Changeset 757 in ETALON for papers/2016_HDR_ND/Advanced_diags/smithpurcell.tex

Timestamp:

Mar 31, 2018, 10:37:05 PM (6 years ago)

Author:

delerue

Message:

HDR - manuscript a la soutenance

File:

• papers/2016_HDR_ND/Advanced_diags/smithpurcell.tex (modified) (4 diffs)

papers/2016_HDR_ND/Advanced_diags/smithpurcell.tex

@@ -3 +3 @@
 
 \chapter{Bunch length measurement}
+\label{chap:SP}
 
 The length of a particle bunch is rather difficult to measure at lepton accelerators where this length is typically less than a millimeter,
@@ -409 +410 @@
 where $\rho(\omega)$ is the amplitude and $\Theta(\omega)$ the phase associated to this form factor. The function $\rho(\omega)$ is thus the result of the measurement and  $\Theta(\omega)$  the information that needs to be recovered.
 
-The Hilbert transform gives then the relation
+The Hilbert transform then gives  the relation
 \begin{equation}
 \Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.
 \end{equation}
 
-and the Kramers Kronig relation gives:
+and using the definition given in~\cite{KK} the Kramers Kronig relation gives:
 \begin{equation}
 \Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega
 \end{equation}
 
-In most cases the value given by these two methods will be very close.
-
-Before I started working on CSPR an algorithm based on KK was already available~\cite{KK} for all coherent radiation phenomena. However its implementation for CSPR was sometimes leading to non-sensical results. With a student, Vitalii Khodnevych, we implemented and studied a new phase recovery algorithm based on the Hilbert transform~\cite{delerue:in2p3-01020128}. We extended this work by doing a large number of simulations to also study the precision of this algorithm, how to optimise the position and number of detectors and how noise can affect the quality of the measurement~\cite{reco_paper}. The main results of this paper are presented below.
+As one can see these two relations are very similar and in most cases the reconstructed phase will be identical, differences will only appear in pathological cases.
+
+Before I started working on CSPR an algorithm based on~\cite{KK} was already available for all coherent radiation phenomena. However its implementation for CSPR was sometimes leading to non-sensical results. With a student, Vitalii Khodnevych, we implemented and studied a new phase recovery algorithm based on the Hilbert transform available in Matlab~\cite{delerue:in2p3-01020128}, allowing faster and more reliable computation of the phase. We extended this work by doing a large number of simulations to also study the precision of this algorithm, how to optimise the position and number of detectors and how noise can affect the quality of the measurement~\cite{reco_paper}. The main results of this paper are presented below.
 
 To study how profiles shapes match each other we defined a variable by analogy to the standard Full-Width at Half-Maximum (FWHM), this variable called Full-Width at X of the Maximum (FWXM)  allows to study the width of a pulse at a certain fraction X of the maximum. The FW0.5M is similar to the usual FWHM. To compare the shape of a reconstructed profile (deco) and its orignal (orig) we define the variable $\Delta_{FWXM}$ as follow:
@@ -431 +432 @@
 
 
-Figure~\ref{fig:good_bad_profiles} for example shows example of simulated profiles correctly reconstructed but also profiles poorly reconstructed. Figure~\ref{fig:sampling_frequency} studies the impact of the number of detectors of the quality of the reconstruction, showing that there is an optimum at about $3 \times 11$ detectors (3 sets of 11 detectors covering different frequency ranges) and figure~\ref{fig:detectors_positionning_comparison} shows that positioning them at constant angle is better than positioning them linearly or logarithmically (in frequency), examples of possible detectors positioning are shown on figure~\ref{fig:detectors_positionning}. Finally we also studied the effect of noise on the measured signal on the quality of the reconstruction~\ref{fig:reco_noise}.
+Figure~\ref{fig:good_bad_profiles} for example shows example of simulated profiles correctly reconstructed but also profiles poorly reconstructed (in that figure and the followings, ``Hilbert'' denotes the computation done using the Matlab implementation of the Hilbert transform and ``Kramers-Kronig'' denotes the phase computed by another code based on~\cite{KK}). Figure~\ref{fig:sampling_frequency} studies the impact of the number of detectors of the quality of the reconstruction, showing that there is an optimum at about $3 \times 11$ detectors (3 sets of 11 detectors covering different frequency ranges) and figure~\ref{fig:detectors_positionning_comparison} shows that positioning them at constant angle is better than positioning them linearly or logarithmically (in frequency), examples of possible detectors positioning are shown on figure~\ref{fig:detectors_positionning}. Finally we also studied the effect of noise on the measured signal on the quality of the reconstruction~\ref{fig:reco_noise}.
 
 
@@ -440 +441 @@
      \includegraphics*[width=65mm]{Advanced_diags/plots_12231.eps} &    \includegraphics*[width=65mm]{Advanced_diags/plots_12667.eps}
 \end{tabular}
-  \caption{Example of profiles correctly reconstructed (upper row) and poorly reconstructed (lower row) as presented in~\cite{reco_paper}. The original profile is in blue and the profiles reconstructed with the Hilbert transform and the full Kramers-Kronig procedures are in red and black respectively.}
+  \caption{Example of profiles correctly reconstructed (upper row) and poorly reconstructed (lower row) as presented in~\cite{reco_paper}. The original profile is in blue and the profiles reconstructed with the Matlab Hilbert transform and the code implementing Kramers-Kronig relation~\cite{KK} are in red and black respectively.}
    \label{fig:good_bad_profiles}
 \end{figure}