# Asgard/NOTT: L-band nulling interferometry at the VLTI
II. Warm optical design and injection system

###### Abstract

Asgard/NOTT (previously Hi-5) is a European Research Council (ERC)-funded
project hosted at KU Leuven and a new visitor instrument for the Very Large
Telescope Interferometer (VLTI). Its primary goal is to image the snow line
region around young stars using nulling interferometry in the L’-band (3.5 to
4.0) µm, where the contrast between exoplanets and their host stars is
advantageous. The breakthrough is the use of a photonic beam combiner, which
only recently allowed the required theoretical raw contrast of $10^{-3}$ in
this spectral range. Nulling interferometry observations of exoplanets also
require a high degree of balancing between the four pupils of the VLTI in
terms of intensity, phase, and polarization. The injection into the beam
combiner and the requirements of nulling interferometry are driving the design
of the warm optics and the injection system. The optical design up to the beam
combiner is presented. It offers a technical solution to efficiently couple
the light from the VLTI into the beam combiner. During the coupling, the
objective is to limit throughput losses to 5 % of the best expected efficiency
for the injection. To achieve this, a list of different loss sources is
considered with their respective impact on the injection efficiency. Solutions
are also proposed to meet the requirements on beam balancing for intensity,
phase, and polarization. The different properties of the design are listed,
including the optics used, their alignment and tolerances, and their impact on
the instrumental performances in terms of throughput and null depth. The
performance evaluation gives an expected throughput loss $<$6.4 % of the best
efficiency for the injection and a null depth of $\sim$2.$10^{-3}$, mainly
from optical path delay errors outside the scope of this work.

Germain<EMAIL_ADDRESS>Azzurra Bigiolia, Romain
Laugiera, Gert Raskina, Johan Morrena, Jean-Philippe Bergerb,
Colin Dandumontc, Harry-Dean Kenchington Goldsmithd, Simon Grosse, Michael
Irelandd, Lucas Labadief,
Jérôme Loicqc,g, Stephen Maddend, Guillermo Martinb, Marc-Antoine Martinoda,
Alexandra Mazzolic,
Ahmed Sannye,f, Hancheng Shaod, Kunlun Yand, and Denis Defrèrea

aKU Leuven, Institute of Astronomy, Leuven, Belgium

bUniversity of Grenoble Alpes/CNRS, IPAG, Grenoble, France

cUniversity of Liège, STAR Institute, Liège, Belgium

dAustralian National University, Research School of Astronomy and
Astrophysics, Canberra, Australian Capital Territory, Australia

eMacquarie University, School of Mathematical and Physical Sciences, MQ
Photonics Research Centre, Macquarie Park, New South Wales, Australia

fUniversität zu Köln, I. Physikalisches Institut, Köln, Germany

gDelft University of Technology, Faculty of Aerospace Engineering, Delft, The
Netherlands

_Keywords_ instrumentation; integrated optics; exoplanets; nulling
interferometry; Very Large Telescope Interferometer; Asgard/NOTT

##  1 Introduction

Imaging planetary systems close to the snow line is an important step in
astronomy. This region is known to have a statistically rich exoplanet
distribution (Fulton et al., 2021). Long baseline interferometry already
manages to probe exoplanets using high angular resolution ($\sim$50 mas) and
high contrast ($\sim 10^{-5}$) with the VLTI/GRAVITY instrument (GRAVITY
Collaboration et al., 2019) at the Very Large Telescope Interferometer (VLTI).
However, to go below the 50 mas and image inside the coronagraphic regime
(Serabyn et al., 2022), a nulling interferometer becomes advantageous in the
mid-infrared range. Nulling interferometry was first proposed by Bracewell
(Bracewell, 1978) and has already been implemented for example with the
Palomar Fiber Nuller (PFN) (Serabyn et al., 2019), the Keck Interferometer
Nuller (Colavita et al., 2009), the Nulling and Imaging Camera (NIC) (Hinz et
al., 2008) at the Large Binocular Telescope (LBT) and the Guided-Light
Interferometric Nulling Technology (Norris et al., 2020; Martinod et al.,
2021) (GLINT) on the Subaru telescope. Asgard/NOTT (previously Hi-5) is a
nulling interferometer (Defrère et al., 2018, 2022) dedicated to high-contrast
imaging. It is part of the Asgard instrumental suite (Martinod et al., 2023)
which is the new visitor instrument of the VLTI. The instrument would be the
first to implement nulling interferometry at the VLTI, aiming to achieve a
contrast of $10^{-5}$ after post-processing.

The instrument will use an integrated optics beam combiner to recombine the
light from the four Unit Telescopes (UTs) or Auxiliary Telescopes (ATs).
Photonic beam combiners have already been successfully implemented at shorter
wavelengths in other interferometric instruments like in VLTI/PIONIER (Berger
et al., 2010; Le Bouquin et al., 2011) or Subaru/GLINT (Norris et al., 2020;
Martinod et al., 2021). It enables a more stable and more compact design as
well as an easier alignment procedure than with free-space optics. The
innovation brought by Asgard/NOTT is the use of such integrated optics in the
L’-band (3.5 to 4.0) µm (Sanny et al., 2022), a spectral region where guided
optics is still emerging.

The optical design of the instrument includes the cold optics located inside a
cryostat and cooled at 90K (Dandumont et al., 2022). They comprise the
injection system, the photonic chip, the spectrograph, and the detector.
Upstream of the cryostat are the warm optics at ambient temperature. This
paper focuses on the warm optics and the injection system up to the
waveguides’ inputs of the photonic chip. The coupling between free-space beams
with imperfect wavefronts and the guided modes inside a photonic chip requires
careful engineering for the optical design. On top of this challenge, nulling
interferometry is also particularly demanding when it comes to the balancing
of the different beams in terms of intensity, phase, and polarization.

This work presents the warm optical design and the injection system of
Asgard/NOTT up to the photonic beam combiner. The proposed design offers a
technical solution to maximise the coupling of the light inside the guided
modes while enhancing the contrast achievable with nulling interferometry.
Section 2 first introduces the instrumental requirements which drive the
overall optical design. In Sec. 3, a complete description of the optical
design is given. Finally, Sec. 4 estimates its performances in terms of null
depth and coupling efficiency between the light and the guided modes of the
beam combiner.

##  2 Instrumental Requirements

In this section, we define the objective as well as the requirements for the
optical design of the instrument.

###  2.1 Requirements on Coupling Efficiency

The studied chip beam combiner (or “chip") is an integrated optic fabricated
from a dielectric substrate where an ultrafast laser inscription (ULI) changed
the refractive index of the material locally to create optical waveguides
(Labadie et al., 2018). The use of couplers inside the chip enables to
generate interferences by evanescent coupling between the different waveguides
(see Fig. 1) . The chip planned to be used in the Asgard/NOTT instrument is
manufactured at Macquarie University. It is made of gallium lanthanum sulfide
(GLS) chalcogenide glass with propagation losses as low as 0.22 dB/cm at 4 µm
wavelength (Gretzinger et al., 2019). This chip has four inputs corresponding
to the four telescopes of the VLTI with the following properties for each
input:

$\left\\{\begin{array}[]{c}\text{Numerical Aperture (NA)}=0.1\\\ \text{Core
physical diameter}=19.4\times 19.8\,\text{µm}\\\ \text{$1/e^{2}$ Mode Field
Diameter (MFD)}=22.3\pm 0.1\times 22.6\pm 0.1\,\text{µm\,@$\lambda$=4\,µm}\\\
\text{Input Separation}=125\,\text{µm}\end{array}\right..$ (1)

Figure 1: Schematic layout of the photonic chip designed for Asgard/NOTT. On
the left, the reproduction of the entire chip with inputs coming from the VLTI
telescopes. The S-bend used to reduce stray light after the input is visible.
On the right, a closer view of the couplers and the eight outputs. The
photometric outputs P1, P2, P3, and P4 emit 20 % of the injected light. The
bright interferometric outputs I1 and I4 mainly emit the on-axis stellar
signal. The dark interferometric outputs I2 and I3 emit the off-axis signal
corresponding to possible exoplanets or disks. Source: Ref. (Sanny et al.,
2022).

The performance evaluation in Sec. 4 considers a circular MFD of 22.45 µm. The
impact of the MFD error and ellipticity is estimated in Sec. 4.5. The eight
outputs of the chip shown in Fig. 1 are spectrally dispersed using grisms with
spectral resolutions of 20, 400, or 2000. They are then imaged on a HAWAII
2RG® detector array (Dandumont et al., 2022). Different substrates were
available for the chip, for example with lithium niobate (Heidmann et al.,
2011; Martin et al., 2014a, b) or silicon. These alternatives have benefits:
internal fringe locking for lithium niobate or propagation losses $<<$1 dB/cm
for silicon. They may be tested as future prospects using the optical design
described in this paper.

The coupling efficiency $\rho$ corresponds to the energy ratio between the
incoming electric field and the light that actually couples and propagates
into the waveguide. When simplifying the problem by considering a Gaussian
approximation for the single-mode waveguide and an Airy pattern from a
diffraction-limited regime for the incoming electric field, a simple
expression for the coupling efficiency is obtained (Ruilier, 1999) with

$\displaystyle\rho=2$
$\displaystyle\left[\frac{e^{-\beta^{2}}-e^{-\beta^{2}\alpha^{2}}}{\beta\sqrt{1-\alpha^{2}}}\right]^{2},$
(2) and
$\displaystyle\beta=\frac{\pi}{2}\frac{\omega_{0}}{\lambda}\frac{1}{f_{\\#}},$
(3)

with $\alpha$ the central obstruction, $\omega_{0}$ the mode radius of the
waveguide, $\lambda$ the wavelength of the beam and $f_{\\#}$ the f-number of
the injection optic. The resolution of this equation with no obstruction
($\alpha=0$) gives a maximum value of $\rho\simeq 81\,\%$ for $\beta=1.1209$,
which corresponds to $f_{\\#}\simeq 3.93$ @$\lambda$=4 µm according to the MFD
given in Eq. (1).

One main objective of the design presented in this paper is to maximize the
coupling efficiency of each beam into the photonic chip. This value is very
sensitive to optical aberrations, alignment errors, vignetting, phase
wavefront perturbations, diffraction, and manufacturing errors of the optics.
Some optical aberrations are inherent to the optical design and cannot be
minimized or corrected without adding other surfaces to the design. This is
the case for instance of coma produced by off-axis parabolas (OAPs). In this
paper, the maximum coupling efficiency is defined as the efficiency obtained
in a situation where only irreducible aberrations are considered (see Sec.
4.1). The requirement is therefore to limit the impact from all other sources
at $5\,\%$ of the maximum coupling efficiency, assuming a perfect wavefront at
the entrance. The number of 5 % is chosen arbitrarily to be less than the
losses from the wavefront correction upstream (see Sec. 4.6.3). Section 4
lists the estimated impact from individual sources and the resulting
throughput loss that can be expected. The Fresnel losses of the system are
also estimated but they are not impacting the coupling efficiency. Therefore,
they are not accounted in the 5 % criterion.

###  2.2 Requirements on the Null Depth

####  2.2.1 Null depth definition

The null depth of the instrument corresponds to the ratio between the stellar
flux transmitted to the null output (or stellar leakage) and the total stellar
flux collected by the telescopes. A lower null depth enables the detection of
fainter companions around the host star. To achieve its science goals,
Asgard/NOTT targets a raw null depth of $N_{\text{raw}}\leq 10^{-3}$ and a
calibrated null depth of $N_{\text{calibrated}}\leq 10^{-5}$ over the full
L’-band (Defrère et al., 2018). These values are very sensitive to imbalances
between the different beams regarding intensity, polarization and phase. To
achieve $N_{\text{raw}}\leq 10^{-3}$, the sources of errors should each aim
for a null depth $\leq 10^{-4}$ as a safe margin.

####  2.2.2 Impact of the phase error

The relative phase difference between the four beams may have various sources:
mainly the Optical Path Difference (OPD) and the longitudinal dispersion from
water vapor (Absil, O. et al., 2006) and CO2. The OPD represents the
difference of optical path $\Delta x$ between two light beams. The associated
phase difference $\Delta\phi$ scales with the optical frequency $\nu$ as

$\Delta\phi=2\pi n\nu\frac{\Delta x}{c}=2\pi n\frac{\Delta x}{\lambda},$ (4)

with c, the speed of light in vacuum and n, the refractive index of the medium
of propagation. After correction of the static phase difference, the total
relative phase fluctuation $\sigma_{\phi}$ is limiting the null depth
$N_{\phi}$ (Serabyn, 2000)

$N_{\phi}>\frac{\sigma_{\phi}^{2}}{4}.$ (5)

Current levels of OPD errors with the GRAVITY fringe tracker are estimated at
200 nm (0.335 rad) root mean square (RMS) for the UTs (Lacour et al., 2019).
Ongoing efforts within the GRAVITY+ consortium aim at improving this value
down to 100 nm (0.168 rad) RMS (Bigioli et al., 2022; Courtney-Barrer et al.,
2022). The adaptive optics of Asgard/HEIMDALLR (Ireland et al., 2018) are
expected to reduce it down to 50 nm (0.084 rad) RMS (Martinache and Ireland,
2018). Using Eqs. (4) and (5), the raw null depth from OPD fluctuations is
computed and presented in Fig. 2.

The static OPD is corrected by the internal delay lines of Asgard/NOTT up to
$\sim$12 mm range.

Figure 2: (a) Impact of OPD RMS errors on the raw null depth for the UTs. The
effects of OPD RMS correction from GRAVITY+ and Asgard/HEIMDALLR are given
with the two vertical lines. (b) Impact of the polarization angle error on the
raw null depth and comparison with the impact of phase error from OPD RMS
after correction by Asgard/HEIMDALLR.

####  2.2.3 Impact of the relative intensity error

The relative intensity error between the inputs $\sigma_{I}$ is limiting the
null depth $N_{I}$ (Martin et al., 2014a)

$N_{I}>\frac{\sigma_{I}^{2}}{16}.$ (6)

The aimed relative intensity error is then $\sigma_{I}\leq 4$ % for a null
depth of $N_{I}<10^{-4}$ according to Eq. (6). Section 4 presents the expected
intensity error at the injection from propagation through the optical design
only. Strategies to control errors from other sources are discussed in Sec.
5.2. Their performance evaluation is part of future work.

####  2.2.4 Impact of polarization errors

Relative polarization errors, both static and dynamic, are hampering the null
depth of the instrument. In the optical system, this error may arise from a
relative polarization rotation angle difference $\alpha_{rot}$ between the
beams. $\alpha_{rot}$ is limiting the null depth of the instrument $N_{P}$
with (Serabyn, 2000)

$N_{P}>\frac{\alpha_{rot}^{2}}{4}.$ (7)

This angle should be kept at $\alpha_{rot}\leq 0.02$ rad in order to have a
null depth $N_{P}<10^{-4}$ according to Eq. (7) (see Fig. 2).

Instrumental birefringence may also impact the null depth limit. When light
propagates in a birefringent material, it generates a differential TE/TM OPD.
If this OPD is different between two beams, that will limit the null depth
according to Eq. (5).

An estimation for these two polarization errors is given is Sec. 4.7 for the
optical design and for the VLTI. We expect only the instrumental birefringence
to have a significant impact on the instrumental null depth. The optical
design thus includes a correction setup (see Sec. 3.6) to maintain the OPD
from birefringence at $<$10 µm ($<$0.02 rad), i.e., $<10^{-4}$ null depth.

##  3 Optical Design

In this section, we describe the warm optical design, the injection system,
and the motivations for their architecture. The optical designs presented in
this section were entirely realized using OpticStudio-Zemax.

###  3.1 Objective

The light from the telescopes of the VLTI reaches the Visitor Instrument table
with diameters of 18 mm and separated by 240 mm. These beams are first
expected to go through the Asgard/HEIMDALLR fringe tracker with adaptive
optics mirrors reducing the beam diameter down to 12 mm. For GRAVITY or
PIONIER, the injection inside integrated optics is performed using optical
fibers (Lippa et al., 2018; Le Bouquin et al., 2011). However in the context
of Asgard/NOTT, optical fibers working in the L’-band that resist cryo-
temperatures and also maintain polarization are not mature enough to be
implemented here.

The objective is then to use a free-space optical system to inject these 12
mm-diameter beams into the four inputs of the chip combiner while respecting
the requirements on coupling efficiency and null depth stated in Sec. 2.

###  3.2 Step-by-step Description of the Main Optics

Figure 3 gives a complete view of the optical design. The four beams from the
VLTI are first directed toward the Asgard/HEIMDALLR fringe tracker. Using
dichroics optimized to transmit only the L’-band, collimated beams with a
diameter of 12 mm are then reflected along the edge of the optical table. The
light beams then encounter shutters, longitudinal dispersion correctors (LDCs)
and, the delay lines. These LDCs are presented in Sec. 3.6. The delay lines
consist of two plane mirrors with a 45 $\deg$ angle reflecting the beams in
the same direction but 40 mm lower (going from 125 mm to 85 mm height). The
two mirrors are moved in parallel to the beams to change their optical path
length without affecting the alignment of the system downstream.

Figure 3: Warm optical design and injection system of the Asgard/NOTT
instrument up to the photonic chip. After the dichroics of Asgard/HEIMDALLR,
TTMs1 reflect the light throughout the shutters and LDCs. The delay lines then
reflect back the beams 40 mm lower toward the TTMs2. The beams are then
focused by the OAPs1 with mirrors reflecting the light on the slicer. The
slicer is located in an IIP and recombines the four beams which are then
collimated by the OAP2 in the direction of the chip. After traversing the
window of the cryostat, the light is injected into the chip using a
telecentric lens.

Since optical fibers cannot be used to inject into the chip, a free-space
optical system is chosen. A multi-lens array like the one used in GLINT
(Norris et al., 2020) is considered too expensive to be compatible with cryo-
temperatures in the L’-band. The solution of a single telecentric lens made of
zinc selenide (ZnSe) is then adopted for the injection. By implementing a cold
stop in the focal plane of the injection lens, the magnification of the image
becomes independent from the incidence angle so that mirrors at different
angles can be used to recombine the four telescope’s pupils. The use of mirror
slices with different angles and located at an intermediate image plane (IIP)
enables this recombination; these slices will be assembled in a monolithic
optic called “slicer". OAPs are implemented to focus the incoming beams on the
slicer, collimate, and direct them toward the chip and the injection lens.

Finally, two sets of Tip/Tilt Mirrors (TTMs) are added to the design to ensure
the alignment of both the pupils and the images of the beams (see the
alignment procedure in Sec. 3.4).

###  3.3 Design of the Main Optics

The separation between the slices is chosen to be 1.67 mm which needs to be
demagnified to match the 125 µm chip’s inputs separation. Therefore, the
magnification $M$ of the OAP2+lens system needs to be $M=0.075$. At the chip
inputs, the beams also need to have $f_{\\#\text{, chip}}\simeq 3.93$ to match
with the result from Eq. (3) to obtain the optimal coupling efficiency with
the guided mode of the chip. A pupil diameter D requires an injection lens
with a focal length of

$f_{\text{lens}}=\text{D}\times f_{\\#\text{, chip}}.$ (8)

The focal length of the OAP2 is then given as

$f_{\text{OAP${}_{2}$}}=\frac{f_{\text{lens}}}{M}.$ (9)

But the diameter D also constraints $f_{\text{OAP${}_{2}$}}$ from the
divergence of the beams at the IIP $f_{\\#\text{, IIP}}$ which gives

$f_{\text{OAP${}_{2}$}}=\text{D}\times f_{\\#\text{, IIP}}.$ (10)

The solution for matching Eqs. (9) and (10) is then to set

$f_{\\#\text{, IIP}}=\frac{f_{\\#\text{, chip}}}{M}.$ (11)

$f_{\\#\text{, IIP}}$ is defined by the focal length of OAPs1
($f_{\text{OAP${}_{1}$}}$) and the diameter of the entrance pupil
(D${}_{\text{en}}$=12 mm)

$f_{\\#\text{, IIP}}=\frac{f_{\text{OAP${}_{1}$}}}{\text{D}_{\text{en}}}.$
(12)

From Eq (11), we can derive the value of $f_{\text{OAP${}_{1}$}}$

$f_{\text{OAP${}_{1}$}}={\text{D}_{\text{en}}}\times\frac{f_{\\#\text{,
chip}}}{M}\simeq 629.2\,\text{mm}.$ (13)

The only free parameter left for the system is the diameter D of the beam
collimated by OAP2. It determines the values for the remaining parameters
defined above. D is set to 5 mm to enable manufacturing and optical alignment
easiness, giving the following values:

$D=5\,\text{mm}\Rightarrow\left\\{\begin{array}[]{c}f_{lens}\simeq
19.66\,\text{mm}\\\ f_{\text{OAP${}_{2}$}}\simeq
262.17\,\text{mm}\end{array}\right..$ (14)

###  3.4 Alignment Procedure of the Optical System

The alignment of the system is realized by two TTMs for each beam: the first
ones (TTMs1) are placed after the dichroics; the seconds (TTMs2) are placed
after the delay lines (see Fig. 3). These mirrors are used to align the beams
both at the pupil position (i.e., at the cold stop) and at the image position
(i.e., at the chip inputs). The TTMs1 and TTMs2 are responsible for the
alignment at the pupil and image plane respectively. However, since both
mirrors are not placed in a conjugate plane, they will have a combined effect
in the alignment.

Using a coordinated motion between TTMs1 and TTMs2 (see Sec. 4.2.2), it is
possible to change the alignment of the pupils without impacting the alignment
at the chip inputs and vice versa. Changing only the pupil alignment with the
cold stop may therefore be used to regulate the injected intensity for each
beam. This would enable to actively reduce relative intensity errors mentioned
in Sec. 2.2.3 using the photometric outputs of the chip (see Sec. 5.2).

###  3.5 Pupil Matching between Asgard/HEIMDALLR and the Cold Stop

To maintain a stable injection throughout the cold stop, the alignment of the
beams at this position needs to be independent of any tip/tilt generated by
Asgard/HEIMDALLR and especially their deformable mirrors (see Fig. 3). The
solution is to reimage the deformable mirrors at the position of the cold
stop. This can be done without altering the light path by imposing convex
spherical surfaces to the slicer. With an appropriate radius of curvature, a
pupil matching between the cold stop and the deformable mirrors can be
achieved, thus providing a stable alignment at the cold stop position.

###  3.6 Longitudinal Dispersion Correctors (LDC)

Figure 4 shows the current opto-mechanical design of the LDCs. The objective
of this subsystem is to correct the instrumental birefringence, the water
vapor dispersion and the CO2 dispersion altogether for each beam individually.

For the instrumental birefringence, a solution is to use a waveplate and
rotate it on a stage until the instrumental birefringence is canceled. The
technique was proposed and tested in PIONIER (Lazareff et al., 2012), and
achieves a correction down to $<0.1$ µm OPD precision. However, to achieve a
null depth of $10^{-4}$ a correction down to $\sim 10$ nm is required. A
comprehensive study is thus necessary to assess the possibility of reaching
such performance.

The correction of water vapor dispersion has already been explored in the case
of the Keck Interferometer Nuller (Koresko et al., 2003). A similar design
with two wedges of LiNbO3 is planned here: the method is to translate one of
the wedges relative to the other to change the light dispersion impacting the
transmitted beam. The characteristics of the two wedges can be computed using
the performance simulator of Asgard/NOTT: SCIFYsim (Laugier et al., 2023). A
wedge angle of 11.5 deg and a stroke range of 25 mm are estimated for the
current design.

For the CO2 dispersion, its effect on the phase dispersion is observed near
and after 4 µm for example in the VLTI/MATISSE instrument (Lopez et al.,
2022). The technical solution proposed in Asgard/NOTT is to use a chamber
filled with CO2 at a constant pressure and change its length with a
translation stage to control the transmitted beam’s dispersion. SCIFYsim
enables estimation of the range of the translation stage at 50 mm.

After correction of the water vapor and CO2 dispersions using the LDCs, the
chromatic phase dispersion is estimated to be $<$$0.01$ rad by SCIFYsim. This
corresponds to a null depth of $2.5\times 10^{-5}$ according to Eq. (5).

Figure 4: Current opto-mechanical design for the LDC showing a) the
birefringence control system, b) the water vapor dispersion control system,
and c) the chamber for CO2 dispersion control. The water vapor dispersion
control system consists of two LiNbO3 plates: one of them is fixed and used as
the window for the chamber, the other plate is on a translation stage. The CO2
chamber is connected to a reservoir to maintain the same pressure of CO2 gas
inside while a translation stage moves the left side of the chamber to change
its volume.

###  3.7 Discussion on an Additional IIP for the Optical System

During the design phase of the instrument, the option of adding a second IIP
to the optical system was considered. On paper, this option has one crucial
advantage: it would enable the alignment of the pupils at the cold stop and
the images independently without combining the movements of two TTMs like in
the previously described system. Indeed, a tip/tilt movement generated at an
image plane will change the transversal position of the beam in the pupil
plane but not in the image plane. Using the additional IIP as a TTM thus
allows an independent alignment and relative intensity error correction at the
cold stop. The additional IIP also allows more flexibility on the position of
the pupil matching mentioned in Sec. 3.5. Using again convex-shaped spherical
surfaces, it is possible to reimage the deformable mirrors with the new IIP.
At the same time, the slicer can define the entrance pupil position of the
cold stop. They can thus be placed at the same position everywhere along the
optical axis. If a TTM is then placed at the same location, its movement will
only affect the position of the images without disturbing the pupil alignment
at the cold stop.

To realize this system, two additional OAPs are needed compared to the optical
system in Fig. 3. The new IIP is defined at the focus of the OAPs1. The two
new OAPs are then used to collimate and refocus the beams on the slicer. A TTM
is inserted in between these two OAPs, where pupil matching is done. If the
effective focal length of both OAPs is equal, this ensures conservation of the
beam divergence. In this case, the optical design in Sec. 3.3 stays valid for
the optics downstream of the slicer. A complete view of this system is drawn
in Fig. 5.

This solution is indeed efficient to perform an independent alignment between
the two TTMs. However, its realization implies an increase in cost and
complexity. Considering these factors of risk, it has been decided to not
pursue this option anymore in the final design.

##  4 Performance Evaluation

Using the design presented in Sec. 3 and realized in Zemax, this section aims
to give a performance evaluation of the system. It is then compared to the
Asgard/NOTT requirements listed in Sec. 2.

###  4.1 Estimation of the Maximum Coupling Efficiency

The coupling efficiency is used to evaluate the performance of the injection
inside the chip. Using the tools available in Zemax, it is computed for each
beam at different wavelengths in the L’-band. When the alignment and the
manufacturing of the optics are ideal, the coupling efficiency obtained
corresponds to the maximum coupling efficiency of the system as defined in
Sec. 2.1. Table 1 shows that a throughput loss of $<$23 % (i.e., maximum
coupling efficiency of 77 %) can be expected during the injection even in an
ideal case. This maximum coupling efficiency is different from the one
announced by Eq. (2) ($\sim$81 %). The main reason is the use of OAP2 and the
injection lens with off-axis beams. This generates aberrations that hamper the
coupling efficiency even for perfect alignment and the manufacturing
precision.

Table 1 also shows that the injection results in an irreducible relative
intensity error even in an ideal case. It translates into a null depth limit
from Eq. (6) for each wavelength that is well within the requirement from Sec.
2.2.3.

###  4.2 Tip/tilt Mirrors Evaluation

####  4.2.1 Tip/tilt angular resolution

The Asgard/NOTT requirement stated in Sec. 2.1 mentions that the optical
design should provide at least 95 % of the maximum coupling efficiencies (see
Table 1). This gives a criterion on the angular resolution needed for the
TTMs. The coupling efficiency computing in Zemax also considers the impact of
vignetting upstream from the inputs. The value thus gives a realistic idea of
the throughput loss due to misalignments both at the cold stop and chip
inputs.

Figure 6 shows the sensitivity of coupling efficiency with respect to the
pitch angle of TTMs1. Using the 95 % criterion defined above, an angle
accuracy of 46$\pm$0.4 µrad is required to stay within this criterion.
Assuming a safe margin of 1/10 to find the ideal alignment, an angular
resolution of 4 µrad can be used for this mirror.

Figure 5: Warm optical design and injection system of the Asgard/NOTT
instrument with an additional IIP. After the delay lines, OAPs1 are focusing
the beams on TTMs1 located in an IIP. OAPs are then collimating the light
toward TTMs2 located at the pupil matching plane. Additional OAPs are focusing
the light on a slicer located at an IIP. The slicer is recombining the four
beams which are then collimated by the OAP2 in the direction of the chip.
After traversing the window of the cryostat, the light is injected into the
chip using a telecentric lens.

Using the same method for the yaw angle and for TTMs2, the results are
summarized in Table 2. Such angular resolution can be reached using piezo
optical mounts, for example with Agilis${}^{\text{TM}}$ from Newport.

Table 1: Maximum coupling efficiencies computed by Zemax and null depths at 3.5, 3.8 and 4.0 µm wavelengths. The colors stated for the beams correspond to the colors in Fig. 3. | Wavelength
---|---
Coupling efficiency | 3.5 µm | 3.8 µm | 4.0 µm
Input 1 (blue beam) | 76.7 % | 78.6 % | 77.0 %
Input 2 (green beam) | 77.0 % | 78.6 % | 77.1 %
Input 3 (red beam) | 77.0 % | 78.5 % | 77.2 %
Input 4 (yellow beam) | 76.7 % | 78.2 % | 77.1 %
Null depth$\times 10^{-7}$ | 2.4 | 4.1 | 1.1
Figure 6: Zemax simulation of the coupling efficiencies in the four chip inputs as a function of the pitch angle from TTMs1. The colors correspond to the beams from Fig. 3. Table 2: Simulation results of angular resolution required for the two TTMs to ensure at least 95 % of maximum coupling efficiency. A safe margin of about 1/10 is assumed for the final resolution requirement. | TTM1 | TTM2
---|---|---
Pitch angle at 95 % [µrad] | 47$\pm$0.9 | 40$\pm$2.0
Yaw angle at 95 % [µrad] | 46$\pm$1.4 | 38$\pm$1.7
Required resolution [µrad] | 4 | 4

####  4.2.2 Tip/tilt angular range

The lower limit for the angular range of the TTMs depends on the sky coverage
necessary for the instrument. Indeed, the movement from TTMs1 can be used to
change the part of the sky injected into the chip inputs, enabling a scan of
the field of view. The alignment between the target and the center of the
field of view is realized in the H-band by the Asgard/Baldr (Martinod et al.,
2023) instrument. However, the light dispersion from the atmosphere generates
an angular offset between the H-band and the L’-band where Asgard/NOTT
operates, as shown in Fig. 7 (Laugier et al. in prep.). When observing with
the UTs at 30 deg elevation, this offset is at most 150 mas projected on-sky.
As a safe margin, the system should then be able to have a sky coverage of
$\pm$300 mas around the center of the field of view to ensure a good alignment
with the target in the L’-band.

Figure 7: Simulation of the Asgard’s field of view projected on-sky with the
diffraction pattern’s size and position for each instrument (Hi-5 being the
older name of NOTT). The simulation corresponds to an observation with the UTs
at 30 deg elevation. The dispersion from Earth’s atmosphere shifts the light
from the L’-band by an angle of at most 150 mas compared to the light from the
H-band.

To calculate the resulting physical angle from this offset in the design, the
étendue $U$ can be used. $U$ is given by the surface of the pupil
$S_{\text{pupil}}$ and the solid angle $\Omega$:

$U=S_{\text{pupil}}\times\Omega,$ (15)

and remains constant in an optical system (Fischer et al., 2008). The exit
pupil of the telescope shares the same étendue as the entrance pupil of the
telescope. The physical angle $\theta$ at the exit pupil can thus be
calculated from the angle on the sky $\theta_{\text{sky}}$ and the pupil
diameters $D_{\text{entrance}}$ and $D_{\text{exit}}$

$\displaystyle\theta$
$\displaystyle=\arcsin{\left(\frac{D_{\text{entrance}}}{D_{\text{exit}}}\times\sin{\left(\theta_{\text{sky}}\right)}\right)}.$
(16)

The objective is to adapt the system in order to ensure
$\theta_{\text{sky}}=\pm$300 mas in every direction on the sky. For the UTs,
$D_{\text{entrance}}=$ 8.2 m and $D_{\text{exit}}=$ 18 mm.

To compensate for $\theta_{\text{sky}}$, the TTMs1 need to be tilted by an
angle $\theta_{\text{TTM${}_{1}$}}=\pm$0.03 deg. But because TTMs1 are not at
the pupil matching position (see Sec. 3.4), this results in a misalignment at
the cold stop as shown in Fig. 8, generating vignetting. The second step is
then to use TTMs2 to realign the beams at the cold stop. This can be done with
an angle $\theta_{\text{TTM${}_{2}$}}=\pm$0.011 deg but it also simultaneously
moves the spot by $\sim$17 µm from the chip inputs. The solution is therefore
to combine the movements of both TTMs until the four beams are aligned at the
chip and the cold stop. Table 3 summarizes the values for the angles
$\theta_{\text{TTM${}_{1}$}}$ and $\theta_{\text{TTM${}_{2}$}}$ for which an
optimal alignment is achieved in the case of $\theta_{\text{sky}}=\pm$300 mas
in both directions on the sky. Table 3 also shows the ratio of vignetted light
when using only TTMs1 for the sky coverage. This ratio goes up to $\sim$20 %
in every direction. The conclusion is that even when only using TTMs1 to
search for a target in the field of view, $\sim 80\,\%$ of the flux should
always be transmitted through the cold stop, enabling the detection of the
target.

The angular range required from Table 3 for both TTMs can be reached using the
piezo optical mounts Agilis${}^{\text{TM}}$ from Newport for example.

Figure 8: Misalignment at the cold stop when using only the movements of TTMs1 for sky coverage with $\theta_{\text{sky}}=$ 300 mas. The colors correspond to the beams from Fig. 3. Table 3: Zemax results of angular range required for the two tip/tilt mirrors. The required range is taken with a safe margin compared to the results of the simulation. | $\theta_{\text{TTM${}_{1}$}}$ (µrad) | $\theta_{\text{TTM${}_{2}$}}$ (µrad) | Vignetting (%)
---|---|---|---
Pitch angle for $\theta_{\text{sky}}$=+300 mas | -812$\pm$8.7 | -310$\pm$14 | 19$\pm$1.1
Pitch angle for $\theta_{\text{sky}}$=-300 mas | 812$\pm$8.7 | 314$\pm$17 | 18$\pm$1.0
Yaw angle for $\theta_{\text{sky}}$=+300 mas | -807$\pm$7.6 | -314 | 18$\pm$0.6
Yaw angle for $\theta_{\text{sky}}$=-300 mas | 807$\pm$38 | 314$\pm$25 | 19$\pm$0.6
Required angular range [µrad] |  $\pm$1000 |  $\pm$500 |

###  4.3 Impact of Fresnel Diffraction

####  4.3.1 Definition

The Fresnel diffraction model can be used to estimate the encircled energy
loss at the pupil plane in the near field. This diffraction may have a
significant impact especially when the beam diameter is small ($D=5$ mm). The
distance along the optical axis z where this approximation is valid is between
$z_{\text{min}}=0.62\sqrt{\frac{(D/2)^{3}}{\lambda}}$ and
$z_{\text{max}}=\frac{2(D/2)^{2}}{\lambda}$ (Balanis, 2005).

From the Fresnel approximation, the encircled energy $\eta$ is estimated with
the relation:

$\displaystyle\eta(z)$
$\displaystyle=\frac{\iint_{-D/2}^{+D/2}E(x,y,z)^{2}dxdy}{\iint_{-\infty}^{+\infty}E(x,y,z)^{2}dxdy}.$
(17) $\displaystyle\text{with:\quad}E(x,y,z)$
$\displaystyle=\frac{e^{ikz}}{i\lambda
z}\iint_{-\infty}^{+\infty}E(x^{\prime},y^{\prime},0)e^{\frac{ik}{2z}\left[(x-x^{\prime})^{2}+(y-y^{\prime})^{2}\right]}dx^{\prime}dy^{\prime}.$
(18)

$E(x,y,z)$ represents the electric field at every spatial point and $k$ its
wavenumber.

####  4.3.2 Results

To simplify the problem, $\eta$ from Eq. (17) is computed in 1D (with $y=0$)
at $\lambda=4$ µm. Figure 9 shows the results for the two beam diameters in
the system: $D=12$ mm before OAPs1 and $D=5$ mm between OAP2 and the injection
lens. The results in energy losses for each diameter are presented in Table 4.

###  4.4 Impact of Diffraction from the Slicer and the Cold Stop

The diffraction pattern at the chip’s inputs is expected to be altered by the
cropping of the intensity distribution at the image plane (from the slicer’s
segments) and at the pupil plane (from the cold stop). This alteration may
impact the maximum coupling efficiency described in Sec. 2.1. Figure 10 shows
the resulting intensity distribution at the chip’s input after the cropping
from the slicer’s segment and the cropping from the cold stop. Computing the
coupling efficiency $\rho$ between the altered diffraction pattern and the MFD
gives an estimated loss of 0.08 % due to the diffraction from the slicer’s
segment and the cold stop. This impact is well within our requirement (see
Sec. 2.1).

Figure 9: Simulated encircled energy from Fresnel diffraction with respect to propagation length $z$ at $\lambda=4$ µm. The figure shows the propagation length in the optical design for the beams at 12 and 5 mm diameter. Table 4: Estimated energy loss $\eta$ from Fresnel diffraction for beam diameters of 12 mm and 5 mm at $\lambda=4$ µm. The propagation length $z$ in both cases is indeed inside the validity range between $z_{\text{min}}$ and $z_{\text{max}}$. | $z_{\text{min}}$ [m] | $z_{\text{max}}$ [m] | $z$ [m] | $\eta(z)$
---|---|---|---|---
D=12 mm | 0.14 | 18 | 10.8 | 1.5 %
D=5 mm | 0.04 | 3.1 | 0.85 | 0.29 %
Figure 10: (a) Logarithmic intensity distribution at the slicer for one mirror
segment. (b) Linear intensity distribution at the cold stop with diffraction
from the slicer segment. The red circles indicate the cold stop diameter and
the central obstruction. (c) Logarithmic intensity distribution at the cold
stop with diffraction from the slicer segment. The red circles indicate the
cold stop diameter and the central obstruction. (d) Logarithmic intensity
distribution at the chip’s input with diffraction from the slicer segment and
the cold stop. The dotted circle indicates the MFD of the chip’s waveguide.

###  4.5 Impact of Mode Field Diameter (MFD) Error and Ellipticity

In this section, the MFD is assumed to be circular with a value of 22.45 µm.
Measurements (Gretzinger et al., 2019) indicate an elliptical shape of
22.3x22.6 µm with an error of $\pm$0.1 µm. The MFD ellipcticity and error may
have an impact on the maximum coupling efficiency that can be achieved. This
impact is estimated by considering two extreme cases: a circular MFD of 22.2
µm and 22.7 µm respectively. In an ideal alignment situation, the worst case
scenario is a MFD of 22.2 µm with a loss of 0.5 % on the maximum coupling
efficiency at 3.5 µm wavelength. The impact of waveguide ellipticity is thus
assumed to be $<$0.5 %, which is well within our requirement (see Sec. 2.1).

###  4.6 Fabrication Errors Evaluation

####  4.6.1 Definitions

We identify two types of fabrication errors that affect the quality of optics:
the figure errors and the surface errors. The first one represents the
deviation of the full-figure form of the surface from the ideal one. This
induces a deviation from the theoretical radius of curvature and conic. The
second one represents the local deviations of the surface compared to the
full-figure form inducing local wavefront errors.

####  4.6.2 Impact of figure errors

The figure errors on the radius of curvature of a surface are estimated to be
less critical as they do not generate aberrations. The change of focal length
from these errors can be measured by the manufacturer and later compensated in
the optical design. However, this error needs to be constrained as it also
changes the value of $NA$ and therefore can impact the performance of the
injection into the chip. The tolerances of the curvature for the aspherical
optics of the system are summarized in Table 5. The four OAPs1 are assumed to
share the same radius of curvature if they are manufactured from the same
parabola.

####  4.6.3 Impact of surface errors

Surface errors generate wavefront errors which result in aberrations when
focusing the beams at the chip inputs. These aberrations are hampering the
expected coupling efficiencies. This energy loss from wavefront errors can be
approximated by the Strehl ratio.

For small aberrations (normalized wavefront error $\omega\leq 1/10$ RMS), this
Strehl ratio S can be estimated using Maréchal’s approximation

$S\sim e^{-(2\pi\omega)^{2}},$ (19)

with the normalized wavefront error RMS $\omega$ corresponding to

$\omega=\frac{\lambda_{\text{ref}}}{N}.\frac{1}{\lambda_{0}},$ (20)

with $\lambda_{\text{ref}}$ the reference wavelength (usually
$\lambda_{\text{ref}}=633$ nm), N the surface quality index and $\lambda_{0}$
the central wavelength ($\lambda_{0}=3.75$ µm in the L’-band).

For a multiple optics combination, the total Strehl ratio of a number of
optics $N_{\text{optics}}$ is

$\displaystyle S$ $\displaystyle\sim e^{-(2\pi\omega_{t})^{2}},$ (21)
$\displaystyle\text{with:}\quad\omega_{t}$
$\displaystyle=\sqrt{\sum_{i}^{N_{\text{optics}}}\omega_{i}^{2}}.$ (22)

Table 5: Simulation results of radius of curvature tolerances for the curved optics to ensure at least 95 % of maximum coupling efficiency across the L’-band. A safe margin of 1/10 is assumed for the final tolerances. | OAPs1 | OAP2 | Lens
---|---|---|---
Tolerances at 95 % [mm] |  $\pm$42 |  $\pm$17 |  $\pm$0.06
Tolerances at 95 % [%] |  $\pm$3.4 |  $\pm$3.3 |  $\pm$0.2
Final tolerances [%] |  $\pm$0.3 |  $\pm$0.3 |  $\pm$0.02

For reflective optics, the surface errors $\delta$ and wavefront errors
$\omega$ are related by a factor 2

$\omega=2\times\delta.$ (23)

For refractive optics, the surface errors $\delta$ and wavefront errors
$\omega$ are related by a factor depending on the glass index of refraction n

$\omega=(n-1)\times\delta.$ (24)

In some cases, the surface errors are given by manufacturers in peak-to-valley
(P-V). The RMS equivalents are typically approximated to a range going from
$\frac{1}{5}$ to $\frac{1}{3}$ of the P-V values (Fischer et al., 2008).

####  4.6.4 Estimation of the maximum surface errors

The objective is to find the maximum surface errors for each optic that reach
the objective of a Strehl ratio $S_{\text{NOTT}}>95$ % while minimizing the
manufacturing challenges. In Fig. 3, the wavefront errors from optics prior to
the dichroics are corrected by the Zernike Wavefront Sensor in Asgard/Baldr
using the deformable mirrors of Asgard/HEIMDALLR. Only the optics after and
including the dichroics are considered to estimate $S_{\text{NOTT}}$. This
comprises the OAPs1, OAP2 and slicer (three reflective surfaces), the cryostat
window and injection lens (four refractive surfaces), the dichroics (two
refractive surfaces) and the flat mirrors (five reflective surfaces). This
estimation does not include the wavefront errors generated by the LDCs.

The OAPs1, OAP2 and slicer are custom optics designed with high surface
quality $\left(\frac{\lambda_{\text{ref}}}{40}\,\text{RMS}\right)$. The
injection lens is designed with an expected surface quality of
$\frac{\lambda_{\text{ref}}}{20}$ RMS from ZnSe ($n=2.4178$ at $\lambda=3.75$
µm). The window of the cryostat is chosen with a transmitted wavefront error
of $\frac{\lambda_{\text{ref}}}{10}$ P-V. Finally the dichroics are made of
CaF2 ($n=1.4120$ at $\lambda=3.75$ µm) with an expected surface quality of
$\frac{\lambda_{\text{ref}}}{10}$ RMS. These hypotheses allow to study whether
the use of commercially available off-the-shelf optics for the flat mirrors is
possible. Figure 11 shows that a Strehl ratio $S_{\text{NOTT}}=97.5\,\%$ is
expected with a surface quality of $\frac{\lambda_{\text{ref}}}{20}$ P-V for
the flat mirrors. This surface quality is indeed obtainable, e.g., from Edmund
Optics.

The wavefront correction done by Asgard/Baldr is expected to reach a Strehl
ratio $S_{\text{Baldr}}$ of $>$40%̇ and $>$50 % for the UTs and the ATs,
respectively, in the J-band (Martinod et al., 2023). This corresponds to a
Strehl ratio of $>$90.3 % and $>$92.6 %, respectively, in the L’-band (or a
throughput loss of $<$9.7 % and $<$7.4 %, respectively, during the injection
in the chip). Using Eq. (21), the Strehl ratio expected at the chip’s inputs
$S$ corresponds to

$S=S_{\text{Baldr}}\times S_{\text{NOTT}},$ (25)

where $S$ is thus estimated to be $>$88.0 % and $>$90.3 % for the UTs and the
ATs respectively in the L’-band.

Figure 11: Strehl ratio estimation at the chip input as a function of the
number of flat surfaces and their surface quality. The error bars come from
the uncertainty in the translation from P-V to RMS values. The estimation is
made at 3.75 µm wavelength with $\lambda_{\text{ref}}=633nm$.

###  4.7 Impact of Polarization Errors

From the optical design simulated on Zemax, the different polarization errors
can be estimated with their impact on the null depth (see Sec. 2.2.3). This
estimation is done considering three linear polarization states at the
beginning of the design: $|V\rangle$, $|H\rangle$, and $|D\rangle$ for
“vertical", “horizontal", and “diagonal" polarization states respectively. At
the inputs of the chip, the final polarization angle is calculated for the
$|V\rangle$ and $|H\rangle$ cases and the difference between the four beams
gives a null depth limit using Eq. (7). In the $|D\rangle$ case, the phase
delay between $|V\rangle$ and $|H\rangle$ is also calculated at the inputs of
the chip and for each beam. This phase delay is due to the birefringence of
the optical system and also contributes to limiting the null depth. Table 6
summarizes the results for the three cases. As first expected, birefringence
represents the main null depth limitation from polarization errors
($<3.7\times 10^{-3}$), which confirms the need for a birefringence correction
(see Sec. 3.6).

For the polarization errors induced by the VLTI optics, a recent study
estimated that the differential birefringence between the UTs generates a
phase error of $<$$0.0175$ rad at $2.2$ µm wavelength (Widmann et al., 2023).
This corresponds to an OPD error of $<$$6$ nm. In the L’-band, the phase error
is $<$$0.01$ rad. This limits the null depth to $2.6\times 10^{-5}$, which is
within our requirements. For the ATs, larger differential birefringence is
expected and can be corrected given the model of its Jones matrix. Other
polarization errors (degree of polarization, polarization angle, etc.) may
arise from the VLTI optics but a more extensive study is needed to estimate
them.

###  4.8 Final Throughput Loss and Null Depth Budget

This section summarizes the impact of the different sources impairing the
optical throughput (Table 7) and the null depth from the optical system (Table
8). Table 8 highlights that OPD errors and birefringence are expected to be
the main limitations for the null depth of the instrument. After correction by
the LDCs, the null depth from birefringence is expected to be $\leq 10^{-4}$,
which leaves the OPD errors as the main limitation (assuming that the null
depth is not limited by the photonic chip). This confirms that reducing OPD
errors at the VLTI is the best way to improve high-contrast imaging with
nulling interferometry.

The impact of alignment and figure errors is calculated with Monte Carlo
method on Zemax. Using the values from Table 2 and 5, less than 1 % throughput
loss is expected across the bandwidth and for the four beams.

The throughput loss during the injection is estimated at $<$6.4 % of the best
coupling efficiency. While it is above the objective, it remains less than the
impact from wavefront errors. The optical design is thus assumed to have an
optimized injection efficiency.

An estimation of the Fresnel losses gives a throughput loss of 40 %. It
represents the losses due to reflections and transmissions from the
Asgard/NOTT optics (including the LDCs). The global throughput loss of the
instrument is then estimated at $<$60 %.

Table 6: Simulated polarization errors after propagation through the optical system. $|V\rangle$, $|H\rangle$, and $|D\rangle$ represent the “vertical", “horizontal", and “diagonal" linear polarization states at the beginning of the system. For the $|V\rangle$ and $|H\rangle$ cases, the simulation gives the polarization angle difference at the chip’s inputs. For the $|D\rangle$ case, the simulation gives the phase delay between $|V\rangle$ and $|H\rangle$ at the chip’s inputs. The colors stated for the beams correspond to the color in Fig. 3. | Angle at the chip [mrad] | Phase delay at the chip [mrad]
---|---|---
Starting polarization state | $|V\rangle$ | $|H\rangle$ | $|D\rangle$
Input 1 (blue beam) | $4.775$ | $4.611$ | $<87.3$
Input 2 (green beam) | $4.761$ | $4.564$ | $<122$
Input 3 (red beam) | $4.753$ | $4.559$ | $<122$
Input 4 (yellow beam) | $4.784$ | $4.625$ | $<87.3$
Null depth | $2.5\times 10^{-10}$ | $10^{-9}$ | $<3.7\times 10^{-3}$
Table 7: List of phenomena impairing the optical throughput. A first summary is given considering only the phenomena impacting the injection losses that we minimize (see Sec 2.1). A second one including all the combined losses. The Fresnel losses correspond to the reflectivity and the transmittance of the Asgard/NOTT optics (including the LDCs). | Throughput loss [%]
---|---
Fresnel diffraction | $1.8$
Slicer & Cold stop diffractions | $0.8$
MFD ellipticity | $<0.5$
Alignment & figure errors | $<1$
Surface errors | $2.5$
Total 1 | $<6.4$
Maximum coupling efficiency | $<23$
Wavefront errors (after Asgard/Baldr correction) |  $<9.7$ [UTs] and $<7.4$ [ATs]
Fresnel losses | $40$
Total 2 | $<60$
Table 8: List of phenomena impairing the null depth performance. The birefringence value before correction includes the errors from the UTs and from Asgard/NOTT (the errors from the UTs being negligible). The values corresponding to the intensity error and the birefringence after correction are goals and their estimation is part of future work. The total is calculated assuming the goals for intensity errors and birefringence are met. | Null depth
---|---
OPD error (after Asgard/HEIMDALLR) | $2\times 10^{-3}$
Intensity error (objective) | $\leq 10^{-4}$
Longitudinal dispersion from H2O & CO2 | $2.5\times 10^{-5}$
Polarization angle difference | $10^{-9}$
Birefringence before [after] LDCs correction |  $<3.7\times 10^{-3}$ [$10^{-4}$]
Total | $2.2\times 10^{-3}$

##  5 Discussion

###  5.1 Beam Swapping Capacity

Because Asgard/NOTT uses a two-stage nulling recombination scheme, its
sensitivity map depends on the order in which the telescopes are recombined
(Laugier et al., 2023). Therefore, the optical design must enable the swap of
the light beams two-by-two before the couplers inside the chip. This swap
would allow to ensure more flexibility to detect an exoplanet.

Several internal solutions have been considered, including beam swapping with
different photonic chips, with the slicer, or with mirror permutation, but
none of these sufficiently fit the instrumental constraints. An alternative
solution is to perform beam swapping using the M16 mirrors of the VLTI and
moving the VLTI delay lines accordingly to correct for the OPD. This solution
would make Asgard/NOTT simpler and easier to implement at Paranal but also
less automatic.

###  5.2 Intensity Matching

Section 2.2.3 presents the requirement in terms of relative intensity error
$\sigma_{I}$ between the different injected beams. Table 1 shows that the
expected contribution from the optical design is well within the null depth
requirement of $10^{-4}$. However, other sources of intensity mismatch may
contribute, like the difference of throughput between the telescopes of the
VLTI. Measurements done in April 2020 show a difference of throughput between
the four ATs which gives a relative intensity error of almost 20 % in the
K-band, i.e., a null depth of $2.5\times 10^{-3}$ (X. Haubois, personal
communication, December 14, 2023). This example motivates the need for a
strategy to maintain $\sigma_{I}<4\,\%$.

The intensity mismatch can be directly measured using the photometric outputs
of the chip (see Fig. 1).To actively reduce the mismatch, several strategies
are considered for the instrument:

  * •

Using motorized iris diaphragms, the pupil diameter of each beam can be
actively adjusted to minimize $\sigma_{I}$.

  * •

Using the TTMs, misalignments between the pupil planes and the cold stop can
generate vignetting to minimize $\sigma_{I}$ (see Sec. 3.4).

However both of these solutions cannot provide a chromatic correction of the
intensity injected into the chip. In order to enable a wavelength-dependent
control of the intensity as well as the phase, an adaptive nuller (Peters et
al., 2010) is considered. The system would be a future improvement for the
instrument installed in between OAP2 and the cryostat.

##  6 Conclusion

The finalized version of the Asgard/NOTT warm optical design and injection
system is presented in this paper. This work gives a complete description of
the optical layout as well as explanations for the choices made during the
design process. The performance section lists different phenomena that can
hamper the throughput of the instrument and/or the null depth. The impact for
each of these is estimated and an overall null depth limit of $2.2\times
10^{-3}$ can be given for the instrument with a throughput loss of
$<$$60\,\%$. The null depth limit is expected to be dominated by the OPD error
from the atmosphere and the vibrations at the VLTI, even after correction by
the adaptive optics of Asgard/HEIMDALLR. The loss of throughput is dominated
by the Fresnel losses and wavefront errors after the Asgard/Baldr correction.
For the injection into the chip, the throughput loss is estimated to be $<$6.4
% of the maximum coupling efficiency. Even if the estimation is above the
requirement of 5 % stated in Sec. 2.1, it remains below the loss from
wavefront errors. The optical system and its tolerancing are thus assumed to
be adapted for efficiently injecting the VLTI beams into the beam combiner of
the instrument.

Some aspects of the design are still risky and subject to change in the future
like the LDCs for which a prototype demonstration is required. Combining the
two TTMs also needs a comprehensive study to allow for alignment and possible
intensity matching. These will be implemented in the test bench of the
Asgard/NOTT instrument located at KU Leuven.

### Acknowledgments

SCIFY has received funding from the European Research Council (ERC); Award no.
CoG – 866070 under the European Union’s Horizon 2020 research and innovation
program. M-A.M. has received funding from the European Union’s Horizon 2020
Research and Innovation Program; Award No. 101004719. Acknowledgment to J.B.
Le Bouquin (IPAG/OSU) for discussions about the injection alignment.

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* $\begin{cases}\phantom{+}x^{*(3)}+x^*z^{*(2)}+y^*z^{*(2)}-2x^{*(2)}w^*-2x^*y^*w^*\\-2y^{*(2)}w^*+3x^*w^{*(2)}+3y^*w^{*(2)}-3w^{*(3)}\end{cases}$

* $-x^{*(2)}y^*-x^*y^{*(2)}-y^{*(3)}+x^*z^*w^*+y^*z^*w^*+w^{*(3)}$

* $\begin{cases}\phantom{+}x^{*(3)}+x^*z^{*(2)}+y^*z^{*(2)}-x^{*(2)}w^*-x^*y^*w^*-y^{*(2)}w^*\\+2x^*w^{*(2)}+2y^*w^{*(2)}-3w^{*(3)}\end{cases}$

* $\begin{cases}\phantom{+}7x^{*(3)}+6x^{*(2)}y^*+6x^*y^{*(2)}+6y^{*(3)}+2x^{*(2)}z^*+2x^*y^*z^*\\+2y^{*(2)}z^*+2x^{*(2)}w^*+2x^*y^*w^*+

§.§.§ The corresponding Gorenstein ideal has 6 minimal generators.

*       $-3x^{*(3)}-2x^{*(2)}y^*+2x^{*(2)}z^*+x^*z^{*(2)}-x^{*(2)}w^*-x^*y^*w^*+x^*z^*w^*+z^{*(2)}w^*$

* $\begin{cases}-2x^{*(3)}-2x^{*(2)}y^*+2x^{*(2)}z^*+x^*z^{*(2)}-x^{*(2)}w^*-x^*y^*w^*+x^*z^*w^*\\+z^{*(2)}w^*+x^*w^{*(2)}+w^{*(3)}\end{cases}$

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