11institutetext:  H. Akamatsu 22institutetext: SRON Netherlands Institute for
Space Research, Niels Bohrweg 4 2333 CA Leiden, The Netherlands 22email:
<EMAIL_ADDRESS>33institutetext: W.B. Doriese 44institutetext: National
Institute of Standards and Technology (NIST), United States Department of
Commerce, Boulder, CO, USA 44email<EMAIL_ADDRESS>55institutetext:
J.A.B. Mates 66institutetext: National Institute of Standards and Technology
(NIST), United States Department of Commerce, Boulder, CO, USA 66email:
<EMAIL_ADDRESS>77institutetext: B.D. Jackson 88institutetext: SRON
Netherlands Institute for Space Research, Landleven 12, 9747 AD Groningen, The
Netherlands 88email<EMAIL_ADDRESS>
# Signal readout for Transition-Edge Sensor X-ray imaging spectrometers

H. Akamatsu corresponding author W.B. Doriese  J.A.B. Mates  B.D. Jackson

###### Abstract

Arrays of low-temperature microcalorimeters provide a promising technology for
X-ray astrophysics: the imaging spectrometer. A camera with at least several
thousand pixels, each of which has an energy-resolving power ($E/\Delta
E_{\mathrm{FWHM}}$) of a few thousand across a broad energy range (200 eV to
10 keV or higher), would be a revolutionary instrument for the study of
energetic astrophysical objects and phenomena. Signal readout is a critical
enabling technology. Multiplexed readout, in which signals from multiple
pixels are combined into a single amplifier channel, allows a kilopixel-scale
microcalorimeter array to meet the stringent requirements for power
consumption, mass, volume, and cooling capacity in orbit. This chapter
describes three different multiplexed-readout technologies for transition-
edge-sensor microcalorimeters: time-division multiplexing, frequency-domain
multiplexing, and microwave-SQUID multiplexing. For each multiplexing
technique, we present the basic method, discuss some design considerations and
parameters, and show the state of the art. The chapter concludes with a brief
discussion of future prospects.

Keywords X-ray spectrometer, microcalorimeter, cryogenic electronics, signal
readout, multiplexed readout, semiconductor calorimeter, transition-edge
sensor, metallic-magnetic calorimeter, SQUID

##  1 The cryogenic X-ray imaging spectrometer

High-energy-resolution X-ray spectroscopy is a powerful probe of elemental and
chemical composition that is used across many scientific disciplines, from
materials science to the physics of hot plasma in the Universe. Wavelength-
dispersive (WD) X-ray spectrometers, such as gratings, Bragg crystals, and
multi-layer materials, can achieve the highest energy-resolving powers
($E/\Delta E\sim 1,000$ to 10,000). WD spectrometers have been deployed
previously for X-ray astronomy, such as the Reflection Grating Spectrometer
(RGS) on board the XMM-Newton satelliteden Herder et al. (2001) and Chandra’s
High Energy Transmission Grating SpectrometerMarkert et al. (1994). However,
WD spectrometers have important drawbacks for astronomy, such as a limited
simultaneous energy range, a low collecting efficiency (solid-angle coverage
times quantum efficiency), and a lack of any inherent imaging capability.
Another common technology, the solid-state energy-dispersive (ED) detector
(e.g., the silicon-drift detector, or SDD, and the charged-coupled device, or
CCD), which measures the X-ray energy directly during detection, can combine
high collecting efficiency with imaging capability across a wide energy band;
however the energy resolution is limited ($E/\Delta E\leq 50$ @ 6 keV for the
best SDDs and somewhat lower for X-ray CCD cameras).

Figure 1: The microcalorimeter array as a cryogenic imaging spectrometer.
Left: comparison of cryogenic spectrometers to wavelength-dispersive and
solid-state energy-dispersive spectrometers on a plane of energy-resolving
power vs. collecting efficiencyFriedrich (2006). Here, collecting efficiency
is the solid angle subtended by the spectrometer multiplied by the cumulative
quantum efficiency of all components of the spectrometer. Middle: cartoon
diagram of a single cryogenic microcalorimeter pixel. An X-ray absorber is
connected to a thermal bath via a weak thermal link, and also to a
thermometer. Measurement of a pulsed change in the absorber temperature yields
the energy of the X-ray. Right, top to bottom: $R$ vs. $T$ of a semiconductor
calorimeter, $R$ vs. $T$ of a transition-edge sensor, and $B$ vs. $T$ of a
metallic-magnetic calorimeter.

A technology that has been developed over the last few decades for a variety
of present and future astronomical missions is the microcalorimeter
arrayMoseley et al. (1984); McCammon (2005) (see Fig. 1). A microcalorimeter
array is an imaging spectrometer that combines some of the best features of WD
spectrometers and ED detectors: $E/\Delta E\geq 2,400$ @ 6 keV and a high
collecting efficiency. A microcalorimeter pixel consists of a sensitive
thermometer and an X-ray absorber connected weakly to a cryogenic thermal
bath. When an X-ray strikes the X-ray absorber, the thermometer measures the
magnitude of the resulting temperature pulse to determine the energy of the
X-ray.

The energy resolution of a cryogenic spectrometer is estimated as follows:

$\Delta E\sim\sqrt{\frac{k_{\mathrm{B}}T^{2}C}{\left|\alpha\right|}}.$ (1)

where $k_{\mathrm{B}},T,C$, and $\alpha\equiv d\mathrm{ln}R/d\mathrm{ln}T$ are
the Boltzmann constant, detector temperature, heat capacity, and thermometer
sensitivity, respectively. The keys to achievement of high energy-resolving
power (low $\Delta E$) are a low operating temperature, a small heat capacity,
and a high sensitivity.

Table 1: Summary of the three main types of X-ray microcalorimeters: semiconductor, transition-edge sensor (TES), and metallic-magnetic calorimeter (MMC). The upper section lists some basic characteristics of each type of microcalorimeter. Semiconductor microcalorimeters use junction-gate field-effect transistors (JFETs) to read out their voltage, while TESs and MMCs employ superconducting quantum-interference devices (SQUIDs) to read out the device current. The lower section gives the best energy resolution achieved in a single pixel and the largest operating detector array (as of the publication of this book).  | Semiconductor | TES | MMC
---|---|---|---
$T$ of sensor bath [K] | 0.05 | 0.05 | 0.03
resistance [$\rm\Omega$] | 106 | 0.001 to 1 | 0.1
readout amplifier | JFET | SQUID | SQUID
$T$ of readout amplifier [K] | 140 | 0.05 | 0.03
multiplexing | difficult | highly developed | possible
amp. bandwidth | NA | 10 MHz (SQUID) | 10 MHz
|  |  4 to 8 GHz (HEMT) |
best $\Delta E$ @ 6 keV [eV] | 3.7 Kelley et al. (2008); Porter et al. (2009, 2010) | 1.6 Smith et al. (2012); de Wit et al. (2022) | 1.6 Kempf et al. (2018)
largest array [pixels] | 36 Ishisaki et al. (2018) | 992 Szypryt et al. (2021b, 2022) | 64 Mantegazzini et al. (2022)
TES: transition-edge sensor
MMC: metallic-magnetic calorimeter
JFET: Junction Field Effect Transistor
SQUID: superconducting quantum interference device (see Sect.2.2)
HEMT: High Electron Mobility Transistor (see Sect.3.5)

While many types of cryogenic thermometers have been proposed and studied,
only a few are developed enough to be considered for space-borne
instrumentation: semiconductor calorimeters, transition-edge sensors (TESs),
and metallic-magnetic calorimeters (MMCs). Table 1 summarizes the basic
properties of each type of microcalorimeter, while Table 4 in Sec. 4 lists
present and future missions for each.

The semiconductor calorimeter has a negative thermal coefficient (thermistor
$R$ decreases as $T$ increases), a high impedance ($\sim$M$\rm\Omega$), and a
moderate sensitivity ($|\alpha|\sim 7$). Semiconductor calorimeters have been
used in many projects, including the X-ray Quantum Calorimeter (XQC) sounding
rocket experimentMcCammon et al. (2002) and the ASTRO-E111ASTRO-E was the
first satellite to be equipped with a cryogenic X-ray spectrometer, but it was
lost during launch., ASTRO-E2 (SuzakuMitsuda et al. (2007)/ XRSKelley et al.
(2007))222ASTRO-E2/ XRS was the first microcalorimeter instrument to reach
orbit on a satellite, but was lost due to rapid evaporation of the cryogens.,
and ASTRO-H (HitomiTakahashi et al. (2012)/ SXSMitsuda et al.
(2014))333Hitomi/ SXS was a pioneer in low-temperature X-ray microcalorimetry
and successfully observed X-ray emission from astronomical objectsHitomi
Collaboration et al. (2016). Unfortunately, however, the satellite was lost
due to a malfunction in its attitude-control system. satellites. They will
also be employed for the ResolveIshisaki et al. (2018) instrument on
XRISMTashiro et al. (2018). McCammonMcCammon (2005) gives a detailed review of
the technology.

The transition-edge sensor (TES) works in the superconducting transition from
the normal to superconducting states, which provides a high temperature
sensitivity ($\alpha\sim 100$). The operating impedance of a TES
microcalorimeter ranges from $\sim$ 1 to 10 m$\rm\Omega$, depending on the TES
design and the material chosen for the superconducting film. The sensor is
voltage biased and its current is read out via a SQUID. There are several
excellent reviews of the TES X-ray microcalorimeter in the literatureIrwin and
Hilton (2005); Ullom and Bennett (2015); Gottardi and Nagayashi (2021). TES
calorimeters are in use in or planned for many present and future X-ray
astronomy projects, including Micro-X (the first TES array and readout system
operated in space)Adams and others. (2020), the X-IFU instrumentBarret et al.
(2018) onboard the ESA Athena mission, HUBSCui et al. (2020), Super-DIOSSato
et al. (2020), and LEM444https://lem.physics.wisc.edu/, and the TES is one of
the proposed technologies for the LynxThe Lynx Team (2018) imaging
spectrometerBandler et al. (2019).

Unlike other resistance calorimeters, the metallic-magnetic calorimeter (MMC)
utilizes a non-resistive, paramagnetic temperature sensor; a MMC converts a
temperature change into a change in magnetization, which is then measured as a
change in magnetic flux in a dc SQUIDFleischmann et al. (2005). MMCs have two
interesting differences from resistive microcalorimeters: (1) no power is
dissipated in the sensor and (2) the readout makes no galvanic contact to the
sensor. MMCs also exhibit excellent energy resolution, high dynamic range, and
high linearity in their energy-gain scale. MMCs will be used in the
International Axion ObservatoryAbeln et al. (2021) and are also one of the
proposed detector technologies for the Lynx imaging spectrometerStevenson et
al. (2019).

Because cryogenic X-ray microcalorimeters are formed as small patches of thin
films on a silicon substrate, they are naturally fabricated in a two-
dimensional array of pixels to create an imaging spectrometer. An
international effort is underway to develop large arrays of low-temperature
X-ray microcalorimeters. The following two technologies are particularly
active areas of research and development:

  1. 1.

arrays of the scale of 104 pixels via microfabrication technology;

  2. 2.

signal-multiplexing techniques that read out multiple pixels per readout
channel, which is necessary to reduce the volume of electronics on the low-
temperature (typically about 50 mK to 100 mK) stage of the spectrometer and
the number of wires to that low-temperature stage (see Sec. 3.1).

This chapter presents issues and recent progress in signal-readout technology
for arrays of X-ray microcalorimeters.

##  2 Basic concepts of signal readout

###  2.1 Impedance matching

Figure 2: Left: Electrical schematic showing the output impedance, $r$, of an
X-ray microcalorimeter and the input impedance, $R$, of its readout. Right:
Power transmitted to the readout vs. $R$ for $r=100$ ${\mathrm{\Omega}}$.
Transmitted power is maximized when the output impedance of the
microcalorimeter and the input impedance of the readout electronics are equal.

When two circuit elements are connected, such as a microcalorimeter pixel to
its readout, a difference in impedance will cause some signal power to be
reflected. This loss of signal in the readout decreases the signal-to-noise
ratio. This can be compared to exchanging water between two hoses of different
diameters: different hose diameters result in water loss.

The following simple calculation illustrates the concept of impedance
matching. A signal voltage, $V$, drives a current, $I$, through the circuit
shown in Fig. 2(left), given by:

$I=\frac{V}{\textit{R}+\textit{r}},$ (2)

where $r$ and $R$ are the output impedance of an X-ray microcalorimeter pixel
and input impedance of its readout, respectively. Therefore, the voltage drop
across $R$ is:

$V_{R}=RI=V\frac{R}{R+r}.$ (3)

The power dissipated in $R$, $P_{R}\equiv IV_{R}$, and its first derivative
with respect to $R$ are:

$P_{R}=V^{2}\frac{R}{(R+r)^{2}}\hskip 20.0pt{\rm and}\hskip
20.0pt\frac{dP_{R}}{dR}=V^{2}\frac{r-R}{(R+r)^{3}}.$ (4)

Thus, power transmission to the readout is maximized ($dP_{R}/dR=0$) when the
output impedance of the microcalorimeter pixel is equal to the input impedance
of the readout (when the impedances are matched). Fig. 2(right) plots $P_{R}$
vs. $R$ for $r=100$ ${\mathrm{\Omega}}$.

As shown in Table 1, the various microcalorimeter types have different output
impedances, and so require different readout circuits.

###  2.2 dc Superconducting quantum interference device (dc-SQUID)

Superconducting quantum interference devices (SQUIDs) are a class of highly
sensitive magnetic sensors that use ring-shaped superconductors containing
Josephson junctions to measure extremely weak magnetic fieldsAnderson and
Rowell (1963); Jaklevic et al. (1964); Chesca et al. (2004). When two
superconductors are brought close enough together that their wave functions
overlap, a tunnel current flows that is proportional to the phase difference
between the two superconducting wave functions. A modern Josephson junction is
a sandwich of a few-nanometer-thick insulator or normal-conductive metal
between two superconducting films. SQUIDs can reach magnetic field
sensitivities as low as $\sim 10^{-15}$ T(Drung et al., 2007); by contrast,
the strength of a refrigerator magnet is about 0.02 T, while the magnetic
fields of the human heart and brain are orders of magnitude lower ($\sim
10^{-12}$ T)Cohen and Givler (1972). There are two main types of SQUIDs:
direct current (dc) and radio frequency (rf). The latter are easier to
manufacture but are less sensitive because they operate with a single
Josephson junction.

Figure 3: Left: sketch of a dc-SQUID, consisting of two Josephson junctions in
a superconducting loop. Middle: simulated $V$ vs. $I$ curves in a dc-SQUID for
the cases of an integral ($\Phi=n\Phi_{0}$) and a half-integral
($\Phi=[n+1/2]\Phi_{0}$) number of applied flux quanta ($n$ is an integer).
Right: simulated $V$ vs. $\Phi$ curve for a dc-SQUID.

The dc-SQUID is a sensitive magnetometer with a wide inherent frequency
bandwidth from dc up to the gigahertz range. It has two Josephson junctions in
a washer-shaped superconducting loop (see Fig. 3-left). An external magnetic
flux applied to the loop drives a screening current that maintains the total
flux enclosed by the loop as an integral number of flux quanta (the magnetic-
flux quantum is $\Phi_{0}=h/2e=2.07\times 10^{-15}$ Wb, where $h$ is the
Planck constant and $e$ is the charge of the electron). This modulates the
maximum supercurrent that can flow across the loop periodically, with a period
of $\Phi_{0}$. When the current through either Josephson junction exceeds its
critical current, a voltage develops. The voltage-vs.-current ($V$-$I$) curve
(see Fig. 3-middle) is thus modulated as a periodic function of the external
magnetic field. When the external flux is zero, the phase difference of the
superconducting wave functions in the two sides of the SQUID is small (high
critical current and thus lower voltage; $\Phi=n\Phi_{0}$ case, where $n$ is
an integer), while when the external magnetic field is close to
$(n+1/2)\Phi_{0}$, the critical current is suppressed and the generated
voltage is higher. When the current bias applied to the device is higher than
its highest critical current, the output voltage shows a sinusoidal response
to applied magnetic flux.

Series arrays of dc-SQUIDsWelty and Martinis (1991) are used to increase the
voltage gain. Some designs (typically operating at a temperature of 4 K or
lower) can approach quantum-limited energy resolution (the energy resolution
per bandwidth approaches $\hbar$).

Parasitic inductances and capacitances in the circuitry of a dc-SQUID can lead
to resonances that amplify gigahertz-frequency Josephson oscillations in the
SQUID. These resonances can cause undesirable kinks in the otherwise
sinusoidal response of the SQUID and excess noise. Careful design, including
the use of resistive structures to damp microwave resonances, can reduce or
eliminate such behaviorHuber et al. (1997, 2001).

##  3 Principles of multiplexed readout of X-ray TES microcalorimeters

###  3.1 Why is multiplexed readout necessary?

X-ray astronomy differs significantly from other fields in that X-ray signals
from celestial objects do not reach the ground due to atmospheric absorption.
Thus, X-ray observatories (and also, generally, gamma-ray observatories) must
be placed on space-borne platforms such as rockets and satellites.

Figure 4: Schematic of the readout of a TES via a single-stage dc-SQUID.
Cryogenic components are inside the dashed box. This is the simplest
reasonable implementation of (non-multiplexed) readout of a TES, and requires
three cryogenic wire pairs per TES (for TES bias, SQUID bias/output, and SQUID
feedback). However, typically, a two-stage-SQUID architecture is needed to
achieve the required gain without dissipation of too much Joule power on the
coldest cryogenic stage (see Fig. 7); in this case, at least five cryogenic
wire pairs per TES are needed.

Low-temperature microcalorimeters make excellent X-ray imaging spectrometers,
as described by Gottardi & Smith in this book. However, readout of large
arrays of microcalorimeters is nontrivial due to the electrical power and
cooling capacity available in a satellite. For example, the total electrical
draw of a typical communication satellite is about 1 kW to 1.5
kW555https://www.esa.int/Enabling_Support/Space_Engineering_Technology/Power_Systems
(about the same as a standard household microwave oven), while the Hubble
Space Telescope uses about 2.8
kW666https://www.nasa.gov/content/goddard/hubble-space-telescope-electrical-
power-system. For X-ray astronomy, the Athena satellite has a planned power
draw of $\sim$6 kW, of which the X-IFU instrument will draw about 1.3 kWRavera
et al. (2014).

As discussed in Fig. 4, readout of a single TES microcalorimeter pixel
requires a minimum of three cryogenic wire pairs, while five pairs per TES are
typical. Thus, brute-force readout of X-IFU’s $\sim$2,400 TESsBarret et al.
(2018) would require more than 10,000 wire pairs. The power draw from 2,400
channels of room-temperature electronics plus a cryogenic system capable of
overcoming the thermal conductivity of this many wires would overwhelm any
feasible satellite platform. Future arrays of X-ray microcalorimeters will be
even larger. To solve this problem, signal-multiplexing techniques that read
signals from multiple sensors with fewer wires are essential.

###  3.2 General considerations

Figure 5: Simulated TES-current pulses (left: current vs. time; right:
current$*$time vs. frequency) under various damping conditions. In the
overdamped condition, the relation between the TES time constants is
$\tau_{\mathrm{rise}}<\tau_{\mathrm{decay}}$ (and both are real). In a
critically damped pulse, $\tau_{\mathrm{rise}}=\tau_{\mathrm{decay}}$. In an
underdamped pulse, the time constants are complex conjugates.

Multiplexed readout should introduce minimal noise and signal degradation. The
noise budget is application-dependent: e.g. in Athena X-IFU, about 3% of the
energy-resolution budget is allocated to readout. Successful multiplexing
generally entails these three considerations:

1\. The signal and noise bandwidths of the sensor should be made as narrow as
feasible for the astronomical application. The signal and noise currents of a
TES X-ray microcalorimeters generally obey two low-pass time constants: the
electrical time constant (determines behavior on the leading edge of an X-ray
pulse) and the thermal time constant (determines the pulse’s decay behavior).
Irwin and HiltonIrwin and Hilton (2005) discuss the theory of TES signals,
including how electrothermal feedback (ETF) governs the interaction between
the electrical and thermal time constants. Here, we refer to the TES’s ETF-
modified time constants on the leading and falling edges of a pulse as
$\tau_{\mathrm{rise}}$ and $\tau_{\mathrm{decay}}$, respectively. The
achievable X-ray count rate is governed by $\tau_{\mathrm{decay}}$: a faster-
decaying detector exhibits less pulse pileup for a given input rate of X-rays.
The signal bandwidth of the TES is governed by $\tau_{\mathrm{rise}}$. The
ratio between $\tau_{\mathrm{rise}}$ and $\tau_{\mathrm{decay}}$, is generally
adjusted via inductance in the TES-bias loop (see, e.g., $L_{\mathrm{Ny}}$ in
Fig. 7). Fig. 5 shows simulated TES-current pulses under various damping
conditions. The critically damped (or just a bit overdamped) condition is
often the design goal for the TES-bias circuit: for a given TES model,
critical damping provides the largest $\tau_{\mathrm{rise}}$, and thus the
narrowest overall signal bandwidth, and so is often preferred for multiplexed
readout. Also, because $\tau_{\mathrm{rise}}$ cannot be longer than
$\tau_{\mathrm{decay}}$, per-pixel X-ray counting capability is in natural
tension with multiplexing factor (and thus with array size). Finally,
filtering limits sensor noise from being aliased within a multiplexer pixel or
from leaking into neighboring multiplexer pixels: the TES-current noise due to
phonon exchange with the bath is low-pass filtered with the time constants
$\tau_{\mathrm{decay}}$ and $\tau_{\mathrm{rise}}$, while the TES-Johnson
noise is low-pass filtered with the time constant $\tau_{\mathrm{rise}}$.

2\. The readout system should have a wide bandwidth. Multiplexed readout is
essentially an exercise in stuffing many low-bandwidth TES signals into a
high-bandwidth readout system. Higher readout bandwidth generally allows more
TESs per readout channel.

Table 2: Summary information for four multiplexed-readout technologies for TESs X-ray calorimeters: time-division (TDM), code-division (CDM), MHz-frequency-division (FDM), and microwave-SQUID ($\upmu$mux) multiplexing. The top section lists the type of TES bias and the type of modulation. The second section discusses the front-end SQUID. The third section discusses the 2nd-stage amplifier; in $\upmu$mux readout, a high-electron-mobility transistor (HEMT) amplifier is traditionally used, but the recently developed kinetic-inductance traveling-wave parametric amplifier (KITWPA) offers an attractive lower-power alternative. Finally, the fourth section lists the state of the art in the readout of X-ray-TES arrays. |  TDM Chervenak et al. (1999); Durkin et al. (2021) |  CDMIrwin et al. (2010); Morgan et al. (2016) |  MHz FDMYoon et al. (2001); Kiviranta et al. (2002) |  $\upmu$muxIrwin and Lehnert (2004); Mates et al. (2008)
---|---|---|---|---
TES Bias | dc | dc | ac | dc
how modulated? | SQUIDs on/off |  Walsh codes Walsh (1923) | resonating TESs | resonating SQUIDs
front-end SQUID | dc | dc | dc | rf
FE SQ: $P_{\mathrm{Joule}}$ |  450 pWDurkin et al. (2020) |  450 pWDoriese et al. (2019) |  450 pWKiviranta et al. (2021) | 20 pW
2nd-stage amplifier | SQUID array | SQUID array | SQUID array |  HEMTDuh et al. (1988)
2nd stage: $P$ @ $T$ | $\sim$300 nW @ 2 K | $\sim$300 nW @ 2 K | $\sim$300 nW @ 2 K | $\sim$5 mW @ 4 K
|  |  |  |  KITWPA Malnou et al. (2022)
|  |  |  | (0.2 to 20) $\mu$W @ 4 K∗
demonstrated $\Delta E_{\mathrm{FWHM}}$ @ $E$ | 2.23 eV @ 6 keV | 2.77 eV @ 6 keV | 2.23 eV @ 6 keV | 3.3 eV @ 6 keV
@ channels $\times$ mux factor | @ $1\times 40$ Durkin et al. (2019) | @ $1\times 32$ Morgan et al. (2016) | @ $1\times 37$ Akamatsu et al. (2021) | @ $1\times 37$ Nakashima et al. (2020)
| 1.98 eV @ 6 keV |  | 2.14 eV @ 6 keV | 3.4 eV @ 6 keV
| @ $8\times 32$ Smith et al. (2021) |  | @ $1\times 31$ Akamatsu et al. (2021) | @ $1\times 28$ Yoon et al. (2018)
|  |  |  | 2.04 eV @ 1.25 keV
|  |  |  | @ $1\times 88$ Bennett et al. (2019)
|  |  |  |  10 eV @ 6† keV
|  |  |  | @ $1\times 116$ [pre-pub. comm.]
$*$ KITWPA power dissipation depends on pump power and therefore scales with
compression point of the amplifier.
$\dagger$ TOMCATSzypryt et al. (2021a) TESs were optimized for very high count
rates and not for energy resolution.
Figure 6: Multiplexer modulation functions vs. time for the basis vectors of
four-channel versions of time-division (TDM), code-division (CDM), MHz-band
frequency-domain (FDM), and microwave-SQUID multiplexing ($\upmu$mux). Colors
indicate the different pixels. Modulation in TDM is via switching SQUIDs on
and off, so the basis set is the identity matrix. In CDM, TESs are coupled to
different SQUIDs with different polarities to create orthogonal Walsh codes.
In FDM, different TESs are biased at different megahertz frequencies. In
$\upmu$mux, different rf SQUIDs are biased at different gigahertz frequencies.
Both FDM and $\upmu$mux thus use sine waves to form their orthogonal basis
sets.

3\. The multiplexer should be built from an orthogonal basis set to prevent
signal leakage from pixel to pixel. Fig. 6 shows the basis sets used by the
main multiplexing techniques for TESs to be discussed in Secs. 3.3 to 3.5:
time-division multiplexing (TDM), frequency-domain multiplexing (FDM), and
microwave-SQUID multiplexing ($\upmu$mux). In addition, this figure shows the
basis set of code-division multiplexing (CDM)Irwin et al. (2010, 2012); Morgan
et al. (2016); Doriese et al. (2019); Yu et al. (2020). Irwin, et al.Irwin
(2009) discuss, in the context of information theory, the number of pixels
that can be multiplexed via the various techniques. A summary of these
techniques is given in Table 2.

TDM and $\upmu$mux for TESs are under development by the Quantum Sensors Group
at NIST’s Boulder Labs. FDM for TESs was initially demonstratedCunningham et
al. (2002) by a team from Lawrence Livermore National Lab, UC Berkeley, and
Lawrence Berkeley National Lab, and is under development by UC Berkeley,
McGill University, and SRON. CDM for TESs was pursued by NIST as an early
option for Athena X-IFU and Lynx, but is not presently under development for
any mission and so is not discussed further in this chapter. MMCs can be read
out via $\upmu$muxWegner et al. (2018).

###  3.3 Time-division multiplexing (TDM)

In the time-division-multiplexing (TDM) technique, each TES has its own first-
stage dc-SQUID (SQ1). The TESs are dc-biased and are always on, while the SQ1s
are turned on and off such that one SQ1 is on at a time per readout
column777In TDM, an amplifier chain is usually referred to as a readout
“column.” Throughout this chapter, we use the terms “amplifier chain,”
“readout channel,” “multiplexer channel,” and “readout column” interchangeably
to refer to the logical readout unit that is divided into detector pixels via
multiplexing.. Only the signal current from the TES whose SQ1 is on is read
out, instantaneously, by the column.

The TDM scheme was proposed in 1999Chervenak et al. (1999) for the readout of
arrays of TES bolometers and microcalorimeters for various astronomical
applications. In the 20+ years since, several TDM-based bolometric
arraysHolland et al. (2013); Swetz et al. (2011); Harper et al. (2018);
Gandilo et al. (2016), each of the kilopixel to multi-kilopixel scale, have
been deployed for far-IR, millimeter-wave, and submillimeter-wave astronomy.
In addition, about 15 TDM-based X-ray spectrometersDoriese et al. (2017);
Yamada et al. (2021); Lee et al. (2019), each of the few-hundred-pixel scale
(to be discussed further in Sec. 3.3), have been deployed to X-ray-science
facilities around the world. A 128-pixel TDM X-ray array flew on the Micro-X
sounding rocket in 2018Adams et al. (2021) and again in August 2022.
Presently, TDM is undergoing refinementSmith et al. (2021); Durkin et al.
(2021) for the multi-kilopixel X-IFU imaging spectrometer for ESA’s Athena
mission (see Sec. 3.3).

How does TDM compare to the other main multiplexing technologies for TES
microcalorimeters, philosophically and at the systems level? Philosophically,
in TDM, the SQ1s are the modulated elements, while the TESs themselves are not
modulated; this is the same as in the microwave-multiplexing ($\upmu$mux; Sec.
3.5) scheme, while the frequency-domain scheme (FDM; Sec. 3.4) takes the
opposite approach. At the systems level, in TDM the cryogenic electronics are
the most complicated, followed by those in $\upmu$mux and then FDM, while the
TDM room-temperature electronics are simpler than those of either FDM or
$\upmu$mux.

#### Principles of TDM operation

Figure 7: Readout of a dc-biased TES by a dc-SQUID chain. (a) A single TES
under dc bias is read out by a first-stage dc SQUID (SQ1) and a series array
of dc SQUIDs (SSA). The signal of interest, the TES current, is converted to
flux in SQ1 via its input coil ($M_{\mathrm{in1}}$). The SSA reads out the SQ1
current in the same manner. The elements in the grey box are located on the
same temperature stage as the TES, which for X-ray TESs is typically around 50
mK. TES-related circuit elements (the TES shunt resistor, or
$R_{\mathrm{sh}}$, and the bandwidth-limiting Nyquist inductor, or
$L_{\mathrm{Ny}}$) are blue, while the SQUID-related circuit elements are in
black. The SSA is usually located at a higher-temperature cryogenic stage to
accommodate its Joule-power dissipation. (b) The same circuit diagram as (a),
but with the addition of a superconducting switch (red) to turn on and off the
SQ1. When the switch is closed, the SQ1-bias current bypasses SQ1, so the TES
signal current is not transmitted up the readout chain. When the switch is
open, circuit (b) behaves like circuit (a).

Fig. 7 shows circuit diagrams that contain the building blocks of a TDM
readout system. Fig. 7(a) shows readout of a single dc-biased TES by two
stages of dc SQUIDs (see Fig. 3). The signal of interest in a TES X-ray
microcalorimeter is its current, which is modulated as a pulsed decrement by
each incident X-ray; the amplitude of the TES-current pulse is roughly
proportional to the energy of the X-ray. A change in the TES current modulates
the flux in the first-stage dc-SQUID (SQ1), which in turn modulates the SQ1
bias current. The current-biased SQUID-series-arrayHuber et al. (1997, 2001)
(SSA) amplifier transduces the SQ1 current to a voltage at the SSA output,
which is amplified further by a room-temperature low-noise amplifier (LNA).

Because the (quasi-sinusoidal) transfer function of the combined $V$-$\Phi$
curve of the SQ1 and the SSA is highly nonlinear, the readout is run as a
flux-locked loop (FLL) to linearize the response. Thus, the “output” signal of
this readout system is the feedback voltage, $V_{\mathrm{FB}}$, that is
applied to null the TES-signal current to maintain a constant flux in its SQ1.

Fig. 7(b) shows the addition of a superconducting switch to turn on and off
the SQ1. The switch chosen for modern TDM implementationsBeyer and Drung
(2008); Doriese et al. (2016) is based on the dc-SQUID-like Zappe
interferometerZappe (1977). An interferometer element consists of several
Josephson junctions in parallel (with the number of junctions varying across
implementations). The “row-address” (RA) current ($I_{\mathrm{RA}}$) is
coupled to the SQUID loops such that with $I_{\mathrm{RA}}=0$ the
interferometer acts as a superconducting wire (the switch is “closed”), while
with $I_{\mathrm{RA}}=\Phi_{\mathrm{0}}/2$ the junctions are perfectly out of
phase and the nominal current-carrying capacity of the interferometer is zero
(the switch is “open”). In the NIST implementationIrwin et al. (2012), each
interferometer element contains four junctions with equal critical currents
for ease of fabrication and wide operating margins. Tens of these
interferometric elements are wired in series to create an operating dynamic
resistance of this “flux-actuated switch” (FAS) that is much larger than that
of the signal SQ1.

Figure 8: (left) Schematic of a two-column by two-row TDM. Each colored block
is a unit pixel and contains the same cryogenic components as shown in the
grey box in Fig. 7(b). Rows of SQ1s are sequentially turned on via application
of a row-address current ($I_{\mathrm{RA}}$) to their flux-actuated switches.
Thus, only one TES at a time per column is read out. To keep the non-linear,
two-stage SQUID amplifier in a quasi-linear range, each column is run as a set
of digitally interleaved flux-locked loops. (right) Color-coded, simulated
data streams showing the interleaved $V_{\mathrm{FB}}$ signals for Columns 1
and 2 as the four TESs respond in current pulses to X-ray events.

Fig. 8(left) shows a cartoon circuit schematic of a 2-column $\times$ 2-row
TDM system. In a general $M$-column $\times$ $N$-row array of TESs, both the
count of cryogenic wires and the number of amplifier chains would scale as
$MN$ under brute-force (non-multiplexed) readout; with TDM readout these scale
only as $\sim(M+N)$ and $M$, respectively. These savings in channels and wires
are the main reason to use a multiplexed-readout scheme.

Rows of SQ1s are sequentially turned on via application of a RA current
($I_{\mathrm{RA}}$) to their FASs. TDM columns are read out in parallel. The
signal from one TES at a time per column is passed to that column’s SSA.
During a row, the LNA’s output voltage, or error signal ($V_{\mathrm{er}}$),
is digitally sampled, and then the proportional-integral $(P,I)$ flux-feedback
signal ($V_{\mathrm{FB}}$) that would servo the row’s $V_{\mathrm{er}}$ to a
constant value is digitally calculated and stored to be applied inductively
(via $M_{\mathrm{FB}}$) on the digital FLL’s next visit to the row. The
digital-readout electronics (DRE; to be discussed further in Sec. 3.3)
synchronize the RA and D-FLL signals and stream the TES-current data to the
back-end processor. The TDM system samples each TES once per frame time
($t_{\mathrm{fr}}=t_{\mathrm{row}}N_{\mathrm{rows}}$); if the frame time is
short compared to the time constants of the TES-current signals, then the
system can reconstruct each TES’s current with high fidelity. Fig. 8(right)
shows the multiplexed time streams of the flux-feedback signals of the two
two-pixel TDM columns.

#### Circuit parameters, multiplexing factor, and noise scaling

For the dimensioning of a TDM system, an important characteristic of the TES
signal-current ($I_{\mathrm{TES}}$) is its slew rate. The highest required
TES-current slew-rate, $\dot{I}_{\mathrm{max}}$, will occur at the leading
edge of an X-ray pulse and is a function of the TES design and its bias-
circuit inductance and the highest X-ray energy of interest. If the D-FLL
system is to function correctly, the flux difference in any SQ1 from one
sample to the next, injected into $M_{\mathrm{in1}}$ by $I_{\mathrm{TES}}$,
must result in an error signal that remains within a quasi-linear region of
the combined SQ1-SSA $V$-$\Phi$ curve (see Fig. 3 for an example curve). Thus,
the flux difference must obey $\Delta\Phi\leq F\Phi_{\mathrm{0}}$. The
fraction, $F$, of the flux quantum over which the SQUID curve is roughly
linear depends on the details of the SQUID designs and is usually between 0.2
and 0.5. The flux difference between frames is
$\Delta\Phi=\dot{I}M_{\mathrm{in1}}t_{\mathrm{fr}}$. ThusDoriese et al.
(2006),

$\dot{I}_{\mathrm{max}}M_{\mathrm{in1}}t_{\mathrm{row}}N_{\mathrm{rows}}\leq
F\Phi_{\mathrm{0}}$ (5)

sets a joint condition on $M_{\mathrm{in1}}$ and $N_{\mathrm{rows}}$. If all
else is equal, a larger multiplexing factor requires a proportionally smaller
input coupling of TES current to SQ1 flux.

The second important consideration in dimensioning a TDM system is readout
noise. A given TES X-ray microcalorimeter design will have a value of the
spectral density of (white) multiplexed readout noise ($\sqrt{S_{I\mathchar
45\relax\mathrm{mux}}}$; referred to the TES current; units of A/$\surd$Hz)
above which it is unable to meet the mission/experimental specification on
energy resolution. A goal of the system design, then, is to keep
$\sqrt{S_{I\mathchar 45\relax\mathrm{mux}}}$ below this maximum-allowed value.
The SQUID-amplifier system’s noise (including contributions from both stages
of dc-SQUIDs and the room-temperature electronics) is approximated as being
white (with spectral density $\sqrt{S_{\Phi}}$; referred to SQ1 flux; units of
$\Phi_{\mathrm{0}}/\surd$Hz) out to a high-frequency, with a single-pole, low-
pass rolloff with time constant $\tau_{\mathrm{OL}}$. In a modern TDM
systemDoriese et al. (2016) values of
$\sqrt{S_{\Phi}}=0.2\mu\Phi_{\mathrm{0}}/\surd$Hz and $f_{\mathrm{OL}}\sim 8$
MHz to 10 MHz are reasonable. The large open-loop bandwidth,
$f_{\mathrm{OL}}$, needed to switch quickly from row to row in TDM, combined
with the readout strategy that samples each row only once per frame, means the
Nyquist anti-aliasing criterion, $1/t_{\mathrm{fr}}\geq 2f_{\mathrm{OL}}$, is
not met for the readout noise. As a result, high-frequency amplifier noise is
aliased into the TES-signal band. This noise aliasing is an inherent condition
of the TDM method and is generally what limits the multiplexing factor for a
given application. The TES-referred, multiplexed readout-noise level is given
byDoriese et al. (2006):

$\sqrt{S_{I\mathchar
45\relax\mathrm{mux}}}=\sqrt{AN_{\mathrm{rows}}S_{\Phi}}/M_{\mathrm{in1}}.$
(6)

Here, $A\geq 1$ is an alias-scaling factor that depends on the digital-
sampling strategy and the ratio $t_{\mathrm{row}}/\tau_{\mathrm{OL}}$. For the
common operational case in which $t_{\mathrm{row}}\sim 2\pi\tau_{\mathrm{OL}}$
and the digital sampling occupies a small fraction of $t_{\mathrm{row}}$ at
the end of the row periodDoriese et al. (2006), $A\sim\pi$. In the limit of
$f_{\mathrm{OL}}\rightarrow\infty$, in which the digital sampling can occupy
the full row period, $A\rightarrow 1$. In both the modern lab systems (see
Sec. 3.3) and the proposed design of Athena X-IFU (see Sec. 3.3), the row time
is chosen to be $t_{\mathrm{row}}=160$ ns; this is well matched to the
achievable open-loop bandwidth in standard dc-coupled twisted-pair and
flexible-microstrip circuits of $f_{\mathrm{OL}}\sim 8$ MHz to 10 MHz.

Eq. 6 shows that the TES-referred readout noise scales as
$\sqrt{N_{\mathrm{rows}}}/M_{\mathrm{in1}}$, while Eq. 5 shows that
$M_{\mathrm{in1}}$ scales as $1/N_{\mathrm{rows}}$. Thus, in a TDM system,
TES-referred readout noise scales with the multiplexing factor as
$\sqrt{S_{I\mathchar 45\relax\mathrm{mux}}}\propto N_{\mathrm{rows}}^{3/2}$.

#### Room-temperature electronics

The room-temperature electronics for TDM readout separate neatly into two
functional pieces: the warm-front-end electronics (WFEE) and the digital-
readout electronics (DRE).

The WFEE contains the LNA and generates various low-frequency analog-bias
signals for the SQUIDs and TESs. Because the pre-LNA SSA-output signals are
small (a few millivolts) and thus EMI-sensitive, the WFEE often shares a
Faraday cage with the cryostat. For high-performance TDM readout of TES X-ray
microcalorimeters, the LNA is challenging to implement. Its needed
specifications are $f_{\mathrm{OL}}\geq 10$ MHz and input-voltage noise below
1 nV/$\surd$Hz; this combination is on the edge of what is available in
commercial op-amps, so the front end of the LNA is often assembled from
discrete transistors. Various groups have produced WFEE modules that have been
used for TDM readout over the last 20+ yearsBattistelli et al. (2008); Drung
et al. (2005); Reintsema et al. (2003); Doriese et al. (2016); Sakai et al.
(2022); Gonzales et al. (2022).

The DRE generates the fast RA signals to drive the FASs and runs the D-FLL for
each column. In most ground-based applications, raw D-FLL data are streamed to
a host computer for demultiplexing, triggering, filtering, and other signal
processing; however, on Athena X-IFU, these higher-level functions will be
performed within the DRE. Various versions of TDM-DRE have been produced and
used over the last 20+ yearsBattistelli et al. (2008); Reintsema et al. (2003,
2009); Sakai et al. (2022).

#### Laboratory TDM systems

Between 2010 and 2022, the NIST Quantum Sensors Group (Boulder, CO, USA) has
deployed about 15 TDM-based TES-microcalorimeter arrays to various X-ray-
spectroscopy experiments and facilities around the worldDoriese et al. (2017).
This development effort has allowed TDM technology to be tested in real-world
conditions beyond the traditional environment of the detector lab. Application
areas have included time-resolved absorption and emission spectroscopy with a
tabletop, broad-band sourceUhlig et al. (2013); Miaja-Avila et al. (2016);
O’Neil et al. (2017), synchrotron-based X-ray-emission and absorption
spectroscopyYamada et al. (2021); Lee et al. (2019), synchrotron-based energy-
resolved scatteringLee et al. (2022), particle-induced X-ray-emission
spectroscopyPalosaari et al. (2016), spectroscopy of hadronic and muonic
atomsHashimoto et al. (2022); Okumura et al. (2021), the metrology of X-ray-
line energies Fowler et al. (2021); Szypryt et al. (2019), and X-ray
tomography of integrated circuitsSzypryt et al. (2021b). Similar arrays of
gamma-ray TESs have been used for the assay of special-nuclear
materialsBennett et al. (2012).

Figure 9: The 50 mK snout package. (a) Photograph of the snout. The circular
pedestal at the bottom of the image has a diameter of 58.4 mm and the snout
has a height of 67 mm. (b) CAD image of the snout, with various parts
highlighted in different colors. Yellow: The TES chip and X-ray aperture sit
on top. Green: on flexible circuits, a total of 512 Al traces (256 pairs) run
from the detector plane to the four side panels. Purple: The “interface chips”
contain the TES-shunt resistors and the Nyquist inductors; there is one of
these chips per TDM column, or 8 chips total. Cyan: The TDM multiplexer chips
contain the SQ1s and FASs. Orange: A rigid-flexible printed-circuit board
(PCB) connects to the interface and TDM chips via Al wirebonds and wraps
around the snout to carry signal lines from panel to panel. A trio of 65-lead
“Nano-D” connectors on the inside of the rigid-flex PCB connect to twisted-
pair cables that carry signals to and from higher-temperature cryogenic
circuitry. From Doriese, et al., Rev. Sci. Instrum. (2017)Smith et al. (2021);
re-printed with permission.

The NIST lab TDM system for X-ray TESs accommodates up to 8 TDM columns and up
to 32 TDM rows (up to 256 TES pixels). The heart of each spectrometer is the
50 mK “snout” package (see Fig. 9). This package contains the TES array, the
biasing circuitry for the TESs (blue components in Fig. 7), and the SQ1s and
FASs.

Figure 10: Coadded 8-column by 32-row TDM spectra for (a) Ti-K$\alpha$, (b)
Mn-K$\alpha$, (c) Co-K$\alpha$, (d) Cu-K$\alpha$, and (e) Br-K$\alpha$; all
are measured from pure-metal foils except Br-K$\alpha$, which is measured from
a KBr salt. The red dots in the main plots are the histogrammed spectra data,
the light-blue lines are the natural line shapes (not broadened by the finite
resolution of the spectrometer), and the dark blue line is the best fit to the
data. Below each main plot, the red dots show residuals (data minus fit). (f)
$\Delta E_{\mathrm{FWHM}}$ vs. X-ray energy for these 2019 measurements (red
squares) and a previous data set acquired with a less mature version of the
SQ1s (black circles). The black line shows the X-IFU instrument-level
requirements on energy resolution, while the blue line shows the requirements
considering only the detector and readout subsystem and excluding margin. From
Smith, et al., IEEE Trans. Appl. Supercond. (2021)Smith et al. (2021); re-
printed with permission.

The 8x32 TDM snout system has also been used to develop TDM readout to meet
the stringent performance requirements for Athena’s X-IFU. As Fig. 10
illustrates, by late 2019 TDM had achieved the needed performance for X-IFU.

#### Optimizations for space flight: Athena X-IFU

TDM has been improved in three significant ways from its 8x32-lab
configurationDoriese et al. (2017, 2016) to optimize it for Athena X-IFU.

The first optimization was to increase the open-loop bandwidth in the link
between the SQ1s and the SSA. This bandwidth is $R_{\mathrm{dyn}}/(2\pi
L_{\mathrm{tot}})$, and for X-IFU, the desired value is 15 MHz (12 MHz
specification plus a 25% margin). Here, $R_{\mathrm{dyn}}\equiv dV/dI$ is the
dynamic resistance of the components in the SQ1-bias loop (SQ1,
$R_{\mathrm{ser}}$, $R_{\mathrm{b1}}$; see Figs. 7 and 8), while
$L_{\mathrm{tot}}$ is the series inductance in the SQ1-bias loop (dominated by
the self inductance of the SSA input coil and the wiring that connects the
TES/SQ1 stage to the SSA stage). In X-IFU, it is desired to allow
$L_{\mathrm{wiring}}\leq 350$ nH, which is higher than in the 8x32 TDM-snout
systems. In early 2022, $R_{\mathrm{dyn}}\geq 35$ ${\mathrm{\Omega}}$ was
demonstrated (Durkin, et al.Durkin et al. (2021) give an intermediate report
of progress toward this milestone).

The second set of optimizations was to the packaging of the TDM componentry.
Cryogenic components, such as the TES-shunt resistorsDoriese et al. (2019) and
SQ1s, were shrunk in area. The various wire coils (SQ1-input transformer and
Nyquist inductors) were designed in “even” and “odd” versions to allow close-
packing of TDM cells in a 2-dimensional grid without an increase in cell-to-
cell inductive crosstalk. Most importantly, a cold-indium bump-bonding
processLucas et al. (2022) was developed to allow attachment of the X-IFU TDM
chips to the FPA side panels without the need for a challenging number of
wirebonds.

The third optimization was a change in the architecture of the flux-actuated
switches. X-IFU has baselined a “two-level-switching” scheme in which each TDM
pixel retains its own “pixel-select” switch and a second layer of switch, a
“cluster-select” switch, shorts out a larger cluster of pixels. This scheme’s
chief advantage is a significant reduction in the number of wires needed for
row addressing. Dawson, et al.Dawson et al. (2019) describe the idea further
and provide some preliminary demonstrations.

###  3.4 MHz frequency-domain multiplexing (FDM)

---
Figure 11: Cartoon of the distribution of signals in frequency space in FDM.
(top panel) In the TDM and $\upmu$mux readout schemes, the TESs are biased via
a dc current so the signals are distributed around zero frequency. (bottom
panel) In the FDM scheme, each TES is biased by its own MHz ac current and
therefore the signals are distributed in frequency space. The bias frequencies
of the TESs should be separated enough to avoid cross-talk between TESs. The
signals are demodulated by the room temperature electronics.

In the MHz-FDM scheme, signals from the TESs are multiplexed in frequency
space and the TESs are biased with megahertz-frequency alternating currents
(ac bias)Yoon et al. (2001); Cunningham et al. (2002); Kiviranta et al.
(2002); Ullom et al. (2003); Dobbs et al. (2008); Takei et al. (2009). Each
TES is connected to a passive $LC$ resonator. The resonators are sufficiently
separated in frequency space to avoid interferences (e.g. Fig. 11). Signals
from the different TESs are summed in the front-end SQUID amplifier. There are
three major advantages of the MHz-FDM architecture. First, the cold
electronics are relatively simple (there is one dc-SQUID front-end amplifier
per multiplexer chain). Second, MHz-FDM is relatively robust to
electromagnetic interference (EMI) because the signals are transported at
frequencies well above the electromagnetic disturbances due to the cryocoolers
and the frequency dependence of the skin depth makes shielding easier and
lighter. Third, MHz-FDM has lower environmental susceptibilityVaccaro et al.
(2022b) (see also Gottardi&Smith in this book). The X-IFU demonstration model
(Sec. 3.4), on which we base this presentation of MHz-FDM, will use a second-
stage series-array SQUIDKiviranta et al. (2021) located on the 2 K stage.
Multiplexed signals are demodulated per TES (per assigned frequency) in the
room temperature electronics.

The sum of signals can be achieved in several ways in SQUID readout: current
summingKiviranta et al. (2004), flux summingMitsuda et al. (1999); Kimura et
al. (2008), and voltage summingYoon et al. (2001); Cunningham et al. (2002).
Traditionally, voltage summing has been performed via summing loops; however,
because SQUIDs are inherently sensitive to changes in magnetic flux, and
current is straightforwardly transduced to flux via a wire coil, it is more
straightforward to sum the signals via current or magnetic flux. The flux-
summing approach requires a separate input coil for each TES. This complicates
the design of the SQUID and may cause electrical crosstalk between input coils
via mutual inductance. On the other hand, the current-summing method does not
require a complicated SQUID design, and crosstalk can be avoided via careful
implementation of the coils. This section focuses mainly on the current-
summing method.

The multiplexed signal is demodulated (at the assigned frequency) for each TES
in the room-temperature electronics. The modulation (bias) frequency must be
much higher than the signal bandwidth of the TES response ($\sim$50 kHz).
Therefore, a readout bandwidth of several megahertz is necessary to ensure a
reasonable multiplexing factor. As mentioned in the previous section, dc-SQUID
amplifiers combine sufficient bandwidth ($\sim 10$ MHz) with a high flux
dynamic range ($\sim 10^{8}\rm or\sim
0.01$$\upmu\mathrm{\Phi}_{0}/\sqrt{\mathrm{Hz}}$). This allows multiplexing of
tens of TES X-ray microcalorimeter signals, or more than 100 lower-bandwidth
bolometric TESs, in a single readout chain.

In the FDM method, the signal-to-noise ratio is not degraded by an increased
multiplexing factor as it is in the TDM method (see Eq. 6 in Sec. 3.3).
However, in FDM each TES must be activated by a megahertz-frequency ac bias,
which introduces additional physical phenomena into the TES, such as the ac
Josephson effect due to the lateral proximity effect of the superconducting
leadsSadleir et al. (2010, 2011); Gottardi et al. (2014a, 2018) (see also the
chapter by Gottardi & Smith in this book for details). Subsections 3.4 and 3.4
below detail some important technologies needed to realize FDM readout, while
subsections 3.4 and 3.4 describe some experimental demonstrations of FDM
readout of X-ray-TES arrays.

#### Room temperature electronics

The FDM room-temperature electronics consist of two main components: the
analog electronics (Front-End Electronics, or FEE) and the digital electronics
(DEMUX board).

---
Figure 12: Left: photograph of the dual FEE with 2-channel LNA and SQUID
controller. Right: photograph of the DEMUX board for FDM readout. Both of
these electronics were developed at SRON.

FEE: The primary role of the FEE is to amplify the signal from the cryogenic
SQUID array so its signal size and output-voltage noise are compatible with
the room-temperature ADC. A low-noise amplifier (LNA) is thus needed. The FEE
developed at SRON, pictured in Fig. 12, has an input-voltage noise of $\sim
300~{}\rm pV/\sqrt{Hz}$. Information about the SRON FEE can be found in the
literatureWang et al. (2020).

Figure 13: Block diagram of the FDM readout architecture. The ac-bias carrier
is generated in the FPGA and converted to an analog signal by a DAC in the
digital electronics (DEMUX board). Each TES in the readout channel is equipped
with an $LC$ filter with a unique frequency; the two-stage SQUID is read out
by an LNA (low-noise amplifier) at room temperature. Signals are converted
from analog to digital by an ADC and demodulated by an FPGA. The demodulated
signal is decimated by a 4-stage filter to reduce the data volume. The final
sampling rate is 156.3 kSample/s. The feedback signal is calculated and fed
back to the system after compensating for phase delays caused by electronics
and the length of the wiring harness in the system. From den Hartog, et al.,
(2009)den Hartog et al. (2009); re-printed with permission.

DEMUX board: The DEMUX board handles many functions, including demodulation of
signals and generation of the TESs’ ac-bias carrier combs and the feedback
signal applied to the front-end SQUID. In addition, the DEMUX board must be
capable of handling delays and phase shifts introduced by the environment. In
the presence of the long wiring harnesses that are expected in
satellites,888For Athena X-IFU instrument, the length of the harness between
the 50 mK stage and the room-temperature digital electronics may be as long as
5 m. For other similar cryogenic instruments such as SPICA/SAFARIRoelfsema et
al. (2012), even longer harnesses ($\sim$12 m) have been proposed. standard
feedback methods (e.g., the conventional flux-locked-loop, or FLL) do not
properly compensate the input signal to the SQUIDs amplifier due to the phase
difference between the input signal and the feedback signal. The lack of phase
margin in a standard FLL would thus make the feedback loop unstable above some
maximum frequency, $f_{\mathrm{max}}$. The gain-bandwidth product,
$f_{\mathrm{max}}*G_{\mathrm{FB}}$, of a standard FLL controller is limited by
the phase shift due to signal travel time through a total travel length in the
cable, $l$, asden Hartog et al. (2009)

$f_{\mathrm{max}}*G_{\mathrm{FB}}=\displaystyle{\frac{c}{8l\sqrt{\eta}}},$ (7)

where $G_{\mathrm{FB}}$, $c$, and $\eta$ are the loop gain of the feedback,
the speed of light, and the dielectric constant of the cable’s dielectric
material, respectively. Assuming $G_{\mathrm{FB}}=10$ (which is required to
keep the total flux in the linear regime of SQUID response) and $\eta$=3, for
a 1 m cable harness ($l=2$ m for the round-trip travel distance of signal and
feedback) the maximum frequency will be about 1 MHz. With even longer cables
and FDM’s higher bias frequencies, standard FLLs cannot work.

Baseband feedback (BBFBden Hartog et al. (2009); Fig. 13) is one way to
overcome this limitation999Another method is digital-active nulling (DANde
Haan et al. (2012)). The main difference is that in BBFB there is feedback to
the summing junction shared by the carrier voltage. DAN is or will be
implemented in the SPT-3G instrumentMontgomery et al. (2022) and the LiteBird
satelliteHazumi et al. (2020); de Haan et al. (2020). The BBFB method sends
the signal from the TES back to the SQUID after correcting for delay and phase
rotation at each carrier frequency. BBFB significantly improves the bandwidth,
linearity, and dynamic range of the SQUID amplifier. At SRON, the DEMUX board
consists of an AD9726 16-bit DAC and a Xilinx XC7V585T Virtex 7 Field
Programmable Gate Array (FPGA). Signals are measured at 20 MSample/s and
decimated to 156.3 kSample/s by a 4-stage filter. The DAC performance is
described by den Hartog, et al.den Hartog et al. (2018). The firmware of the
DEMUX board will allow up to 64 X-ray microcalorimeters or up to 172
bolometers.

#### Lithographic $LC$ filter

Figure 14: Left: photograph of a lithographic $LC$ filter chip containing 2
$\times$ 40 resonators. The chip size is 36 mm $\times$ 60 mm2. Upper right:
an enlarged photograph of one capacitor (left) and one 2 $\upmu$H spiral coil
with back-spiraling return line (right). Bottom Right: Effective series
resistance (blue diamonds) and quality factor (red squares) of the $LC$
resonators vs. frequencyBruijn et al. (2014, 2018). The blue-dashed and red-
solid lines indicate the contribution of the shunt resistance and the $Q$
specification of this design.

High-quality $LC$ resonators/filters are one of the key cryogenic components
to achieve high multiplexing factors in FDM readout; a higher density of $LC$
filters allows more efficient use of the available bandwidth. However, there
is a trade-off between fabrication accuracy due to technology limitations and
performance metrics such as crosstalk and stability. For a single-pole $LC$
filter, the resonator bandwidth ($\Delta f_{\mathrm{BW}}$), resonant frequency
($f_{\mathrm{c}}$), and quality ($Q$) factor are described as follows:

${\Delta f_{\mathrm{BW}}}=\frac{R}{2\pi
L},\,\,\,\,\,\,\,\,\,\,f_{\mathrm{c}}=\frac{1}{2\pi\sqrt{LC}},\,\,\,\,\,\,\,\,\,\,\rm
and\,\,\,\,\,\,\,\,\,\,Q=\frac{1}{\textit{R}}\sqrt{\frac{\textit{L}}{\textit{C}}},$
(8)

where $R$ and $L$ represent the total resistance and inductance, respectively.
Typically, the same inductance value is used for different $LC$ filters and
the resonant frequencies are tuned via the capacitance. In this way, the
design and uniformity of the $LC$ filter can be well controlled. It is common
for proposed satellite-borne FDM systems to have a minimum bias frequency of
about 1 MHz. This is because at lower frequencies, the area required for the
capacitors begins to dominate the overall area of the cryogenic components. At
the higher-frequency end of the readout band, challenges arise due to the
equivalent series resistance (ESR) generated in the dielectric medium within
the capacitor, the analog bandwidth in the full chain, and power dissipation
in the warm electronics.

SRON uses an $LC$ filter with a 2 $\upmu$H coil. A superconducting transformer
is also used to adjust the detector bandwidth via the damping inductance.
SRON’s $LC$ filter is fabricated via lithographic technologyBruijn et al.
(2014, 2018).

---
Figure 15: Upper left: photograph of the 2-column FDM-readout demonstrator.
The gray cylinder is the Nb superconducting magnetic shield. Upper right:
Photograph of the focal plane showing the SRON kilo-pixel TES arrayNagayoshi
et al. (2020), superconducting transformers, $LC$ filters, and FE SQUIDs. Two
FDM columns are connected. Lower left: co-added energy spectrum of the
Mn-K$\alpha$ complex from 37 TES pixels under FDM readout with a combined
energy resolution of 2.2 eV at 6 keVAkamatsu et al. (2021). The spectral data
were fitted to the Mn-K$\alpha$ line model of Hölzer, et al.Hölzer et al.
(1997) convolved with the Gaussian detector response. Cash statistics Cash
(1979); Kaastra and Bleeker (2016) are employed to avoid fitting bias. Lower
right: resonator distribution vs. frequency for column-1 (top) and column-2
(bottom) of the 2-column FDM system.

#### Demonstrations

Achievement of the best energy resolution in TES X-ray microcalorimeters under
FDM readout has required resolution of several deep and difficult issues
related to the fundamental physics of superconductivity and SQUID
readoutAkamatsu et al. (2020). The three main areas of research over the last
10+ years have been 1) the physics of superconducting thin films under ac
biasSadleir et al. (2010, 2011); Akamatsu et al. (2014); Gottardi et al.
(2014b, 2018), 2) performance degradation due to carrier leakagevan der Kuur
et al. (2004); Akamatsu et al. (2020), and 3) performance degradation due to
intermodulation productsvan der Kuur et al. (2016); Vaccaro et al. (2021); van
der Hulst et al. (2021). As of late 2021, all issues were resolved; Akamatsu,
et al.Akamatsu et al. (2021) give more information on these issues and their
resolution.

Fig. 15 (upper) shows a 2-column FDM demonstrator developed at SRON. It
employs magnetic shields of both high-µ and superconducting (Nb) materials
against the earth’s magnetic fieldBergen et al. (2016). The system is
installed in a dry dilution-refrigerator with a vibration-reduction
mechanismGottardi et al. (2019). Forty $LC$ resonators, each consisting of a 2
$\upmu$H coilBruijn et al. (2014, 2018), a capacitor, and a superconducting
transformer, were implemented to tune the electrical circuits seen by selected
TESs in a kilopixel array of SRON X-ray TESsNagayoshi et al. (2020); Taralli
et al. (2020); D’Andrea et al. (2021). The effective inductance seen by the
TES (equivalent to $L_{\mathrm{Ny}}$ in TDM) is tuned to be 60% of the
critical inductance at a TES resistance of 15% of $R$n. The resonator centers
span the 1 MHz to 5 MHz readout bandwidth with 100 kHz separation, meaning
there are 40 resonators per column. The rms scatter of the frequency centers
is about 4 kHz and is dominated by lithographic accuracy. A 2-stage SQUID is
employed, which is provided by VTT/FinlandKiviranta et al. (2018, 2021). Fig.
15 (lower left) shows a co-added X-ray spectrum of the Mn-K$\alpha$ complex
from 37 multiplexed pixels (two resonators did not yield and one pixel was
turned off); the achieved energy resolution was $\Delta
E_{\mathrm{FWHM}}=2.23$ eV at 6 keV. In a different experiment performed in a
different cryo-moduleAkamatsu et al. (2014), 31 multiplexed TESs achieved a
summed spectral resolution of $\Delta E_{\mathrm{FWHM}}=2.14$ eV. The level of
the thermal cross-talk is evaluated to be $\sim 10^{-4}$Vaccaro et al. (2022c)
for pixels that are physically nearest neighbors, which surpasses the Athena
X-IFU instrument requirement of $<10^{-3}$.

The performance degradation due to readout is estimated to be 0.9 eV in
quadrature (see Fig. 3 in Akamatsu et al.Akamatsu et al. (2021)) and is
dominated by known and non-fundamental problems such as thermal gradients in
the cryogenic stage. In other words, there is ample room for further
improvement and increasing multiplexing factor in future experiments. A two-
column FDM demonstration is currently underway (see Fig.15 bottom right).

#### Demonstration Model of Focal Plane Assembly of Athena X-IFU

---
Figure 16: Top left: CAD-rendered image of the X-IFU FPA DM. Inside the gold-
plated-copper housing are two magnetic shields (high-µ metal and
superconducting Nb) and a prototype of the 50 mK detector assembly. The FPA is
suspended by Kevlar wires to damp micro-vibrations from the mechanical
cryocoolers and to thermally isolate the 50 mK stage from the 2 K environment.
Top right: Photograph of some disassembled components of the X-IFU FPA DM. The
triangular component is the mechanical supporting structure based on Kevlar
wires. Bottom left: photograph of the flexible-interconnect chipvan Weers
(2013). Bottom right: photograph of the 50 mK detector assembly. The central
kilopixel TES array is connected to 4 FDM columns with 20 pixels per column.

An important application of FDM readout of TES X-ray microcalorimeters is
verification of the the Demonstration model (DM) of the Athena X-IFU Focal
Plane Assembly (FPAJackson et al. (2016); see Fig. 16). The X-IFU FPA DM is
designed to investigate various technological challenges. The X-IFU FPA DM
hosts a kilo-pixel TES array that was extensively characterized under ac
biasTaralli et al. (2020); D’Andrea et al. (2021); Taralli et al. (2021), an
anti-coincidence detectorMacculi et al. (2016); D’Andrea et al. (2017, 2020)
and 4 FDM columns with 20 pixels per column connected (i.e., 80 TESs in
total). The X-IFU FPA DM has three main purposes:

  1. 1.

A test-bed for key technology items that have not yet been demonstrated at
all, or have not been demonstrated in combination with other parts of the FPA.
Examples are the dual-stage SQUID amplifiers which are divided over two
temperature stages, the magnetic shields, and the Kevlar suspension (see Fig.
16 top left).101010Micro-vibration can degrade the performance of the
instrument significantly. It is of importance to isolate the instrument from
the source of the vibration. See Takei et al.Takei et al. (2018) for more
details of a similar investigation for Hitomi/SXS.

  2. 2.

A physical reference for thermal, magnetic, and mechanical simulations that
will guide the design of the engineering model and eventually the flight
model. Measurements obtained from the FPA DM allow the simulations to be
checked and, when necessary, can guide improvements to the simulations.

  3. 3.

A demonstrator for the environments in which the TES sensor array needs to
perform. As the main task of the FPA is to shield the TES array from adverse
thermal, magnetic, and mechanical influences from outside, the FPA DM will
provide a test in the environment of a cryostat. In particular, it will serve
as a demonstrator of the 2K core in the CEA Cryostat XIII, where the FPA will
be operated in proximity to an adiabatic demagnetization refrigerator and
flight-representative Joule-Thomson coolers.

As of August 2022, following milestones had been met:

  1. 1.

the thermal loads on the various cryogenic stages were measured;

  2. 2.

temperature stability of the detector stage was 0.9 $\upmu$K with 1 s sampling
repetition over the required period of more than one hour; and

  3. 3.

on the first cool-down of the system, energy resolution better than 2.5 eV at
6 keV was demonstrated.

These results verify that the thermal and mechanical environment provided by
the FPA is suitable for the needed performance in X-IFU.

###  3.5 Microwave SQUID multiplexing ($\upmu$mux)

Figure 17: Survey of microwave transmission $\left|S_{21}\right|$ through a
microwave-SQUID multiplexer operating in the 4 GHz to 6 GHz band, with 256
channels distributed across 4 discrete $\upmu$mux chips.

Multiplexing techniques utilizing gigahertz of bandwidth became possible with
the development of low-thermal-conductance cryogenic microwave cables, in the
form of narrow gauge Cu-Ni and Nb-Ti coaxial cables, and a low-noise cryogenic
microwave amplifier, in the form of the High Electron-Mobility Transistor
(HEMT) amplifierDuh et al. (1988).

Figure 18: Left $\upmu$mux readout circuitry on the backside of a 1,820 pixel
CMB polarimeter array, read out by 28 $\upmu$mux chips on two pairs of coaxial
cables. Right “microsnout” readout package for 256 X-ray microcalorimeters.
The microsnout is modular: microsnouts can be tiled within a focal plane and
electrically connected in series within a multiplexer channel.

Microwave SQUID multiplexing ($\upmu$mux)Mates et al. (2008); Hirayama et al.
(2013); Mates et al. (2017); Ahrens et al. (2019) is a form of frequency-
domain multiplexing that allocates this bandwidth between input channels by
the use of distinct, high-$Q$ microwave resonances, each coupled to its own
rf-SQUID and reading out the current signal from its own detector. As a
superposition of microwave tones passes through the circuit, each tone is
modulated by its own SQUID/resonator circuit before being amplified by the
HEMT and brought to room temperature on a single coaxial cable.

Figure 19: Schematic of a 3-pixel microwave-SQUID multiplexer, consisting of
quarter-wave resonators inductively coupled to rf-SQUIDs. Resonance
frequencies are set by the lengths of the transmission lines, resonance
bandwidths by $C_{\mathrm{c}}$, and resonance frequency shifts by
$M_{\mathrm{c}}$, $L_{\mathrm{\mathrm{S}}}$, and $L_{\mathrm{J}}$.

An example circuit is shown in Figure 19. In this circuit, distributed
quarter-wave resonators of different lengths are capacitively coupled to a
common microwave “feedline” at one end and inductively coupled to their
individual SQUIDs at the other end. Two notable variations utilize lumped-
element resonatorsAhrens et al. (2019) rather than quarter-wave resonators and
direct incorporation of the SQUID into the resonatorHirayama et al. (2013)
rather than inductive coupling to the SQUID.

In any implementation, the multiplexer works by loading the resonator with the
flux-variable inductance of a Josephson junction:

$L_{\mathrm{J}}(\phi)=\frac{\Phi_{0}}{2\pi I_{\mathrm{c}}\mathrm{cos}(\phi)}$
(9)

where $I_{\mathrm{c}}$ is the critical current of the Josephson junction,
$\Phi_{0}$ is the magnetic flux quantum, and $\phi\equiv 2\pi(\Phi/\Phi_{0})$
is the difference in phase of the superconducting wave function across the
junction.

Figure 20: Illustration of the principle of operation of a microwave SQUID
multiplexer channel. a) The resonance frequency shifts in response to magnetic
flux. b) For a fixed microwave frequency, the complex transmission parameter
$S_{21}$ describes an arc on a circle. c) The SQUID response is periodic with
a magnetic flux quantum $\Phi_{0}$.

As the flux in the rf-SQUID varies, the inductive load it applies to the
resonator varies, both positively and negatively, causing the resonance
frequency $f_{0}$ to change without significant change in microwave loss
(Figure 20). For a fixed microwave probe tone the complex transmission
traverses an arc on a circle with the angular position on this arc being a
periodic function of magnetic flux.

SQUID coupling: | Inductive | Direct
---|---|---
$\Phi_{\mathrm{rf}}$ | $M_{\mathrm{c}}I_{\mathrm{rf}}$ | $L_{\mathrm{S}}I_{\mathrm{rf}}$
$\Delta L_{\mathrm{pp}}$ | $\frac{2{M_{\mathrm{c}}}^{2}}{L_{\mathrm{J}}\left(1-\lambda^{2}\right)}$ | $\frac{2\lambda^{2}L_{\mathrm{J}}}{1-\lambda^{2}}$
Resonator type: | Quarter-wave | Lumped-element
$\frac{\mathrm{d}f_{0}}{\mathrm{d}L}$ | $-\frac{4{f_{0}}^{2}}{Z_{1}}$ | $-\frac{\pi}{\sqrt{L/\left(C+C_{\mathrm{c}}\right)}}$
$\Delta f_{\mathrm{BW}}$ | $8\pi{f_{0}}^{3}{C_{\mathrm{c}}}^{2}Z_{0}Z_{1}$ | $2\pi^{2}{f_{0}}^{3}{C_{\mathrm{c}}}^{2}Z_{0}\sqrt{L/\left(C+C_{\mathrm{c}}\right)}$
$P_{\mathrm{feed}}$ | $\frac{\pi Z_{1}\Delta f_{\mathrm{BW}}}{16f_{0}}{I_{\mathrm{rf}}}^{2}$ | $\frac{\Delta f_{\mathrm{BW}}}{4f_{0}}\sqrt{L/\left(C+C_{\mathrm{c}}\right)}$
Table 3: Table of equations describing the behavior of a SQUID-resonator pair.
Most variables are defined in Figure 19. $\Phi_{\mathrm{rf}}$ is the amplitude
of microwave flux excitation in the SQUID, $\Delta L_{\mathrm{pp}}$ is the
peak-to-peak variation in load inductance with flux, $\Delta f_{\mathrm{BW}}$
is the resonator bandwidth, and $P_{\mathrm{feed}}$ is the power per tone on
the feedline. For a lumped-element design, $L$ and $C$ are the lumped
inductance and capacitance of the resonator, respectively.

One key parameter of the rf-SQUIDs is the “screening-parameter”Likharev
(2022), sometimes denoted $\beta_{L}$, which gives the ratio of SQUID self-
inductance to Josephson inductance:

$\lambda\equiv\frac{L_{\mathrm{S}}}{L_{\mathrm{J}}}=\frac{2\pi
I_{\mathrm{c}}L_{\mathrm{S}}}{\Phi_{0}}.$ (10)

For $\lambda<1$ the SQUID response is single-valued, while for $\lambda>1$ the
SQUID may switch hysteretically between multiple statesLikharev (2022). An rf-
SQUID may be operated in the hysteretic regime, but it requires damping to
make its behavior predictable. Because this damping would limit resonator
$Q$-factors, microwave-SQUID multiplexers operate in the non-hysteretic
regime, typically targeting $\lambda\approx 1/3$.

Table 3 lays out some of the equationsMates (2011) describing the behavior of
an rf-SQUID and resonator pair. Although these equations are only strictly
valid in the regime of both low-power slow-flux-ramp, they provide a useful
guide to device design.

For example, the equations show how $M_{\mathrm{c}}$ and $C_{\mathrm{c}}$ may
be adjusted in tandem to achieve designs with different pixel bandwidths while
holding the ratio of peak-to-peak frequency shift to bandwidth
($\eta\equiv\Delta f_{\mathrm{pp}}/\Delta f_{\mathrm{BW}}$) constant. They
further show that the microwave-flux excitation (on-resonance and in the
inductive coupling, quarter-wave-resonator configuration of Figure 19) in the
SQUID is related to the power-per-tone on the feedline as:

$P_{\mathrm{feed}}={\Phi_{\mathrm{rf}}}^{2}\frac{\pi^{2}f_{0}I_{\mathrm{c}}}{\eta\Phi_{0}\left(1-\lambda^{2}\right)}.$
(11)

In Section 3.5 we will use this information to derive expected readout noise
and discuss the trade-off between noise and maximum signal slew-rate. In
Section 3.5 we will discuss the trade-off between frequency packing and
crosstalk. This analysis will show what capabilities we can expect of a
$\upmu$mux system designed for any particular application.

#### Flux-ramp modulation

With microwave-SQUID multiplexing there is no practical way to apply feedback
without the addition of a large number of wires. Instead, a flux-ramp
modulation schemeMates et al. (2012) is typically employed that sweeps rapidly
across the SQUID response such that the input signal is transduced into a
phase shift of the response function (Figure 21). This phase may then be
extracted at room temperature either with a fit or with Fourier techniques
over an integral number of oscillations. The initial section of the flux-ramp
response is usually contaminated with a transient from the reset and must be
discarded, with analysis only over a useful fraction $\alpha$.

Figure 21: Illustration of the principle of flux-ramp modulation, which
transforms (a) the input flux signal into (b) a phase-shift of the SQUID
response, shown as blue lines with the baseline response as dashed-red lines.
The flux-ramp reset transient is visible at the start of each ramp response
window.

This both linearizes the SQUID readout and upconverts it above the significant
low-frequency noise that is due to two-level systems, while reducing the
readout signal power by approximately $\frac{\alpha}{2}$, where the factor of
$\frac{1}{2}$ arises from averaging across the roughly sinusoidal SQUID
response curve and the factor of $\alpha$ arises from discarding transient-
contaminated data.

The flux-ramp repetition rate $f_{\mathrm{r}}$ thus becomes the sampling rate
of the input signal. To fit the SQUID modulation within the resonator
bandwidth it must obey $f_{\mathrm{r}}n_{\Phi_{0}}<\Delta f_{\mathrm{BW}}/2$,
where $n_{\Phi_{0}}$ is the number of flux quanta per ramp. It must also
substantially exceed the frequency content of the input signals in order to
read them out accurately, as the fidelity of the modulation scheme begins to
degrade at an input signal slew rate of order $f_{\mathrm{r}}\Phi_{0}/2$.
These considerations determine the resonator bandwidth necessary for readout
of a detector system.

####  $\upmu$mux readout noise

There are multiple sources of noise in a microwave-SQUID multiplexer: HEMT-
amplifier noise is generally dominant, two-level system noise becomes
significant for narrow resonator bandwidths, and digital noise limits the
number of tones we can cleanly generate and digitize in a set of room
temperature electronics.

The HEMT amplifier produces broadband white noise with typical noise
temperatures on the order of $T_{\mathrm{N}}\approx 4$ K
($k_{\mathrm{B}}T_{\mathrm{N}}\approx-193$ dBm/Hz). To convert this noise to
units of magnetic flux we must find the power in the microwave probe tone,
which should be as large as possible, up to the limits of SQUID non-linearity.

Figure 22: Non-linear, non-equilibrium simulation results showing various ways
in which the SQUID response departs from the simple theory in practical
operation. a) Low power, slow flux-ramp, but with a significant flux-ramp-
reset transient. b) Slow flux-ramp but with high microwave-drive power that
suppresses the amplitude of the SQUID response. c) High-power, fast flux-ramp
so that the resonator never has time to fully equilibrate and instead follows
a “figure-8” trajectory along the resonance circle.

Determination of the optimal value of $P_{\mathrm{feed}}$ requires analysis
beyond the linear, small-signal approximations of the equations in Table 3,
including non-linear and non-equilibrium dynamics. As $\Phi_{\mathrm{rf}}$
grows to a significant fraction of $\Phi_{0}$, the SQUID inductance stops
being effectively constant over the microwave excitation and siphons power
from the microwave-probe frequency into higher harmonics. This behavior can be
analyzed in expansionWegner et al. (2022), but to include all complicating
phenomena generally requires simulation.

Figure 22 shows the results of simulations that include different sets of
complicating factors, and in which we find the optimal microwave flux
amplitude to be 0.25 $\Phi_{0}$ to 0.4 $\Phi_{0}$, depending on design
parameters. For an example set of design parameters ($\eta=1$, $f_{0}=6$ GHz,
$I_{\mathrm{c}}=5$ $\mu$A, $\lambda=1/3$), this yields a tone power on the
feedline of -73 dBm to -70 dBm, or approximately 120 dB above the 1 Hz noise
power of the HEMT, and an arc of approximately $\pm 1$ radians.

This gives a noise spectrum in $\theta$, the angle on the resonance arc
(Figure 20) of:

$S_{\theta}=\left(\frac{1-Q/Q_{i}}{2}\right)^{-2}\frac{1}{2}\frac{k_{\mathrm{B}}T_{\mathrm{N}}}{P_{\mathrm{feed}}}\approx
2\frac{k_{\mathrm{B}}T_{\mathrm{N}}}{P_{\mathrm{feed}}}$ (12)

where the first term gives the radius of the resonance circle in terms of the
total quality factor $Q$ and internal quality factor $Q_{i}$. The
approximation assumes $Q_{i}\gg Q$, but for narrow resonance applications
($<$$100\text{\,}\mathrm{kHz}$) $Q_{i}$ may approach the total $Q$, which
increases this noise contribution.

We can then assume a sinusoidal SQUID response, as shown in Figure 21, with an
amplitude of $\theta_{\mathrm{max}}\approx 1$ and apply the flux-ramp
modulation penalty to get:

$S_{\Phi}^{\mathrm{HEMT}}=S_{\theta}\left(\frac{\Phi_{0}}{2\pi\theta_{\mathrm{max}}}\right)^{2}\frac{2}{\alpha}\approx\frac{k_{\mathrm{B}}T_{\mathrm{N}}}{P_{\mathrm{feed}}}\frac{2}{{\pi}^{2}}{\Phi_{0}}^{2}$
(13)

where in the final approximation we have assumed a 2$\mathrm{\Phi}_{0}$ flux
ramp and have discarded the first oscillation for $\alpha=\frac{1}{2}$. This
implies a HEMT contribution to readout noise of approximately 0.5
$\upmu\mathrm{\Phi}_{0}/\sqrt{\mathrm{Hz}}$. Although this theoretical limit
is rarely achieved, total readout noise of (1.0 to 2.0)
$\upmu\mathrm{\Phi}_{0}/\sqrt{\mathrm{Hz}}$ is common and sufficient for most
low-temperature detector applications (see chapter by Gottardi & Smith in this
book).

Two-level-system (TLS) noise arises from the coupling of the microwave fields
in the resonator to two-level systems in the dielectrics. This produces a
fractional-frequency noise that depends on temperature, internal power,
material, and geometryGao et al. (2008). The power spectrum of this noise has
a characteristic $1/\sqrt{f}$ frequency dependence, which the flux-ramp
modulation of Section 3.5 helps to avoid.

The TLS noise is not stationary over the period of the SQUID response, but we
may bound its contribution by considering the equivalent flux noise at the
steepest slope of the SQUID response, where:

$\frac{\mathrm{d}\theta}{\mathrm{d}\Phi}=\theta_{\mathrm{max}}\frac{2\pi}{\Phi_{0}}\,\,\,\,\,\,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\,\,\,\,\frac{\mathrm{d}\theta}{\mathrm{d}f}=\frac{4}{\Delta
f_{\mathrm{BW}}}=4Q/f_{0}$ (14)

so that (again assuming $\theta_{\mathrm{max}}\approx 1$) the implied
contribution to flux noise is:

$S_{\Phi}^{\mathrm{TLS}}=\frac{S_{\delta
f_{0}}}{{f_{0}}^{2}}\left(\frac{2Q}{\pi}\right)^{2},$ (15)

which scales with $Q$ or inversely with the resonance bandwidth.

The number of factors determining the TLS noise makes it difficult to make a
general statement about its level, but measured fractional-frequency noise for
the geometry, materials, and internal power of a typical $\upmu$mux device is
less than $S_{f_{0}}/{f_{0}}^{2}\approx 10^{-19}$ Hz-1 at 1 kHz. For example,
in a 2 MHz resonator at 6 GHz, modulated at 1 MHz, we therefore expect a TLS
noise contribution of less than 0.1
$\upmu\mathrm{\Phi}_{0}/\sqrt{\mathrm{Hz}}$. For multiplexers optimized for
typical X-ray applications, with 1 MHz to 10 MHz resonator bandwidths, this
contribution tends to be substantially less than the contribution of the
amplifier chain.

Finally, any system for synthesizing and digitizing a large number of
microwave tones adds its own noise: digital-quantization noise, clipping
noise, and a pseudo-white-noise of intermodulation products due to the
imperfect linearity of the DACs and ADCs. This is usually described by a
dynamic range between the tone powers and the digital-noise floor. A careful
analysis of this noiseBennett et al. (2014) is beyond the scope of this
chapter, except to say that multiplexing factors of $\sim$1000 push the limits
of existing digitizers, which requires multiple digitizers to cover the full
band with sufficient dynamic range, but that digitizer capabilities are
advancing rapidly and can be expected to continue to both improve in
capability and fall in cost.

####  $\upmu$mux crosstalk

There are three main mechanisms of crosstalk in the microwave-SQUID
multiplexerMates et al. (2019): Lorentzian-tail crosstalk, coupled-harmonic-
oscillator crosstalk, and broadband-non-linearity crosstalk. Each constrains
the multiplexer design in a different way.

Lorentzian-tail crosstalk arises from the quadratic falloff of the Lorentzian
resonance shape, which allows one resonance to affect the transmission of a
probe tone at the frequency of a neighboring resonance. For inductive
coupling:

$\chi_{\mathrm{Lorentz}}\approx\frac{|S_{21}|^{2}}{2}\left(\frac{\Delta
f_{\mathrm{BW}}}{f_{2}-f_{1}}\right)^{2}$ (16)

where $\chi$ gives the fractional crosstalk between the flux-ramp-modulated
signals.

This mechanism dictates the minimum frequency-spacing between resonances as a
multiple of their bandwidths and therefore the ultimate bandwidth efficiency.
For part-per-thousand crosstalk between frequency neighbors, a target
frequency spacing of (7 to 10) $\Delta f_{\mathrm{BW}}$ is typically
necessary.

Coupled-harmonic-oscillator crosstalk arises when resonances are both
electromagnetically coupled and close in frequency, such that the eigen modes
of the system are actually linear combinations of the uncoupled-resonance
modes:

$\chi_{\mathrm{CHO}}\approx\left(\frac{4\bar{f}^{2}}{f_{2}-f_{1}}\frac{M_{\mathrm{x}}}{Z_{0}}\right)^{2}$
(17)

where $M_{\mathrm{x}}$ is the mutual inductance between resonance
terminations. Other mechanisms of electromagnetic coupling add similar terms.

This mechanism motivates a design feature seen in all multiplexers that
utilize superconducting resonators, which is the placement of frequency-
adjacent resonators far apart in physical space and of physically-adjacent
resonators far apart in frequency spaceNoroozian et al. (2012).

Broadband-non-linearity crosstalk occurs within the broadband microwave
components, such as the HEMT amplifier and room-temperature mixers, whose
slight non-linearity allows a 3rd-order mixing process to transfer a fraction
$\chi_{\mathrm{nonlin}}$ of the modulation sidebands of one carrier tone to
another:

$\chi_{\mathrm{nonlin}}\approx
4\frac{P_{\mathrm{feed}}|S_{21}|^{2}}{P_{\mathrm{IP3}}}$ (18)

where $P_{\mathrm{IP3}}$ is the 3rd-order intercept point (here, referred to
the input of the HEMT), which is the standard measure of cubic non-linearity
in amplifiers and mixers.

While this crosstalk is typically less than that of the other mechanisms, it
is all-into-all and therefore remains of serious design concern. To reduce
this crosstalk we must use broadband components with high $P_{\mathrm{IP3}}$
or reduce the microwave power per tone at the cost of readout noise.

####  $\upmu$mux optimization for X-ray applications

For each application, the following optimization should be performed:

  1. 1.

Design the input coupling $M_{\mathrm{in}}$ such that the readout noise will
not significantly degrade the detector signal.

  2. 2.

Calculate the flux-ramp rate necessary to accommodate the signal slew rate,
which determines the resonance bandwidth.

  3. 3.

Space the resonance frequencies by a multiple of their bandwidths that is
sufficient to meet the required crosstalk limits.

For a 2$\mathrm{\Phi}_{0}$ flux-ramp and $10\,\,\Delta f_{\mathrm{BW}}$
spacing for part-per-thousand crosstalk, this implies a multiplexing factor
of:

$N\approx\frac{\Delta f_{\mathrm{BW-
tot}}}{80}\frac{\Phi_{0}}{M_{\mathrm{in}}\left|\frac{\mathrm{d}I}{\mathrm{d}t}\right|_{\mathrm{max}}}$
(19)

which is the number of detectors that can be read out using one HEMT, a pair
of coaxial cables, and a flux-ramp line.

As an abstract example, a signal with a maximum slew-rate of 0.05
$\mathrm{\Phi}_{0}$/$\text{\,}\mathrm{\SIUnitSymbolMicro s}$ would require a
flux-ramp rate of $100\text{\,}\mathrm{kHz}$, a resonance bandwidth of
$400\text{\,}\mathrm{kHz}$, and achieve a multiplexing factor of $\sim$1,000
in a $4\text{\,}\mathrm{GHz}$ to $8\text{\,}\mathrm{GHz}$ HEMT. This is more
than an order of magnitude higher than can be provided by conventional
multiplexing technologies ($N\sim 40$ by TDM and FDM).

Traditional X-ray applications with relatively high count-rates are generally
well-matched to the practical range of $\upmu$mux designs. As the count rate
falls substantially below $\sim$$100\text{\,}\mathrm{Hz}$, it becomes
difficult to match the $\upmu$mux bandwidth (to less than
$100\text{\,}\mathrm{kHz}$) and resonance frequency placement (to better than
$1\text{\,}\mathrm{MHz}$). Reliable $\upmu$mux designs have been proven for a
range of bandwidths from $100\text{\,}\mathrm{kHz}$ to
$30\text{\,}\mathrm{MHz}$.

#### Example $\upmu$mux systems

The first fielded application of $\upmu$mux readout for TES microcalorimetry
was the SLEDGEHAMMER instrument at Los Alamos National LaboratoryMates et al.
(2017), in which 128 gamma-ray microcalorimeters were read out in 1 GHz of
bandwidth in the initial demonstration. The system is used to perform high-
resolution gamma-ray spectroscopy of samples of nuclear materials. To
appropriately sample the microcalorimeter pulses, the resonators were designed
to have bandwidths of $300\text{\,}\mathrm{kHz}$ at a spacing of
$3\text{\,}\mathrm{MHz}$. The typical readout noise is approximately
$30\text{\,}\mathrm{p}\mathrm{A}\mathrm{/}\sqrt{\mathrm{Hz}}$, a factor of
$\sim$5 below the noise of the TES. The pixel count has since been expanded to
256.

A large X-ray spectrometer is presently being assembled at NIST as part of the
TOMCAT tomographic imaging systemSzypryt et al. (2021a), which has
demonstrated readout of $\sim$1000 detectors and will expand to 3000 detectors
in the near future. To match the
$\sim$$1\text{\,}\mathrm{A}\text{\,}{\mathrm{s}}^{-1}$ slew rate of the
detectors, the resonators have bandwidths of $1\text{\,}\mathrm{MHz}$ on a
$7.5\text{\,}\mathrm{MHz}$ spacing. The detector assembly utilizes multiple
modular “microsnouts” (Figure 18), with 248 detectors per microsnout in 2 GHz
of bandwidth.

Most relevant to the topic of cryogenic X-ray imaging spectrometers for
astronomy is that $\upmu$mux readout has been proposed for the Lynx LXMBandler
et al. (2019) and the planning documents provide a useful guide to expected
readout capability. The proposal calls for readout of $\sim$100,000 pixels via
a combination of thermal and electrical multiplexing, so that the main array
would require 4,000 $\upmu$mux channels with $1.4\text{\,}\mathrm{MHz}$
resonator bandwidths and 400 resonators per HEMT amplifier. Other sub-arrays
have different readout requirements, which are laid out by BandlerBandler et
al. (2019) and BennettBennett et al. (2019). While Lynx is not yet a funded
mission, technology advancement is ongoing to increase the readiness level of
the readout and to simplify its application to future large-scale cryogenic
X-ray astronomy instruments.

##  4 Summary and future prospects

The operation of large-scale arrays of cryogenic X-ray microcalorimeters in a
satellite environment requires the minimization of heat loads and cryogenic
complexity of the readout wiring. There are three main multiplexing
technologies for accomplishing this: TDM, MHz-FDM, and $\upmu$mux. All of
these technologies are currently deployed to multiple ground-based instruments
and planned for use in additional future satellite and ground-based
instruments, as shown in Table 4.

Table 4: An overview of missions that have flown (upper section) or will fly/employ (lower section) cryogenic X-ray spectrometers. Mission | Launch | Detector | Total # of | Readout | # of sensors
---|---|---|---|---|---
| date |  | sensors | type | per readout
XQC McCammon et al. (2002) | 2002 | semiconductor | 18 | JFET | 1
Hitomi/SXS Mitsuda et al. (2014) | 2016 | semiconductor | 36 | JFET | 1
Micro-X Adams and others. (2020) | 2018, 2022 | TES | 128 | TDM | 16
XRISM/Resolve Ishisaki et al. (2018) | 2023 | semiconductor | 36 | JFET | 1
Athena/X-IFU Barret et al. (2018) | mid-2030 | TES |  $\sim$2,500 | TDM | 33
BabyIAXO∗ Abeln et al. (2021) | 2025 | MMC | TBD |  $\upmu$mux | NA
HUBS∗ Cui et al. (2020) |  $\sim$2030 | TES |  $\sim$3,500 | TDM or FDM | TBD
LEM∗ | mid-2030s | TES |  $\sim$14,000 | TDM | 60
Lynx∗ Bandler et al. (2019); Stevenson et al. (2019) | early 2040s | TES or MMC |  $\sim$100,000 |  $\upmu$mux | TBD
Super-DIOS∗ Sato et al. (2020) | early 2040s | TES |  $\sim$30,000 |  $\upmu$mux |  $\sim$400
$\ast$: Missions in the conceptual or proposing phase. |  |

In the near future, work on TDM will focus on the achievement of technology-
readiness level (TRL)-6 and beyond for Athena X-IFU. TRL-6 involves
verification of the needed performance in a high-fidelity prototype that is
tested in a relevant environment. The X-IFU Engineering Model (EM) is intended
to provide such verification of all X-IFU systems. Aspects of TDM to be tested
in the X-IFU EM will include a fully differential wiring architecture, X-IFU’s
long cable harnesses and how they impact the open-loop bandwidth needed to
operate with 160 ns row times, and the yield of the 50 mK TDM components.
Beyond X-IFU, TDM may hybridized to $\upmu$mux as discussed further below.

Future work on MHz-FDM will target increasing the multiplexing factor per
column, via narrowing the detector bandwidth without losing performance,
increasing the upper limit of the usable bandwidth, and stabilizing the TES
performance under high-frequency ac bias (see also Gottardi & Smith in this
book for the state-of-art single-pixel performance under ac bias). It will
also require improvements to the firmware and electronics to realize a lower-
mass and simpler system. A multiplexing factor of $\sim$50 to 60 seems
feasible improvement in the near future(Vaccaro et al., 2022a). In parallel,
the linearity of the energy-gain scale will be demonstrated in a full
spectrometer under multi-column readout.

Future work on $\upmu$mux will continue to advance its technological readiness
to match TDM and MHz-FDM. It will also attempt to improve fabrication
capability to reliably achieve $Q_{i}$ values above 200,000 and to define
resonance frequencies more accurately than $\pm$0.02%, as well as improve
consistency of the fabrication quality and yield. Finally, it will require a
microwave packaging solution (e.g. bump-bonded unit cells that confine the
microwave fields) that allow it to be readily integrated into a TES focal
plane without risking degradation of microwave performance.

For space and ground projects, TDM and FDM are planned for arrays of up to
about 104 pixels, while $\upmu$mux is planned for arrays of more than 104
pixels (or for smaller arrays of very fast detectors). However, while
$\upmu$mux provides multiple gigahertz of output bandwidth, because of the
current limitations on quality factor and frequency placement it is difficult
to optimize for very large numbers ($>$2,000) of low-bandwidth input signals.
The bandwidth-utilization efficiency of $\upmu$mux with resonator bandwidths
of greater than $\sim$$300\text{\,}\mathrm{kHz}$ is comparable to that of TDM
and MHz-FDM, but it cannot maintain that efficiency for slower input signals
that require bandwidths narrower than can currently be fabricated.

For this reason several groups have begun working on hybrid multiplexing
solutionsSchuster et al. (2022); Yu et al. (2020), in which TDM, CDM, or MHz-
FDM are used as a front end to combine multiple slow input signals into a
single $\upmu$mux “pixel.” These hybrid solutions could potentially enable
multiplexing factors in the tens of thousands, which will probably be
necessary for instruments measuring faint astronomical X-ray objects, in which
the optimization will tend toward large (megapixel-scale) arrays of slow TESs
to tile the focal plane.

---
Figure 23:  Number of pixels per microcalorimeter or microbolometer array vs.
time. In 1995, the modern scheme for TES bias and readout (voltage bias to
provide negative electrothermal feedback and current readout via a SQUID
ammeter) was proposedIrwin (1995), which led quickly to the use of TESs in
real instruments. Filled markers indicate deployed instrumentation, while open
markers indicate future projects. Circles and stars represent arrays
semiconductor calorimeters and TES microcalorimeters, respectively, while
squares represent arrays of sub-millimeter or microwave TES bolometers. Grey
markers represent terrestrial experiments (bolometers for astronomy and X-ray
microcalorimeters for laboratory science), while black markers represent
space-borne missions. Two of the future X-ray missions show the effective
number of TES absorbers in the “Hydra” configurationSmith et al. (2020): for
LEM and LynxBandler et al. (2019), the number of Hydra pixels per TES is 4 and
5 respectively (not all TESs are configured as Hydras). TES bolometers require
much less bandwidth than do microcalorimeters, and thus have higher
multiplexing factors and larger arrays. For both space and ground projects,
TDM and FDM are used up to $\sim 10^{4}$ pixels, while $\upmu$mux will be used
for larger arrays. References for projects (not perfect list): FIBREBenford
(2002), APEX-SZ(Mehl, 2008), MUSTANG(Dicker, 2006),
BICEP2(BICEP2Collaborationand others, 2014), POLARBEAR2(Kermish, 2012; Inoue,
2016), MUSTANG2(Dicker, 2014), ACT-Pol(Grace, 2014), Adv-ACTPol(Henderson,
2016), SPT-Pol(Austermann, 2012), SPT-3GHenning (2015), SCUBA-2(Holland et
al., 2013), Simons ObservatoryAde et al. (2019), CMB S4(Abazajian, 2022),
LiteBirdLiteBIRDCollaborationand others (2022), CMB HDSehgal (2019)
TOMCAT(Szypryt et al., 2021b, 2022)

The development time for new technologies to be space ready is about 20 to 30
years, as seen in the gap between terrestrial and space projects in Fig. 23.
Beyond X-ray astronomy, these developments will also have terrestrial X-ray
spectroscopic applications in basic physics and materials analysis.

## References

  * Abazajian (2022) Abazajian Ko (2022) CMB-S4: Forecasting Constraints on Primordial Gravitational Waves. The Astrophysical Journal926(1):54, DOI 10.3847/1538-4357/ac1596, 2008.12619
  * Abeln et al. (2021) Abeln A, et al. (2021) Conceptual Design of BabyIAXO, the intermediate stage towards the International Axion Observatory. Journal of High Energy Physics 2021(5):1–80
  * Adams and others. (2020) Adams JS, et al. (2020) First Operation of TES Microcalorimeters in Space with the Micro-X Sounding Rocket. Journal of Low Temperature Physics 199(3-4):1062–1071, DOI 10.1007/s10909-019-02293-5, 1908.09689
  * Adams et al. (2021) Adams JS, et al. (2021) First operation of transition-edge sensors in space with the Micro-X sounding rocket. In: Holland AD, Beletic J (eds) X-Ray, Optical, and Infrared Detectors for Astronomy IX, International Society for Optics and Photonics, SPIE, vol 11454, pp 195 – 203, DOI 10.1117/12.2562645
  * Ade et al. (2019) Ade P, et al. (2019) The Simons Observatory: science goals and forecasts. Journal of Cosmology and Astroparticle Physics 2019(2):056, DOI 10.1088/1475-7516/2019/02/056, 1808.07445
  * Ahrens et al. (2019) Ahrens F, Wegner M, Paluch P, Fleischmann A, Enss C, Kempf S (2019) Superconducting ghz resonators for microwave squid multiplexing of metallic magnetic calorimeters. Verhandlungen der Deutschen Physikalischen Gesellschaft
  * Akamatsu et al. (2014) Akamatsu H, et al. (2014) Performance of TES X-ray Microcalorimeters with AC Bias Read-Out at MHz Frequencies. Journal of Low Temperature Physics 176:591–596, DOI 10.1007/s10909-014-1130-8
  * Akamatsu et al. (2020) Akamatsu H, et al. (2020) Progress in the Development of Frequency-Domain Multiplexing for the X-ray Integral Field Unit on Board the Athena Mission. Journal of Low Temperature Physics 199(3-4):737–744, DOI 10.1007/s10909-020-02351-3
  * Akamatsu et al. (2021) Akamatsu H, et al. (2021) Demonstration of MHz frequency domain multiplexing readout of 37 transition edge sensors for high-resolution x-ray imaging spectrometers. Applied Physics Letters 119(18):182601, DOI 10.1063/5.0066240, 2111.01797
  * Anderson and Rowell (1963) Anderson PW, Rowell JM (1963) Probable Observation of the Josephson Superconducting Tunneling Effect. PRL10(6):230–232, DOI 10.1103/PhysRevLett.10.230
  * Austermann (2012) Austermann JEo (2012) SPTpol: an instrument for CMB polarization measurements with the South Pole Telescope. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VI, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 8452, p 84521E, DOI 10.1117/12.927286, 1210.4970
  * Bandler et al. (2019) Bandler SR, et al. (2019) Lynx x-ray microcalorimeter. Journal of Astronomical Telescopes, Instruments, and Systems 5:021017, DOI 10.1117/1.JATIS.5.2.021017
  * Barret et al. (2018) Barret D, et al. (2018) The ATHENA X-ray Integral Field Unit (X-IFU). In: SPIE, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 10699, p 106991G, DOI 10.1117/12.2312409
  * Battistelli et al. (2008) Battistelli ES, et al. (2008) Functional description of read-out electronics for time-domain multiplexed bolometers for millimeter and sub-millimeter astronomy. Journal of Low Temperature Physics 151(3):908–914
  * Benford (2002) Benford DJo (2002) First astronomical use of multiplexed transition edge bolometers. In: Porter FS, McCammon D, Galeazzi M, Stahle CK (eds) Low Temperature Detectors, American Institute of Physics Conference Series, vol 605, pp 589–592, DOI 10.1063/1.1457715
  * Bennett et al. (2019) Bennett DA, Mates JA, Bandler SR, et al. (2019) Microwave squid multiplexing for the lynx x-ray microcalorimeter. Journal of Astronomical Telescopes, Instruments, and Systems 5(2):021007
  * Bennett et al. (2012) Bennett DA, et al. (2012) A high resolution gamma-ray spectrometer based on superconducting microcalorimeters. Review of Scientific Instruments 83(9):093113
  * Bennett et al. (2014) Bennett DA, et al. (2014) Integration of tes microcalorimeters with microwave squid multiplexed readout. IEEE Transactions on Applied Superconductivity 25(3):1–5
  * Bergen et al. (2016) Bergen A, et al. (2016) Design and validation of a large-format transition edge sensor array magnetic shielding system for space application. Review of Scientific Instruments 87(10):105109, DOI 10.1063/1.4962157
  * Beyer and Drung (2008) Beyer J, Drung D (2008) A squid multiplexer with superconducting-to-normalconducting switches. Superconductor Science and Technology 21(10):105022
  * Bruijn et al. (2014) Bruijn MP, et al. (2014) Tailoring the High-Q LC Filter Arrays for Readout of Kilo-Pixel TES Arrays in the SPICA-SAFARI Instrument. Journal of Low Temperature Physics 176(3-4):421–425, DOI 10.1007/s10909-013-1003-6
  * Bruijn et al. (2018) Bruijn MP, et al. (2018) LC Filters for FDM Readout of the X-IFU TES Calorimeter Instrument on Athena. Journal of Low Temperature Physics 193(5-6):661–667, DOI 10.1007/s10909-018-1951-y
  * Cash (1979) Cash W (1979) Parameter estimation in astronomy through application of the likelihood ratio. The Astrophysical Journal228:939–947, DOI 10.1086/156922
  * Chervenak et al. (1999) Chervenak JA, et al. (1999) Superconducting multiplexer for arrays of transition edge sensors. Applied Physics Letters 74(26):4043, DOI 10.1063/1.123255
  * Chervenak et al. (1999) Chervenak JA, et al. (1999) Superconducting multiplexer for arrays of transition edge sensors. Applied Physics Letters 74(26):4043–4045
  * Chesca et al. (2004) Chesca B, Kleiner R, Koelle D (2004) SQUID Theory, John Wiley & Sons, Ltd, chap 2, pp 29–92. DOI https://doi.org/10.1002/3527603646.ch2
  * Cohen and Givler (1972) Cohen D, Givler E (1972) Magnetomyography: magnetic fields around the human body produced by skeletal muscles. Applied Physics Letters 21(3):114–116, DOI 10.1063/1.1654294
  * Cui et al. (2020) Cui W, et al. (2020) HUBS: a dedicated hot circumgalactic medium explorer. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 11444, p 114442S, DOI 10.1117/12.2560871
  * Cunningham et al. (2002) Cunningham MF, et al. (2002) High-resolution operation of frequency-multiplexed transition-edge photon sensors. Applied Physics Letters 81(1):159, DOI 10.1063/1.1489486
  * D’Andrea et al. (2017) D’Andrea M, et al. (2017) The Cryogenic AntiCoincidence detector for ATHENA X-IFU: a scientific assessment of the observational capabilities in the hard X-ray band. Experimental Astronomy 44(3):359–370, DOI 10.1007/s10686-017-9543-4
  * D’Andrea et al. (2020) D’Andrea M, et al. (2020) The Demonstration Model of the ATHENA X-IFU Cryogenic AntiCoincidence Detector. Journal of Low Temperature Physics 199(1-2):65–72, DOI 10.1007/s10909-019-02300-9
  * D’Andrea et al. (2021) D’Andrea M, et al. (2021) Single Pixel Performance of a 32 × 32 Ti/Au TES Array With Broadband X-Ray Spectra. IEEE Transactions on Applied Superconductivity 31(5):3065303, DOI 10.1109/TASC.2021.3065303
  * Dawson et al. (2019) Dawson CS, et al. (2019) Two-level switches for advanced time-division multiplexing. IEEE Transactions on Applied Superconductivity 29(5):2500205
  * de Haan et al. (2012) de Haan T, Smecher G, Dobbs M (2012) Improved performance of TES bolometers using digital feedback. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VI, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 8452, p 84520E, DOI 10.1117/12.925658
  * de Haan et al. (2020) de Haan T, et al. (2020) Recent Advances in Frequency-Multiplexed TES Readout: Vastly Reduced Parasitics and an Increase in Multiplexing Factor with Sub-Kelvin SQUIDs. Journal of Low Temperature Physics 199(3-4):754–761, DOI 10.1007/s10909-020-02403-8
  * de Wit et al. (2022) de Wit M, et al. (2022) Performance of the SRON Ti/Au Transition Edge Sensor X-ray Calorimeters. arXiv e-prints arXiv:2208.12556, 2208.12556
  * den Hartog et al. (2009) den Hartog R, et al. (2009) Baseband Feedback for Frequency-Domain-Multiplexed Readout of TES X-ray Detectors. In: Young B, Cabrera B, Miller A (eds) American Institute of Physics Conference Series, vol 1185, pp 261–264, DOI 10.1063/1.3292328
  * den Hartog et al. (2018) den Hartog R, et al. (2018) Performance of a state-of-the-art DAC system for FDM readout. In: den Herder JWA, Nikzad S, Nakazawa K (eds) Space Telescopes and Instrumentation 2018: Ultraviolet to Gamma Ray, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 10699, p 106994Q, DOI 10.1117/12.2312793
  * den Herder et al. (2001) den Herder JW, et al. (2001) The Reflection Grating Spectrometer on board XMM-Newton. Astronomy & Astrophysics 365:L7–L17, DOI 10.1051/0004-6361:20000058
  * Dicker (2006) Dicker SRo (2006) A 90-GHz bolometer array for the Green Bank Telescope. In: Zmuidzinas J, Holland WS, Withington S, Duncan WD (eds) Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 6275, p 62751B, DOI 10.1117/12.672166
  * Dicker (2014) Dicker SRo (2014) MUSTANG2: a large focal plan array for the 100 meter Green Bank Telescope. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9153, p 91530J, DOI 10.1117/12.2056455
  * Dobbs et al. (2008) Dobbs M, Bissonnette E, Spieler H (2008) Digital Frequency Domain Multiplexer for Millimeter-Wavelength Telescopes. IEEE Transactions on Nuclear Science 55(1):21–26, DOI 10.1109/TNS.2007.911601
  * Doriese et al. (2006) Doriese WB, et al. (2006) Progress toward kilopixel arrays: 3.8 ev microcalorimeter resolution in 8-channel squid multiplexer. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 559(2):808–810
  * Doriese et al. (2016) Doriese WB, et al. (2016) Developments in time-division multiplexing of x-ray transition-edge sensors. Journal of low temperature physics 184(1):389–395
  * Doriese et al. (2017) Doriese WB, et al. (2017) A practical superconducting-microcalorimeter x-ray spectrometer for beamline and laboratory science. Review of Scientific Instruments 88(5):053108
  * Doriese et al. (2019) Doriese WB, et al. (2019) Optimization of time-and code-division-multiplexed readout for athena x-ifu. IEEE Transactions on Applied Superconductivity 29(5):2500305
  * Drung et al. (2005) Drung D, et al. (2005) dc squid readout electronics with up to 100 mhz closed-loop bandwidth. IEEE Transactions on Applied Superconductivity 15(2):777–780
  * Drung et al. (2007) Drung D, et al. (2007) Highly Sensitive and Easy-to-Use SQUID Sensors. IEEE Transactions on Applied Superconductivity 17(2):699–704, DOI 10.1109/TASC.2007.897403
  * Duh et al. (1988) Duh KG, et al. (1988) Ultra-low-noise cryogenic high-electron-mobility transistors. IEEE transactions on electron devices 35(3):249–256
  * Durkin et al. (2019) Durkin M, et al. (2019) Demonstration of Athena X-IFU Compatible 40-Row Time-Division-Multiplexed Readout. IEEE Transactions on Applied Superconductivity 29(5):2904472, DOI 10.1109/TASC.2019.2904472
  * Durkin et al. (2020) Durkin M, et al. (2020) A Predictive Control Algorithm for Time-Division-Multiplexed Readout of TES Microcalorimeters. Journal of Low Temperature Physics 199(1-2):275–280, DOI 10.1007/s10909-020-02342-4
  * Durkin et al. (2021) Durkin M, et al. (2021) Mitigation of finite bandwidth effects in time-division-multiplexed SQUID readout of TES arrays. IEEE Transactions on Applied Superconductivity 31(5):1600905
  * Fleischmann et al. (2005) Fleischmann A, Enss C, Seidel G (2005) Metallic Magnetic Calorimeters, Springer, pp 151–216
  * Fowler et al. (2021) Fowler JW, et al. (2021) Absolute energies and emission line shapes of the l x-ray transitions of lanthanide metals. Metrologia 58(1):015016
  * Friedrich (2006) Friedrich S (2006) Cryogenic x-ray detectors for synchrotron science. Journal of synchrotron radiation 13(2):159–171
  * Gandilo et al. (2016) Gandilo NN, et al. (2016) The primordial inflation polarization explorer (piper). In: Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VIII, vol 9914, pp 372–379
  * Gao et al. (2008) Gao J, et al. (2008) Experimental evidence for a surface distribution of two-level systems in superconducting lithographed microwave resonators. Applied Physics Letters 92(15):152505
  * Gonzales et al. (2022) Gonzales M, et al. (2022) Fully differential broadband lna with active impedance matching for squid readout. Journal of Low Temperature Physics p in review
  * Gottardi and Nagayashi (2021) Gottardi L, Nagayashi K (2021) A review of x-ray microcalorimeters based on superconducting transition edge sensors for astrophysics and particle physics. Applied Sciences 11(9), DOI 10.3390/app11093793, URL https://www.mdpi.com/2076-3417/11/9/3793
  * Gottardi et al. (2014a) Gottardi L, et al. (2014a) Josephson effects in an alternating current biased transition edge sensor. Applied Physics Letters 105(16):162605, DOI 10.1063/1.4899065
  * Gottardi et al. (2014b) Gottardi L, et al. (2014b) Weak-Link Phenomena in AC-Biased Transition Edge Sensors. Journal of Low Temperature Physics 176:279–284, DOI 10.1007/s10909-014-1093-9
  * Gottardi et al. (2018) Gottardi L, et al. (2018) Josephson Effects in Frequency-Domain Multiplexed TES Microcalorimeters and Bolometers. Journal of Low Temperature Physics 193(3-4):209–216, DOI 10.1007/s10909-018-2006-0
  * Gottardi et al. (2019) Gottardi L, et al. (2019) A six-degree-of-freedom micro-vibration acoustic isolator for low-temperature radiation detectors based on superconducting transition-edge sensors. Review of Scientific Instruments 90(5):055107, DOI 10.1063/1.5088364
  * Grace (2014) Grace Eo (2014) ACTPol: on-sky performance and characterization. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9153, p 915310, DOI 10.1117/12.2057243
  * Harper et al. (2018) Harper DA, et al. (2018) Hawc+, the far-infrared camera and polarimeter for sofia. Journal of Astronomical Instrumentation 7(04):1840008
  * Hashimoto et al. (2022) Hashimoto T, et al. (2022) Measurements of strong-interaction effects in kaonic-helium isotopes at sub-ev precision with x-ray microcalorimeters. Physical review letters 128(11):112503
  * Hazumi et al. (2020) Hazumi M, et al. (2020) LiteBIRD satellite: JAXA’s new strategic L-class mission for all-sky surveys of cosmic microwave background polarization. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 11443, p 114432F, DOI 10.1117/12.2563050
  * Henderson (2016) Henderson SWo (2016) Advanced ACTPol Cryogenic Detector Arrays and Readout. Journal of Low Temperature Physics 184(3-4):772–779
  * Henning (2015) Henning Jo (2015) SPT-3G: The third generation camera and survey for the South Pole Telescope. In: American Astronomical Society Meeting Abstracts #225, American Astronomical Society Meeting Abstracts, vol 225, p 220.02
  * Hirayama et al. (2013) Hirayama F, et al. (2013) Microwave squid multiplexer for tes readout. IEEE transactions on applied superconductivity 23(3):2500405–2500405
  * Hitomi Collaboration et al. (2016) Hitomi Collaboration, et al. (2016) The quiescent intracluster medium in the core of the Perseus cluster. Nature535(7610):117–121, DOI 10.1038/nature18627
  * Holland et al. (2013) Holland WS, et al. (2013) SCUBA-2: the 10,000 pixel bolometer camera on the James Clerk Maxwell telescope. Monthly Notices of the Royal Astronomical Society 430(4):2513–2533
  * Hölzer et al. (1997) Hölzer G, et al. (1997) K$\alpha$1,2 and K$\beta$1,3 x-ray emission lines of the 3d transition metals. Phys. Rev. A56(6):4554–4568, DOI 10.1103/PhysRevA.56.4554
  * Huber et al. (1997) Huber ME, et al. (1997) Dc squid series arrays with intracoil damping to reduce resonance distortions. Applied Superconductivity 5(7-12):425–429
  * Huber et al. (2001) Huber ME, et al. (2001) Dc squid series array amplifiers with 120 mhz bandwidth (corrected). IEEE Transactions on Applied Superconductivity 11(2):4048–4053
  * Inoue (2016) Inoue Yo (2016) POLARBEAR-2: an instrument for CMB polarization measurements. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VIII, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9914, p 99141I, DOI 10.1117/12.2231961, 1608.03025
  * Irwin (1995) Irwin KD (1995) An application of electrothermal feedback for high resolution cryogenic particle detection. Applied Physics Letters 66(15):1998–2000, DOI 10.1063/1.113674
  * Irwin (2009) Irwin KD (2009) Shannon Limits for Low-Temperature Detector Readout. In: Young B, Cabrera B, Miller A (eds) The Thirteenth International Workshop on Low Temperature Detectors - LTD13, American Institute of Physics Conference Series, vol 1185, pp 229–236, DOI 10.1063/1.3292320
  * Irwin and Hilton (2005) Irwin KD, Hilton GC (2005) Transition-edge sensors, Springer, pp 63–150
  * Irwin and Lehnert (2004) Irwin KD, Lehnert KW (2004) Microwave SQUID multiplexer. Applied Physics Letters 85(11):2107, DOI 10.1063/1.1791733
  * Irwin et al. (2010) Irwin KD, et al. (2010) Code-division multiplexing of superconducting transition-edge sensor arrays. Superconductor Science Technology 23(3):034004, DOI 10.1088/0953-2048/23/3/034004
  * Irwin et al. (2012) Irwin KD, et al. (2012) Advanced code-division multiplexers for superconducting detector arrays. Journal of Low Temperature Physics 167(5):588–594
  * Ishisaki et al. (2018) Ishisaki Y, et al. (2018) Resolve Instrument on X-ray Astronomy Recovery Mission (XARM). Journal of Low Temperature Physics 193(5-6):991–995, DOI 10.1007/s10909-018-1913-4
  * Jackson et al. (2016) Jackson BD, et al. (2016) The focal plane assembly for the Athena X-ray Integral Field Unit instrument. In: den Herder JWA, Takahashi T, Bautz M (eds) Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9905, p 99052I, DOI 10.1117/12.2232544
  * Jaklevic et al. (1964) Jaklevic RC, et al. (1964) Quantum Interference Effects in Josephson Tunneling. PRL12(7):159–160, DOI 10.1103/PhysRevLett.12.159
  * Kaastra and Bleeker (2016) Kaastra JS, Bleeker JAM (2016) Optimal binning of X-ray spectra and response matrix design. Astronomy & Astrophysics 587:A151, DOI 10.1051/0004-6361/201527395, 1601.05309
  * Kelley et al. (2007) Kelley RL, et al. (2007) The Suzaku High Resolution X-Ray Spectrometer. Publications of the Astronomical Society of Japan59:77–112, DOI 10.1093/pasj/59.sp1.S77
  * Kelley et al. (2008) Kelley RL, et al. (2008) Ion-Implanted Silicon X-Ray Calorimeters: Present and Future. Journal of Low Temperature Physics 151(1-2):375–380, DOI 10.1007/s10909-007-9663-8
  * Kempf et al. (2018) Kempf S, et al. (2018) Physics and Applications of Metallic Magnetic Calorimeters. Journal of Low Temperature Physics 193(3-4):365–379, DOI 10.1007/s10909-018-1891-6
  * Kermish (2012) Kermish Zo (2012) The POLARBEAR experiment. In: Holland WS, Zmuidzinas J (eds) Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VI, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 8452, p 84521C, DOI 10.1117/12.926354, 1210.7768
  * Kimura et al. (2008) Kimura S, et al. (2008) Performance Measurement of the 8-Input SQUIDs for TES Frequency Domain Multiplexing. Journal of Low Temperature Physics 151(3-4):946–951, DOI 10.1007/s10909-008-9771-0
  * Kiviranta et al. (2018) Kiviranta M, Grönberg L, van der Kuur J (2018) Two SQUID amplifiers intended to alleviate the summing node inductance problem in multiplexed arrays of Transition Edge Sensors. arXiv e-prints arXiv:1810.09122
  * Kiviranta et al. (2002) Kiviranta M, et al. (2002) SQUID-based readout schemes for microcalorimeter arrays. In: Porter FS, McCammon D, Galeazzi M, Stahle CK (eds) Low Temperature Detectors, American Institute of Physics Conference Series, vol 605, pp 295–300, DOI 10.1063/1.1457649
  * Kiviranta et al. (2004) Kiviranta M, et al. (2004) Design and performance of multiloop and washer SQUIDs intended for sub-kelvin operation. Superconductor Science Technology 17(5):S285–S289, DOI 10.1088/0953-2048/17/5/038
  * Kiviranta et al. (2021) Kiviranta M, et al. (2021) Two-Stage SQUID Amplifier for the Frequency Multiplexed Readout of the X-IFU X-Ray Camera. IEEE Transactions on Applied Superconductivity 31(5):3060356, DOI 10.1109/TASC.2021.3060356
  * Lee et al. (2022) Lee S, et al. (2022) Generic character of charge and spin density waves in superconducting cuprates. Proceedings of the National Academy of Sciences 119(15):e2119429119
  * Lee et al. (2019) Lee SJ, et al. (2019) Soft x-ray spectroscopy with transition-edge sensors at stanford synchrotron radiation lightsource beamline 10-1. Review of Scientific Instruments 90(11):113101, DOI 10.1063/1.5119155
  * Likharev (2022) Likharev KK (2022) Dynamics of Josephson junctions and circuits. Routledge
  * Lucas et al. (2022) Lucas TJ, et al. (2022) Indium bump process for low-temperature detectors and readout. Journal of Low Temperature Physics pp 1–6
  * Macculi et al. (2016) Macculi C, et al. (2016) The Cryogenic AntiCoincidence detector for ATHENA X-IFU: a program overview. In: den Herder JWA, Takahashi T, Bautz M (eds) Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9905, p 99052K, DOI 10.1117/12.2231298
  * Malnou et al. (2022) Malnou M, et al. (2022) Performance of a kinetic inductance traveling-wave parametric amplifier at 4 kelvin: Toward an alternative to semiconductor amplifiers. Physical Review Applied 17(4):044009
  * Mantegazzini et al. (2022) Mantegazzini F, et al. (2022) Metallic magnetic calorimeter arrays for the first phase of the ECHo experiment. Nuclear Instruments and Methods in Physics Research A 1030:166406, DOI 10.1016/j.nima.2022.166406, 2111.09945
  * Markert et al. (1994) Markert TH, et al. (1994) High-Energy Transmission Grating Spectrometer for the Advanced X-ray Astrophysics Facility (AXAF). In: Siegmund OH, Vallerga JV (eds) EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 2280, pp 168–180, DOI 10.1117/12.186812
  * Mates et al. (2008) Mates J, et al. (2008) Demonstration of a multiplexer of dissipationless superconducting quantum interference devices. Applied Physics Letters 92(2):023514
  * Mates et al. (2012) Mates J, et al. (2012) Flux-ramp modulation for squid multiplexing. Journal of Low Temperature Physics 167(5):707–712
  * Mates et al. (2017) Mates J, et al. (2017) Simultaneous readout of 128 x-ray and gamma-ray transition-edge microcalorimeters using microwave squid multiplexing. Applied Physics Letters 111(6):062601
  * Mates et al. (2019) Mates J, et al. (2019) Crosstalk in microwave squid multiplexers. Applied Physics Letters 115(20):202601
  * Mates (2011) Mates JAB (2011) The microwave squid multiplexer. PhD thesis
  * McCammon (2005) McCammon D (2005) Semiconductor Thermistors, Springer, pp 35–62
  * McCammon (2005) McCammon D (2005) Thermal Equilibrium Calorimeters - An Introduction, vol 99, Springer, p 1
  * McCammon et al. (2002) McCammon D, et al. (2002) A High Spectral Resolution Observation of the Soft X-Ray Diffuse Background with Thermal Detectors. The Astrophysical Journal576(1):188–203, DOI 10.1086/341727
  * Mehl (2008) Mehl Jo (2008) TES Bolometer Array for the APEX-SZ Camera. Journal of Low Temperature Physics 151(3-4):697–702, DOI 10.1007/s10909-008-9738-1
  * Miaja-Avila et al. (2016) Miaja-Avila L, et al. (2016) Ultrafast time-resolved hard x-ray emission spectroscopy on a tabletop. Phys Rev X 6:031047
  * Mitsuda et al. (1999) Mitsuda K, et al. (1999) Multi-pixel readout of transition-edge sensors using a multi-input SQUID. Nuclear Instruments and Methods in Physics Research A 436(1-2):252–255, DOI 10.1016/S0168-9002(99)00630-0
  * Mitsuda et al. (2007) Mitsuda K, et al. (2007) The X-Ray Observatory Suzaku. Publications of the Astronomical Society of Japan59:1–7, DOI 10.1093/pasj/59.sp1.S1
  * Mitsuda et al. (2014) Mitsuda K, et al. (2014) Soft x-ray spectrometer (SXS): the high-resolution cryogenic spectrometer onboard ASTRO-H. In: Takahashi T, den Herder JWA, Bautz M (eds) Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 9144, p 91442A, DOI 10.1117/12.2057199
  * Montgomery et al. (2022) Montgomery J, et al. (2022) Performance and characterization of the SPT-3G digital frequency-domain multiplexed readout system using an improved noise and crosstalk model. Journal of Astronomical Telescopes, Instruments, and Systems 8(1):014001, DOI 10.1117/1.JATIS.8.1.014001, 2103.16017
  * Morgan et al. (2016) Morgan KM, et al. (2016) Code-division-multiplexed readout of large arrays of TES microcalorimeters. Applied Physics Letters 109(11):112604, DOI 10.1063/1.4962636
  * Moseley et al. (1984) Moseley SH, Mather JC, McCammon D (1984) Thermal detectors as x-ray spectrometers. Journal of Applied Physics 56(5):1257–1262, DOI 10.1063/1.334129
  * Nagayoshi et al. (2020) Nagayoshi K, et al. (2020) Development of a Ti/Au TES Microcalorimeter Array as a Backup Sensor for the Athena/X-IFU Instrument. Journal of Low Temperature Physics 199(3-4):943–948, DOI 10.1007/s10909-019-02282-8
  * Nakashima et al. (2020) Nakashima Y, et al. (2020) Low-noise microwave SQUID multiplexed readout of 38 x-ray transition-edge sensor microcalorimeters. Applied Physics Letters 117(12):122601, DOI 10.1063/5.0016333
  * Noroozian et al. (2012) Noroozian O, et al. (2012) Crosstalk reduction for superconducting microwave resonator arrays. IEEE Transactions on Microwave Theory and Techniques 60(5):1235–1243
  * Okumura et al. (2021) Okumura T, et al. (2021) Deexcitation dynamics of muonic atoms revealed by high-precision spectroscopy of electronic k x rays. Physical Review Letters 127(5):053001
  * O’Neil et al. (2017) O’Neil GC, et al. (2017) Ultrafast time-resolved x-ray absorption spectroscopy of ferrioxalate photolysis with a laser plasma x-ray source and microcalorimeter array. The Journal of Physical Chemistry Letters 8(5):1099–1104
  * BICEP2Collaborationand others (2014) BICEP2Collaborationand others (2014) Detection of B-Mode Polarization at Degree Angular Scales by BICEP2. PRL112(24):241101, DOI 10.1103/PhysRevLett.112.241101, 1403.3985
  * LiteBIRDCollaborationand others (2022) LiteBIRDCollaborationand others (2022) Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey. arXiv e-prints arXiv:2202.02773, 2202.02773
  * Palosaari et al. (2016) Palosaari MRJ, et al. (2016) Broadband ultrahigh-resolution spectroscopy of particle-induced x rays: extending the limits of nondestructive analysis. Physical Review Applied 6(2):024002
  * Porter et al. (2009) Porter FS, et al. (2009) The Astro-H Soft X-ray Spectrometer (SXS). In: Young B, Cabrera B, Miller A (eds) The Thirteenth International Workshop on Low Temperature Detectors - LTD13, American Institute of Physics Conference Series, vol 1185, pp 91–94, DOI 10.1063/1.3292564
  * Porter et al. (2010) Porter FS, et al. (2010) The detector subsystem for the SXS instrument on the ASTRO-H Observatory. In: Arnaud M, Murray SS, Takahashi T (eds) Space Telescopes and Instrumentation 2010: Ultraviolet to Gamma Ray, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 7732, p 77323J, DOI 10.1117/12.857888
  * Ravera et al. (2014) Ravera L, et al. (2014) The X-ray Integral Field Unit (X-IFU) for Athena. In: Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, SPIE, vol 9144, p 91442L, DOI 10.1117/12.2055884
  * Reintsema et al. (2003) Reintsema CD, et al. (2003) Prototype system for superconducting quantum interference device multiplexing of large-format transition-edge sensor arrays. Review of Scientific Instruments 74(10):4500–4508
  * Reintsema et al. (2009) Reintsema CD, et al. (2009) Electronics for a next-generation squid-based time-domain multiplexing system. In: AIP Conference Proceedings, vol 1185, pp 237–240
  * Roelfsema et al. (2012) Roelfsema P, et al. (2012) The SAFARI imaging spectrometer for the SPICA space observatory. In: Clampin MC, Fazio GG, MacEwen HA, Oschmann J Jacobus M (eds) Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 8442, p 84420R, DOI 10.1117/12.927010
  * Sadleir et al. (2010) Sadleir JE, et al. (2010) Longitudinal Proximity Effects in Superconducting Transition-Edge Sensors. PRL104(4):047003, DOI 10.1103/PhysRevLett.104.047003
  * Sadleir et al. (2011) Sadleir JE, et al. (2011) Proximity effects and nonequilibrium superconductivity in transition-edge sensors. PRB84(18):184502, DOI 10.1103/PhysRevB.84.184502
  * Sakai et al. (2022) Sakai K, et al. (2022) Developments of laboratory-based transition-edge sensor readout electronics using commercial-off-the-shelf modules. Journal of Low Temperature Physics p in review
  * Sato et al. (2020) Sato K, et al. (2020) Super DIOS mission for exploring “dark baryon”. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 11444, p 114445O, DOI 10.1117/12.2561681
  * Schuster et al. (2022) Schuster C, et al. (2022) Flux ramp modulation based hybrid microwave squid multiplexer. Applied Physics Letters 120(16):162601
  * Sehgal (2019) Sehgal No (2019) CMB-HD: An Ultra-Deep, High-Resolution Millimeter-Wave Survey Over Half the Sky. In: Bulletin of the American Astronomical Society, vol 51, p 6, 1906.10134
  * Smith et al. (2012) Smith SJ, et al. (2012) Small Pitch Transition-Edge Sensors with Broadband High Spectral Resolution for Solar Physics. Journal of Low Temperature Physics 167(3-4):168–175, DOI 10.1007/s10909-012-0574-y
  * Smith et al. (2020) Smith SJ, et al. (2020) Toward 100,000-Pixel Microcalorimeter Arrays Using Multi-absorber Transition-Edge Sensors. Journal of Low Temperature Physics 199(1-2):330–338, DOI 10.1007/s10909-020-02362-0
  * Smith et al. (2021) Smith SJ, et al. (2021) Performance of a broad-band, high-resolution, transition-edge sensor spectrometer for x-ray astrophysics. IEEE Transactions on Applied Superconductivity 31(5):2100806
  * Stevenson et al. (2019) Stevenson TR, et al. (2019) Magnetic calorimeter option for the Lynx x-ray microcalorimeter. Journal of Astronomical Telescopes, Instruments, and Systems 5(2):1 – 9, DOI 10.1117/1.JATIS.5.2.021009
  * Swetz et al. (2011) Swetz DS, et al. (2011) Overview of the Atacama Cosmology Telescope: receiver, instrumentation, and telescope systems. The Astrophysical Journal Supplement Series 194(2):41, DOI 10.1088/0067-0049/194/2/41
  * Szypryt et al. (2021a) Szypryt P, Bennett DA, et al. (2021a) Design of a 3000-pixel transition-edge sensor x-ray spectrometer for microcircuit tomography. IEEE Transactions on Applied Superconductivity 31(5):1–5
  * Szypryt et al. (2019) Szypryt P, et al. (2019) A transition-edge sensor-based x-ray spectrometer for the study of highly charged ions at the national institute of standards and technology electron beam ion trap. Review of Scientific Instruments 90(12):123107
  * Szypryt et al. (2021b) Szypryt P, et al. (2021b) Design of a 3000-pixel transition-edge sensor x-ray spectrometer for microcircuit tomography. IEEE Transactions on Applied Superconductivity 31(5):2100405
  * Szypryt et al. (2022) Szypryt P, et al. (2022) A tabletop x-ray tomography instrument for nanometer-scale imaging: demonstration of the 1,000-element transition-edge sensor subarray. IEEE Transactions on Applied Superconductivity (in preparation)
  * Takahashi et al. (2012) Takahashi T, et al. (2012) The ASTRO-H X-ray Observatory. In: Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol 8443, DOI 10.1117/12.926190
  * Takei et al. (2009) Takei Y, et al. (2009) SQUID multiplexing using baseband feedback for space application of transition-edge sensor microcalorimeters. Superconductor Science Technology 22(11):114008, DOI 10.1088/0953-2048/22/11/114008