mirror of
https://gitlab.science.ru.nl/mthesis-edeboone/m.internship-documentation.git
synced 2025-10-31 04:06:37 +01:00
Thesis: Radio Interferometry: WIP
This commit is contained in:
parent
89ddf4cc81
commit
475d34a28c
1 changed files with 122 additions and 66 deletions
|
|
@ -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}
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue