The rear of the vicarage in Nuenen. Vincent's studio was in the low outhouse on the right.
The letter in which Vincent informs Theo he has already arrived in Paris. It was written hurriedly on a sheet torn from the artist's sketchbook.


text:
Boris Johnson has 'a bit of an Eddie Mair'
In what is doubtless his most damaging interview to date, Boris Johnson's reputation was eviscerated this Sunday by the softly-spoken (and lethal) Eddie Mair: 'Making up quotes, lying to your party leader, wanting to be a part of somebody getting beaten up... You're a nasty piece of work, aren't you?'
by Rebecca Burn-Callander
Mayor of London Boris Johnson was expecting a walk in the park on BBC 1's Andrew Marr show yesterday. Firstly, Marr is still off sick, recovering from a stroke, so a stand-in interviewer had to be provided in the man of Eddie Mair. Who'd have predicted that this last-minute replacement, with his lilting Scottish brogue, would turn out to be even more sharp-tongued than Marr.
The interview started off innocuously enough: talking about immigration and the West Ham stadium deal. But then Mair gets to the juicy stuff. 'I want to talk about you, ' he says. 'Well, that's just what I would like to avoid,' responds Johnson. And then it begins: a succession of hard-hitting questions, inspired by the forthcoming Michael Cockerell documentary for BBC2 - which Mair has seen but Johnson has not, and claims to show a corrupt, unwholesome side to the Latin-spouting, jovial, self-styled buffoon.
The material seems pretty damning. Evidence of Johnson's extra-marital goings on, which he then lies about to his party leader at the time Michael Howard. The fact that he was fired from The Times for making up quotes, 'I merely sandpapered,' whines Johnson. And, Mair's shocking finale. The fact that Johnson was willing to share the home address and contact details of a journalist to his old friend Darius Guppy, who wanted to beat him up: 'the phrase 'broken ribs' was bandied around,' says Mair.
Watch the whole interview below:
Leadership Reputation Government MT Break
BoE demands 'honesty' from banks over empty coffers
Profits at major UK banks down 40% on 2011
Prince Charles kicks up a stink over closure of 'smelly' kipper smokehouse
G4S 'on the verge' of closing Olympic compo deal
Stop accusing firms of tax-dodging, says E&Y

redpajama_set_name: RedPajamaCommonCrawl


Every so often here on The Niche I list science and news articles worth a look. Below is my list for this weekend. Enjoy! The first article doesn't really fit into the other main categories, but the title "herding brains" caught my eye.
Stem cells and regenerative capacity in aging brain, article in Cell. Quiescence Modulates Stem Cell Maintenance and Regenerative Capacity in the Aging Brain. Does this kind of research have relevance to humans? See interesting, complex graphical abstract from the paper, Kalamakis, et. al.
William Wan and Laurie McGinley at WaPo do a deep dive into controversial umbilical cord "stem cell" supplier Liveyon including quotes from the company's founder, John Kosolcharoen. Many patients are reporting serious adverse events. I'm hoping to have an interview with Kosolcharoen here on The Niche soon.
"Scientists at the University of Bath have identified how a mutant gene in fish is involved in controlling stem cells." When I first saw a newsy note on this and it said something like "scientists in Bath…identify mutant gene in fish" I kind of imagined the scientists and the fish in a bath together.
One type of the base editor system is very error prone so in many cases likely won't be a viable alternative to CRISPR gene editing without some kind of major improvement (Science). Here's the research article, also in Science.
Is CRISPR patent battle finally over?
Did the Chinese government fund He Jiankui's "rogue" CRISPR of babies and then later try to make him a scapegoat?


Jodi Sheets started CrossFit in May 2012 after seeing the CrossFit games on ESPN. She was completely intrigued the minute she witnessed the physical fitness that the athletes exhibited and had to find out what this program was about and how she could learn to do all of that fun stuff. After googling CrossFit in Ascension Parish she found two locations and began her journey. After one WOD with Cecil Piazza at CrossFit Virtus, that was all she wrote. She competed in her first competition in October 2012 at the Deep South Shootout and had a blast, as she is just a little competitive. With her background in group fitness and personal training, she knew that she had to get more involved in this incredible program and went to get her CrossFit Level 1 certificate in February 2013 and most recently her CrossFit Level 2 certificate in September 2018. Exercising is a hobby for Jodi, but coaching is a passion. There is nothing more satisfying for a coach, than to witness someone accomplish something that they never believed they could accomplish.
Jodi was born and raised in Ascension Parish. She graduated from East Ascension High School and Southeastern Louisiana University where she received a Bachelor's of Science in Nursing. She has been nursing since December of 1995 and has been a nurse for the Ascension Parish School Board since 1999. Jodi became the President of the Louisiana School Nurse Organization in March of 2018. She also became a Nationally Certified School Nurse in November of 2018. She has been married to Cade Sheets, yes, Cade's Frozen Custard, and yes, that does seem a little strange to most people that she owns an ice cream shop, since 1996. She has three beautiful girls that are all very involved athletes: soccer, softball and basketball.
CrossFit has changed my entire outlook on health and fitness. I have exercised my entire life. I danced until I graduated from high school. I started working out in a gym when I was sixteen. I've done many different types of exercise, including: group fitness classes (step aerobics, kickboxing, spin classes, MMA training), weightlifting, adventure racing, running and biking. All things that I truly enjoyed, or at least I thought I did, until I found CrossFit. CrossFit is the most challenging and rewarding program I have ever been involved in. The community is unlike anything I have ever experienced besides my immediate family. The overwhelming support that is found in every box I have ever visited is beyond what words can express. I have certainly found a new passion that I crave daily. If I could package it up in a jar and keep it with me all of the time I would. The continuous encouragement that is given and received gives me a sense of belonging and acceptance. As a "Masters" CrossFitter, I can honestly say that I am physically and mentally in the best shape of my life and am so grateful to have found my CrossFit family. I would like to give a special Thank You to Cecil and Sara Piazza for allowing me the honor and privilege to be a part of the CrossFit Virtus team.


When you're a kid and close to the ground, you have a natural affinity with the smallest creatures, the tiny strivers that tunnel their way through the foundations of the world. I was always poking into spaces between grassroots and bricks, charmed to find them occupied by mantises, ants, grass-snakes and assorted goggas. Once, I watched a swarm of exuberantly furry orange caterpillars overwhelm a tree in my mother's garden. It was a startling image that burrowed its way into my brain and lay there for decades, pupating, until it emerged on the opening page of my novel Nineveh.
These little guys, Promeces longipes, encountered on Table Mountain, were the models for the mystery beetle that infests Nineveh. They're a gorgeous iridescent blue, which you can't see in my photo.
It's lowly creatures like this that my protagonist Katya (proprietor: Painless Pest Relocations, the humane eradication service) traffics in. She's employed by the powerful Mr Brand, developer of a glitzy housing estate outside Cape Town, to rid his property of a plague of rare and troublesome beetles. It doesn't help that the whole place is sinking into a swamp – helped along by an array of subversive forces, of both human and insect varieties.
This is the rejoicing city that dwelt carelessly, that said in her heart, I am, and there is none beside me: how is she become a desolation, a place for beasts to lie down in!
This is meant as an image of utter despair, but I rather like the idea of wild creatures moving in and repurposing our ruins, at least until something else is built in their place. The power of the smallest to undermine the greatest is heartening; the sense that what seems oppressively monumental can be toppled – or at least tweaked – by the most despised. What results is not "a desolation", but new, unbidden constellations of life. "Nineveh" stands for this rise and fall, this flux.
It's where most of us live in now: rapidly changing urban worlds, where every inhabitant – large and small, powerful and weak, human and other – must perpetually adjust to shifts in circumstance, negotiating life with our neighbours in complex and unpredictable ecosystems. There's anxiety in that, but also possibility.
Me in front of the famous lion-hunt friezes from Nineveh, now in the British Museum. I've always loved these wrenching, beautiful images from the ancient city, and was determined to call one of my novels Nineveh, come what may.
Nineveh the novel is not about destruction. Although it deals with in part with rootlessness, fear and insecurity, it's also the funniest and in some ways the most hopeful thing I've written. There's comedy (for me!) in a vision of grubs and tadpoles battling with single-malt-swilling businessmen for control of a real-estate empire. That comic energy is personified in the anarchic, malodorous figure of Len, Katya's disreputable dad, indestructible as a cockroach.
I just hope that not too many of my readers suffer from entomophobia, excessive squeamishness with regard to frogs, or intolerance of pigeons and meerkats; and that their memories of being knee-high to a beetle remain as fond as mine.
Nineveh comes out in the US and in the UK in November 2016 – pre-order now.
Do you have a great picture of an insect or some other bug, critter or creepy-crawly? post it with the hashtag #ninevehbugs on Facebook, instagram or twitter for a chance to win a copy of Nineveh from Aardvark Bureau. More details here.


The Path to your Purpose is not behind youBy Dr. Jay Polmar Stuck in the problems of the past or past lives? This is the script of Dr. Jay Polmar's live workshop on clearing out the past from this life, and others, to clearly know you purpose in life, and accelerate using clear visualization to see it, and your own self-empowering abilities … to create it.
The Secret Metabolic CodeBy Dr. Jay PolmarIf you are one of the millions who spend hundreds to thousands of dollars every year desperately trying to shed those excess pounds, we can help with just a few secret tips. Stop what you're doing right now and learn these secret techniques!
2012 – The World Isn't Coming To An EndBy Jay.C PolmarThere certainly was a big hullabaloo from Roland Emmerich's film 2012, and all the doomsday disaster films we've seen for the last few years and it's often scaring the living heck out of viewers.
So badly are they scared of the end of the world that they are watching the same movies of disaster, over and over, with different actors, different titles, and different producers – all aimed to scare you and empty your pockets.


text:
\section{Introduction}
The formation of the first stars in the universe, and associated events, lead to a small but distinct global cosmological 21-cm signature \cite{Furlanetto06}. The key quantity in deciding this is the spin temperature (quantifying the relative population of hyperfine levels of a neutral hydrogen atom), which in turn is determined by the CMB temperature, gas temperature and Wouthuysen-Field effect \cite{wouthuysen52}\cite{field58}. The latter is caused by the Lyman-$\alpha$ photons produced by the first stars at Cosmic Dawn \cite{Mittal21}.

The main feature of this signal is a trough visible against the cosmic microwave background (CMB), Figure \ref{T21}. Theoretically the depth of this trough is expected to be between 50 mK and 250 mK considering conventional cooling mechanisms \cite{fialkov19}. The detection by EDGES instead placed the depth at 500 mK requiring more exotic cooling mechanisms than previously expected \cite{Bowman2018}.

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.6\linewidth]{figures/T21.png}
	\end{center}
	\caption{The magnitude of the expected sky temperature, modeled as a -2.5 power law, against the absolute magnitude of an example global 21~cm signal. There is expected to be at least four orders of magnitude difference between the maximum 21~cm signal and the sky temperature at the frequency band of interest here.}
	\label{T21}
\end{figure}

The aim for the Radio Experiment for the Analysis of Cosmic Hydrogen experiment (REACH) \cite{acedo19} is to provide a confident detection of the 21~cm hydrogen signal from the Cosmic Dawn and the Epoch of Re-ionization. The background spectrum being up to five orders of magnitude larger than the signal itself means that to make a confident detection a detailed understanding is required for the combined effects of between the expected power-spectrum of the sky and the beam of the antenna is required. A number of experiments aim to measure the 21~cm signal, notably EDGES which reported a detection of the signal in 2018 \cite{Bowman2018}. Many other single-instrument global 21~cm signal experiments, including Experiment to Detect the global EoR Signature \cite{Bowman2018}, Shaped Antenna measurement of the background RAdio Spectrum \cite{singh18}, Probing Radio Intensity at high-Z from Marion \cite{Philip2019}, Mapper of the IGM Spin Temperature (\url{http://www.physics.mcgill.ca/mist/}), and Broadband Instrument for global Hydrogen Reionisation Signal \cite{sokolowski15} are also searching to make a detection. These instruments lack the angular resolution required to easily perform sky source calibrations and so require a more detailed understanding of the antenna beam and receiver for the calibration process, as compared to interferometers such as the Square Kilometer Array \cite{dewdney09}, Hydrogen Epoch of Reionization Array \cite{DeBoer_2017}, Large Aperture Experiment to Detect the Dark Ages \cite{Price18} and Low Frequency Array \cite{vanHaarlem13}.

This paper discusses the use of a quantitative figure of merit based antenna design method, using physically parameterised computer simulations to produce a polynomial fit to describe the individual and combined figures of merit, allowing the identification of an optimal design in an unbiased fashion from a range of initial designs.

The section \textit{Design Considerations} discusses the merits of differing antenna styles and their interactions with sky maps used for data analysis. The section \textit{Antenna Design Considerations and Figures of Merit} details the figures of merit used for the design of the REACH dipole antenna divided into generic antenna figures of merit and those specifically relevant to global 21~cm experiments. An overview of the quantitative figure of merit based antenna refinement process is given in the section \textit{Design Process}. This design process is then implemented in section \textit{Example case: REACH dipole} to produce a dipole for the REACH global 21~cm detection experiment.

\section{Design Considerations}
Sky-averaged 21~cm cosmology experiments are based on detecting very weak spectral (across frequency, $\nu$) perturbations of the power measured by a single antenna radiometer, which integrates the sky signal across angle in the sky ($\Omega$) and time ($t$) during an observation. For simplicity, this paper will refer to equivalent temperature quantities instead of power.

The antenna temperature at a frequency $\nu$ measured by a single antenna is given as,
\begin{equation}
T_{ant}(\nu) = \int_t\biggl(\biggl(\frac{1}{4\pi}\int_\Omega D(\nu,\Omega)\eta(\nu)(T_{\text{sky}}(t,\nu,\Omega)+T_\text{21}(\nu))d\Omega)\biggr)(1-|\Gamma_A(\nu)|^2)G_{RX}(t,\nu) + N(t,\nu,\Gamma_A)\biggr)dt
\label{T_ant}
\end{equation}
and is composed of the antenna's beam directivity, $D(\nu,\Omega)$, weighted by the radiation efficiency of the antenna, $\eta(\nu)$, integrated over all angles with the background sky temperature, $T_{\text{sky}}(t,\nu,\Omega)$, and the global 21~cm signal, $T_{21}(\nu)$. This temperature power is then amplified by the receiver gain (typically dominated by the first stage Low Noise Amplifier) when matched to the antenna impedance (thus weighted by the antenna's reflection coefficient), $(1-|\Gamma_A(\nu)|^2)G_{RX}(t,\nu)$, and a, post-amplification, system noise term is added, $N(t,\nu,\Gamma_A)$. The angular scale of fluctuations in $T_\text{21}$ is expected to be of the order 1 degree on the sky, compared to the tens of degrees of the antenna beam, and so $T_\text{21}$ can be approximated as having no angular dependence \cite{Liu_2013}. It is worth noting that the directivity and the antenna reflection coefficient terms are considered here to be time independent. In reality, they are not, but the assumption made here is that changes in these parameters will mostly occur at time scales much longer than the time the data is integrated for. This assumption should be carefully considered for specific cases (eg. observations from locations with rapidly changing atmospheric conditions).

\begin{figure}[ht]
	\begin{center}
		\includegraphics[trim = 0 100 20 80, width=\linewidth]{figures/RadiometerDiagram.pdf}
	\end{center}
	\caption{Diagram of a basic sky-averaged radiometer highlighting the main components of (\ref{T_ant}). The noise term, $N(t,\nu)$, is to be taken in this paper as the result of a combination of noise components generated at different stages of the chain depicted by this diagram, and before the time integration takes place.}
	\label{GlobalRadiometer}
\end{figure}

The REACH dipole antenna has been designed using a figure of merit driven refinement of computer simulations, with additional mock detection pipeline and prototyping for more detailed testing. This allowed for a wide range of initial designs to be tested and attempts to preserve superior features of each antenna within the final design. These computational models will also be used for the final data analysis, allowing for compensation for the sky chromaticity and informing where this chromaticity in the beam may have the largest effect on the probability of a detection. The remainder of this section discusses some initial considerations, fundamental to the remaining of the design process: the choice of design for calibratability against spectral smoothness; the choice of narrow versus wide beam antennas; and the effect of the uncertainty in the sky models.

\FloatBarrier
\subsection{Calibration versus Spectral smoothness}

There are three fundamental options in sky-averaged 21~cm experiments. Either the telescope is designed to introduce small enough spectral perturbations in the final antenna temperature, or later stage calibration implemented to compensate for these effects, or a combination of both. In this paper the assumption is made that the receiver effects can be calibrated using a combination of measurements and receiver models \cite{Roger12}, however this assumption is not made for the antenna structure itself. Accurate measurements of a single antenna operating at these wavelengths (1.5 - 6 m) are extremely difficult and unlikely to be sufficient on their own to calibrate receiver effects. On the other hand, in order to achieve a large enough bandwidth (at the very least 2:1) to be able to leverage on the spectral differences between the 21~cm signal and the foregrounds, it becomes an almost impossible task to design an antenna that will not introduce any (or small enough) spectral perturbations. Thus, it shall be argued throughout this paper that designs for a good radio antenna for this type of experiments, need to consider both aspects (ability to calibrate the antenna and inherent spectral smoothness), with the calibratability based on a realistic data analysis pipeline being the dominant aspect.

Furthermore, as discussed in the next subsection, the ultimate effect of the antenna design in the antenna temperature does not depend only on the antenna alone, but as indicated by (\ref{T_ant}), on how the antenna beam couples to the sky signal across time, frequency and angle. Therefore, it is required to take into account the lack of knowledge of the foreground component of the sky when designing a radio antenna for this application.

Typically, 1st generation global 21~cm experiments, such as SARAS2 \cite{singh18}, have aimed to construct a global 21~cm antenna with an achromatic beam. REACH will compensate for the beam chromaticity using a Bayesian analysis jointly fitting the cosmological signal, with a modelled sky temperature and an antenna beam model \cite{anstey20}. This means that rather than directly designing the antennae for an achromatic beam, the simplicity of modelling the antenna and its beam is a critical factor in the design of the REACH antennae. Another advantage of this consideration in the data analysis is allowing a broader frequency range to be considered than typically used with low chromaticity antennas, with EDGES-low observing over 50-100 MHz. REACH in contrast will aim to operate a similar dipole over 50-135 MHz. This larger observation range allows for the easier separation of foreground and 21~cm signal and is required by the large variety of possible shapes and locations for the 21~cm signal \cite{fialkov19}. Using a single continuous range of frequencies for the antenna increases the likelihood of the entire signal being contained within the observed band so increasing the chance of a confident detection. For the REACH system a conical log-spiral antenna will also be used, over a 50-170 MHz band with a typically less chromatic beam. However, the increased physical complexity of the antenna means modelling is more difficult than for the dipole antenna, resulting in more difficult calibration. Both antennas will observe from a site in the Karoo Radio Astronomy Reserve near the town of Carnarvon in the Great Karoo semi-desert in South Africa.

\FloatBarrier
\subsection{Wide beams versus Narrow beams}

For the detection of the 21~cm signal the approach of taking a sky averaged measurement to observe a 'global' signal is adopted. This application can lend itself well to a single wide beam antenna, so the averaging is done by the instrument itself. This approach allows for a smaller instrument and reduction in complexity of the required physical equipment. While the physical equipment is simpler the understanding of the antenna temperature, (\ref{T_ant}), is made more complex if larger portions of the sky are included in the observation. There is also the possibility that extremely wide beams, $>60^{\circ}$, will start to view the horizon. This introduces additional ionospheric effects, radio-frequency interference and increased impact from the ground and soil around the antenna.

To understand better the effect of the antenna beam width on detection of the global EoR signal five different antennas are evaluated. In particular, the theoretical model of a horizontal dipole over an infinite ground plane, \cite{balanis}, a simulated realistic bow-tie like dipole, \cite{bowtie}, the SKALA1 antenna that is a log periodic antenna, \cite{deLeraAcedo2015}, the HERA antenna which is a reflector antenna, \cite{DeBoer_2017}, and a capacitively loaded antenna array - CLDA which is a simplified design based on the previous work, \cite{kolitsidas}. The selected antennas are representative candidates for the EoR signal with two of these, the HERA and SKALA, to be directly involved in instruments that are being deployed to detect the fluctuations of the EoR signal. Bow-tie like dipoles have also been used in the Murchison Widefield Array (MWA), an SKA precursor instrument, \cite{mwa}. The selected antennas are classified into two categories based on their respective capabilities of angular resolution. For this section focus is placed on the chromatic beam effects of each antenna. The fundamental descriptive properties of the antennas are summarized in Table \ref{tab:antennas}.

\begin{table}
\centering
\caption{Summary of the properties of the evaluated antennas. The indicated properties are the min and max directivity $D$ on a dB scale, the voltage standing wave ratio VSWR, and the angular resolution.}
\label{tab:antennas}
\begin{tabular}{l*{4}{c}}
\hline
& \multicolumn{4}{c}{Antenna Property} \\
\hline
\cmidrule(lr){2-5}
Antenna & min $D$ & max $D$ & VSWR & Ang. res.  \\
\hline
\midrule
Ideal dip. & 3 & 7 & - & Low  \\
Bow-tie dip. & 3 & 8 & 2.5 & Low \\
SKALA & 3 & 9 & 2.6 & Low \\
HERA & 6.5 & 29 & 2.2 & High \\
CLDA & 7 & 30 & 2.2 & High \\
\botrule
\end{tabular}
\end{table}

An advantage of a high resolution antenna would be the ability to avoid the complex modelling of the foreground sky signals required with a wider beam antenna, and so increasing the likelihood of a detection \cite{Liu_2013}. High resolution antennas are achieved through either an array or large antenna structure, which means that the beam will have a higher chromaticity than a lower resolution antenna due to additional structures introduced towards the horizon. Array structures also introduce mutual coupling effects in addition to more complex calibration requirements compared to a single small low-resolution antenna. The differing advantages and drawbacks for high and low resolution antennas motivates classification into two different groups for a study of their respective effects on antenna temperature.

The realistic antennas have been simulated with the time domain finite integral method solver from Computer Simulation Technology - Microwave Studio, \cite{cst}, with the exception of the ideal dipole that is analytically described. Standard sensitivity testing was performed to verify that during the simulations no numerical resonances occur and a smooth convergence is obtained. The antennas are assumed to be located in the South African Karoo Astronomy Reserve, with coordinates approximately $30.71^{\circ} S$ and $21.45^{\circ}E$. In all simulations a infinite perfect electric conductor is used as a ground plane to remove edge effect reflections and so consider only the chromaticity from the antenna structure. For a finite ground plane the edge reflections will induce a sinusoidal type ripple into the antenna beam, which for a more detailed analysis should be added to the infinite ground plane simulation, or finite ground plane simulations used.

\FloatBarrier
\subsubsection{Low Angular Resolution Antennas}
\label{subsec:lowang}

Global 21~cm experiments have, so far, been utilizing primarily wide beam antennas such as dipoles. This results in sampling the radio sky with low angular resolution. Defining low angular resolution antennas in this study as antennas that have half power beamwidth - HPBW$>20^{\circ}$ and for this analysis this includes the ideal dipole, the simulated dipole and the SKALA.

The directivity of the simple horizontal dipole over an infinite ground plane can be described by the analytical formula based on \cite{balanis}:
\begin{equation}
D(\theta,\phi) = 4 \sin ^2 (kh \cos \theta) (1- \sin ^2 \theta \sin ^2 \phi) / R,
\label{eq:dirdip}
\end{equation}
\begin{equation}
R = \left[\frac{2}{3} - \frac{\sin(2kh)}{2kh} - \frac{\cos(2kh)}{(2kh)^2} + \frac{\sin(2kh)}{(2kh)^3}   \right],
\label{eq:resdip}
\end{equation}
where $\theta,\phi$ are the zenith and azimuthal angles, $\lambda$ is the wavelength, $h$ is the height above the ground plane and $k =2\pi/\lambda$ is the wavenumber. A schematic representation of the dipole is depicted in Figure \ref{fig:lowangularantennas} (left). The beam chromaticity is a slowly varying function based on the frequency dependent ratio $kh$. It is worth noting, that an approximation is not taken for \eqref{eq:resdip} since it will not hold for the entire frequency band over which the dipole is evaluated. The parameter $h$ is selected within the range $\lambda_{h\nu}/3 < h < \lambda_{h\nu}/2$, where $h\nu$ denotes the high end of the frequency band. This constraint ensures that only one lobe is achieved within this frequency range minimizing the antenna chromaticity. The obtained beam pattern is illustrated in Figure \ref{fig:dipolebeams} (left) at 150 MHz projected into the celestial sphere in equatorial coordinates. This model is included in the present study as a reference case for slowly varying chromatic beams compared to a realistic antenna model.

\begin{figure}[htp!]
	\centering
		\subfigure{\includegraphics[width=0.33\textwidth,height=7cm,keepaspectratio]{figures/dipoleid.pdf}}
        \hfil
        \subfigure{\includegraphics[width=0.33\textwidth,height=3.5cm,keepaspectratio]{figures/dipolebowtie.pdf}}
		\hfil
        \subfigure{\includegraphics[width=0.33\textwidth,height=6cm,keepaspectratio]{figures/skala.pdf}}
	\caption{Antenna schematics for ideal dipole  (left) simulated dipole (center) and  SKALA (right).}
	\label{fig:lowangularantennas}
\end{figure}

\begin{figure}[htp!]
\centering
\subfigure{\includegraphics[width=0.32\textwidth]{figures/dipole_ideal_beam.png}}
\hfil
\subfigure{\includegraphics[width=0.32\textwidth]{figures/dipoley.png}}
\hfil
\subfigure{\includegraphics[width=0.32\textwidth]{figures/skala.png}}
\caption{Normalized antenna directivity pattern at 150 MHz projected into the celestial sphere in equatorial coordinates for the (left) ideal dipole over a ground plane (center) dipole simulated in CST and (right) SKALA.}
\label{fig:dipolebeams}
\end{figure}

\begin{figure}[htp!]
\centering
\subfigure{\includegraphics[width=0.32\textwidth]{figures/dipolewf.png}}
\hfil
\subfigure{\includegraphics[width=0.32\textwidth]{figures/dipolecstwf.png}}
\hfil
\subfigure{\includegraphics[width=0.32\textwidth]{figures/skalawf.png}}
\caption{Waterfall diagram of the simulated antenna temperatures for a single day (22/07/2018) located in the Karoo of (left) ideal dipole (center) simulated dipole and (right) SKALA. Showing the antenna temperature over frequency and time over one day, the galaxy rising over the course of the day causing the rise in antenna temperature seen over time. Note the different scales for the antenna temperature.}
\label{fig:wfdipoles}
\end{figure}

The second model is a realistic bow-tie like dipole antenna. Bow-tie dipoles utilize tapering to improve the bandwidth performance over the standard dipole. This antenna will be benchmarked directly with the ideal dipole to observe the differences in the performance as global EoR probes. For a comparison at the same frequencies, the same dimensions are used with the bow-tie dipole and ideal dipole and only optimized the width and the tapering. As previously discussed, differences in the distance over the ground plane will result in different chromaticity in the upper half of the frequency band. The schematic of the designed dipole antenna is depicted in Figure \ref{fig:lowangularantennas} (center). The resulting beam at 150 MHz, as projected into the celestial sphere in equatorial coordinates, is depicted in Figure \ref{fig:dipolebeams} (center).

The final antenna evaluated in the subclass of the low angular resolution antennas is a single SKALA antenna. This is a variation of a dual-polarized log-periodic dipole antenna array. This type of antenna is potentially able to provide an appropriate large bandwidth to probe the EoR with very good inherent impedance matching. An illustration of the SKALA is depicted in Figure \ref{fig:lowangularantennas} (right). The benefit of the SKALA antenna is that part of the reflector is integrated in the antenna itself for most of the frequency band thereby minimizing the soil and ground effects. The SKALA antenna has not been designed to be used as a stand alone antenna but as part of phased array and furthermore it has been designed to cover a large frequency band (7:1), and to provide high sensitivity, which inherently introduces some level of chromaticity on the antenna performance. An analysis of the chromaticity requirements for SKA and the SKALA design can be found in \cite{deleraSKALA}. This effect is clearly visible with the sidelobes appearing in the radiation pattern as depicted in Figure \ref{fig:dipolebeams} (right). The SKALA choice can be considered as a more advanced but realistic implementation of the ideal dipole as better impedance matching can be achieved.

The waterfall diagrams for the low angular resolution antennas are illustrated in Figure \ref{fig:wfdipoles} noting the different scales for each antenna. There is approximately an 800 K difference between the ideal dipole model and the simulated bow-tie dipole at the peak temperature. The antenna temperature of the SKALA is approximately 2000 K higher due to the enhanced directivity.

\FloatBarrier
\subsubsection{High Angular Resolution Antennas}

An alternative to the wide beam antennas described in the previous section are the high gain antennas with a narrower beamwidth. These antennas can offer higher angular resolution and have not yet been explored extensively in the literature for this type of experiments. Antennas that naturally fall into this category are reflector antennas, lens antennas, large horns and antenna arrays. Lens antennas are typically based on dielectrics and this will add to the thermal noise of the antenna hence will not be explored further here. The main benefit of a well designed highly directive antenna is that the field of view can potentially be aligned with cold regions of the sky.

As a first model to evaluate from the class of high angular resolution antennas comes from the reflector type antennas and specifically a single HERA reflector antenna, \cite{DeBoer_2017}. Reflector antennas have a long and successful history in radio astronomy. The HERA antenna is a reflector with no mechanically moving parts and the beam oriented always to the antenna zenith. The feeder for the HERA dish in this analysis is the dipole initially devoted to Precision Array for Probing the Epoch of Reionization (PAPER, \cite{Parsons2010}). The design is depicted in Figure \ref{fig:herarrdes} (left). This design has the benefit that the majority of the collected sky signal is reflected from a metallic mesh reflector to the focal point. Therefore the effects of the surroundings and the ground are minimized. In addition, HERA's high gain narrow beamwidth with low sidelobes will provide better receiver sensitivity. The simulated normalized beam as projected into the celestial sphere at 150 MHz is illustrated in Figure \ref{fig:heraarraybeams} (left).

Finally a dense fully populated capacitively loaded dipole array is evaluated as an alternative solution. In Figure \ref{fig:herarrdes} (right) the designed array's unit cell is illustrated. The array is dual polarized and consists of $25\times25$ elements per polarization totaling an overall 1250 element ports. The corresponding normalized beam as projected into the celestial sphere at 150 MHz is illustrated in Figure \ref{fig:heraarraybeams} (right) where the $\sin x / x $ pattern of the uniform aperture illumination is clearly visible. The proposed CLDA can provide similar bandwidth to the SKALA and HERA but in addition it can add certain flexibility, i.e. the beam could scan a cold region of the sky. This reduces the integration time and improves the required stability from the receiver electronics. One could also obtain a reduced chromaticity of the array beam by employing wavelength scale sub-arraying to have an almost constant beamwidth across the frequency band. The CLDA was designed to have almost the same physical area as the HERA dish. The obvious disadvantage of the CLDA is the required control points, hence it is an expensive solution. Another possibility that can be considered is a mechanically steerable reflector or a reflector with a phased array feed.

\begin{figure}[htp!]
	\centering
		\subfigure{\includegraphics[width=0.45\textwidth]{figures/hera.pdf}}
        \hfil
        \subfigure{\includegraphics[width=0.45\textwidth,height=6cm,keepaspectratio]{figures/unitcell.pdf}}
	\caption{Illustration of the antenna design of (left) HERA and (right) the unit cell of the CLDA.}
	\label{fig:herarrdes}
\end{figure}

\begin{figure}[htp!]
\centering
\subfigure{\includegraphics[width=0.4\textwidth]{figures/hera_beam.png}}
\hfil
\subfigure{\includegraphics[width=0.4\textwidth]{figures/array_beam.png}}
\caption{Normalized antenna directivity pattern in dB scale at 150 MHz for (left) simulated HERA and (right) simulated CLDA with uniform aperture illumination.}
\label{fig:heraarraybeams}
\end{figure}

\begin{figure}[htp!]
	\centering
		\subfigure{\includegraphics[width=0.45\textwidth]{figures/herawf.png}}
        \hfil
        \subfigure{\includegraphics[width=0.45\textwidth]{figures/arraywf.png}}
	\caption{Waterfall diagram of the simulated antenna temperatures of (left) HERA and (right) CLDA.}
	\label{fig:herarrwf}
\end{figure}

In Figure \ref{fig:herarrwf} the waterfall diagrams for the antenna temperature are depicted. Observe that when the galactic centre is at the antenna zenith the CLDA a has double antenna temperature compared to the HERA antenna. On the contrary, during the putative observation time (0:00-4:00 LST) the CLDA and the HERA antenna temperature is similar and approximately 1500 K. It is also observed that in the low frequencies ($\nu \in [50-75] MHz$) the HERA antenna has higher antenna temperatures than the CLDA.

Based on the waterfall diagrams of all the antennas it is chosen to simulate virtual antenna temperatures between 0:00-4:00 LST for every 15 min for an 8 day span. This data is then used for all calculations referred to in the following analysis. The 8 day span is selected to simulate the conditions of a realistic experiment and further smoothing of the antenna temperature.
\FloatBarrier
\subsubsection{Foreground modelling for different antenna types}

At the frequencies where the global 21~cm signal is expected (50-200 MHz) the radio sky is dominated by synchrotron emission \cite{Jelic2010}. The synchrotron emission has been proven to follow a simple power law relation, \cite{Jelic2010}, and the scaling factor is usually referred to as the spectral index $\beta$. Equation \eqref{eq:sky} reads the frequency scaling relation of the radio sky referenced to the entire sky survey of the 408 MHz Haslam map \cite{Haslam1982}. The CMB is not scaled and the Planck data, \cite{planckcmb}, is used for the CMB performing the operation at every pixel. The scaled map for each frequency $\nu$ is given as:

\begin{equation}
T_{sky} (\nu,\mathbf{s}) = \left( T_{Sky~Model}(\mathbf{s}) - T_{CMB} \right) \left(\frac{\nu}{\nu_{sky}}\right)^{-\beta} + T_{CMB}.
\label{eq:sky}
\end{equation}

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/beta.png}
	\caption{Spectral Index $\beta$ in galactic coordinates.}
	\label{fig:beta}
\end{figure}
\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/log150haslam.png}
	\caption{150 MHz scaled Haslam map in equatorial coordinates using the angular dependent spectral index $\beta$ from Figure \ref{fig:beta}.}
	\label{fig:150has}
\end{figure}

In order to do an initial evaluation of the sky-antenna spatial-spectral chromaticity in the context of analyzing antennas with different beam widths (wide or narrow) and its impact on the calibration of foregrounds for global 21-cm experiments, here two different sky models are used. The first sky model is the unfiltered - NF version of the Haslam map, \cite{Haslam1982} with a constant spectral index $\beta=2.5$, as has been extensively used in the literature \cite{Matteo2004}, \cite{Rogers2008} and \cite{Bowman2009}. The second sky model is the widely used destripped desourced - DSDS updated Haslam map from \cite{Remazeilles2015}. To account for the angular dependency a similar technique to \cite{Bernardi2015} and \cite{Mozdzen2017} is adopted, where the spectral index is extracted according to equation:
\begin{equation}
\beta = - \left[ \ln \left( \frac{T_{sky~\nu_{\beta}} - T_{CMB}}{T_{sky~408MHz} - T_{CMB}} \right) / \ln \left( \frac{\nu_{\beta}}{408}\right) \right],
\label{eq:beta}
\end{equation}
where $T_{sky~\nu_{\beta}}$ refers to the the sky as extracted from \cite{DeOliveiraCosta2008} whereas the $T_{sky~408MHz}$ is the DSDS Haslam map. The result of this ratio \eqref{eq:beta} is depicted in Figure \ref{fig:beta} in galactic coordinates, where it is observed that the spectral index is not uniform across the entire sky when calculated pixel by pixel. In this work, the spectral index is calculated only for a single frequency at $\nu_{\beta}=100$ MHz. The scaled Haslam map at 150 MHz with varying spectral index is illustrated in Figure \ref{fig:150has} in equatorial coordinates. A straightforward extension to this method is to obtain the spectral index for each frequency following the same procedure using \eqref{eq:beta}. This will also account for the frequency variants of the spectral index. From the illustration of the spectral index in Figure \ref{fig:beta}, it is expected that the varying spectral index is more suitable for the high angular resolution antennas as a smaller part of the sky is sampled with the antenna beam whereas a constant spectral index can be utilized in the low angular resolution antennas as the effect will be averaged out to a larger extent.

For an almost achromatic antenna, if it is possible to assume a small contribution of the spectral index distribution (e.g. one single spectral index), a logarithmic polynomial model of the antenna temperature when only foregrounds are observed, as in (\ref{eq:poly}), should be suitable (similar to the one proposed in the works of \cite{Pritchard2010} and \cite{hu}).

\begin{equation}
\log \hat{T}_{ant} (\nu) = \sum_{n=0} ^{m} \alpha_n (\log \nu)^n,
\label{eq:poly}
\end{equation}
where the $\hat{}$ denotes modelled data.

Then, to assess the impact of the antenna chromaticity on foreground calibration the figure of merit $SFoM(\nu)_{i}$ is used, as in equation:

\begin{equation}
SFoM(\nu)_{i} = \sqrt{\left| T_{ant}(\nu)_{i} - \hat{T}_{ant}(\nu)\right|^2}, ~i \in \lbrace NF,DSDS\rbrace ,
\label{eq:sfom}
\end{equation}
where the root mean square (rms) error for each antenna model of the estimated value and the polynomial fit in the entire band for each sky model is evaluated. In the following analysis the two sets $\{NF~ \textnormal{map},\beta=2.5 \}$ and $\{DSDS~ \textnormal{map},\beta= \textnormal{Figure \ref{fig:beta}}\}$ are used and the corresponding set is indicated by the map choice as a subscript. It is noted that in this exercise the data is integrated for 4 hours and that here at present only the foreground calibration is under assessment and not any potential signal loss and thus the order of the polynomial model is kept low.

Also defined is the chromaticity correction, $C(\nu)$:
\begin{equation}
\ C(t,\nu) = \frac{\int_\Omega D(\nu,\Omega)T_{\text{sky}}(t,\nu_0,\Omega)d\Omega}{\int_\Omega D(\nu_0,\Omega)T_{\text{sky}}(t,\nu_0,\Omega)d\Omega}\,\,
.\label{chromaFoM}
\end{equation}

This is used as a scaling factor to reduce errors in antenna temperature due to chromaticity in an antenna beam, directivity $D(\nu,\Omega)$, with a sky temperature, $T_{\text{sky}}$, to a reference frequency, $\nu_0$. The selection of an appropriate sky map is necessary for an accurate correction to an antenna temperature. With a consistent sky map this figure can be used to quantify and compare the chromaticity present in various antenna beams.

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/dipoleskyres.png}
	\caption{Residuals for the theoretical dipole model and the simulated bow-tie dipole model.}
	\label{fig:resdip}
\end{figure}
\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/skalaskyres.png}
	\caption{Residual for the SKALA model for the different sky models.}
	\label{fig:resskalaskymod}
\end{figure}

The ideal dipole and the simulated bow-tie dipole are evaluated and the results are presented in Figure \ref{fig:resdip}. It is observed that for the ideal dipole the rms error is very small $(\leq 5$ mK) in both sky models evaluated as well as for only $4^{th}$ order polynomial fit. The results are illustrated in the zoomed area in Figure \ref{fig:resdip}. The reason for this behavior is traced to the slowly varying function that represents the ideal dipole. In contrast, the simulated bow-tie like dipole has a more severe impact on the rms error between the fitted data and the calculated one. The average $\langle SFoM(\nu)_{DSDS} \rangle = 17$ mK whereas is about $\langle SFoM(\nu)_{NF} \rangle = 36$ mK for $6^{th}$ order polynomial fitting. This is an indication of the impact of sky model on the calibration of the instrument. In addition, even though the ideal dipole has very low residuals it is hardly the case for a representative realistic antenna model. This implies that in the design of the global EoR experiment the chromatic antenna behavior should be taken into account.

The impact of the antenna chromaticity in the SKALA case is illustrated in Figure \ref{fig:resskalaskymod},  with the data fitted per frequency over the entire interval. The averaged $\langle SFoM(\nu)_{NF} \rangle = 104$ mK and $\langle SFoM(\nu)_{DSDS} \rangle = 82$ mK for a $7^{th}$ order polynomial fitting. These differences stem from the antenna beam chromaticity of the SKALA as well as the limited simulated time interval of observation.

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/heraskyres.png}
	\caption{Residual for the HERA model for the different sky models.}
	\label{fig:resheraskymod}
\end{figure}
\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/arrayskyres.png}
	\caption{Residual for the CLDA model for the different sky models.}
	\label{fig:rescldaskymod}
\end{figure}

The residuals for the case of the HERA reflector antenna are presented in Figure \ref{fig:resheraskymod}. Here
it is noted that the scale on the $SFoM(\nu)_{i}$ axis is K. Observe that the $\langle SFoM(\nu)_{i} \rangle$ becomes $\langle SFoM(\nu)_{NF} \rangle = 1.14$ K and $\langle SFoM(\nu)_{DSDS} \rangle = 0.5$ K for a $7^{th}$ order polynomial fitting. The reason for these high values is that the HERA antenna radiation pattern, despite being sensitive to a smaller region of the sky compared to wide beam antennas is also fairly chromatic, resulting in a potentially challenging foreground modelling. This chromaticity is induced due to the reflector being electrically small and substantial diffraction effects being present, in addition to standing wave effects between the feed and the dish. The impact of a noisy, with no angular resolution sky model is visible in Figure \ref{fig:rescldaskymod} for the case of CLDA. In this case the antenna has been tuned to present a smooth beam across frequency. The average sky $FoM$ becomes $\langle SFoM(\nu)_{DSDS} \rangle = 8$ mK for a $6^{th}$ order polynomial fitting that is the lowest for all realistic evaluated antenna models in this study.

Following this discussion, it is concluded that a high angular resolution antenna, sensitive to a smaller region of the sky when compared to a wide beam antenna, does not necessarily translate into an easier foreground modelling, and the beam chromaticity seems, in principle, to be a more important factor. It is also important to use a sky model that provides the correct angular resolution. With this in mind a focus will now be placed on wide beam antennas, which are typically also easier to model and cheaper to build, as well as potentially less chromatic than high gain antennas owing to their smaller electrical size and the smaller number of current modes they can support.

\FloatBarrier
\subsection{Uncertainty in the sky model}

A further complication in the calibration process of a radio antenna beam for global experiments is that much of it relies on some knowledge of the foregrounds. Several approaches have been proposed and used to do this, including no correction at all, a correction based on a chromaticity correction factor \eqref{chromaFoM} \cite{Monsalve17} or a Bayesian fitting of a a parametric foreground model procedure \cite{anstey20}, as used by REACH. It shall be argued in section \textit{Antenna Design Consideration and Figures of Merit}, that the chromaticity correction factor is a good, and straightforwardly calculated, indicator of the chromaticity present in an antenna beam. An improved method for performing this calculation is to construct a pipeline for injection and attempted detection of a signal through the candidate antenna beam (in the case of REACH, this is the Bayesian foreground calibration algorithm described in \cite{anstey20}), this however is too time consuming to be done to thousands of antennas. One of the reasons to do this is the lack of good foreground maps at these frequencies, the sensitivity of figures of merit and less flexible beam corrections to these sky maps. In order to illustrate the impact of the uncertainties introduced by the lack of knowledge of the sky, Figure \ref{fig:sky_unc} shows a simulation of the antenna temperature for a simple square dipole. This antenna temperature, $T_{A,corr}$, has been corrected using the chromaticity correction factor \cite{Monsalve17} using a different sky map than the one used to generate the simulated data. This is then compared to an antenna temperature calculated using the 50 MHz beam at all frequencies, $T_{A,noChr}$. If the chromaticity correction is perfect the, plotted, value $T_{A,corr}-T_{A,noChr}$ should be 0, with the same pair of sky maps used for antenna temperature calculation and correction.

In Figure \ref{fig:sky_unc} the sky model used to generate the antenna temperature was the GSM(200) map \cite{DeOliveiraCosta2008} and the Haslam map \cite{Haslam1982} was used as the model. This is instructive as the Haslam map scaled with a spectral index of $-2.5$ was the model used by EDGES in their chromaticity correction \cite{Bowman2018}. The residuals of the chromaticity correction for different locations of the antenna on the Earth surface are shown. All curves show a high degree of non smooth behaviour and residual features of up to $-0.6$ K in the frequency range 50-100 MHz.

The other example case shown here (see Figure \ref{fig:sky_uncPS}) is on the influence of point sources. For this, the sky is assumed to be the Haslam map with point sources in it while the de-sourced Haslam map is taken as the model. One can see that in this case the residuals are much smaller and only on the order of 10 mK in the range from 50-100 MHz.

We also note that similar dependencies are observable when changing the reference frequency for the correction, the integration time, etc., as well as many other factors that would make the sky model different from the real one, such as a per pixel spectral index, polarization, etc. For a more detailed analysis, or confident detection, to be made these additional features should be included in the used sky map over what has been included for this initial exploration of antenna designs.

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/sky_uncert.png}
	\caption{Residuals when correcting the antenna temperature data generated with a GSM sky using a HASLAM sky model.}
	\label{fig:sky_unc}
\end{figure}

\begin{figure}
	\centering
		\includegraphics[width=0.45\textwidth]{figures/sky_uncertPS.png}
	\caption{Residuals when correcting the antenna temperature data generated with a HASLAM sky with point sources using a de-sourced HASLAM sky model.}
	\label{fig:sky_uncPS}
\end{figure}
\FloatBarrier
\section{Antenna Design Considerations and Figures of Merit}
For the design process of the REACH antenna a quantitative refinement approach was taken, using a selection of figures of merit to describe and optimise features of candidate antennas. Some of these measures are suitable for many antenna used, others are more specialised for a global 21~cm antenna. These figures of merit focus on reducing the chromaticity within the beams and matching the dipole impedance to reduce reflections and improve calibratability of the instrument, this is done through optimisation of the physical geometry of the antenna and accompanying balun structures. Refinement of the physical dimensions of the antenna was undertaken; accompanied by prototyping principal component analysis and mock detection pipeline analysis of promising designs, as part of a recursive process detailed in Figure \ref{flow chart}.

The two main aspects of the antenna to be optimised are the input for the antenna (when described in the receiving sense), that is the electromagnetic beam pattern which interacts with the global signal, along with the output interaction with the antenna receiver structure, quantified through the antenna impedance.

\FloatBarrier
\subsection{Generic}

\subsubsection{Antenna impedance figures of merit}
For the REACH system a 50 $\Omega$ coaxial cable is used to connect the antenna structure to the receiver system. The antenna is required to be well matched to the receiver to aid calibration.
The suitability of the one-port scattering parameter, $\text{S}_{11}$, which is the impedance reflection coefficient into 50 $\Omega$, is considered through a collection of figures of merit. As the phase of the $\text{S}_{11}$ is of low importance for this case the absolute value in dB is considered here.

As calibration error is proportional to the value of $|\text{S}_{11}|$ the mean value of this quantity is considered to be minimised as the first figure of merit $\alpha$:

\begin{equation}
    \alpha = \langle \text{dB}( |\text{S}_{11}| )\rangle
    \label{eqn: alpha}
\end{equation}

To assess the possible calibratable bandwidth of the instrument the figure of merit $\beta$ is considered, defined as:

\begin{equation}
    \beta = \nu|^+_{\text{S}_{\text{11}} = -10\text{dB}} - \nu|^-_{\text{S}_{\text{11}} = -10\text{dB}}
    \label{eqn: beta}
\end{equation}

where $\nu|^-_{\text{S}_{\text{11}} = -10\text{dB}}$ and $\nu|^+_{\text{S}_{\text{11}} = -10\text{dB}}$ are the minimum and maximum frequencies at which $|\text{S}_{11}|$ crosses -10 dB. If this value is crossed multiple times the lowest pair of frequencies is selected. Theoretical predictions of the 21~cm signal, and the EDGES detection, suggest that this range should cover from at least 60 MHz to 130 MHz to ensure that the large trough of the signal is included within the operating range of the antenna.

To give an approximation of the shape of the reflection coefficient an integrated area between threshold values and the reflection coefficient magnitude is considered, $\gamma$:

\begin{equation}
    \bigg\langle \sum_{i=0}^6 \int f_i(\nu) d\nu\bigg\rangle
    \label{eqn: gamma}
\end{equation}
with
\begin{equation}
    f_i(\nu) = \max\{6+2i-|\text{dB}(\text{S}_{11})|,0\}
\end{equation}

It is expected that a steeply varying change in the reflection will reduce the accuracy of calibration of the final data and should be avoided, especially in regions where a 21~cm signal is expected to reside. A lower value of $\gamma$ will provide a flatter and lower $\text{S}_{11}$ curve \cite{Roger12}.

As bandwidth, $\alpha$, can be increased through scaling the antenna it is important to act to keep the majority of bandwidth in the relevant section of frequency space, 50-150 MHz. So the fourth figure of merit, to be minimised, $\delta$ is defined as:

\begin{equation}
    \delta = \nu|^-_{\text{S}_{\text{11}} = -10\text{dB}}
    \label{eqn: delta}
\end{equation}

Effective calibration of an antenna can be assisted with an understanding of the phase of $\text{S}_{11}$. The main contributions for this effect vary between antennas, for example with a simple dipole the main contribution is from the length of cable connecting the antenna blades and receiver. When physically controllable in this way it is not required to compensate for this effect within the blade design. For other antenna designs, such as a conical log-spiral, the large and comparatively complex balun structure required for feeding and matching the antenna means that the phase of $\text{S}_{11}$ would require a figure of merit during the design process.
\subsubsection{Engineering and complexity considerations}

In order to allow for an understanding of the antenna as detailed as possible, a simple design is of high importance. For the initial design and final analysis stages this pertains to the computational models used for simulations. The complexity of these models is best quantified in the number of mesh cells required for accurate modelling of the antenna (given antennas of roughly equal size, operating over similar frequencies).

Additionally engineering constraints for the construction and maintenance of the antenna are considered, with this aspect of the antenna investigated through the use of prototypes constructed in a lab setting. These prototypes are used to verify simulation results for the antenna, indicating areas of simulation requiring increased attention.
\FloatBarrier
\subsection{Global 21~cm experiment}
\subsubsection{Antenna beam chromaticity figure of merit}

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/beam_on_sky_up.png}
	\end{center}
	\caption{An overlay of the beams of a sample dipole at 50 MHz (black), 100 MHz (red) and 150 MHz (green) with the contour showing the 3dB gain contour. Here it is seen that even a relatively narrow beam covers a large section of the sky. This area of the sky will also at times include the galaxy resulting in a complex interaction between beam and sky. A beam smooth over frequency and space will allow for an easier calculation of this interaction.}
	\label{beam_sky_up}
\end{figure}

To quantify the chromaticity of the antenna beam a method following from the EDGES chromaticity correction is used \cite{Monsalve17}. To match to the final data analysis pipeline, the sky map used for calculation of the chromaticity figure of merit is generated using a nine region spectral index sky map, where each region of the sky has a spectral index calculated using the method detailed previously, based on \eqref{eq:sky} and \eqref{eq:beta}; giving an expected antenna temperature $T_{\text{sky}}$ for the candidate antennas.

This sky map serves as a close approximation of the structures seen within the sky at the frequencies to be observed, allowing for a good representation of the interaction between the beam structure and the structure of the sky for design purposes. For the REACH antenna a broad beam antenna will be used meaning that a large area of the sky will be observed at all times, so averaging out details within the sky.

 The chromaticity correction factor \eqref{chromaFoM} provides an approximation of the chromaticity present within an antenna beam. For the figure of merit, $\sigma$ the variance of this time-frequency grid is considered:
 \begin{equation}
    \sigma = \langle C^2 \rangle -\langle C \rangle ^2
    \label{eqn: sigma}
 \end{equation}

 The value of $\sigma$ is sought to be minimised for an optimal antenna, thus giving the smoothest beam response; as during the data analysis it is required to recover the expected sky signal following its interaction with the antenna beam.

\subsubsection{Mock detection pipeline}
For the most promising designs a more stringent, and time consuming, analysis is carried out using a mock detection pipeline. For this a selection of possible 21~cm signals are injected into a constructed sky map \cite{anstey20}. These skies are then treated as the data input for a Bayesian process during which the pipeline will attempt to detect the input signals, or lack thereof. This method allows for a more detailed verdict on the beams and using the full interaction between the beams, sky map, signal and analysis technique. This method is expected to provide the most accurate representation of the effectiveness of an antenna for detecting a global 21~cm signal. However due to the long run time of evaluations, above seven days, this method is only suitable for a final decision between a small number of the most promising designs identified other figures of merit \cite{anstey2021}
\FloatBarrier
\section{Design process}
\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/flow_chart.png}
	\end{center}
	\caption{The design process used for the design of the REACH antennas, designed to be unbiased towards antenna design styles, Figure \ref{initial_antennas}. Starting with the consideration of many styles before refining the chosen design style with an iterative quantitative figure of merit based refinement.}
	\label{flow chart}
\end{figure}

For the quantitative figure of merit design process, an example for REACH shown in Figure \ref{flow chart}, an iterative refinement process was used. This approach was designed to start with as little prejudice as possible towards any particular antenna design, instead focusing on producing an antenna to best synergise with the calibration and analysis stages of the system. Following the analysis discussed previously in this paper it was chosen to focus upon wide beam antennas, as any advantage gained from a narrower beam in an array is negated by the increased cost and complexity required in modelling. To do this the first stage required a large collection of different antenna types for an initial rough analysis, narrowing down a dozen designs to one or two. First an example of each of these designs was optimised using the figures of merit and then examined using the mock detection pipeline to identify the likelihood of a successful detection being made with each candidate antenna design

Following the selection of the overall design type the commercial electromagnetic field simulation software CST microwave studio is used to generate a large initial design set. These designs are all processed using numerical figures of merit. This set is then used for a preliminary analysis allowing for expansion into other promising areas of the parameter space.

Once promising design sets have been selected several more approaches can be taken to test the designs and inform further revisions. Prototypes are used to verify the computational simulations and provide data to inform data analysis. Additionally prototyping is important to assess the engineering aspects of the designs, and ensure that the final design is constructed accurately compared to the computer simulations.

When a smaller sample set is decided upon the next step is to again apply the mock detection pipeline, which due to time constraints is not usable on large design numbers. This acts as a strong test for the expected interaction of the antenna beams with the sky and 21~cm signal. As the interaction between beam chromaticity and ability to detect a 21~cm signal is less well understood, than the effect of impedance on calibration, this step is used to assist the selection of limits to place on figures of merit

After a sufficiently capable design is arrived at the refinement loop ceases and a preliminary design is made and deployed to the field to attempt a detection. It is expected that with a first design further refinements will be required due to complications arising from taking data.

\FloatBarrier
\subsection{CST simulation}
The antenna simulations where carried out predominantly using the finite element method based frequency domain solver inbuilt to CST. Typical calculations take an hour of server (16 cores at 2.6 GHz, 387 GB RAM) time to compute, with some variance due to the complexity of design

For each simulation the inbuilt CST mesh refinement method, based upon $\text{S}_{11}$ convergence, routine was used.
Figure \ref{conv_exmple} shows the convergence for simulation time, meshcells and $\text{S}_{11}$ for a hexagonal dipole. Each simulation was conducted with a minimum of 4 refinement stages, and is terminated once three consecutive simulations have less than 0.02 fractional difference between $\text{S}_{11}$ calculations.

\begin{figure}
\centering
  \centering
  \includegraphics[width=0.8\linewidth]{figures/mesh_refinement_process.png}
  \caption{The convergence of an example simulation for a dipole, showing the near exponential growth in solver time and mesh cells. Along with the reduction in $\Delta\text{S}_{11}$ as the mesh grows finer, for halting the simulation and mesh refinement $\Delta\text{S}_{11}<0.02$ for 5 consecutive simulations.}
  \label{conv_exmple}
\end{figure}
\begin{figure}
  \centering
  \includegraphics[width=0.8\linewidth]{figures/antenna_mesh.PNG}
  \caption{Example of a final meshing for antenna blades, showing the grouping of smaller mesh cells at the edges of the metal sheet. Adding smaller scale structure to simulations, has a large impact on the required simulation time.}
  \label{mesh_example}
\end{figure}

Figure \ref{mesh_example} shows the final mesh used for a simulation of an example pair of antenna blades. With the addition of the 10 m radius ground plane, this simulation required $4.25\times10^5$ tetrahedral cells. As is shown in the figures the size of these cells varies greatly depending upon the surrounding model structure. Given the presence of quasi-singular currents on the edges of the blades, a finer mesh on the blades contour is desired. With the more detailed areas, such as the edges of the blades, requiring a far more detailed mesh to compute. Shown in Figure \ref{antenna_s11_buildup_comparison} are the corresponding reflection coefficients for the different levels of completeness of the antenna, showing the impact of including the balun and ground plane in the simulation, encouraging the optimization of all components at the same time, despite the longer computation times.

\begin{table}[ht]
    \centering
    \begin{tabular}{ccc}
        \hline
        \textbf{Component} &\textbf{1st pass mesh cells} & \textbf{Final pass Mesh cells}\\
        \colrule
        Blades only & $1.0\times10^4$ &$2.5\times10^4$ \\
        Balun only & $5.0\times10^4$& $1.0\times10^5$\\
        Blades and 5~m ground plane &$8.0\times10^4$ & $1.5\times10^5$\\
        Blades and 10~m ground plane &$2.5\times10^5$ & $4.25\times10^5$\\
        Full antenna and 10~m ground plane& $2.5\times10^5$&$1.0\times10^6$\\
        \botrule
    \end{tabular}
    \caption{The number of mesh cells required to simulate components of a broad band dipole. The simulation with only the antenna blades forming the smallest and simplest composition therefore requiring the least mesh cells to accurately simulate. The addition of the large ground plane increases the starting number of mesh cells by an order of magnitude. Adding the smaller scale balun structure to the simulation required double the number of mesh cells for the final pass.
    }
    \label{mesh comp}
\end{table}

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/antenna_S11_buildup_comparison.png}
	\end{center}
	\caption{The accompanying scattering parameters for table \ref{mesh comp}, demonstrating the need to simulate the entire antenna structure for an accurate measurement of the reflection coefficient.}
	\label{antenna_s11_buildup_comparison}
\end{figure}

\FloatBarrier
\subsection{Figure of merit scaling}
To allow comparison between the various figures of merit scaling is required. Two methods are used for this step, one for an initial exploration of the unknown parameter space and the second is applicable once quantified requirements for the figures of merit are established. The values of $\text{S}_{11}$ are taken from each antenna design from which each of the $\text{S}_{11}$ figures of merits are calculated.

For initial parameter space exploration a large and sparse initial sampling set is used. These values are then scaled linearly over all samples, with the most preferential result given a 0 value and the least preferential 1. So in this case the scaled figures of merit ($\phi$) are:

\begin{equation}
    \phi_i = \frac{F-F_{\textrm{min}}}{F_{\textrm{max}}-F_{\textrm{min}}}
\end{equation}

When limits of figures of merit are more accurately understood a sigmoid function can be used for a more efficient and targeted refinement, the function used for the REACH dipole is:
\begin{equation}
    \Phi_i = \bigg(1+\bigg(\frac{F}{\text{y}}\bigg)^{\frac{\log(9)}{\log(\frac{\text{y}}{\text{z}})}}\bigg)^{-1},
\end{equation}
where $y$ is the minimum acceptable value of the figure of merit, so maximised gradient of $\Phi_i$, at $\Phi_i(y)=0.5$, and $z$ is selected such that $\Phi_i(\text{z}) = 0.1$. An example of this scaling function with areas of interest is shown in Figure \ref{figure_of_merit_scaling} using the $\sigma$ figure of merit.

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/fgure_of_merit_scaling.png}
	\end{center}
	\caption{Showing the scaling of the $\sigma$ figure of merit divided into four sections. A: A good value of the figure of merit, improvements focused on other figures of merit. B: Beyond an acceptable value of the figure of merit, diminishing returns for improvements compared to other figures of merit. C: Between unacceptable and good value of figure of merit, increasing gains for improvement in the figure of merit. D: Unacceptable figure of merit}
	\label{figure_of_merit_scaling}
\end{figure}

\FloatBarrier
\subsection{Antenna figure of merit}
The overall antenna figure of merit, $\Psi$, is constructed using the individual figures of merit using a weighted sum of squares:

\begin{equation}
 \Psi = \sqrt{W_{\sigma}\Phi_{\sigma}^2+W_{\alpha}\Phi_{\alpha}^2+W_{\beta}\Phi_{\beta}^2+W_{\gamma}\Phi_{\gamma}^2+W_{\delta}\Phi_{\delta}^2}
.\label{AFoM}
\end{equation}
where $\Phi_i$ are the scaled figures of merit discussed previously and $W_i$ are selected weights for each figure of merit.

\FloatBarrier
\section{Example case: REACH dipole}

\subsection{Initial design types}
The initial design stage required the choice of antenna style. For this a selection of antennas either already used in the 21~cm experiments, shown in Figure \ref{initial_antennas}, monopoles \cite{singh18_1}, dipoles \cite{Bowman2018} or conical log-spirals \cite{dyson65} \cite{sokolowski15}, in addition to antennas used within other frequency bands, such as sinuous \cite{buck08} and ridged horn antennas \cite{Lee04}, were considered. For each of these design types a candidate antenna was simulated with a 5 m circular ground plane using CST and FEKO \cite{FEKO}. The 5 m ground plane is smaller than expected to be deployed in the final instrument, however when backed by vacuum this size is large enough to correctly guide the shape of the beam while maintaining usable simulation times

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/antenna_types.png}
	\end{center}
	\caption{Renders of example conical log-spiral (left), monopole (top center), ridged horn (top right), inverse conical sinuous (bottom center) and dipole antennas (bottom right). For the antennas, other than the monopole where it is pictured, a 5 m circular ground plane was used for the simulations but is omitted from these images for clarity.}
	\label{initial_antennas}
\end{figure}

\begin{table}[ht]
    \centering
    \begin{tabular}{cccc}
        \hline
        \textbf{Design style}  &\textbf{mean $\text{S}_{11}$/dB} & \textbf{$\beta$}  &\textbf{$\sigma$}\\
        \colrule
        Conical log-spiral & -25 & 145 & 0.0003\\
        Wide band Dipole & -12 & 90 & 0.02\\
        Horn & -15 & 150 & 0.02\\
        Monopole & -10 & 80 &0.06\\
        Sinuous & -5 & 0 & 2 \\
        \botrule
    \end{tabular}
    \caption{A selection of the figures of merit for candidate antenna styles. Due to the preliminary stage the exact shape of $\text{S}_{11}$ is of less importance and so only the mean value and -10 dB range is considered.
    These figures of merit do not account for the complexity of the instruments, where the conical log-spiral and horn antennas perform worse than the dipole. It should be noted that the dipole design considered here did not use a balun which would improve the reflection coefficient. The other antenna simulations include simple simulated baluns.}
    \label{candidate_fom}
\end{table}

These initial designs were first roughly optimized using the previously discussed figures of merit for a global 21~cm antenna, the most promising of these where then run through a mock detection pipeline to assess the ability of the designs to perform a confident detection.

An example of the figures of merit for the antenna design styles is shown in Table \ref{candidate_fom}. The monopole design was discounted due to the low inclination of beams, greater than $30^{\circ}$ from zenith and their expected interaction with both the terrestrial environment and bright sources passing through the horizon.

Of the other two designs the conical log-spiral had better figures of merit compared to the dipole. For the dipole the possible observation range was reduced from a 4:1 to 5:2 band as the formation of multiple lobes in the beam at higher frequencies dramatically increases beam chromaticity. For the expected operational range of the dipole a detection in the mock pipeline was possible. The dipole, however, is geometrically more simple to model and manufacture; additionally the use of a dipole antenna in the EDGES detection supports this style of antennas ability to detect the global 21~cm signal.
For these reasons the decision was made to proceed with a dipole design initially to establish and test the methodology used for the REACH experiment. Once the additional understanding has been gained from the deployment and operation of a low band dipole a conical log-spiral design will be added using the refined methods from the initial deployment.
\FloatBarrier
\subsection{Dipole parameter selection}
The design of the dipole can be divided into three selections of parameters. The first section to govern the size and shape of the antenna blades. The second to describe the balun structure required for converting to a single ended feed and matching to the receiver structure. Finally the ground planes size, distance between the blades and antenna height are accounted for.

\begin{table}[ht]
    \centering
    \begin{tabular}{cc}
        \hline
        \textbf{Name} & \textbf{Antenna section}\\
        \colrule
        A$_{\text{x}}$, B$_{\text{x}}$, B$_{\text{y}}$, C$_{\text{x}}$, C$_{\text{y}}$, D$_{\text{x}}$, D$_{\text{y}}$, E$_{\text{x}}$ & Blade corners\\
        $l_{\text{X}}$, $l_{\text{O}}$, $R_{L}$ & Balun and leg structure \\
        $R_{\text{G}}$ & Circular ground plane size\\
        h & Blade height\\
        gap & Blade separation\\
        \botrule
    \end{tabular}
    \caption{The parameters used for the design of the REACH dipole.}
    \label{param_selection}
\end{table}

Eight parameters are used to define the seven pointed blade designs for the REACH dipole, as seen in Figure \ref{fig:schematic}. This choice was made to imitate an elliptical antenna while simplifying the manufacturing and computational process by removing the curved surfaces.

A Roberts balun was used following from its use in the EDGES2 experiment. This structure requires a further three parameters, governing the open-ended short, radius of tubes shielding the coaxial cables, and the connecting bar between these shields. Detailed in Table \ref{param_selection} are all 14 parameters used to define and optimize the physical dimensions of the REACH dipole.
For the initial sample generation a latin hypercube is used to evenly sample over all of the parameters required to define the antenna. For the antenna blade parameter ranges were chosen to be arranged around a quarter wavelength blade size at 75 MHz, as this is the most promising area for the 21~cm signal. To identify possible interactions between ground plane dimensions and the dipole blades the radius is allowed to vary between 5 m and 15 m, with the upper limit chosen to maintain reasonable simulation time. The baluns parameters are centered at the expected quarter wavelength value for the center of the band.

\begin{table}[ht]
    \centering
    \begin{tabular}{ccc}
        \hline
        \textbf{Name} & \textbf{minimum}/mm & \textbf{maximum}/mm\\
        \colrule
        A$_{\text{x}}$ & 16 & 19\\
        B$_{\text{x}}$ & 68 & 85\\
        B$_{\text{y}}$ & 565 & 640\\
        C$_{\text{x}}$ & 265& 460\\
        C$_{\text{y}}$ & 495& 595\\
        D$_{\text{x}}$ & 620& 780\\
        D$_{\text{y}}$ & 445& 550\\
        E$_{\text{x}}$ & 850 & 920\\
        $l_{\text{X}}$ & 415 & 479 \\
        $l_{\text{O}}$ & 430 & 440\\
        $R_{L}$ & 22 & 31 \\
        $R_{\text{G}}$ & 5000 & 15000\\
        h & 650 & 900\\
        gap & 11 & 14\\
        \botrule
    \end{tabular}
    \caption{The parameter ranges for the initial hypercube sampling for figure of merit analysis used for the REACH dipole}
    \label{param_ranges}
\end{table}

\begin{figure}
    \centering
    \includegraphics[width = \linewidth]{figures/blades_and_dipole.png}
    \caption{Left: The design of the Roberts balun used for the REACH dipole. Using a coaxial cable connection to the receiver through one of two shielded legs. The second leg, in addition to providing symmetry, contains an open ended coaxial cable used to refine the balun impedance transformation.
    Right: The generic seven pointed shape used to describe the shape of the dipole, designed to emulate an elliptical dipole while removing the need for curved edges which increase simulation complexity and time.}
    \label{fig:schematic}
\end{figure}
\FloatBarrier
\subsection{Balun design}
The REACH antenna uses a Roberts style balun \cite{Roberts57}, similar to that used in the EDGES experiment. A schematic for the structure of this balun is shown in Figure \ref{fig:schematic}, comprising of a coaxial feed from which the outer section is connected to the antenna blade and center wire is then connected to an open short of the other blade via a printed circuit board bridge. The length of this open stub has the largest tuneable impact upon the impedance of the balun.

The balun parameters are optimised during the same sequence as the antenna blades, to allow as much synergy as possible between the sections of the design. A resonant dipole presents an absolute impedance of 73 $\Omega$ which is frequency dependent. The addition of a balun to the dipole structure allows the balun to act as a transformer and may be tuned to better match the antenna impedance to the output 50 $\Omega$ cable

\begin{figure}[ht]
	\begin{center}
		\includegraphics[width=0.8\linewidth]{figures/balun_circuit_mk2.png}
	\end{center}
	\caption{A circuit diagram for the Roberts balun, showing the main contributions to the impedance transformation, the open short and parallel transmission line formed within the leg structure}
	\label{balun_ciruict}
\end{figure}
\FloatBarrier
\subsection{REACH figures of merit}
For the REACH antenna five figures of merit are used for the design process, detailed in Table \ref{FoMs}. Table \ref{reach_scaling_goals} gives the values of y and z used for scaling the respective figures of merit for combination into the overall figure of merit $\Psi$. Due to diminishing returns or physical limitations the 'optimal value' of some of the figures of merit is not the same as the value taken for y, for example $\sigma$ where there is expected to be some chromaticity present in the beam. As the analysis pipeline is able to compensate for a small level of chromaticity it becomes preferable to optimise other figures of merit. Figure \ref{figure_of_merit_scaling} shows the scaling function for $\sigma$ divided into four rough regions demonstrating a good, acceptable, improving and unacceptable figures of merit.

\begin{figure}
	\begin{center}
		\includegraphics[width=\linewidth]{figures/Function_with_colourbar_large_style_mean_cut4-1-1-1-1.png}
	\end{center}
	\caption{Full figure of merit compared to parameters, using $W=[4,1,1,1,1]$, for this plot the parameters not relevant to each subplot are set to their mean value. Here the individual figure of merit values generated for each antenna of the initial hypercube have been fitted using a 6th order polynomial over the physical parameters. For this plot each square is a slice through the mean value of the other parameters in the hyperspace. This plot allows the identification of promising regions within the parameter space, shown in blue. Using numerical methods a minimum point of these polynomials is found as the expected best parameter set for the given constraints. The overall figure of merit is shown ranging from blue to red.}
	\label{4-1-1-1-1}
\end{figure}

\begin{table}
    \centering
    \begin{tabular}{>{\centering\arraybackslash}p{0.1\linewidth}>{\centering\arraybackslash}p{0.1\linewidth}>{\centering\arraybackslash}p{0.3\linewidth}>{\centering\arraybackslash}p{0.2\linewidth}}
        \hline
        \textbf{Figure of Merit} & \textbf{Defining equation} & \textbf{Design aspect} & \textbf{Optimal value}\\
        \colrule
        $\sigma$ & \eqref{eqn: sigma} & Smoothness of antenna beam & minimize to 0\\
        $\alpha$ & \eqref{eqn: alpha} & Mean value of reflection caused by antenna impedance & minimize to -10 dB\\
        $\beta$ & \eqref{eqn: beta} & Range of values with an acceptable $\text{S}_{11}$ value & maximise to 90 MHz\\
        $\gamma$ & \eqref{eqn: gamma} & Flatness and depth of reflection coefficient & minimise to 0\% \\
        $\delta$ & \eqref{eqn: delta} & Minimum frequency for -10 dB $\text{S}_{11}$ & minimise to 50 MHz\\
        \botrule
    \end{tabular}
    \caption{The five numerical figures of merit used for the design of the REACH antenna, with the smoothness of the antenna beam and the matching of the antenna impedance to a standard 50 $\Omega$ receiver}
    \label{FoMs}
\end{table}

\begin{table}[ht]
    \centering
    \begin{tabular}{ccc}
        \hline
        \textbf{Figure of merit} & \textbf{y} & \textbf{z}\\
        \colrule
        $\sigma$ & 0.0015 & 0.001\\
        $\alpha$ & -10 & -12\\
        $\beta$ & 50 & 70\\
        $\gamma$ & 50 & 30\\
        $\delta$ & 60 & 55\\
        \botrule
    \end{tabular}
    \caption{The preferential values, for calculation of $\Phi_i$ used for scaling the figures of merit for the REACH antenna.}
    \label{reach_scaling_goals}
\end{table}

For the design of each antenna the priority of the previously described figure of merits will vary depending on the intended use case. The weighting of the overall figure of merit and choice of scaling functions allow for tuning of the prioritised figures of merit. The weightings allow for balancing of aspects of the antenna within the figure of merit, for instance the initial weighting of $\Psi$ used for the REACH antenna is $[W_{\sigma},W_{\alpha},W_{\beta},W_{\gamma},W_{\delta}]=[4,1,1,1,1]$, with a weight for each of the previously discussed figures of merit. This was chosen initially to balance the contributions from the beam chromaticity and the antenna impedance reflections.

To help visualise the antenna figure of merit corner plots such as Figure \ref{4-1-1-1-1} are used. These plots show the value of the figure of merit function evaluated over grids between pairs of physical parameters. For this figure the mean value of the other 12 parameters is used. This allows for an understanding to be gained of possible interactions between the many parameters of the antenna design, with lower values of figure of merit preferable.

Using the weightings it is informative to consider the two separate parts of the figure of merit, shown in Figure \ref{FoM_tradeoff} using $W = [1,0,0,0,0]$ and $W=[0,1,1,1,1]$ to consider only the beam chromaticity and the overall impedance of the antenna. The weighting $W = [0,0,0,0,1]$ is also highly informative indicating the key components influencing lowest antenna operating frequency.

\begin{figure}
\centering
\begin{minipage}{.45\textwidth}
  \centering
  \includegraphics[width=\linewidth]{figures/h_E_x_CFoM.png}
 \captionsetup{labelformat=empty}
  \caption{$W=[1,0,0,0,0]$.}
  \label{h_Ex_CFoM}
\end{minipage}%
\begin{minipage}{.45\textwidth}
  \centering
  \includegraphics[width=\linewidth]{figures/h_E_x_S11.png}
 \captionsetup{labelformat=empty}
  \caption{$W=[0,1,1,1,1]$}
  \label{h_Ex_S11}
\end{minipage}
\addtocounter{figure}{-2}
\caption{Two slices of figure of merit comparing the chromaticity (left) and combined S$_{11}$ (right) weightings. Both slices are taken with all other parameter values at their mean for the full parameter range. A demonstration of the differing preferences of the figure of merit, with an increased Ex, and height assisting S$_{11}$ whereas chromaticity benefits from a lower height and middling Ex around 850 mm}
\label{FoM_tradeoff}
\end{figure}

Seen in Figures \ref{FoM_tradeoff} is an example of one trade off required to be made for a global 21~cm antenna. Here it is seen that the reduced antenna height is favoured by the chromaticity figure of merit, but unfavourable by the combined $\text{S}_{11}$ figures of merit. These plots also demonstrate the possibility of regional preference for the figures of merit. When considering these plots it is important to recall that they are 2D cut through slices of a fourteen dimensional space, parameterised by the physical parameters of the antenna.

Figure \ref{4-1-1-1-1}, allows for an idea of the relationship between the various physical parameters and the figures of merit. This information is expected to be highly valuable for further field refinements of the antenna and optimization  once observation and data analysis is able to begin.

A functional minimisation is used to find the minima of the polynomial fitted to the figure of merit to identify the most promising candidate; which is then to be evaluated in more detail using prototypes and a mock detection pipeline.

\FloatBarrier

\subsection{Prototype}
For verification of the computational methods in addition to manufacturing and construction procedures to be established, a preliminary antenna design was constructed at Stellenbosch University South Africa, shown in Figure \ref{SA_prototype}. The reflection coefficient of this prototype was measured using a 910 mm radius circular ground plane with 48 300 mm triangular serrations. For this measurement an extra length of cable was required below the antenna. For comparison to simulations an ideal coaxial transmission line of length 240 mm was added in series after the antenna for calculation of S$_{11}$.

\begin{figure}
	\begin{center}
		\includegraphics[width=\linewidth]{figures/SA_prototype.png}
	\end{center}
	\caption{A prototype antenna constructed at Stellenbosch University, right, and the CST simulation model of the same antenna, used for verification of the simulations used for the design of the dipole antenna. Due to materials and space available a smaller ground plane was used than in standard simulations or the final deployment.}
	\label{SA_prototype}
\end{figure}

\begin{figure}
	\begin{center}
		\includegraphics[width=\linewidth]{figures/prototype_antenna_vs_simulation_comparison.png}
	\end{center}
	\caption{The impedance of a prototype antenna, compared to simulations of the same antenna with a 2 m circular serrated ground plane and a 10 m smooth circular ground plane. There is negligible difference between the simulations using different sizes for ground planes. There is some difference seen at the peaks of resistance for the measured antenna, at 55 MHz and 90 MHz, however overall the simulations and measurement are sufficiently close.}
	\label{SA_impedance}
\end{figure}

The ground plane used for the measurement was smaller than the expected final ground plane. Ground plane size is expected to have negligible effect upon the antenna impedance. To verify this the antenna was simulated with a 10 m circular ground plane for additional comparison to the smaller ground plane simulation and measurement. As shown in Figure \ref{SA_impedance}, the difference between the two simulations is smaller than the difference between either simulation and the measurement.

Figure \ref{SA_impedance} shows the impedance for the prototype antenna. This figure shows that there is very close agreement between the two simulated antennas of varying ground plane sizes and the measured prototype. The difference in impedance for the simulations including different ground planes is negligible. Here it is seen that there is some deviation with a lag of 10 MHz between the real and imaginary parts of the impedance, with the largest deviation occurring at 65 MHz. This deviation is small enough to continue using the computational models for the antenna for suggestions to the design. In the field an active calibration will be used to measure the reflection coefficient for the antenna, so general features of the simulated reflection are of more import than the exact values.
\FloatBarrier

\subsection{Mock detection pipeline results}
To assess the ability of this style of antenna to perform a possible detection of the global 21~cm signal a CST model of the prototype antenna, Figure \ref{SA_prototype}, was placed through a mock analysis pipeline. Although this simulation uses only the small ground plane and a vacuum backing the addition of these features to a final calculation are expected to improve the ability of the antenna to perform a possible detection.

This analysis was calculated using the same method as detailed in \cite{anstey20}\cite{anstey2021}. In Figure \ref{Prototype_mock_detection} are shown the returned fit using no injected signal and three injected Gaussian signals with $\sigma$ 15 MHz, depth 155 mK and center frequencies 80 MHz, 100 MHz, 120 MHz. The sky map was taken  from 00:00 to 01:00 on 01-01-2019 as viewed from the Karoo Astronomy Reserve and the frequency range observed over 50 MHz to 135 MHz resolution of 1 MHz. With real data the frequency resolution and observation time will be increased, which is expected to reduce the uncertainty seen in Figure \ref{Prototype_mock_detection}. Using the sample Gaussian signal the antenna was able to detect all center frequencies with confidence and with a 2.5:1 odds confidence reject a signal when there is none present. This ability of the antenna to confidently detect signals across the operational frequency range with both the data analysis and physical construction of the antenna still to be improved is a good verification of the antennas capability.

\begin{figure}
    \centering
    \includegraphics[width=0.8\linewidth]{figures/2x2signal_detection.png}
    \caption{Attempted reconstructions of four different injected signals using the prototype antenna. Top left: Null signal showing no false detection is provided by the antenna. Top right, bottom left, bottom right: Three injected Gaussian signals, green line, reconstructed, blue, the variance seen in the reconstructed signal is expected to be reduced with increased frequency resolution and observation time.}
    \label{Prototype_mock_detection}
\end{figure}

\FloatBarrier
\section{Conclusions}
This paper considered the general characteristics of candidate antennas for a global 21~cm experiment noting benefits and trade-offs for the design styles. Showing the importance of coupling to the sky model used and effect of beam width on the accuracy of this measurement. Following this the figures of merit required for an effective global 21~cm antenna were discussed.

Also was detailed a quantitative, physically based, figure of merit approach for the design of a global 21~cm antenna. Starting with multiple design styles before selecting and refining a dipole design. The current prototype design shows a 2.5:1 observation band from 55 MHz and antenna gain patterns able to detect a range of injected signals across the 80 MHz to 130 MHz center frequency at 155 mK amplitude and correctly identify a null signal. A deployment is expected to take place in the Karoo Radio Astronomy Reserve South Africa during 2021, with observations beginning shortly afterwards.

\section{Acknowledgements}
The authors would like to thank the Kavli Foundation for their support of REACH. John Cumner was supported by EPSRC iCASE in partnership with BT. Eloy de Lera Acedo was supported by the Science and Technologies Facilities Council. Dirk de Villiers  was supported in part by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Technology (Grant Number: 75322). The work of Girish Kulkarni and Shikhar Mittal was supported by the Department of Atomic Energy (Government of India) research project under Project Identification Number RTI 4002 and also in part by the Max Planck Society via a partner group grant.

\bibliographystyle{ws-jai}

redpajama_set_name: RedPajamaArXiv