# Design of an Ultra-Wideband Antenna Feed and Reflector for use in Hydrogen
Intensity Mapping Interferometers

John Podczerwinski1 and Peter Timbie1 1Department of Physics, University of
Wisconsin Madison, Madison, WI 53703, USA

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###### Abstract

This paper describes the design of a 5.5:1 bandwidth feed antenna and
reflector system, intended for use in hydrogen intensity mapping experiments.
The system is optimized to reduce systematic effects that can arise in these
experiments from scattering within the feed/reflector and cross-coupling
between antennas. The proposed feed is an ultra wideband Vivaldi style design
and was optimized to have a smooth frequency response, high gain, and minimal
shadowing of the reflector dish. This feed can optionally include absorptive
elements which reduce systematics but degrade sensitivity. The proposed
reflector is a deep parabolic dish with $f/d=0.216$ along with an elliptical
collar to provide additional shielding. The procedure for optimizing these
design choices is described.

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##  1 Introduction

This paper describes the design of a 5.5:1 bandwidth feed antenna and
reflector intended for applications in radio astronomy, particularly for the
proposed Packed Ultra-wideband Mapping Array (PUMA) telescope Ansari _et al._
(2019). This feed will be referred to as the “proposed feed” in the paper. The
proposed PUMA telescope is a radio-interferometer consisting of 32K tightly
packed antennas. Each antenna will consist of a feed suspended over a 6 m
diameter parabolic reflector. These antennas are planned to have a bandwidth
from 200 to 1100 MHz, with extra emphasis placed on the performance from 200
to 475 MHz.

The paper also describes a version of the feed, referred to as the “minimum
systematics feed”. This feed is similar to the proposed PUMA feed but includes
absorptive elements that decrease systematic effects but increase the noise
temperature of the antenna system. It will be argued that this design may be
useful for telescopes operating at lower frequencies than PUMA.

A major goal of observational cosmology is to measure the statistical
properties of the spatial distribution of matter in the Universe. Such
measurements are of interest as they could tell us about dark energy,
inflationary physics, the growth of structure, early star formation and more.
The distribution of matter in the Universe is traced by neutral hydrogen (HI).
Thus, observations of the 21 cm line from HI can probe the matter distribution
(up to a bias term). Using 21 cm emission or absorption to measure these
density fluctuations without resolving individual galaxies is a technique
referred to as 21 cm (or HI) intensity mapping Chang _et al._ (2008). Because
of the expansion of the Universe, the 21 cm line will be redshifted by an
amount that scales with the distance between the emitting region and Earth.
For this reason, different epochs of the Universe can be probed based on the
range of observing frequencies chosen. The information that can be extracted
from the distribution varies with cosmological redshift, denoted by $z$.
Measurements of the distribution at redshifts $z\leq 6$ could probe the growth
of structure, the physics of dark energy, and inflationary physics.
Measurements during the Epoch of Reionization (EOR) ($6<z<20$) would tell us
about the formation of the first stars. Lastly, measurements during the cosmic
dark ages ($20<z<1100$), an epoch preceding the first stars and relatively
unaffected by astrophysical phenomena, would probe the physics of inflation.
The 200 to 1100 MHz bandwidth of PUMA is chosen to capture redshifts $0.3\leq
z\leq 6$.

For reasons of sensitivity and cost, radio-interferometers are typically used
for HI intensity mapping. In interferometry, an array consisting of many
antenna elements is used. The signals from pairs of antennas are correlated to
form “visibilities”. In the absence of mutual coupling, correlating these
signals eliminates the offset from system noise one sees in single dish
telescopes. This in turn relaxes requirements for gain stability. Moreover,
combining signals from many pairs of antennas allows the same angular scale to
be measured many times, providing a distinct advantage over single dish
experiments.

Such interferometers are optimized to measure the spatial power spectrum
$P(k)$ of the 21 cm line. Cosmologists are particularly interested in large
spatial scales, corresponding to small values of the wavenumber $k$. This
interest in large spatial scales leads to interferometers with tightly packed
antenna elements. In the case of PUMA, at the longest wavelength, dishes are
planned to be about $4\lambda$ wavelengths in diameter with centers located
about $5\lambda$ from each other. This spacing is small enough to probe large
spatial scales at all wavelengths, while also being large enough to allow
dishes to be repointed to latitudes away from zenith without beam blockage.
Moreover, the HI signal being measured is small, with a brightness temperature
of roughly 1 mK, while system temperatures are the order of $\sim 25-100$ K in
the post EOR universe and up to $\sim 1000$ K in the case of EOR measurements.
The small magnitude of the signal being measured motivates interferometers
with large collecting areas. These requirements lead to ‘large N, small D’
designs, which consist of large numbers of small diameter antennas. Although
this paper focuses on the proposed PUMA telescope, other designs exist as
well. Table 1 provides summaries of several interferometers intended for
measuring the EOR and post EOR Universe.

[h] Summary of a few interferometers intended for 21 cm intensity mapping.
Fiducial values for PUMA’s aperture efficiency and amplifier temperature are
used. Antenna Bandwidth Array Geometry Effective Aperture per Element
PUMAAnsari _et al._ (2019) 6 m diameter parabolic dish, feed on-axis 0200-1100
MHz $6\geq z\geq 0.3$ (Post EOR) 32K tightly packed antennas.
$\hphantom{0}\approx 20\mathrm{m}^{2}$ HIRAXSaliwanchik _et al._ (2021) 6 m
diameter parabolic dish, feed on-axis 0400-800 MHz $2.55\geq z\geq 0.78$ (Post
EOR) 1024 elements in a $32\times 32$ tightly packed array.
0$15.5\mathrm{m}^{2}$ at 600 MHz CHORDVanderlinde _et al._ (2019) 6 m diameter
parabolic dish, feed on-axis. Also includes 90 m long 10 m wide cylindrical
reflectors with a focal ratio of $0.25$. 0300-1500 MHz $3.7\geq z\geq 0$ (Post
EOR) 512 tightly packed dishes at the central station. Two outrigger stations
placed 1500 km and 3000 km away. These outrigger stations have 64 dishes and 1
cylinder each. HERADeBoer & HERA (2015) 14 m diameter parabolic dish, feed on-
axis 50-250 MHz $27\geq z\geq 4.7$ (EOR) 350 dishes tightly packed in a
hexagonal array. 32 outrigger antennas placed several hundred meters from the
center of the array. $\hphantom{0}\approx 100\mathrm{m}^{2}$ at 150 MHz

In an HI intensity mapping instrument, candidate antennas must be designed to
minimize systematic effects. The primary systematic effect is contamination
from Galactic synchrotron radiation. The Galactic foregrounds are of concern
as they are about 5 orders of magnitude brighter than the predicted HI signal.
However, the foregrounds have a smooth spectrum, while the spectrum of the HI
signal is quite ‘spikey’, corresponding to clumps of hydrogen gas along the
line of sight. Thus, one can confine the Galactic foregrounds into a small
region of the delay domain by ensuring that the antennas have a smooth
response as a function of frequency. In 21 cm intensity mapping, delay is the
Fourier dual to spectral frequency Ewall-Wice _et al._ (2016). Moreover, one
needs to minimize mutual coupling between antennas, which becomes challenging
due to the close element spacing required in HI interferometers. Such mutual
coupling can degrade delay spectrum performance, introduce correlated noise,
and make certain calibration methods more difficult. Designs that minimize
these systematic effects can be seen in Fagnoni _et al._ (2021) as well as
Saliwanchik _et al._ (2021). Of course, one also needs to consider the
sensitivity of the instrument to the HI power spectrum. Sensitivity will be
given less priority during optimization than the previously mentioned
systematics effects, since sensitivity can be improved by increasing the
number of elements or observing for longer.

Lastly, polarization leakage also should be considered. The desired HI signal
is unpolarized, while the foregrounds do have a polarized component. Leakage
of polarized foregrounds into estimates of unpolarized emission can be caused
by beam ellipticity and by cross-polarization in the beam. Such leakage is an
additional systematic effect one could consider when designing an antenna for
HI intensity mapping. This design was not optimized with polarization leakage
in mind, but cross-polarization and leakage estimates are still provided in
Section 4.4.

This paper is organized as follows. Section 2 describes the performance
requirements of the instrument and the motivation for the requirements.
Section 3 describes the optimization process for the feed antenna along with
the resulting design. In addition, it shows simulations of the S-parameters
and beam patterns. Section 4 summarizes the performance of both the proposed
and minimum systematics antenna systems. Finally, Section 5 describes the
fabrication of a $4/13$ scale version of the minimum systematics feed and
provides some measured results.

##  2 Performance Requirements

The feed antennas and reflector were designed with HI intensity mapping
instruments such as PUMA in mind. PUMA is a large, tightly packed antenna
array. In instruments such as PUMA, mitigation of systematic effects is of
primary importance. In addition, one must also consider the sensitivity of the
instrument to the signal being measured.

In Sections 2.1 and 2.2, aspects of performance most crucial to minimizing
systematic effects are considered. In Section 2.3, sensitivity of the
instrument to the HI signal is considered.

###  2.1 Spectral Smoothness and the Foregrounds

Due to their smooth frequency spectrum, the foregrounds encountered in
instruments such as PUMA are naturally contained in a region of 2-dimensional
k-space (with k components parallel to and perpendicular to the line of sight)
referred to as “the wedge”. When taking measurements, the wedge is convolved
with the response of the instrument, causing foregrounds to leak out into
otherwise signal-dominated modes. In 21 cm intensity mapping, it is the case
that delay ($\tau$) is proportional to line of sight wavenumber $k_{||}$. It
is thus desirable for the antenna response in the delay domain be concentrated
at 0 delay and decay rapidly at larger delays.

In order to test the smoothness of the antenna’s response, we will excite the
antenna with a planewave coming from some direction $\hat{n}$. The voltage at
the terminals of the antenna is then given by

$v(\hat{n},\nu)=\mathbf{r}(\hat{n},\nu)\cdot\mathbf{E}(\hat{n},\nu).$ (1)

In this formula, $\mathbf{r}$ is referred to as the “voltage beam” of the
antenna. We then quantify the delay performance of the antenna is via the
Fourier transform of the power kernel. The power kernel is defined as

$R(\hat{n},\nu)=|\mathbf{r}(\hat{n},\nu)|^{2},$ (2)

In particular, we will characterize the delay response of the instrument using
the delay spectrum of the power kernel at zenith. This is denoted via
$\tilde{R}(\theta=0,\phi=0,\tau)$. The goal is to have $\tilde{R}(0,0,\tau)$
drop by 50 dB as quickly as possible as a function of $\tau$. Refer to Ewall-
Wice _et al._ (2016) for more details.

###  2.2 Mutual Coupling

Another concern in HI intensity mapping instruments is mutual coupling between
the antennas in the array. It has been found that mutual coupling changes the
beam patterns of antenna elements as described in Fagnoni _et al._ (2020) and
Kern _et al._ (2019). This effect on the beam patterns will likely degrade
spectral smoothness Fagnoni _et al._ (2020). Moreover, it causes beam patterns
to vary from element to element. Such beam pattern variation is undesirable as
it would prevent the use of redundant calibration techniques which have been
proposed to overcome the computational challenge of calibrating large,
redundant, arrays. In redundant calibration, one assumes that antennas in the
array have identical beam patterns. This assumption would not be valid in the
case of significant mutual coupling Liu _et al._ (2010).

Mutual coupling can also cause receiver noise radiated from one antenna to be
absorbed by a different antenna. This coupling creates “correlated noise”,
which causes non-zero visibilities. These correlated noise contributions can
masquerade as sky signals Kwak _et al._ (Submitted). In principle, this
coupling varies slowly in time, but it sets a requirement on the gain
stability of the instrument.

During the design process, mutual coupling was characterized by simulating the
$S_{21}$ coupling between two adjacent antennas. The antennas are placed 8 m
apart and pointed at zenith. Such a setup can be thought of as the worst case
scenario for coupling between two antennas in arrays such as PUMA. Simulations
were performed with antennas co-located in each other’s E-planes and H-planes.
Using the model for coupled signals described in Kern _et al._ (2019), one
should find that the “correlated noise” between two antennas follows Eq 3.

$T_{coupled}\propto|S_{21}|T_{ampl}.$ (3)

Since $T_{ampl}$ is typically 4 to 5 orders of magnitude larger than $T_{HI}$,
the $S_{21}$ must be kept below $-100$ dB to prevent correlated noise from
overpowering the signal. It should be acknowledged that the model presented in
Eq 3 is an approximation. A more sophisticated model of this phenomenon is
presented in Kwak _et al._ (Submitted).

###  2.3 Sensitivity

In addition to minimizing systematic effects, we require this antenna (and
array) to provide enough sensitivity to detect the HI signal. In such an
measurement, the contribution of system noise to the power spectrum goes like
Liu & Shaw (2020); Parsons _et al._ (2014)

$P_{N}\propto\frac{\Omega_{p}^{2}}{\Omega_{pp}}T_{sys}^{2},$ (4)

Let $P$ be the power pattern of the antenna. We define $\Omega_{p}$ and
$\Omega_{pp}$ in terms of $P$ via

$\Omega_{p}\equiv\int d\Omega P(\hat{\mathbf{n}})$ (5)

and

$\Omega_{pp}\equiv\int d\Omega P(\hat{\mathbf{n}})^{2}.$ (6)

In these formulas, the system temperature $T_{sys}$ is estimated via Ansari
_et al._ (2019)

$T_{sys}\approx\frac{T_{ampl}}{\eta_{sp}\eta_{c}}+\frac{1-\eta_{sp}}{\eta_{sp}}T_{ground}+T_{sky}.$
(7)

In this formula, $T_{ampl}$ is the noise temperature of the amplifier and
$\eta_{sp}$ is defined as the fraction of the antenna’s beam that shines on
the sky. One can compute $\eta_{sp}$ using

$\eta_{sp}=4\pi\int_{0}^{\frac{\pi}{2}}\int_{0}^{2\pi}d\theta d\phi
G(\theta,\phi).$ (8)

The optical efficiency $\eta_{c}$ is defined as

$\eta_{c}=\eta_{S11}\eta_{rad},$ (9)

where $\eta_{S11}$ is the fraction transmitted through the feed antenna’s
input terminals and $\eta_{rad}$ is the radiation efficiency.

For the nominal PUMA design Ansari _et al._ (2019), the proposed values of
these parameters for achieving adequate sensitivity are: $\eta_{sp}=0.9$,
aperture efficiency $\eta_{A}=0.7$, $\eta_{c}=0.9$ and $T_{ampl}=50$ K. Thus,
the designs proposed in this paper ought to provide comparable sensitivity to
these fiducial parameters.

The performance requirements for the antennas proposed in this paper are
summarized in Table 2.3. [h] The primary performance requirements for the
antenna described in this paper. Aspect of Performance Goal Delay Response The
antenna response at zenith decays by 50 dB as quickly as possible in the delay
domain. As a realistic goal, we wanted to keep the $50\,dB$ half width
$\lesssim 0.10hMpc^{-1}$ when described in k-space. Mutual Coupling The
$S_{21}$ of two antennas separated by 8 m is at or below -100 dB when the
antennas are co-located in each other’s E-plane or H-plane. Impedance Match
The feed has $S_{11}\leq-10\,$dB. The final design presented will be relative
to $200\,\Omega$. Power Spectrum Sensitivity Power spectrum sensitivity is
comparable to that achieved by the fiducial PUMA parameters Ansari _et al._
(2019). This is achieved via a low $S_{11}$ and very high spillover efficiency
($\eta_{sp}\approx 0.99$). Bandwidth All of these requirements are satisfied
over the entire PUMA band,200 to 1100 MHz.

##  3 The Proposed Antenna and its Optimization

This section describes the electrical and mechanical design of the proposed
system, and motivates the choices made to achieve the performance
requirements. The proposed antenna system was designed with the
electromagnetic simulation software package CST Microwave Studio
111http://www.cst.com. The work described in this paper primarily used CST’s
time domain solver. However, the frequency domain solver and integral equation
solver were used at various times to double check results. All beam patterns
presented in this work were computed using the “open (add space)” boundary
condition in CST.

###  3.1 Dish Reflector

In this paper, $6$ m diameter parabolic dishes with $f/d=0.216$ and $0.375$
are shown. These dishes are on-axis reflectors. Off-axis reflectors were also
considered, but are not favored since they provide limited shielding against
mutual coupling between antennas. The diameter of the dishes corresponds to
$4\lambda$ at the longest wavelength. This small dish size was chosen since
smaller dishes allow for a more tightly packed array, which in turn provides
access to larger angular scales on the sky. This small dish size drove many
design choices in the feed. In particular, it was one of the main reasons a
Vivaldi-style feed was chosen, as opposed to a quad-ridge horn. The use of a
horn antenna would cause too much shadowing when placed directly over a small
dish.

A low $f/d$ ratio is favored in order to provide shielding for the feeds,
which reduces mutual coupling. An example of the benefits of using a deeper
dish can be seen in Saliwanchik _et al._ (2021). In addition, outfitting the
deep dish with a collar to provide extra shielding was considered. In this
particular paper, a simple elliptical collar was used. Collars providing a
smoother junction with the dish can be designed using the methods shown in
Gupta _et al._ (1990). Using a curved collar, whether elliptical or through
Gupta’s method, is desirable for two reasons. The first is that these methods
minimize the discontinuity at the boundary between the parabola and the
collar. This in turn minimizes diffracted fields emanating from that boundary.
The second reason is that the curvature of the collar minimizes standing waves
between the feed and collar. If this were not the case, then the presence of
the collar would degrade the delay spectrum significantly.

Some of the dishes considered can be seen in Figure 1.

Figure 1: The three dishes considered in this paper. The top image shows the
$f/d=0.216$ dish with the elliptical collar included. The middle image shows
the $f/d=0.216$ dish without the collar. The bottom image shows the
$f/d=0.375$ dish.

Decreasing $f/d$ increases reflections between the dish and feed, which in
turn degrades delay spectrum performance. Fortunately, the reduced mutual
coupling provided by a deep dish improves delay spectrum performance by
cutting down on inter-dish reflections Saliwanchik _et al._ (2021). It was
found in the Saliwanchik et al that these inter-dish reflections dominate over
intra-dish reflections. Thus, $f/d$ should be chosen as low as possible while
still being mechanically feasible and providing adequate sensitivity.

Figure 2: Directivities in the E and H-planes for the proposed antenna system

. These patterns were simulated using the $f/d=0.216$ reflector with an
elliptical collar. Note that the beam patterns of the minimum systematics feed
look similar to these. However, the directivity at the horizon tends to be
lower.

Some example beam patterns can be seen in Figure 2. These patterns were
computed using an $f/d=0.216$ dish with an elliptical collar. One will note
that the value of the directivity at $90^{\circ}$ is about $40\,$dB to
$50\,$dB below its peak value. This behavior is desirable as low directivity
towards the horizon corresponds to lower mutual coupling between antenna
elements.

One will also notice however, that a there is a dip in gain at zenith for the
$1000$ MHz and $1100$ MHz plots. This feature is not particularly concerning
with respect to power spectrum measurements, as one is still sampling the same
patch of the uv plane whether the null is present or not. So, as long as the
beam is calibrated well, it should not be an issue. However, the complexity of
these beams is concerning for calibration, since a complicated beam pattern
will be more difficult to measure whether using drones or point source
references.

It should be noted that this dip is due to phase center variation with respect
to frequency. It can be fixed by adjusting the location of the feed with
respect to the focus of the dish. However, such adjustments will come at the
expense of phase efficiency at low frequencies.

[h] Performance when the minimum systematics feed is paired with different
reflector designs. $\mathbf{S_{11}}$ (dB) Directivity (dBi) Maximum Sidelobe,
E-plane (dB) Maximum Sidelobe, H-plane (dB) $\mathbf{\eta_{sp}}$ f/d=0.375
-7.6 at 200 MHz -14.5 at 600 MHz -35.8 at 1100 MHz 20.2 at 200 MHz 28.3 at 600
MHz 31.1 at 1100 MHz -24.4 at 200 MHz -15.8 at 600 MHz -20.1 at 1100 MHz -20.0
at 200 MHz -15.6 at 600 MHz -18.7 at 1100 MHz 0.929 at 200 MHz 0.932 at 600
MHz 0.927 at 1100 MHz f/d=0.216 without collar -7.0 at 200 MHz -15.5 at 600
MHz -18.1 at 1100 MHz 18.7 at 200 MHz 24.8 at 600 MHz 24.0 at 1100 MHz -16.9
at 200 MHz -16.2 at 600 MHz -9.4 at 1100 MHz -13.9 at 200 MHz -14.7 at 600 MHz
-14.3 at 1100 MHz 0.985 at 200 MHz 0.996 at 600 MHz 0.998 at 1100 MHz
f/d=0.216 with collar -7.0 at 200 MHz -15.0 at 600 MHz -18.3 at 1100 MHz 19.4
at 200 MHz 24.2 at 600 MHz 24.0 at 1100 MHz -19.3 at 200 MHz -15.8 at 600 MHz
-10.1 at 1100 MHz -14.5 at 200 MHz -15.7 at 600 MHz -15.0 at 1100 MHz 0.989 at
200 MHz 0.998 at 600 MHz 0.998 at 1100 MHz

###  3.2 Feed

The feed designs presented in this paper are of a Vivaldi style. Such a design
was chosen for its wide bandwidth, high directivity, and ‘openness’. It is
easier to achieve high spectral smoothness with an open design since there are
no enclosed spaces in which resonances might form. Moreover, the Vivaldi
geometry minimizes aperture blockage, which is especially important in this
context where the dish size is only $D\approx 4\lambda$ at the longest
wavelength.

Figure 3 presents a drawing of the minimum systematics feed. The proposed feed
is identical to the minimum systematics feed, except it does not include the
rings. All feed designs considered in this paper are shown in Figure 12. In
particular, the proposed feed is shown in the bottom left panel of the figure.
Although not presented, horn antennas were also simulated but provided poor
performance due to aperture blockage.

####  3.2.1 Ridges

This section describes the ‘ridges’ of the feed, which are shown in Figure 4.

Figure 3: Top shows the minimum systematics feed with the cone and rings
removed in order to make certain dimensions easier to see. All dimensions are
in mm. Note that the ridges of the feed are $3.3$ mm thick. The bottom image
shows the minimum systematics feed with certain important pieces labeled. The
proposed feed is identical to the minimum systematics feed, but with the rings
removed. Figure 4: A drawing of one of the four ridges used on the proposed
feed. $r=168\,$mm refers to the radius of the circular flare used at the
aperture. Points denote the locations $(z_{i},x_{i})$ used in the
optimization.

As one can tell from the figure, the proposed feed was designed with a
numerical optimizer.

In particular, the CMA-ES optimization algorithm was used Hansen _et al._
(2003). This algorithm is built-in to CST. The design of these ridges began as
a more traditional Vivaldi shape, below called the ‘vanilla design’, with an
exponential flare inside and flat section outside. The profile was then broken
into line segments connecting points $(x_{1},z_{1}),(x_{2},z_{2}),...$. The
CMA-ES optimizer was then used to find the optimal values for the points
$(z_{i},x_{i})$. Note that only $x_{i}$ was optimized for certain points and
only $z_{i}$ for others.

Allowing the ridge geometry to vary in this way was found to minimize
sidelobes and backlobes, increasing spill efficiency ($\eta_{s}$). Changing
the ridge geometry from the vanilla to jagged design was found to degrade
delay spectrum performance to some extent. However, it was found that this
difference is negligible compared to the non-smooth contribution from dish-
feed reflections. Moreover, the addition of the absorber-lined rings described
in Section 3.2.5 helps somewhat with spectral-smoothness. So, the final feed
design performs comparably to the vanilla design (shown in the bottom right of
Figure 12) as far as spectral smoothness is concerned.

####  3.2.2 Differential Line

The output signals are carried from the antenna to amplifiers via a
differential line, which can be seen in Figure 5. Differential lines have been
used on designs such as the horn described in Beukman _et al._ (2014), and the
HERA Vivaldi described in Fagnoni _et al._ (2021). This choice eliminates the
need for baluns, thus reducing losses and lowering $T_{sys}$. It has also been
shown to improve beam patterns Beukman _et al._ (2014). Moreover, it
introduces the possibility of using common modes for beam pattern shaping
Beukman _et al._ (2014).

This differential line is filled with a PTFE “puck” of diameter $88\,$mm in
order to decrease the impedance. The pins have diameter $4.2\,$mm. The line
was chosen to be $26\,$mm long. These values were determined by the optimizer.
The resulting line has impedance $Z_{0}\approx 200\,\Omega$.

Such an impedance is quite high and not ideal for impedance matching to the
amplifier. The reason the impedance is high comes from the wide bandwidth of
the feed. A lower impedance requires a larger diameter differential line. Such
a line would allow $TE$ and $TM$ modes to appear at the higher frequencies in
the band. It should be noted, though, that one could lower this impedance by
replacing the PTFE with a dielectric with an even higher index. For instance,
using Taconic RF-35, which has $\epsilon=3.5$, would reduce the impedance to
$\approx 150\,\Omega$. However, it would be best to pair this antenna with an
impedance matching circuit between the feed and amplifier. This device would
be similar to the HERA’s front end module (FEM) Fagnoni _et al._ (2021).

Figure 5: The differential line used on the feed. The line is $25.7$ mm long.

####  3.2.3 Tabs

Figure 6: An illustration of the transition from the differential line to the
ridges. Figure 7: A drawing of one of the tabs used to transition from the
differential line to the ridges, as seen in Figure 6. Dimensions are in mm and
the tabs are $4$ mm thick. Figure 8: A drawing of the brick and pin used in
the transition section and differential line respectively. One can see these
pieces in the context of the feed in Figure 6. The brick fits into the slot of
the tab, while the pin is used as part of the differential line.

During optimization, it was found that the ideal thickness for the ridges on
the feed was around $33\,$mm. In order to reduce weight and cost, the
thickness of the ridges was reduced to about $3.3$ mm. Reducing the thickness
of the ridges increased the impedance of the feed, leading to a poor impedance
match. In order to fix the issue, tabs were added, giving a smooth impedance
transition from the differential line to the throat of the feed. A similar
approach was used in Fagnoni _et al._ (2021). An illustration of this
transition can be seen in Figure 6. In this transition, the pins of the
differential line are connected to rectangular “bricks”. These bricks are
$4.2$ mm $\times$ $4.2$ mm $\times$ $68$ mm and are attached to the tabs. The
tabs then connect to the ridges, providing a smooth impedance transition.
Another view of the tabs, with dimensions, can be seen in Figure 7. By using
the tabs, a good impedance match was achieved with only slight degradation in
spillover efficiency compared to when thicker ridges were used.

####  3.2.4 Cone

Figure 9: A drawing of the feed’s cone, with dimensions. Our simulations used
$1.7$ mm as the thickness for the cone, but this dimension is not important
for performance.

To further improve beam patterns, a cone was introduced behind the throat of
the ridges. This cone can be seen in Figures 3 and 9. The cone is meant to
prevent energy from radiating away from the differential line before reaching
the ridges. The taper of the cone is intended to give a smooth impedance
transition from the line to the throat section of the ridges. As shown below
in Section 3.4, the cone degrades the $S_{11}$ at low frequencies by
increasing the impedance. However, it also increases spill efficiency
($\eta_{s}$), which is a more important feature, since it corresponds to lower
mutual coupling levels. Moreover, the presence of the cone was found to
degrade spectral smoothness to some extent. However, as with the jagged ridge
geometry, this is no longer the case once the rings are added to the design.

####  3.2.5 Rings

Lastly, rings were introduced to the design to improve its beam patterns. The
incorporation of the rings came from the desire to decrease cross-coupling. In
order to decrease cross-coupling, one would like minimize the beam pattern at
angles toward the horizon. This sort of behavior is achieved by lowering
illumination at the edge of the dish, i.e. making the feed’s beam pattern
narrower. In order to make the beam narrower, it was necessary to increase the
size of the feed or add additional structure. It was found that the increased
aperture blockage from the larger feed degraded performance. Wrapping a metal
sheet around the feed to turn it into a quad-ridge horn was found to greatly
improve the feed’s beam patterns. However, as expected, this aperture blockage
degraded performance. The compromise reached was to introduce the rings seen
in Figure 3. These rings reduce the edge taper on the reflector while
introducing minimal additional aperture blockage.

The geometry of the feed had already been optimized before the rings were
introduced, so rings were added to locations on the ridges that were amenable
to them. The ideal number and placement of the rings was found through trial
and error.

Note, however, that an absorber was added to the rings in order to prevent a
resonance from occurring. For the full-sized feed, our proposed design uses a
flexible ferrite absorber. In particular, model M6 from Fair-
Rite222http://www.fair-rite.com/flexible-ferrite/ was used in the simulations.
The simulations described were performed using a layer of absorber $1.5$ mm
thick on the largest ring and $0.5$ mm thick on the other two rings. The
absorber does degrade radiation efficiency, as can be seen in Figure 10.

Figure 10: Radiation efficiency when the feed with rings is placed over a dish
with $f/d=0.216$ and an elliptical collar.

Although not shown, simulated radiation efficiency was nearly perfect in the
absence of the absorber. Note that the radiation efficiency is poorest at the
lowest end of the band, but increases to $-1$ dB by $250$ MHz.

It should also be noted here that the presence of the absorber increases the
noise temperature of the antenna system. The noise contribution was measured
for the fabricated 4/13 scale minimum systematics feed. In particular, the
noise of the feed was measured at room temperature and submerged in liquid
nitrogen. The noise temperature was then obtained by taking the difference of
the two measurements while accounting for amplifier gain and impedance
matching. The results of these measurements are presented in Figure 11. Note
that only frequencies up to 1 GHz are presented due to the limitations of the
low noise amplifiers available to us. Also, the lowest frequency measured was
650 MHz, as this corresponds to the lowest frequency in the band of the 4/13
scale feed. From these measurements, it was found that the typical noise
contribution of the absorbers was $\approx 120$ K. Although these measurements
were taken for the scaled down feed, it is safe to assume results would be
similar for the full scale feed, as simulations showed similar beam patterns
and S-parameters for the two versions.

Figure 11: Noise contribution of the absorber in the fabricated minimum
systematics feed.

###  3.3 Optimization of the Feed

Since this feed was optimized with PUMA in mind, the dimensions shown in the
following sections are set to optimize performance from $200\,{\rm MHz}$ up to
$1100\,{\rm MHz}$.

The optimizer’s error functions were defined as

$\mathcal{E}=a\mathcal{E}_{s}+b\mathcal{E}_{S11}+c\mathcal{E}_{d}.$ (10)

In this equation, $\mathcal{E}_{s}$ considers spill efficiency,
$\mathcal{E}_{S11}$ considers impedance matching, and $\mathcal{E}_{d}$ is
used to keep track of delay spectrum performance. The weights $a,b,c$ were
typically chosen to be $\approx 1$, but their values were sometimes adjusted
if some aspect of performance was lagging behind the others.

The error for the $S_{11}$ term is defined via

$\mathcal{E}_{S11}=\frac{1}{N}\Sigma_{n=1}^{N}\Theta(S_{11}[n]+10)[S_{11}[n]+10]^{2}.$
(11)

In this formula, $n$ indexes the frequency samples in the simulation, $S_{11}$
is given in dB and $\Theta$ is the Heaviside step function. In other words,
the goal was to keep $S_{11}$ below $-10\,$dB.

The error in the spill term $E_{S}$ is defined as

$\mathcal{E}_{s}=\Sigma_{n}\Theta(0.93-\eta_{s}[n])(0.93-\eta_{s}[n]).$ (12)

In this formula, $\eta_{s}$ is defined as the portion of the feed’s beam
illuminating the dish. (Note that this is not the same as $\eta_{sp}$, which
is defined as the portion of the dish’s beam illuminating the sky.) The spill
$\eta_{s}$ is calculated assuming a dish with an $f/d$ ratio of 0.25. Also,
note that $n$ is an index used to label the frequencies at which $\eta_{s}$
was estimated. During optimization, it was found that the fine-tuned ridge
geometry was not capable of improving $\eta_{s}$ over the entire $200$ to
$1100\,{\rm MHz}$ bandwidth. As such, $\eta_{s}$ was only considered for
frequencies $200\,{\rm MHz}<f<475\,{\rm MHz}$. This choice was made to
maximize performance at the frequencies most important for PUMA’s science
goals. Finally, the delay error was defined as

$\mathcal{E}_{d}=\Theta(\Delta t-17)(\Delta t-17),$ (13)

where $\Delta t$ is the time in nanoseconds that it takes for the energy in
the system to drop by $30$ dB when using the CST time domain solver. This
ensures that no resonances are present in the feed. The $17$ ns value was
chosen based on a candidate design that was considered to have adequate delay
spectrum performance.

###  3.4 A Performance Comparison

In this subsection, plots comparing the simulated performance for different
feed designs are shown to illustrate the benefits of the design features
added. These feed versions can be seen in Figure 12.

Figure 12: Pictures of feed versions. In reading order: vanilla feed, jagged
ridges no cone, proposed feed and minimum systematics feed.

In particular, $\eta_{s}$, which is the fraction of the feed’s beam that
illuminates the dish, $\eta_{ap}$, the $S_{11}$ (of the feed alone), and some
beam patterns are shown in Figures 13, 14,15, and 16, respectively.

Figure 13: Comparisons of simulations of $\eta_{s}$ as a function of frequency
for different versions of the feed. Figure 14: Comparison of simulated
aperture efficiencies on a $f/d=0.25$ dish as a function of frequency. Figure
15: Simulated $S_{11}$ values for different versions of the feed. The values
are simulated without the dish, and assuming a reference impedance of
$200\,\Omega$. Figure 16: Simulated peak-normalized H-plane directivities for
different versions of the feed.

In Figure 13, $\eta_{s}$ is plotted for various iterations of the feed.
“vanilla” (Figure 12) refers to a version of the feed with no cone and a
simple exponential taper and straight lines for the outside and back part of
the profile. One can see that the introduction of the jagged profile improves
spillover efficiency at low frequencies but degrades it above $\approx 400$
MHz. As mentioned in Section 3.3, this is because jagged ridges can’t improve
the beam patterns for all frequencies. So, the jagged ridges were optimized to
improved performance in the lower part of the band where the most cosmological
information is contained.

Introducing the cone (the proposed feed) reduces spillover at all frequencies,
and is especially effective for $f<300$ MHz. Introducing the rings (minimum
systematics) improves the spillover efficiency even further, and fixes the
problems caused by the jagged ridges at higher frequencies.

Figure 14 shows simulated aperture efficiencies for the various versions of
the feed design. This plot shows that the introduction of the curved ridges
greatly improves the aperture efficiency at the lower end of the band. On the
other hand, the introduction of the rings seems to degrade the efficiency of
the minimum systematics feed. One will also notice that the aperture
efficiency decreases significantly at frequency increases. This is due to the
fact that the feed geometry is large compared to such short wavelengths,
meaning that the dish is located in the nearfield of the feed. Although not
shown, predictions of aperture efficiency based on feed farfield patterns
agree with Figure 14 at low frequencies but begin to disagree significantly as
frequency increases. Despite the low aperture efficiency, these feed designs
can still achieve adequate sensitivity at high frequencies due to their high
spillover efficiencies.

Figure 15 shows simulated $S_{11}$ values for different versions of the feed.
This plot shows that versions of the feed without the cone have
$S_{11}\lessapprox-15$ dB for the entire band. The introduction of the cone
raises the $S_{11}$ above $-10$ dB for the frequencies below $f\approx 250$
MHz. This is due to the fact that the cone raises the impedance of the antenna
at these frequencies. This slightly higher $S_{11}$ should not cause
significant problems, however, since receiver noise is subdominant to the sky
temperature at these frequencies Ansari _et al._ (2019). Aside from that, the
$S_{11}$ stays below the goal of $-10$ dB for the rest of the band. Lastly,
note that introducing the rings only makes a slight difference to the
$S_{11}$.

Figure 17: A plot showing the boresight coordinate of the phase center as a
function of frequency.

Figure 17 shows how the phase center varies with frequency for different feed
designs. These phase centers were computed by CST. The “boresight” setting in
CST was used, with a $90\,^{\circ}$ degree angle around the z-axis. The values
shown here are the average of the phase centers computed in the E-plane and
H-plane.

First, note that the phase center Beukman _et al._ (2014) of the minimum
systematics design varies by about $400$ mm across the band. Such variation
means that achieving good phase efficiency for the minimum systematics feed
may not be possible over the whole band. Also, note that the cone helps keep
the phase center stable at low frequencies. This is due to the fact that the
cone prevents a resonance from occurring in the backshort, which has a
perimeter close to $\lambda/4$ at the longest wavelengths of operation.
Moreover, the feed with rings seems to have three different regimes of
behavior. At low frequencies, the phase center is located close to the largest
ring. In the middle of the band, the phase center is closer to the middle
ring. At low frequencies the phase center is close to the smallest ring.

This phase center variation is responsible for the dip in gain at zenith seen
in figure 2 at high frequencies. For these beam patterns, the feed was placed
$1500$ mm above the dish, meaning that the phase center is located $200$ mm
from the focus. At low frequencies, this means that the phase center will be
quite close to the focus. However, at higher frequencies the phase center will
be about $150\,$mm from the focus, leading to higher phase errors. These
errors are responsible for the dip in gain seen at zenith in Figure 2. In this
paper, we have chosen the placement in order to optimize for performance at
lower frequencies.

Figure 18: Simulated peak normalized E-plane directivities for different
versions of the feed.

Finally, consider the beam patterns of the feeds. H-plane patterns for the
proposed feed, the jagged feed with a cone, and the vanilla feed can be seen
in Figure 16. Note first that the presence of the rings causes the beam
pattern to taper off more quickly and smoothly. This feature is important
since it will help suppress mutual coupling in the H-plane, where it tends to
be most severe. Moreover, note that the proposed feed has suppressed the
backlobes. The smaller backlobes are a result of the rings used in that
design. E-plane beam patterns can be seen in Figure 18. For the E-plane, note
that the red proposed feed performs the best at the very lowest end of the
band. Moreover, it can be seen that the different designs perform similarly as
frequency increases.

##  4 Simulated Performance and Comparison to Requirements

In this section, relevant aspects of the antennas’ performance are presented
and compared to the requirements described in Section 2. Note that these
metrics were all evaluated with the feed placed on a $6$ m diameter dish with
$f/d=0.216$. Since no such dishes were available for testing, estimated
results based on simulations will be shown instead. Measured beam patterns and
S-parameters for the proposed feed will be shown in Section 5.

###  4.1 Delay Spectrum Performance

Recall from Section 2.1 that the delay response of the instrument can be
characterized using $\tilde{R}(0,0,\tau)$, the Fourier transform of the power
kernel at zenith. In order to compute this, the antenna is excited with a
plane wave propagating along the boresight of the antenna. This simulation was
performed using the time domain solver in CST, with a planewave excitation
coming from above, the differential line of the feed terminated by a waveguide
port, and a voltage monitor placed between two pins on the differential line.
The time domain solver simulated up until $400\,$ns after the initial
excitation. This $400\,$ns time was chosen in order to capture delay spectrum
performance at large delays. The voltage present at the antenna’s terminals is
recorded in order to obtain values for ${R}(0,0,\nu)$. Then, two Blackman-
Harris filters are applied in order to smoothly bring the power kernel to 0 at
the edges of the band. These filters will be denoted as $W(\nu)$. This leads
to a delay space power kernel

$\tilde{R}(0,0,\tau)=\int d\nu R(0,0,\nu)W(\nu)^{2}e^{2\pi i\nu\tau}.$ (14)

Results for minimum systematics, proposed and vanilla feeds are shown in
Figure 19 in terms of $k_{||}$, which is the spatial wavenumber along the line
of sight of the telescope. In cosmological contexts, one finds that
$k_{||}\propto\tau$.

Figure 19: Delay spectrum: comparison of the power kernel of the full antenna
for the minimum systematics, vanilla, and proposed feed. The top plot is for
the full band. The middle plot was made using a bandpass that selects for
frequencies $200\,{\rm MHz}<\nu<500\,{\rm MHz}$. The bottom plot shows the
power kernel of the feeds without the reflector present.

First, note from Figure 19 that all three designs provide similar results in
all three plots. So, it can be concluded that the presence of the cone, rings,
and jagged ridges does not significantly worsen delay spectrum performance.
Next, note that the top plot of the figure shows the power kernel dropping by
$50\,$dB after a delay corresponding roughly to $k_{||}\approx 0.1\,{\rm
hMpc^{-1}}$. Thus, roughly speaking, modes within $0.1\,{\rm hMpc^{-1}}$ of
the wedge would be avoided in order to be sure that the foregrounds are not
overpowering the HI signal. Readers should also note that lumps appear in the
delay spectrum as $k_{||}$ increases. These lumps are due to reflections
bouncing between the feed and the dish. In the middle plot, one can see that
the lower end of the band performs a bit better, dropping by $50\,$dB after
about $0.065\,{\rm hMpc^{-1}}$. This result is of interest as the band
corresponds to redshifts $2<z<6$, which are particularly important for the
PUMA experiment. Lastly, the bottom plot shows the delay spectrum for the
feeds only. Readers will note that the kernel drops off quite quickly in this
case, reaching $-50\,$dB within $k_{||}\approx 0.03\,{\rm hMpc^{-1}}$. This is
due to the absence of dish-feed reflections, which worsen performance in the
middle and top plots. As mentioned in Section 2, the power kernel should drop
$50\,$dB as quickly as possible. However, the presence of dish-feed
reflections slows down this process, giving a $-50\,$dB width that corresponds
to $0.1\,{\rm hMpc^{-1}}$. The use of a higher $f/d$ or off-axis design would
help reduce these reflections. However, as seen in Saliwanchik et al. (2021),
the increased mutual coupling from such a design would be worse than the dish-
feed reflections one incurs with a deep dish Saliwanchik _et al._ (2021).

###  4.2 Mutual Coupling Performance

In order to evaluate the proposed antenna’s mutual coupling performance, the
S-parameters for two antennas were considered. These antennas both point at
zenith and are taken to be separated by $5.3\lambda$ at the longest
wavelength. Section 2.2 provides motivation on this setup. For these
simulations, the vanilla, proposed and minimum sytematics feeds were
considered. The results of these simulations are shown in Figure 20.

In the E-plane, the $f/d=0.216$ no-collar dish has mutual coupling $20\,$dB to
$40\,$dB lower than the shallower $f/d=0.375$ dish. When the collar is used,
the mutual coupling is $20\,$dB to $40\,$dB lower than the shallow dish when
frequencies are between $200\,$MHz and $400\,$MHz. For the rest of the band,
the $S_{21}$ is $40\,$dB to $60\,$dB lower when the deep with-collar dish is
used rather than the shallow dish.

In the H-plane, the use of a deeper dish also helps with mutual coupling. The
deep, no-collar dish provides about a $10\,$dB to $20\,$dB improvement in the
lower part of the band. Note that the presence of the collar doesn’t help
quite as much in the H-plane, providing only about a $5\,$dB improvement.

Now, the various feed options will be compared. First, note that the proposed
feed provides an advantage over the other two options in the E-plane at low
frequencies for the deep dish with collar case. On average, we find the
proposed feed providing about a $10\,$dB improvement over the vanilla feed for
frequencies $\lesssim 300\,$MHz. Moreover, the proposed feed also provides
some improvement over the vanilla feed in the H-plane at low frequencies,
giving about a $10\,$dB improvement over the vanilla feed for frequencies
$\lesssim 400\,$MHz. Notably, the minimum systematics feed does provide a
coupling level about $20\,$dB better than the vanilla feed in the H-plane at
the lower end of the band (from $200\,$MHz up to $400\,$MHz). Although these
may sound like modest improvements at first, recall the lower end of the band
is critical for PUMA’s science goals. Thus, any improvement in that part of
the band is especially important.

Recall also that the goal was to keep $S_{21}$ levels below $-100\,$dB. This
goal is achieved in the E-plane for all designs for frequencies above
approximately $600\,$MHz, when the deep dish with collar is used. Results do
not quite meet the goal in the H-plane. The $S_{21}$ of the minimum
systematics feed stays below about $-70\,$dB for the entire band, which,
although an improvement, does not meet the goal.

Recall that this simulation is meant to cover the worst case scenario, where
the dishes are very close. For longer baselines, this coupling would certainly
go down. However, shorter baselines cannot be ignored, since they provide
information about large angular scales.

Since the mutual coupling level is still not quite low enough at PUMA’s
redshifts of interest, it will be necessary to remove this effect during the
data analysis stage. So far, techniques have been developed for the removal of
mutually coupled signals from the data Kern _et al._ (2019) Parsons _et al._
(2010). These techniques rely on the fact that coupled receiver noise varies
slowly with time compared to the sky signal. However, this type of approach
doesn’t work so well for NS-oriented baselines, for which the sky signal also
varies slowly with time. Moreover, how to handle the effects of mutual
coupling on antenna beam patterns is still an open question. Thus, new
techniques are likely needed to control mutual coupling.

Figure 20: Mutual coupling: $S_{21}$ values for various antenna options. The
top plot shows $S_{21}$ values calculated when each antenna is located in the
E-plane of the other. The bottom plot shows $S_{21}$ values when the antennas
are located in each other’s H-plane. Colors label different dish choices.
Dotted lines correspond to the proposed feed, solid lines correspond to the
minimum systematics feed, while dashed lines are used for the vanilla feed.

###  4.3 Sensitivity Performance

In order to evaluate the sensitivity of the antennas, the PUMA noise
calculator 333https://puma.bnl.gov was used. This calculator takes in
parameters describing the properties of a fiducial PUMA design. These
parameters describe beam size, observing time, and fraction of the sky
observed. It then produces an estimate for the power spectrum due to noise via
Equation D4 given in Ansari _et al._ (2019). This equation scales as
$\Omega_{p}$ as in Eq 5. This contribution is computed for one $\mathbf{k}$
mode at a time. The calculator was edited, however, to use the formula for the
noise power spectrum given in Eq 4. Sensitivity estimates are presented in
Figure 21. Note that all of these curves assume an $f/d=0.216$ dish with a
collar included. Calculations were performed assuming a 32,000 element
hexagonal array with $1.25$ years of integration time. The integration time
was set assuming that PUMA would run for 5 years with 6 hours per day of
observation. A value of 50 K for was assumed for $T_{ampl}$. For the minimum
systematics feed, we accounted for the noise contributed by the absorber by
adding an additional term to Eq 7. This term is given by

$\frac{\eta_{s_{11}}T_{abs}}{\eta_{c}\eta_{sp}}.$ (15)

For these calculations, we assumed that $T_{abs}$, the noise temperature
provided by the absorber, is $117$ K. This value was obtained by averaging the
data points we measured for Figure 11. Note that this is a rough estimate,
since we were unable to accurately measure the noise contribution at higher
frequencies.

The beam patterns used for the various antenna designs come from CST
simulations with a single antenna pointing at zenith. Note that sensitivity
will decrease slightly for pointings away from zenith due to increased noise
pickup from the ground. This plot shows the performance of the proposed feed
when paired with different reflector designs. The performance of the idealized
PUMA antenna described in Ansari _et al._ (2019) is also shown. For the
fiducial PUMA antenna, an Airy beam pattern and the parameters given in
Section 2.3 were used.

Note in Figure 21 that the proposed feed and the vanilla feed do provide
comparable sensitivity to the fiducial PUMA antenna, with somewhat worse
performance at lower redshifts and somewhat improved performance at higher
redshifts.

Lastly, one will also surely notice that the sensitivity provided by the
minimum systematics feed is significantly lower than the other feed designs
presented. This is due to the decrease in radiation efficiency and increase in
noise caused by the absorber present on the rings. In the context of PUMA,
this leads to significantly decreased sensitivity to smaller scale modes.
However, this increase in noise temperature may not be so severe in the
context of EOR telescopes, where the sky temperature can be up to $1000\,$K,
which is significantly higher than the $\approx 100\,$K provided by the
absorber.

Figure 21: Maximum $|\mathbf{k}|$ for which ${\rm SNR}>1$. These calculations
were performed assuming $\hat{k}\cdot\hat{n}=0.5$.

###  4.4 Cross-Polarization

Although not considered for the performance requirements of the antenna, the
cross-polarization properties will now be presented. Figure 22 shows the peak-
normalized cross-polarized pattern of the proposed feed in the D-plane, where
it is most severe. The peak-normalized patterns are given in dB, as

$20\mathrm{log_{10}}\left(\frac{|E_{xp}(\theta,\phi)|}{\mathrm{max}(|E_{co}|)}\right),$
(16)

where the $E$ functions refer to the electric field patterns of the feed. For
this paper, we use the Ludwig-3 convention for cross-polarization Ludwig
(1973) Beukman _et al._ (2014). These values vary significantly with $\theta$,
with the highest levels being about 10 dB below the peak value. These peak
values are comparable to those shown in Beukman _et al._ (2014).

Figure 22: Peak-normalized cross-polar beam patterns of the proposed feed in
the D-plane.

Figure 23 shows peak-normalized cross-polar beam patterns for the proposed
antenna system, including both dish and feed.

Figure 23: Peak-normalized cross-polar beam patterns of the proposed antenna
system in the D-plane.

In this figure, one finds cross-polar beam patterns approaching $0$ at angles
close to the boresite These patterns peak at angles close to the edge of the
main-lobe, with the sharpest peaks located at the upper end of the band.

The most important aspect to consider for this paper, however, is how this
cross-polarized beam pattern may contribute to polarization leakage in the
context of an intensity mapping interferometer. One can naively try to
eliminate polarized emission from one’s data by averaging visibilities from
perpendicularly polarized feeds:

$V_{I}=\frac{1}{2}(V_{XX}+V_{YY}).$ (17)

In this formula, $V$ refers to visibility and $X,Y$ refer to polarizations of
the feeds in the baseline. We approximate contributions of unpolarized and
polarized emissions to $V_{I}$ in the manner of Shaw _et al._ (2015)

$R_{I\rightarrow I}=(E_{x}^{a}E_{x}^{b*}+E_{y}^{a}E_{y}^{b*})P_{ab}^{I}.$ (18)

In this formula, $P_{ab}^{I}$ is a matrix selecting for the $I$ Stokes
parameter. $E_{x}^{a}$ refers to polarization component $a$ of the electric
field pattern of an x-polarized feed. Similarly, the contribution of polarized
emission is given by

$R_{P\rightarrow
I}^{2}=\sum_{p\in\\{Q,U,V\\}}|(E_{x}^{a}E_{x}^{b*}+E_{y}^{a}E_{y}^{b*})P_{ab}^{p}|^{2}.$
(19)

In Figure 24, we present results of leakage calculations for a few frequency
channels. These curves are shown in the D-plane and are peak normalized to the
maximum value of $R_{I\rightarrow I}$.

Figure 24: Peak-normalized $R$ values for three frequency channels. Solid
lines show $R_{P\rightarrow I}$ and dashed lines show $R_{I\rightarrow I}$.
These curves show leakage in the D-plane.

In all plots, one finds no polarization leakage along the boresite. However,
there do exist significant levels off-axis, with the most severe case being
$1100$ MHz with leakage values around 0.4 of the peak copolar value. These
levels of polarization leakage are comparable to those estimated for the CHIME
telescope in Shaw _et al._ (2015).

##  5 Fabrication and Testing of the Feed Antenna

After completing the design phase of the project, we fabricated a version of
the minimum systematics feed. This feed is roughly $0.9$ m long and $1.2$ m in
diameter at its widest point. Such a large design would not fit in the near-
field antenna testing chamber available to us, so a $\frac{4}{13}$ scale
version was fabricated instead. In simulations, we found no significant
performance difference between the $\frac{4}{13}$ scale design and the
original design.

In this section, the fabrication of this scaled down feed is described. We
also include some discussion on what would be done differently for the
fabrication of the full-scale feed. Lastly, we present measurements of the
feed’s performance and compare these results to simulations.

###  5.1 Fabrication of the Scaled Feed

Figure 25: A picture of the $\frac{4}{13}$ scale model of the minimum
systematics feed. The balun and coaxial cables are present just for testing.
In the future, an impedance matching and amplification module would be used
instead.

A photo of the fabricated feed antenna can been seen in Figure 25. During the
fabrication of this feed, a $0.1$ mm tolerance was requested on all dimensions
given. This includes the positions $x_{i},z_{i}$ of points on the profile,
differential line pin radius $a_{pin}$, etc.

The ridges were cut from $1$ mm sheet aluminum using a laser cutter. Such an
approach may work for the full-sized feed, where the ridges are $\approx 1/8"$
thick. However, it may be better to use a water-jet cutter for the full-sized
feed.

The cone and differential line shielding were cut from a single block of
aluminum. The cone was cut from an aluminum block rather than rolled due to
the difficulty of rolling such a thin sheet. Since welding was not an option,
the shielding for the differential line was cut from the same block to avoid
having to attach the two pieces. Such difficulties would not be present in the
fabrication of the full-sized feed. In that case, the use of thicker aluminum
would allow for the welding together of pieces and easier rolling of the cone.

An additional aluminum bracket was added to the feed in order to hold the
ridges in place. This piece is located on the back of the ridges close to the
differential line. It can be seen in Figure 25. Since it is located in the
backlobe of the antenna, this piece makes a negligible difference to the
$S_{\rm 11}$ and beam patterns of the feed.

Another aspect to note here is the different absorber used. In the full scale
version, Fair-Rite flexible ferrite tile was intended to be used. This
material was mentioned in Subsubsection 3.2.5. However, data in the frequency
range of interest for the scaled down feed could not be found. So, Eccosorb
NS-1000 was used instead. In this scaled down design, a layer of absorber
$1.2$ mm thick was used on the largest ring. A $0.6$ mm thick layer absorber
was used on the other two rings.

###  5.2 Fabricating a Balun

Figure 26: A drawing of the balun used to transition to SMA. These SMA ports
then plug into the input ports of a 180 degree hybrid.

This feed was designed with a differential line as its output. In order to
test it, a balun was required to transition from the differential line to SMA
connectors. This was done by using a metal disk shown in Figure 26. Note that
this disk has holes for each pin of the differential line. The diameter of
these holes is chosen to form a $50\,\Omega$ transmission line. SMA connectors
were then placed on the surface of the disk. The center pins of the SMA
connectors touch the pins of the differential line, forming a connection.
These SMA connectors are then connected to a $180^{\circ}$ hybrid, converting
the differential signal to an unbalanced one.

This simple balun works, and impacts the beam patterns negligibly. However, it
does not provide a particularly good impedance match to the feed. This is
acceptable for now, since the main goal was to test whether or not the CST
simulations are accurate. In the future an impedance matching circuit similar
to HERA’s FEM could be usedFagnoni _et al._ (2021).

###  5.3 Testing

In order to make sure our CST simulations were accurate, the $S_{11}$ of the
proposed feed was measured using a network analyzer. The beam patterns were
then measured using a StarLab Multiprobe System444Manufactured by Satimo
http://www.mvg-world.com/en/products/fieldproductfamily/antenna-
measurement-2/starlab located in an anechoic chamber at UW-Madison.

Figure 27: Comparison of the measured $S_{11}$ to the simulated $S_{11}$

The resulting $S_{11}$ can be seen in Figure 27. The simulation and
measurements here were both done using the $4/13$ scale version of the feed
and the simple balun. This $S_{11}$ is reasonably close to the simulated
values, being within $1-2\,$dB over the entire band. One will note that these
values for the $S_{11}$ are quite high. This is due to the presence of the
simple balun in both the simulation and the measurement, which degrades the
impedance match. This balun connects a $200\,\Omega$ differential line to a
$50\,\Omega$ coaxial line without any attempts at impedance matching. These
values would surely improve if one connected the antenna to an output better
matched to the differential line.

Figure 28: Measurements of E-plane gains for the scaled down feed. Figure 29:
Measurements of H-plane gains for the scaled down feed.

Some examples of measured beam patterns can be seen in Figure 28 and Figure
29. One feature to note is that the patterns are not quite symmetric about
$\theta=0$. This is expected due to fabrication errors and the presence of the
$180^{\circ}$ hybrid inside the anechoic chamber. The only feature that is
somewhat concerning is the poorer taper at low frequencies in the H-plane.
Having the beam patterns taper off adequately at the rim of the dish is an
important feature for reducing mutual coupling. Otherwise, the measured beams
appear to be quite close to those produced by our simulations.

##  6 Conclusion

In this paper, we proposed an antenna design optimized for use in PUMA, a
proposed HI intensity mapping observatory. The chief performance requirements
were to achieve a smooth frequency response, low levels of mutual coupling
between antennas in the array, and sensitivity comparable to that of the
fiducial PUMA antenna described in Ansari _et al._ (2019).

An important performance goal was to have the delay-space power kernel of the
instrument drop by $50\,$dB as quickly as possible. This goal was set in order
to minimize the leakage of Galactic foregrounds beyond the foreground wedge.
For the proposed design, it was found that the power kernel dropped to the
$-50\,$dB level at a delay corresponding to $k_{||}\approx 0.1\,{\rm
hMpc^{-1}}$. This level was set mostly by feed-dish reflections, with the
power kernel of the feed alone dropping by $50\,$dB after a delay
corresponding to $k_{||}\approx 0.03\,{\rm hMpc^{-1}}$. Reducing these
reflections would require using a shallower dish or perhaps an off-axis
design. However, these design choices would worsen delay spectrum performance
by increasing mutual coupling. Thus, the use of a deep dish with a collar
appears to be the best option for maximizing spectral smoothness.

The next performance goal was to minimize mutual coupling. It is important to
minimize this coupling due to its effects on the delay spectrum, redundant
calibration, and the risk of introducing correlated noise. Based on the simple
model shown in Kern _et al._ (2019), the goal was to keep the level of
coupling below $-100\,$dB for the entire band. This goal was not achieved, but
significant progress was made through optimization. The proposed antenna has
coupling at the $-60\,$dB level at low frequencies for closely packed antennas
located in the each other’s H-plane. Moreover, $-100\,$dB coupling was
achieved in the E-plane for frequencies $\nu>600\,$MHz.

We also sought to achieve sensitivity close to that provided by PUMA’s
fiducial antenna parameters. Our proposed feed satisified this goal, giving
slightly lower sensitivity at lower redshifts and slightly higher sensitivity
at higher redshifts.

The design presented here was optimized with the systematic effects of HI
intensity mapping instruments in mind. This design achieved comparable delay
spectrum performance to the HERA Vivaldi feed Fagnoni _et al._ (2021) and
provided sensitivity comparable to the fiducial parameters provided for in
PUMA telescope proposal. The optimization process for the antenna improved
mutual coupling significantly, but the level achieved for the lower end of the
band still did not meet the goal. It may be possible to reduce this coupling
further using calibration techniques. However, it will likely be necessary to
find a way to eliminate this effect in the data analysis stage.

## Acknowledgments

We thank Trevor Oxholm, Anh Phan, Gage Siebert, David Kwak and Yanlin Wu for
feedback on the feed/antenna design. We also thank Doug Dummer and Sara Yaeger
of the UW Physics Instrument Shop for fabricating the feed. We thank Chad Seys
for computing support. Thanks to Kevin Bandura for helpful feedback on the
paper. Thanks to Craig Podczerwinski for fabrication advice. Finally we thank
Alex Bouvy, Nader Behdad and Mirhamed Mirmozafari for assistance with the
antenna measurements. This work was partially supported by NSF Award
AST-1616554, the University of Wisconsin Graduate School, the Thomas G.
Rosenmeyer Cosmology Fund, and by a student award from the Wisconsin Space
Grant Consortium.

Dimensions of the Ridges

This appendix gives dimensions of the ridges of the full scale design. If one
wishes to make the scaled down version seen in Sec.5, then these values can
simply be scaled by a factor of $4/13$. First, note that the ridges are made
from $1/8$ ” aluminum plate. For the scaled down design, one would use 1 mm
aluminum plate instead. In Figure 4, one can see that the ridge profile is
broken into several sections. The inner edge of the ridge is made up of a
series of nodes connected by straight lines. The locations of these nodes are
provided in the first set of values in Table 1.

Table 1: Coordinates of all nodes on the ridge shown in Figure 4. The top set of values corresponds to the inner edge of the ridge. The bottom set corresponds to the outer edge of the ridge. z (mm) | x (mm)
---|---
-54.2 | 27.7
0 | 27.7
41.1 | 38.4
82.4 | 57.8
123.6 | 62.8
164.8 | 64.9
206.0 | 93.8
247.3 | 138.3
288.5 | 151.2
329.7 | 185.4
370.9 | 199.8
412.1 | 235.1
453.3 | 279.1
494.5 | 351.2
z (mm) | x (mm)
377.7 | 599.5
373.3 | 625.5
334.0 | 625.5
314.4 | 614.2
294.7 | 608.6
275.1 | 614.9
255.4 | 593.4
235.8 | 562.3
216.1 | 531.2
196.5 | 526.5
176.8 | 451.8
157.2 | 171.5
137.5 | 464.6
98.2 | 464.6
78.6 | 370.0
58.9 | 363.1
39.3 | 294.7
0 | 325.0
-54.2 | 325.0
-104.2 | 401.1
-154.2 | 393.8
-204.2 | 385.0
-254.2 | 333.7
-308.8 | 330.2
-308.7 | 264.2
-322.7 | 198.1
-414.6 | 132.1
-322.7 | 198.1
-414.6 | 132.1
-322.8 | 0
-304.2 | 0

Then, there is a circular section connecting the inner edge of the ridge to
the outer edge. This circular section has a radius of $168$ mm. The outer edge
is also made up of nodes connected by straight lines. The node positions of
this edge are provided in the second set of values in Table 1.

Lastly, a quarter circle is used for the backshort section of the ridge. This
quarter circular has a radius of $250$ mm.

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where $I(\Omega_{Z})={\rm Im~}\frac{-\Omega_{Z}}{(\Omega_{Z},v)}$ and $W$ is identity on ${\rm Amp}(M)$ (. <cit.>). By <cit.>, for any $v$-generic  stability condition $\sigma'\in {\rm Stab}^\dagger(X)$, the coarse moduli space $M_{\sigma'}(\srX, v)$ is the birational model of $M$ which corresponds to $\ell(\sigma').$ One can check that $\ell(\tau)$ is proportional to $h$ and  there is a canonical identification $M_\tau(\srX, v)\cong M.$

Finally, since $\phi_*(\Omega_{Z_0})=\Omega_{Z_1}$, we can choose the a derived equivalence $\Phi$ such that $\Phi^{\widetilde\rH}=\phi_*$ and $\Phi(\tau)=\gamma(1)$ by Theorem <ref>, Lemma <ref> and Theorem <ref>. The isomorphism $\varphi$ is induced by $\Phi$, hence $\varphi_*=\Phi^{\widetilde\rH}=\phi_*$.

From the proof, one can see that  if the second statement of Theorem <ref> holds for twisted K3 surfaces,  Proposition <ref> holds for twisted K3 surfaces  as well.

Using the same method, we can show that

For any $\phi \in \mathrm{Bir}(M)$, there exists a derived (anti-)automorphism $\Phi:\rD^b(\srX)\rightarrow \rD^b(\srX)$ such that $\phi^{\widetilde\rH}=\Phi^{\widetilde\rH}$ (resp. $\phi^{\widetilde\rH}=-\Phi^{\widetilde\rH}$) and there is an open subset $U\subset M$ and $\phi([E])=[\Phi(E)]$ for any $[E]\in U$.

According to Lemma <ref> and Theorem <ref>, we can lift the action $\phi_*$ to a derived autoequivalence $\Psi:\rD^b(X)\rightarrow \rD^b(X)$ if the action $\phi_*$ on $A_Y$ is $\id$.   If the action $\phi_*$ on $A_Y$ is $-\id$,  we can lift $\phi_\ast$ to a derived anti-autoequivalence $\Psi:\rD^b(X)\rightarrow \rD^b(X)^{\opp}$ by Lemma <ref>.

Then $\Psi$ (resp. $\Psi[1]$) induces an isomorphism $M\rightarrow M_{\Psi(\sigma)}(v)$. Due to <cit.>, there exists a derived (anti-)autoequivalence $\Psi'$ and a birational map
$$ M_{\Psi(\sigma)}(v)\dashrightarrow M,~E\rightarrow \Psi'(E)$$
defined over an open subset $U\subset M_{\Psi(\sigma)}(v).$
Compositing with the isomorphism $M\rightarrow M_{\Psi(\sigma)}(v)$, we obtain another birational map

$$\varphi:M\to M_{\Psi(\sigma)}(v)\dashrightarrow M,$$
which sends $[E]$ to $[\Phi(E)]$ for $E$ lying in some open subset of $M$, where $\Phi=\Psi'\circ \Psi$ (resp. $\Phi=\Psi'\circ \Psi[1]$).
The actions of $\phi_*$ and $\varphi_*$ on $\rH^2(M,\ZZ)$ differ by products of prime exceptional reflections by <cit.>. Then according to <cit.>, $\phi_*$ and $\varphi_*$ must conincide. Hence $\varphi=\phi$ and $\phi^{\widetilde\rH}=\Phi^{\widetilde\rH}$ (resp. $\phi^{\widetilde\rH}=-\Phi^{\widetilde\rH}$).

Our main theorems have a quick application for twisted (anti-)autoequivalences, which is a strengthen of Theorem <ref>.

Let $\Phi$ be derived (anti-)autoequivalence between a twisted K3 surface $\srX\rightarrow X$. If the action on the transcendental is $\pm \id$ and
assume that $\Phi^{\widetilde\rH}(v)=v$ (resp. $\Phi^{\widetilde\rH}(v)=-v$) for some isotropic vector $v$. If $\Phi^{\widetilde\rH}$ is generated by reflexive involutions, then $\Phi^\CH= \pm\id$ (resp. $\mp\id$). In paricular, this holds when  $\rank(\NS(X))\geq 3$.

The proof is similar to Theorem <ref>. With the same construction and notations, it suffices to show the (anti)-symplectic automorphism $\varphi\in \Aut(Y)$ constructed in the manner of Claim B in Theorem <ref> acts as $\pm \id$ on $\CH_0(Y)_{\hom}$. The assertion then follows from  Theorem <ref> and Theorem <ref>.

§.§ Proof of Theorem <ref>

In both cases,    $Y$ is  a moduli space of $\srX$-twisted sheaves on $X$ for some $\srX\to X$ by Corollary <ref>. According to Proposition <ref>, it suffices to consider the case $Y=M_\sigma(\srX, v)$ and $\phi$ is induced by an (anti-)autoequivalence $\Phi:\rD^b(\srX)\to \rD^b(\srX)$.

For (i), $\srX=X$ is untwisted,  by Proposition <ref>, it suffices to show $\Phi^{\CH}=\pm \id$.  Then the assertion follows from Theorem <ref>.

For (ii),  this follows from Theorem <ref>.

If $\phi\in \Bir(M_\sigma (\srX,v))$ preserves a biratioanl  Lagrangian fibration, then Conjecture <ref> for $\phi$  holds if either $\phi$ is symplectic or $\srX\to X$ is untwisted.

As above, we assume $\phi$ is induced by an (anti)-autoequivalence $\Phi$.    According to Proposition <ref> (i)(iii), it suffices to show $\Phi^{\CH}=\pm\id$.  By our assumption, there exists an isotropic vector $w\in  v^\perp\subseteq \widetilde{\NS}(\srX)$ which is fixed by  $\Phi^{\widetilde{\rH}}$. It follows from  Theorem <ref> that  $\Phi^{\CH}=\id$ if $\phi$ is symplectic and $\Phi^{\CH}=-\id$ if $\phi$ is anti-symplectic.

Let $Y$ be a $K3^{[n]}$-type hyper-Kähler variety. If the Markman-Mukai lattice $\Lambda_{\alg}(Y)$ contains a hyperbolic lattice and $\rank (\Lambda_{\alg}(Y) )\neq 2$ and $3$, then Conjecture <ref> holds.

This follows directly from Theorem <ref> and Theorem <ref>.

§.§ Proof of Theorem <ref>
It suffices to prove $\mathrm{Fix}(\iota)$ is a constant cycle subvariety.
For $n=1$, this is well-known. For $n=2$ and $\rank(\NS(Y)^{\iota_*})=1$, according to <cit.>,  $Y$ is a double EPW sextic and the assertion follows from Proposition <ref>.

In all the remaining cases,  $Y$ is a Bridgeland moduli space on a twisted K3 surface $\srX\to X$ by Theorem <ref>.  According to  Proposition <ref>,
we can assume that there exists an (anti-)autoequivalence $\Phi:\rD^b(\srX)\rightarrow \rD^b(\srX)$ such that $\iota([E])=[\Phi(E)]$.
Since $\rank(\NS(Y))^{\iota_*}\geq 2$, the invariant lattice of $\Phi^{\widetilde{\NS}}$ on $\widetilde{\NS}(\srX)$ is isomorphic to $ \widetilde{\Lambda}_\alg(Y)^{\iota_*}$ which is an indefinite $2$-elementary lattice of rank $>1$.   As seen in the proof of Theorem <ref>, the lattice $\widetilde{\Lambda}_\alg(Y)^{\iota_*}$ is isotropic.  Thus we can get

by applying Theorem <ref>.

For any $[F_1]$ and $[F_2]$  lying in the fixed locus of $\iota$, we have
This gives $c_2(F_1)=c_2(F_2)$ as $\CH_0(Y)$ is torsion free. Then the constantness follows from Proposition <ref>(i).

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