11institutetext: Centro de Astrobiología (CSIC-INTA), ESAC - Camino Bajo del
Castillo s/n, 28692 Villanueva de la Cañada, Spain 22institutetext: Telespazio
UK for the European Space Agency (ESA), ESAC, Spain. 33institutetext:
Instituut voor Sterrenkunde, KU Leuven, Celestijnenlaan 200D, bus-2410, 3000
Leuven, Belgium. 44institutetext: Centro de Astrobiología (CSIC-INTA),
Carretera de Ajalvir, 28850 Torrejón de Ardoz, Madrid, Spain. 55institutetext:
UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh
EH9 3HJ, UK. 66institutetext: Institute of Particle Physics and Astrophysics,
ETH Zurich, Wolfgang-Pauli-Str 27, 8093 Zurich, Switzerland. 77institutetext:
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218,
USA. 88institutetext: Leiden Observatory, Leiden University, P.O. Box 9513,
2300 RA Leiden, The Netherlands. 99institutetext: Faculty of Aerospace
Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The
Netherlands. 1010institutetext: Chalmers University of Technology, Onsala
Space Observatory, S-439 92 Onsala, Sweden. 1111institutetext: SRON
Netherlands Institute for Space Research, PO Box 800, 9700 AV, Groningen, The
Netherlands. 1212institutetext: Center for Astrophysics, Harvard &
Smithsonian, 60 Garden St., Cambridge, MA, 02138, USA. 1313institutetext:
Rutherford Appleton Laboratory, Harwell Campus, Didcot, Oxfordshire, OX11 0QX,
UK.

# Wavelength Calibration and Resolving Power
of the JWST MIRI Medium Resolution Spectrometer

A. Labiano 1122 I. Argyriou 33 J. Álvarez-Márquez 44 A. Glasse 55 A. Glauser
66 P. Patapis 66 D. Law 77 B. R. Brandl8,9 K. Justtanont10 F. Lahuis11 J.R.
Martínez-Galarza12 M. Mueller11 A. Noriega-Crespo7 P. Royer3 B. Shaughnessy13
B. Vandenbussche3

(Received …, 2021; accepted …, 2021)

###### Abstract

Context. The Mid-Infrared Instrument (MIRI) on-board the James Webb Space
Telescope will provide imaging, coronagraphy, low-resolution spectroscopy and
medium-resolution spectroscopy at unprecedented sensitivity levels in the mid-
infrared wavelength range.The Medium-Resolution Spectrometer (MRS) of MIRI is
an integral field spectrograph that provides diffraction-limited spectroscopy
between 4.9 and 28.3 $\mu$m, within a FOV varying from $\sim$13 to $\sim$56
arcsec square. The design for MIRI MRS conforms with the goals of the JWST
mission to observe high redshift galaxies and to study cosmology as well as
observations of galactic objects, stellar and planetary systems.

Aims. From ground testing, we calculate the physical parameters essential to
general observers and calibrating the wavelength solution and resolving power
of the MRS is critical for maximizing the scientific performance of the
instrument.

Methods. We have used ground-based observations of discrete spectral features
in combination with Fabry-Perot etalon spectra to characterize the wavelength
solution and spectral resolving power of the MRS. We present the methodology
used to derive the MRS spectral characterization, which includes the precise
wavelength coverage of each MRS sub-band, computation of the resolving power
as a function of wavelength, and measuring slice-dependent spectral
distortions.

Results. The ground calibration of the MRS shows that it will cover the
wavelength ranges from 4.9 to 28.3 $\mu$m, divided in 12 overlapping spectral
sub-bands. The resolving power, is R$\gtrsim$3500 in channel 1, R$\gtrsim$3000
in channel 2, R$\gtrsim$2500 in channel 3, and R$\gtrsim$1500 in channel 4.
The MRS spectral resolution optimises the sensitivity for detection of
spectral features with a velocity width of $\sim$ 100 km s-1 which is
characteristic of most astronomical phenomena JWST aims to study in the mid-
IR. Based on the ground test data, the wavelength calibration accuracy is
estimated to be below one tenth of a pixel (0.1 nm at 5 $\mu$m and 0.4 at 28
$\mu$m), with small systematic shifts due to the target position within a
slice for unresolved sources, that have a maximum amplitude of about 0.25
spectral resolution elements. The absolute wavelength calibration is presently
uncertain at the level of 0.35 nm at 5 $\mu$m and 46 nm at 28 $\mu$m, and will
be refined using in-flight commissioning observations.

Conclusions. Based on ground test data, the MRS complies with the spectral
requirements for both the R and wavelength accuracy for which it was designed.
We also present the commissioning strategies and targets that will be followed
to update the spectral characterisation of the MRS.

###### Key Words.:

instrumentation: detectors; instrumentation: spectrographs; methods: data
analysis; infrared: general; Astrophysics - Instrumentation and Methods for
Astrophysics

##  1 Introduction

The James Webb Space Telescope (JWST) is a space infrared observatory
developed by the National Aeronautics and Space Administration (NASA), the
European Space Agency (ESA) and the Canadian Space Agency (CSA). JWST
comprises the Optical Telescope Element (OTE), an Integrated Science Module
(ISIM), a Sun shield, and a Spacecraft Bus. The OTE collects and feeds the
light to the scientific instruments in the ISIM. The main component of the OTE
is the JWST segmented 6.5-meter primary mirror, made of 18 gold-plated
beryllium, deployable, hexagonal mirrors. The ISIM is the structure that
harbors the 4 scientific instruments of JWST: the Near Infrared Camera
(NIRCam, Rieke et al., 2005) the Near Infrared Spectrograph (NIRSpec, Bagnasco
et al., 2007), the Near Infrared Imager and Slitless Spectrograph (NIRISS,
Maszkiewicz, 2017) and the Mid-Infrared Instrument (MIRI, Rieke et al., 2015).
The JWST 5-layered sun shield is designed to block the light from the Sun,
Earth and Moon, and allow the OTE and ISIM to passively cool to $\sim$40 K.
MIRI then has a dedicated cryo-cooler to allow it to reach its operating
temperature of about 7 K. The spacecraft bus carries all the ambient
temperature equipment necessary to operate the JWST, and communicate with the
ground stations. The expected launch date JWST is late 2021, and it will take
about one month to reach its halo orbit around the second Earth-Sun Lagrangian
point L2. It has a nominal mission time of 5 years, with a goal of 10 years.

Figure 1: Left: MRS SW detector (channels 1 and 2) illumination caused by an
extended source filling the MRS FOV. Right: MRS SW detector illumination
caused by a compact source. The illumination pattern along the slices follow
the iso-$\alpha$ lines (see text for details). The iso-$\alpha$ and
iso-$\lambda$ lines are highlighted with red (iso-$\alpha$) and white
(iso-$\lambda$) lines overlaid on the detector image.

With a spectral coverage from 4.9 to 28.3 $\mu$m, MIRI is the only mid-
infrared instrument on board JWST. It has four scientific operation modes:
photometric imaging, coronagraphy, low resolution (R$\sim$100) long-slit
spectroscopy, and medium resolution (R $\sim$1500 to 3500) integral field
spectroscopy. The imaging is performed by the MIRI Imager (MIRIM) in nine
bands, within a field of view (FOV) of up to 2.3 square arcminutes (Bouchet et
al., 2015), depending on the subarray configuration selected. Coronagraphy is
performed by three four-quadrant phase masks (4QPMs) at 10.65, 11.4, and 15.5
$\mu$m, with a FOV of 24$\arcsec\times$24$\arcsec$, plus a Lyot coronagraph at
23 $\mu$m with a FOV of 30$\arcsec\times$30$\arcsec$ (Boccaletti et al.,
2015). The Low Resolution Spectrometer is a long-slit
(0.5$\arcsec\times$4.7$\arcsec$) spectrometer with both slit and slitless
capabilities from 5 to 14 $\mu$m (Kendrew et al., 2015). The Medium Resolution
Spectrometer (MRS, Wells et al., 2015) is an integral field spectrograph (IFS)
that provides diffraction-limited spectroscopy between 4.9 and 28.3 $\mu$m,
within a FOV varying from $\sim$12 to $\sim$56 arcsec square.

MIRI will provide about 50 times the sensitivity and 7 times the angular
resolution of Spitzer. It is expected to make important contributions to all
four of the JWST science pillars (first light of the Universe, assembly of
galaxies, star and planet formation, and evolution of planetary systems plus
conditions for life, e.g. Rieke et al., 2015) and astronomy in general. In
fact, 41$\%$ of the submitted Cycle 1 General Observer proposals requested to
use MIRI, and $31\%$ had MIRI as the Prime Instrument Mode. Half of them
($14.5\%$) requested the MIRI MRS as
Prime111https://www.stsci.edu/jwst/science-execution/approved-
programs/cycle-1-go. The MRS was designed to provide medium resolution (R
$=\lambda/\Delta\lambda>1000$) 3D spectroscopy in the whole MIRI range. It
will be able to resolve (or separate) lines that have a velocity widths
(separations) of about 100 km s-1. The scientific goals behind this
requirement include (e.g., Gardner et al., 2006): to study the evolution of
the Hydrogen (Lyman, Balmer, Paschen) lines ratios and derive the ionizing
continuum of galaxies and ionization state of the Universe as a function of
redshift (this measurement requires R = 1000 spectroscopy of of the Hydrogen
lines, with intensities down to $2\times 10^{-19}$ erg cm-2s-1); to measure
the relative motions between (and within) galaxies, and scaling laws of their
basic properties (luminosity, size, kinematics, metallicity, etc) from
redshifts 1 to 7, revealing the physical processes responsible for those; to
search for the redshifts and power sources of the high redshift Ultra Luminous
Infrared Galaxies using narrow emission lines such as [NeVI] 7.66 $\mu$m or
rest-frame optical lines at high redshift, and comparing them with the PAH
emission (e.g., a R $>$ 1000 will allow separating the [NII] and Hα lines,
plus detection and in some cases resolution of the [NeVI] 7.66$\mu$m and PAH
lines up to redshifts z$\sim$6, based on the extrapolation of the Arp220
spectra and Circinus galaxy, at 10$\sigma$, in less than 10s); to map and
detect the molecular gas in galaxies through e.g., the H2 line at 28.22 $\mu$m
with adequate contrast of the emission line relative to the continuum (high
spectral resolution is essential to obtain an adequate contrast of the line
over the continuum), where we expect the largest contributions from telescope
and zodiacal backgrounds; to study, map, and trace the evolution of the
physical properties, composition, and structure of young stars, brown dwarfs,
faint companions, circumstellar disks, and protoplanets immediately adjacent
to their much brighter neighbors (to detect a 10 $\mu$ Jy source, such as a
typical Class 0 protostar in Taurus-Auriga, we need a R = 2000 at 15 $\mu$m,
plus a continuum sensitivity of$\sim 7\times 10^{22}$ W m-2 per resolution
element); to study and characterise the atmospheres and properties of
exoplanets, combined with direct imaging of these (e.g., a 7 MJUP object is
bright enough that spectra at R $\sim$ 3000 can be collected, enabling
atmospheric structure and composition to be inferred); to resolve the spectral
signatures of key ices and silicates, both in crystalline and amorphous form,
in the disks and observe planets perturbing the disk structures (R = 3000
allows the detection of broad features such as absorption bands against the
spectra of background sources, or emission tracers from warm material in the
inner regions of disks), and study the mineralogy of the dust in nearby debris
disks and of their precursors around young stars; to determine the basic
physical parameters of a brown dwarf atmosphere such as gravity, composition,
the temperature-pressure profiles (gradient, inversions), mass, and the effect
of clouds (a R $\sim$ 1000 is required to resolve most of the spectral
features tracing the atmosphere properties); to study the gas phase processes
in the inner comae of active comets, revealing their composition (comets will
have extremely rich spectra, with up to thousands of overlapping lines, thus
JWST spectroscopy of comets requires R up to 3000 for some isotopic ratios);
to extend spectroscopic studies from the best ground-based studies of Pluto
and Triton to more distant Kuiper Belt objects (a R = 3000 will allow to seek
isotopic ratios in the water ice and other components, as well as monitor
surface temperature through the nitrogen overtone band), provide compositional
and isotopic data, and even separate the size and albedo of such objects,
thereby constraining their physical size.

The MRS consists of two optical modules: the Spectrometer Pre-Optics (SPO) and
the Spectrometer Main Optics (SMO), plus the MRS internal calibration source.
The SPO carries two dichroic and grating wheels assemblies (DGA) and a set of
mirrors that split the light entering the MRS into 4 spectral channels, each
one with a different optical path. The DGA dichroics also divide the light
into three spectral sub-bands per channel. The SPO then feeds the light of the
four channels into four dedicated integral field units (IFUs), each of which
produces a 2D spectrum of the FOV. These spectra are recorded by two
detectors: one for the short wavelength channels (SW detector), one for the
long wavelength channels (LW detector).

The four spectral channels of the MRS are named 1, 2, 3, and 4 in sequential
order from the shortest wavelength to the longest, each consistent of sub-
bands SHORT (or A), MEDIUM (or B), and LONG (or C), from short to long
wavelengths in each channel (see Table 1). The fields of view of the four
channels are roughly concentric, with the smaller FOV corresponding to channel
1 and the largest to channel 4. Hence, to obtain a spectrum of a target for
the whole MRS wavelength range, the observer needs to stitch together 12 sub-
band spectra, obtained in three different exposures. The first exposure would
cover the 1A-2A-3A-4A bands, the second one bands 1B-2B-3B-4B, and the third
one bands 1C-2C-3C-4C. By default, the JWST pipeline will produce a spectral
cube (two axes with spatial information plus a third axis with the spectral
information) per MRS band, although it is also capable of producing a spectral
”super-cube” in which all 12 bands are combined (see, e.g., early descriptions
by Labiano et al., 2016; Gordon et al., 2015).

Each IFU of the MRS divides its FOV in a different number of slices whose
widths (0.176$\arcsec$, 0.277$\arcsec$, 0.387$\arcsec$, 0.645$\arcsec$ from
channel 1 to channel 4) are designed to be the same fraction (roughly the
FWHM) of the diffraction-limited PSF for all channels. Thus, the channel 1 IFU
contains 21 slices (FOV = $3.2\arcsec\times 3.7\arcsec$), channel 2 (FOV =
$4.0\arcsec\times 4.8\arcsec$) has 17 slices, 16 slices for channel 3 (FOV =
$5.5\arcsec\times 6.2\arcsec$), and 12 slices for channel 4 (FOV =
$6.9\arcsec\times 7.9\arcsec$). The IFUs of channels 1 and 2 disperse their
light on the SW detector, while channels 3 and 4 are dispersed on the LW
detector. All slices of a single IFU cover roughly one half of a detector.
Figure 1 shows the MRS detectors illuminated by an extended (left panel) and
point source (right panel), which fills the full FOV of the four channels. A
full description of the MRS design and operations can be found in Wells et al.
(2015); Rieke et al. (2015). Argyriou et al. (2020) and Glauser et al. (2010)
include a detailed description of the MRS FOV coordinate system, and the
detector – sky projection and reconstruction.

Within each channel FOV, we use two coordinates to define the spatial
directions: the $\alpha$-axis is defined parallel to the IFU image slicer
along-slice direction, the $\beta$-axis is defined perpendicular to the
$\alpha$-axis and describes the slicer across-slice direction. Due to the
optical design of the MRS, the region illuminated by each slice on the
detector follows a curved line; the spatial ($\alpha$) and spectral
($\lambda$) directions of the dispersed spectrum are not parallel to the x and
y coordinate directions of the detector pixels. Within a single slice, we
define as isoalpha (or iso-$\alpha$) the lines on the detector where the
$\alpha$ coordinate remains constant. Thus, an iso-$\alpha$line traces
constant spatial position at various wavelengths. In the same way, isolambda
(or iso-$\lambda$) lines are defined by the coordinates of constant wavelength
on the detector. The data from the MIRI test campaigns show that the
iso-$\alpha$lines are well described by third order polynomials while the
iso-$\lambda$lines are described by second order polynomials. Thus, both the
iso-$\alpha$ and iso-$\lambda$ are functions of (x,y) detector coordinates,
with a unique and discontinuous function for each slice (Patapis et al. in
prep.). An example of iso-$\alpha$ and iso-$\lambda$ lines on the detector are
shown in Fig. 1.

In this contribution, we discuss the methods used to calibrate the wavelength
solution and spectral resolving power of the MRS using ground test data.
Unlike at optical wavelengths, in the mid-infrared there are no arc lamps
available that can provide a well sampled grid of bright, unresolved emission
lines throughout the wavelength range of the MRS. We therefore use a hybrid
approach that combines Zemax optical models, Fabry-Perot etalons, and a
variety of absolute wavelength reference points in order to derive a
comprehensive wavelength solution.

In §2 we describe the ground-test data that forms the backbone of the MRS
wavelength solution, in particular the Fabry-Perot etalons that produce an
interference pattern with a well-defined peak-to-peak separation. We then
describe how we define a small number of absolute wavelength reference points
in each of the twelve MRS bands using features in the spectral response
function of a variety of filters. By combining these reference points with the
Fabry-Perot data, we demonstrate how we are able to expand the wavelength
solution to cover the entire MRS spectral range in §4, with additional
adjustments to the relative solutions in each detector pixel using super-
sampled observations of the line profiles. We expand upon this analysis to
estimate the spectral resolving power in §5. We note a few caveats to this
ground-based solution in §6, and conclude with a discussion in §7 of our
strategy for future refinement of our solutions using in-flight observations.
We summarize our conclusions in §8.

##  2 Ground-Test Observations

Before its delivery to NASA in 2012, MIRI underwent two ground test campaigns:
one for the verification model (VM), and one for the flight model (FM). The VM
was a fully operational version of MIRI, with reduced scientific
functionalities, built as a pathfinder for the test and calibration campaigns,
to de-risk the opto-mechanical concepts and assembly integration and
verification program. The main goal of the VM campaign was to verify the
instrument optical performance in realistic conditions to detect possible
problems that could affect the FM. It also served as a test bed for the FM
campaign planning and test designs (Wright et al., 2015; Eccleston et al.,
2012; Glasse et al., 2010; Lim et al., 2008). The FM campaign was much more
extensive than the VM campaign, and aimed to more fully characterize the
performance of MIRI under flight conditions. All the data relevant for the
spectral characterization of the MRS were obtained during the FM campaign, and
were processed through the standard MIRI ramps-to-slopes routines, background
subtraction and flat-field correction of the resulting slope images. No fringe
flats were applied since not all fringe flats were available by the time of
the analysis. Our tests show that at the signal to noise ratio (SNR) obtained,
not including the fringe flats does not affect the measured centroids of the
etalon lines by more than a few percent of the resolution element (Argyriou et
al., 2020). Throughout this paper we will refer to these measurements as FM
results or FM data.

To recreate the MIRI input signal from JWST, both campaigns used the MIRI
Telescope Simulator (MTS, Colina et al., 2004; Herrada et al., 2007; Belenguer
et al., 2008). The MTS was an infrared cryogenic telescope simulator developed
by the Instituto Nacional de Técnica Aeroespacial (INTA) that delivered a
diffraction-limited illumination beam reproducing the main particularities
introduced by the JWST optics, including both point source and extended source
illumination capabilities. The MTS contained a filter wheel which included
four Fabry-Perot etalons and two wave-pass filters. The etalons were designed
to provide unresolved spectral features across the entire MIRI wavelength
range222Each of the four etalons was optimized for a single MRS channel. For
convenience, we will name them: ET1 for Channel 1, ET2 for Channel 2, ET3 for
channel 3, and ET4 for Channel 4., while the long (LWP) and short (SWP) wave-
pass filters have sharp cut-offs at 6.6 and 21.6 $\mu$m respectively (see Fig.
2) that provide well-defined wavelength reference points at a temperature of
35 K.

Figure 2: Transmission curves for the SWP filter (top) and the LWP filter, as
measured in the laboratory by INTA, before mounting them on the MTS filter
wheel. Sharp spectral features in the transmission curves are used for
defining absolute wavelength reference points for the MRS.

Figure 3:  Top: transmission of ET1 measured with the FTS for the wavelength
range of MRS band 1A (FTS data), before mounting it on the MTS for the MIRI
test campaigns. Multiple well-separated (unblended) resonance lines can be
seen. Middle: fit of a single etalon line in the ET1 FTS data (black profile).
The line shape in the data is impacted by the non-zero divergence angle of the
beam (deviation from a perfectly collimated beam), and shows up as here as the
asymmetry of the line. We account for this in the fit (see Sect. §2, as shown
by the red profile. The green profile shows the same etalon line if the beam
was perfectly collimated (Eqs. 5 and 6). Bottom: fit of different line
profiles from the (MTS mounted) ET1 etalon, as observed with the MRS (black
points), and residuals of the fits computed by subtracting the fitted model
from the data. The MTS+MIRI setup is not affected by collimation issues. Its
spectral resolution, as expected, is clearly lower than in the FTS
measurements.  Table 1:  Spectral characteristics of the MRS from Zemax
modeling.

MRS | MRS | Nominal coverage∗ | Pixel scale ${}^{*}*$
---|---|---|---
Channel | Band | ($\mu$m) |  $\mu$m/pixel
| SHORT (A) | 4.87 – 5.81 | 0.0009
1 | MEDIUM (B) | 5.62 – 6.73 | 0.0011
| LONG (C) | 6.49 – 7.76 | 0.0012
| SHORT (A) | 7.45 – 8.90 | 0.0014
2 | MEDIUM (B) | 8.61 – 10.28 | 0.0016
| LONG (C) | 9.91 – 11.87 | 0.0019
| SHORT (A) | 11.47 – 13.67 | 0.0021
3 | MEDIUM (B) | 13.25 – 15.80 | 0.0025
| LONG (C) | 15.30 – 18.24 | 0.0029
| SHORT (A) | 17.54 – 21.10 | 0.0035
4 | MEDIUM (B) | 20.44 – 24.72 | 0.0042
| LONG (C) | 23.84 – 28.82 | 0.0049
333$*$$*$footnotetext: These ranges correspond to the Zemax model results, not
the final, wavelength calibrated ranges (Wells et al.,
2015).$**$$**$footnotetext: Nominal wavelength coverage / 1024 pixels. The
full details of the MRS characteristics can be found in Wells et al. (2015)
and JDox pages.

Table 2:  Location of relevant features used for MRS spectral characterisation. |  | Relative wavelength calibrators | Absolute wavelength calibrators (RPT)
---|---|---|---
MRS | MRS | ET1 | ET2 | ET3 | ET4 | LWP | SWP | X-Dichroic
Channel | Band | lines | lines | lines | lines | features | features | features
1 | SHORT (A) | X | X |  |  |  |  |
1 | MEDIUM (B) | X | X |  |  |  | X |
1 | LONG (C) | X | X |  |  | X |  |
2 | SHORT (A) | X | X |  |  |  |  | X
2 | MEDIUM (B) | X | X | X |  |  |  | X
2 | LONG (C) | X | X | X |  | X |  |
3 | SHORT (A) | X | X | X |  |  |  |
3 | MEDIUM (B) | X | X | X | X |  |  |
3 | LONG (C) | X | X | X | X |  |  |
4 | SHORT (A) | X |  | X | X |  |  | X
4 | MEDIUM (B) |  |  |  | X |  | X | X
4 | LONG (C) |  |  |  | X |  |  | X

The pure transmission (with no MTS or MIRI involved) of the etalons at
cryogenic temperatures ($\sim$80 K) was measured at Rutherford Appleton
Laboratory (RAL) using a Fourier Transform Spectrograph (FTS). Each etalon was
taken out of the MTS and mounted on a block cooled with continuously flowing
liquid nitrogen, and illuminated with quasi-collimated beam (divergence angle
$<1\deg$) at normal incidence. Throughout the paper we will refer to these
measurements as FTS data. The uncertainties associated to these measurements
are not available. Figure 3 shows the transmission curve of ET1, as an example
of the etalon FTS data, in the MRS band 1A wavelength range. Constructive
interference within the etalon layers results in clear transmission peaks,
whilst destructive interference results in no line appearing in the spectrum.

The middle panel of Fig. 3 shows a zoom into one etalon line ET1. There is a
prominent skewness of the line, which substantially complicates the
determination of both the wavelength of the line center and its FWHM. This
skewness is in fact present in all the lines, and it is due to the finite
diameter of the FTS pupil giving a cone angle of a few degrees at each point
on the etalon. What we observe is our reference line integrated over a cone
angle (range of incidence angles). Given that the requirement on the spectral
calibration accuracy is 10% of a resolution element444The 1/10th resolution
element calibration accuracy was driven by the acceptance that mapping
velocity structure below 30 km/s at more than a single grating setting was not
a goal. Mass and space constraints required MIRI/MRS to break the spectrum
into bands which were selected using moving mechanisms. 1/10th spectral
resolution element was then seen as a credible design goal, with the
mechanisms designed specifically to provide optimum stability and
reproducibility in the wavelength dimension., we need to account for the cone
angle variation.

Mathematically, the etalon line model is then described by the following set
of equations. Equation 1 defines the ”coefficient of finesse”, a quantity
linked to the reflectivity of an optical material. The higher the
reflectivity, the higher the coefficient of finesse, and the larger the
contrast of an etalon line. Equation 2 defines the phase difference between
successive emission peaks. We use Eq. 1 and Eq. 2 in the Fabry-Pérot
transmittance function given by Eq. 3 (Lipson & Lipson, 1969). This function
describes the transmission spectrum of an etalon based on its reflectivity,
the refractive index of the material used, and the optical thickness of the
etalon as a function of wavelength and incidence angle.

$F_{R}(\lambda^{\prime})=\frac{\pi\sqrt{R_{\lambda}(\lambda^{\prime})}}{1-R_{\lambda}(\lambda^{\prime})}$
(1)

$\delta(\lambda^{\prime},\theta)=\frac{4\pi\cdot n_{R}\cdot
D_{gap}\cdot\cos(\theta)}{\lambda^{\prime}}$ (2)

$T_{\theta}(\lambda^{\prime},\theta)=\frac{1}{1+\left(\frac{2F_{R}(\lambda^{\prime})}{\pi}\right)^{2}\cdot\sin^{2}\left(\frac{\delta(\lambda^{\prime},\theta)}{2}\right)}$
(3)

The above set of equations describe an etalon line produced by a ray, yielding
a symmetric line. If the etalon is illuminated by rays with a finite cone
angle, Eq. 3 has to be integrated over the profile of the incident light. To
do so we multiply Eq. 3 by Eq. 4 (where we assume an intensity which has a
Gaussian dependence on angle of incidence on the etalon), and then we
integrate over all incidence angles. As a last step we normalize the line
profile, as shown by Eq. 5 and 6.

$f(\theta)=2\cdot\pi\cdot\theta\cdot\exp^{-\left(\frac{\theta}{\theta_{cone}}\right)^{2}}$
(4)

$f_{norm}(\theta_{cone})=\int^{\theta_{cone}}_{\theta_{off}}f(\theta^{\prime})d\theta^{\prime}$
(5)

$T_{\lambda}(\lambda^{\prime},\theta_{cone})=\frac{\int^{\theta_{cone}}_{\theta_{off}}f(\theta^{\prime})\cdot
T_{\theta}(\lambda^{\prime},\theta^{\prime})d\theta^{\prime}}{f_{norm}(\theta_{cone})}$
(6)

We find a best fit to the observed etalon profile using $\theta_{cone}$ =
0.016 radians, as illustrated in Fig. 3, where the offset between the
centroids of the best fit (red) and the intrinsic etalon profile (green) is
found to be $10^{-5}\mu m$. This translates to an offset of about a tenth of a
pixel in MRS band 1A. Hence, we fitted the collimated beam line profiles to
obtain the line wavelengths and FWHM for the 4 etalons.

An (MTS-mounted) etalon line, as observed with the MIRI MRS, is a convolution
between the intrinsic etalon line profile (take for example the green curve in
Fig. 3) and a Gaussian profile with a width defined by the spectral resolution
of the MRS. An etalon line profile is given by Eq. 3. This line profile is
mathematically equivalent to a Lorentzian line profile. Convolving a
Lorentzian line profile with a Gaussian line profile yields a Voigt profile.
Due to the limited number of data points in the MRS data, to fit the FM etalon
lines we define a ”pseudo-Voigt” profile by taking the fractional
contributions of a Gaussian and a Lorentzian line shape and summing them. The
process is portrayed by Eq. 7, 8, and 9.

$G(\mu,\sigma)=\frac{1}{\sigma}\cdot\sqrt{\frac{4\leavevmode\nobreak\
ln(2)}{\pi}}\cdot e^{-4\leavevmode\nobreak\
ln(2)\left(\frac{z-\mu}{\sigma}\right)^{2}}$ (7)

$L(\mu,\sigma)=\frac{2}{\pi\sigma}\cdot\frac{1}{1+4\left(\frac{z-\mu}{\sigma}\right)^{2}}$
(8)

$V(A,\mu,\sigma)=A\left[f\cdot L(\mu,\sigma)+(1-f)\cdot G(\mu,\sigma)\right]$
(9)

In the above expressions, $z$ can be wavelength or pixels, and the free-
varying parameters are A, $\mu$, $\sigma$, and $f$. In the case where $f$ = 0
the above formalism reduces to a Gaussian line profile approximation.
Alternatively, in the case where $f$ = 1 the above formalism reduces to a
Lorentzian line profile approximation.

Figure 4: From left to right: observation of the MTS black body emission
through the SWP (top) and LWP (bottom) filters, and with no MTS filter
selected (center). The right panels show the division of the first (left)
image by the second (center) image, producing a detector plane image of the
wave pass filter throughput curves.

In Fig. 3 we show the result of fitting an FM etalon spectrum, extracted from
an iso-$\alpha$line centered in a slice on the detector. The fitting is done
with three different line profiles, a Gaussian, a Lorentzian, and a pseudo-
Voigt profile. Each spectral line is fitted separately with each model, and
then the fitted profiles are plotted together. The residuals of the fit of
each model are shown in the bottom plot. We note that the data have been
background-subtracted. Examining the residuals, we clearly see that the
pseudo-Voigt model fits the line core and line wings best. The Gaussian model
underestimates the line wing contribution to the lines, while the Lorentzian
model overestimates it.

Having accurately determined the etalon line centroids in the FM data in pixel
space, and the FTS data in wavelength space, the next step is to match the two
sets of centroids.

##  3 Absolute Calibration

The MTS etalons provide a plethora of unresolved, identifiable spectral
features for the wavelength calibration of the MRS. However, the etalon lines
are largely indistinguishable from one another so we cannot tell which is
which when observing a line in the detector pixel space of the MRS without an
external, absolute wavelength reference. Furthermore, a number of factors make
the matching of the etalon lines of the FTS and FM data extremely difficult:

  * •

The spectrophotometric response curves of the instruments used (FTS and
MTS+MIRI/MRS) are different.

  * •

The optical chain before the etalons in the FTS bench and the MTS bench were
not the same. As described in the previous section, we accounted for this by
estimating the chief ray collimated line centroid in the FTS case, and assumed
(based on the MTS design and the ground test data) that the MTS yielded a
perfectly collimated beam.

  * •

The FTS and FM data were taken at different temperatures (80 K for the FTS
versus 35 K for the MTS), causing the refractive index of the etalon internal
surfaces and the thermal coefficients of the material (which are temperature
dependent) to be different during the FTS and FM measurements. Even though the
effect is not expected to be large at these temperatures, it adds an
additional, small uncertainty on the FTS-FM line centroids transformation.

To resolve these issues, an unambiguous reference point (RPT) is needed that
matches a known wavelength determined from the high-resolution FTS data with
an (x,y) pixel coordinate observed on the MRS detector. This reference point
can be used to identify the nearest etalon line, and by extrapolation all of
the other etalon lines on the detector. The FM campaign included several such
tools that can be used to get the needed RPTs for the wavelength
characterisation of most of the MRS spectral sub-bands. As described in §3.1
\- 3.2 (and Table 2), in order of preference these include spectral features
in specially-constructed wave-pass filters (used for Channels 1B, 1C, 2C, and
4B), features observed in the MRS order-selection dichroic filters when in a
crossed-configuration (used for Channels 2A, 2B, 4A, 4B, and 4C), and spectral
overlaps between adjacent MRS bands (used for Channels 1A, 3A, 3B, and 3C). In
this Section, we describe how we obtained the RPT for one pixel (x,y)
coordinates on the central iso-$\alpha$of each slice, for each channel.
Section 4 shows how these RPTs were extrapolated to the rest of
iso-$\alpha$lines and slices in the channel.

###  3.1 Spectral calibration reference points

As described in Section §2, the MTS filter wheel included two wave-pass
filters. The transmission of the wave-pass filters was measured in a
laboratory at the University of Reading (henceforth UoR). The measurements
were performed at room temperature and at 35 K and in the latter case an
average uncertainty of $\pm$2% was reported for the transmission values across
the entire wavelength range. No uncertainties were reported on the wavelength
measurements. One wave-pass filter was designed to give a RPT for the short
wavelengths (SWP) and has a sharp cut-off at 21.6 $\mu$m. The second wave-pass
filter was built to produce a RPT for the long wavelengths (LWP), and has a
sharp cut-off at 6.6 $\mu$m. Both filters also show spectral features in other
wavelengts that can be used as RPTs (see Table 2). Measurements of the wave-
pass filters with the MRS were performed at 7 K using the MTS filter wheel
during the FM campaign. To determine the filter transmission, mounted on the
MTS under flight conditions, the MRS was first flood-illuminated using the MTS
calibration source at 800 K. Thereafter the same measurement was performed by
placing the wave-pass filter in the beam path. By dividing the measurements
with and without the filter, we compute the filter transmission in the
detector plane. The result is shown in Fig. 4 for MRS bands 1B and 1C as an
example.

For band 1C the cut-off is not sharp enough to accurately determine the
reference point wavelength-pixel pair. To address this we assumed that the
wavelength scale of the Zemax model is correct and attempt to determine a
global offset between the UoR data and the MRS data. Initially the MRS data
are wavelength-calibrated based on the Zemax model. The global offset is
computed by minimizing the transmission difference between the lab and MRS
data below 6.6 $\mu m$, as shown in Fig. 5. An offset of +0.05 $\mu m$ is
found. Translating this back to pixels, the cut-off of 6.6$\mu m$ is located
at (x,y,$\lambda$) = (76pix,117.11pix,6.6$\mu m$). Sub-bands 2C and 4C were
wavelength calibrated using the minima and maxima of the spectral features in
the LWS and SWP. Sub-bands 2A, 2B, 4A, and 4B are wavelength calibrated
similar to band 1C, namely by minimizing the difference in transmission
between the UoR data and the MRS data.

###  3.2 Overlapping spectral regions

Adjacent bands in the MRS show an overlap in wavelength coverage. This allowed
us to 1) calibrate bands with no RPT defined, and 2) check consistency in the
wavelength calibration of adjacent bands.

Band 1A, for example, has no spectral features to define a RPT (Table 2), but
we had a RPT (and therefore a wavelength solution) for band 1B. In that case,
we used the beating pattern of ET1 and ET2 to find the overlap region with
band 1B. Once defined, we assigned the (x,y,$\lambda$) triplets to the etalon
lines in the overlap region using the wavelength solution from band 1B. Then,
we cross correlated the fitted peaks of the lines in wavelength (FTS) and
pixels (FM data) for the rest of 1A, as we did with any other band, and
continued the process normally to obtain a wavelength calibration for all
slices in band 1A.

To check for the calibration consistency in different bands, we used the fact
that, as shown in Table 2, all sub-bands except 4B and 4C include lines from
at least two etalons. The order in which the lines of the different etalons
appear, combine, and overlap (beating pattern), and the wavelengths of their
centroids, must be the same in the overlaping region of two adjacent bands.
Thus, by comparing the wavelength solution of two adjacent bands and the
beating pattern of the etalons in the overlap regions, we confirmed that the
wavelength solutions of all sub-bands, up to 4A were consistent with each
other.

Channel 3, where there are no RPTs present, was calibrated using this method.
In this case we had the advantage of using the overlaps with band 2C and 3A,
and band 4A and 3C. In a similar manner, we calibrated 3B using the
overlapping regions with 3A and 3C. Thus, we obtained a wavelength solution
for all bands in channel 3, and also checked the consistency of the
calibration of channels 2 and 4, as the solutions derived for channel 3 using
wither of them had to be consistent.

##  4 Relative Calibration

###  4.1 Expanding to a full solution

The identified (x,y,$\lambda$) triplets are used to identify the etalon lines
around the RPT. By extrapolation, all the other etalon lines in the
iso-$\alpha$line are spectrally identified. Figure 6 shows the principle of
matching the etalon lines in wavelength and in pixel space using the
previously-determined reference point. The MRS line closest to the RPT is the
same as the FTS line closest to the reference wavelength. As such, all the
lines to the left and to the right of that line are also matched, and the
pixel-to-wavelength mapping is complete for that iso-$\alpha$line.

Figure 5: Determination of global wavelength offset between UoR data and MRS
data. The MRS data are wavelength calibrated using the Zemax optical model
predictions. Figure 6: A reference wavelength/pixel pair is used to match the
MRS and FTS etalon spectra.

Since the etalon is flood-illuminated, the etalon lines span the entire width
of a slice. Once we had the central iso-$\alpha$calibrated for a slice, we
were then able to calibrate the entire slice following the iso-$\lambda$lines
on the detector for that slice. The analysis of determining the reference
point position, however, had to be repeated for the other slices.

The wavelength assignation to etalon line peaks created a map of
(x,y,$\lambda$) coordinate triplets for $\sim$10$\%$ of the pixels in the
detector. The last step of the wavelength calibration was to fit a 2D
polynomial function of second order to these triplets, yielding a
parametrization of wavelength as a function of detector coordinates,
$\lambda(x,y)$, for every point on the detector:

$\lambda_{s}(x,y)=\sum_{i,j=0}^{N,N}K_{\lambda s}(i,j)\cdot(x-x_{s})^{j}\cdot
y^{i}$ (10)

where s is the slice number, $N$ is the order of the polynomial, $K_{\lambda
s}$ are the polynomial coefficients, and $x_{s}$ is the x-coordinate on the
detector for the center of the slice in the middle of the detector (y=512). We
produced one set of polynomials for each slice. Figure 7 shows the resulting
2D fit of the (x,y,$\lambda$) triplets (i.e. the final, calibrated wavelength
solution) of slice 10 of sub-band 1C.

Figure 7:  2D polynomial fit of the ET1 line centroids in slice 10 of MRS sub-
band 1C. The scatter points show the ET1 centroids fitted on the detector
(x,y) pixel space. The respective wavelength values matched using the FTS data
are shown on the z-axis and in the color scale of the dots. X and y axes show
the respective pixel coordinates on the detector plane. The light blue plane
shows the fitted 2D polynomial.

###  4.2 Refining the Relative Solution

If all iso-$\alpha$lines and slices of a sub-band were perfectly calibrated, a
single etalon line should have the same transmission profile over wavelength
anywhere on the detector. Thus, representing on one single plot the line
profile of a line from all the spatial locations where it is present should
produce the same transmission versus wavelength curve. Basically, the method
is similar to ”collapsing” all spatial direction on a single sub-band to
produce one spectrum of the etalon, to produce a ”super-sampled” spectrum with
the different pixel phases along a given slice. With this method we are able
to mitigate the uncertainties in defining the RPT in the different slices, as
well as the uncertainties in the MRS etalon line centroid.

The collapsing of the two spatial dimensions onto the spectral axis (i.e.
wavelength space) is done by omitting the spatial coordinate of a given pixel
in ($\alpha,\beta$) and plotting it based solely on its spectral coordinate
($\lambda$, in $\mu$m). This effectively creates a point cloud in two
dimensions, or equivalently, an oversampled 1d spectrum, as shown in Fig. 8.
The top plot showcases a sub-optimal initial wavelength calibration of MRS
band 2A. The calibration is impacted by the inaccurate determination of the
reference point in some slices as well as the noise in the data. In the bottom
plot we use the fact that all etalon lines should fall at the same wavelengths
as an internal consistency check, correcting previous inaccuracies. Producing
a similar plot as Fig. 8 for all spectral sub-bands immediately shows if there
are inconsistencies in the wavelength calibration within any sub-band. The
method works as follows:

  1. 1.

We define a reference slice on the detector. The reference slice is chosen on
the basis of a plot similar to that shown in the top plot of Fig. 8. The
criterion is that the oversampled spectrum resulting from the reference slice
should not deviate from the majority of the oversampled spectra in the other
slices. The discrepancy is very clear as only a couple of slices show large
offsets.

  2. 2.

We use the new wavelength solution to define a reference etalon spectrum in
wavelength space. The reference spectrum is an oversampled spectrum made by
compiling all the pixels in the reference slice. This oversampled spectrum is
used as the absolute wavelength calibration reference.

  3. 3.

We fit the etalon lines of the oversampled MRS spectrum in the reference slice
using pseudo-Voigt profiles.

  4. 4.

We fit the spectra extracted from all the other slices.

  5. 5.

We substitute the wavelength centroids of the etalon lines in the other slices
(same for all iso-$\alpha$lines in the slice) by the centroid of the etalon
line that is found closest in the reference spectrum.

  6. 6.

We re-fit the 2D polynomial to the boot-strapped etalon line centroids.

Figure 8: Internal consistency check of the wavelength solution within a sub-
band. The top panel shows a sub-optimal wavelength calibration of sub-band 2A.
The misalignment of lines introduced by the uncertainties in the cross
calibration of a few iso-$\alpha$lines and slices is clearly seen as a shift
to the right from the main ”oversampled” lines. The bottom panel shows the
resulting etalon spectrum once the uncertainties in the calibration have been
addressed.

By recognizing that the same feature in different iso-$\alpha$lines and slices
should have the same wavelength centers, the spectral calibration is improved
drastically and allowed us to achieve the MIRI-required spectral precision
down to 1/10th of a resolution element. Table 3 shows the resulting MRS
wavelength coverage based on our ground calibration.

##  5 Spectral Resolving Power

Figure 9: Best estimation of the of the MRS resolving power, based on ground
data. Black line: median MRS resolving power. Colored area: 1$\sigma$
confidence level. RMRS(FM) is really a lower limit of the expected MRS
resolving power, which could be probably up to a 10% higher (see text for
details). Table 3:  Calibrated spectral characteristics of the MRS derived
from ground data.

MRS | MRS | Wavelength coverage | Pixel scale∗ | R best estimate | R${}_{(FM+FTS)}^{**}$ | Spectral sampling
---|---|---|---|---|---|---
Channel | Band | ($\mu$m) |  $\mu$m/pixel | RMRS(FM) | RMRS(max) | LSF FWHM (pixels)
| SHORT (A) | 4.885 – 5.751 | 0.0009 | 3400 – 4000 | 3630 | 1.70 – 1.70
1 | MEDIUM (B) | 5.634 – 6.632 | 0.0010 | 3420 – 3990 | 3620 | 1.69 – 1.71
| LONG (C) | 6.408 – 7.524 | 0.0011 | 3330 – 3840 | 3590 | 1.77 – 1.80
| SHORT (A) | 7.477 – 8.765 | 0.0013 | 3190 – 3620 | 3520 | 1.86 – 1.92
2 | MEDIUM (B) | 8.711 – 10.228 | 0.0015 | 3040 – 3530 | 3290 | 1.86 – 1.89
| LONG (C) | 10.017 – 11.753 | 0.0017 | 2890 – 3374 | 3250 | 2.04 – 2.05
| SHORT (A) | 11.481 – 13.441 | 0.0019 | 2450 – 3010 | 3010 | 2.45 – 2.33
3 | MEDIUM (B) | 13.319 – 15.592 | 0.0022 | 2300 – 2460 | 2370 | 2.61 – 2.86
| LONG (C) | 15.400 – 18.072 | 0.0026 | 2020 – 2790 | 2470 | 2.92 – 2.48
| SHORT (A) | 17.651 – 20.938 | 0.0032 | 1400 – 1960 | – | 3.33 – 3.93
4 | MEDIUM (B) | 20.417 – 24.220 | 0.0037 | 1660 – 1730 | – | 3.31 – 3.77
| LONG (C) | 23.884 – 28.329 | 0.0043 | 1340 – 1520 | – | 4.11 – 4.29

555$*$$*$footnotetext: Calculated as wavelength coverage / 1024
pixels.$**$$**$footnotetext:  RMRS(FM+FTS) given for central wavelength of the
sub-band. It could not be determined for channel 4 due to artificial widening
of the etalon lines introduced by the FTS. See text for details.

The resolving power (R) is defined as $\lambda/\Delta\lambda$, where
$\Delta\lambda$ is the minimum distance to distinguish two features in a
spectrum. The shape of the etalon emission lines seen by the MRS during the FM
campaign is well defined by a Voigt profile (see Section 2). It is the result
of the convolution of the intrinsic etalon line described by Lorentzian
profile, and the MRS Line Spread Function (LSF), that has been approximated to
a Gaussian. To avoid dealing with deconvolution artifacts, we use an
approximate relation linking the widths of a Voigt, a Lorentzian, and a
Gaussian profiles (Olivero, 1977):

$\displaystyle FWHM_{FM}\approx 0.5346\cdot FWHM_{ET}+$ (11)
$\displaystyle+\sqrt{0.2166\cdot FWHM_{ET}^{2}+FWHM_{MRS}^{2}}$

where, FWHMFM is the FWHM of the etalon line in the FM data (Voigt profile),
FWHMET is the FWHM of the etalon line in the FTS results (Lorentzian profile),
and FWHMMRS is the FWHM of the MRS LSF, which we assume can be described by a
Gaussian profile. Eq. 11 have an accuracy of 0.02%.

If we define the FWHMMRS as the minimum distance to distinguish two features
in the spectrum ($\Delta\lambda$), the MRS resolving power can be defined as:

$R_{MRS(FM+FTS)}=\frac{\lambda}{FWHM_{MRS}}$ (12)

where $\lambda$ is the central wavelength of the etalon line and FWHMMRS is
obtained from Eq. 11.

The determination of the intrinsic etalon line profile is significantly
complex due to the uncertainties on the FTS measurements. The etalons were
illuminated by a quasi-collimated beam generating an asymmetric line profile
(Section 2). When the FTS etalon line is corrected by the cone angle effect,
the FWHM is reduced up to 50$\%$ from the observed FTS values. Additionally,
FTS FWHM values do not agree with the ones simulated by INTA, taking into
account all the fabrications parameters achieved by the manufacturer. The
simulated FTS FWHMs of ET1 are lower by a factor of 2-3, however etalons ET2,
ET3, and ET4 present higher FTS FWHMs by a factor of 1.2-1.1, 1.8-1.1, and
5.6-2.2, respectively. If the intrinsic FWHM of the etalon emission lines are
not consistent with the FTS determinations, the MRS resolving power could be
overestimated. Then, assuming that the intrinsic width of the etalon lines is
negligible (i.e. FWHMET=0), we calculated a lower limit to the MRS resolving
power:

$R_{MRS(FM)}=\frac{\lambda}{FWHM_{FM}}$ (13)

We used the wavelength calibration results, together with the intrinsic width
of the etalon lines obtained from the FTS data, to derive both estimations of
the MRS resolving power, RMRS(FM+FTS) and RMRS(FM). The estimation was done
for each etalon line present in each slice and sub-band. To estimate the
average resolving power of each sub-band as a function of wavelength, we
divided the sub-band into seven wavelength bins to calculate the median, and
the $\pm 1\sigma$ confident level. The median and confidence level of all bins
have been fitted with a polynomial function of second order. The confidence
level gives the variations as a function of location throughout the MRS field
(Wells et al., 2015; Law et al., 2021) for each sub-band. These variations
will be measured in detail during the commissioning and in-flight calibration
campaigns.

RMRS(FM+FTS) tends to overestimate the MRS resolving power at longer
wavelengths due to the large uncertainty in the FTS data. We found that the
intrinsic width of the etalon emission lines from the FTS data is higher than
the width in the simulations provided by INTA ( based on the fabrication
parameters given by the manufacturer) in channels 2 and 3, and even higher
than the widths in the FM data in channel 4. This suggests that the FTS is
artificially widening the etalon emission lines on those channels,
overestimating therefore the MRS resolving power. However, that is not the
case for channel 1, where the FTS data are consistent with the INTA
simulations, and the RMRS(FM+FTS) represents the best estimate of the MRS
resolving power. In this case, the RMRS(FM+FTS) is in average 10% higher than
the RMRS(FM).

Figure 9 shows the RMRS(FM) which, based on the discussion above, is the best
achievable estimation of the MRS resolving power with the current ground data.
RMRS(FM) is really a lower limit, as the intrinsic width of the etalon
emission line has been assumed negligible, but it is consistent with the
estimations based on Zemax optical models (Wells et al., 2015), suggesting
that the real resolving power of the MRS is close to RMRS(FM). Extrapolating
the RMRS(FM+FTS) results for channel 1 to longer channels, the RMRS(FM)
estimates should represent the MRS resolving power within a 10% shift to
higher values. The final resolving power of the MRS will be further
investigated and updated during commissioning and in-flight calibration
campaigns.

##  6 Caveats

###  6.1 Etalon collimation

The main source of uncertainties in our the wavelength calibration analysis
are the etalons. The distance between the peaks of two adjacent lines is given
by the free spectral range (FSR):

$FSR=\frac{\lambda^{2}_{0}}{2nd\leavevmode\nobreak\ cos(\theta)}$

where $\lambda_{0}$ is the wavelength of the peak of the line, n is the
refraction index of the medium in the etalon cavity, d is the width of the
etalon spacing, and $\theta$ is the angle of incidence of the illumination
beam. The FTS measurements described above were done at 80K while the FM data
was obtained at operating temperature of 40K. That difference in temperature
affects the size of the spacing ($d$) and thus the distance between the peaks
of the etalons. That means the distance between etalon lines (in micron) in
the FTS data is different to the distance between etalon lines in FM data (in
micron). To estimate the error introduced by the temperature difference in the
etalon spacing, we used literature information (Browder & Ballard, 1969, 1972;
Smith & White, 1975) on the thermal expansion coefficient for the etalon
materials (ZnSe for ET1,2, and 3, and CdTe for ET4). Our estimations yield a
negligible variation of 4–8% of a pixel for all bands (Martínez-Galarza et
al., 2010; Martínez-Galarza, 2012).

The transmission of the 4 etalons were measured with a quasi-collimated beam
at normal incidence. Even though the uncertainties of these measurements are
not available, we dealt with them with the pseudo Voigt profile and a fit of
the incidence angle in the fit of the etalon line shape and position. Our fits
show that the divergence angle of the beam is $\psi<1\deg$. This angle created
a small shift ($\lesssim 0.001$$\mu$m) on the FTS data, as $\theta$ in the FSR
equation is in this case all angles from $-\psi/2$ to $+\psi/2$ for each
etalon line. By fitting this divergence angle and how it affects the beam, we
corrected the positions of the FTS etalon lines in wavelength before using
them to obtain for the MRS wavelength dispersion solution.

Another source of uncertainty is the definition of the RPTs for the cut-off
filters and dichroics. The fitting methods used to define those RPTs have
uncertainty $\lesssim$8 pixels, which produces offsets in wavelength for the
whole band where the RPT is used. The beating pattern used in the overlap
regions should correct for small variations between bands. However, there is
room for a total offset in wavelength from the ground calibration to the real
one.

Both the total offset in wavelength (if present) and the variations in pixel
scale ($\mu$m/pixel) described above will be accounted for and corrected
during commissioning, as we will use as RPTs known emission lines of the
targets.

The main caveat on the resolving power determination with the ground data is
the unknown intrinsic profile of the etalon emission lines. The lack of
uncertainties of FTS etalon measurements and the effect of the non-collimated
beam, together with the possible effect on the different temperature used in
the FTS and FM measurements do not allow us to well characterize the intrinsic
etalon lines profile. The uncertainties presented in Fig. 9 are directly
related with the uncertainty on the determination of the intrinsic etalon line
profile. It could be up to 40$\%$ in channels 2 and 3, and in channel 4 only a
lower limit could be given. The commissioning activities will allow to reduce
the current uncertainty with the observations of the emission lines of sources
well characterized in the literature.

###  6.2 Across-Slice Wavelength Correction

The MRS spectra are expected to show a small shift in the wavelength direction
of the spectral image formed on the detector, which arises due to (sub slice
width) spatial offsets of a compact target in the across-slice direction. This
is a known effect of slicer-based IFUs (as opposed to fibers or lenslets which
scramble the light and avoid degeneracy between the cross-slice and dispersion
directions). The spatial offset gives rise to a shift in the centroid of the
illumination pattern across the slice; this then translates into a shift in
the dispersion direction of the output spectrum. The shift, which has a
typical value of up to 20 km s-1, is only seen from compact targets and has an
amplitude which varies with the source’s across-slice location. For extended
sources, the effect from each iso-$\alpha$averages out over the slice. Thus,
this effect is only present when an individual source is extracted (the
generation of a single spectrum from the multiple spectral elements measured
using the MRS’ multiple spatial samples).

The optimal spectral extraction routine for point sources in the MRS pipeline
(Labiano et al., 2016) has a step dedicated to correct the wavelength scale
for multiple adjacent slices. With the pipeline step correction we produce a
spectral correction accurate to better than $\pm$0.02 spectral resolution
elements.

##  7 In-Flight Calibration Strategy

###  7.1 Overview

To optimize our use of time during commissioning, we will just confirm and
adjust the wavelength calibration rather than re-deriving it from scratch. The
main goal of the commissioning is then to check the ground calibration and
update or improve it where needed. For the MRS spectral characterization, the
strategy will be to observe several targets with multiple, bright, unresolved
emission lines in the 12 sub-bands. These targets are chosen to be visible
during the expected observing time for this commissioning activity (towards
the end of the JWST commissioning campaign).

The target list includes unresolved (”point”) sources, as well as sources that
fill the MRS FOV (”extended”). We need to observe point sources to obtain an
accurate estimation of the resolving power of the MRS. However, the point
source data will be taken for very few slices (and positions in those slices)
in the MRS detectors. We will therefore use the extended source data to
extrapolate the wavelength calibration given by the point source to the whole
detector.

The point source data will also be used to characterize the across slice
wavelength variations, taking advantage of the different slice locations of
the target in each of the dither pointings. Discrepancies between the ground
calibration and the known wavelength values of the emission lines (from atomic
measurements) will be used to update the ground calibration where needed. This
update will be done in a similar way as the ground calibration, where the
emission lines of the commissioning targets will be used as RPTs. We will also
use the overlap regions of the spectral bands to cross check the consistency
of the in-flight calibration

The density of spectral lines in the target sources should be sufficient to
establish a good dispersion relation for all spectroscopic modes. Since many
of the lines will be unresolved (the best velocity resolution $\Delta v$,
equivalent to $FWHM_{MRS}$ in wavelength space, is about 80 km/sec, see
Section 5), the observed line widths can also be used to update the spectral
resolution. Some lines are expected to be closely spaced and can also provide
an estimates of the spectral resolution.

###  7.2 Target selection

Figure 10 show the Spitzer InfraRed Spectrograph (IRS) and ISO Short Wave
Spectrometer (SWS) spectra of the targets for this commissioning activity.
Below, we list a few of the properties of the targets selected.

Point sources: HD57150 (K=4.516 mag, 2MASS) and HD50083 (K=6.292 mag, 2MASS)
are bright Be stars with plenty of bright, unresolved emission lines in the
MRS wavelength coverage. HD57150 is the brightest target in the list, and very
close to the saturation limit given by the JWST Exposure Time Calculator
(ETC).

We recommend to observe both stars. Indeed, the available MIR spectra for
those sources have a much lower spectral resolution than MIRI-MRS’.
Consequently, the MRS spectra could unveil unexpected line structures,
asymmetries and velocities, resolved emission from discs, higher fluxes than
expected (risking saturation), etc., invalidating the use of one of the
sources for the purpose of wavelength calibration.

Extended source: NGC7027: a close-by planetary nebula with many bright,
unresolved emission lines. For the ETC estimations, we used the ISO-SWS
spectrum to create a 20”x20” source (the ISO-SWS average field). For this
commissioning activity, we will use a pointing in the ring of the nebula,
where the emission lines are generated. We are also considering observations
of NGC6302, which shows similar emission line fluxes, but a fainter continuum.

Figure 10: Top: ISO-SWS spectrum of NGC7027. This spectrum covers a much
larger field of view than the MRS. Thus, the fluxes expected in the MRS data
should be much lower. Bottom: IRS spectra of HD57150 and HD50083.

The IRS spectra of the stars shows that all sub-bands will include several
emission lines to update the wavelength calibration. There is no high-
resolution spectra, at MIRI wavelength ranges for this objects, so there is a
slight uncertainty on the sharpness of the lines. Some of these could be
resolved by the MRS. That is not a problem for the wavelength calibration, but
could affect the resolving power estimations. We do have, however, enough
lines all over the MIRI range in the stars and the planetary nebula to
guarantee unresolved emission features. Some regions of the nebula are
expected to show intrinsic velocities, which will yield shifted line
centroids. We plan to address those shifts by comparing the spectra, and the
wavelength calibration produced by the three targets independently. The cross
calibration among adjacent bands should also correct for isolated shifts in
the lines. Also, we are currently working on MIRISim (Klaassen et al., 2021)
simulations of the targets with the flight pipeline to improve the analysis
strategy if needed (e.g., Álvarez-Márquez et al., 2019). Taking all these into
account, and based in the instrument models and extensive analysis of the
data, we expect variations between the ground and on-orbit wavelength
calibration below 1-5 pixels at most. Our results over the years suggest that
this variation will most likely happen as a wavelength offset, without varying
significantly the dispersion (arsec/pixel) on the spectral direction. If we
needed to update the dispersion, we can do it with the emission lines from the
celestial targets (three per sub-band should be enough), and also reuse the
etalon transfer curves to fine tune it, based on the RPs obtained on orbit.

The observed data will be corrected for radial velocity (v = 9 km s-1 for NGC
7027) and heliocentric velocity before assigning wavelengths. Also, the FM
data shows no skewness dependence on position on the FOV. Even though we don’t
expect a skewness dependency on FOV position, we will need to confirm its
absence using the extended source commissioning data.

The MIRI optics were designed with dithering in mind. On non-dithered
observations of point sources, we observe clearly the effect of the
pixel/slice phase. The convolution of the input PSF with the top-hat
pixel/slice response function results in an effective combined PSF profile
(FWHM and minor/major axis ratio) that vary strongly within the field of view
depending on whether a source is located in the middle of a sampling element
or between two elements (see, e.g., Fig 6 of Law et al., 2015, for a similar
effect on fiber-type IFUs). The dithering strategy is designed to mitigate
this sampling issue and provide as uniform a PSF profile as possible. While
this is an interesting discussion in its own right, it is well beyond the
scope of this paper.

We also expect some variations in the resolution with FOV position, as is the
case in most IFUs (e.g., Law et al., 2021). These variations will be
characterised during commissioning. They are not expected to be significant
for most science cases, as most of the deviations in R we see in the ground
data are mainly caused by the uncertainties in the etalon calibration and FTS
measurements.

##  8 Summary

The ground test campaigns, carried under flight conditions, produced a large
amount of the data to calibrate the MRS in detail. Concerning the spectral
characterization, the MTS provided all tools necessary to measure the spectral
dispersion and resolution of the MRS. The wave-pass filters on the MTS, and
the cross-dichroic configurations of the MRS showed clear spectral features at
known wavelengths, providing RPTs to establish an absolute wavelength scale.
Four MTS etalons (each optimized for each MRS channel) were designed to
produce enough unresolved transmission lines to measure the wavelength values
for enough pixels all across the detector. A 2D polynomial fit per slice
completed the wavelength calibration for the whole detector. Four MRS bands
had no RPTs defined.

By construction, all adjacent MRS bands have an overlap in wavelength.
Combining the etalon transmission curves, and the absolute wavelength
calibration of bands with RPTs, we were able to extrapolate the absolute
calibration of the RPT calibrated bands to those with no RPTs. The overlap
regions were also used to cross-check the calibration of adjacent bands. An
oversampling method was used to refine the wavelength calibration down to the
required precision of 1/10 of a resolution element.

The etalon transmission curves were also used to estimate the resolving power
of the 12 bands. Some problems on the ground measurements produced an
uncertainty of 10$\%$ in the final values. Taking into account this
uncertainty, our results suggest a better resolving power than the initial
MIRI/MRS requirements. Table 3 shows the wavelength coverage and resolving
power based on the ground calibration.

We also presented the commissioning strategies that will be followed to update
the spectral characterisation of the MRS, and confirm the resolving power
measured. The target selection was for the commissioning campaign was also
discussed the different targets proposed. These targets can further be used
for the calibration campaigns during the routine operations of the JWST

###### Acknowledgements.

We thank the anonymous referee for her/his very useful recommendations. A.L.
acknowledges the support from Comunidad de Madrid through the Atracción de
Talento grant 2017-T1/TIC-5213. A.L and J.A.M. acknowledge support from the
Spanish State Research Agency (AEI) under grant numbers ESP2017-83197, and
PID2019-106280GB-I00, and the support from CSIC through ILINK grant
LINKB20003. This research has been partially funded by the Spanish State
Research Agency (AEI) Project MDM-2017-0737 Unidad de Excelencia María de
Maeztu - Centro de Astrobiología (INTA-CSIC). Ioannis Argyriou thanks the
European Space Agency (ESA) and the Belgian Federal Science Policy Office
(BELSPO) for their support in the framework of the PRODEX Programme.
The work presented is the effort of the entire MIRI team and the enthusiasm
within the MIRI partnership is a significant factor in its success. MIRI draws
on the scientific and technical expertise of the following organisations: Ames
Research Center, USA; Airbus Defence and Space, UK; CEA-Irfu, Saclay, France;
Centre Spatial de Liége, Belgium; Consejo Superior de Investigaciones
Científicas, Spain; Carl Zeiss Optronics, Germany; Chalmers University of
Technology, Sweden; Danish Space Research Institute, Denmark; Dublin Institute
for Advanced Studies, Ireland; European Space Agency, Netherlands; ETCA,
Belgium; ETH Zurich, Switzerland; Goddard Space Flight Center, USA; Institute
d’Astrophysique Spatiale, France; Instituto Nacional de Técnica Aeroespacial,
Spain; Institute for Astronomy, Edinburgh, UK; Jet Propulsion Laboratory, USA;
Laboratoire d’Astrophysique de Marseille (LAM), France; Leiden University,
Netherlands; Lockheed Advanced Technology Center (USA); NOVA Opt-IR group at
Dwingeloo, Netherlands; Northrop Grumman, USA; Max-Planck Institut für
Astronomie (MPIA), Heidelberg, Germany; Laboratoire d’Etudes Spatiales et
d’Instrumentation en Astrophysique (LESIA), France; Paul Scherrer Institut,
Switzerland; Raytheon Vision Systems, USA; RUAG Aerospace, Switzerland;
Rutherford Appleton Laboratory (RAL Space), UK; Space Telescope Science
Institute, USA; Toegepast- Natuurwetenschappelijk Onderzoek (TNO-TPD),
Netherlands; UK Astronomy Technology Centre, UK; University College London,
UK; University of Amsterdam, Netherlands; University of Arizona, USA;
University of Bern, Switzerland; University of Cardiff, UK; University of
Cologne, Germany; University of Ghent; University of Groningen, Netherlands;
University of Leicester, UK; University of Leuven, Belgium; University of
Stockholm, Sweden; Utah State University, USA. A portion of this work was
carried out at the Jet Propulsion Laboratory, California Institute of
Technology, under a contract with the National Aeronautics and Space
Administration. We would like to thank the following National and
International Funding Agencies for their support of the MIRI development:
NASA; ESA; Belgian Science Policy Office; Centre Nationale D’Etudes Spatiales
(CNES); Danish National Space Centre; Deutsches Zentrum fur Luft-und Raumfahrt
(DLR); Enterprise Ireland; Ministerio De Economiá y Competividad; Netherlands
Research School for Astronomy (NOVA); Netherlands Organisation for Scientific
Research (NWO); Science and Technology Facilities Council; Swiss Space Office;
Swedish National Space Board; UK Space Agency. This research has made use of
ESASky, developed by the ESAC Science Data Centre (ESDC) team and maintained
alongside other ESA science mission’s archives at ESA’s European Space
Astronomy Centre (ESAC, Madrid, Spain). We take this opportunity to thank the
ESA JWST Project team and the NASA Goddard ISIM team for their capable
technical support in the development of MIRI, its delivery and successful
integration.

## References

  * Álvarez-Márquez et al. (2019) Álvarez-Márquez, J., Colina, L., Marques-Chaves, R., et al. 2019, A&A, 629, A9
  * Argyriou et al. (2020) Argyriou, I., Wells, M., Glasse, A., et al. 2020, arXiv e-prints, arXiv:2007.16143
  * Bagnasco et al. (2007) Bagnasco, G., Kolm, M., Ferruit, P., et al. 2007, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6692, Overview of the near-infrared spectrograph (NIRSpec) instrument on-board the James Webb Space Telescope (JWST), 66920M
  * Belenguer et al. (2008) Belenguer, T., Alcacera, M. A., Aricha, A., et al. 2008, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7010, MIRI Telescope Simulator, 701039
  * Boccaletti et al. (2015) Boccaletti, A., Lagage, P.-O., Baudoz, P., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 633
  * Bouchet et al. (2015) Bouchet, P., García-Marín, M., Lagage, P.-O., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 612
  * Browder & Ballard (1969) Browder, J. S. & Ballard, S. S. 1969, Appl. Opt., 8, 793
  * Browder & Ballard (1972) Browder, J. S. & Ballard, S. S. 1972, Appl. Opt., 11, 841
  * Colina et al. (2004) Colina, L., Diaz, E., Aricha, A., et al. 2004, in Optical, Infrared, and Millimeter Space Telescopes, ed. J. C. Mather, Vol. 5487, International Society for Optics and Photonics (SPIE), 804 – 813
  * Eccleston et al. (2012) Eccleston, P., Glasse, A., Grundy, T., et al. 2012, 8449, 09
  * Gardner et al. (2006) Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci. Rev., 123, 485
  * Glasse et al. (2010) Glasse, A. C. H., Bauwens, E., Bouwman, J., et al. 2010, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7731, The throughput and sensitivity of the JWST mid-infrared instrument, 77310K
  * Glauser et al. (2010) Glauser, A. M., Glasse, A., Morrison, J. E., et al. 2010, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7731, Space Telescopes and Instrumentation 2010: Optical, Infrared, and Millimeter Wave, 77313K
  * Gordon et al. (2015) Gordon, K. D., Chen, C. H., Anderson, R. E., et al. 2015, PASP, 127, 696
  * Herrada et al. (2007) Herrada, F. J., Eiriz, V., Alcacera, M. A., et al. 2007, New Developments in Optomechanics, 6665, 66650Y
  * Kendrew et al. (2015) Kendrew, S., Scheithauer, S., Bouchet, P., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 623
  * Klaassen et al. (2021) Klaassen, P. D., Geers, V. C., Beard, S. M., et al. 2021, MNRAS, 500, 2813
  * Labiano et al. (2016) Labiano, A., Azzollini, R., Bailey, J., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9910, Observatory Operations: Strategies, Processes, and Systems VI, 99102W
  * Law et al. (2021) Law, D. R., Westfall, K. B., Bershady, M. A., et al. 2021, AJ, 161, 52
  * Law et al. (2015) Law, D. R., Yan, R., Bershady, M. A., et al. 2015, AJ, 150, 19
  * Lim et al. (2008) Lim, T., Alvarez, J. L., Bauwens, E., et al. 2008, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7010, First results from MIRI verification model testing, 70103A
  * Lipson & Lipson (1969) Lipson, S. G. & Lipson, H. 1969, Optical physics.
  * Martínez-Galarza (2012) Martínez-Galarza, J. R. 2012, PhD thesis, Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, the Netherlands
  * Martínez-Galarza et al. (2010) Martínez-Galarza, J. R., Glauser, A. M., Hernán-Caballero, A., et al. 2010, in Space Telescopes and Instrumentation 2010: Optical, Infrared, and Millimeter Wave, ed. J. M. O. Jr., M. C. Clampin, & H. A. MacEwen, Vol. 7731, International Society for Optics and Photonics (SPIE), 1313 – 1324
  * Maszkiewicz (2017) Maszkiewicz, M. 2017, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10564, Proc. SPIE, 105642Q
  * Olivero (1977) Olivero, J. 1977, J. Quant. Spec. Radiat. Transf., 17, 233
  * Rieke et al. (2015) Rieke, G. H., Ressler, M. E., Morrison, J. E., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 665
  * Rieke et al. (2015) Rieke, G. H., Wright, G. S., Böker, T., et al. 2015, PASP, 127, 584
  * Rieke et al. (2005) Rieke, M. J., Kelly, D., & Horner, S. 2005, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 5904, Overview of James Webb Space Telescope and NIRCam’s Role, ed. J. B. Heaney & L. G. Burriesci, 1–8
  * Smith & White (1975) Smith, T. F. & White, G. K. 1975, Journal of Physics C: Solid State Physics, 8, 2031
  * Wells et al. (2015) Wells, M., Pel, J.-W., Glasse, A., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 646–664
  * Wright et al. (2015) Wright, G. S., Wright, D., Goodson, G. B., et al. 2015, Publications of the Astronomical Society of the Pacific, 127, 595