Design and Performance of a 30
/
40GHz Diplexed Focal Plane for the BICEP Array
Corwin Shiu
1
, Ahmed Soliman
2
,
3
, Roger O
’
Brient
2
,
3
, Bryan Steinbach
2
, James J. Bock
2
,
3
, Clifford F. Frez
3
, William C. Jones
1
,
Krikor. G. Megerian
3
, Lorenzo Moncelsi
2
, Alessandro Schillaci
2
, Anthony D. Turner
3
, Alexis C. Weber
3
, Cheng Zhang
2
, and
Silvia Zhang
2
1
Princeton University, Princeton, NJ 08544, USA
2
California Institute of Technology, Pasadena, CA 91125, USA
3
Jet Propulsion Laboratory, Pasadena, CA 91109, USA
Received 2024 January 27; revised 2024 March 4; accepted 2024 March 15; published 2024 April 29
Abstract
We demonstrate a wideband diplexed focal plane suitable for observing low-frequency foregrounds that are
important for cosmic microwave background polarimetry. The antenna elements are composed of slotted bowtie
antennas with 60% bandwidth that can be partitioned into two bands. Each pixel is composed of two interleaved
12
×
12 pairs of linearly polarized antenna elements forming a phased array, designed to synthesize a symmetric
beam with no need for focusing optics. The signal from each antenna element is captured in-phase and uniformly
weighted by a microstrip summing tree. The antenna signal is diplexed into two bands through the use of two
complementary, six-pole Butterworth
fi
lters. This
fi
lter architecture ensures a contiguous impedance match at all
frequencies, and thereby achieves minimal re
fl
ection loss between both bands. Subsequently, out-of-band rejection
is increased with a bandpass
fi
lter and the signal is then deposited on a transition-edge sensor bolometer island. We
demonstrate the performance of this focal plane with two distinct bands, 30 and 40 GHz, each with a bandwidth of
∼
20 and 15 GHz, respectively. The unequal bandwidths between the two bands are caused by an unintentional
shift in diplexer frequency from its design values. The end-to-end optical ef
fi
ciency of these detectors is relatively
modest, at 20%
–
30%, with an ef
fi
ciency loss due to an unknown impedance mismatch in the summing tree. Far-
fi
eld beam maps show good optical characteristics, with edge pixels having no more than
∼
5% ellipticity and
∼
10%
–
15% peak-to-peak differences for A
–
B polarization pairs.
Uni
fi
ed Astronomy Thesaurus concepts:
Astronomical instrumentation
(
799
)
;
CMBR detectors
(
259
)
;
Polarimeters
(
1277
)
1. Introduction
Measurements of the spatial anisotropies of the cosmic
microwave background
(
CMB
)
provide fundamental tests of
cosmological theories. Measurements of a cosmological,
degree-scale, B-mode polarization would provide a constraint
on the tensor-to-scalar ratio
r
and place limits on the energy
scale of in
fl
ation
(
Seljak & Zaldarriaga
1997
; Kamionkowski &
Jaffe
2001
)
. The BICEP experiment has placed the tightest
constraints on
r
, with an upper limit of
r
0.05
<
0.036 at 95%
con
fi
dence
(
Ade et al.
2021
)
.
These measurements are complicated by the fact that
cosmological signals at large angular scales are highly
contaminated by astrophysical foregrounds. Thermal emission
from spinning dust grains produces a polarized emission that
dominates at high frequencies
(
Finkbeiner et al.
1999
; Planck
Collaboration et al.
2020
)
. Galactic synchrotron emission is
emitted by electrons gyrating in magnetic
fi
elds and dominates
at low frequencies
(
Ginzburg
1969
; Bennett et al.
2013
)
. CMB
polarimetry experiments must have several frequency bands to
remove contamination from foreground emission and uncover
the underlying cosmological signal
(
Brandt et al.
1994
)
.
Much remains to be learned about characterizing low-frequency
foregrounds
’
spectral and spatial behaviors
(
Planck Collaboration
et al.
2016
)
. WMAP has observed substantial amounts of
polarized foreground emission due to synchrotron radiation, even
at high Galactic latitudes
(
Page et al.
2007
)
, and this has been
con
fi
rmed by other experiments
(
Krachmalnicoff et al.
2018
;
Eimer et al.
2024
)
. Kogut et al.
(
2007
)
have observed a
fl
attening
of the synchrotron spectral index closer to the Galactic plane;
however, Choi & Page
(
2015
)
have attributed this effect to the
spatial correlation of dust and synchrotron.
A cross-correlation analysis of BICEP 95 and 150 GHz data
up to 2018, when combined with publicly available WMAP K
+
Ka bands and Planck NPIPE 30 and 44 GHz data, did not
yield any statistically signi
fi
cant evidence supporting the
detection of synchrotron radiation
(
Ade et al.
2021
)
. Therefore,
while synchrotron contamination is not a driving source of
uncertainty for current upper limits on
r
, as experiments
become more sensitive, this foreground will play a more
signi
fi
cant role in an unbiased recovery of
r
.
Furthermore, although synchrotron radiation is the primary
source of low-frequency, polarized foregrounds, it may not be
the exclusive source of low-frequency contamination. Magnetic
dipole emissions resulting from thermal
fl
uctuations of
ferromagnetic interstellar grains have been proposed
(
Draine
& Lazarian
1999
; Draine & Hensley
2013
)
. These emissions
are expected to primarily occur
ν
100 GHz, but the behavior
likely differs depending on whether they are from iron
inclusions in dust grains or free-
fl
ying iron nanoparticles
(
Hoang & Lazarian
2016
)
. While there is currently no
statistically signi
fi
cant detection of polarized anomalous
microwave emission
(
AME; Planck Collaboration et al.
2016
;
Herman et al.
2023
)
, increasingly precise measurements have
the potential to improve the understanding of low-frequency
foregrounds.
The Astrophysical Journal Supplement Series,
272:12
(
13pp
)
, 2024 May
https:
//
doi.org
/
10.3847
/
1538-4365
/
ad34d8
© 2024. The Author
(
s
)
. Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the
Creative Commons Attribution 4.0 licence
. Any further
distribution of this work must maintain attribution to the author
(
s
)
and the title
of the work, journal citation and DOI.
1
Advances in millimeter- and submillimeter-wave bolometer
arrays directly drive improvements in characterizing the
millimeter-wave sky. Ground-based CMB experiments have
long been limited by photon noise
(
Ade et al.
2014
)
, and
therefore the sensitivity of an experiment scales with the square
root of the number of detectors on the sky. For a given optical
system and
fi
xed focal plane area, the sensitivity would be
improved by taking advantage of a wide-bandwidth antenna
partitioned into multiple spectral bands. Various technologies
exist to couple electromagnetic radiation onto bolometer arrays
ranging from lenslet-coupled broadband sinuous antennas to
multichroic horn antennas
(
McMahon et al.
2012
;O
’
Brient
et al.
2013
)
. Several experiments have successfully deployed
multichroic detector arrays, including ACT
(
Thornton et al.
2016
)
and SPT
(
Benson et al.
2014
)
, and future-generation
experiments will all be multichroic
(
Suzuki et al.
2016
; Kiuchi
et al.
2020
; Walker et al.
2020
)
.
This paper describes the design and performance of a fully
lithographed, diplexed, low-frequency focal plane that will be
powerful for characterizing synchrotron emission. The design
builds upon previous BICEP instruments
’
successful phased
array design
(
BICEP2 Collaboration et al.
2015
)
. The phased
array con
fi
guration gives us several advantages. The detectors
are entirely planar and fully lithographed in thin
fi
lms, which
signi
fi
cantly simpli
fi
es the fabrication of these devices. The
phased array con
fi
guration naturally synthesizes the beam
without any need for focusing optics.
Antenna arrays have
fl
exibility in their illumination pattern
and can match feedhorn arrays
’
aperture ef
fi
ciencies of
∼
0.70,
at smaller pixel sizes than their feedhorn counterparts
(
Grif
fi
n
et al.
2002
)
. Consequently, pixel densities from antenna arrays
can be higher, owing to both geometric tiling of square pixels
and an increased number of detectors within the same focal
plane area. These theoretical advantages in ef
fi
ciency arise
from the ability for antenna arrays to have greater directivity.
An extended discussion can be found in Appendix
C
, which
outlines the theoretical contributions to focal plane mapping
speed.
2. Focal Plane Overview
Our detector design is entirely planar and requires no
contacting optics such as lenslets or horn antennas. Incoming
optical power is coupled to polarized planar antenna arrays, the
details of which are provided in Section
2.3
. The power is then
directed through a microstrip summing network, as explained
in Section
2.4
. The power subsequently passes through on-chip
band-de
fi
ning
fi
lters, which partition the power by frequency;
the design of these
fi
lters is elaborated upon in Section
2.5
.
Finally, the energy is dissipated on a bolometer island and
detected by a superconducting transition-edge sensor
(
TES;
Irwin & Hilton
2005
)
, the design of which is explained in
Section
2.6
. Variations in the TES current are read out by a
time-domain multiplexing system based on SQUIDS
(
de Korte
et al.
2003
)
.
2.1. Overview of Detector Wafer
The Microdevices Laboratory at the Jet Propulsion Labora-
tory
(
JPL
)
fabricates the detector wafers, with a schematic of
the detector module shown in Figure
1
. Antenna arrays are built
on 6 inch, 625
μ
m thick silicon wafers
(
ò
r
=
11.8
)
. The
millimeter wave circuit consists of four distinct layers,
described in order from the layer closest to the silicon to the
layer farthest away. First, a niobium ground-plane
fi
lm is
deposited, and then slotted antenna arrays are patterned through
a liftoff process. Subsequently, we grow a 0.3
μ
m SiO
2
inner-
layer dielectric
(
ILD
)
on top. Following that, we de
fi
ne the
resistive termination for the bolometers. Finally, in the last
fi
lm,
we pattern the upper niobium conductor, which is responsible
for shaping the antenna feed network and in-line
fi
lters.
Fabrication details can be found in BICEP2 Collaboration et al.
(
2015
)
. Figure
2
shows a photograph of the antenna elements.
Because the antenna arrays are backside illuminated, a fused-
quartz
(
ò
r
=
3.9
)
antire
fl
ection layer is applied to the bottom of
the entire stack, serving as the topmost layer facing the sky. As
a result, the antenna elements face a superconducting niobium
re
fl
ective backshort, positioned at a distance of
λ
/
4 away. The
entire detector module comprises the quartz antire
fl
ection
wafer, silicon detector array, niobium backshort, amumetal 4k
magnetic shielding, and readout printed circuit board cards, all
enclosed in a compact niobium frame. Additional information
on the focal plane engineering, measurements, and hybridiza-
tion can be found in other publications
(
Schillaci et al.
2023
;
Soliman
2023
)
.
2.2. Mitigating Polarized Frame-edge Effects
Unwanted electromagnetic interactions between the niobium
frame and antennas degrade the quality of the antenna beams.
Figure 1.
(
Left
)
A cross-sectional 3D render of the focal plane module. The focal plane module consists of a detector tile hybridized with its readout chain, all housed
in a superconducting niobium box and frame for magnetic shielding.
(
Right
)
A photograph of the machined corrugated frames designed to minimize undesired
electrical interactions between the edge-pixel antenna elements and the structure. The visible tile is the antire
fl
ection coating, facilitating the backside illumination of
the detectors through the silicon.
2
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)
, 2024 May
Shiu et al.
Employing a solid metal frame can lead to polarization-
dependent deformations of the antenna beams, especially for
edge pixels. Such deformations can introduce
(
1
)
differential
beam centers for orthogonal polarizations and result in dipole
artifacts or
(
2
)
polarized beam ellipticities and result in
quadrupole artifacts. These beam systematics leak temperature
to polarization and potentially introduce a false B-mode signal
(
Hu et al.
2003
; Ade et al.
2015
)
. Although BICEP has
developed a deprojection technique to mitigate low-order beam
systematics effectively, controlling higher-order beam effects is
challenging
(
Ade et al.
2016
)
.
At lower frequencies, there are far more edge pixels than
center pixels. Soliman et al.
(
2020
)
has devised a novel
corrugated frame that minimizes polarized beam steering.
These grooves serve to smooth out impedance discontinuities,
effectively reducing the abrupt boundaries caused by the
presence of a conductive frame. The frame relies on quarter-
wave corrugations with quarter-wave pitches. This design
ensures that surface waves re
fl
ecting off the corrugated frame
become out of phase and destructively interfere. To account for
the wide bandwidth of this antenna, Soliman
(
2023
)
designed a
doubly corrugated frame with depths tailored for both 30 and
40 GHz. The peak-to-peak polarization subtraction for the
broadband corrugated frame shows a minimized differential
offset
(
<
10%
–
15%
)
. The remaining residuals are effectively
fi
ltered out using a deprojection technique developed by the
BICEP team. These results show that the corrugated frame
successfully minimizes the differential offset compared to
earlier focal plane unit detector designs; Ade et al.
(
2019
)
report a differential offset of
∼
40%. Readers interested in the
design speci
fi
cations of this corrugated frame are referred to a
different publication
(
Soliman
2023
)
.
2.3. Antenna Design
A single pixel consists of two interleaved phased antenna
arrays. To prevent grating lobes, a phased antenna array must
be spaced to Nyquist sample the focal plane
(
Kuo et al.
2008
)
.
The antenna elements are arranged in a square lattice rotated by
45
°
, which sets the Nyquist condition as the following
inequality:
⎛
⎝
⎞
⎠
()
l
-
s
N
1
1
.1
r
0,min
In this equation,
s
is the antenna spacing and
l
0,min
represents
the minimum operating wavelength of the band. The term in
the parenthesis suppresses the array factor
’
s end-
fi
re beam
response where
N
is the number of antenna elements along a
single axis of the square lattice.
This relationship imposes an upper limit on the size of the
antenna element, which must not exceed a wavelength. This
constraint can be challenging because wideband antennas are
typically larger structures spanning multiple wavelengths
(
Chu
1948
)
. Another common strategy to achieve large
bandwidths is designing self-complementary antennas, which
have the property of frequency-independent input impedance
(
Mushiake
1992
)
. This concept is the driving principle behind
sinuous antennas, as detailed in O
’
Brient et al.
(
2008
,
2010
,
2013
)
. However, this approach is not suitable for this
architecture, because of the space requirements for including
orthogonal antenna elements. Consequently, implementing this
phased antenna array scheme necessitates designing an antenna
with a broad
fi
rst resonant peak, which is then operated in that
mode. This is the fundamental principle behind this design.
Like the slot dipole antenna, the slotted bowtie antenna is
linearly polarized. This design incorporates a wide
fl
are angle,
which broadens the
fi
rst resonance and increases the available
bandwidth
(
Brown & Woodward
1952
)
. While a bowtie
antenna is sometimes described as a traveling wave antenna
parameterized solely by its
fl
are angle, this is not true for a
fi
nite antenna lacking a resistive termination. An impedance
discontinuity leads to the formation of standing waves and
therefore resonant modes. As discussed by Stutzman & Thiele
(
2012
)
as well as Balanis
(
2016
)
, increasing the
fl
are angle
reduces the antenna
’
s standing-wave characteristics by mini-
mizing the phase mismatch between the voltage and currents
along the slot. This results in a smoother impedance response
and effectively widens the
fi
rst resonance. This phenomenon is
evident in full-
fi
eld FEM simulations of the slotted bowtie
antenna using Ansys High-Frequency Structure Simulator
(
HFSS
)
. Figure
3
illustrates that an increase in
fl
are angle
shifts the
fi
rst resonance to lower frequencies and smoother
reactance variations.
Additionally, rounding out the ends of the slotted bowtie
modestly reduces the driving-point impedance at no cost of
space for our feed network
(
Qu & Ruan
2006
)
.A
fl
are angle of
Figure 2.
Bowtie tile.
(
a
)
Photograph of a full tile showing a 4
×
4 grid of pixels.
(
b
)
Zoom-in on a pixel. We have a 12
×
12 pairs of bowtie antennas comprising the
antenna array with four bolometers capturing A
/
B polarization at our two bands.
(
c
)
Zoom-in on a single antenna element. Dark green indicates the ground-plane
cutout showing both the subantenna and the coplanar waveguide driving the antenna.
3
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Shiu et al.
50
°
was determined to meet our bandwidth requirements while
still adhering to space constraints.
In this design, depicted on the left side of Figure
4
, a pixel
consists of a square lattice rotated by 45
°
, highlighted in red.
The antenna spacing between nearest-neighbor elements is
s
=
1315
μ
m. Although this spacing is
∼
15% smaller than
required by the Nyquist criterion
(
Equation
(
1
))
, we observed
improved impedance characteristics with the smaller spacing.
However, a more convenient mathematical representation
would be the array in blue: a single polarization consists of a
12
×
12 square grid of resonant bowtie antenna pairs. These
pairs of antennas are arranged in a Bravais lattice with vectors
()
=
a
a
2,0
, and
()
=
a
a
0, 2
2
, where the lattice spacing is
a
=
930
μ
m. Individual pairs are located at
(
)
=
a
0, 0
and
(
)
=
a
aa
,
, with the sign originating from the polarization.
This alternative parameterization of the lattice is useful for the
beam model in a later discussion in Section
3.2
. The lattice for
the orthogonal polarization is achieved by a simple translation
of
(
a
,0
)
.
2.4. Microstrip Feed Network
We collect power by combining signals from 12
×
12 bowtie
pairs in the array with a microstrip feed network. All antennas
are coherently fed with uniform power division. A cartoon
diagram of a simpli
fi
ed microstrip network is shown by the
right side of Figure
4
. We achieve coherent summation by
summing waves coherently by row and then as a column using
microstrip T-junctions. The power is subsequently split by a
diplexer and
fi
ltered before terminating onto a bolometer
island.
We refer to uniform and in-phase summation as
top-hat
illumination. We chose top-hat illumination for its simplicity.
Figure 3.
The simulated resistance and reactance of the bowtie antenna, calculated using Ansys HFSS. The simulation shows that the antenna reactance diminish
es as
the
fl
are angle increases and the
fi
rst resonance widens. The solid line, labeled with a
fl
are angle of 50
°
, is the nominal con
fi
guration. We drive this antenna at 100
Ω
using a quasi-CPW line in our design.
Figure 4.
(
Left
)
An abbreviated schematic of a single-polarization antenna array, depicted in two colors to illustrate two possible descriptions. The red array repr
esents
this design as a square array rotated by 45
°
, establishing the Nyquist criterion for antenna spacing. Alternatively, the blue array describes our con
fi
guration as a
product of a square array with two elements per square, offering a convenient analytical description of the total beam.
(
Right
)
An abbreviated schematic of a single
pixel
’
s microstrip summing tree network. Antenna subelements are located at the ends of all horizontal brackets. Power is coherently summed by the horizont
al
network, and then summed by the vertical network. Different polarizations are directed left and right. The signal is partitioned by a diplexer, as dep
icted in blue, and
fi
ltered by a bandpass
fi
lter, as depicted in red, and terminated on a bolometer island.
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)
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Shiu et al.
However, it is important to note that tapered illumination could
improve beam quality
(
O
’
Brient et al.
2012
)
, as top-hat
illumination tends to result in larger side lobes.
A relatively high-impedance line of around
∼
100
Ω
is
required to drive the bowtie antenna ef
fi
ciently. This
impedance is unachievable with microstrip lines with a thin
ILD
(
0.3
μ
m of SiO
2
)
. Instead, we can create a high-impedance
line by removing the ground plane directly underneath the
conductor, so that the
fi
eld lines mimic those of a coplanar
waveguide
(
CPW; Arbabi et al.
2006
)
. We found that a
substantial gap, 20
μ
m side-to-side, allowed us to achieve our
desired impedance. Then, adjacent quasi-CPW lines join at a
T-junction and transition, with minimal re
fl
ection loss, to a
microstrip line for the remainder of the summing tree.
Furthermore, ground bridges are strategically placed to short
the two outer ground conductors. These bridges eliminate
potential slotline modes that may radiate. Their spacing has
determined to ensure that no slotline modes are excited within
the operating bandwidth of this detector array.
Crosstalk can be a concern with phased array antennas.
Coupling between neighboring transmission lines can introduce
phase errors across the pixel, leading to beam steering
(
BICEP2
Collaboration et al.
2015
)
. The larger footprint of the bowtie
antenna restricts the spacing available for routing the summing
tree. We mitigate crosstalk by two design principles. First, we
opt for a thin ILD to enhance the con
fi
nement of the
fi
eld lines
of the microstrip mode. Second, we route the microstrip lines to
fan in and out wherever space allows. Coupling between
microstrip lines relies on excitations of even and odd modes
(
Pozar
2005
)
.By
fl
aring the microstrip lines, no signi
fi
cant
sections of the summing tree remain in close contact, and the
two modes continually change impedances along the lines. This
prevents coupling even when lines must compress within a few
line widths of distance. The spacing of the antennas
necessitates lines as close as 5
μ
m. However, these modes
are con
fi
ned within a thin ILD
(
0.3
μ
m
)
and remain in close
proximity for no more than 500
μ
m, about 10% of a
wavelength, resulting in minimal coupling. HFSS simulations
show crosstalk levels of
−
50 dB between neighboring lines at
our tightest junctions. We expect crosstalk to be a minor
contributor to beam effects, with frame-edge coupling assum-
ing the predominant role.
2.5. On-chip Filtering
Each detector band is synthesized from a combination of the
diplexer and the wideband bandpass
fi
lter. The bandpass
fi
lter
de
fi
nes the upper-band edge for the high-frequency band and
the lower-band edge for the low-frequency band. The diplexer
splits the power contiguously between the two bands.
The diplexer is composed of a sixth-order high-pass and a
low-pass Butterworth
fi
lter. In this topology, the speci
fi
c values
for all the components have been carefully selected to achieve
equal and opposite reactance for the two inductance
–
capaci-
tance
(
LC
)
ladder circuits. Therefore, the reactance is matched
across all frequencies and identically cancels out at the antenna
input, ensuring a continuous power split without any re
fl
ec-
tions. The design is described in detail in Matthaei et al.
(
1980
)
.
We chose this design for its simplicity and robustness to
fabrication nonuniformity.
This circuit was realized by
fi
nding lithographic approxima-
tions to lumped-element circuit elements through extensive
simulations using Sonnet. Lumped inductors are synthesized by
subwavelength high-impedance transmission lines, while
lumped capacitors are synthesized by parallel plate capacitors.
In practice, lithographic elements only approximate lumped
elements at microwave frequencies. Frequency dispersion
reduces their effectiveness. We selected a three-pole Butter-
worth
fi
lter to strike a balance between achievable lithographic
elements and
fi
lter sharpness. In particular, series capacitors
become more dispersive with larger capacitance values,
nullifying the bene
fi
ts of higher-order
fi
lters.
The bandpass
fi
lter design concept was adapted from the
previous BICEP style of
fi
lters
(
BICEP2 Collaboration et al.
2015
)
. In its most basic description, the bandpass
fi
lter
architecture consists of three-series LC tanks joined by
impedance inverters
(
Matthaei et al.
1980
)
. The Kuroda
identities convert the impedance K-inverter into a physically
realizable capacitor network
(
Pozar
2005
)
. Additionally, we
found it advantageous to perform a
Y
-to-
π
transformation of the
capacitor network at these microwave frequencies to reduce the
series capacitances necessary for lithography. The design table
is tabulated in Table
1
, accompanied by a corresponding circuit
diagram in Figure
5
. The left side of Figure
6
shows the
simulated performance of these
fi
lters, and we leave the
discussion of the performance to Section
3.1
.
2.6. Bolometer Design
Power captured by the antennas propagates through the
microstrip network, passes through on-chip
fi
lters, and is
ultimately dissipated as heat by a lossy gold termination on a
suspended bolometer island. This heat is ultimately detected by
a TES bolometer
(
Irwin & Hilton
2005
)
.
TES bolometers are widely adopted in many CMB
experiments because they have excellent noise properties and
are a suitable choice to make large-format arrays. They can be
fabricated using standard thin-
fi
lm lithography techniques, and
importantly, they are compatible with multiplexing schemes
(
de Korte et al.
2003
; Irwin & Lehnert
2004
; Dobbs et al.
2012
)
. Additionally, they have a strong negative electrothermal
feedback, increasing the device
’
s linearity
(
Irwin & Hil-
ton
2005
)
. A TES bolometer is operated by voltage biasing
the detector between superconducting and normal states. When
an incident photon is absorbed and heats the bolometer, slight
temperature changes will result in signi
fi
cant variations in the
TES resistance. The electrical current
fl
owing through the
device is inductively coupled to the SQUID readout scheme,
enabling a highly sensitive measurement of relative energy
changes.
Table 1
Circuit Design Table for On-Chip Filters
L
1
C
1
L
2
C
2
L
3
C
3
Bandpass
1.85
0.25
2.70
0.41
L
0.51
Diplexer
1.55
1.76
1.55
1.20
0.758
0.259
Note.
The corresponding circuit elements are in Figure
5
. To scale from the
design table to physical values: inductor values are scaled by
w
-
Z
0
1
0
, and
capacitor values are scaled by
()
w
-
Z
00
1
, where
ω
0
=
2
π
f
0
is the desired
−
3dB
transition between the two bands and
Z
0
is the port impedance. For this
particular design,
f
0
=
35
GHz
and
Z
0
=
25
Ω
. Simulations are then performed
to convert the electrical inductance and capacitance to a lithographic element,
the details of which are outlined in Appendix
B
.
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)
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Shiu et al.
The detector design has two TES in series with different
superconducting transition temperatures
(
T
c
)
. One is an
aluminum TES with a
T
c
=
1.2 K. This TES is designed to
handle signi
fi
cantly higher optical loading in laboratory
settings, in particular during calibration and ground-based
optical characterization. The titanium TES has a much lower
T
c
=
450 mK for science observation with signi
fi
cantly reduced
loading conditions. This TES offers superior detector stability
and substantially higher sensitivity. The saturation power is a
key design parameter of a bolometer and is the total power
needed to bring the TES temperature to
T
c
:
()
()
()
()
==
-
+
+
PPTGT
TT
n
1
1
,2
ccc
c
n
sat
bath
1
where
T
bath
∼
280 mK is the surrounding heat bath temper-
ature,
G
c
is thermal conductance of the bolometer isolation legs
at
T
=
T
c
, and
n
corresponds to the nature of phonon transport
in the legs. For our bolometer design, the isolation legs are thin,
so phonons are reduced to two dimensions and
n
has been
experimentally determined to be
∼
2.
G
c
is carefully designed
so that
P
sat
is above the optical loading we expect when
doing science observations at the South Pole while keeping it
as low as possible in order to minimize the noise-equivalent
power
(
NEP
)
.
The noise in the TES detector consists of several
components, including photon noise, phonon noise, SQUID
readout noise, and Johnson noise from the shunt resistor
(
Zmuidzinas
2003
)
. The latter two sources of noise can be
deliberately minimized to be subdominant to the
fi
rst two,
which are intrinsic and establish the fundamental sensitivity of
the bolometer. Thermal
fl
uctuations across the silicon nitride
legs of the bolometers contribute the following NEP:
()
()
=
NEP
k T G F T T
4,, 3
B
c
cc
phonon
22
bath
where
F
(
T
c
,
T
bath
)
is a function of bath and TES island
temperature and accounts for the nonequilibrium effects
(
Mather
1982
)
. Typical values for our bolometers are
F
∼
0.5.
Because phonon NEP is proportional to the square root of
thermal conductance, the value of
G
c
has a direct impact on
mapping speed. The bolometer island is suspended by four
legs, forming the weak thermal link between the island and the
thermal bath. The
G
c
, tunable by leg lengths, is optimally
designed to maximize detector sensitivity, while avoiding
saturation during science operation. The bolometer legs are
constructed from 1
μ
m thick low-stress nitride
(
LSN
)
and are
9
μ
m wide for the leg that bridges the wire connecting the
microstrip summing tree to the resistive termination, 6
μ
m wide
for the two legs that carry the DC bias lines, and 4
μ
m thick for
Figure 5.
On-chip
fi
lters that de
fi
ne the bandpass of the detectors. On the left is the bandpass
fi
lter that de
fi
nes the extremities of the bands. The bandpass
fi
lter is
equivalent to a three-pole LC series resonator with impedance inverters. On the right is the diplexer composed of a Butterworth high-pass and low-pas
s
fi
lter. The
image above is a Sonnet schematic of the diplexer, where port 1 connects to the antenna network, port 2 is the low passband, and port 3 is the high passband
. Below
the image is a lumped-element equivalent network for this diplexer. The design table is in Table
1
, where the inductor values must be scaled by
Z
0
/
ω
0
and capacitor
values scaled by
()
w
-
Z
00
1
.
Figure 6.
Comparison between the design spectrum
(
in the dashed lines
)
and measured spectrum
(
in solid lines
)
. The left panel highlights the simulated spectra, while
the right panel highlights the simulated spectra.
(
Left
)
Simulations of the antenna transmission spectrum and the intended bands from the on-chip
fi
lters. Measured
spectra have been scaled vertically for visual comparison. A comparison of simulated and measured values shows that the diplexer shifted up 4 GHz from
intended
design values, causing the lower band to encroach on the bandwidth of the higher band. This leads to unequal bandwidths between the two bands.
(
Right
)
An
unknown impedance step causes a
∼
3 GHz ringing. The atmospheric transmission at the South Pole is shown in gray, with molecular oxygen responsible for the line
at 60 GHz
(
Tretyakov et al.
2005
)
.
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)
, 2024 May
Shiu et al.
the one remaining support leg. The legs are 800
μ
m long. Note
that these differ from the
G
c
values used for the monochromatic
30
/
40 GHz bolometer arrays as described in other publications
(
Zheng et al.
2020
; Zhang
2023
)
. A tabulation of the bolometer
properties and performance is shown in Figure
7
. While
detector sensitivity can be enhanced by reducing phonon noise
with a lower
T
bath
=
100 mK, the incremental gains are
outweighed by the increased cryogenic complexity, given that
these detectors are photon-noise limited. Consequently, a lower
100 mK design was not pursued
(
Zheng et al.
2020
)
.
3. Optical Characterization
This focal plane was cooled down and tested in a testbed
cryostat that mimics the
fi
ltering present on a BICEP-style
receiver but lacks the imaging optics of the complete telescope
insert. A series of optical
fi
lters, including absorbing plastic
fi
lters and re
fl
ective metal-mesh
fi
lters, was used to limit the
radiative thermal load on the focal plane
(
Ade et al.
2006
)
. The
on-chip
fi
ltering ultimately de
fi
nes the bands of the detectors
for science observation. All laboratory measurements were
performed using the aluminum superconducting transition
designed for higher optical load.
To minimize radio-frequency interference
(
RFI
)
, archival
measurements were conducted at night with WiFi routers and
personal wireless devices powered off. Laboratory testing has
demonstrated that the majority of RFI for these styles of
receivers couples in through the readout chain rather than the
optical chain
(
Soliman
2023
)
. To reduce RF pick-up, a Faraday
cage made of 1
/
8 inch wire metal mesh enclosed the readout
electronics and housekeeping electronics. Additionally, alumi-
nized Mylar tape was used between readout modules for RF
shielding.
3.1. Spectral Bandpass Characterization
The spectral response of these antennas,
S
(
ν
)
, was
characterized using a Martin
–
Puplett Fourier transform
spectrometer
(
FTS
)
. The input light is generated by an HR-
10 source submerged in liquid nitrogen for thermal stability of
the source. First, the millimeter-wave radiation passes through
a wire grid, selecting for linear polarization. Subsequently, it is
directed to a collimating mirror and onto a beam splitter
consisting of a wire grid oriented 45
°
to the incident
polarization. The beam is split into two: one path is a
fi
xed
rooftop mirror, and the other is a variable rooftop mirror
continuously driven by a stepper motor. The two paths interfere
when returning to the beam splitter, re
fl
ecting off an output
wire grid. This wire grid is oriented at 45
°
so that only the
desired linear polarization leaves the FTS box, thereby cleaning
up any undesired polarized systematics. Finally, the beam is
focused by an HDPE lens designed to illuminate a single pixel
in the optical testbed. We do not believe that this source is
beam-
fi
lling, so the throughput on the single detector is
expected to be
A
Ω
=
f
λ
2
, where
f
is a fractional value.
However, it is important to note that the FTS source operates
in the Rayleigh
–
Jeans limit:
I
(
ν
)
=
k
b
T
λ
−
2
. Therefore, the
reported detector response
S
(
ν
)
is equivalent to the response to
a source with constant spectral radiance.
The spectral response of the detectors is de
fi
ned in the band
center by
()
()
()
ò
ò
n
nn n
nn
º
Sd
Sd
4
0
and bandwidth by
()
()
(())
()
ò
ò
n
nn
nn
Dº
Sd
Sd
.5
2
2
The normalization of our spectra was computed by measuring
the optical response of our detectors. This was performed by
comparing the optical loading of the detectors under a beam-
fi
lling blackbody source at room temperature and at liquid
nitrogen temperatures. Therefore, the spectra in Figure
6
represent the end-to-end optical ef
fi
ciency of the antennas to in-
band photons, accounting for losses through the optical
elements such as the window and millimeter-wave
fi
lters, as
well as electrical losses through impedance mismatches and
dielectric loss in the microstrip summing tree.
The left side of Figure
6
shows the simulated antenna
bandwidth as the dashed black line, along with simulated on-
chip
fi
lters de
fi
ning the two bands. The simulated antenna
spectrum suggests additional usable bandwidth; however, the
oxygen line at 60 GHz must be rejected using on-chip
fi
lters.
The diplexer continuously distributes power between the two
bands. There is spillover between the two bands, as the
fi
lter is
optimized to minimize re
fl
ected power between the bands
rather than band separation. This design choice was made for
two reasons:
(
1
)
avoiding power loss between the two bands, as
there are no atmospheric lines between 30 and 40 GHz, and
(
2
)
opting for a conservative
fi
ltering scheme. This design avoids
reliance on two electrically interacting on-chip
fi
lters, where
drift in one would interfere with the other through three-port
interactions. Ultimately, this spectral spillover reduces our
lever arm in measuring
β
s
, as the effective frequency centers of
both bands are closer in frequency space.
The right side of Figure
6
shows the full focal plane
averaged spectrum of both bands in blue and orange. The sum
of the two bands is the solid black line, indicating that physical
measurements show a consistent bandwidth between measure-
ments and our simulated expectations. Tabulated band centers
and widths are in Table
2
. High fractional bandwidth is
achieved, with both the high and low bands having
substantially larger fractional bandwidth than the traditional
slot design
(
Zheng et al.
2020
)
.
Power is split between the two contiguous channels,
indicating that the diplexer performs as intended. The upper-
Figure 7.
(
Left
)
A microscope image of a suspended bolometer island on the detector wafer. In the photograph, on-sky power propagates from the upper right leg and
is deposited on the lossy gold meander on the right side of the island. Two DC bias lines come in from the two left legs and bias the Al and Ti TES bolometers i
n
series.
(
Right
)
A tabulation of measured bolometer properties.
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)
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Shiu et al.
and lower-band edges agree with our simulated results. The
diplexer appears to have shifted from the design frequency by
10%, which causes more spillover from the lower band to the
upper band than intended.
The achieved optical ef
fi
ciency is modest, with end-to-end
measurements of 20%
–
30% optical ef
fi
ciency. A portion of
these quoted lower optical ef
fi
ciencies occurs due to spectral
spillover; a sum of the bands shows 35% ef
fi
ciency in the
passband. While this ef
fi
ciency remains lower than the
monochromatic slot design, these detectors remain competitive,
with similar optical responsivity, due to larger overall
bandwidth.
Additionally, there is an observed
∼
3 GHz ringing in the
frequency bands, suggesting the presence of an impedance
mismatch associated with this length scale or one of its
harmonics. The effective dielectric constant of microstrip
modes in a 0.3
μ
m SiO
2
is approximately
ò
MS, eff
∼
3.4, which
results in a corresponding wavelength of
∼
54 mm. Unfortu-
nately, many possible junctions within the summing tree have
this length scale, making it challenging to identify the source of
this impedance mismatch.
In future iterations of this design, we would adjust the
diplexer to shift the frequency down so that the two bands have
a more balanced distribution. Alternatively, we can employ a
different
fi
ltering scheme to achieve a sharper separation
between the two bands, at the expense of some loss of photons
between the two bands. A preliminary design has been
explored in Appendix
A
. Additionally, more investigative
work must be done to identify the source of impedance
mismatches in the microstrip network, to identify the source of
re
fl
ections. Highlighting that this design meets the baseline
requirement within BICEP Array to achieve its science goals,
there remains signi
fi
cant potential for enhancing the end-to-end
ef
fi
ciency of these detectors to meet simulated expectations.
3.2. Far-
fi
eld Beam Patterns
The antenna beam of a phased antenna array is, in principle,
highly tunable. The con
fi
guration of the radiators, along with
the amplitude and phase applied to each antenna element,
collectively determine the beam shape. The gain of an antenna
array is modeled as the following:
() ()∣()∣
()
qf
qf qf
=
GGAF
,,,, 6
array
0
2
where
G
0
is the gain of an individual element, and
AF
is short for
the array factor. The array factor is the discrete Fourier transform
of the antenna elements. Generically, the array factor can be
expressed as
(
)
()
ˆ
·( )
qf
=
å
p
l
A
FAirxnm
,
exp
,
nm
nm
,
2
,where
ˆˆ
ˆ
qf qf
=+
r
xy
sin cos
sin sin
,where
θ
and
f
are the polar and
azimuthal angles, respectively,
A
nm
is the excitation amplitude of
each radiator, and
()
xnm
,
is the location of each radiator.
The number of subradiators was chosen to match the
telescope optics of f
/
1.5 in BICEP Array
(
Hui et al.
2018
)
, but
in principle, this can be tunable to any optical system. For a
single polarization, a pixel comprises of a 12
×
12 square grid
of resonant bowtie antenna pairs. This parameterization is
convenient, as it allows us to analytically separate the array
factor into the product of two array factors:
⎛
⎝
⎞
⎠
()()
()()
∣( )∣
()
()
qf
qf
qf
qf
qf
p
l
qf f
=
́
p
l
p
l
p
l
p
l
AF
M
a
,
sin
sin cos sin
sin sin
sin
sin cos sin
sin sin
cos
2
2
sin cos
sin .
7
Ma
Ma
aa
22
2
22
The
fi
rst term represents the
M
=
12 square array factor. The
second term represents the two
-element antenna pair, with
the
±
for the A and B polarizations respectively. The sign choice
originates from antenna pairs being located at quadrants 1 and 3
for polarization A and quadrants 2 and 4 for polarization B. This
beam model implies an inherent A
–
B mismatch between the two
polarization pairs, arising from the two-element component
portion of the array factor. Indeed, the A
–
B mismatch for this
component increases with larger polar angles from the antenna
boresight. However, the square array factor concentrates the
antenna beam and minimizes this source of A
–
B mismatch. A
calculation shows that, for a 12
×
12 array, this pairwise array
factor contributes only a 0.7% peak-to-peak A
–
Bmismatch.The
intrinsic antenna element beam,
G
0
, is determined through HFSS
simulations and contributes an additional 2%, leading to a
theoretical limit of a 2.7% peak-to-peak A
–
B mismatch.
Previous antenna arrays have used an 8
×
8 array. This is
expected to have a 1.6% peak-to-peak A
–
B mismatch purely
from the pairwise array factor, and a 5% peak-to-peak A
–
B
mismatch when accounting for individual beams. The
increased number of elements for this antenna array design
suppresses differential ellipiticity and leads to overall better
beam performance.
Far-
fi
eld beams were characterized by an in-house beam
mapper, which consists of a Thorlabs MC2000 optical chopper:
a variably controlled
(
2
–
20 Hz
)
thermally chopped source that
is mounted on an
X
–
Y
translation stage. The thermal source is a
heated ceramic, reaching temperatures of several hundreds of
degrees Celsius, and is chopped with respect to a re
fl
ective
room-temperature blade. The re
fl
ected beams terminate within
the cold optics of the testbed. To prevent multiple re
fl
ections in
our measurement setup, we took care to blacken the beam
Table 2
Summary Statistics of the Optical Performance of the Diplexed Focal Plane
Spectra
Beams
ν
0
Δ
ν
n
n
D
0
FWHM
M
(
deg
)
FWHM
m
(
deg
)
Δ
(
M
−
m
)
(
deg
)
Low Band
33.7 GHz
21.2 GHz
63%
18.5
±
1.3
16.9
±
0.9
1.6
±
1.2
High Band
41.5 GHz
15.4 GHz
37%
15.1
±
1.6
14.1
±
1.4
0.8
±
0.4
Note.
A shift in the diplexer central frequency has caused spillover of the lower-band bandpass, encroaching on the bandwidth of the upper band. The beam-el
lipticity
statistics are driven by undesired polarized frame and edge-pixel electromagnetic interaction, which make up the majority of the pixels on a 4
×
4 focal plane. Notably,
center pixels and edge pixels with the orthogonal polarization to the frame, show signi
fi
cant improvements to their beam ellipticities.
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)
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Shiu et al.
mapper facing the detectors with Eccosorb HR-10. Since there
are no focusing optics in the testbed cryostat, the chopped
source was located in the far
fi
eld of the detector array.
We demodulate the time-ordered data using the chopper
optical encoder as reference, so that only variations at the chop
frequency and in phase are interpreted as signal. This reference
signal was read out using the same readout electronics
employed for the detectors.
Out-of-band photons, rather than being captured by the
antennas and
fi
ltered out by the on-chip
fi
lters, can inadver-
tently directly illuminate the bolometer island. The exact
mechanism for this response is not fully understood, but a
plausible mechanism is that it caused by potential differences in
the bolometer island relative to the ground plane, which drive
currents in the resistive termination and heat the bolometer
island
(
Zhang
2023
)
.
In order to control for the baseline response of the detectors,
we measured the direct island response using out-of-band,
high-frequency photons. This response is mapped out and
subtracted from our nominal beam maps. These data were
acquired by
fi
ltering the source with a specially designed thick-
grill
fi
lter, a one cm thick metallic plate with a dense array of
circular waveguide apertures. The
fi
lter is designed to permit
only power above the waveguide cutoff, 61 GHz, allowing
illumination solely through direct island stimulation.
Figure
8
shows the resulting beam map, corrected for direct
island illumination. Each of the beam maps are
fi
tted to a 2D
Gaussian with full width at half maximum
(
FWHM
)
in the plot
inset and
−
3,
−
5, and
−
10 dB contours plotted on top. We also
include the
f
/
1.5 optics on top of the beam map. The theoretical
expectations for 12
×
12 pairs of antennas, convolved with the
beam of a single element bowtie antenna, match well with the
measured results. For a
δ
-function response at 35 and 45 GHz,
we compute an FWHM of 19
°
.6 and 15
°
.8, respectively. Real
beam maps are an average over the entire band.
In the rightmost panel of Figure
8
, we show A
–
B differences
for this representative edge pixel. A simulated
fi
nite antenna
array, with no edge effects, is expected to have
∼
3% peak-to-
peak differences in a quadrupole pattern.
Edge effects caused by a solid frame are expected to cause
strong differential pointing and result in peak-to-peak varia-
tions of 30%
–
40%, depending on the precise distance between
the frame and the antenna array
(
Soliman
2023
)
. An optimal
spacing and corrugated frame is expected to achieve polarized
beam residuals of less than 10% from peak to peak. Our
measurements show that a representative edge pixel has 15%
peak-to-peak beam residuals. This is similar to beam maps
from center pixels, suggesting that the polarized beam residuals
are not dominated by electromagnetic interactions with the
corrugated frame.
4. Conclusion
At any frequency, dichroic detectors offer twice the number
of detectors in the same focal plane footprint, allowing a
2
improvement in noise-equivalent temperature
(
NETs
)
. How-
ever, at these lower CMB observation frequencies, pixel sizes
Figure 8.
Far-
fi
eld beam-map measurements of a representative low-band detector pair
(
upper row
)
and a high-band detector pair
(
lower row
)
. These four detectors are
low- and high-band A
/
B polarization pairs from an edge pixel. All plots show the beam map and a 2D Gaussian
fi
t with
−
3,
−
5,
−
10 dB black dashed contour lines
overlaid on top. The white line overlays the optical footprint of an
f
/
1.5 system. The polarized frame edge affects the Pol. A antennas and causes an ellipticity of the
main beam. Pol. B is unaffected by the frame. The 2D Gaussians are differenced in the last column, showing A
–
B differences. The peak-to-peak differences for the
low band are 9% and
−
6%, and for the high band they are 10% and
−
5%. This detector shows typical performance. Peak-to-peak variations as low as 10% have been
measured on edge pixels; this is near the
∼
8% theoretical limit.
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)
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Shiu et al.
must scale accordingly with the wavelengths, resulting in
naturally smaller values for
N
. The NETs show steeper
improvement in this regime, and therefore there is a larger
absolute bene
fi
t from increasing
N
at these lower frequencies.
We have developed a compact, linearly polarized antenna
with a broad
fi
rst resonance. This
fi
rst resonance can effectively
be divided into two distinct frequency bands, operating at
30
/
40 GHz.
This antenna is compact and well-suited for integration into
BICEP
’
s phased antenna array architecture. The detectors are
entirely planar and fully lithographed in thin
fi
lms without
the need for any focusing optics. The top-hat illumination
maximizes the directivity of the antenna beam within the
smallest area. It exhibits beam characteristics on par with the
slot antenna arrays, making it a suitable choice for CMB
polarimetry. Overall, the antenna array architecture enables the
highest pixel density for a
fi
xed focal plane area.
This detector array deployed in the inaugural season of
BICEP Array, and it is the
fi
rst demonstration of a dichroic
planar array system ever used for CMB measurements at the
South Pole
(
Soliman
2023
)
. This work enhances the sensitivity
of BICEP array toward low-frequency foregrounds. The dual-
frequency measurement enables simultaneous measurement of
both the synchrotron amplitude and its spectral index, offering
a more powerful constraint on this source of foreground for
CMB polarimetry. Further on-sky synchrotron data, recorded
by this detector, are being analyzed for a potential publication
in the near future.
Immediate future work would be to improve the optical
ef
fi
ciency of the detectors by identifying the source of
impedance mismatch in the microstrip summing tree. Further-
more, the potential for higher-frequency focal planes
(
220
–
270 GHz
)
would allow for ever more precise constraints
of thermal dust emission. However, the scalability of the
current design poses certain challenges. While the antenna
elements themselves would scale in proportion to wavelength,
the microwave feed network encounters nontrivial scaling
issues. The impedance of microstrip lines and desired spacing
between these lines may not necessarily decrease with higher
frequencies. The existing design, already constrained by space
at these frequencies, poses a challenge in scaling down the
antenna elements while maintaining similar microstrip trace
widths. These challenges have been preliminary addressed in
this publication
(
Soliman
2023
)
for two CMB observation
bands at 90
/
150 GHz, simultaneously. Additionally, dielectric
loss is larger at higher microwave frequencies, leading to
expected degradations in performance. These challenges
underscore the considerable efforts required in order to extend
the design to higher frequencies. Addressing these issues would
allow us to place more detectors in the sky; this is especially
important at higher frequencies, where detectors are more
easily limited by photon background.
Acknowledgments
This research was carried out
(
in part
)
at the Jet Propulsion
Laboratory, California Institute of Technology, under a
contract with the National Aeronautics and Space Administra-
tion and funded through JPL
ʼ
s Strategic University Research
Partnerships
(
SURP
)
program. This publication is also
supported by the National Aeronautics and Space Administra-
tion grant No. NNX17AC55G.
Appendix A
Improving Diplexer Separation
The diplexer design in Section
2.5
is optimized to minimize
re
fl
ections in band but exhibits signi
fi
cant spillover between the
two bands. The band overlap diminishes our ability to measure
the synchrotron SED slope. In a separate fabrication run, we
optimized the diplexer for band separation.
We designed a diplexer shown in Figure
9
, consisting of two
parallel bandpass
fi
lters, each centered at 30 and 40 GHz with a
10 GHz bandwidth. The architecture is identical to previously
designed bandpasses: a three-pole LC-tank joined with shunt
capacitors acting as impedance inverters.
In order to minimize electrical interactions between the two
bands, we found it advantageous to have a steep cutoff for the
bandpass
fi
lters. When the bandpass
fi
lter is out of band, the
resistance drops to zero and the reactance diverges. A sharper
bandpass results in less interaction between the two
fi
lters. We
found it advantageous to use a diplexer with 0.5 dB ripples or
have variations of up to 12% in the passband.
Achieving this steep cutoff required a substantial increase in
inductor values. However, achieving these inductor values
lithographically would be challenging at our operational
frequencies. Instead, we found it advantageous to decrease
the diplexer
’
s operating impedance,
Z
0
, from 25
Ω
to 10
Ω
to
decrease inductor values while keeping the capacitors within
achievable values. Additionally, unlike our previous bandpass
fi
lter, we did not need to perform a
Y
-to-
π
transformation for
the capacitive network. Therefore, Figure
9
shows the circuit
topography optimized for our design. This is then translated to
lithography on the right side of Figure
10.
The measured spectra in Figure
10
show that this diplexer
scheme achieves sharper band de
fi
nitions between the two
pixels. However, strong ringing in the passband degrades pixel
Figure 9.
A diplexer composed of two bandpass
fi
lters. Each bandpass
fi
lter is composed of a three-pole LC resonator circuit. The design table for the circuit network
is tabulated on the left. To scale from design table to physical values: inductor values are scaled by
w
-
Z
0
1
0
and capacitor values are scaled by
()
w
-
Z
00
1
, where
ω
0
=
2
π
f
0
is the desired central frequency of the bandpass
fi
lters
(
30 and 40 GHz
)
, and
Z
0
is the port impedance. For this design,
Z
0
=
10
Ω
has been lowered in order
to achieve smaller and more physically realizable inductor values. Simulations are then performed to convert the electrical inductance and capacit
ance to a lithographic
element.
10
The Astrophysical Journal Supplement Series,
272:12
(
13pp
)
, 2024 May
Shiu et al.
performance, resulting in lower optical response. This design
requires further investigation to identify the source of the
impedance mismatch
’
s source.
Appendix B
Procedure to Convert Lumped Elements to Lithography
with Sonnet
This appendix documents our procedure for converting
lumped elements into lithographic designs. The fundamental
building block of a bandpass
fi
lter is the LC resonator,
consisting of a series inductor and series capacitor as shown in
Figure
11
.
The output of a Sonnet simulation is an S-matrix of the two-
port network. These can be converted to an impedance matrix:
()( )
()( )
()
=
+-+
---
zZ
ssss
ssss
11
11
,B1
11
0
11
22
12 21
11
22
12 21
()( )
()
=
---
zZ
s
ssss
2
11
,B2
12
0
12
11
22
12 21
()( )
()
=
---
zZ
s
ssss
2
11
,B3
21
0
21
11
22
12 21
()( )
()( )
()
=
-++
---
zZ
ssss
ssss
11
11
.B4
22
0
11
22
12 21
11
22
12 21
The equivalent circuit for the
Z
-matrix of any reciprocal two-
port network is shown in Figure
12
. Therefore, a generic series
impedance from a two-port network can be calculated by the
impedance parameters as demonstrated in Figure
12
:
()
=
-
Z
ZZ
Z
Z
.B5
11 22
12
2
12
Additionally, the circuit model of an LC resonator, shown in
Figure
11
, has the impedance
⎛
⎝
⎞
⎠
()
w
w
w
w
=+ = -
ZiL
iC
iL
C
22
.B6
The derivative of this expression gives us an additional
constraint equation:
⎛
⎝
⎞
⎠
()
p
w
p
w
w
w
¶
¶
=
¶
¶
=+
Z
f
Z
iL
C
2
22
.B7
These may be expressed as a system of equations:
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎞
⎠
⎟
()
()
w
p
w
w
¶
¶
=
-
Z
Z
f
i
L
C
2
12
12
1
,B8
with the corresponding inverse
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
()
w
w
w
p
=-
-
¶
¶
L
C
i
Z
Z
f
1
2
11
1
2
1
2
2
.B9
This allows us to solve for the inductance and capacitance,
where
Z
determined by Equation
(
B5
)
:
⎜⎟
⎛
⎝
⎡
⎣
⎢
⎤
⎦
⎥
⎞
⎠
[]
(
)
w
w
p
=+
¶
¶
LImZIm
Z
f
1
22
,B10
⎜⎟
⎛
⎝
⎡
⎣
⎢
⎤
⎦
⎥
⎞
⎠
[]
(
)
w
w
p
=- +
¶
¶
-
CImZIm
Z
f
1
42
.B11
1
Additionally, the parasitic capacitance is
[]
[]
()
ww
=
-
-
=
-
-
C
i
ZZ
ZZ
Z
Im Z
Im Z
ZZ
Z
11
.B12
p
22
11
11 22
12
2
22
11
11 22
12
2
To develop a lithographed
fi
lter, we
fi
rst compile a
comprehensive library of LC resonator simulations for a
fi
xed
dielectric stack. We use the niobium metal model in Sonnet,
which has zero resistance in DC and AC and accounts for
kinetic inductance with
L
s
=
0.11 pH
/
sq. We conduct
simulations with various lengths of capacitors and inductors,
systematically tabulating the lumped-element equivalent mod-
els, including the frequency dispersions of both elements.
Subsequently, when aiming for a speci
fi
c lumped-element
model, we interpolate the precise geometric parameters
necessary for achieving the desired
fi
lter characteristics. Then,
a realized lithographic
fi
lter is fully simulated and tweaked if
necessary. This iterative process allows us to
fi
ne-tune and
optimize the lithographic design to meet speci
fi
c performance
criteria and minimize frequency dispersion.
Figure 10.
(
Left
)
Measured spectra of the double-bandpass diplexer from a pair of characteristic pixels. We have a stronger band de
fi
nition between the two bands.
However, an impedance mismatch degrades the performance of these pixels. The photograph on the right shows the newly designed diplexer consisting of
two
bandpass
fi
lters. Each bandpass comprises a three-pole series LC resonator circuit joined by shunt capacitors.
11
The Astrophysical Journal Supplement Series,
272:12
(
13pp
)
, 2024 May
Shiu et al.