# Potential of commercial SiN MPW platforms for developing mid/high-resolution
integrated photonic spectrographs for astronomy

Pradip Gatkine  1,* Nemanja Jovanovic  1  Christopher Hopgood2  Simon Ellis2
Ronald Broeke3  Katarzyna Ławniczuk3  Jeffrey Jewell4 and J. Kent Wallace4
Dimitri Mawet  1,4  1Department of Astronomy, California Institute of
Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
2Australian Astronomical Optics, Macquarie University, 105 Delhi Rd, North
Ryde, NSW 2113, Australia
3Bright Photonics BV, Horsten 1, 5612 AX Eindhoven, The Netherlands
4Jet Propulsion Laboratory, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
<EMAIL_ADDRESS>
###### Abstract

Integrated photonic spectrographs offer an avenue to extreme miniaturization
of astronomical instruments, which would greatly benefit extremely large
telescopes and future space missions. These devices first require optimization
for astronomical applications, which includes design, fabrication and field
testing. Given the high costs of photonic fabrication, Multi-Project Wafer
(MPW) SiN offerings, where a user purchases a portion of a wafer, provide a
convenient and affordable avenue to develop this technology. In this work we
study the potential of two commonly used SiN waveguide geometries by MPW
foundries, i.e. square and rectangular profiles to determine how they affect
the performance of mid-high resolution arrayed waveguide grating spectrometers
around 1.5 $\upmu$m. Specifically, we present results from detailed
simulations on the mode sizes, shapes, and polarization properties, and on the
impact of phase errors on the throughput and cross talk as well as some
laboratory results of coupling and propagation losses. From the MPW-run
tolerances and our phase-error study, we estimate that an AWG with R $\sim$
10,000 can be developed with the MPW runs and even greater resolving power is
achievable with more reliable, dedicated fabrication runs. Depending on the
fabrication and design optimizations, it is possible to achieve throughputs
$\sim 60\%$ using the SiN platform. Thus, we show that SiN MPW offerings are
highly promising and will play a key role in integrated photonic spectrograph
developments for astronomy.

††journal: osajournal††articletype: Research Article

##  1 Introduction

The ability to spectroscopically characterize objects is one of the most
powerful diagnostic tools in an astronomer’s arsenal. It can be used to
determine the chemical composition and abundances of constituents, as well as
the relative velocity of the object with respect to the observer, which can
constrain its distance. By combining this data with imaging, it is possible to
get a global understanding of a wide variety of astrophysical targets spanning
from planets all the way up to the structure of the Universe.

In the ultraviolet (UV) to mid-infrared spectral regions, astronomers
typically rely on spectrographs constructed with classical optics such as
lenses, mirrors, gratings, and prisms. Prior to the invention of adaptive
optics (AO), all ground-based telescopes operated in the seeing limit,
providing a large point spread function (PSF) to the slit of a spectrograph.
The size of a seeing-limited spot is dependent on atmospheric turbulence,
which varies temporally within a night and from site-to-site. The resolving
power (R) of a seeing-limited spectrograph depends on the input spot size,
which means the design and slit size of such a spectrograph needs to be
optimized from site-to-site. In addition, the diameter of a seeing-limited
spectrograph collimator increases as a function of the telescope diameter,
which has the knock on effect of increasing the size of subsequent optics and
hence the volume, mass and cost of the spectrograph [1]. The larger volume of
a spectrograph adversely affects its thermal and mechanical stability,
reducing the accuracy of the spectroscopic measurement. Therefore, instrument
size and cost are critical issues for the next-generation extremely large
telescopes (ELTs), which have primary mirror diameters of $>24$ m.

On the other hand, the size of a diffraction-limited instrument is independent
of the site and telescope characteristics. This has two important properties:
1) a diffraction-limited spectrograph will have the same performance (R) at
all telescopes and sites and 2) the instrument will be the most compact it can
be for a given set of specifications [2]. These characteristics mean that
diffraction-limited spectrographs could be replicated reducing the development
costs. In addition, very high R spectrographs (R > 100,000), designed for
large telescopes (> 8 m), that would otherwise be extremely large ($>$ 15 m3)
and costly ($>$ $10 million) [3], can now be made compact and cost-effective.
The advantages of diffraction-limited spectrographs have been noticed in the
community. RHEA [4], iLocator [5], PARVI [6], and HISPEC/MODHIS [7] have all
adopted this architecture. Operating across $y$–$K$ bands, HISPEC/MODHIS will
conduct $R>100,000$ science at both Keck and TMT Observatories (10 m and 30 m
apertures respectively), while costing a fraction of first-light seeing-
limited spectrographs under development for ELTs.

To enable a diffraction-limited instrument architecture however, the telescope
itself must operate at the diffraction-limit, which is possible in three ways:
1) operating in space free from the effects of the atmosphere, 2) having a
telescope diameter smaller than the Fried parameter, which implies a small
diameter ($<0.5$ m) telescope, and/or 3) using an AO system to correct for the
atmosphere. The third option is the most practical approach to combining a
high-resolution spectrograph with a large collecting area telescope. Since AO
systems are imperfect, and its beneficial to mount the spectograph off
telescope, all solutions that rely on a diffraction-limited spectrograph
design typically use single-mode fibers (SMF) to collect the light from the
focal plane and route it to the instrument. The SMF acts as a spatial filter
cleaning up any residual imperfections in the wavefront at the expense of
coupling efficiency. As the coupling efficiency into a SMF is directly
proportional to the Strehl ratio, the slowly improving Strehl ratio over the
past 25 years has limited the applicability of this approach in the past. For
low Strehl ratios, it is more efficient to capture the focused light into a
multi-mode fiber (MMF) and use a photonic lantern to adiabatically split the
MMF into multiple diffraction-limited SMFs. However, this option requires
using multiple diffraction-limited spectrographs for each SMF [8, 9]. With the
advent of “extreme AO" systems [10, 11, 12], the Strehl ratio has finally
reached $90\%+$ in the near IR (NIR) in better than median seeing conditions
on bright stars, which means efficient direct coupling to an SMF (without a
photonic lantern) is finally a reality [13, 14].

When operating at the diffraction-limit, it is possible to consider replacing
the bulk optics with photonic circuits. In terms of spectrographs, there are
numerous technologies that could be used including planar concave gratings,
arrayed waveguide gratings (AWGs), and Fourier transform spectrometers, for
example [15]. The additional benefit of using these devices is an even more
compact instrument than can be achieved with bulk optics, reducing volume and
mass, and increasing thermal and mechanical stability. At the same time, the
single-mode regime offers immense flexibility to manipulate the light (before
the spectroscopy step) such as adding in-line notch filters to specifically
suppress atmospheric emission lines using waveguide/fiber Bragg gratings [16,
17], precision wavelength calibrators [18], interferometers [19, 20], and
photonic nullers to suppress unwanted starlight in a binary or a star-planet
system [21]. Thus, the benefits of miniaturization and additional light-
manipulation capabilities promote applications for both ground-based and
space-borne instruments.

Among the photonic spectrograph technologies, AWGs are the most promising for
astronomy. AWGs have been extensively developed by the telecommunications
industry and as such are the most mature. For this reason there have been
previous efforts to characterize, optimize and build such devices into
prototypical instruments for astronomy applications [22, 23, 24, 25],
culminating in the demonstration of a highly efficient single-object
spectrograph on an 8-m telescope [26]. These efforts have highlighted the
potential for this technology in astronomy. The next steps are to customize
the spectrograph characteristics to better match astronomical applications.

There are many properties that can be optimized in such circuits including the
R, the free-spectral range (FSR), the throughput, sensitivity to phase errors,
spectral dropout, numerical aperture (NA) and mode-field diameter (MFD) for
coupling into/out of devices, and polarization sensitivity. Although the
science drives the requirements on these parameters, there are strong
constraints imposed by the material platform and the manufacturing
limitations. So it is critical to design the circuits with a holistic view of
all of these terms in mind.

To achieve the highest level of performance with the least amount of
iterations, it is pragmatic to lean on commercial vendors with significant
fabrication expertise with a given material platform. Commercial vendors
understand that photonic prototyping can be very costly ($\gtrsim$ USD$50k)
and some offer Multi-Project Wafers (MPWs), which allow a user to purchase a
section of a wafer, thus greatly reducing the cost of accessing the platform
($\sim$ USD$10k to $20k). This is ideal for developing circuit architectures
and trading parameters of the design. These MPWs are offered on platforms like
silicon nitride (SiN), Silicon-on-insulator (SOI), Indium phosphide (InP) and
so on, but not silica-on-silicon (SOS), to the best of our knowledge. Of these
platforms, SiN offers a moderate index contrast, which is ideal for coupling
to/from [27], while still being able to maintain small bend radii and hence
compact circuits. There are two main classes of waveguide geometries offered
by SiN MPW foundries, namely square and rectangular core shapes and both show
great promise for enabling astronomers to rapidly prototype devices.

In this body of work, we demonstrate through simulation, the potential of
developing mid-to-high resolving power (R = 5k to 10k) AWGs based on SiN
waveguides using the square and rectangular geometries of commercial MPWs. We
explore the properties of the devices with realistic waveguide geometries and
constraints imposed by the platforms in order to provide a potential future
designer with a sense of the possibilities. In Section 2, we outline in detail
some of the attributes of the two platforms. In Section 3, we describe the
details of the design and simulation process. Section 4 summarizes the main
properties of the devices, while Section 5 highlights the impacts of these
properties. Section 6 provides some ideas for future development of these
platforms to enhance astronomical applications.

##  2 Commercial platform attributes

Exploiting commercial foundries for prototyping of photonic circuits brings
significant experience and process control, but can be a costly approach. As
outlined in the previous section, this is where MPWs can help. SiN is the
optimal platform to consider for astronomical applications thanks to its
favorable throughput compared to others (e.g. SOI, InP, etc.). In addition,
SiN also has the following beneficial properties: a large spectral
transparency window allowing a broad operational band ($\lambda$ = 400$-$2400
nm) [28], a higher index contrast compared to silica-on-silicon allowing tight
bends with $<$ 100 $\upmu$m radius with low radiation losses [29], low
propagation losses, adjustable MFDs to efficiently couple to optical fibers,
and a relatively high $dn/dT$ for realizing compact thermal phase shifters
[30, 31]. The high index contrast/small bend radius makes it possible to
design very compact AWGs, which in turn minimizes propagation losses as well.
Having the flexibility to adjust the MFD of the guides at the interfaces to
the chip is also critical to minimizing coupling losses and maximizing the
overall system throughput.

There are numerous commercial foundries that can support MPW runs in SiN
including but not limited to Ligentec 111https://www.ligentec.com/, LioniX
222https://www.lionix-international.com/, LETI 333https://europractice-
ic.com/, and AMF. One differentiating feature of the offerings is the
waveguide geometry. For example, Ligentec specializes in making square
waveguides with dimensions of $800\times 800$ nm, while most other vendors
focus on rectangular waveguides with single and dual stripes with typical
thicknesses of 50$-$250 nm/stripe. Indeed, making thick layers of SiN is very
challenging as the structures typically suffer large stresses and crack, but
Ligentec have developed a proprietary approach that circumvents this allowing
for square waveguides with ultra-low propagation losses ($\sim$0.1 dB/cm).
Square waveguides offer polarization insensitivity and hence, are better-
suited for the unpolarized sources in astronomy. We will explore both options
in the simulations that follow.

##  3 AWG Design and Simulations

In this section, we introduce the AWG properties and describe our design and
simulation schemes for the AWG spectrographs.

###  3.1 AWG properties

The general scheme and terminology of an AWG-based spectrograph are explained
in [32] and Fig. 1. To reiterate, an input waveguide illuminates an array of
waveguides through a region called the input free propagation region (FPR).
The ends of the FPR follow a Rowland curvature. The output of this FPR has an
array of single-mode waveguides that collect the diffracting beam. The array
of waveguides is designed to introduce a phase slope, with a constant path
length difference between adjacent waveguides. The resultant phased array
creates an interference pattern with constructive interference (i.e. peak) for
different wavelengths at different spatial points along the Rowland curvature
of the output FPR (i.e. the focal plane). This is where the dispersed spectrum
is formed that is then sampled by discrete output waveguides or by directly
imaging the focal plane on a detector [23]. In the case of discrete sampling,
the spectral separation between the neighboring output waveguides
($\delta\lambda$) is called the channel spacing.

Figure 1: The structural components of an AWG. Top left: AWG schematic showing
the full CAD. Top center: The input FPR simulated with 3D EIM. There are
tapers at the FPR-waveguide interface on both sides. The length of waveguide
array simulated with EIM was up to the point where the lateral separation of
the waveguides was $>$ 2$\times$MFD. Top right: The electric field amplitude
obtained through the 3D EIM beam propagation simulation of the input FPR.
Bottom right: The electric field amplitude at the end of the simulated region
showing the Gaussian envelope of the illumination pattern of the input
waveguide (plotted down to 1/10th of the maximum field amplitude) as sampled
by the waveguide array (seen as narrower peaks). Bottom left: The interface
between the FPR and the array of waveguides showing the linear tapers and the
300 nm gap that has been introduced in to simulate the MPW fabrication
tolerance. Bottom center: The interface between the output FPR and the output
waveguides (receiver channels) showing the linear tapers touching each other
in order to minimize the spectral dropout.

The key properties of an AWG transmission response include: FSR, insertion
loss, crosstalk, and spectral dropout. They are briefly described below. For
further details about each of these properties, we direct the reader to [33].

Free spectral range: Figure 2 shows the transmission response of an AWG with a
R=1000 (i.e. channel spacing = 1.55 nm) and a FSR = 20 nm. Each peak within a
FSR indicates one output waveguide. Note that multiple spectral orders (i.e.
FSRs) across the range of operation of the AWG (say, 1450$-$1650 nm) spatially
overlap at the output FPR (and thereby, at the output waveguides). For
instance, the central output waveguide will have the spectral response shown
in black (i.e. peaks for $\lambda$ = 1530 nm, 1550 nm, 1570 nm and so on) due
to the spatially overlapping spectral orders.

Figure 2: A representative AWG spectral response showing the key spectral
features of a R=1000, FSR=20 nm device with a center wavelength of 1550 nm.
Left: The transmission response of the AWG showing two spectral orders
(1520-1560 nm) with spectral orders spanning 20 nm (i.e. the FSR). The
corresponding central and edge channels in the two spectral orders are
highlighted in black and green, respectively. Center: The crosstalk and
insertion loss for the central channel are shown here. Right: The dashed black
line is the summed spectrum over all output channels. The spectral dropout
around the central channel is indicated by the red arrow.

Insertion loss: The insertion loss includes the losses at the waveguide-FPR
interfaces (due to mode-field mismatch at the FPR-taper interfaces, gaps
between the tapers, and the truncation of the illumination pattern of the
input waveguide) and the light lost to higher spatial orders in an AWG (the
spatial orders beyond the region sampled by the output waveguides). The
insertion loss is shown in Fig. 2b. Note that the insertion losses of the edge
channels for a given order are higher than those of the central channels as
can be seen in Fig. 2a. The envelope of AWG peaks is determined by the far
field pattern of the waveguides illuminating the output FPR.

Crosstalk: We define crosstalk as the difference between the peak transmission
for a given channel and the transmission of its neighboring channel at that
peak wavelength. It is a measure of the contrast ratio or sharpness of the AWG
peaks. The electric field at an output channel is the image of the electric
field of the input waveguide modulated by the sampling by the array of
waveguides. The sampling function introduces sidelobes in the spectral
profile, which contribute to a higher spectral leakage. In addition, broader
tapers at the output FPR (e.g. touching tapers shown in Fig. 1, bottom-center
panel) broaden the spectral profile of an output channel, thus leading to a
higher crosstalk. In astronomical spectroscopy, this is analogous to the line
spread function [34]. For astronomical sources, it is more critical to collect
all the light at the output FPR than the spectral purity of the channels as
long as the spectral leakage is $<$ 30-40% (or crosstalk $>$ 4$-$5 dB) and the
line spread function is well-characterized. Therefore, touching tapers at the
output FPR are better suited for astronomy.

Spectral dropout: Another requirement in astronomical spectroscopy is the
ability to reconstruct a continuous spectrum to ensure a complete picture of
the target and that lines of interest are not missed. However, in a typical
AWG, the dispersed spectrum at the output FPR is sampled by discrete output
waveguides (or receiving channels), thus giving rise to a discreteness in the
spectral response. This can be seen in Fig. 2b where each channel (indicated
with a different color) has a peak at one wavelength and the response tapers
off as we go away from that peak wavelength. If we sum all channels together,
we get an idea of the total throughput of the device. This gives rise to a
wavy response as shown in Fig. 2c. Thus, there is an increased loss in the
spectral regions between the peak wavelengths of any two spectral channels. We
define this loss as the spectral drop-out, which occurs due to the
discretization of the output FPR (or the focal plane). There is no such
dropout if we directly image the focal on a detector. Here we have only
defined the magnitude of the drop-out, but the width is equally as important
and depends on the device design. We only focus on the amplitude in this work.

###  3.2 Waveguide Geometry

Given the range of possibilities, we will focus our simulation efforts here on
the two main kinds of the commercial MPW geometries available, namely a square
(Ligentec-like) and rectangular (LioniX-like) geometry and simulate a range of
AWG devices using these to understand how this impacts the various properties
of the spectrographs. Figure 3 shows the cross-sectional geometries of the
waveguides utilized in all the simulations. The left panel describes the
square waveguide and is representative of the Ligentc MPW offering. In reality
the side walls of the guides are inclined at $89^{\circ}$. However, we
simulate them as perfectly vertical (i.e. $90^{\circ}$) for ease of simulation
without altering the mode or the effective index in any significant way. The
right panels show the rectangular (Lionix-like) waveguide geometry. LioniX’s
MPW offering uses a dual stripe approach with a top and bottom stripe
thickness of 175 and 75 nm respectively, and a separation between the two
stripes of 100 nm (as shown in the center panels of Fig 3; for a detailed
summary, see the Triplex platform in [35]). However, to eliminate the need for
full 3D simulation, we identified an equivalent rectangular waveguide geometry
providing a similar mode size and effective index through an iterative
process. The rectangular analog is 900 nm wide and 263 nm thick as shown in
Fig. 3 \- right panels. This single stripe rectangular analog was used in all
further simulations. Note that the advantage of a double stripe geometry is
that it offers a relatively high confinement factor without requiring thick
SiN deposition for either stripes, thus avoiding the material stresses (and
stress-induced microcracks [36]).

Figure 3: Waveguide cross-sectional geometries and mode profiles of the a)
square waveguide (Ligentec-like), b) double-stripe waveguide (Lionix MPW
profile), c) rectangular waveguide (equivalent to Lionix MPW profile from the
perspective of effective index and the mode profiles of both polarizations).
The black arrows indicate orientation of electric field (Ex and Ey) in the
profiles shown here. Dashed lines indicate the 1/e2 intensity locations and
hence the size of the MFD.

###  3.3 AWG Design: Spectral parameters

In order to explore AWG designs employing the two geometries on equal footing,
we fixed the channel spacing (i.e. a discrete resolution element) of the AWG
and the FSR. Astronomical spectrographs come in a range of resolving powers
depending on the science case. For instance, a lower resolving power (R $\sim$
1000) is useful for taking the spectra of faint objects or for measuring
galaxy redshifts, while high resolving power (R $>$ 10,000) is useful for
measuring the composition and kinematics of galactic flows or composition of
exoplanet atmospheres [37]. We therefore choose three resolutions (R $\sim$
1000, 5000, 10000) to explore the parameter space of interest. Even higher R
is of interest in astronomy, but we start our efforts at more modest values
for simplicity. Here, we define the resolving power as $\lambda/\delta\lambda$
where $\delta\lambda$ is the discrete channel spacing of the AWG (discretized
due to output waveguides).

We choose the near-IR band around 1550 nm for simulating these AWGs for
several reasons. First, the SiN platform and foundry processes are optimized
around 1550 nm since this is the waveband of interest for telecommunication
applications. Second, for galaxies and quasars in the early universe
(cosmological redshift $>$ 6), the rest-frame optical and ultra-violet
wavebands which carry the most useful spectral lines tracing the multi-phase
gas (e.g. C3+, Mg+, Fe+) are all redshifted to the near-IR [38]. Third,
terrestrial exoplanet atmospheres should contain molecular species such as
H2O, CH4, CO2, which have spectral features in the near-IR [37]. Therefore we
choose the center wavelength of the AWG as 1550 nm.

A FSR of 20 nm is chosen to limit the number of waveguides in the array, and
therefore, keep the footprint of the AWG within a standard reticle area
(typically 100$-$200 mm2). At the same time, this FSR is large enough to cover
the whole astronomical H-band (1490$-$1800 nm) in 15 spectral orders, thus
requiring only a low-resolution cross-dispersion [24, 39].

###  3.4 AWG Design: Structural parameters

Given the spectral parameters and the waveguide geometry, we design the AWGs
for each case using the design algorithm described in [40]. As explained
earlier, we fix the center wavelength ($\lambda_{0}$) at 1550 nm and the FSR
at 20 nm. The spectral order ($m$) is determined in each case using the
equation: $FSR$ $\approx$ $\lambda_{0}/m$ $\times$ $n_{\rm eff}/n_{\rm
group}$, where $n_{\rm eff}$ and $n_{\rm group}$ are the effective and group
indices of the waveguide, respectively. The path difference between the
adjacent waveguides in the array is determined as: ($\Delta L$ =
$m\lambda_{0}/n_{\rm eff}$). The number of output channels (or discrete
resolution elements) is determined by the FSR and channel spacing
($\delta\lambda$, or the spectral width of one resolution element) as $N_{\rm
chan}$ = $FSR/\delta\lambda$.

The AWG is designed to keep the nominal crosstalk level (see definition above)
to less than $-$20 dB for the central channel, which dictates both the number
of waveguides in the array and the length of the FPR. Given the different
waveguide geometries we are considering here, it is important to ensure that
we maintain the same number of waveguides in the array for a similar resultant
resolving power and a fair comparison. This was achieved by adjusting the
length of the FPR for each geometry until the outermost waveguide in the array
captured no more than $\mathrm{\sim 1/10^{th}}$ of the flux of the central
waveguide (refer to the right panels of Fig. 1). This criterion also ensured
that most of the light was captured by the array. The key structural
properties of the AWGs are summarized in Table 1.

We further added adiabatic linear tapers at all the four slab-waveguide
interfaces (two on the input and two on the output FPRs) with a width of
3$\times$W and a length of 10$\times$W (where W is the waveguide width) to
ensure a smooth transition of the slab mode to the waveguide mode (and vice
versa). A spacing of 300 nm was added between the tapers at the interface
between input/output FPRs and the waveguide array to account for the smallest
feature size limitation of the lithography process used in the commercial MPW
runs. We do not enforce this limitation at the interface of the output FPR and
the output waveguides (see Fig. 1 bottom-center panel) since the output
waveguides could be polished off to directly image the continuous spectrum on
a detector. The touching tapers along the Rowland curvature of the output FPR
ensure that most of the spectrum is captured without significant dropout.

The minimum allowed radius of curvature for the waveguides in the array
depends on the specific waveguide geometry to maintain low bend losses.
Foundries have experimentally determined that if the radii of the waveguides
are kept above this minimum criterion, the bend losses can be $<$0.5 dB/cm. To
operate in this low-loss regime, a minimum bend radius of 20 $\upmu$m if only
the TE polarization is considered or 150 $\upmu$m if both polarizations are
required, is needed for our square waveguides (Ligentec-like; 0.8 $\times$ 0.8
$\upmu$m). Similarly, a minimum bend radius of 80 $\upmu$m is needed for the
TE polarization for our rectangular waveguides (Lionix-like, 0.9 $\times$ 0.26
$\upmu$m; only TE polarization is considered since the rectangular waveguides
are birefringent).

Geometry = 0.8 $\times$ 0.8 $\upmu$m |  No. of waveguides |  Length of FPR ($\upmu$m) |  $\Delta$L ($\upmu$m)
---|---|---|---
Res 1000 |  33 |  99.9 |  57.3
Res 5000 |  163 |  506.6 |  58.0
Res 10000 |  323 |  1013.3 |  58.0
Geometry = 0.9 $\times$ 0.26 $\upmu$m |  |  |
Res 1000 |  33 |  110.7 |  65.9
Res 5000 |  163 |  553.5 |  65.9
Res 10000 |  323 |  1104.3 |  64.7
Table 1:  Structural properties of the AWGs discussed in this paper for both
the square and rectangular geometry. The spectral order ($m$) is 64 in both
the designs.

###  3.5 Simulations

We simulate the designs obtained in the step above using RSoft CAD[41] and the
AWG utility of the BeamPROP module[42]. We use the 3D effective index method
(EIM) to simulate the propagation from the input waveguide to the FPR and into
the waveguides in the array until the modes in the array of waveguides are
well-separated (by at least 2 $\times$ the MFD) to ensure proper decoupling of
the electric field. After this simulation, the phase increment and phase
errors are added to the waveguides in the array. Finally, the beam propagation
in the output FPR is simulated using the 3D EIM to construct the spectrum
received at the output waveguides. The AWGs are designed for the TE mode and
are simulated by illuminating both TE and TM modes to understand the
polarization-dependent response of the designs.

###  3.6 Simulation of phase errors

The array of waveguides needs to be phased precisely before illuminating the
second FPR in order to produce a spectrum. Any phase errors can deteriorate
the sharpness of the peaks and will therefore, determine the contrast between
the spectral peaks and the side lobes. This in turn determines how high a
resolution spectrograph can be built using a fabrication platform, as
discussed below.

Phase errors in the array could arise from side wall imperfections, defects,
thickness variations in the deposition processes, stitching errors in case of
e-beam lithography or if using multiple reticles for a single device in case
of stepper lithography, and so on [43]. To simulate the impact of these
imperfections, we added phase errors to our simulations. This is achieved by
altering the phase in each waveguide of the array by a random number uniformly
distributed between 0 and a maximum phase error value ($\Delta\phi$ in
degrees). The resulting simulated spectral response for each AWG (with
different spectral parameters and waveguide geometries) is discussed in
section 4 through various metrics such as throughput and crosstalk as a
function of maximum phase error. These results are helpful in determining the
limits on the spectral parameters that can be achieved given a phase error
tolerance for the fabrication process and the desired performance. In section
4.6, we estimate how these phase-error tolerances translate into geometrical
tolerances of width and height fluctuations for different AWG devices and
thereby discuss the potential limitations imposed by the fabrication processes
on building high-resolution devices.

##  4 Results

###  4.1 Spectral profile dependence on phase errors

We start our investigation by examining the impact of phase errors on AWG
properties. Figures 4 and 5 show the profile of a single spectral channel (the
central channel in an FSR), as a function of resolving power and phase error,
for TE (in the top row) and TM polarizations (in the middle row) for the
square and rectangular waveguides respectively. The dashed profiles show the
neighboring output channels (only for zero phase error) to provide an estimate
of the spectral degradation and crosstalk with phase error. The first thing to
notice is that in the cases where there are no phase errors, a discrete
spectral channel is formed, indicating that both waveguide geometries can
support AWGs with a range of resolving powers assuming perfect fabrication.

Figure 4: Spectral response of a single channel of AWGs designed using square
waveguides (Ligentec-like), as a function of resolving power, for TE and TM
polarization’s. The impact of phase errors on the spectral profile is shown in
all cases. The bottom panel shows the wavelength offset between the TE and TM
polarizations.  Figure 5: Spectral response of a single central channel of
AWGs designed using rectangular waveguides (LioniX-like), as a function of
resolving power, for TE and TM polarization’s. The impact of phase errors on
the spectral profile is shown in all cases. The bottom panel shows the
wavelength offset between the TE and TM polarizations.

It can be seen that the profiles of the spectral channels for TE and TM
polarizations are near identical in the case of the square waveguides (Fig.
4). In addition, there is only a slight spectral offset of 0.2 nm between the
wavelength of the central channels for TE and TM polarizations. This is
unsurprising given the symmetry of the waveguide geometry. In comparison, for
the rectangular waveguides, the sidelobes of TM polarization are broader than
those for TE (in Fig. 5, compare the side lobes of the red channels in each
plot), which also results in a slightly greater crosstalk with the adjacent
channels for the TM mode. Unlike the square waveguides, the rectangular ones
have a large central wavelength offset between the two polarizations ($\sim$
5.7 nm $+$ 2$\times$FSR). This is expected, given the significantly broader
and lower effective-index mode profile for the TM mode as opposed to the TE
mode in the rectangular waveguide (as shown in Fig. 3). In other words, the TE
and TM modes at a single wavelength will land in different regions of the
output FPR and couple into different output waveguides. Therefore, the
rectangular waveguides are well-suited in the cases where only a single
polarization is fed to the AWG, or the polarizations are separated and then
fed to two different AWGs (with rectangular waveguides) optimized for the
respective polarizations.

Finally, as the phase error present in the array of waveguides is increased,
the peak transmission of the spectral channel reduces (i.e. the insertion loss
increases), and the side lobes get stronger. This is because as the phase
error increases, the sharpness (or the coherence) of the constructive
interference peaks diminishes. The reduced efficiency is a result of the light
of a given wavelength being spread over a larger region at the output of the
second FPR and hence coupling into a larger number of output waveguides,
increasing the strength of the side lobes at the same time. It can be seen
that for phase errors up to 120∘ (green curves), despite the elevated side
lobes, the primary spectral channel profile is retained with little reduction
in peak efficiency. In section 4.6, we estimate the corresponding width or
height errors permitted for devices designed with each waveguide geometry to
result in the level of phase error shown in the plots above.

###  4.2 Insertion loss and crosstalk dependence on phase errors

Figure 6 shows the throughput of the AWG devices as a function of the phase
error and resolving power, for TE and TM modes, and for both the square and
rectangular waveguides. In each plot, we show the throughput for the central
(circle) and edge (triangle) channel of the spectral order around 1550 nm.
From the plot it can be seen that the throughput of the central channel does
not significantly vary with the resolving power. On the other hand, the
throughput of the edge channels is substantially lower at higher resolutions.
This is likely due to the sidelobes that arise given the asymmetric
illumination received at the output edge channels and become more significant
at higher resolutions (discussed later in this section).

With no phase error present the devices could achieve throughputs of $-$1 to
$-$2 dB (80$-$63%). The throughput reduces monotonically as a function of
increasing phase error, which as outlined above, is because the light is
spatially smeared out across the output of the second FPR and thus less light
is coupled into a given output waveguide. It can be seen that with phase
errors of up to 120∘ there is minimal additional loss ($<$ 1.5 dB or $<$ 30%),
which is consistent with Figures 4 and 5 above. To maintain high efficiency,
it is critical to minimize phase errors in the array to the greatest extent
possible.

The throughput of the edge channels is lower than the central channel. As
explained in Section 3.1, this is determined by the far field pattern of the
waveguides illuminating the output FPR. It is clear that the square waveguides
have a higher loss for the edge channels as compared to the rectangular
guides, thus reducing the overall device efficiency. This should be considered
in the overall loss budget while designing a photonic spectrograph.

Finally, the throughput of the AWGs does not appear to vary significantly
between the TE and TM modes for both waveguide geometries.

Figure 6: Throughput of AWGs with square (Ligentec-like) and rectangular
(LioniX-like) waveguides as a function of phase errors, for various resolving
powers for both TE and TM polarizations. Figure 7: Crosstalk of AWGs with
square (Ligentec-like) and rectangular (LioniX-like) waveguides as a function
of phase errors, for various resolving powers for both TE and TM
polarizations.

Figure 7 shows the crosstalk of the AWG devices as a function of phase error
and resolving power, for TE and TM modes, and for both the square and
rectangular waveguides. In each plot, we show the crosstalk for the central
and edge channel for a spectral order around 1550 nm. It can be seen that in
nearly all cases, the cross-talk increases as a function of increasing phase
error. However, for the edge channels, this trend gets flatter at higher
resolutions. This is because the edge channels see an additional one-sided
sidelobe that arises due to the asymmetric illumination that they receive from
the waveguide array. This is shown in Fig. 8 for the square waveguide in TE
polarization. As seen in this figure, this additional sidelobe dominates the
crosstalk for the edge channels. The transmission of the peak and the sidelobe
degrade in a similar fashion with increasing phase errors, therefore the
variation of edge-channel crosstalk appears flatter (as compared to the
central channels).

Figure 8: Spectral response of a single ‘edge channel’ of the AWG with square
waveguides for TE polarization, as a function of resolving power and phase
errors. The asymmetric sidelobe arising from the asymmetric illumination of
the edge channels is clearly shown. The dotted lines show the neighboring
channels at zero phase error. It can be clearly seen that at higher
resolutions, the asymmetric sidelobe dominates the crosstalk.

The dashed $-$3 dB line (50% spectral purity) in Fig. 7 is a threshold, which
indicates one would have a poorly defined spectral channel with $>$ 50%
contamination from the neighboring channels, thus making the device
ineffective. The edge channels approach this threshold sooner as compared to
the central channels because of the additional asymmetric sidelobes (see Fig.
8).

If the sole requirement was to ensure all devices stayed above this threshold,
then phase errors should typically be maintained below 120∘. In this case,
<$-5$ dB of crosstalk can be achieved at all resolutions (for both central and
edge channels), with both waveguide geometries and in both TE and TM
polarizations with phase errors of <$120^{\circ}$. In general, one will most
likely not disperse the output of these waveguide channels any further and
hence get a single flux measurement per channel (or per discrete resolution
element). In that case, minimizing crosstalk is ideal to minimize the
wavelength confusion in reconstruction of the spectrum. If a $-$20 dB
crosstalk was achieved, then 99% of the light in a given channel would be from
a specific wavelength range (or a resolution element) alone, thus minimizing
wavelength confusion and improving the quality of the final reconstructed
spectrum. This level is shown in dotted lines in Fig. 7. This goal (for the
central channels) could be achieved with <$60^{\circ}$ of phase error for the
square waveguides, but only with $\sim 0^{\circ}$ phase error for the TE
polarization when rectangular guides are used.

###  4.3 Spectral drop out

The spectral drop-out, defined in Section 3.1, as a function of R for both the
TE and TM polarizations is shown in Fig. 9. For square waveguides, the
spectral dropout is similar for TE and TM polarizations, with 3$-$4 dB at
R=1,000 dropping monotonically towards 0$-$3 dB at R=10,000. Similarly, for
the rectangular waveguides, the dropout decreases monotonically with
increasing R in the TE polarization up to R=5,000. However, in the TM
polarization, they show no spectral drop-out. This is primarily due to the
broad TM mode profile which leads to a broader spectral profile compared to
the TE mode for a given AWG (as shown in Fig. 5), resulting in a higher
crosstalk with the neighboring waveguides, but at the same time, a lower
spectral dropout.

Figure 9: Spectral drop-out as a function of resolving power for both the
square and rectangular waveguides for TE (Left) and TM (Right) polarizations.

###  4.4 Coupling to/from AWGs

Understanding the waveguide properties, namely the MFD and NA at the facets,
is critical to efficiently interfacing the fibers/detectors with the AWGs. To
efficiently couple to standard NIR SMFs, which have typical MFDs of $\sim$10
$\upmu$m, requires the use of tapers. Two main types of tapers can be used: 1)
inverted tapers within the plane of the circuit and 2) 3D tapers (sometimes
refered to as ‘spot size converters’). Inverted tapers can be implemented by
reducing the width of the waveguide towards the facet of the chip [27]. As the
aspect ratio of the waveguide is modified during the taper, the shape of the
mode and the polarization properties change in this region of the device. 3D
tapers alter the waveguide width and height resulting in a more isotropic
expansion of the waveguide mode [44]. Another advantage of the 3D tapers is
that a larger MFD range can be achieved with better reproducibility (i.e. the
mode can be expanded more) than by using an inverted taper. As such, 3D tapers
offer lower losses when coupling to the SMF28 fiber. Besides offering lower
losses, they are also less sensitive to mis-alignments. On the other hand,
inverted tapers are easier to fabricate (without requiring additional etching
steps), and hence are cheaper and widely used. Ligentec offers both types of
tapers, but the 3D versions are not standard as part of MPW runs and have an
additional cost. For Lionix, both inverted tapers as well as 3D tapers, which
can match to the 10 $\upmu$m SMF mode are part of the MPW offering.

To get a sense of the properties of inverted tapers, Fig. 10 shows the MFD and
NA as a function of the waveguide width (for a fixed waveguide height). As
outlined above, the inverted taper is formed by shrinking the width of the
waveguide (i.e. the regions to the left of the vertical black dashed lines
indicating the nominal waveguide widths utilized throughout this paper).

Figure 10: MFD and NA versus waveguide width, with a fixed waveguide height,
for both square (Ligentec-like) and rectangular (LioniX-like) geometries, for
TE and TM polarizations in x and y.

Figure 10 shows the range of possible output MFDs/NAs with inverted tapers. As
can be seen, at very small widths, the MFD blows up. Given the steepness of
the curve at this regime it is extremely difficult to get a precise output
MFD. Hence it is challenging to reliably create an inverted taper that can
enlarge the MFD to optimize coupling with SMF28 (although LioniX has done
this). Therefore, the ultra-high NA fibers, which have smaller MFDs can be
used as an intermediate step. In this case the UHNA fiber can be spliced to
SMF28 and by applying a short taper to the spliced region a low loss
conversion can be easily realized. In addition to UHNA fibers, lensed fibers
can also be used. The benefit of lensed fibers is that the output MFD is
continuously selectable between 2 to 6 $\upmu$m at the time of manufacturing,
unlike with the UHNAs which only come in discrete values, allowing for more
precise mode matching. Also, the lensed fiber can be fabricated on almost any
fiber, which means no splicing taper is needed. One challenge of the lensed
fibers is bonding them to the chip as the tip can not be immersed in glue
(since the spot is usually formed at some distance from the fiber).

Figure 11 shows the experimentally measured propagation and coupling losses to
Ligentec and LioniX (dual stripe) MPW waveguides. The Lionix and Ligentec
measurements were done with SMF28 in different labs. The Lionix and Ligentec
reference waveguides had inverted tapers for coupling. The propagation losses
were difficult to quantify with tight error bars due to their small values (<1
dB/cm) and the limited reference waveguides we had access to. In the future,
we will test more devices and refine these values with better strategies for
quantifying the propagation loss [45]. The coupling losses for the Ligentec
waveguides (with inverted tapers for UHNA4 fiber), are as low as 2.3 dB/facet
around 1550 nm, but increase at shorter wavelengths. On the other hand, as a
result of the larger MFDs of the LioniX inverted tapers, they can be
efficiently coupled to SMF28 directly, with losses <1.5 dB/facet around 1550
nm. A summary of the options for coupling from fibers to waveguides for the
two platforms is shown in Table 2.

Figure 11: Measured propagation and coupling losses to Ligentec and LioniX MPW waveguides for the TE polarization only. The coupling loss is measured with an SMF28 fiber. The fainter shades show the uncertainty in measurements.  Coupling type | Square/Ligentec | Rectangular/LioniX
---|---|---
Inverted taper coupling to SMF28 |  5.4 dB/facet1 |  as low as $0.5$ dB/facet4
Inverted taper coupling to SMF28 |  $\sim$2.3 dB/facet2 (2 dB/facet3) | -
3D taper coupling to SMF28 |  $<1$ dB/facet3 |  $<0.5$ dB/facet4
Table 2: Coupling efficiencies around 1550 nm. 1. Simulated. 2. Measured by
the authors on a batch of recent Ligentec waveguides. 3. Value provided by
Ligentec. 4. Value provided by LioniX.

Besides coupling to/from fibers, it may be desirable to image the output beam.
To do this the optical system must be able to capture all of the light
emanating from the waveguide, which is easiest when the NA is $<0.3$, since
most inexpensive commercial optics operate below this level. Figure 10 can be
used to determine the properties of the inverted taper needed to be able to
have a given NA to support the imaging system. If the focal plane (FPR) of the
AWG is imaged directly (i.e. not sampled by discrete waveguides), then it is
difficult to use tapers. This means the output facet is a slab cross-section
(of the FPR). This will result in the smallest MFD and largest possible output
NA, placing extreme demands on the imaging system in the back end. Careful
consideration must be given in this case to build a suitable imaging system. .

###  4.5 Footprint

Figure 12 shows the area of the device as a function of the resolving power
and the waveguide geometry. It is clear that the area grows monotonically with
increasing R, which is because both the number of waveguides and the
incremental path length in the array increase with R, thus requiring a larger
footprint. For the square waveguide, it can be seen that using tighter bend
radii as low as 20 $\upmu$m brings only marginal gains compared to a device
with bend radii $>150~{}\upmu$m. However, square waveguides (eg: 0.8 $\times$
0.8 $\upmu$m), which produce smaller modes, offer significant advantages in
terms of footprint over larger modes provided by rectangular waveguides (eg:
0.9 $\times$ 0.26 $\upmu$m), particularly at R > 10,000. The area of a reticle
offered by Ligentec is 10$\times$5 mm2 versus LioniX which offers 16$\times$16
mm2. Based on these cell sizes, the plot indicates that an R=10,000 device
could fit onto a Ligentec cell, while a R$\sim$15,000 device could fit within
a LioniX cell. It should be noted that there are more optimal ways to arrange
AWG circuits which could be used to push these limits, but this gives a sense
of what can be accommodated within a reticle. Another thing to note is that
larger devices are more susceptible to process variation and accumulating
phase errors and this could be the dominant error source limiting the
performance of the device. This should be carefully considered with the
foundry and tested experimentally.

Figure 12: Footprint of the AWG chip (i.e. the area) as a function of
resolving power for square and rectangular waveguide geometries considered in
this paper. The Rmin is the minimum radius of curvature allowed in order to
minimize the radiation losses due to bends. For the square geometry (Ligentec-
like), Rmin = 20 $\mathrm{\upmu m}$ is recommended if only TE mode is of
interest and Rmin = 150 $\upmu$m is recommended to support both polarizations
(to minimize the bend losses).

###  4.6 Phase errors and fabrication tolerances

In order to put the simulation results described below into perspective, it is
important to understand how the fluctuations in the width and height of a
waveguide lead to phase errors in an AWG. In our treatment of phase errors, we
have assigned a randomly selected phase error for each waveguide in the array
($\Delta\phi_{\rm wg}$), and thereby a randomly selected $\Delta n_{\rm
eff,wg}$. In other words, each waveguide in the array is assumed to have a
constant (but randomly selected) error in the effective index ($\Delta n_{\rm
eff,wg}$ along the entire length of the waveguide, $L_{\rm wg}$). In reality
the error in the effective index of a waveguide will vary along the length of
the waveguide, with some parts of the waveguide having a larger fluctuation
than the others, thus giving rise to an equivalent net error in the $n_{\rm
eff}$ of that waveguide. Hence, the current treatment is a reasonable
approximation to study the effect of total phase errors. The following
equations show how the phase error in angle ($\Delta\phi_{\rm wg}$) relates to
the error in $n_{\rm eff}$, which in turn depends on the fluctuations in
waveguide width or height.

$\phi_{\rm wg}=\frac{2\pi L_{\rm wg}}{\lambda_{0}/n_{\rm eff}}\\\ $ (1)

$\therefore\Delta\phi_{\rm wg}=\left(\frac{2\pi L_{\rm
wg}}{\lambda_{0}}\right)\Delta n_{\rm eff,wg}$ (2)

It was previously shown experimentally in silica platforms that the phase
errors grow linearly with the length of the waveguides [43]. If we convert the
$\Delta n_{\rm eff,wg}$ to $\Delta W_{\rm wg}$ (fluctuation in the width of
the waveguide) assuming a fixed height or, $\Delta H_{\rm wg}$ (fluctuation in
the height) assuming a fixed width, we get an estimate of the allowable
fabrication tolerances for a given AWG. Note that this conversion depends on
the fractional length of the waveguide affected by such a fluctuation
(dictated by the $L_{\rm wg}$ term in equation 2). Assuming $\Delta\phi_{\rm
wg}$ = 180∘ in equation 2, we first obtain the $\Delta n_{\rm eff,wg}$. Then
using the effective index calculations of the waveguide geometries as a
function of width and height (for the TE mode), we calculate the relevant
$\Delta W_{\rm wg}$ or $\Delta H_{\rm wg}$ that would give rise to the
corresponding $\Delta n_{\rm eff,wg}$.

Figure 13: Magnitude of the fluctuations in the width (Left) and height
(Right) of the waveguide as a function of the length of the fluctuation that
would induce a 180∘ of error. Square waveguides are shown in solid lines,
while rectangular waveguides are highlighted with dashed lines.

Figure 13 shows what fluctuation in the waveguide width (left panel) or height
(right panel) could be tolerated for AWGs with various resolving powers
designed with both waveguide geometries, as a function of the fractional
length of the fluctuation (= section of the waveguide affected by the
fluctuation / total length of the waveguide) to achieve an overall maximum
phase error of 180∘. A fractional length fluctuation of 1 means that the
waveguide width or height error uniformly affects the entire length, whereas
smaller values mean that the error is localized to only a section of the
waveguide. This error only refers to the waveguides in the array.

It is clear from Fig. 13 that the shorter the fractional length of the width
or height fluctuation, the larger the tolerance to keep the phase error under
a given threshold. The figure shows that lower resolving power AWGs can handle
larger width or height fluctuations by up to an order of magnitude. This is a
result of the waveguides in the array being longer for higher resolution
devices, which makes them more sensitive to fluctuations in dimensions.

In addition, the rectangular waveguides (0.9 $\times$ 0.26 $\upmu$m) are more
tolerant to width variations compared to the square waveguides (0.8 $\times$
0.8 $\upmu$m), while being less tolerant to height variations, as one would
expect given their aspect ratio. As an example, with a phase error threshold
of 180∘, width errors of 10 nm and 1 nm could be tolerated $-$ assuming the
entire length of the waveguides (in the array) was affected $-$ for devices
with R=1000 and 10000, respectively. This corresponds to a fractional width
error ($\Delta W/W$) of 1.3% and 0.13% for the square waveguides,
respectively. However, if the defects were localized to say 1/10th of the
waveguide length, then the tolerated waveguide width variation would be 40-50
nm and 4-5 nm respectively for R=1000 and 10000. The allowable fractional
width errors in this case have increased to 5.6% and 0.56%, respectively.
Keeping the width fluctuation below 4-5 nm within a reticle ($\sim$ 100 mm2)
is achievable with the Ligentec444private communication, MPW manual available
from Ligentec upon request., process for its square waveguides. Similarly, in
terms of height, there is a slow variation $\sim$3 nm in the waveguide
thickness across a 400 mm2 span of a typical Ligentec wafer. For an AWG with
R=10000, which has a typical footprint of 30 mm2 ($\sim 6\times 6$ mm), a
thickness fluctuation of less than 1 nm can be achieved, thus producing a
phase error of less than 120∘ (from Eqn. 2 and Fig. 13).

Note that better fabrication processes are available in the dedicated runs
with the commercial foundries with better optimized tolerances. These
processes are preferable for building higher resolution AWGs (R $>$ 10000)
which also have larger footprints ($>$ 100 mm2, from Fig. 12).

##  5 Discussion

To summarize, we presented a scan of AWG spectral characteristics as a
function of resolution, phase errors, and polarization for two distinct
waveguide geometries used by commercial SiN platforms. While the presented
designs do not make use of any specialized treatment to optimize a given
spectral characteristic of interest (say, throughput or crosstalk), they
provide the community a realistic insight, particularly relevant to
astronomical spectroscopy, prior to making fabrication decisions.

###  5.1 Overall Throughput

As astronomy is a photon-starved field, high throughput is the key
requirement. The losses in an AWG spectrograph consist of the following terms:
fiber-to-chip coupling (can be tackled by mode matching to a large extent),
Fresnel reflection losses (can be tackled by index matching and anti-
reflective coating), propagation and bending losses, insertion loss, and chip-
to-free-space loss (in the case of detector imaging of the spectrum; need to
match NA of the imaging optics).

The results presented above can be combined with vendor/lab data to get a
sense of overall throughput. Given the complexity of different factors giving
rise to the phase errors, we can only make a crude estimate of the phase
errors one might expect. As discussed at the end of Sec 4.6, for an AWG with
R=5000 (typical footprint of 30 mm2), the expected height fluctuation would be
less than 1 nm for the Ligentec MPW process, thus producing a phase error of
less than 120∘ (from Eqn. 2 and Fig. 13). Conservatively, we assume a 120∘
maximum phase error in the waveguide array to determine the overall throughput
for a realistic device.

Table 3 shows the losses of each element of an AWG (assuming R$\sim$5000)
along with the total loss, for TE polarization only. The results for both the
square and rectangular designs are included along with values for optimized
devices with future developments (or existing techniques available in
dedicated fabrication runs of the commercial foundries). It can be seen that
with the current state of technology, that AWGs based on square waveguides
shown here would have losses of 5.9 dB (a throughput of 26$\%$), while
rectangular devices would have losses of 4.7 dB (a throughput of 34$\%$) for
the TE polarization. These values are a little low, but note that these are
estimates for an MPW run without any special optimizations. Significant gains
can be made by currently available optimizations in the dedicated runs. For
instance, Ligentec offers specialized tapers (e.g. 3D tapers) at the fiber-
waveguide interface to match the mode of a SMF28 fiber (MFD: 10 $\upmu$m).
This immediately reduces the coupling losses to $<$0.5 dB/facet. By simply
improving the phase errors from 120∘ to 60∘, which is achievable with the
current commercial platforms, the insertion losses are suppressed by nearly
1.25 dB (see Fig. 6). The insertion losses can be further improved by
optimizing the FPR-waveguide tapers. An improvement in taper efficiency from
the 92% for the linear tapers to 98% for optimized tapers can further reduce
the insertion losses by 1 dB (since there are four such interfaces). We
revisit the areas where throughput can be improved in the future in section 6.

As shown in Table 3, a throughput of $\sim 60\%$ can be obtained for the AWG
chip. For an estimate of the total throughput on a telescope, we take the case
of SCExAO (Subaru Coronagraphic Extreme Adaptive Optics, installed on an 8-m
telescope) feeding an SMF which then illuminates the photonic spectrograph
[46]. Assuming an injection throughput of $\sim 45\%$ (from sky to the SCExAO
focus) as described in [47] and a capture throughput from the focal spot to
the SMF of $\sim$ 50% (corresponding to a Strehl ratio of 65% in H-band), the
total sky-to-detector throughput of the instrument will be $\sim 0.45\times
0.50\times 0.6$ = 13.5%. Thus, with these improvements in the AWG, the
optimized throughput of a photonic spectrograph is poised to be comparable to
the current bulk optics near-IR spectrographs (typically $\sim$15% sky-to-
detector throughput which depends on the specific instrument and configuration
[48, 49]).

We remind the reader that these values represent the throughput estimate at
the peak of the FSR and that there will be additional losses at the edges of
the orders. Further efforts are required to engineer the waveguide structures
at the FPR-waveguide interface to suppress this extra loss. In addition, due
to discretization of the output field into waveguides, there is also spectral
drop-out not included in the table. However, there is no spectral drop-out if
the output waveguides are removed and the spectrum is imaged directly onto a
detector from the output FPR (eg. see [24]).

Loss term |  Square (dB) |  Optimized (dB) |  Rectangular (dB) |  Optimized (dB) |  Improvement approach
---|---|---|---|---|---
Insertion loss |  3.2 (48%) |  1.2 (76%) |  3.0 (50%) |  1.0 (79%) |  Phase-error minimization; Array-FPR taper optimization
Coupling loss |  2.5 (56%) |  1.0 (79%) |  1.5 (71%) |  1.0 (79%) |  3D tapers
Propagation loss (over 1 cm) |  0.2 (95%) |  0.2 (95%) |  0.2 (95%) |  0.2 (95%) |  Already optimized
Total loss |  5.9 (26%) |  2.4 (58%) |  4.7 (34%) |  2.2 (60%) |
Table 3:  Components of losses in a R $=$ 5000 AWG shown in dB-scale. The
numbers in parentheses show the corresponding throughput for each term in
percentages. The numbers under square and rectangular show the losses assuming
a phase error of 120∘ and no specific optimizations for slab-waveguide tapers
and with the coupling loss measured using simple 2D tapers. The optimized
columns show the improved losses after improving the phase errors to 60∘ and
incorporating optimized tapers for fiber-waveguide (3D spot-size converters)
and waveguide-slab (i.e. waveguide-FPR) interfaces.

###  5.2 Fabrication tolerances

In terms of tolerances, it is clear that tighter process control is required
for high-resolution devices. Due to the longer lengths of the waveguides in
the array required for the high-resolution AWGs, they are more susceptible to
fluctuations in the height or width dimensions. Therefore, the larger the
footprint of the device, the greater the susceptibility and hence tighter the
required tolerances.

From Figs. 4-7, it is clear that we should not exceed the $\Delta\phi$ beyond
180∘ in any case, but even tighter tolerances can help with improved
throughput and reduced cross-talk. Despite that fact we have demonstrated that
AWGs with a range of resolutions can be designed, but manufacturing
limitations, which induce phase errors and degrade performance, need to be
carefully understood to determine what can be realized. Although some
properties such as uniformity of waveguide thickness and width versus location
on a wafer can be measured and fed into simulations, ultimately devices need
to be fabricated to fully assess the actual phase errors that are present.
This is where a close working relationship with the foundry, and several
iterations of device design, fabrication and characterization will be needed.
Indeed for best results, the foundry should be involved from the beginning of
the design stage.

###  5.3 Polarization sensitivity

Since most astronomical sources are unpolarized, a polarization insensitive
spectrograph is preferred to eliminate the need for polarization splitting. We
explored the throughput and crosstalk characteristics of AWGs with both the
square and rectangular geometries. The insertion losses are not significantly
different between the two polarizations in either geometries. The crosstalk is
higher in TM mode for the rectangular waveguides, but still within acceptable
limits for astronomical spectrographs.

However, the most important issue is the birefringence. The shift in the
spectrum due to birefringence is given by : ($n_{\rm eff,TE}~{}-~{}n_{\rm
eff,TM}$)/$n_{\rm eff,mean}$ $\times$ $\lambda$. Due to a nearly 4% variation
in the effective index between TE and TM mode in the rectangular waveguides
(from Fig. 3), they can only be used for a single polarization. For square
waveguides, the difference is only 0.015%, thus giving a wavelength shift of
0.23 nm, which is acceptable for R = 1000 ($\Delta\lambda$ = 1.55 nm) and 5000
($\Delta\lambda$ = 0.31 nm), but for R=10,000 ($\Delta\lambda$ = 0.155 nm) the
offset exceeds the bandwidth of a single resolution element and hence will
lead to a degradation in the spectral resolution. Therefore, it is more
suitable to split the polarization and acquire separate spectra for TE and TM
polarizations for high-resolution AWGs (R $>$ 10,000). Alternatively, ultra-
low birefringence waveguides could be constructed, which would however require
a tighter fabrication control to maintain the low birefringence. Nonetheless,
for low-resolution spectrographs, polarization insensitive AWGs can be
constructed using square waveguides.

##  6 Future outlook

From Table 3 it is clear that the losses of current generation SiN devices are
higher than desirable for astronomy. The table also includes columns that show
the expected losses/throughput if critical elements of the circuit were
optimized (e.g. coupling). Broadly speaking, these optimizations could include
optimized coupling tapers for fiber-waveguide interfaces (such as 3D spot-size
converters), improved taper designs to minimize losses at the four waveguide-
FPR (i.e. slab-to-waveguide) interfaces on the chip, and minimizing phase
errors by exploring tolerant designs as well as choosing optimized fabrication
processes. By exploiting these approaches, AWGs based on square waveguides
could have losses of 2.4 dB (a throughput of 58$\%$), while rectangular
devices could have losses of 2.2 dB (a throughput of 60$\%$) for the TE
polarization.

A significant contribution to the loss is associated with the insertion loss
element, which could be addressed in the future (apart from improving phase
errors) with innovative design methodologies such as inverse design to acquire
high efficiency over a broad band [50]. Inverse design is based on setting up
a metric, for example to maximize throughput, and then allowing the algorithm
to explore the parameter space of waveguide widths, heights, and lengths of
smaller elements of the taper to optimize the design for the metric. This
approach can result in non-intuitive architectures, which can provide superior
performance (as demonstrated in [50]). It may be possible to use these design
approaches to engineer waveguide cross sections at the waveguide-FPR
interfaces to flatten the envelope of AWG peaks in a spectral order (see Fig.
2A), thus reducing losses towards the edges of a spectral order.

In addition, optimizing spectral drop-out is also critical. This term could be
optimized through shaping of the input field fed to the first FPR to be, for
instance, similar to a box profile, which will create flatter peak profiles at
the output waveguides (since the output field profile is an image of the
input). Methods such as using parabolic tapers [51] or 1$\times$2 MMIs [52]
can be used at the input waveguides to achieve this. Another alternative would
be to modulate a sinc profile on the illumination of the waveguides in the
array to create a flat (rectangular) wavelength response for each spectral
channel (instead of a Gaussian) [53], since the field profile at the output
waveguide is a Fourier transform of the illuminating field from the array.
Both of these methods will effectively reduce the spectral drop-out without
significantly increasing the crosstalk (see Sec 9.5.1 in [33] for more
details).

Currently the fabrication of astrophotonic AWG spectrographs with Ligentec and
Lionix MPW processes is underway. These tests will further inform on the
manufacturing limitations and the specific improvements that need to be
undertaken to enhance the spectral properties of these AWGs.

##  7 Conclusion

In this paper, we explored the key structural and spectral properties of AWG
spectrographs using commercial MPW geometries from an astronomical
perspective. We discuss a range of concerns critical to astronomy including
the throughput, injection and imaging of light from the chip, and polarization
sensitivity. We also discussed realistic limitations of fabrication by
studying the effect of phase errors and exploring how high a resolution
spectrograph can be constructed using commercial MPW processes. We find that R
$\sim$ 10,000 are easily achievable in the commercial MPW runs and reiterate
that more dedicated runs will be able to provide greater fabrication stability
and tolerances, and thereby the ability to construct even higher resolution
AWGs without accumulating phase errors $>$ 180∘. We provide a breakdown of
loss terms in an AWG spectrograph and specify avenues $-$ coupling loss, FPR-
waveguide tapers and advanced designs $-$ where greatest improvements can be
made. We also estimate the expected AWG throughputs through currently
available optimized strategies at these foundries (in dedicated fabrication
runs). The expected throughput from these strategies ($\sim$ 60%) would
position these devices to be competitive with the classical astronomical
spectrographs. Finally, we provide a future outlook for the astrophotonics
community on the thrust areas to make astrophotonic spectrographs an
attractive alternative to the classical astronomical spectrographs.

Acknowledgments Pradip Gatkine was supported by the David and Ellen Lee
Postdoctoral Fellowship at the California Institute of Technology. Simon Ellis
acknowledges the help and expertise of LioniX International in the fabrication
of one of the chips tested in this work as part of their MPW service. Nemanja
Jovanovic acknowledges the help and expertise of BRIGHT Photonics and Ligentec
in the design and fabrication of one of the chips tested in this work as part
of their MPW service. This work was supported by the Wilf Family Discovery
Fund in Space and Planetary Science, funded by the Wilf Family Foundation.
This research was carried out at the California Institute of Technology and
the Jet Propulsion Laboratory under a contract with the National Aeronautics
and Space Administration (NASA) and funded through the President’s and
Director’s Research $\&$ Development Fund Program.

Disclosures The authors declare no conflicts of interest.

## References

  * [1] J. Bland-Hawthorn, J. Lawrence, G. Robertson, S. Campbell, B. Pope, C. Betters, S. Leon-Saval, T. Birks, R. Haynes, N. Cvetojevic, and N. Jovanovic, “PIMMS: photonic integrated multimode microspectrograph,” in _Ground-based and Airborne Instrumentation for Astronomy III,_ vol. 7735 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ I. S. McLean, S. K. Ramsay, and H. Takami, eds. (2010), p. 77350N.
  * [2] N. Jovanovic, C. Schwab, N. Cvetojevic, O. Guyon, and F. Martinache, “Enhancing Stellar Spectroscopy with Extreme Adaptive Optics and Photonics,” Publications of the Astronomical Society of the Pacific 128, 121001 (2016).
  * [3] A. Quirrenbach, P. Amado, J. Caballero, R. Mundt, A. Reiners, I. Ribas, W. Seifert, M. Abril, J. Aceituno, F. Alonso-Floriano _et al._ , “Carmenes instrument overview,” in _Ground-based and airborne instrumentation for astronomy V,_ vol. 9147 (International Society for Optics and Photonics, 2014), p. 91471F.
  * [4] T. Feger, M. J. Ireland, C. Schwab, J. Bento, C. Bacigalupo, and D. W. Coutts, “Attaining m s-1 level intrinsic Doppler precision with RHEA, a low-cost single-mode spectrograph,” Experimental Astronomy 42, 285–300 (2016).
  * [5] J. R. Crepp, J. Crass, D. King, A. Bechter, E. Bechter, R. Ketterer, R. Reynolds, P. Hinz, D. Kopon, D. Cavalieri _et al._ , “ilocater: a diffraction-limited doppler spectrometer for the large binocular telescope,” in _Ground-Based and Airborne Instrumentation for Astronomy VI,_ vol. 9908 (International Society for Optics and Photonics, 2016), p. 990819.
  * [6] R. K. Gibson, R. Oppenheimer, C. T. Matthews, and G. Vasisht, “Characterization of the c-red 2: a high-frame rate near-infrared camera,” Journal of Astronomical Telescopes, Instruments, and Systems 6, 011002 (2019).
  * [7] D. Mawet, M. Fitzgerald, Q. Konopacky, C. Beichman, N. Jovanovic, R. Dekany, D. Hover, E. Chisholm, D. Ciardi, E. Artigau, R. Banyal, T. Beatty, B. Benneke, G. A. Blake, A. Burgasser, G. Canalizo, G. Chen, T. Do, G. Doppmann, R. Doyon, C. Dressing, M. Fang, T. Greene, L. Hillenbrand, A. Howard, S. Kane, T. Kataria, E. Kempton, H. Knutson, T. Kotani, D. Lafrenière, C. Liu, S. Nishiyama, G. Pandey, P. Plavchan, L. Prato, S. P. Rajaguru, P. Robertson, C. Salyk, B. Sato, E. Schlawin, S. Sengupta, T. Sivarani, W. Skidmore, M. Tamura, H. Terada, G. Vasisht, J. Wang, and H. Zhang, “High-resolution infrared spectrograph for exoplanet characterization with the keck and thirty meter telescopes,” Bulletin of the AAS 51 (2019).
  * [8] S. G. Leon-Saval, A. Argyros, and J. Bland-Hawthorn, “Photonic lanterns,” Nanophotonics 2, 429–440 (2013).
  * [9] I. Spaleniak, S. Gross, N. Jovanovic, R. J. Williams, J. S. Lawrence, M. J. Ireland, and M. J. Withford, “Multiband processing of multimode light: combining 3d photonic lanterns with waveguide bragg gratings,” Laser & Photonics Reviews 8 (2014).
  * [10] L. M. Close, J. R. Males, K. M. Morzinski, S. Esposito, A. Riccardi, R. Briguglio, K. B. Follette, Y.-L. Wu, E. Pinna, A. Puglisi, M. Xompero, F. Quiros, and P. M. Hinz, “Status of MagAO and review of astronomical science with visible light adaptive optics,” in _Adaptive Optics Systems VI,_ vol. 10703 of _Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series_ L. M. Close, L. Schreiber, and D. Schmidt, eds. (2018), p. 107030L.
  * [11] J.-L. Beuzit, A. Vigan, D. Mouillet, K. Dohlen, R. Gratton, A. Boccaletti, J.-F. Sauvage, H. M. Schmid, M. Langlois, C. Petit _et al._ , “Sphere: the exoplanet imager for the very large telescope,” Astronomy & Astrophysics 631, A155 (2019).
  * [12] B. Macintosh, J. R. Graham, P. Ingraham, Q. Konopacky, C. Marois, M. Perrin, L. Poyneer, B. Bauman, T. Barman, A. S. Burrows, A. Cardwell, J. Chilcote, R. J. De Rosa, D. Dillon, R. Doyon, J. Dunn, D. Erikson, M. P. Fitzgerald, D. Gavel, S. Goodsell, M. Hartung, P. Hibon, P. Kalas, J. Larkin, J. Maire, F. Marchis, M. S. Marley, J. McBride, M. Millar-Blanchaer, K. Morzinski, A. Norton, B. R. Oppenheimer, D. Palmer, J. Patience, L. Pueyo, F. Rantakyro, N. Sadakuni, L. Saddlemyer, D. Savransky, A. Serio, R. Soummer, A. Sivaramakrishnan, I. Song, S. Thomas, J. K. Wallace, S. Wiktorowicz, and S. Wolff, “First light of the Gemini Planet Imager,” Proceedings of the National Academy of Science 111, 12661–12666 (2014).
  * [13] N. Jovanovic, C. Schwab, O. Guyon, J. Lozi, N. Cvetojevic, F. Martinache, S. Leon-Saval, B. Norris, S. Gross, D. Doughty _et al._ , “Efficient injection from large telescopes into single-mode fibres: Enabling the era of ultra-precision astronomy,” Astronomy & Astrophysics 604, A122 (2017).
  * [14] A. J. Bechter, J. Crass, J. Tesch, J. R. Crepp, and E. B. Bechter, “Characterization of Single-mode Fiber Coupling at the Large Binocular Telescope,” Publications of the Astronomical Society of the Pacific 132, 015001 (2020).
  * [15] P. Gatkine, S. Veilleux, and M. Dagenais, “Astrophotonic spectrographs,” Applied Sciences 9, 290 (2019).
  * [16] T. Zhu, Y. Hu, P. Gatkine, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Arbitrary on-chip optical filter using complex waveguide bragg gratings,” Applied Physics Letters 108, 101104 (2016).
  * [17] S. Ellis, J. Bland-Hawthorn, J. Lawrence, A. Horton, R. Content, M. Roth, N. Pai, R. Zhelem, S. Case, E. Hernandez _et al._ , “First demonstration of oh suppression in a high-efficiency near-infrared spectrograph,” Monthly Notices of the Royal Astronomical Society 492, 2796–2806 (2020).
  * [18] E. Obrzud, M. Rainer, A. Harutyunyan, M. H. Anderson, J. Liu, M. Geiselmann, B. Chazelas, S. Kundermann, S. Lecomte, M. Cecconi _et al._ , “A microphotonic astrocomb,” Nature Photonics 13, 31–35 (2019).
  * [19] L. Labadie, J.-P. Berger, N. Cvetojevic, R. Haynes, R. Harris, N. Jovanovic, S. Lacour, G. Martin, S. Minardi, G. Perrin _et al._ , “Astronomical photonics in the context of infrared interferometry and high-resolution spectroscopy,” in _Optical and Infrared Interferometry and Imaging V,_ vol. 9907 (International Society for Optics and Photonics, 2016), p. 990718.
  * [20] K. Perraut, L. Jocou, J. Berger, A. Chabli, V. Cardin, G. Chamiot-Maitral, A. Delboulbé, F. Eisenhauer, Y. Gambérini, S. Gillessen _et al._ , “Single-mode waveguides for gravity-i. the cryogenic 4-telescope integrated optics beam combiner,” Astronomy & Astrophysics 614, A70 (2018).
  * [21] B. R. Norris, N. Cvetojevic, T. Lagadec, N. Jovanovic, S. Gross, A. Arriola, T. Gretzinger, M.-A. Martinod, O. Guyon, J. Lozi _et al._ , “First on-sky demonstration of an integrated-photonic nulling interferometer: the glint instrument,” Monthly Notices of the Royal Astronomical Society 491, 4180–4193 (2020).
  * [22] N. Cvetojevic, J. S. Lawrence, S. C. Ellis, J. Bland-Hawthorn, R. Haynes, and A. Horton, “Characterization and on-sky demonstration of an integrated photonic spectrograph for astronomy,” Optics Express 17, 18643–18650 (2009).
  * [23] N. Cvetojevic, N. Jovanovic, J. Lawrence, M. Withford, and J. Bland-Hawthorn, “Developing arrayed waveguide grating spectrographs for multi-object astronomical spectroscopy,” Optics Express 20, 2062 (2012).
  * [24] N. Cvetojevic, N. Jovanovic, C. Betters, J. Lawrence, S. Ellis, G. Robertson, and J. Bland-Hawthorn, “First starlight spectrum captured using an integrated photonic micro-spectrograph,” Astronomy & Astrophysics 544, L1 (2012).
  * [25] P. Gatkine, S. Veilleux, Y. Hu, J. Bland-Hawthorn, and M. Dagenais, “Arrayed waveguide grating spectrometers for astronomical applications: new results,” Optics Express 25, 17918 (2017).
  * [26] N. Jovanovic, N. Cvetojevic, B. Norris, C. Betters, C. Schwab, J. Lozi, O. Guyon, S. Gross, F. Martinache, P. Tuthill, D. Doughty, Y. Minowa, N. Takato, and J. Lawrence, “Demonstration of an efficient, photonic-based astronomical spectrograph on an 8-m telescope,” Optics Express 25, 17753 (2017).
  * [27] T. Zhu, Y. Hu, P. Gatkine, S. Veilleux, J. Bland-Hawthorn, and M. Dagenais, “Ultrabroadband high coupling efficiency fiber-to-waveguide coupler using Si3N4/SiO2 waveguides on silicon,” IEEE Photonics Journal 8, 1–12 (2016).
  * [28] D. J. Blumenthal, R. Heideman, D. Geuzebroek, A. Leinse, and C. Roeloffzen, “Silicon nitride in silicon photonics,” Proceedings of the IEEE 106, 2209–2231 (2018).
  * [29] J. F. Bauters, M. J. Heck, D. John, D. Dai, M.-C. Tien, J. S. Barton, A. Leinse, R. G. Heideman, D. J. Blumenthal, and J. E. Bowers, “Ultra-low-loss high-aspect-ratio si3n4 waveguides,” Optics express 19, 3163–3174 (2011).
  * [30] A. Arbabi and L. L. Goddard, “Measurements of the refractive indices and thermo-optic coefficients of si 3 n 4 and sio x using microring resonances,” Optics letters 38, 3878–3881 (2013).
  * [31] A. W. Elshaari, I. E. Zadeh, K. D. Jöns, and V. Zwiller, “Thermo-optic characterization of silicon nitride resonators for cryogenic photonic circuits,” IEEE Photonics Journal 8, 1–9 (2016).
  * [32] P. Gatkine, S. Veilleux, Y. Hu, T. Zhu, Y. Meng, J. Bland-Hawthorn, and M. Dagenais, “Development of high-resolution arrayed waveguide grating spectrometers for astronomical applications: first results,” in _Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation II,_ vol. 9912 (International Society for Optics and Photonics, 2016), p. 991271.
  * [33] K. Okamoto, _Fundamentals of optical waveguides_ (Academic press, 2010).
  * [34] R. Casini and A. de Wijn, “On the instrument profile of slit spectrographs,” JOSA A 31, 2002–2010 (2014).
  * [35] C. G. Roeloffzen, M. Hoekman, E. J. Klein, L. S. Wevers, R. B. Timens, D. Marchenko, D. Geskus, R. Dekker, A. Alippi, R. Grootjans _et al._ , “Low-loss si3n4 triplex optical waveguides: Technology and applications overview,” IEEE journal of selected topics in quantum electronics 24, 1–21 (2018).
  * [36] E. Irene, “Residual stress in silicon nitride films,” Journal of Electronic Materials 5, 287–298 (1976).
  * [37] J. Wang, D. Mawet, G. Ruane, R. Hu, and B. Benneke, “Observing exoplanets with high dispersion coronagraphy. i. the scientific potential of current and next-generation large ground and space telescopes,” The Astronomical Journal 153, 183 (2017).
  * [38] Y. Shen, J. Wu, L. Jiang, E. Bañados, X. Fan, L. C. Ho, D. A. Riechers, M. A. Strauss, B. Venemans, M. Vestergaard _et al._ , “Gemini gnirs near-infrared spectroscopy of 50 quasars at z $\gtrsim$ 5.7,” The Astrophysical Journal 873, 35 (2019).
  * [39] P. Gatkine, M. Sitaram, S. Veilleux, M. Dagenais, and J. Bland-Hawthorn, “Development of an integrated near-ir astrophotonic spectrograph,” in _Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation IV,_ vol. 11451 (International Society for Optics and Photonics, 2020), p. 114516L.
  * [40] M. K. Smit and C. Van Dam, “Phasar-based wdm-devices: Principles, design and applications,” IEEE Journal of Selected Topics in Quantum Electronics, 2 (2) (1996).
  * [41] C. RSoft Photonics, “Layout user guide, rsoft design group, inc,” Physical Layer Division 200 (2020).
  * [42] BeamPROP, https://https://optics.synopsys.com. AWG Utility from Synposys.
  * [43] A. Stoll, K. Madhav, and M. Roth, “Performance limits of astronomical arrayed waveguide gratings on a silica platform,” Optics Express 28, 39354–39367 (2020).
  * [44] X. Mu, S. Wu, L. Cheng, and H. Fu, “Edge couplers in silicon photonic integrated circuits: A review,” Applied Sciences 10, 1538 (2020).
  * [45] Y.-W. Hu, Y. Zhang, P. Gatkine, J. Bland-Hawthorn, S. Veilleux, and M. Dagenais, “Characterization of low loss waveguides using bragg gratings,” IEEE Journal of Selected Topics in Quantum Electronics 24, 1–8 (2018).
  * [46] N. Jovanovic, F. Martinache, O. Guyon, C. Clergeon, G. Singh, T. Kudo, V. Garrel, K. Newman, D. Doughty, J. Lozi, J. Males, Y. Minowa, Y. Hayano, N. Takato, J. Morino, J. Kuhn, E. Serabyn, B. Norris, P. Tuthill, G. Schworer, P. Stewart, L. Close, E. Huby, G. Perrin, S. Lacour, L. Gauchet, S. Vievard, N. Murakami, F. Oshiyama, N. Baba, T. Matsuo, J. Nishikawa, M. Tamura, O. Lai, F. Marchis, G. Duchene, T. Kotani, and J. Woillez, “The Subaru Coronagraphic Extreme Adaptive Optics System: Enabling High-Contrast Imaging on Solar-System Scales,” Publications of the Astronomical Society of the Pacific 127, 890 (2015).
  * [47] N. Jovanovic, N. Cvetojevic, B. Norris, C. Betters, C. Schwab, J. Lozi, O. Guyon, S. Gross, F. Martinache, P. Tuthill _et al._ , “Demonstration of an efficient, photonic-based astronomical spectrograph on an 8-m telescope,” Optics express 25, 17753–17766 (2017).
  * [48] “Gemini telescope GNIRS slit properties,” http://www.gemini.edu/sciops/instruments/ipm/data-products/GNIRS/gnirs-throughput.html.
  * [49] “MOIRCS spectrograph specifications,” https://subarutelescope.org/Observing/Instruments/MOIRCS/spec_sensitivity.html.
  * [50] C. Sideris, E. Garza, and O. P. Bruno, “Ultrafast simulation and optimization of nanophotonic devices with integral equation methods,” ACS Photonics 6, 3233–3240 (2019).
  * [51] K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electronics letters 32, 1661–1662 (1996).
  * [52] M. Amersfoort, J. Soole, H. LeBlanc, N. Andreadakis, A. Rajhel, and C. Caneau, “Passband broadening of integrated arrayed waveguide filters using multimode interference couplers,” Electronics letters 32, 449–451 (1996).
  * [53] K. Okamoto and H. Yamada, “Arrayed-waveguide grating multiplexer with flat spectral response,” Optics letters 20, 43–45 (1995).