% vim: fdm=marker fmr=<<<,>>>
\documentclass[../thesis.tex]{subfiles}

\graphicspath{
	{.}
	{../../figures/}
	{../../../figures/}
}

\begin{document}
\chapter{Air Shower Radio Interferometry}
\label{sec:interferometry}
The radio signals emitted by an \gls{EAS} (see Section~\ref{sec:airshowers}) can be recorded by radio antennas.
For suitable frequencies, an array of radio antennas can be used as an interferometer.
Therefore, air showers can be analysed using radio interferometry.
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.
\\
In \cite{Schoorlemmer:2020low}, a technique was developed to obtain properties of an air shower using interferometry.%
\footnote{
	Available as a python package at \url{gitlab}.
}
As shown in Figure~\ref{fig:radio_air_shower}, the shower axis and particle densities along that axis can be observed.
From these, the energy, composition and direction of the cosmic particle can be derived.
\\

The accuracy of the technique is primarily dependent on the timing accuracy of the detectors.
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.
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 $2\ns$.
\\

\begin{figure}
	\centering
	\begin{subfigure}[t]{0.47\textwidth}
		\includegraphics[width=\textwidth]{2006.10348/fig01.no_title}%
		\caption{
			From \protect \cite{Schoorlemmer:2020low}.
			Radio interferometric power analysis of an \gls{EAS}.
			\protect \Todo{describe and expand caption, remove title}
		}
		\label{fig:radio_air_shower}
	\end{subfigure}
	\hfill
	\begin{subfigure}[t]{0.47\textwidth}
		\includegraphics[width=\textwidth]{2006.10348/fig03_b}%
		\caption{
			From \protect \cite{Schoorlemmer:2020low}.
			$\Xmax$ resolution as a function of detector-to-detector synchronisation.
			A typical noise (gaussian) background is simulated.
			\protect \Todo{describe and expand}
		}
		\label{fig:xmax_synchronise}
	\end{subfigure}
\end{figure}

\section{Radio Interferometry}
% interference: (de)coherence
Radio interferometry exploits the coherence of wave phenomena.
\\
In a radio array, each radio antenna records its ambient electric field.
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 $\vec{x}$,
\begin{equation}\label{eq:interferometric_sum}%<<<
	\phantom{.}
	S(\vec{x}, t) = \sum_i S_i(t + \Delta_i(\vec{x}))
	.
\end{equation}%>>>

% time delays: general
The time delays $\Delta_i(\vec{x})$ are dependent on the finite speed of the radio waves.
Being an electromagnetic wave, the instantaneous velocity $v$ depends solely on the refractive~index~$n$ of the medium as $v = \frac{c}{n}$.
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.
\\
In many cases, the refractive index can be taken constant over the trajectory to simplify models.
As such, the time delay due to propagation can be written as
\begin{equation}\label{eq:propagation_delay}%<<<
	\phantom{,}
	\Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_\mathrm{eff}
	,
\end{equation}%>>>
where $n_\mathrm{eff}$ is the effective refractive index over the trajectory of the signal.
\\
% time delays: particular per antenna
Note that unlike in astronomical interferometry, the source of the signal is not in the far-field (see Figure~\ref{fig:rit_schematic}).
Thus, instead of introducing a geometric phase, this requires us to compute the time delays for each antenna location separately.
\\

% Features in S
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.
Figures~\ref{fig:trace_overlap:best} and~\ref{fig:trace_overlap:bad} show examples of this effect for the same recorded waveforms.
At the true source location, the recorded waveforms are aligned.
The combined waveform therefore shows the 
Meanwhile, at a far away location, the waveforms add up incoherently resulting in a low amplitude combined waveform.
\\
% Noise suppression
An additional effect of the summing is the suppression of noise particular to individual antennas as this is adds up incoherently.
\Todo{rephrase}
\\

\begin{figure}% fig:trace_overlap %<<<
	\centering
	\begin{subfigure}[b]{0.47\textwidth}
		\includegraphics[height=8cm, width=\textwidth]{radio_interferometry/rit_schematic_far.pdf}%
		\caption{}
		\label{fig:rit_schematic}
	\end{subfigure}
	\hfill
	\begin{minipage}[b][7cm][s]{.47\textwidth}
		\begin{subfigure}{\textwidth}
			\includegraphics[height=2.5cm, width=\textwidth]{radio_interferometry/trace_overlap_best.png}
			\caption{}
			\label{fig:trace_overlap:best}
		\end{subfigure}
		\vfill
		\begin{subfigure}{\textwidth}
			\includegraphics[height=2.5cm, width=\textwidth]{radio_interferometry/trace_overlap_bad.png}
			\caption{}
			\label{fig:trace_overlap:bad}
		\end{subfigure}
	\end{minipage}
	\caption{
		\textit{Left:}
		Schematic of radio interferometry.
		The antennas the time delays for a location $\vec{x}$ not trained on the source $S_0$.
		\protect \Todo{describe}
		\textit{Right:}
		Overlap between the recorded waveforms for the source location~\subref{fig:trace_overlap:best} and a far away location~\subref{fig:trace_overlap:bad}.
		\protect\Todo{include sum}
	}
	%\hfill
	%\begin{subfigure}[t]{0.3\textwidth}
	%	\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_medium.png}
	%	\label{fig:trace_overlap:medium}
	%\end{subfigure}
	%\hfill
	%\begin{subfigure}[t]{0.3\textwidth}
	%	\includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_best.png}
	%	\label{fig:trace_overlap:best}
	%\end{subfigure}
	%\label{fig:trace_overlap}
\end{figure}% >>>


% Spatial mapping of power
In the technique from \cite{Schoorlemmer:2020low}, the air shower is identified using the power in the combined waveform.
An example of this power distribution of $S\vec{x}$ is shown in Figure~\ref{fig:radio_air_shower}.
\\
Here, 

Computing the combined waveform $S$ for multiple locations, and analysing the power in it, a source region can be identified as a maximum
At locations with high power, the recorded waveforms interfere constructively while for low power locations, the interference is destructive.
\end{document}
