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Thesis: Radio Interferometry: WIP
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\begin{document}
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\chapter{Air Shower Radio Interferometry}
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\label{sec:interferometry}
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The radio signals emitted by the air shower (see Section~\ref{sec:airshowers}) can be recorded by radio antennas.
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An array of radio antennas can be used as an interferometer.
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The radio signals emitted by an \gls{EAS} (see Section~\ref{sec:airshowers}) can be recorded by radio antennas.
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For suitable frequencies, an array of radio antennas can be used as an interferometer.
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Therefore, air showers can be analysed using radio interferometry.
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Note that since the radio waves are mainly caused by processes involving electrons (see Section~\ref{sec:airshowers}), any derived properties are tied to the electromagnetic component of the air shower.
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\\
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In \cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using interferometry.%
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\footnote{
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Available as a python package at \url{gitlab}.
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}
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As shown in Figure~\ref{fig:radio_air_shower}, the shower axis and particle densities along that axis can be observed.
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From these, the energy, composition and direction of the cosmic particle can be derived.
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\\
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%
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Unlike, astronomical interferometry, the source of the signal is closeby.
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The accuracy of the technique is primarily dependent on the timing accuracy of the detectors.
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In Figure~\ref{fig:xmax_synchronise}, the estimated atmospheric depth resolution as a function of detector synchronisation is shown as simulated for different inclinations of the air shower.
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According to Figure~\ref{fig:xmax_synchronise}, to be able to distinguish the iron and proton showers from Figure~\ref{fig:airshower_depth} ($\Delta\Xmax \sim 40\;\mathrm{g/cm^2}$), we need a synchronisation better than $2\ns$.
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\\
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\begin{figure}
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\centering
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\includegraphics[width=0.5\textwidth]{radio_interferometry/rit_schematic_true.pdf}%
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% \includegraphics[width=0.5\textwidth]{radio_interferometry/Schematic_RIT_extracted.png}
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% \caption{From H. Schoorlemmer}
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\begin{subfigure}[t]{0.47\textwidth}
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\includegraphics[width=\textwidth]{2006.10348/fig01.no_title}%
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\caption{
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From \protect \cite{Schoorlemmer:2020low}.
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Radio interferometric power analysis of an \gls{EAS}.
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\protect \Todo{describe and expand caption, remove title}
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}
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\label{fig:radio_air_shower}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.47\textwidth}
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\includegraphics[width=\textwidth]{2006.10348/fig03_b}%
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\caption{
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From \protect \cite{Schoorlemmer:2020low}.
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$\Xmax$ resolution as a function of detector-to-detector synchronisation.
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A typical noise (gaussian) background is simulated.
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\protect \Todo{describe and expand}
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}
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\label{fig:xmax_synchronise}
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\end{subfigure}
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\end{figure}
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\begin{equation}\label{eq:propagation_delay}%<<<
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\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff}
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\end{equation}%>>>
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\section{Radio Interferometry}
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% interference: (de)coherence
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Radio interferometry exploits the coherence of wave phenomena.
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\\
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In a radio array, each radio antenna records its ambient electric field.
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A simple interferometer can be achieved by summing the recorded waveforms $S_i$ with appropriate time delays $\Delta_i(\vec{x})$ to compute a coherent\Todo{word} waveform for a location $\vec{x}$,
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\begin{equation}\label{eq:interferometric_sum}%<<<
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\phantom{.}
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S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
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.
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\end{equation}%>>>
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\begin{figure}
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\centering
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\begin{subfigure}[t]{0.3\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
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\label{fig:trace_overlap:bad}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.3\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
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\label{fig:trace_overlap:medium}
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\end{subfigure}
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\hfill
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\begin{subfigure}[t]{0.3\textwidth}
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\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
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\label{fig:trace_overlap:best}
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\end{subfigure}
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\caption{
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Trace overlap due to wrong positions
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}
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\label{fig:trace_overlap}
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\end{figure}
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\begin{figure}
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\centering
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\includegraphics[width=0.7\textwidth]{2006.10348/fig03_b.png}%
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\caption{
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From \protect \cite{Schoorlemmer:2020low}.
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$\Xmax$ resolution as a function of detector-to-detector synchronisation.
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}
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\label{fig:xmax_synchronise}
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\end{figure}
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\section{Time Synchronisation}
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\label{sec:timesynchro}
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The main method of synchronising multiple stations is by employing a \gls{GNSS}.
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This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}).
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% time delays: general
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The time delays $\Delta_i(\vec{x})$ are dependent on the finite speed of the radio waves.
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Being an electromagnetic wave, the instantaneous velocity $v$ depends solely on the refractive~index~$n$ of the medium as $v = \frac{c}{n}$.
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In general, the refractive index of air is dependent on factors such as the pressure and temperature of the air the signal is passing through, and the frequencies of the signal.
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\\
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In many cases, the refractive index can be taken constant over the trajectory to simplify models.
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As such, the time delay due to propagation can be written as
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\begin{equation}\label{eq:propagation_delay}%<<<
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\phantom{,}
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\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_\mathrm{eff}
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,
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\end{equation}%>>>
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where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
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\\
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% time delays: particular per antenna
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Note that unlike in astronomical interferometry, the source of the signal is not in the far-field (see Figure~\ref{fig:rit_schematic}).
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Thus, instead of introducing a geometric phase, this requires us to compute the time delays for each antenna location separately.
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\\
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Need reference system with better accuracy to constrain current mechanism (Figure~\ref{fig:reference-clock}).
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% Features in S
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Features in the combined waveform $S(\vec{x})$ are enhanced according to the coherence of that feature in the recorded waveforms with respect to the time delays.
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Figures~\ref{fig:trace_overlap:best} and~\ref{fig:trace_overlap:bad} show examples of this effect for the same recorded waveforms.
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At the true source location, the recorded waveforms are aligned.
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The combined waveform therefore shows the
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Meanwhile, at a far away location, the waveforms add up incoherently resulting in a low amplitude combined waveform.
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\\
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% Noise suppression
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An additional effect of the summing is the suppression of noise particular to individual antennas as this is adds up incoherently.
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\Todo{rephrase}
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\\
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%\begin{figure}
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% \centering
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% \includegraphics[width=0.5\textwidth]{clocks/reference-clock.pdf}
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% \caption{
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% Using a reference clock to compare two other clocks.
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% \protect \todo{
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% redo figure with less margins,
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% remove spines,
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% rotate labels
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% }
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% }
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% \label{fig:reference-clock}
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%\end{figure}
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\begin{figure}% fig:trace_overlap %<<<
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\centering
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\begin{subfigure}[b]{0.47\textwidth}
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\includegraphics[height=8cm, width=\textwidth]{radio_interferometry/rit_schematic_far.pdf}%
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\caption{}
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\label{fig:rit_schematic}
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\end{subfigure}
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\hfill
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\begin{minipage}[b][7cm][s]{.47\textwidth}
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\begin{subfigure}{\textwidth}
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\includegraphics[height=2.5cm, width=\textwidth]{radio_interferometry/trace_overlap_best.png}
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\caption{}
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\label{fig:trace_overlap:best}
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\end{subfigure}
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\vfill
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\begin{subfigure}{\textwidth}
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\includegraphics[height=2.5cm, width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
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\caption{}
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\label{fig:trace_overlap:bad}
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\end{subfigure}
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\end{minipage}
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\caption{
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\textit{Left:}
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Schematic of radio interferometry.
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The antennas the time delays for a location $\vec{x}$ not trained on the source $S_0$.
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\protect \Todo{describe}
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\textit{Right:}
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Overlap between the recorded waveforms for the source location~\subref{fig:trace_overlap:best} and a far away location~\subref{fig:trace_overlap:bad}.
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\protect\Todo{include sum}
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}
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%\hfill
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%\begin{subfigure}[t]{0.3\textwidth}
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% \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
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% \label{fig:trace_overlap:medium}
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%\end{subfigure}
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%\hfill
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%\begin{subfigure}[t]{0.3\textwidth}
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% \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
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% \label{fig:trace_overlap:best}
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%\end{subfigure}
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%\label{fig:trace_overlap}
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\end{figure}% >>>
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% Spatial mapping of power
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In the technique from \cite{Schoorlemmer:2020low}, the air shower is identified using the power in the combined waveform.
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An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
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\\
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Here,
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Computing the combined waveform $S$ for multiple locations, and analysing the power in it, a source region can be identified as a maximum
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At locations with high power, the recorded waveforms interfere constructively while for low power locations, the interference is destructive.
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\end{document}
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