From 475d34a28ca2aaa3ec9daf7665510757af0e25ca Mon Sep 17 00:00:00 2001 From: Eric Teunis de Boone Date: Fri, 27 Oct 2023 12:20:32 +0200 Subject: [PATCH] Thesis: Radio Interferometry: WIP --- .../thesis/chapters/radio_interferometry.tex | 188 ++++++++++++------ 1 file changed, 122 insertions(+), 66 deletions(-) diff --git a/documents/thesis/chapters/radio_interferometry.tex b/documents/thesis/chapters/radio_interferometry.tex index 7a8d237..ac9b9f4 100644 --- a/documents/thesis/chapters/radio_interferometry.tex +++ b/documents/thesis/chapters/radio_interferometry.tex @@ -10,85 +10,141 @@ \begin{document} \chapter{Air Shower Radio Interferometry} \label{sec:interferometry} -The radio signals emitted by the air shower (see Section~\ref{sec:airshowers}) can be recorded by radio antennas. -An array of radio antennas can be used as an interferometer. +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. \\ -% -Unlike, astronomical interferometry, the source of the signal is closeby. - +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 - \includegraphics[width=0.5\textwidth]{radio_interferometry/rit_schematic_true.pdf}% -% \includegraphics[width=0.5\textwidth]{radio_interferometry/Schematic_RIT_extracted.png} -% \caption{From H. Schoorlemmer} + \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} -\begin{equation}\label{eq:propagation_delay}%<<< - \Delta_i(\vec{x}) = \frac{ \left|{ \vec{x} - \vec{a_i} }\right| }{c} n_{eff} -\end{equation}%>>> - - +\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}%>>> - -\begin{figure} - \centering - \begin{subfigure}[t]{0.3\textwidth} - \includegraphics[width=\textwidth]{radio_interferometry/trace_overlap_bad.png} - \label{fig:trace_overlap:bad} - \end{subfigure} - \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} - \caption{ - Trace overlap due to wrong positions - } - \label{fig:trace_overlap} -\end{figure} - - - -\begin{figure} - \centering - \includegraphics[width=0.7\textwidth]{2006.10348/fig03_b.png}% - \caption{ - From \protect \cite{Schoorlemmer:2020low}. - $\Xmax$ resolution as a function of detector-to-detector synchronisation. - } - \label{fig:xmax_synchronise} -\end{figure} - -\section{Time Synchronisation} -\label{sec:timesynchro} -The main method of synchronising multiple stations is by employing a \gls{GNSS}. -This system should deliver timing with an accuracy in the order of $10\ns$ \cite{} (see Section~\ref{sec:grand:gnss}). +% 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. \\ -Need reference system with better accuracy to constrain current mechanism (Figure~\ref{fig:reference-clock}). +% 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} -% \centering -% \includegraphics[width=0.5\textwidth]{clocks/reference-clock.pdf} -% \caption{ -% Using a reference clock to compare two other clocks. -% \protect \todo{ -% redo figure with less margins, -% remove spines, -% rotate labels -% } -% } -% \label{fig:reference-clock} -%\end{figure} +\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}