# Superconducting microstrip losses at microwave and sub-mm wavelengths

S. Hähnle<EMAIL_ADDRESS>SRON - Netherlands Institute for Space Research,
Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  Department of
Microelectronics, Faculty of Electrical Engineering, Mathematics and Computer
Science (EEMCS), Delft University of Technology, Mekelweg 4, 2628 CD Delft,
The Netherlands  K. Kouwenhoven  SRON - Netherlands Institute for Space
Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  Department of
Microelectronics, Faculty of Electrical Engineering, Mathematics and Computer
Science (EEMCS), Delft University of Technology, Mekelweg 4, 2628 CD Delft,
The Netherlands  B. Buijtendorp  Department of Microelectronics, Faculty of
Electrical Engineering, Mathematics and Computer Science (EEMCS), Delft
University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands  A. Endo
Department of Microelectronics, Faculty of Electrical Engineering, Mathematics
and Computer Science (EEMCS), Delft University of Technology, Mekelweg 4, 2628
CD Delft, The Netherlands  Kavli Institute of NanoScience, Faculty of Applied
Sciences, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The
Netherlands.  K. Karatsu  SRON - Netherlands Institute for Space Research,
Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands  Department of
Microelectronics, Faculty of Electrical Engineering, Mathematics and Computer
Science (EEMCS), Delft University of Technology, Mekelweg 4, 2628 CD Delft,
The Netherlands  D. J. Thoen  Department of Microelectronics, Faculty of
Electrical Engineering, Mathematics and Computer Science (EEMCS), Delft
University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands  V.
Murugesan  SRON - Netherlands Institute for Space Research, Sorbonnelaan 2,
3584 CA Utrecht, The Netherlands  J. J. A. Baselmans  SRON - Netherlands
Institute for Space Research, Sorbonnelaan 2, 3584 CA Utrecht, The Netherlands
Department of Microelectronics, Faculty of Electrical Engineering, Mathematics
and Computer Science (EEMCS), Delft University of Technology, Mekelweg 4, 2628
CD Delft, The Netherlands

###### Abstract

We present a lab-on-chip experiment to accurately measure losses of
superconducting microstrip lines at microwave and sub-mm wavelengths. The
microstrips are fabricated from NbTiN, which is deposited using reactive
magnetron sputtering, and amorphous silicon which is deposited using plasma-
enhanced chemical vapor deposition (PECVD). Sub-mm wave losses are measured
using on-chip Fabry-Pérot resonators (FPR) operating around
$350\text{\,}\mathrm{GHz}$. Microwave losses are measured using shunted half-
wave resonators with an identical geometry and fabricated on the same chip. We
measure a loss tangent of the amorphous silicon at single-photon energies of
$\tan\delta=3.7\pm 0.5\times 10^{-5}$ at $\sim$6\text{\,}\mathrm{GHz}$$ and
$\tan\delta=2.1\pm 0.1\times 10^{-4}$ at $350\text{\,}\mathrm{GHz}$. These
results represent very low losses for deposited dielectrics, but the sub-mm
wave losses are significantly higher than the microwave losses, which cannot
be understood using the standard two-level system loss model.

##  I Introduction

Low-loss transmission lines are a fundamental requirement for integrated
superconducting devices operating at both microwave and sub-mm wavelengths. At
microwave frequencies, the primary driver for low-loss transmission lines is
the development of qubits[1, 2] and microwave kinetic inductance detectors
(MKIDs)[3, 4] based on high quality factor resonators[5, 6], as well as
parametric amplifiers based on very long transmission lines
$>$100\text{\,}\lambda$$ [7]. Additionally, high-impedance transmission lines
can be used to further integrate microwave electronics onto the device chip
[8, 9]. All these devices are predominantly realized using coplanar waveguide
(CPW) technology, achieving losses corresponding to $Q_{i}>10^{6}$[5, 6].
However, the planar nature of CPWs leads to large and complicated designs
which can be difficult to scale. Multi-layer structures such as
microstrips[10] and parallel-plate capacitors[11] are preferable in order to
create smaller devices and easily obtain high-impedance transmission lines,
but these structures suffer from increased losses in the additional dielectric
layer.

At sub-mm wave frequencies, astronomical applications rely increasingly on
integrated devices, such as multi-color/multi-polarization pixels [12], phased
array antennas [13] and on-chip filterbanks [14, 15, 16]. These usually use
microstrips, as common mode excitation and radiation loss become serious
issues at higher frequencies [17, 18]. However, these devices are usually
based on Nb/SiO2 technology, which has a $690\text{\,}\mathrm{GHz}$ cutoff due
to the critical temperature of Nb and relatively high losses due to the SiO2.

The losses of microstrips at these frequencies and sub-Kelvin temperatures are
generally attributed to the existence and excitation of two-level tunneling
systems (TLS) in the bulk of the amorphous dielectric. While the macroscopic
behaviour of TLS is relatively well understood, their microscopic origin is
still for the most part unclear. Due to the lack of a microscopic
understanding, development and investigation of low-loss dielectrics relies
heavily on iterative cycles of deposition and measurement. These loss
measurements are usually only carried out at microwave frequencies, under the
assumption that material properties are comparable at sub-mm frequencies based
on TLS theory [19]. As a result, limited data is available at
$f>$100\text{\,}\mathrm{GHz}$$ and $T<$1\text{\,}\mathrm{K}$$, where both
signal generation and detection become increasingly challenging.

Measurements comparing microwave and sub-mm wave loss were performed by Chang
et al.[20] and Gao et al.[21] on Nb/SiO2/Nb microstrips at
$220\text{\,}\mathrm{GHz}$ and $110\text{\,}\mathrm{GHz}$ respectively, and by
Endo et al. [22] on NbTiN/SiN/NbTiN microstrips at $650\text{\,}\mathrm{GHz}$.
These experiments combined quasi-optical techniques with different on-chip
test devices exploiting either path length differences (Chang et al.) or
resonant structures (Gao et al. and Endo et al.). However, these approaches
suffer from large intrinsic uncertainties and are not sufficiently precise to
study materials with a lower loss tangent. In the case of resonant structures,
this can be mitigated by using long resonators at higher mode numbers, where
coupler effects are suppressed.

In this paper, we present loss measurements of superconducting
NbTiN/a-Si/NbTiN microstrips at microwave and sub-mm wavelengths and at sub-
Kelvin temperatures. We show TLS-like behaviour at microwaves, but find a
significant increase in loss at sub-mm wavelengths which is inconsistent with
the TLS standard model.

##  II Device Design

Figure 1: a) Chip schematic. NbTiN CPW structures are given in light blue, the
Al/NbTiN CPW of the hybrid MKIDs is given in white, a-SI is shown in red and
the NbTiN microstrip lines are shown in dark blue. One of the CPW microwave
resonators (µWR5) is also shown in dark blue to highlight the use of the upper
NbTiN layer for the CPW center line. b) FPR coupler schematic with two cross
sections A-A’ and B-B’. The schematic is shown top-down with semi-transparent
layers, in order to clearly show the layout of the bottom NbTiN layer which is
fully covered by a-Si. c) Optical microscope image of FP2. d) Angled scanning
electron microscope (SEM) image of the FPR coupler. e) High-resolution SEM
image of the microstrip open end. Overetch into the a-Si layer is visible,
with the dotted white line indicating the border between NbTiN and a-Si. f)
Picture of the device mounted in the cryostat.

We have designed a chip which combines shunted microwave resonators (µWR) and
in-line sub-mm wave Fabry-Pérot resonators (FPR). The µWR are coupled to a co-
planar waveguide (CPW) transmission line (henceforth called readout line)
which runs over the entire chip, terminating in bond pads connecting the chip
to a sample holder via Al bond wires. The loss as a function of photon number
can be obtained by measuring the readout line transmission around the
resonance frequency of the shunted resonator and retrieving the loaded quality
factor $Q_{L}$ and minimum transmission $S_{21,min}$ of the lorentzian dip.
The internal loss factor $Q_{i}$ and coupling strength $Q_{c}$ then follow
from the dip depth $S_{21,min}=Q_{L}/Q_{i}$ and the loaded quality factor,
which can be expressed as

$Q_{L}(n)=\frac{nQ_{c,1}Q_{i}}{nQ_{c,1}+Q_{i}}$ (1)

where $n$ is the mode number of the resonance peak and $Q_{c,1}$ is the
coupling strength at $n=1$, with $Q_{c}=nQ_{c,1}$ (see supplementary
material).

For the sub-mm wave case, we excite the FPRs using a quasi-optical scheme,
coupling radiation from a coherent source via an antenna and CPW line to the
on-chip FPR. The relative power transmitted is measured as a function of
source frequency using a microwave kinetic inductance detector (MKID) coupled
to the far end of the FPR and read out using the same readout line as the
µWRs. This approach was previously used to measure CPW FPRs[17].

The design of the chip is given in Fig. 1a). Six µWRs with resonance
frequencies around $6\text{\,}\mathrm{GHz}$ are coupled to the readout line.
Of these, four are half-wave microstrip resonators using NbTiN/a-Si/NbTiN
microstrips with a dielectric thickness of $h=$250\text{\,}\mathrm{nm}$$ and
line width $w=$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, and two are quarter-
wave CPW resonators with a linewidth of $6\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ and slot width of $16\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The CPW
resonators are deposited directly on the Si substrate, with µWR6 fully
implemented in the lower NbTiN layer, while the center line of µWR5 is
implemented using the upper NbTiN layer.

We implement four FPRs on the chip, where each FPR is connected to its own
separate feeding network via two identical couplers, using the design shown in
Fig. 1b). All four resonators are made with identical line and coupler
geometry, but have different lengths $l_{FP}$ (FPR1: $5\text{\,}\mathrm{mm}$;
FPR2: $10\text{\,}\mathrm{mm}$; FPR3: $20\text{\,}\mathrm{mm}$; FPR4:
$50\text{\,}\mathrm{mm}$). This design was chosen to obtain resonators which,
for the same frequency, have different mode numbers but identical $Q_{i}$ and
$Q_{c,1}$.

We use a CPW with $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ line and gap width
between the antenna and the FPR, since we know its loss to be negligible at
sub-mm wave frequencies [17]. The MKIDs are quarter-wave resonators of a
standard Al/NbTiN hybrid CPW design[4], where incoming power is absorbed in a
short Al central line, leading to changes in the MKIDs microwave properties
which are measured via the microwave readout line.

##  III Device Fabrication

We start the fabrication process with a
$375\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick 4-inch Si wafer (dielectric
constant $\epsilon_{r}=11.44$[23]) coated on both sides with a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick, low tensile stress
($\sim$250\text{\,}\mathrm{MPa}$$) SiN layer ($\epsilon_{r}=7$), deposited
using low pressure chemical vapor deposition (LPCVD). The SiN on the device
side is etched away almost everywhere, except for small patches below the MKID
Al center lines [24]. For the FPR and µWR fabrication, we now start by
depositing a $220\text{\,}\mathrm{nm}$ thick NbTiN layer
($T_{c}=$15.1\text{\,}\mathrm{K}$$,
$\rho_{n}=$138\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$)
on the device side using reactive sputtering of a NbTi target in a nitrogen-
argon atmosphere [25]. This layer is patterned and etched to contain the
microstrip ground plane and all CPW elements of the chip. A
$250\text{\,}\mathrm{nm}$ thick a-Si layer $\epsilon_{r}\approx 10$, deposited
using plasma enhanced chemical vapor deposition (PECVD), serves as dielectric
layer of the microstrip[26]. We then define the microstrip lines in a second
NbTiN layer of $300\text{\,}\mathrm{nm}$ ($T_{c}=$15.0\text{\,}\mathrm{K}$$,
$\rho_{n}=$104\text{\,}\mathrm{\SIUnitSymbolMicro\SIUnitSymbolOhm}\text{\,}\mathrm{cm}$$),
using the same process as the first NbTiN layer. With the FPR and µWR
finalized, we finish fabrication of the MKIDs and microwave readout line using
a $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick layer of polyimide LTC9505
and a $50\text{\,}\mathrm{nm}$ thick layer of Al
($T_{c}=$1.25\text{\,}\mathrm{K}$$) [27]. Finally, a $40\text{\,}\mathrm{nm}$
thick layer of $\beta$-phase Ta ($T_{c}=$0.7\text{\,}\mathrm{K}$$) is
deposited on the backside and patterned into an absorbing mesh for stray light
control[28].

As NbTiN and a-Si require the same dry etch agents, an overetch in the order
of $40\text{\,}\mathrm{nm}$ is present for the lower layers, as highlighted in
Fig. 1e). The thickness of the lower NbTiN layer was increased accordingly to
maintain designed antenna and MKID properties.

##  IV Experimental Setup

The µWRs are characterized in a dark cryogenic setup, which is optimized for
low-background MKID experiments as discussed in [4, 29], and using a standard
homodyne technique enabled by a commercial vector network analyzer (VNA). In
order to achieve acceptable noise levels at the low microwave powers required
to reach the single-photon regime, a $-36\text{\,}\mathrm{dB}$ attenuation on
the input, and an amplifier with $36\text{\,}\mathrm{dB}$ gain on the output
were added to the microwave readout chain at room temperature.

For measurements of the FPR transmission, a $2\times 2$ lens-array is mounted
on the chip backside and aligned to the double-slot antennas. Each lens has a
hyper-hemispherical shape with $2\text{\,}\mathrm{m}\mathrm{m}$ diameter,
creating a diffraction limited beam [30]. The chip is then placed in a Cu
sample holder, which is surrounded by a tight-fitting mu-metal magnetic shield
(see Fig. 1f)). This assembly is mounted on the cold stage of a dilution
refrigerator operated at $120\text{\,}\mathrm{mK}$. A TERABEAM 1550 (TOPTICA
Photonics AG) photomixer continuous wave (CW) source at room temperature emits
a tunable signal in the range of $0.1...$1\text{\,}\mathrm{THz}$$ with a step
size of $10\text{\,}\mathrm{MHz}$ and a bandpass filterstack inside the
cryostat that defines a frequency band centered at
$f=$346\text{\,}\mathrm{GHz}$$. The source signal is attenuated with a beam
splitter to keep the MKIDs in a linear operating regime. Frequency
multiplexing readout electronics are used to enable simultaneous measurements
of all FPRs[31]. Details on the setup can be found in [17], which is identical
except for the cryogenic unit.

##  V Results and Discussion

###  V.1 Microwave resonators

We measure transmission dips of the µWRs at $50\text{\,}\mathrm{mK}$ as a
function of readout power in the range of
$P_{read}=$-65\text{\,}$...$-157\text{\,}\mathrm{dB}\text{\,}\mathrm{m}$$, as
shown exemplary in Fig. 2a) for µWR1. The internal quality factor $Q_{i}$ of
the resonator is then determined at each power from a Lorentzian fit [32]. We
measure $S_{21}(f)$ from $4.7\text{\,}\mathrm{GHz}$ to
$6.7\text{\,}\mathrm{GHz}$ in 4001 points at $T=$60\text{\,}\mathrm{mK}$$,
excluding any points falling on a resonance feature, to create a
$0\text{\,}\mathrm{dB}$ transmission reference. Figure 2b) shows the resulting
$Q_{i}$ as a function of the average photon number in the resonator per
$\lambda/2$, given by

$<n_{ph}>=\frac{P_{int}}{hf^{2}}$ (2)

where $P_{int}$ is the internal power, given for the µWRs by
$P_{int}=mQ^{2}P_{read}/\pi Q_{c}$ with $m=2$ or $m=1$ for quarter-wave and
half-wave resonators respectively. All µWRs show a characteristic behaviour
for TLS loss and can be fitted using [33]

$\frac{1}{Q_{i}}=\frac{\tanh(\hbar\omega/2k_{b}T)}{Q_{i,0}(1+(<n_{ph}>/n_{s}))^{\beta/2}}+\frac{1}{Q_{r}}$
(3)

where $Q_{i}$ has a minimum value $Q_{i,0}$ at photon numbers below the
saturation value $n_{s}$, increases with a slope given by $\beta$ until other
loss source dominate at high photon numbers, saturating at $Q_{r}$.

Figure 2: a) Transmission for µWR1 at three different readout powers, with the
respective Lorentzian fits shown by the black dotted lines. b) Measured
internal quality factor $Q_{i}$ as function of $<n_{ph}>$ for all µWRs shown
as symbols, with fits of Eq. 3 shown by dotted lines with corresponding
colors.

As the CPW resonators do not contain a-Si, their TLS losses are dominated by
the metal-substrate and substrate-air interfaces, with crystalline Si as
substrate[34]. Both $Q_{i,0}$ and $Q_{r}$ values (µWR5: $Q_{i,0}\approx
2.0\times 10^{5}$, $Q_{r}\approx 9.5\times 10^{6}$. µWR6: $Q_{i,0}\approx
1.2\times 10^{5}$, $Q_{r}\approx 9.9\times 10^{6}$) are comparable with the
state of the art for CPW resonators made of sputtered NbTiN [5], indicating an
excellent film and interface quality for both the upper and lower NbTiN layers
(sampled by µWR5 and µWR6 respectively) with no significant degradation due to
the fabrication process. We find values of $\beta$ between $0.7$ and $0.8$ for
all resonators, which is consistent with literature[33].

Losses in amorphous microstrips are generally dominated by TLS in the bulk
dielectric layer[35]. We assume that losses in the metal-substrate interface
can be neglected here, due to the surface layer quality shown by the CPW
resonators. Consequently, the internal loss factor of the a-Si film, can be
obtained as $Q_{i,aSi}($6\text{\,}\mathrm{GHz}$)=27\pm 4\times 10^{3}$ from an
average over all microstrip resonators and using

$Q_{i,aSi}=FF\times Q_{i}$ (4)

with the filling factor $FF=0.96$ of the microstrip line, taking the a-Si
layer overetch into account (see Fig. 1e). This value corresponds to a loss
tangent $\tan\delta=Q_{i}^{-1}=3.7\pm 0.5\times 10^{-5}$, which is expected
for a-Si films [36]. We calculate the filling factor $FF$ by simulating a
short microstrip line in CST and varying the loss tangent of the dielectric.
The relation between simulated line loss and the set loss tangent is then
given by $FF$.

###  V.2 Fabry-Pérot resonators

We measure the transmission through the four FPRs as a function of source
frequency and apply a correction for directly coupled stray light, using the
response of the blind MKID as introduced in [17]. Figure 3b) shows the
resulting spectra containing clear transmission peaks with an average
frequency spacing $dF$ corresponding to the different resonator lengths.
Variations in peak height, as well as secondary peaks visible in FPR1, can be
explained qualitatively by standing waves before the FPR modifying the
transmission through the first coupler with a wavelength corresponding roughly
to the electrical distance between antenna and FPR coupler. For each resonance
peak, we obtain $Q_{L}$ from a Lorentzian fit and the mode number from the
resonance frequency $F_{n}$ using $n=F_{n}/dF$. The standing waves before the
FPR also introduce an oscillation in the measured $Q_{L}$, due to a modified
$Q_{c}$.

Figure 3: (a) Averaged loaded quality factor $Q_{L}$ as a function of average
mode number in the four FPRs. The fit using Eq. 1 is shown by the black line.
(b) Figure supporting figure a): Measured spectra of the FPRs, offset along
the y-axis for better comparison. Line colors correspond to the markers in
figure a).

In this setup, we do not have an absolute reference for unity transmission and
can therefore not use the height of the FPR peaks to obtain $Q_{i}$ and
$Q_{c}$ independently. However, as we are measuring different resonators, each
designed with different mode numbers $n$ at the same frequency, we can fit
$Q_{L}(n)$ using Eq. 1 to retrieve $Q_{c,1}$ and $Q_{i}$. For this purpose, we
take the mean value of $Q_{L}$ for each resonator, as plotted in Fig. 3a),
while assuming that $Q_{L}$ is linear with frequency and that the standing
wave oscillations are sampled well enough to be averaged out. The resulting
mean value then corresponds to the value at the center frequency of the
measurement band, $350\text{\,}\mathrm{GHz}$, while the error bars are a
measure of both the statistical scatter and the linear slope over the averaged
frequency range. A fit to Eq. 1 then results in $Q_{c,1}=12.4\pm 0.7$ and
$Q_{i}=4.9\pm 0.3\times 10^{3}$.

Following Eq. 4, we then obtain the internal loss factor of a-Si at
$350\text{\,}\mathrm{GHz}$ as $Q_{i,aSi}($350\text{\,}\mathrm{GHz}$)=4.7\pm
0.3\times 10^{3}$, corresponding to a loss tangent of
$\tan\delta($350\text{\,}\mathrm{GHz}$)=2.1\pm 0.1\times 10^{-4}$ which
represents a significantly higher loss compared to the microwave regime, where
$\tan\delta($6\text{\,}\mathrm{GHz}$)=3.7\pm 0.5\times 10^{-5}$. Furthermore,
we estimate the maximum incident power at the FPR to be less than
$5\text{\,}\mathrm{pW}$, placing the FPR in the single-photon regime with
$n_{ph}<8$. The increased loss at sub-mm wave frequencies is in disagreement
with the TLS standard model, where a constant loss is predicted in the single-
photon regime when $\hbar\omega>>k_{B}T$, which is the case for both
measurements.

A possible explanation of this discrepancy, could be a modified TLS model with
a frequency dependent density of states $N(f)$. Such a model has been recently
used to describe dephasing and noise due to TLS, combining interacting TLS
with the power-law of $N(f)\propto(f^{\mu})$ with $\mu\approx 0.3$ obtained
from noise measurements[37]. We find that a value of $\mu\approx 0.4$ would
resolve the discrepancy in our measurement. Note, that the power dependence of
$\beta<1$ we find at microwave frequencies is also often identified with
interacting TLS. Alternatively, it is possible that we observe the low-
frequency tail of resonant phonon absorption peaks at far-infrared
wavelengths. Room temperature measurements of SiOx and SiN in the far-infrared
regime indicate that such a low-frequency tail could have the relevant
magnitude [38, 39]. Losses in the NbTiN film due to disorder effects should
also be considered. However, we consider this highly unlikely to be the
dominant effect, since measurements on a NbTiN CPW line with a similar film
yield a $Q_{i}\approx 17\times 10^{3}$ limited fully by radiation losses [17].
We also find no indication in DC- or microwave-measurements that the NbTiN
film quality of either the top- or bottom layer is degraded. The two CPW
datasets in Fig. 2 represent a CPW line from each NbTiN layer. Further
experiments in the $100-1000$\text{\,}\mathrm{GHz}$$ frequency range, in
combination with microwave measurements, are needed to clarify this issue.

##  VI Conclusion

In conclusion, we measured the losses of high-Q resonators made of
NbTiN/a-Si/NbTiN microstrips in the single-photon regime at
$6\text{\,}\mathrm{GHz}$ and $350\text{\,}\mathrm{GHz}$. Particularly, we
demonstrate an effective method to independently measure $Q_{i}$ and $Q_{c}$
at sub-mm wave frequencies without an absolute power calibration, using
multiple Fabry-Pérot resonators on a single chip. This method can be easily
extended up to the bandgap frequency of NbTiN around $1\text{\,}\mathrm{THz}$.
We measure a very low loss tangent for the a-Si film at microwave frequencies,
with a power dependence that is consistent with TLS theory. At sub-mm wave
frequencies, we find an unexpected increase in the loss tangent, which
requires further investigation to identify the root cause.

##  Appendix A Mode Number Dependence of Qc

In Goeppl et al. [40], an equation for the coupling Q-factor of a FPR
($Q_{ext}=Q_{c}$) is derived as

$Q_{ext}=\frac{C}{2n\omega_{0}R_{L}C_{k}^{2}}$ (5)

with the real load impedance $R_{L}=$50\text{\,}\mathrm{\SIUnitSymbolOhm}$$,
the coupling capacitance $C_{k}$ and the resonator capacitance $C=C_{l}l/2$
where $C_{l}$ is the capacitance per unit length and $l$ is the resonator
length. This equation apparently shows a mode number dependence
$Q_{ext}\propto 1/n$, which is seemingly in conflict with the dependence
$Q_{c}\propto n$ used in this paper, which is based on the equation

$Q_{c}=\frac{n\pi}{|S_{21,c}|^{2}}$ (6)

where $S_{21,c}$ is the transmission through a single coupler. In the
following section we will show the derivation of Eq. 6, the equivalence
between Eq. 5 and Eq. 6, the underlying reason for the apparent conflict in
mode number dependence and finally why $Q_{c}\propto n$ should be used when
considering the properties of a FPR.

Equation 6 is based on the definition of the quality factor

$Q=\frac{\omega E_{stored}}{P_{loss}}$ (7)

where $E_{stored}$ is the total energy stored in the resonator, while
$P_{loss}$ is the power loss per full resonator cycle.

The power loss per cycle through a single coupler of the FPR is given by

$P_{loss}=NfE_{stored}|S_{2^{\prime}1^{\prime}}|^{2}$ (8)

where $S_{2^{\prime}1^{\prime}}$ is the transmission through the coupler and
$N$ is the number of times the coupler is encountered per resonator cycle. For
a half-wave resonator, $N$ is given by the mode number as $N=1/n$. Combining
Eq. 7 and 8 then results in

$Q_{c1}=\frac{n2\pi}{|S_{2^{\prime}1^{\prime}}|^{2}}$ (9)

The second coupler can then be included to obtain the total coupling Q-factor
as

$\frac{1}{Q_{c}}=\frac{1}{Q_{c1}}+\frac{1}{Q_{c2}}=\frac{n\pi}{|S_{2^{\prime}1^{\prime}}|^{2}}$
(10)

where the latter relation retrieves Eq. 6 for identical coupling strength on
both sides ($Q_{c1}=Q_{c2}$).

In order to show the equivalence of $Q_{ext}=Q_{c}$, we start by taking the
ABCD matrix of an isolated coupler as given by Goeppl et al.

$\left(\begin{matrix}A&B\\\
C&D\end{matrix}\right)=\left(\begin{matrix}1&\frac{1}{i\omega_{n}C_{k}}\\\
0&1\end{matrix}\right)$ (11)

with $\omega_{n}=n\omega_{0}$ and retrieve $S_{21,c}$ from it as

$S_{21,c}=\frac{2}{2+1/(i\omega_{n}C_{k}R_{L})}$ (12)

where we use

$S_{21}=\frac{2}{A+B/R_{L}+CR_{L}+D}$ (13)

under the assumption that the characteristic line impedances $Z_{0}$ on either
side of the coupler are matched to the load impedance $R_{L}$.

The absolute square $|S_{21,c}|^{2}$ then follows as

$|S_{21,c}|^{2}=\frac{4\omega_{n}^{2}C_{k}^{2}R_{L}^{2}}{1+4\omega_{n}^{2}C_{k}^{2}R_{L}^{2}}\approx
4\omega_{n}^{2}C_{k}^{2}R_{L}^{2}$ (14)

where the approximation follows from the condition of a small capacitance
($\omega_{n}C_{k}R_{L}<<1$) as used by Goeppl et al. Note here the
proportionality $|S_{21,c}|^{2}\propto\omega_{n}^{2}$.

Combining Eqns. 5 and 14 then results in

$Q_{ext}=\frac{2\omega_{n}CR_{L}}{|S_{21,c}|^{2}},$ (15)

which is not quite identical to Eq. 6, but already follows its proportionality
of $Q_{ext}\propto n$ due to $\omega_{n}=n\omega_{0}$.

We then use $v_{ph}=1/\sqrt{(L_{l}C_{l})}$, the line impedance
$Z_{0}=\sqrt{\frac{L_{l}}{C_{l}}}$ and $L_{l}$ the inductance per unit length
to obtain the identity of

$\frac{\omega_{n}C_{l}l}{2}=\frac{\omega_{n}l}{v_{ph}Z_{0}}$ (16)

which we can rewrite, using $\omega_{n}/v_{ph}=2\pi/\lambda$, $l/\lambda=n/2$
(for a half-wave resonator) and the matched impedance condition $Z_{0}=R_{L}$,
as

$\frac{\omega_{n}C_{l}l}{2}=\frac{n\pi}{R_{L}}.$ (17)

Substituting Eq. 17 in 15 we finally obtain the form of Eq. 6

$Q_{ext}=\frac{n\pi}{|S_{21,c}|^{2}}=Q_{c}.$ (18)

Note here, that the matched impedance condition was only used to easily show
the equivalence, but is not necessary for either equation to be used on its
own. For Eq. 5 it is explicitly included due to the presence of $R_{L}$, while
Eq. 6 includes it implicitly as $S_{21,c}$ is obtained from EM-simulations and
can therefore account for any impedance mismatch. Additionally, any impedance
mismatch does not factor into the mode number dependence.

In order to explain the apparently conflict in mode number dependence, we
suggest, that while Eq. 5 is based on a flawless derivation and provides
accurate fits to measurements[40], it conflates mode number dependence with
frequency dependence of the coupler and therefore does not provide an accurate
physical interpretation of the mode number dependence.

The frequency dependence of the coupler can be understood from Eqns. 14 and
15: The couplers used in both of the papers considered here are fundamentally
high-pass filters operated in the stop-band regime and their transmission is
therefore proportional as $|S_{21,c}|^{2}\propto\omega^{2}$. Note, that we
omit the subscript $n$ here, as this proportionality exists independent from
the resonant behaviour of the FPR and must therefore be considered separate
from the mode number.

Goeppl et al. argue, that Eq. 5 shows $Q_{ext}\propto 1/n$. However, its
primary dependence should be understood as $Q_{ext}\propto 1/\omega_{n}$
which, when taking the coupler transmission into account, becomes
$Q_{ext}\propto\omega_{n}/\omega^{2}\propto n/\omega^{2}$ (see Eq. 15). In
contrast, Eq. 6 shows a clear separation of coupler properties ($S_{21,c}$)
and mode number.

The difference shown here is usually not significant for single resonators, as
both equations will yield the same results for $Q_{c}$. However, it needs to
be taken into consideration when one wants to understand and exploit the
fundamental behaviour of a transmission line FPR.

## References

  * Barends  _et al._ [2013] R. Barends, J. Kelly, A. Megrant, D. Sank, E. Jeffrey, Y. Chen, Y. Yin, B. Chiaro, J. Mutus, C. Neill, P. O’Malley, P. Roushan, J. Wenner, T. C. White, A. N. Cleland, and J. M. Martinis, Coherent Josephson Qubit Suitable for Scalable Quantum Integrated Circuits, Phys. Rev. Lett. 111, 080502 (2013).
  * Arute  _et al._ [2019] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, A. Dunsworth, E. Farhi, B. Foxen, A. Fowler, C. Gidney, M. Giustina, R. Graff, K. Guerin, S. Habegger, M. P. Harrigan, M. J. Hartmann, A. Ho, M. Hoffmann, T. Huang, T. S. Humble, S. V. Isakov, E. Jeffrey, Z. Jiang, D. Kafri, K. Kechedzhi, J. Kelly, P. V. Klimov, S. Knysh, A. Korotkov, F. Kostritsa, D. Landhuis, M. Lindmark, E. Lucero, D. Lyakh, S. Mandrà, J. R. McClean, M. McEwen, A. Megrant, X. Mi, K. Michielsen, M. Mohseni, J. Mutus, O. Naaman, M. Neeley, C. Neill, M. Y. Niu, E. Ostby, A. Petukhov, J. C. Platt, C. Quintana, E. G. Rieffel, P. Roushan, N. C. Rubin, D. Sank, K. J. Satzinger, V. Smelyanskiy, K. J. Sung, M. D. Trevithick, A. Vainsencher, B. Villalonga, T. White, Z. J. Yao, P. Yeh, A. Zalcman, H. Neven, and J. M. Martinis, Quantum supremacy using a programmable superconducting processor, Nature 574, 505 (2019).
  * Day  _et al._ [2003] P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, A broadband superconducting detector suitable for use in large arrays, Nature 425, 817 (2003).
  * Janssen  _et al._ [2013] R. M. J. Janssen, J. J. A. Baselmans, A. Endo, L. Ferrari, S. J. C. Yates, A. M. Baryshev, and T. M. Klapwijk, High optical efficiency and photon noise limited sensitivity of microwave kinetic inductance detectors using phase readout, Applied Physics Letters 103, 203503 (2013), https://doi.org/10.1063/1.4829657 .
  * Barends  _et al._ [2010] R. Barends, N. Vercruyssen, A. Endo, P. J. de Visser, T. Zijlstra, T. M. Klapwijk, P. Diener, S. J. C. Yates, and J. J. A. Baselmans, Minimal resonator loss for circuit quantum electrodynamics, Applied Physics Letters 97, 023508 (2010).
  * Megrant  _et al._ [2012] A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O’Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmstrøm, J. M. Martinis, and A. N. Cleland, Planar superconducting resonators with internal quality factors above one million, Applied Physics Letters 100, 113510 (2012).
  * Ho Eom  _et al._ [2012] B. Ho Eom, P. K. Day, H. G. LeDuc, and J. Zmuidzinas, A wideband, low-noise superconducting amplifier with high dynamic range, Nature Physics 8, 623 (2012).
  * Wagner  _et al._ [2019] A. Wagner, L. Ranzani, G. Ribeill, and T. A. Ohki, Demonstration of a superconducting nanowire microwave switch, Applied Physics Letters 115, 172602 (2019), https://doi.org/10.1063/1.5120009 .
  * Colangelo  _et al._ [2021] M. Colangelo, D. Zhu, D. F. Santavicca, B. A. Butters, J. C. Bienfang, and K. K. Berggren, Compact and Tunable Forward Coupler Based on High-Impedance Superconducting Nanowires, Physical Review Applied 15, 024064 (2021).
  * Mazin  _et al._ [2010] B. A. Mazin, D. Sank, S. McHugh, E. A. Lucero, A. Merrill, J. Gao, D. Pappas, D. Moore, and J. Zmuidzinas, Thin film dielectric microstrip kinetic inductance detectors, Applied Physics Letters 96, 102504 (2010).
  * Beldi  _et al._ [2019] S. Beldi, F. Boussaha, J. Hu, A. Monfardini, A. Traini, F. Levy-Bertrand, C. Chaumont, M. Gonzales, J. Firminy, F. Reix, M. Rosticher, S. Mignot, M. Piat, and P. Bonifacio, High q-factor near infrared and visible al2o3-based parallel-plate capacitor kinetic inductance detectors, Opt. Express 27, 13319 (2019).
  * Myers  _et al._ [2008] M. J. Myers, K. Arnold, P. Ade, G. Engargiola, W. Holzapfel, A. T. Lee, X. Meng, R. O’Brient, P. L. Richards, H. Spieler, and H. T. Tran, Antenna-Coupled Bolometer Arrays for Measurement of the Cosmic Microwave Background Polarization, Journal of Low Temperature Physics 151, 464 (2008).
  * Ade  _et al._ [2015] P. A. R. Ade, R. W. Aikin, M. Amiri, D. Barkats, S. J. Benton, C. A. Bischoff, J. J. Bock, J. A. Bonetti, J. A. Brevik, I. Buder, E. Bullock, G. Chattopadhyay, G. Davis, P. K. Day, C. D. Dowell, L. Duband, J. P. Filippini, S. Fliescher, S. R. Golwala, M. Halpern, M. Hasselfield, S. R. Hildebrandt, G. C. Hilton, V. Hristov, H. Hui, K. D. Irwin, W. C. Jones, K. S. Karkare, J. P. Kaufman, B. G. Keating, S. Kefeli, S. A. Kernasovskiy, J. M. Kovac, C. L. Kuo, H. G. LeDuc, E. M. Leitch, N. Llombart, M. Lueker, P. Mason, K. Megerian, L. Moncelsi, C. B. Netterfield, H. T. Nguyen, R. O’Brient, R. W. O. IV, A. Orlando, C. Pryke, A. S. Rahlin, C. D. Reintsema, S. Richter, M. C. Runyan, R. Schwarz, C. D. Sheehy, Z. K. Staniszewski, R. V. Sudiwala, G. P. Teply, J. E. Tolan, A. Trangsrud, R. S. Tucker, A. D. Turner, A. G. Vieregg, A. Weber, D. V. Wiebe, P. Wilson, C. L. Wong, K. W. Yoon, and J. Zmuidzinas, ANTENNA-COUPLED TES BOLOMETERS USED IN BICEP2, Keck Array, AND SPIDER, The Astrophysical Journal 812, 176 (2015).
  * Endo  _et al._ [2019] A. Endo, K. Karatsu, Y. Tamura, T. Oshima, A. Taniguchi, T. Takekoshi, S. Asayama, T. J. L. C. Bakx, S. Bosma, J. Bueno, K. W. Chin, Y. Fujii, K. Fujita, R. Huiting, S. Ikarashi, T. Ishida, S. Ishii, R. Kawabe, T. M. Klapwijk, K. Kohno, A. Kouchi, N. Llombart, J. Maekawa, V. Murugesan, S. Nakatsubo, M. Naruse, K. Ohtawara, A. Pascual Laguna, J. Suzuki, K. Suzuki, D. J. Thoen, T. Tsukagoshi, T. Ueda, P. J. de Visser, P. P. van der Werf, S. J. C. Yates, Y. Yoshimura, O. Yurduseven, and J. J. A. Baselmans, First light demonstration of the integrated superconducting spectrometer, Nature Astronomy 3, 989 (2019).
  * Shirokoff  _et al._ [2012] E. Shirokoff, P. S. Barry, C. M. Bradford, G. Chattopadhyay, P. Day, S. Doyle, S. Hailey-Dunsheath, M. I. Hollister, A. Kovács, C. McKenney, H. G. Leduc, N. Llombart, D. P. Marrone, P. Mauskopf, R. O’Brient, S. Padin, T. Reck, L. J. Swenson, and J. Zmuidzinas, MKID development for SuperSpec: an on-chip, mm-wave, filter-bank spectrometer, in  _Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VI_, Vol. 8452, edited by W. S. Holland, International Society for Optics and Photonics (SPIE, 2012) pp. 209 – 219.
  * Cataldo  _et al._ [2018] G. Cataldo, E. M. Barrentine, B. T. Bulcha, N. Ehsan, L. A. Hess, O. Noroozian, T. R. Stevenson, K. U-Yen, E. J. Wollack, and S. H. Moseley, Second-generation design of micro-spec: A medium-resolution, submillimeter-wavelength spectrometer-on-a-chip, Journal of Low Temperature Physics 193, 923 (2018).
  * Hähnle  _et al._ [2020] S. Hähnle, N. v. Marrewijk, A. Endo, K. Karatsu, D. J. Thoen, V. Murugesan, and J. J. A. Baselmans, Suppression of radiation loss in high kinetic inductance superconducting co-planar waveguides, Applied Physics Letters 116, 182601 (2020).
  * Spirito  _et al._ [2013] M. Spirito, G. Gentile, and A. Akhnoukh, Multimode analysis of transmission lines and substrates for (sub)mm-wave calibration, in  _82nd ARFTG Microwave Measurement Conference_ (2013) pp. 1–6.
  * Phillips [1987] W. A. Phillips, REVIEW ARTICLE: Two-level states in glasses, Reports on Progress in Physics 50, 1657 (1987).
  * Chang  _et al._ [2015] C. L. Chang, P. A. Ade, Z. Ahmed, S. W. Allen, K. Arnold, J. E. Austermann, A. N. Bender, L. E. Bleem, B. A. Benson, J. E. Carlstrom, H. M. Cho, S. T. Ciocys, J. F. Cliche, T. M. Crawford, A. Cukierman, J. Ding, T. De Haan, M. A. Dobbs, D. Dutcher, W. Everett, A. Gilbert, N. W. Halverson, D. Hanson, N. L. Harrington, K. Hattori, J. W. Henning, G. C. Hilton, G. P. Holder, W. L. Holzapfel, J. Hubmayr, K. D. Irwin, R. Keisler, L. Knox, D. Kubik, C. L. Kuo, A. T. Lee, E. M. Leitch, D. Li, M. Mcdonald, S. S. Meyer, J. Montgomery, M. Myers, T. Natoli, H. Nguyen, V. Novosad, S. Padin, Z. Pan, J. Pearson, C. Posada Arbelaez, C. L. Reichardt, J. E. Ruhl, B. R. Saliwanchik, G. Simard, G. Smecher, J. T. Sayre, E. Shirokoff, A. A. Stark, K. Story, A. Suzuki, K. L. Thompson, C. Tucker, K. Vanderlinde, J. D. Vieira, A. Vikhlinin, G. Wang, V. Yefremenko, and K. W. Yoon, Low loss superconducting microstrip development at argonne national lab, IEEE Transactions on Applied Superconductivity 25, 0 (2015).
  * Gao  _et al._ [2009] J. Gao, A. Vayonakis, O. Noroozian, J. Zmuidzinas, P. K. Day, and H. G. Leduc, Measurement of loss in superconducting microstrip at millimeter-wave frequencies, AIP Conference Proceedings 1185, 164 (2009).
  * Endo  _et al._ [2013] a. Endo, C. Sfiligoj, S. J. C. Yates, J. J. a. Baselmans, D. J. Thoen, S. M. H. Javadzadeh, P. P. Van Der Werf, a. M. Baryshev, and T. M. Klapwijk, On-chip filter bank spectroscopy at 600-700 GHz using NbTiN superconducting resonators, Applied Physics Letters 103, 1 (2013).
  * Lamb [1996] J. W. Lamb, Miscellaneous data on materials for millimetre and submillimetre optics, International Journal of Infrared and Millimeter Waves 17, 1997 (1996).
  * Ferrari  _et al._ [2018] L. Ferrari, O. Yurduseven, N. Llombart, S. J. C. Yates, J. Bueno, V. Murugesan, D. J. Thoen, A. Endo, A. M. Baryshev, and J. J. A. Baselmans, Antenna coupled mkid performance verification at 850 ghz for large format astrophysics arrays, IEEE Transactions on Terahertz Science and Technology 8, 127 (2018).
  * Thoen  _et al._ [2017] D. J. Thoen, B. G. C. Bos, E. A. F. Haalebos, T. M. Klapwijk, J. J. A. Baselmans, and A. Endo, Superconducting nbtin thin films with highly uniform properties over a ${\varnothing}$ 100 mm wafer, IEEE Transactions on Applied Superconductivity 27, 1 (2017).
  * Buijtendorp  _et al._ [2020] B. T. Buijtendorp, J. Bueno, D. J. Thoen, V. Murugesan, P. M. Sberna, J. J. A. Baselmans, S. Vollebregt, and A. Endo, Characterization of low-loss hydrogenated amorphous silicon films for superconducting resonators, arXiv e-prints , arXiv:2012.07692 (2020), arXiv:2012.07692 [astro-ph.IM] .
  * Hähnle  _et al._ [2020] S. Hähnle, O. Yurduseven, S. van Berkel, N. Llombart, J. Bueno, S. J. C. Yates, V. Murugesan, D. J. Thoen, A. Neto, and J. J. A. Baselmans, An Ultrawideband Leaky Lens Antenna for Broadband Spectroscopic Imaging Applications, IEEE Transactions on Antennas and Propagation 68, 5675 (2020).
  * Yates  _et al._ [2017] S. J. C. Yates, A. M. Baryshev, O. Yurduseven, J. Bueno, K. K. Davis, L. Ferrari, W. Jellema, N. Llombart, V. Murugesan, D. J. Thoen, and J. J. A. Baselmans, Surface wave control for large arrays of microwave kinetic inductance detectors, IEEE Transactions on Terahertz Science and Technology 7, 789 (2017).
  * de Visser  _et al._ [2014] P. J. de Visser, J. J. A. Baselmans, J. Bueno, N. Llombart, and T. M. Klapwijk, Fluctuations in the electron system of a superconductor exposed to a photon flux, Nature Communications 5, 3130 (2014).
  * Filipovic  _et al._ [1993] D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, Double-slot antennas on extended hemispherical and elliptical silicon dielectric lenses, IEEE Transactions on Microwave Theory and Techniques 41, 1738 (1993).
  * van Rantwijk  _et al._ [2016] J. van Rantwijk, M. Grim, D. van Loon, S. Yates, A. Baryshev, and J. Baselmans, Multiplexed readout for 1000-pixel arrays of microwave kinetic inductance detectors, IEEE Transactions on Microwave Theory and Techniques 64, 1876 (2016).
  * Khalil  _et al._ [2012] M. S. Khalil, M. J. A. Stoutimore, F. C. Wellstood, and K. D. Osborn, An analysis method for asymmetric resonator transmission applied to superconducting devices, Journal of Applied Physics 111, 054510 (2012), https://doi.org/10.1063/1.3692073 .
  * Molina-Ruiz  _et al._ [2020] M. Molina-Ruiz, Y. J. Rosen, H. C. Jacks, M. R. Abernathy, T. H. Metcalf, X. Liu, J. L. Dubois, and F. Hellman, Origin of mechanical and dielectric losses from two-level systems in amorphous silicon (2020), arXiv:2008.07489 [cond-mat.mtrl-sci] .
  * Wenner  _et al._ [2011] J. Wenner, R. Barends, R. C. Bialczak, Y. Chen, J. Kelly, E. Lucero, M. Mariantoni, A. Megrant, P. J. J. O’Malley, D. Sank, A. Vainsencher, H. Wang, T. C. White, Y. Yin, J. Zhao, A. N. Cleland, and J. M. Martinis, Surface loss simulations of superconducting coplanar waveguide resonators, Applied Physics Letters 99, 113513 (2011).
  * Daalman [2014] W. K.-G. Daalman,  _Noise and loss in superconducting resonators_ , Ph.D. thesis, Delft University of Technology (2014).
  * O’Connell  _et al._ [2008] A. D. O’Connell, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, C. McKenney, M. Neeley, H. Wang, E. M. Weig, A. N. Cleland, and J. M. Martinis, Microwave dielectric loss at single photon energies and millikelvin temperatures, Applied Physics Letters 92, 112903 (2008).
  * Faoro and Ioffe [2015] L. Faoro and L. B. Ioffe, Interacting tunneling model for two-level systems in amorphous materials and its predictions for their dephasing and noise in superconducting microresonators, Phys. Rev. B 91, 014201 (2015).
  * Cataldo  _et al._ [2016] G. Cataldo, E. J. Wollack, A. D. Brown, and K. H. Miller, Infrared dielectric properties of low-stress silicon oxide, Optics Letters 41, 1364 (2016).
  * Cataldo  _et al._ [2012] G. Cataldo, J. A. Beall, H.-M. Cho, B. McAndrew, M. D. Niemack, and E. J. Wollack, Infrared dielectric properties of low-stress silicon nitride, Optics Letters 37, 4200 (2012).
  * Göppl  _et al._ [2008] M. Göppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp, J. M. Fink, P. J. Leek, G. Puebla, L. Steffen, and A. Wallraff, Coplanar waveguide resonators for circuit quantum electrodynamics, Journal of Applied Physics 104, 113904 (2008), https://doi.org/10.1063/1.3010859 .