Hybrid Modular Redundancy]Hybrid Modular Redundancy: Exploring Modular Redundancy Approaches in RISC-V Multi-Core Computing Clusters for Reliable Processing in Space

Michael Rogenmoser
Both authors contributed equally to this research.

ETH Zürich

Yvan Tortorella
[1]

Università di Bologna

Davide Rossi

Università di Bologna

Francesco Conti

Università di Bologna

Luca Benini

ETH Zürich

Università di Bologna

The presented work was supported by Thales Alenia Space.

We acknowledge support by the EU H2020 Fractal project funded by ECSEL-JU grant agreement #877056.

We acknowledge support by TRISTAN, which has received funding from the Key Digital Technologies Joint Undertaking (KDT JU) under grant agreement nr. 101095947. The KDT JU receives support from the European Union’s Horizon Europe’s research and innovation programmes.

The authors thank Luca Rufer for his valuable contribution to the research project.

<concept_desc>Hardware Fault tolerance</concept_desc>
<concept_desc>Computer systems organization Reliability</concept_desc>
<concept_desc>Computer systems organization Multicore architectures</concept_desc>

[500]Hardware Fault tolerance
[500]Computer systems organization Reliability
[500]Computer systems organization Multicore architectures

March 8th, 2023
[revised]August 15th, 2023

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Michael Rogenmoser
Michael's fanciest bio.

Yvan Tortorella
received his Master's Degree in Electronic Engineering in October 2021 from the University of Bologna. He is currently pursuing a Ph. D. in Digital Systems Design in the group of Professor Luca Benini at the Department of Electrical and Information Engineering (DEI) of the University of Bologna. His research interests include the design of PULP (Parallel Ultra-Low Power)-based hardware accelerators for ultra-low power Machine Learning and the design of RISC-V-based computer architectures for satellite applications.

Davide Rossi
received the Ph.D. degree from the University of Bologna, Bologna, Italy, in 2012. He has been a Post-Doctoral Researcher with the Department of Electrical, Electronic and Information Engineering “Guglielmo Marconi,” University of Bologna, since 2015, where he is currently an Associate Professor. His research interests focus on energy-efficient digital architectures. In this field, he has published more than 100 papers in international peer-reviewed conferences and journals. He is recipient of Donald O. Pederson Best Paper Award 2018, 2020 IEEE TCAS Darlington Best Paper Award, 2020 IEEE TVLSI Prize Paper Award.

Francesco Conti
received the Ph.D. degree in electronic engineering from the University of Bologna, Italy, in 2016. He is currently a Tenure-Track Assistant Professor in the DEI Department of the University of Bologna. From 2016 to 2020, he held a research grant in the DEI department of University of Bologna and a position as postdoctoral researcher at the Integrated Systems Laboratory of ETH Zurich in the Digital Systems group. His research focuses on the development of deep learning based intelligence on top of ultra-low power, ultra-energy efficient programmable Systems-on-Chip. His research work has resulted in more than 40 publications in international conferences and journals and has been awarded several times, including the 2020 IEEE TCAS-I Darlington Best Paper Award.

Luca Benini
holds the chair of digital Circuits and systems at ETHZ and is Full Professor at the Università di Bologna. He received a PhD from Stanford University. Dr. Benini's research interests are in energy-efficient parallel computing systems, smart sensing micro-systems and machine learning hardware. He has published more than 1000 peer-reviewed papers and five books. He is a Fellow of the IEEE, of the ACM and a member of the Academia Europaea. He received the IEEE Mac Van Valkenburg award in 2016 and the ACM/IEEE A. Richard Newton Award in 2020.


Further author information: (Send correspondence to Ioannis Argyriou)
E-mail<EMAIL_ADDRESS>
# Characterization of the optical properties of the buried contact of the JWST
MIRI Si:As infrared blocked impurity band detectors

Ioannis Argyriou  Institute of Astronomy KU Leuven, Celestijnenlaan 200D - box
2401, 3001 Leuven, Belgium  George H. Rieke  Steward Observatory and the
Department of Astronomy, The University of Arizona, 933 N Cherry Ave, Tucson,
AZ, 85750, USA  Michael E. Ressler  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  András
Gáspár  Steward Observatory and the Department of Astronomy, The University of
Arizona, 933 N Cherry Ave, Tucson, AZ, 85750, USA  Bart Vandenbussche
Institute of Astronomy KU Leuven, Celestijnenlaan 200D - box 2401, 3001
Leuven, Belgium

###### Abstract

The Mid-Infrared Instrument MIRI on-board the James Webb Space Telescope uses
three Si:As impurity band conduction detector arrays. MIRI medium resolution
spectroscopic measurements (R$\sim$3500-1500) in the 5 $\mu m$ to 28 $\mu m$
wavelength range show a 10-30% modulation of the spectral baseline; coherent
reflections of infrared light within the Si:As detector arrays result in
fringing. We quantify the shape and impact of fringes on spectra of optical
sources observed with MIRI during ground testing and develop an optical model
to simulate the observed modulation. We use our optical model in conjunction
with the MIRI spectroscopic data to show that the properties of the buried
contact inside the MIRI Si:As detector have a significant effect on the
fringing behavior.

###### keywords:

James Webb Space Telescope, mid infrared instrument MIRI, spectroscopy, Si:As
impurity band conduction device, fringing buried contact, doping
concentration, refractive index

Copyright 2020 Society of Photo‑Optical Instrumentation Engineers (SPIE).

##  1 INTRODUCTION

The Mid-Infrared Instrument MIRI on-board the James Webb Space Telescope
(JWST) will perform imaging, coronagraphy, low-resolution spectroscopy, and
medium-resolution spectroscopy at unprecedented sensitivity levels in the 5
$\mu m$ to 28 $\mu m$ wavelength range[1]. MIRI will be used by astronomers to
study distant galaxies, stellar populations, exoplanets, and bodies within our
solar system. The instrument employs three 1024x1024 pixel Si:As IBC detector
arrays, which were manufactured by Raytheon Vision Systems (RVS) [2].

Si:As IBC devices have extensive space flight heritage. Similar devices were
used in all three instruments of the Spitzer space telescope, namely the
Infrared Array Camera (IRAC), the Infrared Spectrograph (IRS), and the
Multiband Imaging Photometer (MIPS). The devices were also used in the Wide-
Field Infrared Survey Explorer (WISE), the Midcourse Space Experiment (MSX),
the Infrared Camera (IRC) on-board Akari, and the Short Wavelength
Spectrometer (SWS) on-board the Infrared Space Observatory (ISO). The high
quantum efficiency of the Si:As IBC devices, in combination with the extensive
wavelength range covered (5-28 $\mu m$) gives these devices a unique advantage
[3, 2]. Other detectors, such as the Teledyne Imaging Sensors’ LWIR HgCdTe
detector have a relatively higher quantum efficiency, however, they cover a
shorter wavelength range (the LWIR HgCdTe detector extends out to 13 $\mu m$)
[4]. It is important thus to understand the spectrophotometric response of the
Si:As IBC device across the long wavelength range covered in order to maximize
scientific output.

Two of the three Si:As detector arrays of MIRI are used in the medium-
resolution spectrometer (MRS)[5]. The detectors sample the 5 $\mu m$ to 28
$\mu m$ wavelength range. The MRS spectral resolution varies from
$\lambda/\Delta\lambda\sim$ 3500 at 5 $\mu m$ and $\lambda/\Delta\lambda\sim$
1500 at 28 $\mu m$ (Labiano, Argyriou, et al., 2021, in prep). As is common
for infrared spectrometers, constructive and destructive interference in
different layers of the MRS detector arrays modulate the detected signal as a
function of wavelength; the detector-constituting layers behave as efficient
Fabry-Pérot etalons. The resulting ”fringing” in the MRS spectra varies in
amplitude between 10-30% of the spectral baseline[6]. Similar fringing was
observed in ISO SWS data, in Spitzer IRS data, as well as near-infrared data
taken with the Space Telescope Imaging Spectrograph (STIS) on-board the Hubble
Space Telescope (HST) [7, 8, 9]. If not corrected, fringing can severely
hamper the scientific interpretation of MIRI observations.

As we will show in this paper, fringing in the MRS detectors is measured as an
apparent beating of two fundamental frequencies. The first frequency is
proportional to the geometric thickness between the detector back side
entrance surface and the pixel metalization. The second frequency is
proportional to the geometric thickness between the buried contact,
characteristic of back-illuminated Si:As IBC devices, and the pixel
metalization. The first frequency has been studied extensively in [6]. Similar
to MRS data, [10] identified a second fringe in their study of Si:As IBC
devices. In this paper we characterize the second fringe in the MIRI Si:As
devices using the resolved fringes in MRS spectra. More importantly, we argue
that the physical cause of the second fringe is the high doping concentration
of the buried contact.

##  2 MIRI DETECTOR ARCHITECTURE AND MRS SPECTRAL FRINGING

The Si:As IBC detectors are grown on a silicon substrate. We show a
representation of the architecture of the MIRI detector arrays in Fig. 1,
which is based on [11, 2, 3, 12]. Photons pass through the anti-reflection
coating on the detector back side, into the substrate, and then through the
buried contact, into the infrared-active layer (detection layer). The
infrared-active layer is doped with arsenic to absorb the incoming photons,
elevating photo-excited electrons from the impurity band into the conduction
band[3]. Assuming a low level of minority acceptor impurities in this layer,
an electric field can be maintained across it that causes the photoelectrons
to migrate to the front of the detector. A thin, high purity layer (blocking
layer) is grown over the front of the infrared-active layer. When operated at
sufficiently low temperatures, thermally-generated free charge carriers cannot
penetrate the blocking layer because it lacks an impurity band and the
carriers have insufficient energy to be lifted into the conduction band. The
photo-generated free charge carriers in the conduction band can, however,
traverse the blocking layer, to be collected at the front contact. The
electric field that drives this process is maintained across the infrared-
active layer between the front contact, on the readout side, and a buried
contact, connected via a V-etch buried implant on one side of the detector.

---
Figure 1: Representation of MIRI detector architecture based on [11, 2, 3,
12]. The dimensions are not to scale.

The MIRI MRS has a short-wavelength arm (SW) and a long-wavelength arm
(LW)[5]. The SW and LW arms use one Si:As IBC device each, which detect
photons in the 5-11.8 $\mu m$ range and in the 11.8-28 $\mu m$ range
respectively. We provide the optical and geometric properties of the MRS SW
and LW detector layers in Table LABEL:tab:miri_sw_detector_optical_layers and
Table LABEL:tab:miri_lw_detector_optical_layers111A semi-infinite thickness is
attributed to the ambient medium and the pixel metalization.. These properties
define the optical medium through which the photons propagate.

Table 1:  Optical and geometric properties of MIRI MRS SW detector layers[2,
6].

Layer Material Geometric thickness Comments Ambient Vaccuum inf. Anti-
reflection coating Zinc Sulphide (ZnS) 0.72 $\mu m$ Minimum reflectance at 6
$\mu m$. Inactive layer Silicon (Si) 460.5$\pm$6.6 $\mu m$ Pure silicon (zero
absorption). Buried contact (electrode) Antimony-doped silicon uncertain
Material is assumed based on [10]. Active layer Arsenic-doped silicon 30 $\mu
m$ \- Thickness variation much less than $1\mu m$. \- Arsenic concentration
N=$5\times 10^{17}cm^{-3}$. \- Contingency array (identical to IRAC). Blocking
layer Silicon (Si) 4 $\mu m$ Pure silicon (zero absorption). Pixel
metalization Aluminium (Al) inf.

Table 2: Optical and geometric properties of MIRI MRS LW detector layers[2,
6].

Layer Material Geometric thickness Comments Ambient Vaccuum inf. Anti-
reflection coating Zinc Sulphide (ZnS) 2.08 $\mu m$ Minimum reflectance at 16
$\mu m$. Inactive layer Silicon (Si) 416.8$\pm$5.2 $\mu m$ Pure silicon (zero
absorption). Buried contact (electrode) Antimony-doped silicon uncertain
Material is assumed based on [10]. Active layer Arsenic-doped silicon 35 $\mu
m$ \- Thickness variation much less than $1\mu m$. \- Arsenic concentration
N=$7\times 10^{17}cm^{-3}$. \- Baseline array. Blocking layer Silicon (Si) 4
$\mu m$ Pure silicon (zero absorption). Pixel metalization Aluminium (Al) inf.

Since the infrared-active layer is doped with arsenic, an n-type impurity, a
similar dopant will have been used for the buried contact, i.e. either arsenic
(As), antimony (Sb), or phosphorus (P). Antimony and phosphorus are two
commonly used n-type dopants in semiconductors[13]. In fact, Si:As IBC devices
manufactured by the Rockwell Science Center (USA) used antimony for the buried
contact and arsenic for the front contact[10]. We assume that that is the case
for the MIRI detectors as well.

The MRS acquires spectra in 12 spectral bands, 6 with the SW detector and 6
with the LW detector. In the left panel of Fig. 2 we show MRS measurements
taken with the spectrometer calibration source in all 12 MRS spectral
bands222The MRS calibration source is a spatially flat source which emits as a
grey body at a temperature of 800K[5]. The x-axis shows the wavelength
coordinate and the y-axis gives the signal units in digital numbers (DN) per
second. We note that no spectrophotometric correction has been applied to the
data. Discontinuities in the signal of neighboring spectral bands are due to
jumps in the spatial and spectral resolution of each band, which also causes
the systematic drop in signal within each band.

Focusing on the top plot of the left panel of Fig. 2, we see a prominent
fringe envelope in all 6 of the SW bands showcased. This fringing is caused by
the coherent reflections between the detector back side surface coating and
the pixel metalization. The MRS SW anti-reflection coating is optimized such
that reflections between the ambient medium and the silicon substrate are
least at 6 $\mu m$, however, the reflectance of the coating never drops below
5% (a constant reflectance of 98% can be assumed at the pixel aluminum
metalization). This results in a prominent high-frequency sinusoidal signature
at all wavelengths below 12 $\mu m$. Superimposed on the high-frequency
fringes is a low frequency modulation, a second fringe. The modulation in
question can be discerned most clearly as a function of wavelength when
looking at the lower part of the envelope of the 6 SW spectra. The picture
becomes slightly less clear in the middle plot (LW bands), where different
detector parameters between the SW and LW bands give a discontinuous image
with respect to the top plot. There is, however, a clear beating pattern in
the signal. This beating pattern is no longer resolved in the bottom plot,
where we see mainly the primary fringe frequency, but no statistically
significant sign of the secondary frequency.

---
Figure 2: Left panel: MRS extracted spectra, as a function of wavelength, from
observations of the spectrometer calibration source. The top plot shows the SW
data, and the middle and bottom plots show the two halves of the LW wavelength
range. Right panel: Fringe transmission as a function of wavenumber. The
transmission is determined by normalizing the spectra in the left panel to a
fringe free continuum (spline connecting the fringe peaks). Transmission
discontinuities between nearby spectral bands in the SW data are due to each
band having a different spectral resolution.

A common way to study fringes is to plot them in wavenumber
($\sigma=1/\lambda$) space. This is shown in the right panel of Fig. 2, where
we normalize the signal shown in the left panel to a spline connecting the
fringe peaks, which we define as the ”fringe-free” spectral continuum. Going
from high to low wavenumbers, we see that the fringe beating between the high
and low frequency fringes becomes more prominent from 2050 $cm^{-1}$
($\sim$4.88 $\mu m$) down to 610 $cm^{-1}$ ($\sim$16.4 $\mu m$). At even lower
wavenumbers, below 575 $cm^{-1}$, there is a lack of a clearly resolved
beating pattern.

Assuming that the layers in the MRS detectors are plane parallel, the distance
between interference peaks in a fringe transmission spectrum can be related to
the optical thickness of the Fabry-Pérot etalon as $D=1/\Delta\sigma$. The
optical thickness is related to the geometric thickness of the etalon
$t_{et}=(D/2)/n$, where $n$ is the refractive index of the medium between the
etalon reflective boundaries. We use the optical properties of crystalline
silicon for the Si:As IBC device[14]. The relation for $D$ and $t_{et}$ can be
used to probe the optical parameters of the etalon producing the MRS fringes.
A more advanced method to determining the optical thickness $D$ is to fit the
fringe transmission with a Fabry-Pérot transmittance function, given by Eq. 1
[15, 6]. In this equation $R$ stands for the equivalent reflectance of the
etalon, assuming that both etalon boundaries have the same reflectance. We use
a scanning window with a fixed width (two fringe periods) and fit the fringes
inside the specified window as we scan the window across the spectrum.

$T_{et}(\sigma)=\left(1+\frac{4R}{(1-R)^{2}}\sin^{2}\left(2\pi
D\sigma\right)\right)^{-1}$ (1)

In Fig. 3 we show the resulting etalon thickness $t_{et}$, estimated from the
fringe transmission in the top and bottom right plot of Fig. 2. From fitting
the high-frequency fringe in the SW data, we find that the etalon thickness
matches the sum of the SW detector back side surface (inactive layer),
infrared-active layer, and blocking layer. We assume that the thickness of the
buried contact layer is much smaller compared to the other layers.
Consequently, this means that the fringe is produced between the back side
surface coating and the pixel metalization. Using the low-frequency fringe
from SW data, a rough back-of-the-envelope calculation yields a thickness of
36.9$\pm$3 $\mu m$ (uncertainty based on the standard deviation of the limited
measurements). We find that this is in agreement with the infrared-active
layer plus blocking layer thickness (34 $\mu m$). In the bottom plot of Fig.
3, we find that the frequency of the fringing in the bottom plot of Fig. 2
matches that of the back side surface (inactive layer) of the LW detector; the
fringing occurs between the back side surface coating and the infrared-active
layer.

---
Figure 3: Etalon thickness derived from fitting 9 fringe transmission profiles
(6 in SW data, 3 in LW data) as a function of wavenumber. Each wavenumber
corresponds to a different pixel location on the MRS detectors, which allows
us to probe detector thickness variations.

##  3 MIRI DETECTOR OPTICAL MODEL

Each layer inside the MIRI Si:As IBC devices has optical properties defined by
the materials listed in Table LABEL:tab:miri_sw_detector_optical_layers and
LABEL:tab:miri_lw_detector_optical_layers. We use refractive index values for
zinc-sulphide, silicon, and aluminium, based on the available literature[16,
14, 17]. Absorption inside the infrared-active layer is parametrized using the
absorption coefficient curve of Si:As derived by [10]333We fit the 5-20 $\mu
m$ data with a third-order polynomial and extrapolate the trend to 28 $\mu m$.
The result is in agreement with the values of [18, 12]..

We model the transmitted, reflected, and absorbed intensity of the MIRI
detectors by using the refractive information of each material, and solving
the Fresnel equations analytically for a multilayer planar stack. We do this
using the Transfer-Matrix Method (TMM)[19, 20]. What we measure in MRS data is
the fraction of the light absorbed in the infrared-active layer at each
wavelength. We focus thus on the absorbed intensity prediction of the TMM
simulation.

In Fig. 4 we show the result of a pair of high-resolution TMM simulations
based on the MRS SW and LW detector optical models. The traditional assumption
has been that the buried contact is transparent and does not affect the
optical performance, so for these simulations no buried contact was included
in the models. The simulations show a beating of two frequencies, similar to
the MRS data shown in Fig. 2. However, we find a significant discrepancy
between the TMM simulated fringes and the fringes in the MRS data, namely, the
amplitude of the secondary frequency modulation in the TMM output is much
smaller compared to the MRS data. This is noticeable in the SW regime, as well
as in the LW regime. In the latter case, there is a significant difference in
the 12-18.2 $\mu m$ wavelength range (830-550 $cm^{-1}$). The beating in the
LW data is very prominent already at 12 $\mu m$ (830 $cm^{-1}$), which is not
the case for the simulated absorbed intensity.

---
Figure 4: Absorbed intensity simulated using the Transfer-Matrix Method (TMM)
for the MRS SW and LW detector optical models. The buried contact is not
accounted for in the optical models.

All material properties in the MRS optical models are well constrained, except
for the material properties of the buried contact layer. We can probe the
material properties of the buried contact layer by minimizing the difference
between the fringes in the TMM-simulated absorbed intensity and the MRS
fringes. This is done as follows:

  1. 1.

We assume a thickness of the buried contact layer of 0.2 $\mu m$ (typical
value for an ion implanted layer).

  2. 2.

We define the buried contact layer refractive index as $n_{bc}=n_{Si}+\Delta
n_{bc}$, where $n_{Si}$ is the refractive index of crystalline silicon[14].
$\Delta n_{bc}$ is defined as a free parameter. For simplicity, the value of
$\Delta n_{bc}$ is kept constant in each MRS spectral band.

  3. 3.

We assume that the absorption coefficient of the buried contact layer is the
same as that of the infrared-active layer. Given the small thickness of the
layer (0.2 $\mu m$ versus 30-35 $\mu m$ for the active layer), a difference in
the absorption coefficient between the buried contact and the infrared-active
layer is a second order effect.

  4. 4.

We keep all other refractive and geometric model properties fixed, defined as
per Table LABEL:tab:miri_sw_detector_optical_layers and Table
LABEL:tab:miri_lw_detector_optical_layers.

  5. 5.

We perform a TMM simulation at each evaluation of the $\Delta n_{bc}$
parameter.

  6. 6.

We convolve the simulation results by the MRS spectral resolution.

  7. 7.

We bring the optical model and the MRS fringes on the same comparative
baseline. This is done by determining a running mean through both datasets and
dividing by this running mean.

  8. 8.

We compute the value of $\Delta n_{bc}$ that minimizes the difference between
the TMM-simulated fringes and the MRS fringes in each of the MRS spectral
bands.

##  4 RESULTS AND DISCUSSION

We show the results of our analysis in Fig. 5 and Fig. 6. In Fig. 5 the black
data show the TMM simulation based on the optimized value of $\Delta n_{bc}$.
The colored data show the MRS fringes. The optimized values for $\Delta
n_{bc}$ as a function of wavelength are shown in Fig. 6. We report a
significant change in the refractive index of the buried contact as a function
of wavenumber.

---
Figure 5: Minimization analysis result. The black data show the TMM models
based on an optimized value for $\Delta n_{bc}$.
---
Figure 6: Optimal values for $\Delta n_{bc}$ determined from minimizing the
difference between the MRS TMM simulation fringes and fringes in MRS data.

The index of refraction of a solid such as doped silicon can be expressed
analytically using the classical dispersion theory of solids[21]. [22] gives a
theoretical formulation for this. We note that we found an erratum (sign
error) in equation 12 of [22]. The correct expression for the index of
refraction is given in Eq. 2. In this equation, $\epsilon^{*}$ is the
permittivity of silicon, $\omega_{p}$ is the plasma frequency (in units of
Hz), $\omega$ is the wave angular frequency ($\omega=2\pi c/\lambda$, in units
of Hz, where $c$ is the speed of light), and $g$ is the damping coefficient.
The plasma frequency $\omega_{p}$ is proportional to the doping concentration
N, and the damping coefficient $g$ is inversely proportional to the carrier
mobility $\mu$ (all the relevant equations and parameters can be found in
[22]). The carrier mobility $\mu$ depends on the semiconductor temperature and
on the doping concentration [23, 24]. The MIRI detectors are operated at
cryogenic temperatures ($\sim$7 K). In this regime the carrier mobility is
approximately constant as a function of temperature, as such it is mainly
dependent on the doping concentration. For the doping we consider the carrier
mobility of degenerate electrons in antimony-doped silicon[25, 26, 27].

$\displaystyle n^{2}=$
$\displaystyle\frac{\epsilon^{*}}{2}\left\\{\left(1-\frac{\omega_{p}^{2}}{\omega^{2}+g^{2}}\right)+\left[\left(1-\frac{\omega_{p}^{2}}{\omega^{2}+g^{2}}\right)^{2}+\left(\frac{g\omega_{p}^{2}}{\omega\left(\omega^{2}+g^{2}\right)}\right)^{2}\right]^{1/2}\right\\}$
(2)

In the top plot of Fig. 7 we plot the index of refraction of the buried
contact $n_{bc}$, computed in this work, the best fit to the determined
$n_{bc}$ values based on Eq. 2, and the refractive index for the MRS SW and LW
detectors based on Eq. 2. For the latter we use the doping concentration of
the respective infrared-active layers given in Table
LABEL:tab:miri_sw_detector_optical_layers and
LABEL:tab:miri_lw_detector_optical_layers. The plotted curves are consistent
with [28]. The index of refraction of the MRS SW and LW infrared-active layer
is approximately constant down to 1000 $cm^{-1}$, below which a small
deviation is predicted. For the buried contact we find a best fit model with
doping concentration N=$7.5\times 10^{18}cm^{-3}$ and carrier mobility
$\mu=1000cm^{2}/(V\cdot s)$, which is consistent with [29].

---
Figure 7: Top: Index of refraction of the buried contact $n_{bc}$ computed in
this work, the best fit to the determined $n_{bc}$ values based on Eq. 2, and
the refractive index based on Eq. 2 for the MRS SW and LW detectors, computed
using the doping concentration of the respective infrared-active layers.
Bottom: Residals between $n_{bc}$ and the best fit model.

In the bottom plot of Fig. 7 we show the residuals of subtracting the best fit
model from the $n_{bc}$ values. We identify three contributions to the
residuals:

  * •

Different doping in the MRS SW and LW detectors: Although we perform a single
fit to all the $n_{bc}$ values, there may be a small difference in the doping
of the MRS SW and LW detectors.

  * •

Only a single $\Delta n_{bc}$ value was determined per MRS spectral band. The
variation in index of refraction within a band is expected to be gradual
instead of a step function.

  * •

Uncertainties in the modeling of the MRS high-frequency fringing. A sub-
optimal correction of the high-frequency fringing results in residuals in the
low-frequency fringe profile, which impact the estimation of $\Delta n_{bc}$.

##  5 CONCLUSIONS AND FUTURE WORK

We have shown that JWST MIRI MRS spectra contain two dominant frequencies in
their spectra. By fitting the resolved fringes with a Fabry-Pérot
transmittance function, we find that the high frequency fringing is produced
between the detector back side surface coating and the pixel metalization. The
low frequency fringing is the result of a beating between the high-frequency
fringing and fringing produced between the detector back side surface and the
infrared-active layer. By using the Transfer-Matrix Method and an optical
model for the MRS SW and LW detectors, we show that the optical behavior of
the buried contact is consistent with it having a doping concentration of
N=$7.5\times 10^{18}cm^{-3}$. This doping concentration results in a
significant change in the index of refraction at the buried contact layer. We
argue that this accounts for the behavior of the prominent low-frequency
fringing in MIRI MRS data.

In future work, the impact of the buried contact layer thickness on the MRS
fringes will be investigated. A larger thickness results in more absorption
inside the layer, which affects the fringing. Due to the relatively small size
of the buried contact layer, compared to the infrared-active layer, absorption
in the buried contact is a second order effect. Nevertheless, a complete model
of the detector requires fitting both refraction and absorption effects
simultaneously.

###### Acknowledgements.

Ioannis Argyriou and Bart Vandenbussche thank 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. 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.

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