$\displaystyle=\sum_{0\leq a<b\leq
3}\acute{\rho}_{\sigma}^{ab}[\det\acute{J}_{ab}^{\sigma}].$

This shows that $\int_{\hat{S}}d\acute{\rho}=\int_{S}d\acute{\rho}$ and
$\int_{\hat{S}}d|\acute{\rho}|=\int_{S}d|\acute{\rho}|$.

###### Remark A.4

  1. 1.

When $S$ is a surface in spatial ${{\mathbb{R}}}^{3}$, we see that
$\int_{S}d|\acute{\rho}|=\int_{S}d\rho$, which is the area of the surface $S$.

  2. 2.

Note that $\int_{S}d\acute{\rho}$ is complex valued.

  3. 3.

As a consequence, when $S$ is a space-like surface, $\int_{S}d\acute{\rho}$ is
real; when $S$ is a time-like surface, $\int_{S}d\acute{\rho}$ is purely
imaginary. That is, $\int_{S}d\acute{\rho}=i\int_{S}d|\acute{\rho}|$.

  4. 4.

By definition and from the proof, we see that
$\acute{\rho}_{\sigma}=\acute{\rho}_{\Lambda\sigma+\vec{a}}$, for any Lorentz
transformation $\Lambda$ and any vector $\vec{a}$.

##  Appendix B Lorentz transformation of a space-like vector

###### Lemma B.1

Let $S$ be a space-like rectangular surface contained in a plane. Without loss
of generality, we assume a parametrization of $S$ given in Definition 2.2.
From Definition 2.5, we can define a basis $\\{\hat{f}_{a}\\}_{a=0}^{3}$.

Let $0\neq\vec{v}$ be a space-like directional vector lying in the time-like
plane $\vec{x}+S^{\flat}$, i.e. $\vec{v}=s\hat{f}_{0}+\bar{s}\hat{f}_{1}\in
S^{\flat}$, whereby $|\bar{s}|>|s|$, and $\vec{x}\in S$. There exists a finite
sequence of translations and Lorentz transformations, which depends on
$\vec{x},\vec{v}$, such that
$\vec{v}\in\vec{x}+S^{\flat}\mapsto-\vec{v}\in\vec{x}+S^{\flat}$.

Proof. First, assume that $S$ contains the zero vector. Without any loss of
generality, we assume that $S$ is parametrized according to Definition 2.2,
with $\vec{a}=0$. From this parametrization, we can define
$\\{\hat{f}_{a}\\}_{a=0}^{3}$ as in Definition 2.5, and it is easy to see that
we have a Lorentz transformation $\bar{\Lambda}$ which sends
$\bar{\Lambda}:\hat{f}_{a}\mapsto e_{a}$, for $a=0,\cdots,3$. Hence,
$\bar{\Lambda}S\subseteq S_{0}$.

Then, $\bar{\Lambda}\vec{x}=\sum_{a=0}^{3}x^{a}e_{a}=x^{2}e_{2}+x^{3}e_{3}$.
Let $R_{i}$ be a rotation about $x^{i}$-axis, and $\Lambda$ be boost in the
$e_{1}$ direction, that sends

$\Lambda:se_{0}+\bar{s}e_{1}\mapsto\left[{\rm
sgn}(\bar{s})\sqrt{\bar{s}^{2}-s^{2}}\right]e_{1},$

whereby ${\rm sgn}(\bar{s})$ is the sign of $\bar{s}$.

Then, we have

$\displaystyle\vec{x}+\vec{v}$
$\displaystyle=s\hat{f}_{0}+\bar{s}\hat{f}_{1}+x^{2}\hat{f}_{2}+x^{3}\hat{f}_{3}$
$\displaystyle\longrightarrow_{\bar{\Lambda}}(s,\bar{s},x^{2},x^{3})\equiv
se_{0}+\bar{s}e_{1}+x^{2}e_{2}+x^{3}e_{3}$
$\displaystyle\longrightarrow_{\Lambda}(0,{\rm
sgn}(\bar{s})\sqrt{\bar{s}^{2}-s^{2}},x^{2},x^{3})$
$\displaystyle\longrightarrow_{R_{3}}(0,{\rm
sgn}(\bar{s})\sqrt{x^{2,2}+\bar{s}^{2}-s^{2}},0,x^{3})$
$\displaystyle\longrightarrow_{R_{2}}-(0,{\rm
sgn}(\bar{s})\sqrt{x^{2,2}+\bar{s}^{2}-s^{2}},0,x^{3})$
$\displaystyle\longrightarrow_{R_{3}^{-1}}-(0,{\rm
sgn}(\bar{s})\sqrt{\bar{s}^{2}-s^{2}},x^{2},x^{3})$
$\displaystyle\longrightarrow_{\Lambda^{-1}}-(s,\bar{s},x^{2},x^{3})$
$\displaystyle\longrightarrow_{R_{1}}(-s,-\bar{s},x^{2},x^{3})$
$\displaystyle\longrightarrow_{\bar{\Lambda}^{-1}}-s\hat{f}_{0}-\bar{s}\hat{f}_{1}+x^{2}\hat{f}_{2}+x^{3}\hat{f}_{3}.$

Hence, we have

$\tilde{\Lambda}(\vec{x},\vec{v}):=\bar{\Lambda}^{-1}R_{1}\Lambda^{-1}R_{3}^{-1}R_{2}R_{3}\Lambda\bar{\Lambda},$

which sends $\vec{v}\in\vec{x}+S^{\flat}\mapsto-\vec{v}\in\vec{x}+S^{\flat}$.
Note that all the rotations depend on $\vec{x}$ and $\vec{v}$, except $R_{1}$
and $R_{2}$, which rotate by angle $\pi$ radians.

Suppose $\tilde{S}$ does not contain the origin. Then, we consider
$S=U(-\vec{a},1)\tilde{S}$, whereby $\vec{a}\in\tilde{S}$. For any space-like
vector $\vec{v}\in\vec{x}+S^{\flat}$, we can map $\vec{v}\mapsto-\vec{v}$,
using the above transformations. Finally, apply $U(\vec{a},1)$ to translate
back to $\tilde{S}$.

###### Remark B.2

On the $x^{2}-x^{3}$ plane or $S_{0}$, any two points on it is space-like
separated. Clearly, there is a Lorentz transformation (rotation) which
transforms $\vec{p}\in S_{0}\mapsto-\vec{p}\in S_{0}$. In general, we see that
there is a sequence of translations and Lorentz transformation which
transforms $\vec{p}\in S\mapsto-\vec{p}\in S$, for some space-like plane $S$.

On a time-like plane $S$, we do not have a rotation in the Lorentz group that
maps a time-like vector $\vec{p}\in S\mapsto-\vec{p}\in S$. It is not possible
to do time and space inversion simultaneously for a time-like vector using
Lorentz transformations.

###### Lemma B.3

Refer to Definition 2.5. Let $S^{\flat}$ be a time-like plane spanned by
$\\{\hat{f}_{0},\hat{f}_{1}\\}$. Fix a time-like vector $u\in S^{\flat}$. For
any time-like or space-like vector $v\in S^{\flat}$, there exist a $\phi$,
$\theta$ depending on $u$ and $v$ respectively, such that

$\displaystyle v\cdot u=$
$\displaystyle\sinh(\phi-\theta)=\left(\begin{array}[]{c}\cosh(\phi-\theta)\\\
\sinh(\phi-\theta)\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\
\end{array}\right),$

when $v$ is space-like; and

$\displaystyle v\cdot u=$
$\displaystyle-\cosh(\phi-\theta)=\left(\begin{array}[]{c}\cosh(\phi-\theta)\\\
\sinh(\phi-\theta)\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right),$

when $v$ is time-like.

Proof. Any vector on $S^{\flat}$ can be represented as a 2-vector $(a,b)^{T}$,
which are the coordinates with respect to the basis
$\\{\hat{f}_{0},\hat{f}_{1}\\}$. Define

$\Lambda(\theta)=\left(\begin{array}[]{cc}\cosh\theta&\ \sinh\theta\\\
\sinh\theta&\ \cosh\theta\\\ \end{array}\right).$ (B.1)

For a time-like vector, we will write it as $\Lambda(\phi)(1,0)^{T}$. For any
space-like vector, we will write it as $\Lambda(\theta)(0,1)^{T}$. Then, we
have

$\displaystyle\Lambda(\theta)\left(\begin{array}[]{c}0\\\ 1\\\
\end{array}\right)\cdot\Lambda(\phi)\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)=$ $\displaystyle\left(\begin{array}[]{c}\sinh\theta\\\
\cosh\theta\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}\cosh\phi\\\
\sinh\phi\\\ \end{array}\right)$ $\displaystyle=$
$\displaystyle-\sinh\theta\cosh\phi+\cosh\theta\sinh\phi=\sinh(\phi-\theta)$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{c}\cosh(\phi-\theta)\\\
\sinh(\phi-\theta)\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}0\\\ 1\\\
\end{array}\right).$

A similar calculation will show that

$\Lambda(\theta)\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)\cdot\Lambda(\phi)\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right)=-\cosh(\theta-\phi)=\left(\begin{array}[]{c}\cosh(\phi-\theta)\\\
\sinh(\phi-\theta)\\\ \end{array}\right)\cdot\left(\begin{array}[]{c}1\\\ 0\\\
\end{array}\right).$

From the calculations, when $\theta=\phi$, we see that $e^{-iu\cdot
v}=\cos(0)=1$, which is real. This happens when $v$ is space-like, $u$ is
time-like. However, if both are time-like or both are space-like, then we have
$e^{-iu\cdot v}=\cos(u\cdot v)-i\sin(u\cdot v)\neq 1$.

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First light of the FIRST visible fibered interferometer upgrade at the Subaru
telescope

Kevin Barjot

LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université,
Université de Paris, 5 place Jules Janssen, 92195 Meudon, France

Since the mid-nineties, five thousands exoplanets have been discovered, mainly
by space missions using indirect detection methods. The existing ground-based
8-m class telescopes and the forthcoming 30-m class telescope offer high
angular resolution imaging of planetary system. On the one hand the advantage
of such a direct detection is obtaining a spectrum of the planet which permits
the study of its atmosphere if it exists and seek for a possible presence of
life. On the other hand the caveat is that the atmosphere of the Earth
corrugates the light coming from the target thus an (Extreme) Adaptive Optic
(ExAO) system is mandatory to correct for these perturbations and perform
direct imaging at the diffraction limit regime of the telescope. The pupil
masking technique [1] consists in placing a non-redundant mask on the
telescope pupil such that each spatial frequency in the image is sampled by a
unique pair of sub-pupils. The angular resolution limit is divided by two and
the effects of atmospheric perturbations on the fringes are limited. The
caveat of this technique is that only a fraction of the flux of the telescope
is utilized. It can be coped using the pupil remapping technique [2] which
rearrange non-redundantly the sub-pupils using optical fibers and allows to
use the flux of the entire pupil. These fibers are single-mode and filter the
speckles remaining across each sub-aperture.

The Fibered Imager foR a Single Telescope

The upgraded version of the FIRST instrument [3] is currently installed on the
Subaru Coronagraphic Extreme Adaptive Optic (SCExAO) platform [4, 5], a
second-stage ExAO system providing light corrected from the atmospheric
turbulence. As illustrated on the scheme of the two versions of FIRST on the
figure 1 the telescope pupil is sampled thanks to an array of $250\,\upmu$m
pitched micro-lenses (2) that focus light into single-mode optical fibers (4).
This injection is optimized via a segmented mirror (1) paved with 37 hexagonal
segments of about $600\,\upmu$m of circumscribed diameter that can be
controlled in piston, tip and tilt. Afterwards, optical delay lines (ODL) (3)
were added during the upgrade of the first version to correct for optical
fiber length mismatch in order to equalize the optical path differences (OPD)
between the incoming beams. The integrated optic (IO) chip (5) combines the
light of the sub-pupils pair-wisely as opposed to the first version where the
optical fibers were mounted non-redundantly on a V-Groove to make the light
beams interfere at the focus of a lens. The chip was tested in laboratory [6,
7] prior to its integration in the instrument in 2021. Finally, the chip
outputs are imaged on the camera (7) and spectrally dispersed by a grism
spectrometer (with a spectral resolution of $4000$ @$700$nm) (6).
Polarizations are split by a Wollaston prism in order to avoid contrast loss
due to the birefringence of the polarization maintaining fibers and of the
chip. FIRSTv2 would ultimately utilize all the sub-pupils to maximize the
overall throughput and the spatial frequency coverage of the instrument.

Figure 1: Schematic of the pupil remapping technique implemented in the FIRST
instrument. Top: original version. Bottom: version 2 with the IO chip.

Figure 2: Photonic chip schematic. The five inputs are interferometrically
recombined by pairs. Each of the 10 outputs corresponds to one pair.

Data acquisition and processing

The key part of the FIRSTv2 instrument is the photonic chip which consists in
a block of glass with engraved optical waveguides. As depicted in the
schematic of the figure 2 it has five inputs (bottom) that are split in four
guides to be recombined thanks to Y-junction couplers giving ten outputs
(top). With such a device each of the spatial frequencies that corresponds to
each pair is imaged on a few pixels on the camera as a fringe pattern
spectrally dispersed. Hence, one frame encodes the interferometric flux for a
specific OPD on several spectral channels. In order to obtain the full
interferogram the sub-apertures need to be modulated to sample the fringes as
a function of the OPD.

Data analysis aims at reconstructing the interferogram from a series of
images. A Pixel-To-Visibility Matrix (P2VM) [8] is calculated prior to
observations and is used to fit the fringes and to retrieve the complex
coherence terms for each baseline and each spectral channel:

$\mu_{nn^{\prime}}=|V_{nn^{\prime}}|e^{i\psi_{nn^{\prime}}}A_{n}A_{n^{\prime}}e^{i\Delta\Phi_{nn^{\prime}}}$
(1)

With $|V_{nn^{\prime}}|$ and $\psi_{nn^{\prime}}$ the object’s complex
visibility modulus and phase, $A_{n}$ the flux of sub-pupil $n$ and
$\Delta\Phi_{nn^{\prime}}$ the differential piston (due to different fiber
lengths, Adaptive Optics (AO) residuals and instrumental aberrations).

The phase of this term has two contributions in the context of a binary system
observation: the continuum on the whole range of wavelength and the companion
on a narrow band range. To obtain the companion contribution
$\psi_{compnn^{\prime}}$ the spectral differential phase is computed
subtracting the phase of the continuum $\psi_{contnn^{\prime}}$ (avoiding the
spectral band of the companion) to the phase on the whole spectrum
$\psi_{nn^{\prime}}$, following:

$\displaystyle\upvarphi_{diffnn^{\prime}}(\sigma)$
$\displaystyle=\psi_{nn^{\prime}}+\Delta\Phi_{nn^{\prime}}-(\psi_{contnn^{\prime}}+\Delta\Phi_{nn^{\prime}})$
(2) $\displaystyle=\psi_{compnn^{\prime}}$ (3)

Moreover, this variable has the advantage of being self-calibrated by
canceling out the differential piston part of the complex coherence terms.

Lab and on-sky results

In the lab two sources were injected on the FIRST testbed to simulate a binary
source which spectrum is presented on the bottom left of the figure 3. The
wide band continuum and the peak respectively simulate the central star and a
companion with a narrow band emission with a contrast of $1.96$ (the companion
is $1.96$ times brighter than the central star).

Figure 3: Differential phase measured on the FIRST test bench for a simulated
binary system.

The figure 3 is the spectral differential phase of each baseline averaged over
$4000$ frames of $300\,$ ms integration time. The black dashed lines indicate
the theoretical phase amplitude expected for such a source. Four of them
($37-9$, $37-24$, $9-24$ and $35-7$) show a value of zero because their
baseline is orthogonal to the orientation of the binary system. The peak
appearing around 641nm for all other baselines show that the companion is
detected.

The figure 4 shows the first light of the upgraded FIRST instrument on Spica
at the Subaru telescope, in 2021. It shows the fringes of the ten outputs of
the photonic chip spectrally dispersed on the vertical axis, for an exposure
time of 80ms and a gain factor of 300.

Figure 4: Image of fringes during observations of Spica. Each vertical line
corresponds to one baseline. The vertical axis is the direction of the
spectral dispersion (from 750nm to 800nm).

Conclusion

FIRST is an instrument for high angular and high contrast imaging of binary
systems which was upgraded with an integrated optics chip to improve its
performances in terms of stability and sensitivity. In 2021 we obtained the
first light of the upgraded instrument on the SCExAO platform at the Subaru
Telescope. Data processing is still in progress.

References

## References

  * [1] Tuthill, P. G., et al. 2000, The Publications of the Astronomical Society of the Pacific, Volume 112, Issue 770
  * [2] Perrin, G., et al. 2006, Monthly Notices of the Royal Astronomical Society, 373, 747
  * [3] Huby, E., et al. 2013, Astronomy & Astrophysics, Volume 560, id.A113
  * [4] Jovanovic, N., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 890
  * [5] Vievard, S., et al. 2022, in preparation
  * [6] Barjot, K., et al. 2020, Proc. of the SPIE, Volume 11446
  * [7] Martin, G., et al. 2020, Proc. of the SPIE, Volume 11446
  * [8] Millour, F., et al. 2004, Proc. of the SPIE, Volume 5491

Short CV 2017: BSc in Physics, Paris Cité University, France 2019: MSc in
Physics and astronomy, Paris Cité University, France 2019-present: PhD in
astronomy, Paris Observatory, France