corresponding author ]SRON Netherlands Institute for Space Research, Niels
Bohrweg 4, 2333 CA Leiden, The Netherlands and also ]Optics Group, Department
of Imaging Physics, Delft University of Technology, Delft, 2628 CJ, The
Netherlands

# Demonstration of MHz frequency domain multiplexing readout of 37 transition
edge sensors for high-resolution X-ray imaging spectrometers

H. Akamatsu<EMAIL_ADDRESS>[  D. Vaccaro  SRON Netherlands Institute for
Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  L. Gottardi
SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, The Netherlands  J. van der Kuur  SRON Netherlands Institute for Space
Research, Landleven 12, 9747 AD Groningen, The Netherlands  C.P. de Vries
SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, The Netherlands  M. Kiviranta  VTT, Tietotie 3, 02150 Espoo, Finland
K. Ravensberg  SRON Netherlands Institute for Space Research, Niels Bohrweg 4,
2333 CA Leiden, The Netherlands  M. D’Andrea  SRON Netherlands Institute for
Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands, and also
Institute for Space Astrophysics and Planetology (INAF/IAPS Roma): Roma,
Lazio, Italy  E. Taralli  SRON Netherlands Institute for Space Research, Niels
Bohrweg 4, 2333 CA Leiden, The Netherlands  M. de Wit  SRON Netherlands
Institute for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands
M.P. Bruijn  SRON Netherlands Institute for Space Research, Niels Bohrweg 4,
2333 CA Leiden, The Netherlands  P. van der Hulst  SRON Netherlands Institute
for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  R. H.
den Hartog  SRON Netherlands Institute for Space Research, Niels Bohrweg 4,
2333 CA Leiden, The Netherlands  B-J. van Leeuwen  SRON Netherlands Institute
for Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  A.J. van
der Linden  SRON Netherlands Institute for Space Research, Niels Bohrweg 4,
2333 CA Leiden, The Netherlands  A.J McCalden  SRON Netherlands Institute for
Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  K. Nagayoshi
SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, The Netherlands  A.C.T. Nieuwenhuizen  SRON Netherlands Institute for
Space Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  M.L. Ridder
SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, The Netherlands  S. Visser  SRON Netherlands Institute for Space
Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  P. van Winden
SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, The Netherlands  J.R. Gao  SRON Netherlands Institute for Space
Research, Niels Bohrweg 4, 2333 CA Leiden, The Netherlands  [  R.W.M.
Hoogeveen  SRON Netherlands Institute for Space Research, Niels Bohrweg 4,
2333 CA Leiden, The Netherlands  B.D. Jackson  SRON Netherlands Institute for
Space Research, Landleven 12, 9747 AD Groningen, The Netherlands  J-W.A. den
Herder  SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333
CA Leiden, The Netherlands

###### Abstract

We report on the development and demonstration of a MHz frequency domain
multiplexing (FDM) technology to read out arrays of cryogenic transition edge
sensor (TES) X-ray microcalorimeters. In our FDM scheme, TESs are AC biased at
different resonant frequencies in the low MHz range through an array of
high-$Q$ LC resonators. The current signals of all TESs are summed at
superconducting quantum interference devices (SQUIDs). We have demonstrated
multiplexing for a readout of 31 pixels using room temperature electronics,
high-$Q$ LC filters and TES arrays developed at SRON, and SQUID arrays from
VTT. We repeated this on a second setup with 37 pixels. The summed X-ray
spectral resolutions $@$ 5.9 keV are $\Delta E_{\rm 31pix~{}MUX}=2.14\pm 0.03$
eV and $\Delta E_{\rm 37pix~{}MUX}=2.23\pm 0.03$ eV. The demonstrated results
are comparable with other multiplexing approaches. There is potential to
further improve the spectral resolution and to increase the number of
multiplexed TESs, and to open up applications for TES X-ray microcalorimeters.

††preprint: AIP/123-QED
---
Figure 1:  A) Photograph of the XFDM setup including [1] Ge thermistor, [2]
VTT SQUIDs, [3] LC filter, [4] superconducting transformer, and [5] TES
arrays. The Helmholtz coils [6] are placed between the TES arrays in the final
assembly. B) A schematic representation showing frequency domain multiplexing
circuit. $R_{\rm Bias}$, $R_{\rm Shunt}$, $C_{\rm Main}$ and $C_{\rm Bias}$
represent the load resistances, shunt resistances, capacitances for the LC
resonator, and capacitances for the capacitive divider, respectively. C) 32 LC
resonators are measured by frequency-sweeping a small amplitude bias tone
across the range of carrier frequencies from 1 to 5 MHz. The difference in the
peak amplitude comes from the frequency resolution of the sweep. The scatter
of the frequency separation is $\Delta f=98.8\pm 2.0$ kHz. The scatter is due
to intrinsic spread due to fabrication accuracy and parasitic inductance in
the setup. All resonators have sufficient Q-factor to read out TES
microcalorimeters.

High-resolution X-ray spectroscopy is one of the most powerful techniques to
understand the chemical composition of materials and the ionization state of
plasma. It can be applied to a wide range of scientific fields varying from
material science to the physics of hot plasma in the UniverseEnss (2005).
Conventional X-ray spectrometers used in these applications, such as silicon
based detectors or grating spectrometers, are sub-optimal: silicon based
detectors have a high photon collecting area but moderate spectral resolution,
while grating spectrometers have a high spectral resolution but a low
collecting area and poor imaging capability. Cryogenic X-ray spectrometers
have the potential of combining a high spectral resolution, imaging capability
and photon collecting area by using a large number of pixel arrays. X-ray
microcalorimeters based on transition edge sensors (TESs) are one of the most
promising cryogenic spectrometer techniques for practical applications. Recent
progress of fabrication technologies makes the production of large format TES
arrays possible. This has created the incentive to develop methods to read out
up to several hundred TESs in one amplifier chain with low power dissipation.

One feasible signal multiplexing technology is frequency domain multiplexing
(FDM)Kiviranta  _et al._ (2002). Together with other technologies such as time
division multiplexingDurkin  _et al._ (2019), code division multiplexingMorgan
_et al._ (2016) and micro-wave multiplexingNakashima  _et al._ (2020), the
development of the FDM technology is progressing dramatically. Under the FDM
scheme, TES pixels are individually connected in series to high-$Q$ LC
resonators with different frequencies and AC biased by a comb of frequencies
in the MHz range. Signals from individual TESs are summed and read out by a
SQUID amplifier. The signals are amplified by a room temperature low noise
amplifier (LNA) and subsequently digitized. For space flight applications, the
FDM readout has advantagesvan der Kuur  _et al._ (2016) of low power
dissipation at cryogenic temperatures, individual adjustment of the TES
operation points, ability to specify requirements of the superconducting
quantum interference devices (SQUIDs), and light-weight shielding. We are
developing FDM readout technology for astronomy in space, such as the X-ray
Integral Field Unit (X-IFUBarret  _et al._ (2018)) instrument on board the
European X-ray astronomical satellite Athena, and for future astronomical
projectsCui  _et al._ (2020); Hazumi  _et al._ (2020); Sato  _et al._ (2020).
For the Athena X-IFU instrument, 34 pixels within a frequency band of 1–5 MHz
and spectral resolution $\Delta E=2.5$ eV $@$ 7 keV are required. In this
letter, we report the recent progress for FDM readout of 37 pixels of TES
X-ray microcalorimeters.

For the FDM readout demonstration reported here, we used two setups, called
XFDM (Figure 1 A) and 40-pixel. The names come simply from the development
work at the time and we concentrate here on the XFDM setup. The setups are
mounted on the mixing chamber of a dry dilution cooler. Room temperature
high-u metal and cryogenic superconducting Nb shields are employed to reduce
the impact of environmental magnetic fields. The residual magnetic field,
typically less than 0.2 uT, is cancelled out by the superconducting Helmholtz
coil. A germanium thermistor (Lake Shore GR50) is used to read out and control
the setup temperature. During the measurement, the setups were kept at 50 mK
and 40 mK for XFDM and 40-pixel setups, respectively. The typical temperature
stability is < 1 µKrms.

Since both setups contain almost identical cryo components except for the
number of resonators, we introduce here only the details of the XFDM setup.
Figure 1 B shows the schematic diagram of the cryo-electronics. The
lithographically-made high-Q LC filters contain parallel plate Nb/a-Si:H/Nb
capacitors and Nb-based gradiometric spiral coilsBruijn  _et al._ (2018). The
inductor value is fixed, and the resonance frequencies are defined by the
capacitor values. In this report, we used 32 LC resonator chips with 2 uH
coils (40 resonators for the 40-pixel setup) and resonances between 1 and 5
MHz with 100 kHz separation (Fig.1 C). A shunt resistance $R_{\rm Shunt}$
(0.75 $\Omega$) is implemented at the 50 mK stage. As shown in Fig.1 middle,
the LC filter contains a capacitive bias voltage divider with a ratio of 1:25,
which makes an effective shunt resistance at the TES side to be 1.1 m$\Omega$.
The measured Q-factor when the TES is in the superconducting state is around
$Q\sim 16\,000\times(f/\rm[MHz])$, limited by the effective shunt resistance
in the circuit. To be able to optimize the bandwidth of different TES designs,
a superconducting transformer per pixel is employed. In this work, a
transforming turns ratio of $n=1:1.125$ with a coupling factor $k=0.94$ is
used. Under this configuration, TES are loaded by the effective inductance of
3.6 µH.

Figure 2:  The combined spectrum of 31 (A) and 37 (B) TESs simultaneous
measurements using FDM readout. The residual at the high energy flank is due
to summing up of different resolution spectra.

The SQUIDs-based amplifier is the most important component to read out a large
number of TESs. We are using two-stage SQUID amplifiers, developed at VTT and
which consist of 6 loops first stage and 184 $\times$ 4 loops 2nd stage
SQUIDsKiviranta, Grönberg, and van der Kuur (2018); Kiviranta  _et al._
(2021). Together with a SRON-developed low noise amplifier (LNA)Wang  _et al._
(2020), the typical readout current noise, referred to the SQUID input, is
$\sim 5~{}\rm pA/\sqrt{\rm Hz}$Gottardi  _et al._ (2015), which is much lower
than the typical TES noise ($\sim$30-50 $\rm pA/\sqrt{\rm Hz}$). To avoid any
undesired behaviour of the SQUIDs due to the coupling between the input coil
of the SQUIDs, and parasitic inductance and capacitance in the LC filtervan
der Kuur  _et al._ (2016), an RC filter with components $R=1~{}\Omega$ and $C$
= 5 pF is implemented just before the input coil of the first stage SQUID.

In our system, the multiplexed data is acquired with 20 M samples/s and the
de-multiplexed data streams are decimated down to 156 k samples/s by using 4
stage filters on an SRON "DEMUX" digital board. The DEMUX board, using AD9726
DACs and a Xilinx XC7V585T Virtex 7 FPGA, generates a modulated comb of AC
bias frequencies for the TESs and the signals for the base-band feedback loop.
To increase the linearity and dynamic range of the SQUIDs, the phase delay due
to the harness between the room temperature electronics and cold electronics
is compensated by the baseband feedbackden Hartog  _et al._ (2009). The TES
signals are demodulated and re-modulated with a controlled phase shift by the
FPGA on the DEMUX board before being fed back to the system. The details of
our digital system can be found in den Hartog et al. 2012 and 2018den Hartog
_et al._ (2009, 2018).

For the FDM demonstration reported here, we employed detector arrays
fabricated at SRONNagayoshi  _et al._ (2020). They consist of a high-aspect
ratio TES bilayer 80$\times$13 µm of Ti (35 nm) and Au (200 nm), located on a
0.5 µm thick SixNy membrane and coupled to a 240$\times$240 µm Au absorber.
The elongated shape towards the direction of current flow was employed to
suppress the frequency dependent non-linearityGottardi  _et al._ (2018) by
increasing TES resistance. The thickness of the absorber is 2.35 µm to ensure
high quantum efficiency (83% at 6 keV). The critical temperature ($T_{c}$) of
the bilayer is tuned to be around 83 mK and the thermal conductance at $T_{c}$
is $G\sim$65 pW/K. The heat capacity is designed to be $C\sim$ 0.85 pJ/K. To
reduce frequency dependent detector behaviour under AC bias, we use high
aspect ratio TESs. This device was extensively investigated under AC bias with
a lower inductance ($L_{\rm eff}$=1 µH) and demonstrated an excellent spectral
resolution of 1.8 eV FWHMde Wit  _et al._ (2020).

To characterise the performance of the TES, we used a 55Fe source located on
the cryogenic platform. The count rate was typically 1 count/s/pixel. Each
pulse was processed by the optimal filtering processSzymkowiak  _et al._
(1993) in the frequency space by using the time average pulse weighted by
noise spectra. Long-term gain drift correction was performed based on TES
baseline current and pulse height information. The energy non-linearity was
corrected by using the zero energy (0 keV), Mn-K${\alpha}$ (5.9 keV) and
Mn-K${\beta}$ (6.5 keV) information. The resultant energy spectrum was fitted
with the Mn-K${\alpha}$ line modelHölzer  _et al._ (1997) convolved with the
detector resolution. To avoid fitting bias, we employed Cash statistics Cash
(1979); Kaastra and Bleeker (2016).

Previous demonstrations with FDM readout of TES spectrometers was mainly
limited by three effectsAkamatsu  _et al._ (2020). These are 1.) degradation
of energy resolution due to carrier leakage from neighbouring resonators, 2.)
the degradation due to inter-modulation distortion from the DACs, and 3.)
frequency dependent non-linearity due to the Josephson junction in TESs under
MHz AC bias. We solved or mitigated these effects using the following
approaches.

Carrier leakage from neighbouring pixels degrades the performance. The origin
of this issue is that when the electrical bandwidth ($R/L$) of the devices is
too large with respect to the frequency spacing, TESs become sensitive to the
AC bias voltage of neighbouring frequencies. Therefore it is different from
the electrical cross-talk due to parasitic and common inductance and
capacitance. In the carrier leakage case, the TES voltage is modulated by
$f_{\rm target}-f_{\rm nei}$, where $f_{\rm target}$ is a bias frequency of
the target pixel and $f_{\rm nei}$ are the neighbouring frequencies. Although
it could be avoided by applying large frequency separation or applying a phase
windowvan der Kuur  _et al._ (2003, 2004), these approaches would not be a
practical solution in a real instrument. Instead, we solved the issue of
excessive electrical bandwidth by increasing the inductance of the LC filter.
However, too large an inductance will lead to an instability due to the
electrical-thermal feedbackIrwin (1995) at the critical inductance. Moreover,
large $L$ can lead to worse performance due to non-linear effects such as a
large excursion in the resistance-temperature space (e.g., Fig. 4 of Kilbourne
et al. 2008Kilbourne  _et al._ (2008)). The critical inductance can be
estimated using the equation (7) in Smith et al. 2016Smith  _et al._ (2016)
showing strong dependency on the detector thermal time constant ($\sim C/G$),
sensitivity to temperature ($\alpha$) and current ($\beta$) dependency.
Previously these optimizations were hampered due to different $\alpha,\beta$
and the rather high $G$ of the available devices. In this report, we used a
factor 5 narrower electrical bandwidth configuration compared to the previous
workAkamatsu  _et al._ (2020). The improvement is realized by the combination
of a slower detector time constant and a different $\alpha-\beta$
relationshipde Wit  _et al._ (2020). Typical detector time constants are rise
time $\tau_{\rm rise}\sim 180$ µs and fall time $\tau_{\rm fall}\sim 1.1$ ms.

Inter-modulation products from non-linearity in the system are an issue for
the detector performance and system dynamic range. In this report we refer to
such products as inter-modulation line noise (IMLN). Since more than 30
carriers are generated by the DAC, IMLN will be an additional noise term when
it falls into the detector thermal response band ($<1\rm kHz$). This IMLN will
degrade the performance and limit the number of multiplexed pixels. This issue
can be avoided by using carriers in a frequency arrangement where the
frequency spacing between all subsequent carriers are on a regular gridden
Hartog  _et al._ (2018) (called grid-frequency). However, this will be a
trade-off with the performance degradation due to an additional (virtual)
shunt resistance resulting from operating off-resonance ($2\times 2\pi L\Delta
f$)Akamatsu  _et al._ (2020). Furthermore, LC filter fabrication accuracy puts
a limit on the attainable resonance frequency distribution. To overcome the
problem, we developed a frequency shift algorithm, which allows us to shift a
resonator digitally without losing the performance. The basic concept and
practical demonstration are given in previous worksvan der Kuur  _et al._
(2018); Akamatsu  _et al._ (2020). The multi-pixel behaviour was characterized
carefully and we concluded that we could operate the system with more than 40
pixels, without the current DAC going into saturationVaccaro  _et al._ (2021);
van der Hulst  _et al._ (2021). In this report we used the grid frequency
1.0-2.0 kHz, which is well above the thermal bandwidth.

Table 1: Summary of the FDM demonstrations Setup | Nres | NMUX | $\textless\Delta E_{\rm Single}>$ | $\Delta E_{\rm MUX}$ | $\Delta E_{\rm deg}$
---|---|---|---|---|---
|  |  | [eV] | [eV] | [eV]
XFDM | 32 | 31 | 1.95 | 2.14$\pm 0.03$ | 0.9
40-pixel | 38 | 37 | 2.05 | 2.23$\pm 0.03$ | 1.0
$\Delta E_{\rm deg}=\Delta E_{\rm MUX}^{2}-\Delta E_{\rm single}^{2}$
In the XFDM setup, 1 resonator is affected by multiple IMLN.
In the 40-pixel setup, 1 resonator is affected by multiple IMLN
and 2 resonators are missing due to fabrication yield.

A frequency dependent non-linear behaviour under MHz AC bias was observed in
previous demonstrationsAkamatsu  _et al._ (2014); Gottardi  _et al._ (2014a,
b). The issue results from the so-called proximity effectSadleir  _et al._
(2010, 2011), where the local order parameter of the bilayer is modified by
the connection to the niobium leads. The resulting structure shows similar
behaviour to a weak superconducting link. Under MHz AC bias, this effect shows
several characteristics such as non-linear response, a Fraunhofer-like
dependence of the critical current on magnetic field and steps in the
superconducting transition. The resulting negative effect on the energy
resolution scales with the resistance of the TES and is higher for low-ohmic
TESs. By comparing devices with different impedance and saturation power, the
power needed to drive TES into the normal state, we confirmed we confirmed our
predictionGottardi  _et al._ (2018) that the Josephson current decreases with
the increase of the superconducting phase difference over the TES,
$\Phi\sim\frac{\sqrt{PR}}{\omega}$, where $P$ is the detector power, $R$ is
the resistance, and $\omega$ is the bias frequency. Accordingly, high-power
and high-resistance devices show the least degradation of the energy
resolution. Since the detector power is defined by the application, we
modified the detector resistance by changing the aspect ratio of the TESs.
Despite the large normal resistance $R_{\rm N}$, we have shown that the
internal thermal fluctuation noise remains small for these devicesGottardi and
Nagayashi (2021). Moreover, a high aspect ratio TES minimizes the impact of
the loss due to eddy currents. The average single pixel performance over 1–5
MHz $\textless\Delta E_{\rm Single}>$ is given in Table 1, and is slightly
worse than one with small inductance measurements ( $\Delta E_{\rm
small~{}\it{L}}\sim$1.8 eV). This is likely due to the large inductance, and
the non-linearity and large signal effects associated with that.. For the
demonstration setups, a 8$\times$8 TES array is implemented for the XFDM and a
32$\times$32 TES array for the 40-pixel.

Figure 3:  Average of the single pixel performance ($\Delta E$) and the
multiplexed performance under FDM (blue circles) with results from other
technologies (stars) such as time-domain multiplexing (TDM) (32 TESs + 8
repeats of the last row, $\Delta E=2.23$ eV)Durkin  _et al._ (2019), code-
division multiplexing (CDM) (30 pixel MUX, $\Delta E=2.77$ eV)Morgan  _et al._
(2016), and microwave SQUID multiplexing (umMUX) (38 pixel MUX, $\Delta
E=3.44$ eV)Nakashima  _et al._ (2020). The orange line indicates the estimated
degradation $0.9$ eV in quadrature. The red oval indicates the expected range
of the multiplexed performance with the high-$R$ devicede Wit  _et al._
(2020).

After the implementation of all mitigating measures discussed above, we
performed a demonstration of the FDM readout. The TESs were operated around
20% of the transition. The AC-bias was set to about 50% of the maximum level,
and the signal feedback to about 36%. The crest factors were about 7 in both
signals. The electrical cross-talk events were excluded by rejecting positive
peaks in the TESs baseline. For both setups, one pixel was switched off
because of exceptionally strong IMLN. The results with 31 pixels and 37 pixels
are summarized in the table 1. The summed spectrum shows the spectral
resolution at 5.9 keV of $\Delta E_{\rm 31pix}$=2.14$\pm 0.03$eV and $\Delta
E_{\rm 37pix}$=2.23$\pm$0.03 eV for the XFDM and the 40-pixel setups,
respectively (Fig.2). Fig.3 shows the compilation of recent demonstrations of
FDM readout and comparison with other multiplexing technologies. The degree of
the degradation due to multiplexing is estimated to be $\Delta E_{deg}$=0.9 eV
in the quadrature (as indicated by the orange line in Fig.3). We attribute the
cause of the present degradation to be a combination of 1.) the residual of
the AC Josephson effect, 2.) sub-optimal thermalization in the array, and 3.)
impact of the IMLN. These issues, in particular on 2) and 3), are not
fundamental limitations and can be mitigated with incremental improvements
such as the relocation of the AMP SQUIDs outside the detector temperature
stage, the further optimising of the readout circuit., and the fabrication of
TESs with even higher resistances. Higher resistance devices in single pixel
mode showed a better energy resolution of $\Delta E$=1.6-1.7 eVde Wit  _et
al._ (2020) (the red dotted line in Fig.3). Testing these devices and
optimizations in multiplexed mode requires different cryogenic components that
are not yet available (e.g., LC filters and transformers). Considering the
well-described scaling relation shown in Fig 3, and the fact that the
bottlenecks are identified, we see room for improvement in a future
demonstration together with a better single-pixel performance device.

In summary: we have shown a demonstration of FDM multiplexed readout of 31
pixels from an 8$\times$8 TES array and 37 pixels from a 32$\times$32 TES
array, both fabricated at SRON. Our current best results are $\Delta E_{\rm
MUX31pixels}=2.14\pm 0.03$ eV, $\Delta E_{\rm MUX37pixels}=2.23\pm 0.03$ eV at
5.9 keV. The performance of these demonstrations is comparable to those
reported by other multiplexing technologies (Fig.3). This FDM readout
technology will provide a means to increase the number of pixels in TES X-ray
spectrometers for various fields from ground-based science to astronomy
(X-rayCui  _et al._ (2020); Sato  _et al._ (2020), and from infrared to cosmic
microwave background measurementsHazumi  _et al._ (2020)).

## Data availability

The data that support the findings of this study are available from the
corresponding or contributing authors upon reasonable request.

###### Acknowledgements.

SRON Netherlands Institute for Space Research is supported financially by NWO,
the Dutch Research Council. This work is funded partly by NWO under the
research programme Athena with project number 184.034.002, and partly by the
European Space Agency (ESA) under ESA CTP Contract No. 4000130346/20/NL/BW/os.

## References

  * Enss (2005) C. Enss,  _Cryogenic Particle Detection_, Vol. 99 (2005).
  * Kiviranta  _et al._ (2002) M. Kiviranta, H. Seppä, J. van der Kuur, and P. de Korte, “SQUID-based readout schemes for microcalorimeter arrays,” in  _Low Temperature Detectors_, American Institute of Physics Conference Series, Vol. 605, edited by F. S. Porter, D. McCammon, M. Galeazzi, and C. K. Stahle (2002) pp. 295–300.
  * Durkin  _et al._ (2019) M. Durkin, J. S. Adams, S. R. Bandler, J. A. Chervenak, S. Chaudhuri, C. S. Dawson, E. V. Denison, W. B. Doriese, S. M. Duff, F. M. Finkbeiner, C. T. FitzGerald, J. W. Fowler, J. D. Gard, G. C. Hilton, K. D. Irwin, Y. I. Joe, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, K. M. Morgan, G. C. O’Neil, C. G. Pappas, F. S. Porter, C. D. Reintsema, D. A. Rudman, K. Sakai, S. J. Smith, R. W. Stevens, D. S. Swetz, P. Szypryt, J. N. Ullom, L. R. Vale, N. A. Wakeham, J. C. Weber, and B. A. Young, “Demonstration of Athena X-IFU Compatible 40-Row Time-Division-Multiplexed Readout,” IEEE Transactions on Applied Superconductivity 29, 2904472 (2019).
  * Morgan  _et al._ (2016) K. M. Morgan, B. K. Alpert, D. A. Bennett, E. V. Denison, W. B. Doriese, J. W. Fowler, J. D. Gard, G. C. Hilton, K. D. Irwin, Y. I. Joe, G. C. O’Neil, C. D. Reintsema, D. R. Schmidt, J. N. Ullom, and D. S. Swetz, “Code-division-multiplexed readout of large arrays of TES microcalorimeters,” Applied Physics Letters 109, 112604 (2016).
  * Nakashima  _et al._ (2020) Y. Nakashima, F. Hirayama, S. Kohjiro, H. Yamamori, S. Nagasawa, A. Sato, S. Yamada, R. Hayakawa, N. Y. Yamasaki, K. Mitsuda, K. Nagayoshi, H. Akamatsu, L. Gottardi, E. Taralli, M. P. Bruijn, M. L. Ridder, J. R. Gao, and J. W. A. den Herder, “Low-noise microwave SQUID multiplexed readout of 38 x-ray transition-edge sensor microcalorimeters,” Applied Physics Letters 117, 122601 (2020).
  * van der Kuur  _et al._ (2016) J. van der Kuur, L. G. Gottardi, H. Akamatsu, B. J. van Leeuwen, R. den Hartog, D. Haas, M. Kiviranta, and B. J. Jackson, “Optimising the multiplex factor of the frequency domain multiplexed readout of the TES-based microcalorimeter imaging array for the X-IFU instrument on the Athena x-ray observatory,” in  _Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, edited by J.-W. A. den Herder, T. Takahashi, and M. Bautz (2016) p. 99055R, arXiv:1611.05268 [astro-ph.IM] .
  * Barret  _et al._ (2018) D. Barret, T. Lam Trong, J.-W. den Herder, L. Piro, M. Cappi, J. Houvelin, R. Kelley, J. M. Mas-Hesse, K. Mitsuda, S. Paltani, G. Rauw, A. Rozanska, J. Wilms, S. Bandler, M. Barbera, X. Barcons, E. Bozzo, M. T. Ceballos, I. Charles, E. Costantini, A. Decourchelle, R. den Hartog, L. Duband, J.-M. Duval, F. Fiore, F. Gatti, A. Goldwurm, B. Jackson, P. Jonker, C. Kilbourne, C. Macculi, M. Mendez, S. Molendi, P. Orleanski, F. Pajot, E. Pointecouteau, F. Porter, G. W. Pratt, D. Prêle, L. Ravera, K. Sato, J. Schaye, K. Shinozaki, T. Thibert, L. Valenziano, V. Valette, J. Vink, N. Webb, M. Wise, N. Yamasaki, F. Douchin, J.-M. Mesnager, B. Pontet, A. Pradines, G. Branduardi-Raymont, E. Bulbul, M. Dadina, S. Ettori, A. Finoguenov, Y. Fukazawa, A. Janiuk, J. Kaastra, P. Mazzotta, J. Miller, G. Miniutti, Y. Naze, F. Nicastro, S. Scioritino, A. Simonescu, J. M. Torrejon, B. Frezouls, H. Geoffray, P. Peille, C. Aicardi, J. André, C. Daniel, A. Clénet, C. Etcheverry, E. Gloaguen, G. Hervet, A. Jolly, A. Ledot, I. Paillet, R. Schmisser, B. Vella, J.-C. Damery, K. Boyce, M. Dipirro, S. Lotti, D. Schwander, S. Smith, B.-J. Van Leeuwen, H. van Weers, N. Clerc, B. Cobo, T. Dauser, C. Kirsch, E. Cucchetti, M. Eckart, P. Ferrando, and L. Natalucci, “The ATHENA X-ray Integral Field Unit (X-IFU),” in  _SPIE_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10699 (2018) p. 106991G, arXiv:1807.06092 [astro-ph.IM] .
  * Cui  _et al._ (2020) W. Cui, J. N. Bregman, M. P. Bruijn, L. B. Chen, Y. Chen, C. Cui, T. T. Fang, B. Gao, H. Gao, J. R. Gao, L. Gottardi, K. X. Gu, F. L. Guo, J. Guo, C. L. He, P. F. He, J. W. den Herder, Q. S. Huang, F. J. Li, J. T. Li, J. J. Li, L. Y. Li, T. P. Li, W. B. Li, J. T. Liang, Y. J. Liang, G. Y. Liang, Y. J. Liu, Z. Liu, Z. Y. Liu, F. Jaeckel, L. Ji, W. Ji, H. Jin, X. Kang, Y. X. Ma, D. McCammon, H. J. Mo, K. Nagayoshi, K. Nelms, R. Qi, J. Quan, M. L. Ridder, Z. X. Shen, A. Simionescu, E. Taralli, Q. D. Wang, G. L. Wang, J. J. Wang, K. Wang, L. Wang, S. F. Wang, S. J. Wang, T. G. Wang, W. Wang, X. Q. Wang, Y. L. Wang, Y. R. Wang, Z. Wang, Z. S. Wang, N. Y. Wen, M. de Wit, S. F. Wu, D. Xu, D. D. Xu, H. G. Xu, X. J. Xu, R. X. Xu, Y. Q. Xue, S. Z. Yi, J. Yu, L. W. Yang, F. Yuan, S. Zhang, W. Zhang, Z. Zhang, Q. Zhong, Y. Zhou, and W. X. Zhu, “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 (2020) p. 114442S, arXiv:2101.05587 [astro-ph.IM] .
  * Hazumi  _et al._ (2020) M. Hazumi, P. A. R. Ade, A. Adler, E. Allys, K. Arnold, D. Auguste, J. Aumont, R. Aurlien, J. Austermann, C. Baccigalupi, A. J. Banday, R. Banjeri, R. B. Barreiro, S. Basak, J. Beall, D. Beck, S. Beckman, J. Bermejo, P. de Bernardis, M. Bersanelli, J. Bonis, J. Borrill, F. Boulanger, S. Bounissou, M. Brilenkov, M. Brown, M. Bucher, E. Calabrese, P. Campeti, A. Carones, F. J. Casas, A. Challinor, V. Chan, K. Cheung, Y. Chinone, J. F. Cliche, L. Colombo, F. Columbro, J. Cubas, A. Cukierman, D. Curtis, G. D’Alessandro, N. Dachlythra, M. De Petris, C. Dickinson, P. Diego-Palazuelos, M. Dobbs, T. Dotani, L. Duband, S. Duff, J. M. Duval, K. Ebisawa, T. Elleflot, H. K. Eriksen, J. Errard, T. Essinger-Hileman, F. Finelli, R. Flauger, C. Franceschet, U. Fuskeland, M. Galloway, K. Ganga, J. R. Gao, R. Genova-Santos, M. Gerbino, M. Gervasi, T. Ghigna, E. Gjerløw, M. L. Gradziel, J. Grain, F. Grupp, A. Gruppuso, J. E. Gudmundsson, T. de Haan, N. W. Halverson, P. Hargrave, T. Hasebe, M. Hasegawa, M. Hattori, S. Henrot-Versillé, D. Herman, D. Herranz, C. A. Hill, G. Hilton, Y. Hirota, E. Hivon, R. A. Hlozek, Y. Hoshino, E. de la Hoz, J. Hubmayr, K. Ichiki, T. Iida, H. Imada, K. Ishimura, H. Ishino, G. Jaehnig, T. Kaga, S. Kashima, N. Katayama, A. Kato, T. Kawasaki, R. Keskitalo, T. Kisner, Y. Kobayashi, N. Kogiso, A. Kogut, K. Kohri, E. Komatsu, K. Komatsu, K. Konishi, N. Krachmalnicoff, I. Kreykenbohm, C. L. Kuo, A. Kushino, L. Lamagna, J. V. Lanen, M. Lattanzi, A. T. Lee, C. Leloup, F. Levrier, E. Linder, T. Louis, G. Luzzi, T. Maciaszek, B. Maffei, D. Maino, M. Maki, S. Mandelli, E. Martinez-Gonzalez, S. Masi, T. Matsumura, A. Mennella, M. Migliaccio, Y. Minami, K. Mitsuda, J. Montgomery, L. Montier, G. Morgante, B. Mot, Y. Murata, J. A. Murphy, M. Nagai, Y. Nagano, T. Nagasaki, R. Nagata, S. Nakamura, T. Namikawa, P. Natoli, S. Nerval, T. Nishibori, H. Nishino, F. Noviello, C. O’Sullivan, H. Ogawa, H. Ogawa, S. Oguri, H. Ohsaki, I. S. Ohta, N. Okada, N. Okada, L. Pagano, A. Paiella, D. Paoletti, G. Patanchon, J. Peloton, F. Piacentini, G. Pisano, G. Polenta, D. Poletti, T. Prouvé, G. Puglisi, D. Rambaud, C. Raum, S. Realini, M. Reinecke, M. Remazeilles, A. Ritacco, G. Roudil, J. A. Rubino-Martin, M. Russell, H. Sakurai, Y. Sakurai, M. Sandri, M. Sasaki, G. Savini, D. Scott, J. Seibert, Y. Sekimoto, B. Sherwin, K. Shinozaki, M. Shiraishi, P. Shirron, G. Signorelli, G. Smecher, S. Stever, R. Stompor, H. Sugai, S. Sugiyama, A. Suzuki, J. Suzuki, T. L. Svalheim, E. Switzer, R. Takaku, H. Takakura, S. Takakura, Y. Takase, Y. Takeda, A. Tartari, E. Taylor, Y. Terao, H. Thommesen, K. L. Thompson, B. Thorne, T. Toda, M. Tomasi, M. Tominaga, N. Trappe, M. Tristram, M. Tsuji, M. Tsujimoto, C. Tucker, J. Ullom, G. Vermeulen, P. Vielva, F. Villa, M. Vissers, N. Vittorio, I. Wehus, J. Weller, B. Westbrook, J. Wilms, B. Winter, E. J. Wollack, N. Y. Yamasaki, T. Yoshida, J. Yumoto, M. Zannoni, and A. Zonca, “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 (2020) p. 114432F, arXiv:2101.12449 [astro-ph.IM] .
  * Sato  _et al._ (2020) K. Sato, T. Ohashi, Y. Ishisaki, Y. Ezoe, S. Yamada, N. Y. Yamasaki, K. Mitsuda, M. Ishida, Y. Maeda, Y. Nakashima, I. Mitsuishi, Y. Tawara, R. Fujimoto, T. G. Tsuru, N. Ota, K. Osato, S. Nakashima, Y. Fujita, D. Nagai, K. Yoshikawa, N. Kawai, K. Matsushita, Y. Ichinohe, and Y. Uchida, “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 (2020) p. 114445O.
  * Bruijn  _et al._ (2018) M. P. Bruijn, A. J. van der Linden, L. Ferrari, L. Gottardi, J. van der Kuur, R. H. den Hartog, H. Akamatsu, and B. D. Jackson, “LC Filters for FDM Readout of the X-IFU TES Calorimeter Instrument on Athena,” Journal of Low Temperature Physics 193, 661–667 (2018).
  * Kiviranta, Grönberg, and van der Kuur (2018) M. Kiviranta, L. Grönberg, and J. van der Kuur, “Two SQUID amplifiers intended to alleviate the summing node inductance problem in multiplexed arrays of Transition Edge Sensors,” arXiv e-prints , arXiv:1810.09122 (2018), arXiv:1810.09122 [astro-ph.IM] .
  * Kiviranta  _et al._ (2021) M. Kiviranta, L. Gronberg, T. Puranen, J. van der Kuur, N. Beev, J. Salonen, D. Hazra, and S. Korpela, “Two-Stage SQUID Amplifier for the Frequency Multiplexed Readout of the X-IFU X-Ray Camera,” IEEE Transactions on Applied Superconductivity 31, 3060356 (2021).
  * Wang  _et al._ (2020) Q. Wang, M. D. Audley, P. Khosropanah, J. van der Kuur, G. de Lange, A. Aminaei, D. Boersma, F. van der Tak, and J.-R. Gao, “Noise Measurements of a Low-Noise Amplifier in the FDM Readout System for SAFARI,” Journal of Low Temperature Physics 199, 817–823 (2020).
  * Gottardi  _et al._ (2015) L. Gottardi, M. Kiviranta, J. van der Kuur, H. Akamatsu, M. P. Bruijn, and R. den Hartog, “Nearly Quantum Limited Two-Stage SQUID Amplifiers for the Frequency Domain Multiplexing of TES Based X-ray and Infrared Detectors,” IEEE Transactions on Applied Superconductivity 25, 2369234 (2015).
  * den Hartog  _et al._ (2009) R. den Hartog, D. Boersma, M. Bruijn, B. Dirks, L. Gottardi, H. Hoevers, R. Hou, M. Kiviranta, P. de Korte, J. van der Kuur, B. J. van Leeuwen, A. Nieuwenhuizen, and M. Popescu, “Baseband Feedback for Frequency-Domain-Multiplexed Readout of TES X-ray Detectors,” in  _American Institute of Physics Conference Series_, Vol. 1185, edited by B. Young, B. Cabrera, and A. Miller (2009) pp. 261–264.
  * den Hartog  _et al._ (2018) R. den Hartog, B. J. van Leeuwen, P. Peille, J. van der Kuur, L. Ravera, D. van Loon, B. Jackson, and J. W. den Herder, “Performance of a state-of-the-art DAC system for FDM readout,” in  _Space Telescopes and Instrumentation 2018: Ultraviolet to Gamma Ray_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10699, edited by J.-W. A. den Herder, S. Nikzad, and K. Nakazawa (2018) p. 106994Q.
  * Nagayoshi  _et al._ (2020) K. Nagayoshi, M. L. Ridder, M. P. Bruijn, L. Gottardi, E. Taralli, P. Khosropanah, H. Akamatsu, S. Visser, and J. R. Gao, “Development of a Ti/Au TES Microcalorimeter Array as a Backup Sensor for the Athena/X-IFU Instrument,” Journal of Low Temperature Physics 199, 943–948 (2020).
  * Gottardi  _et al._ (2018) L. Gottardi, S. J. Smith, A. Kozorezov, H. Akamatsu, J. van der Kuur, S. R. Bandler, M. P. Bruijn, J. A. Chervenak, J. R. Gao, R. H. den Hartog, B. D. Jackson, P. Khosropanah, A. Miniussi, K. Nagayoshi, M. Ridder, J. Sadleir, K. Sakai, and N. Wakeham, “Josephson Effects in Frequency-Domain Multiplexed TES Microcalorimeters and Bolometers,” Journal of Low Temperature Physics 193, 209–216 (2018).
  * de Wit  _et al._ (2020) M. de Wit, L. Gottardi, E. Taralli, K. Nagayoshi, M. L. Ridder, H. Akamatsu, M. P. Bruijn, M. D’Andrea, J. van der Kuur, K. Ravensberg, D. Vaccaro, S. Visser, J. R. Gao, and J. W. A. den Herder, “High aspect ratio transition edge sensors for x-ray spectrometry,” Journal of Applied Physics 128, 224501 (2020).
  * Szymkowiak  _et al._ (1993) A. E. Szymkowiak, R. L. Kelley, S. H. Moseley, and C. K. Stahle, “Signal processing for microcalorimeters,” Journal of Low Temperature Physics 93, 281–285 (1993).
  * Hölzer  _et al._ (1997) G. Hölzer, M. Fritsch, M. Deutsch, J. Härtwig, and E. Förster, “K$\alpha$1,2 and K$\beta$1,3 x-ray emission lines of the 3d transition metals,” Phys. Rev. A 56, 4554–4568 (1997).
  * Cash (1979) W. Cash, “Parameter estimation in astronomy through application of the likelihood ratio,” Astrophys. J. 228, 939–947 (1979).
  * Kaastra and Bleeker (2016) J. S. Kaastra and J. A. M. Bleeker, “Optimal binning of X-ray spectra and response matrix design,” Astronomy & Astrophysics  587, A151 (2016), arXiv:1601.05309 [astro-ph.IM] .
  * Akamatsu  _et al._ (2020) H. Akamatsu, L. Gottardi, J. van der Kuur, C. P. de Vries, M. P. Bruijn, J. A. Chervenak, M. Kiviranta, A. J. van den Linden, B. D. Jackson, A. Miniussi, K. Ravensberg, K. Sakai, S. J. Smith, and N. Wakeham, “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, 737–744 (2020), arXiv:2003.11899 [astro-ph.IM] .
  * van der Kuur  _et al._ (2003) J. van der Kuur, P. A. J. de Korte, H. F. C. Hoevers, W. M. B. Tiest, N. H. R. Baars, M. L. Ridder, E. Krouwer, M. P. Bruijn, M. Kiviranta, and H. Seppa, “AC biased TES-based X-ray microcalorimeter with an energy resolution of 6.3 eV at 5.89 keV,” IEEE Transactions on Applied Superconductivity 13, 638–642 (2003).
  * van der Kuur  _et al._ (2004) J. van der Kuur, P. A. J. de Korte, P. de Groene, N. H. R. Baars, M. P. Lubbers, and M. Kiviranta, “Implementation of frequency domain multiplexing in imaging arrays of microcalorimeters,” Nuclear Instruments and Methods in Physics Research A 520, 551–554 (2004).
  * Irwin (1995) K. D. Irwin, “An application of electrothermal feedback for high resolution cryogenic particle detection,” Applied Physics Letters 66, 1998–2000 (1995).
  * Kilbourne  _et al._ (2008) C. A. Kilbourne, W. B. Doriese, S. R. Bandler, R. P. Brekosky, A.-D. Brown, J. A. Chervenak, M. E. Eckart, F. M. Finkbeiner, G. C. Hilton, K. D. Irwin, N. Iyomoto, R. L. Kelley, F. S. Porter, C. D. Reintsema, S. J. Smith, and J. N. Ullom, “Multiplexed readout of uniform arrays of TES x-ray microcalorimeters suitable for Constellation-X,” in  _Space Telescopes and Instrumentation 2008: Ultraviolet to Gamma Ray_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 7011, edited by M. J. L. Turner and K. A. Flanagan (2008) p. 701104.
  * Smith  _et al._ (2016) S. J. Smith, J. S. Adams, S. R. Bandler, G. L. Betancourt-Martinez, J. A. Chervenak, M. P. Chiao, M. E. Eckart, F. M. Finkbeiner, R. L. Kelley, C. A. Kilbourne, A. R. Miniussi, F. S. Porter, J. E. Sadleir, K. Sakai, N. A. Wakeham, E. J. Wassell, W. Yoon, D. A. Bennett, W. B. Doriese, J. W. Fowler, G. C. Hilton, K. M. Morgan, C. G. Pappas, C. N. Reintsema, D. S. Swetz, J. N. Ullom, K. D. Irwin, H. Akamatsu, L. Gottardi, R. den Hartog, B. D. Jackson, J. van der Kuur, D. Barret, and P. Peille, “Transition-edge sensor pixel parameter design of the microcalorimeter array for the x-ray integral field unit on Athena,” in  _Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray_, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, edited by J.-W. A. den Herder, T. Takahashi, and M. Bautz (2016) p. 99052H.
  * van der Kuur  _et al._ (2018) J. van der Kuur, L. Gottardi, H. Akamatsu, A. C. T. Nieuwenhuizen, R. den Hartog, and B. D. Jackson, “Active Tuning of the Resonance Frequencies of LC Bandpass Filters for Frequency Domain Multiplexed Readout of TES Detector Arrays,” Journal of Low Temperature Physics 193, 626–632 (2018).
  * Vaccaro  _et al._ (2021) D. Vaccaro, H. Akamatsu, J. van der Kuur, P. van der Hulst, A. C. T. Nieuwenhuizen, P. van Winden, L. Gottardi, R. den Hartog, M. P. Bruijn, M. D’Andrea, J. R. Gao, J. W. A. den Herder, R. W. M. Hoogeveen, B. Jackson, A. J. van der Linden, K. Nagayoshi, K. Ravensberg, M. L. Ridder, E. Taralli, and M. de Wit, “Frequency Shift Algorithm: Application to a Frequency-Domain Multiplexing Readout of X-ray Transition-Edge Sensor Microcalorimeters,” arXiv e-prints , arXiv:2102.06092 (2021), arXiv:2102.06092 [astro-ph.IM] .
  * van der Hulst  _et al._ (2021) P. van der Hulst, J. van der Kuur, A. Nieuwenhuizen, D. Vaccaro, H. Akamatsu, P. van Winden, B.-J. van Leeuwen, and J.-W. den Herder, “Frequency shift algorithm: Design of a baseband phase locked loop for frequency-domain multiplexing readout of x-ray transition-edge sensor microcalorimeters,” Review of Scientific Instruments 92, 073101 (2021), arXiv:2107.02754 [physics.ins-det] .
  * Akamatsu  _et al._ (2014) H. Akamatsu, L. Gottardi, J. Adams, S. Bandler, M. Bruijn, J. Chervenak, M. Eckart, F. Finkbeiner, R. den Hartog, H. Hoevers, R. Kelley, C. Kilbourne, J. van der Kuur, A. J. van den Linden, F. Porter, J. Sadleir, S. Smith, and M. Kiviranta, “Performance of TES X-ray Microcalorimeters with AC Bias Read-Out at MHz Frequencies,” Journal of Low Temperature Physics 176, 591–596 (2014).
  * Gottardi  _et al._ (2014a) L. Gottardi, A. Kozorezov, H. Akamatsu, J. van der Kuur, M. P. Bruijn, R. H. den Hartog, R. Hijmering, P. Khosropanah, C. Lambert, A. J. van der Linden, M. L. Ridder, T. Suzuki, and J. R. Gao, “Josephson effects in an alternating current biased transition edge sensor,” Applied Physics Letters 105, 162605 (2014a), arXiv:1604.00549 [physics.ins-det] .
  * Gottardi  _et al._ (2014b) L. Gottardi, H. Akamatsu, M. Bruijn, J.-R. Gao, R. den Hartog, R. Hijmering, H. Hoevers, P. Khosropanah, A. Kozorezov, J. van der Kuur, A. van der Linden, and M. Ridder, “Weak-Link Phenomena in AC-Biased Transition Edge Sensors,” Journal of Low Temperature Physics 176, 279–284 (2014b), arXiv:1604.00680 [cond-mat.supr-con] .
  * Sadleir  _et al._ (2010) J. E. Sadleir, S. J. Smith, S. R. Bandler, J. A. Chervenak, and J. R. Clem, “Longitudinal Proximity Effects in Superconducting Transition-Edge Sensors,” Phys. Rev. Lett. 104, 047003 (2010), arXiv:0910.2451 [cond-mat.supr-con] .
  * Sadleir  _et al._ (2011) J. E. Sadleir, S. J. Smith, I. K. Robinson, F. M. Finkbeiner, J. A. Chervenak, S. R. Bandler, M. E. Eckart, and C. A. Kilbourne, “Proximity effects and nonequilibrium superconductivity in transition-edge sensors,” Phys. Rev. B 84, 184502 (2011), arXiv:1108.4632 [cond-mat.supr-con] .
  * Gottardi and Nagayashi (2021) L. Gottardi and K. Nagayashi, “A review of x-ray microcalorimeters based on superconducting transition edge sensors for astrophysics and particle physics,” Applied Sciences 11 (2021), 10.3390/app11093793.