thanks to the combination of fringe tracking, spatial filtering, low spectral
resolution and the use of new modern EMCCD detectors, more than 7000 star
diameters could be measurable with a precision better than 1% by
CHARA/SPICA[14], a new visible instrument that will replace the VEGA[15, 16]
instrument on the CHARA[17] Array. CHARA/SPICA has many similarities with the
NPOI/VISION[18] instrument, but will benefit from the 1 m telescopes of CHARA.
The CHARA/SPICA observing program aims at measuring the diameter of a thousand
of these stars, distributed over the Hertzsprung-Russell (HR) diagram for
spectral types from O to M, over almost 70 nights per year during 3 years. It
will highly enlarge the JMDC with homogeneous and high-precision measurements.
These measurements will be crucial to derive fundamental parameters of stars
and planets, improve evolutionary models, and constrain the SBCR all over the
HR diagram. To complete the spatial measurements provided by the low spectral
resolution $R=140$, a medium and high spectral resolution ($R=3000$,
$R=10000$) will allow spectro-differential interferometry on some targets for
getting crucial information on dynamical processes such as rotation velocity.
With its high angular resolution of tenth of milliarcseconds, CHARA/SPICA can
also measure the apparent orbit of binaries. When completed with the
spectroscopic orbit, the knowledge of their three-dimension orbit permits to
derive their respective masses. The improved precision of the stellar
diameters will finally improve a lot the estimation of the distances in the
Universe and in particular the distances of the neighbour galaxies[12].

For star hosting transiting planets, the ratio $R_{p}/R_{\odot}$ of their
radius is usually estimated by the space missions like CoRoT, Kepler, K2, TESS
or PLATO. Measuring with precision the star radius $R_{\odot}$ from the
combination of its angular diameter measured with interferometric techniques
and its distance measured by parallaxes techniques (Gaia) finally allows us to
estimate the planet radius $R_{p}$ and deriving its properties such that its
density and its position with respect to the habitable-zone of the parent-
star. Furthermore, the high sensitivity and large coverage of the spatial
frequencies permitted by the 6 telescopes and the broadness of the measured
waveband, from 0.6 to 0.9 µ m , will provide precise limb-darkening profiles,
and imaging of surface features when possible. A precise knowledge of the
stellar surface luminance distribution is critical for the characterisation of
the atmosphere of these planets. The large survey planned with CHARA/SPICA
will thus be of major interest for the current spatial missions and in
particular the incoming PLATO survey whose main objective is to characterise
the fundamental parameters of transiting planets by measuring their radius
with a precision of 2%.

###  1.2 Guiding principles for CHARA/SPICA

Designing an interferometric instrument is not only considering the scientific
rationale and its translation into technical specifications but it aims also
at adapting the very particular entrance pupil plane of a ground-based
interferometer to the needs of the cophased focus. In the precedent
descriptions of SPICA[19, 14] we have shown that reaching the required
performance means a spatial filtering of the beams through single-mode fibers
and a low resolution spectrograph. Active controls of the injection in the
fibers and of the fringe stabilization are mandatory to reach the required
limiting magnitude.

These main considerations led to the general design of SPICA-VIS on the basis
of two optical benches showed in Fig. 1. The first one is called Injection
Table (IT) and holds all the modules required for the injection in the fibers:
picking optics, atmospheric refraction correction, pupil plane and image plane
alignment, fast tip/tilt correction for the injection, pupil plane and image
plane sensor, and injection optics. This first table ends with the 6 fibers
glued on a V-groove. This piece is the entrance of the second optical bench,
the Spectrograph Table. It is dedicated to the formation of the image-plane
dispersed fringes at different spectral resolution, the photometric channels
for the calibration of the complex visibility, and the science detector.

Moreover, based on the experience of VEGA[20] and CLIMB[21] combined
operation, it has been decided to install the fast fringe-sensor SPICA-FT in
the near-infrared H-band at 1.65 $\mu$m to keep the whole band for the science
with SPICA-VIS. It is based on a 6T-ABCD integrated optics beam combiner fed
by the MIRC-X fibers and installed in front of the MIRC-X spectrograph [22]
and the C-RED ONE detector[23]. The control loops (group-delay and phase-
delay) use the main CHARA delay lines for the fringe stabilization.

###  1.3 General implementation and CHARA interface

As previously said, SPICA-FT is integrated inside the MIRC-X instrument and
benefits from all its functionalities: the internal delay lines, the off-axis
parabola for the injection, and the compensation of the birefringence of the
fibers. It could be illuminated either by the beams coming from the telescopes
or by the new Six-Telescope-Simulator (STS [22]) coaligned and cophased on the
CHARA reference source.

For what concerns SPICA-VIS, it has been decided to use the two existing VEGA
tables, both for economic reasons and for darkness reasons in this part of the
focal laboratory of CHARA. A new 6-beam periscopic device will replace the old
VEGA periscope and will be installed on the CHARA visible table. With such an
installation, SPICA-VIS could receive light from the 6 telescopes but also
from the STS, thanks to the addition of 6 dichroic plates mimicking the CHARA
beam-sampler for the STS beams. Fig. 1 presents the general layout of the
CHARA laboratory with the new SPICA systems.

Figure 1: General implementation of the three main SPICA elements in the CHARA
Beam Combination Laboratory: the SPICA-VIS feeding optics, the IT and the
spectrograph (green boxes). The light-blue boxes represent the two new LDC
modules.

As will be described in Ref. 24, it has been decided to improve the
longitudinal dispersion compensators (LDC) of CHARA[25]. The main reasons are
the need for improving their transmission in the infrared bands and to permit
a better correction in the low spectral resolution mode of SPICA-VIS. The
combination of the old LDC (but with new glass), an additional LDC in the
visible beams, and the internal delay lines of the different instruments
permits to reach a high fringe contrast and transmission in all bands. This
development will ease the simultaneous use of CHARA instruments and covering
ideally R+I, J, H, and the K bands.

##  2 SPICA-VIS: design study

###  2.1 Injection Table

As rapidly presented in the introduction, the interferometric combination of
beams collected by separated telescopes requires some preparation on each
individual beam. This is not only the transportation and the equalization of
the optical path length but the beams have to be controlled in terms of
alignment and polarization to reach the highest performance. Coupling the
beams into single-mode fibers is now classical in optical interferometry[26,
27]. It presents the great advantage of clearly separating the functions
before and after the injection. After the injection, the beams are transported
by the fibers and could be easily adapted to the requirements of the science
instrument. This part will be described in sub-section 2.2.

It has been demonstrated[14] that, given the performance of the CHARA adaptive
optics stages[28] in the visible (Strehl below 25%) it is critical to add an
additional fast tip/tilt correction before the injection in the fibers. This
stage permits not only to maximize the injected flux but also to highly reduce
the fraction of frames with flux below a certain threshold [29].

The IT (Figure 2) is built around this fast tip/tilt stage. The ideal location
of this tip/tilt mirror is in a pupil plane and the entrance optics of the
table permit to image the distant pupil plane in the CHARA laboratory. An
intermediate image plane is used to allow a correct centering of the pupil.
Two slow-motion mirrors (M2 in the periscope and M3 in the image plane before
the tip/tilt mirror M4) permit to conjugate the CHARA beams and the injection
modules with the required performance (lateral positioning better than 5% of
the diameter of the pupil, residual tip/tilt better than 10 seconds of arc
(lab units)). A fraction of the light (10%) is used to perform a pupil and
image control on each of the 6 beams. The reference positions on the control
detector are recorded with a laser, retro-feeding the fibers and sent to the
control detector by retroreflectors after reflection on the beam splitters. It
should be noted that this table contains also for each beam a module allowing
to compensate for the differential polarization between two beams owing to the
inhomogeneity’s in the fibers[29].

Figure 2: 3D model of the CHARA/SPICA injection table. The six CHARA beams
arrive from the upper-right part of the figure, after beeing picked-up by the
SPICA periscope (mirrors M1 and M2, M2 being dedicated to the slow alignment
of the image plane). Each beam encounters successively the visible shutter,
the polarisation compensator (PDC), the imaging lens, the M3 mirror in image
plane and permitting the alignment of the pupil plane, the fast tip/tilt
mirror M4 (M4/TT), the collimating lens, the beam splitter (BS) sending 10% of
the flux on the control detector, and finally the injection module installed
in the left-bottom corner of the table. In the middle and bottom part of the
table, six retroreflectors (CC) permit to send the retrofeeding laser to the
control camera for the recording of the reference position.

As described in Sec. 1.1, a major part of the science programs of SPICA will
be done in low resolution mode. It means that the whole band between 600 nm
and 900 nm has to be injected into the single-mode fibers. We chose the
single-mode fibers PM630-HP from
Nufern111https://www.nufern.com/pam/optical_fibers/960/PM630-HP/ suited to
this waveband. Therefore the question of the correction of the atmospheric
refraction is important to consider. In Fig. 3, we plot the injection factor
as a function of the wavelength and for different values of an absolute
displacement. From this we can deduce that the chromatic error on the
correction of the refraction has to be done at the level of 10 mas maximum
(sky unit) to limit the loss on the injection factor to less than 1%. A simple
computation of the differential refraction between 700 nm (reference
wavelength) and 600 nm or 850 nm generates a transverse dispersion of 300 mas.
It is thus mandatory to introduce a correction of the differential refraction.
Moreover, because of the field rotation of the CHARA Coudé beams, these
compensators have to be aligned continuously to the field rotation with a
precision of 0.5 °. Finally, our computation shows that the influence of the
field rotation and the change of refraction in 10 min do not generate a
transverse dispersion larger than 10 mas (field rotation) or 15 mas (change of
refraction in the extreme cases). As a conclusion we decided to rotate the
compensators when slewing only.

Figure 3: Injection factor as a function of the wavelength for different
amplitudes of spatial displacement.

Finally, one important development that has been done concerns the injection
module. It is well-known that the theoretical coupling into a single-mode
fiber is limited to $81.8\%$ (in the case of a beam without a central
obstruction). Taking the central obstruction into account leads down to
$72\%$. In theory, a perfectly focused and aligned off-axis parabola could
reach this level with no chromatism. However, surface aberrations and
misalignment would quickly degrade the performance. We studied a lens system
made with an achromatic doublet and a plano-convex lens (Fig. 4). The two
optics are maintained at their adequate position by construction, while focus
is obtained thanks to fine adjustment of the distance between the second lens
and the fiber. All centering are nominal and not adjustable, as the tolerance
for this aspect is not so tight. We developed a test bench for measuring the
coupling efficiency of this module. The light coming from a collimated source
is divided into 2 parts thanks to a beamsplitter. The first part is directly
imaged on the detector, the second part is injected into the fiber. A
motorized Tip Tilt system is placed just before the injection module in order
to optimize the coupling. With this prototype, we reached, in the lab, a
coupling of $75\%$ instead of the theoretical coupling of $81.8\%$.

Figure 4: The opto-mechanical setup of the injection module developed for
SPICA.

###  2.2 Spectrograph

At the entrance of the spectrograph, the 6 fibers (FOP) are linearly arranged
on a silicon V-groove with a pitch of 250 µ m . As illustrated in Fig. 5, 8
additional fibers, 4 single-mode (FCM) and 4 multimode (FCS), are aligned on
free positions to carry the internal source necessary for the spectral
calibration of the spectrometer. The 6 FOP are aligned according to the most
compact non-redundant configuration that makes possible to distinguish the
baselines in the Fourier space while keeping an efficient sampling of the
fringe pattern on the detector. A microlens array (MLA) is glued at the output
of the V-groove such that each microlens of 218 µ m diameter with $NA=0.12$
collimates the Gaussian beam of each fiber.

Figure 5: Arrangement of the scientific and calibration fibers on the V-groove
in front of the MLA. The 6 main scientific fibers (FOP) respect a non-
redundant alignment while the single-mode (FCM) and multimode (FCS) fibers are
distributed according to different spectral calibrations function. The black
rectangle are unusable positions.

To estimate the absolute visibility of each pair of telescopes, it is
necessary to calibrate the photometric fluctuations resulting from the
imperfect fiber injection [30]. On the contrary to a pair-wise recombination,
the all-in-one recombination chosen for SPICA does not permit the estimation
of the individual intensities from the interferometric data. These intensities
are thus directly measured thanks to a dedicated photometric channel. As
illustrated in Fig. 6, a beam splitter separates the light right after the
microlenses: 10% is sent to the photometric channel that reimages the V-groove
on the detector whereas 90% is combined by the interferometric channel on a
common focus and creates the fringe pattern.

Figure 6: Optical principle of the SPICA spectrograph. The red rays symbolize
the diffracting rays of an individual fiber. The two cylindrical lenses (FCL
and SCL) collimate the beam in the spectral direction. The photometric channel
does the same with its three achromatic doublet. The blue rays symbolize the
paraxial rays of each individual fiber output beam. It clearly makes appear on
the detector, side-by-side, the reimaging of the V-groove by the photometric
channel and the focalisation at the common focus by the interferometric
channel. For matters of clarity, only 3 FOP over 6 have been drawn.

The interferometric channel is made of two cylindrical lenses performing the
anamorphosis of factor $AF\simeq 57$ necessary for reaching the sampling
criteria in the spectral and spatial directions. In the spectral direction,
the V-groove is reimaged by the first cylindrical lens (FCL) onto the object
focal plane of the second cylindrical lens (SCL). The SCL and the camera
achromatic doublet (CAO) play the role of the collimating optics of the
spectrograph in the spectral direction with the suitable magnification ratio
for the final spectral resolution. In the spatial direction, the six aligned
beams propagate without being affected, except from their natural diffraction,
until the CAO combines them at its focal plane to create the dispersed fringe
pattern.

Thanks to the presence of three different dispersing components on a
turntable, SPICA offers three different spectral resolution modes. For
achieving the low spectral resolution $R=140$, a mirror reflects the light
towards a pair of prisms made of F2. For reaching the two high spectral
resolutions $R=3000$ and $R=10000$, the light is dispersed by diffraction
gratings with 300 grooves/mm and 900 grooves/mm respectively.

Both interferometric and photometric channels are dispersed by the same
component. Finally, the CAO images the dispersed fringe pattern of the
interferometric channel next to the six dispersed beams of the photometric
channel on the fast and low-noise EMCCD detector Andor iXon 888. The shortest
interfringe at 0.650 µ m is sampled on 3 pixels and each spectral resolution
element is sampled over 2 pixels. The dispersed fringe pattern of the low
spectral resolution are spread over an area of $400\times 1024$ pixels. The
dispersed image of the V-groove lies on $100\times 1024$ pixels next to the
fringe pattern. Only half of the detector ($500\times 80$ in $R=140$ and
$500\times 1024$ in $R=3000$ and $R=10000$) is used. This is required for
reaching the smallest detector integration time (20 ms) that guaranties fringe
acquisition shorter than the atmosphere coherence time, typically 20 ms, when
the fringe tracker doesn’t work properly.

##  3 SPICA-FT

We have already shown [19, 14] that reaching the limiting magnitude (mV around
8 or 9) required for the science program presented in Section 1.1 supposes the
possibility of single exposures longer than the usual value of 20 ms chosen to
correctly freeze the atmospheric piston at Mount Wilson. This opened the way
for the development of the fringe tracker SPICA-FT. Moreover, as mentioned
before, it has been decided to use the H band for SPICA-FT. It is interesting
to note also that doing so permits to benefit from higher visibility’s in the
fringe tracker because of the largest wavelength.

This general idea was turned into an actual implementation of an integrated
optics device combining the 6 CHARA beams on the ABCD principle. The design
was guided by the work done on VLTI/GRAVITY[31] with an adaptation to 6 beams,
to the H band, and to the CHARA Array. It was therefore decided to simplify
the project by using the H-band injection systems of the instrument MIRC-X[22]
as well as its new detector[23] in the low spectral resolution mode ($R=20$,
so 5-6 spectral channels in the H band). With simple calculations based on the
GRAVITY-FT performance and considering the important reduction of the number
of pixels used in the ABCD setup versus the All-In-One combination of MIRC-X,
it is anticipated that SPICA-FT may exhibit a better sensitivity. Knowing that
MIRC-X equipped of its new detector has already reached mH=8.5, we expect to
be able to achieve the required performance ($\lambda/8$ at mH=8) very soon.
The IO device is used as the sensor of the fringe-tracking loop, a dedicated
optical path difference controller has been developed, and the loop is closed
on the existing fast stages of the CHARA main delay lines, also refurbished to
permit a fast dialog.

In Fig. 2, we present the IO device realized for the purpose of SPICA-FT. The
entrance of the chip is directly glued on the output side of a V-groove with
the 6 single-mode fibers connected to the injection systems of MIRC-X. A
dedicated microlens array (pitch 80 µ m ) is glued at the output side of the
chip to collimate each of the 60 beams independently. The beams are then
dispersed and reimaged with a 1x magnification on the C-RED ONE detector, so
that two outputs are separated exactly by 5 pixels on the detector. During the
first commissioning run in January 2020, we succeeded in getting signals with
the 5 available telescopes at that time but we were not ready to close the
loops on the delay lines.

Figure 7: Left: design of the 6-beam ABCD combiner realized by VLC
Photonics222https://www.vlcphotonics.com/on the design developed by the group
on the basis of the initial idea of P. Labeye[32]. Each entrance beam (on the
left) is divided in 5 equal parts through different splitters functions
(60/40, 50/50, 66/33). Then the 15 pairs of beams are combined through the
ABCD cell performing the adequate dephasing. The 60 outputs are then directed
to the right side of the chip. Right: actual image on the sky of the 60
dispersed output of the SPICA-FT chip (dispersion is vertical in the figure
and covers the H band).

The fringe tracker loop is based on the architecture described for the
Gravity[33] fringe tracker. The images of the detector are stored in a shared
memory and are accessible for recording and for processing[22]. The phase
sensor process estimates in real time the 6 fluxes and the 15 complex coherent
fluxes as well as their related variances and stores them into a second shared
memory. From these quantities the optical path difference (OPD) controller
estimates the 15 group delays and the 15 phase delays as well as the best 10
closure phases and the best 10 quantities called group-delay closure phases.
The variances of these quantities are used in real time for the decision
process. From these quantities, the required OPD are estimated and the signals
for the group-delay loop and for the phase-delay loop are estimated by
comparison between the actual measurements and the values of the reference
matrix accounting for the internal or stellar closure phases. The guiding
principle of the state machine is to start the phase-delay closed loop as soon
as the group delay ensures that the delay is within the central fringe. The
current control loop is based on a simple integrator with a gain for the group
delay and a gain for the phase delay. In Fig. 8, we present the actual
performance reached on our testbench in Nice.

Figure 8: Closed-loop (group delay in the first part, phase delay in the
second part) obtained in September 2020 on the Nice testbed. Measurements are
made on 10 spectral channels over the H band, with a detector integration time
of 8 ms, no perturbations except the lab turbulence. The group-delay gain has
been set to 0.05 while the phase-delay gain is set to 0.1. A residual piston
noise of 22 nm is obtained on any baseline.

Although the conditions of the experiment in the lab in Nice were not
representative of the sky, this result demonstrates the excellent behaviour of
the group-delay and phase-delay loops and permitted to qualify the IO chip. In
phase-delay loop we reached a residual rms of less than 22 nm, which is very
encouraging. We noticed residual internal closure phases at the level of 4
$\mu$m that will be corrected in the final fabrication of the chips. We
noticed also some flux unbalance between the outputs of the ABCD chips. In
principle each entrance beam illuminates 20 outputs (5 ABCD channels) and thus
each output should receive 5% of the flux. We can see variation of flux from 3
to 7% typically. Thanks to the different photometric measurements made on the
chips, it has been understood that the 66/33 splitter function was in fact
providing a flux repartition close to 80/20 and the 50/50 one is close to
60/40. The new fabrication considered in the first semester of 2021 will
permit to correct this thanks to an adaptation of the design of these
individual functions.

##  4 FT simulator

The optical bench of the fringe-tracking cannot simulate every disturbance
schemes that are expected on-sky. That is why we developed a fringe-tracker
simulator that makes easier the understanding of the fringe-tracking servo
loop. It is a Python-adapted version of the IDL code developed by E.
Choquet[34] that was used to optimise GRAVITY fringe-tracker[33].

Furthermore, this simulator benefits from the SPICA-FT experience and will be
enriched with many different interferometer configurations and optimised servo
loops in order to study new fringe-tracking logic’s and fringe-sensing
techniques.

###  4.1 Design

As explained before, the fringe sensor provides two levels of measurement.
First, the phase-delay estimator is precise but wrapped over one wavelength.
Using this estimator, the fringe tracker calculates, with a simple integrator
controller, a precise command confined to the horizon of the wavelength.
Second, the group-delay estimator is noisier but sensitive to the many phase
jumps overpassing the wavelength. It enables to compute by another simple
integrator controller a command less precise but with the capacity of
restoring the tracking reference position, the 0 group-delay. The combination
of these two levels of correction is possible only to the means of a non-
linear logic that smartly synchronises them. To get as much freedom as
possible when developing this non-linear logic, we chose to use a temporal
simulator rather than more standard frequency simulation tools suitable for
linear commands.

Fig. 9 illustrates the structure of the fringe-tracking loop as modeled by the
simulator. It is expected to be a modular package where each module accounts
for a distinct function (fringe sensing, fringe tracking, noise, delay lines
model, …), enabling to test many different component architectures.

Figure 9: Structure of the fringe-tracking simulator. The respective coherence
matrices $\Phi_{o}$ and $\Phi_{d}$ of an object and of a realistic disturbance
pattern are created for $NW$ wavelengths and $NB$ baselines. During the
simulation, $\Phi_{o}$ is static whereas $\Phi_{d}$ varies to simulate the
atmosphere. At a time $t$, they are coherently multiplied and propagated into
a fringe sensor that detects the signal on a spectrograph with $NP$ pixels and
$MW$ spectral channels. An estimated image is processed based on a model of
noise and the coherence matrix $\Phi_{est}$ is estimated. From this, the
fringe tracker derives the piston commands $U_{p}$ for the delay lines. The
final coherence matrix $\Phi_{c}$ associated to the delay line correction is
calculated using its unitary response and is coherently multiplied with the
matrices $\Phi_{o}$ and $\Phi_{d}$ at the time $t+1$. The matrix $\Phi_{true}$
of the new residual coherences is propagated.

###  4.2 Results

We simulate the fringe tracking with SPICA-FT of a non-resolved target at
CHARA for typical atmospheric conditions. According to previous measurements
of the atmosphere behaviour at CHARA[35, 36], the outer-scale $L_{0}$ is
estimated to 25 m. This enables us to make the realistic assumption that the
disturbances on all telescopes are totally decorrelated, even though a
correlation is sometimes observed on the shortest baselines. We thus generate
6 independent temporal pistons representing the typical condition of the 80th
percentile of the summer seeing at Mount Wilson ($r_{0}=15$ cm and $t_{0}=10$
ms). The shape of the power spectral distribution of a disturbance with
average wind speed $W=5$ m/s above the telescopes of diameter $d=1$ m shows
three regimes[37, 38, 39, 36]. At low frequency, the atmospheric disturbance
is proportional to $\nu$ until reaching the low-frequency cut-off
$\nu_{0}=0.2W/L_{0}\simeq 0.04$ Hz above which it starts decreasing with
$\nu^{-2/3}$. Above the high-frequency $\nu_{1}=0.3W/d\simeq 1.5$ Hz,
representing the filtering of the highest frequencies by each individual
telescope, it becomes proportional to $\nu^{-8.5/3}$.

The electromagnetic field of a non-resolved object is propagated through the
disturbed atmosphere and the CHARA delay lines. The final flux received by the
detector accounts for a typical total coherent throughput of 2% in H band,
leading to an irradiance per telescope $N=1.66\cdot 10^{5}$ ph/s for $H=7$.
The camera C-RED ONE of SPICA-FT uses an electron avalanche photo-diode
detector of $320\times 256$ pixels and is expected to work most of the time
with the spectral resolution 22, meaning 4 spectral channels between 1.45 and
1.75 µ m . According to Lanthermann et al[40], we can model it with the excess
noise factor $ENF=1.47$ and the dark current and readout noise gathered within
an additive Gaussian noise of dispersion $\sigma_{tot}=0.5$ e-/pix/frame when
used at 300 Hz, its optimal working mode. A latency of 2 frames is chosen.

Fig. 10(a) shows the residual OPD on three of the five baselines involving the
telescope E1, after running a simulation over 30 seconds on a target of
magnitude 6 with all the parameters previously given. The gains of the phase-
delay and group-delay integrators have respectively been optimised to 0.2 and
0.5, the group-delay loop working with an integration time of 40 frames. We
see that the group-delay command occasionally jumps, corresponding to moments
when the OPD variation induced by the atmospheric disturbance is too fast.
Forgetting these group-delay jumps, the residues remain below 50 nm RMS on all
baselines, i.e. $\lambda/33$ at 1.65 µ m , and $\lambda/15$ at 750 nm. We are
still within the SPICA requirement for single integrations longer than 200 ms.
We observe on Fig. 10(b) that it is possible to reach magnitude of 7.5 at the
cost of a degradation of the transfer function of 20%.

This result has been obtained using the model of the first version of the
fringe tracker based on two integrated commands smartly synchronised. Yet, the
experience with GRAVITY[33] demonstrated the interest for a Kalman control
which brings more robustness to predictable fringe losses.

(a) Residual Optical-Path-Delay after a simulated correction with SPICA-FT on
6 telescopes in good observing conditions. To the right are given the residual
OPD RMS.

(b) Evolution of the correction performance with the target magnitude. The
variance of the OPD, in µ m , is the average of the variances of all 50 ms
temporal samples in the last third of the simulation.

Figure 10: Simulations results of SPICA-FT in typical observing conditions.

###  4.3 Perspectives

In addition to the short-term usage of the simulator for optimising SPICA-FT,
it is intended to become a tool for further investigations on the ideal fringe
tracker associated with interferometers with $N\geq 6$ telescopes. The
development of more sensitive fringe sensing techniques and their associated
servo loops could push the capacities of fringe trackers on fainter objects.
SPICA-FT will of course be the first beneficiary of potential improvements
brought by this study.

With the recent generalisation of the spatial filtering offered by optical
fibers, the fringe tracking is partly limited by the photometry drops that the
imperfect injection involves. Indeed it plays a role in two open questions.
First, although a $N$ telescopes interferometers involves only $N-1$ OPDs to
correct, it provides $N(N-1)/2$ independent measured ones. The question of
using only a part of all the baselines to maximise sensitivity has often been
posed in the past[41]. But the photometry drops regularly reduce or null the
visibility of the different baselines. Based on the experience made with
GRAVITY, SPICA-FT is currently equipped with the conservative approach
consisting in using all the baselines for robustness purpose. However, with
its 5 OPDs and 15 baselines, the gap is even bigger than for GRAVITY which
tracks 3 OPDs with 6 baselines. So this question will be further investigated.
Second, these drops take the fringe tracker out of its linear regime,
demanding a non-linear response to be corrected. Both GRAVITY and SPICA-FT are
equipped with non-linear controls but there still remain other logic that can
be studied. The temporal-domain simulator makes possible testing these new
schemes.

Furthermore, to get the information on the phase, SPICA-FT either uses the
MIRC-X all-in-one configuration or the integrated-optics component encoding
the fringes following the ABCD principle. This simulator enables us to study
more sensitive fringe demodulation techniques by getting closer to the Nyquist
criteria limit.

##  5 Conclusion

We have given an overview of the developments of the two main parts of the
CHARA/SPICA instrument made at Observatoire de la Côte d’Azur.

SPICA-FT is based on 6T-ABCD integrated optics beam combiner for the encoding
of the 15 baselines and a servo logic inspired from the fringe tracker of
VLTI/GRAVITY. The preliminary tests on-sky and in laboratory give confidence
on its capacity to track the fringes with residuals lower than $\lambda/33$ at
1.65 µ m up to magnitude 7-8. New architectures and servo logics may come
improving its performance in the next years.

SPICA-VIS is designed with the goal to get a high accuracy on faint targets
and low visibilities necessary for the direct diameter measurement of a large
number of stars, ranging from M to O spectral types. The low spectral
resolution $R=140$ on the bandwidth 0.6 - 0.9 µ m brings the sensitivity
necessary for reaching stars of magnitude 8 and an interesting coverage of the
(u,v) plane for surface imaging of suited stars. Higher spectral modes
$R=$3000 and $R=$10 000 will give access to important knowledge on the stellar
activity. It will measure visibilities of 0.1 with an accuracy of 1%. The high
signal-to-noise necessary for reaching these performance is made possible by
the new fringe tracker and the spatial filtering properties of single-mode
fibers. It is expected to be on sky at the end of 2021 and to start the
science operation by mid 2022.

###### Acknowledgements.

The CHARA/SPICA instrument is funded by CNRS, Université Côte d’Azur,
Observatoire de la Côte d’Azur, and by the Région Sud. The CHARA Array is
supported by the National Science Foundation under Grant No. AST-1636624 and
AST-1715788. Institutional support has been provided from the GSU College of
Arts and Sciences and the GSU Office of the Vice President for Research and
Economic Development. The postdoc fellowship of FP is funded through the
European H2020 OPTICON program, with the grant agreement n °730 890. FC thanks
support from Onera’s Direction Scientifique Générale.

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