source: ETALON/papers/2016_HDR_ND/Advanced_diags/smithpurcell.tex @ 798

\input{./header}

\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,
corresponding to bunch durations much shorter than what can be measured by fast detectors\footnote{Because in lepton accelerators the particles travel at the speed of light there is a direct correlation between bunch length and bunch duration.  It results that abusingly  the bunch length is often quoted with time units. A bunch duration of \SI{1}{ps} corresponds to a bunch length of \SI{0.3}{mm}.}. It is therefore necessary to convert the bunch length into another quantity that can be measured. 
Coherence is both a motivation and a tool to measure such length. Coherent collective effects such as coherent synchrotron radiation can significantly disrupt a beam. Their dependance on the beam current and not the beam charge means that to control them it is necessary to measure the bunch length instead of simply the bunch charge. As a counterpart, coherent radiation is one of the means of measuring a bunch length.

In plasma acceleration with external injection (such as FACET's  E-200~\cite{e200_litos}, AWAKE~\cite{awake} or ESCULAP~\cite{Delerue:2016tcy}) this bunch length measurement is important to estimate which fraction of the bunch will fit in the acceleration buckets created in the plasma. As with the transverse measurements, at a plasma accelerator it is necessary to make the measurement in a single shot for it to be meaningful.

When the resolution is sufficient the bunch length measurement can become a bunch profile measurement as is the case with the methods based on coherent radiation.

I have worked on one of these techniques using Coherent Smith-Purcell Radiation for several years.


\section{Bunch length  and bunch profile measurement techniques}
\sectionmark{Bunch length and profile measurement techniques}

Several other techniques allow for bunch length measurements and I will first review them.

\subsection{Techniques based on current transformers}

Bunch charge can be measured using current transformers (CT). Depending on the application, there are several flavors of current transformers~: AC-Current Transformers (ACCT),  DC-Current Transformers (DCCT),  Integrating Current Transformers (ICT),...  They all measure the current induced by the beam in a gap of the beam pipe to estimate the beam current. Depending on the signal amplification required the measurement device can be a simple wire connected to an ammeter or a coil with several windings connected to an ammeter or to more complicated electronics. 
The working principle and an example of current transformer are shown on figure~\ref{fig:current_transformer}.

When the time resolution of the current measuring device is good enough and the impedance low enough the current transformer can give information about the variation of the current with them and therefore the bunch length, however this is true only for beams that are at least several centimeters long (nanosecond duration).


\begin{figure}[htbp]
\begin{tabular}{cc}
\includegraphics[height=3.6cm]{Advanced_diags/torus_CT.jpg} &
\includegraphics[height=3.8cm]{Advanced_diags/ICT.png} 
\end{tabular}
\caption{(Left) Principle of a current transformer used for beam charge measurement~: when the particle beam passes through a torus it induces a current proportional to the beam charge in the torus coil windings, allowing for beam charge measurement. (Right) Example of current transformer as sold by a well-known manufacturer~\cite{Bergoz}.}
\label{fig:current_transformer}
\end{figure} 


\subsection{Techniques based on longitudinal to transverse exchange}

For beam with a length in the millimeter range  (or below) it is more difficult to measure their length or their profile in the longitudinal direction (that is the direction of propagation) than their transverse properties. Exchanging the longitudinal direction with one of the transverse directions can therefore make this measurement easier.

A {\bf Streak Camera} is a device available commercially that can achieve this fate~: in a streak camera a photon beam hits a photocathode and is converted by photoemission in a low energy (keV) electron beam. This electron beam passes between two electrodes to which are applied a high frequency high voltage electric field that will streak the electron beam transversely, the electron deflection being proportional to their arrival time.
After the electrodes the electron beam hits a luminescent screen where the transverse profile of the electrons after streaking can be observed. Thus it is possible to obtain the longitudinal profile of the photon beam (convoluted with one of its transverse profile). Commercial streak camera can reach a resolution of about \SI{1}{ps} on photon beams.

A {\bf Streak Camera combined with a radiation emitting screen}, such as a sapphire screen emitting Cerenkov radiation can be used to image the longitudinal profile of an electron beam with \si{MeV} energy or higher. Such method was tested for example at the CANDELA photo injector at LAL~\cite{Devanz:1996kc}. However this method is limited by the spread in the longitudinal profile induced by the screen itself and by the photon beam transport (usually to the outside of the accelerator) and therefore it is not available to measure sub-picoseconds beams.

{\bf Streak camera} are often used in rings to measure the bunch length using the synchrotron radiation emitted\cite{Thomas:2006db,Labat:dipac2007}.

This can be mitigated by deflecting directly the high energy  electrons using a {\bf deflecting RF cavity}. This requires the electric field to be high enough to deflect high energy electrons and to have a high enough frequency to streak significantly the beam. The state of the art in this field has been demonstrated at SLAC recently on both LCLS and FACET accelerators~\cite{PhysRevSTAB.17.102801}, reaching resolutions of \SI{10}{fs} at \SI{14}{GeV} and \SI{70}{fs} at \SI{20}{GeV}. However the cost of such device, including the associated RF power source is more than a million euros.

\begin{figure}[htbp]
\center
\includegraphics[height=3.6cm]{Advanced_diags/TCAV} 
\includegraphics[height=4.2cm]{Advanced_diags/TCAV_photo.png} 
\caption{Left: The principle of the Transverse deflecting cavity used at SLAC to measure bunches longitudinal profiles (Image taken from~\cite{TCAV_URL}). Right: A photo of the FACET transverse deflecting cavity (image taken from~\cite{PhysRevSTAB.17.102801}). }
\label{fig:TCAV}
% https://portal.slac.stanford.edu/sites/ard_public/facet/facilities/Pages/TCAV.aspx
\end{figure} 


A cheaper solution to streak the electrons is to use the so-called {\bf 3-phases method}~\cite{1994NIMPA.341...49G,Vinatier:2014dza} in which one of the accelerating section of the accelerator is dephased to provide a longitudinal streaking effect that can be measured as a variation in the beam energy dispersion. This method has the advantage of being much cheaper but is limited in its resolution by the power of the accelerating cavity and the resolution of the dispersion measurement setup. A resolution of a few picoseconds has been demonstrated experimentally.

\subsection{Measures based on radiation emitted by the beam}

Instead of manipulating the beam to measure it, it is possible to measure the radiation it emits and use it to get information on its longitudinal profile. This is what is done with the streak camera based measurements described above but other methods rely on this technique.

In the {\bf electro-optic sampling} method a birefringent crystal is brought close from the electron beam and a chirped ultrafast laser pulse\footnote{This means that the photons of the laser pulse have a correlation between wavelength and longitudinal position. This is a common techniques with lasers and can be done, for example, by using a set of diffraction gratings.} is shone on it. 
The electromagnetic field of the electron beam will induce a change in the refraction index of the crystal in at least one of the polarisation plane. As the laser pulse is chirped a diffraction grating can then be used to measure its longitudinal profile projected on a screen.
 Thus by comparing the longitudinal profile of the two polarisation components it is possible to deduce the changes that occurred in the crystal as fonction of time and therefore the bunch longitudinal profile.
This is a well known technique in optics  (for example to measure and profile THz pulses) that has been applied to accelerators by several groups (including, for example, \cite{PhysRevLett.85.3404,parc2007study,tomizawa_sato_ogawa_togawa_tanaka_hara_yabashi_tanaka_ishikawa_togashi}).
This technique is illustrated on figure~\ref{fig:electro_optic}.

\begin{figure}[htbp]
\center
%\includegraphics[height=3.6cm]{Fig-1-Layout-of-Electro-optic-sampling.png} 
\includegraphics[width=10cm]{Advanced_diags/urn-cambridge_org-id-binary-alt-20160717172124-83101-mediumThumb-S2095471915000092_fig3g.jpg}
\caption{Principle of bunch length measurements using electro-optic sampling  (taken from \cite{tomizawa_sato_ogawa_togawa_tanaka_hara_yabashi_tanaka_ishikawa_togashi}}
\label{fig:electro_optic}
\end{figure} 


Several techniques rely on the emission of {\bf Coherent radiation} which encodes the bunch length: The electromagnetic spectrum emitted by the electron bunch is modulated by the bunch length. When a photon is emitted in an electron bunch, if the wavelength of that photon is larger than the separation between the emitting electron and another electron it will enhance the probability of emission on that other electron. When $N$ electrons contribute to the emission the probability increases as $N(N-1)$.

It is possible to define the ``form factor'' (${\cal F}$) of the radiation emitted by an electron bunch as: 

\begin{equation}
{\cal F}(\nu) = \left| \int_{-\infty}^{+\infty}  S(x) \exp{\left( i\frac{2\pi \nu}{c}x \right) } dx  \right| \times  \left| \int_{-\infty}^{+\infty}  S(y) \exp{\left( i\frac{2\pi \nu}{c}y \right) } dy  \right| \times
 \left| \int_{-\infty}^{+\infty}  S(z) \exp{\left( i\frac{2\pi \nu}{c}z \right) } dz  \right| 
 \label{eq:form_factor}
\end{equation}

where $\nu$ is the photons' frequency, $S(x)$, $S(y)$ and $S(z)$ are the profile distribution along $x$, $y$ and $z$ respectively.
And the total radiation intensity $I_{tot}$ emitted for a given radiative phenomena at a given frequency will be:

\begin{equation}
I_{tot}(\nu) = I_1(\nu) \left( N + N(N-1) {\cal F}(\nu) \right)
\label{eq:coherent}
\end{equation}

where $I_1$ is the single electron yield for the phenomena and $N$ the number of electrons. This radiation can be used in several different manners to measure the bunch length and its longitudinal profile.

{\bf Coherent Transition Radiation (CTR)} is emitted when a bunch of charged particles passes through a thin foil. Several groups~\cite{Maxwell:2013fj,doi:10.1063/1.4790429,Wesch:2011qm} have studied how to measure the spectrum of the CTR emitted and used it to reconstruct bunch length. 
The SLAC group~\cite{Maxwell:2013fj} uses a KRS-5 prism to disperse the infrared radiation collected and focus it on a detector. The Frascati group~\cite{doi:10.1063/1.4790429} uses a Martin-Puplett interferometer to study the THz radiation produced by their CTR screens. These setups are shown on figure~\ref{fig:CTR_setups}. It should be noted that the methods used by these two groups require a scan of the dispersive element or of the interferometer to measure the radiation spectrum. To overcome this difficulty the DESY group~\cite{Wesch:2011qm} uses several dispersive gratings with several detectors to perform this measurement (see figure~\ref{fig:CTR_setups_DESY}). 

\begin{figure}[htbp]
\center
\begin{tabular}{cc}
\includegraphics[width=7.cm]{Advanced_diags/CTR_SLAC.jpg} &
\includegraphics[width=7.cm]{Advanced_diags/Experimental-layout-for-extraction-top-and-detection-bottom-of-THz-radiation.jpg} 
\end{tabular}
\caption{Principle of Coherent Transition Radiation measurements at SLAC on LCLS and FACET (left; taken from~\cite{Maxwell:2013fj}) and at SPARC (right; taken from~\cite{doi:10.1063/1.4790429}). }
\label{fig:CTR_setups}
\end{figure} 



\begin{figure}[htbp]
\center
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{Advanced_diags/1-s2_0-S0168900211020791-gr3.jpg} & 
\includegraphics[width=6.5cm]{Advanced_diags/1-s2_0-S0168900211020791-gr4.jpg} 
\end{tabular}
\caption{The Coherent Transition Radiation single shot setup at DESY (schematic and picture) as reported in~\cite{Wesch:2011qm}. }
\label{fig:CTR_setups_DESY}
\end{figure} 

The SLAC group has published their ability to reconstruct \SI{3}{fs} to \SI{60}{fs}-long bunches.


Another way to measure the radiation emitted by the bunch is to measure the {\bf Coherent Smith-Purcell Radiation (CSPR)} it emits. CSPR is emitted when a metallic diffraction grating is brought close from the beam. I have worked extensively on CSPR for several years and  it is described details in section~\ref{sec:CSPR}.

\subsection{Comparison of longitudinal bunch measurement techniques}

The different longitudinal bunch measurement techniques are summarized in tables~\ref{tab:long_meas_properties} and~\ref{tab:long_meas_capabilities}.


From theses tables it appears that there is a wide range of cost and complexity among longitudinal bunch length measurements.
The use of RF deflecting cavities is both complex and expensive (the price been dependent on the beam energy and the RF band chosen for the measurement) but it is used at several accelerators as a reference to measure the bunch length\footnote{A comparison of Electro-optic sampling, streak cameras and RF deflecting cavities as longitudinal bunch profile diagnostic for FACET has been published in~\cite{Litos:2011zz}.}.
This has triggered research to find cheaper techniques. Electro-optic sampling is a technique that was already available from the optics community, however it is complex as it requires an ultrafast laser that is hardly compatible with an accelerator enclosure and therefore requires to transport the ultrashort laser pulses inside the accelerator enclosure. CTR has the advantage of being even cheaper and less complicated to setup and several groups have investigated it and solved differently the problem of spectrum measurement. CSPR is a less known phenomena and therefore it has been investigated by fewer groups but renewed interest appeared because of its single shot and non-destructive capability. 

The interest for Energy Recovery Linac (ERL) may also increase the need for non destructive single shot longitudinal diagnostics: current ERLs have shown that correct operations require a very good understanding of the bunch longitudinal phase space at different locations along the ERL. The availability of a non destructive fast diagnostics would therefore be an advantage for such machine and in particular for the machine that is planned to be built in Orsay.

\newgeometry{width=200mm,height=280mm,top=5mm,bottom=10mm} 
\begin{landscape}
\pagestyle{empty}
\hspace{-1cm}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|c|c|c|c||}
Name & Main  technology  & Best resolution & Cost  & Availability status \\
\hline
\hline
Current transformer &  Coil & Limited by electronics &  $\mathcal{O}$ 10k\texteuro  & Commercially available\\
& & $\mathcal{O}$ 100 \si{ps} & & Commonly used  \\
\hline
Streak camera& Photocathode, High Voltage  & Limited by high voltage frequency  &  $\mathcal{O}$ 300k\texteuro  & \small Camera commercially available\\ 
 with radiator  & & $\mathcal{O}$  \si{ps} &  & Setup: R\&D\\
\hline
RF deflector & High power RF + cavity &  $\mathcal{O}$ 10 \si{fs} &  $\mathcal{O}$ 500k\texteuro - 2M\texteuro   & R\&D product distributed\\
& C-band or X-band RF & & (energy dependent)  & by one manufacturer \\
\hline
3 phases & RF cavities &  $\mathcal{O}$  \si{ps} & Use linac & Commonly used  \\
& & & infrastructure &  \\
\hline 
Electro-optic sampling & Laser &   $\mathcal{O}$ 50 \si{fs} &   Laser: $\mathcal{O}$ 200k\texteuro & Commonly used  \\
 &  Birefringent crystals  & (better res. in optics) &  Other:  $\mathcal{O}$ 50k\texteuro  & in optics \\
\hline 
Coherent Transition  & Optics  &  $\mathcal{O}$ 10 \si{fs} & $\mathcal{O}$ 50k\texteuro  & In use at \\
Radiation & THz filtering & & &  several facilities \\
\hline 
Coherent Smith-Purcell & Diff. gratings & $\mathcal{O}$ 50 \si{fs}  & $\mathcal{O}$ 50k\texteuro &  R\&D \\
 Radiation &  THz filtering &  & & \\
\hline 
\hline 
\end{tabular}
\caption{Comparaison of the properties of the different techniques of longitudinal bunch length and profile measurement}
\label{tab:long_meas_properties}
\end{center}
\end{table} 

\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|c|c||}
Name  & Single shot  capability & Beam destruction \\
\hline
\hline
Current transformer  & Yes & No \\
\hline
Streak camera with radiator & Yes  & Radiator dependent \\
& & Synchrotron radiation: No; Screen: Yes \\
\hline
RF deflector & Partially & Yes \\
& The phase of the measurement must be checked & \\
\hline
3 phases &  No  &  Yes (beam off-energy)\\
\hline
Electro-optic sampling  & Yes & No \\
\hline
Coherent Transition Radiation & Interferometer: No;   \small Multpl. Gratings: Yes & Yes \\
\hline
Coherent Smith-Purcell Radiation & Yes  & No \\
\hline 
\hline 
\end{tabular}
\caption{Comparaison of the capabilities of the different techniques of longitudinal bunch length and profile measurement}
\label{tab:long_meas_capabilities}
\end{center}
\end{table} 

\end{landscape}
%\restoregeometry 
%\pagestyle{plain}
\newgeometry{a4paper,total={145mm,217mm},  top=45mm,headsep=18mm}    


\section{Smith-Purcell Radiation}
\label{sec:CSPR}

\subsection{First observation of Smith-Purcell Radiation}

The first observation of what is now called Smith-Purcell Radiation dates back to 1953 when Smith and Purcell reported the observation of visible light from localized surface charges moving across a grating~\cite{SP1953}. Their observation was made using \SI{300}{keV} electrons in a continuous beam. An image of this observation is shown on figure~\ref{fig:first_SPR}.

\begin{figure}[htbp]
\center
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{Advanced_diags/smith_purcell_first_image} & 
\includegraphics[width=6.5cm]{Advanced_diags/grating_radiation.pdf} 
\end{tabular}
\caption{Left: Image taken from~\cite{SP1953} showing the observation of radiation by Smith and Purcell in 1953. Right: Schematic of Smith-Purcell radiation. }
\label{fig:first_SPR}
\end{figure} 

The first observation of Smith-Purcell radiation from relativistic electrons was reported only much later in 1992~\cite{PhysRevLett.69.1761} and
the first observation of Coherent Smith-Purcell Radiation~\cite{PhysRevE.51.R5212} came  in 1995 and an image taken from that publication is shown on figure~\ref{fig:first_CSPR}. 


\begin{figure}[htbp]
\center
\includegraphics[width=4cm]{Advanced_diags/PhysRevE_51_R5212_fig_4.png}
\caption{Image taken from~\cite{PhysRevE.51.R5212} showing the first observation of Coherent Smith-Purcell Radiation (CSPR): the intensity of the radiation varies exponentially with the current, not linearly.}
\label{fig:first_CSPR}
\end{figure} 

\subsection{Interpretation of Smith-Purcell Radiation}

The phenomena was further studied and several interpretations were proposed: 
\begin{itemize}
\item According to Ishiguro and Tako's interpretation~\cite{1961AcOpt.8.25I}, Smith-Purcell radiation comes from dipole radiation: when an electron beam passes above a grating it induces a current in the grating; the oscillations of this current in the teeth of  the grating will create dipole radiation that is emitted. This will later be referred to as the ``surface current" interpretation. This interpretation is shown on figure~\ref{fig:SPR_interpretation} (left).
\item Toraldo di Francia proposed a different interpretation~\cite{1960NCim.16.61T}: according to him a charged particle in straight uniform motion generates a field that can be expanded into a set of evanescent waves. It is the diffraction of these waves on a diffraction grating that creates the Smith-Purcell radiation. This will later be referred to as the ``radiation diffraction" interpretation. This interpretation is shown on figure~\ref{fig:SPR_interpretation} (right).
\end{itemize}


\begin{figure}[htbp]
\center
\begin{tabular}{cc}
\includegraphics[width=6.5cm]{Advanced_diags/grating_dipole_radiation.pdf} & 
\includegraphics[width=6.5cm]{Advanced_diags/grating_EM_diffraction.pdf} 
\end{tabular}
\caption{The two interpretation of Smith-Purcell radiation: The surface current interpretation as in~\cite{1961AcOpt.8.25I} (left) and the radiation diffraction interpretation as in~\cite{1960NCim.16.61T} (right).}
\label{fig:SPR_interpretation}
\end{figure} 

Although these interpretations are different, the underlying physics is the same and the interpretations should yield to comparable predictions. A comparison of several Smith-Purcell radiation models has been published in~\cite{Potylitsyn_SP_comparison} and based on that paper I worked with a student, Maksym Malovitsya,  to perform such comparison for parameters relevants to the experiments we were conducting~\cite{malovytsia:in2p3-01322176}. Our conclusion was that within an order of magnitude all models have to comparable single electron yield as shown on figure~\ref{fig:SPR_comparison}.


\begin{figure}[htbp]
\center
\begin{tabular}{cc}
\includegraphics[width=7.5cm]{Advanced_diags/MOPMB004f2.png} & 
\includegraphics[width=7.5cm]{Advanced_diags/MOPMB004f3.png} 
\end{tabular}
\caption{Comparison of the single electron yield of different Smith-Purcell Radiation models (as published in~\cite{malovytsia:in2p3-01322176}):
the solid blue line corresponds to the Resonant Diffraction Radiation (RDR) model~\cite{Potylitsyn_SP_comparison}, the green line to the Resonant Reflection Radiation (RRR) model~\cite{Potylitsyn_Prewave}, the purple line with square marker label GFW to the Surface Current model as described in~\cite{Doucas_Theory_1998} and the blue dashed line also to the Surface Current (SC) model described in~\cite{Doucas_Theory_1998} but with the grating coupling efficiency expression taken from~\cite{Potylitsyn_SP_comparison}. See~\cite{malovytsia:in2p3-01322176} for details.}
\label{fig:SPR_comparison}
\end{figure} 




\subsection{Application of Smith-Purcell Radiation}

Smith-Purcell radiation and later CSPR have been seen by many scholars as a potential source of high power infrared radiation, leading for example to a patent to that purpose~\cite{ekdahl1986high}. 
As early as 1979, Smith-Purcell radiation has also been seen by some scholars as a promising tool to seed Free Electron Lasers~\cite{doi:10.1063/1.325642,PhysRevA.40.876,PhysRevSTAB.7.070701,PhysRevLett.105.224801,Rullhusen:821138} and has been patented~\cite{brau2006smith}.

The use of CSPR as a bunch longitudinal profile diagnostic was proposed soon after the first observation of CSPR and it was also patented~\cite{nguyen1999measuring} (for the US only) at that time. More detailed work was published by a different team in 2002~\cite{DORIA2002263,PhysRevSTAB.5.072802} and several key steps were done in the following years~\cite{Doucas:2006ci,Doucas_ESB}, including in teams in which I was participating~\cite{Delerue:2009gr,SP_E203_first_paper,andrews:in2p3-00830708,E203prstab}.



\subsection{Theoretical aspects of Smith-Purcell Radiation}


The theory of Smith-Purcell radiation has been developed in several publications. The most relevant of them, based on the surface current model,   being~\cite{Doucas_Theory_1998,Doucas_Theory_1997}. The discussion below is based on these articles.

\begin{figure}[htbp]
\center
\includegraphics[width=7.5cm]{Advanced_diags/grating_radiation_coordinates.eps} 
\caption{Coordinates system used in this chapter. The direction $z$ comes out of the figure plane as is the azimuthal angle $\phi$.}
\label{fig:SP_coordinates}
\end{figure} 

A diffraction grating will reflect light in different directions by interferences between the rays reflected by each surface. As shown on figure~\ref{fig:grating_orders}, reflections corresponding to the simple Snell-Descartes law are called ``Order 0". Higher order correspond to interferences between all the teeth (order 1) or part of the teeth.

\begin{figure}[htbp]
\center
\includegraphics[width=7.5cm]{Advanced_diags/GratingSurface.png} 
\caption{The different orders of a grating: order 0 corresponds to a simple reflection, the other orders corresponds to interferences of the wavelengths reflections. Image taken from~\cite{url-diffraction-grating}.}
\label{fig:grating_orders}
\end{figure} 

The coordinate system used in this chapter is shown on figure~\ref{fig:SP_coordinates}.


The relation between the polar angle of observation $\theta$ (in the plane perpendicular to the grating surface and passing by the beam), the grating pitch $l$ and the wavelength $\lambda$ of the emitted radiation is given by

\begin{eqnarray}
\lambda & = & \frac{l}{n} \left( \frac{1}{\beta} - \cos \theta \right)
\label{eq:SP_wavelength}
\end{eqnarray}
where $n$ is the radiation order and $\beta = \frac{v}{c}$ is the ratio between the particle speed $v$ and the speed of light $c$. In lepton accelerator it is often very close to 1.
This relation is purely a consequence of the fact that waves emitted by a grating will interfere and in each direction constructive interferences correspond to specific wavelengths.

The  intensity of radiation emitted by a single electron (single electron yield), per unit solid angle ($\Omega$) is given by 
\begin{eqnarray}
\label{eq:SP_SEY}
\frac{dI_1}{d\Omega} & = & \frac{L}{l^2}\frac{2 \pi e^2 n^2 \beta^2}{\left(1 - \beta \cos \theta\right)^3}R^2 \exp{\frac{-2 x_0 }{\lambda_e}}
\end{eqnarray}
where $e$ is the electron charge, $L$ the grating length, $x_0$ is the beam grating separation, $\lambda_e$ is the evanescent wavelength of the virtual radiation emitted by the beam and $R^2$ is a factor reflecting the coupling of the beam with the grating.



The evanescent wavelength is given by 

\begin{eqnarray}
\lambda_e = \lambda \frac{\beta \gamma}{2 \pi \sqrt{1 + \left(  \beta \gamma \sin \theta \sin \phi   \right)?2 }}
\end{eqnarray}
where $\gamma$ is the Lorentz factor, $\phi$ is the azimuthal angle (the ascension above or below the plane perpendicular to the grating surface and passing by the beam).

An example of values of single electron yield taken from~\cite{delerue:in2p3-01322166}  is shown on figure~\ref{fig:sey2d}.
\begin{figure}[htbp]
  \centering
  \includegraphics[width=80mm]{Advanced_diags/MOPMB003f2.pdf}
  \caption{Single electron yield of Smith-Purcell radiation for a  $40\times180$ \si{mm^2}   grating  with a \SI{8}{mm} pitch and  a $\ang{30}$ blaze angle. Image taken from~\cite{delerue:in2p3-01322166}. }
  \label{fig:sey2d}
\end{figure}



As discussed above, equation~\ref{eq:coherent} applies to Smith-Purcell Radiation and the total radiation intensity emitted from a charged particle bunch of multiplicity $N$ is given by:
\begin{eqnarray}
\frac{dI}{ d\Omega} & = & \frac{dI_1}{ d\Omega} \left[ N + N(N-1) \mathcal{F}(\omega) \right]
\label{eq:SP_coherent}
\end{eqnarray}
where $\mathcal{F}(\omega)$ is the form factor introduced in equation~\ref{eq:form_factor}.

On figure~\ref{fig:compare_profiles} one can see a comparison of different profiles with the same FWHM (\SI{5}{ps}) but with different shapes and how this changes the expected CSPR signal.
 
\begin{figure}[htbp]
 \centering
  \includegraphics*[width=70mm]{Advanced_diags/shape_profile.eps}
  \includegraphics*[width=70mm]{Advanced_diags/shape_comparison.eps}
  \caption{Comparison of several bunch profiles with the same FWHM (\SI{5}{ps}) but with different shapes (left) and predicted CSPR signal for each of these profiles (right). The capability to distinguish internal features of the beam and in particular substructures appears clearly.  The parameters used for these simulations are based on the experiment done at CLIO (see section~\ref{sec:CLIO}).}
   \label{fig:compare_profiles}
\end{figure}




One important difference between CSPR and other radiative methods is that the choice of the grating will change the radiation intensity observed at different frequencies. On figure~\ref{fig:grating_effect} one can see the radiation intensities observed for the same bunch profile but with different gratings pitches.

\begin{figure}[htbp]
\center
\includegraphics[width=7.5cm]{Advanced_diags/SP_signal_different_gratings.eps} 
\caption{Radiation profiles observed for the same pulse but for different grating pitches. The parameters used for these simulations are based on the experiment done at CLIO (see section~\ref{sec:CLIO}).}
\label{fig:grating_effect}
\end{figure} 




\subsubsection{Improving the beam-coupling calculation}


The factor $R^2$ in equation~\ref{eq:SP_SEY} is rather complicated to estimate. It has been discussed in details in~\cite{Brownell:2005my} and an approximate solution has been given for high energy beams in the case of echelette gratings.


The equation~\ref{eq:SP_wavelength} is very similar the the standard grating equation in the Littrow condition (with the same definition of variables as above)~\cite{gratings_handbook}:
\begin{eqnarray}
\lambda = \frac{2 l \sin \theta}{n}
\end{eqnarray}

This has led me to study with two students Solène Le Corre~\cite{rapport_solene} and Clément Duval~\cite{rapport_duval} whether it would be possible to benefit from the advanced work done in the field of grating theory~\cite{diffraction_gratings_Loewen} to estimate $R^2$. We produced some predictions but these are rather close to what has been obtained with simulation codes based on~\cite{Brownell:2005my} and therefore we have not yet been able to distinguish the two approaches. This may become possible at the experiment described in~\ref{sec:CLIO}.


It should be noted that with the advent of fast and powerful Particle-in-cell electromagnetic simulation softwares, it becomes possible to simulate the electromagnetic effects that a charged particles beam induces near a grating without relying on the models described above. This is what has been attempted in~\cite{bakkalitaheri:in2p3-00822933}.

\subsection{Profile recovery}

As we can see in equation~\ref{eq:SP_coherent}, there is a mathematical relation between the form factor and the CSPR intensity at each wavelength. Using relation~\ref{eq:SP_wavelength} we see that this means that the signal intensity measured at different angles depends on the form factor and therefore encodes the bunch length. Calculating the form factor from the measured signal is therefore rather straightforward.

We have also seen in equation~\ref{eq:form_factor} that the form factor is strongly related to the bunch longitudinal profile. However the inverse operation is not so easy: some information is lost due to the absolute values in equation~\ref{eq:form_factor}. The knowledge of the form factor alone requires more complicated mathematics to recover the profile.

The Fourier transform of a physical function satisfies the Cauchy-Riemann equations therefore  a given real function (the measured form factor) can only be matched to a very limited number of imaginary functions and usually by adding extra conditions there will be only one remaining acceptable imaginary (phase)  function. This phase function can be found using a Hilbert transform or the Kramers Kronig (KK) relations.

To recover the phase of a form factor, it must first be written in the following form:
\begin{eqnarray}
log(\mathcal{F}(\omega))=log(\rho(\omega))+i\Theta(\omega)
\end{eqnarray}
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 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 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}

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:

\begin{equation}
\Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right| 
%\Delta_{FWXM} = \sum_{X \in  \{ 0.1 ; 0.2 ; 0.5 ; 0.8 ; 0.9\} } \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right| 
\end{equation}


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}.


\begin{figure}[htbp]
 \centering
 \begin{tabular}{cc}
  \includegraphics*[width=65mm]{Advanced_diags/plots_11541.eps} &  \includegraphics*[width=65mm]{Advanced_diags/plots_11658.eps}\\
     \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 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}



\begin{figure}[htbp]
 \centering
  \includegraphics*[width=70mm]{Advanced_diags/plots_61.eps}
   \includegraphics*[width=70mm]{Advanced_diags/plots_62.eps}
  \caption{Effect of the sampling frequencies (number of detectors) on the $\chi^2$ (left) and $\Delta_{FWHM}$ (right) when comparing original an reconstructed profiles, as presented in~\cite{reco_paper}. }
   \label{fig:sampling_frequency}
\end{figure}



\begin{figure}[htbp]
 \centering
  \includegraphics*[width=70mm]{Advanced_diags/plot1510_31.eps} 
    \includegraphics*[width=70mm]{Advanced_diags/plot1510_32.eps}
  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps} 
  \caption{Comparison of different samplings methods using the $\chi^2$ criterion (left) and $\Delta_{FWHM}$ (right), as presented in~\cite{reco_paper}.}
   \label{fig:detectors_positionning_comparison}
\end{figure}


\begin{figure}[htbp]
 \centering
  \includegraphics*[width=70mm]{Advanced_diags/plots_41.eps} 
    \includegraphics*[width=70mm]{Advanced_diags/plots_42.eps}
  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps} 
  \caption{Examples of detector positions for different types of  sampling for a minimal detector distance of $5^o$ (left) and $10^o$ (right), as presented in~\cite{reco_paper} .} 
   \label{fig:detectors_positionning}
\end{figure}


\begin{figure}[htbp]
 \centering
  \includegraphics*[width=70mm]{Advanced_diags/plot2210_181.eps}
  \includegraphics*[width=70mm]{Advanced_diags/plot2210_182.eps}
  \caption{Mean $\chi^2$ and $\Delta_{FWXM}$  as function of the noise amplitude, as presented in~\cite{reco_paper}. The interpretation of this figure is that with a 50\% noise on our signal we introduce an error on the profile FWHM of about 10\%.}
   \label{fig:reco_noise}
\end{figure}



\subsubsection{Near field, far field and pre-wave effects}

As the Smith-Purcell radiation is formed by a grating, there is a formation length and the number of teeth of the grating, the distance between the grating and the detector have to be taken into account. 
The line width of the radiation is given by the formula:
\begin{equation}
\frac{\delta \lambda}{ \lambda} = \frac{l}{nL}
\end{equation}

The effect of the grating-detector distance has been discussed in details in~\cite{E203prstab}.

With a student I also studied the predictions for the pre-wave effect~\cite{malovystia-2015} and this work shows that corrections might have to be applied, especially at very high energy. However owing to the non observation of this effect in~\cite{1748-0221-6-07-P07004} as discussed in section \ref{sec:OTR_interference}, experimental verification of the effect of pre-wave interferences and accurate calculation are needed. Experimental attempts will be discussed in section~\ref{sec:SPESO}.


\section{Experimental study of Coherent Smith-Purcell radiation}

When I joined the effort at the University of Oxford on CSPR our main goal was to demonstrate that CSPR could be used as a longitudinal profile diagnostic in the sub-picosecond regime. The work would lead to the E-203 collaboration that took data at SLAC. Later, after I moved to Orsay, I decided to extend this effort to  better understand the properties of CSPR using a test setup installed at SOLEIL and called SPESO and to also study how to make CSPR a tool available in accelerators control room, by making experiments on the CLIO accelerator.

\subsection{Smith-Purcell radiation measurement at FACET: E-203}

FACET (Facility for Advanced Accelerator Experimental Tests) was a \SI{20}{GeV} electron accelerator installed on the SLAC campus in Stanford, California~\cite{Seryi:2009zzc}. It had the advantage of being the only test facility offering easy access to sub-picosecond bunches.

In 2010 we proposed to re-use the hardware from a previous experiment~\cite{Doucas_ESB} to use it at FACET to measure sub-picosecond bunches. This led to the formation of the E-203 collaboration.

The experiment has been described in details in~\cite{SP_E203_first_paper}. It consists of a carousel on which 3 different gratings and a blank (a flat grating without teeth) are mounted. This carousel can be moved inside the accelerator vacuum to bring the grating close from the beam. Opposite to the carousel there are 11 silicon windows located every \ang{10} in $\theta$ from \ang{40} to \ang{140}. Outside the movable windows waveguide array plates (WAP) filters are used to filter only the radiation at the wavelength expected from the selected grating. After the filters Winston cones concentrate the radiation within  their acceptance onto pyroelectric detectors. Several pictures of the E-203 experiment are shown on figure~\ref{fig:E_203_setup} and the optical elements are shown on figure~\ref{fig:E_203_setup_optics}.


%\vspace*{1cm}
\begin{figure}[htbp]
   \centering
    \begin{tabular}{cc}
   \vspace*{-0.5cm} & \multirow{3}{*}{\includegraphics*[width=68mm,angle=90]{Advanced_diags/E203_motor_side.JPG}} \\
   \includegraphics*[width=50mm,angle=0]{Advanced_diags/E203_filters_side.JPG} \\
   \includegraphics*[width=50mm]{Advanced_diags/E203_carousel.JPG}    
       \end{tabular}
   \caption{The E-203 experiment (during pre-installation tests) seen from the detectors side (top left), from the motors side (right) and the grating's carousel (bottom left). Images taken from~\cite{andrews:in2p3-00830708}.}
   \label{fig:E_203_setup}
\end{figure}

\begin{figure}[htbp]
   \centering
   \hspace*{-.0cm} \begin{tabular}{ccc}
\includegraphics*[height=60mm]{Advanced_diags/E203_filters.png} \hspace*{-2cm}&
\includegraphics*[height=60mm]{Advanced_diags/E203_Winston_cone.png} &
\includegraphics*[height=60mm]{Advanced_diags/E203_optics.png} 
       \end{tabular}
   \caption{The optical parts of the E-203 experiment: (from left to right) the filters, the optical concentrators (winston cones) and the optical assembly (image taken from~\cite{E203prstab}).}
   \label{fig:E_203_setup_optics}
\end{figure}

The experiment was successful in measuring sub-picosecond bunches~\cite{SP_E203_first_paper,E203prstab,andrews:in2p3-00830708}. and examples of the profiles measured are shown on figure~\ref{fig:E_203_profiles}.



%\vspace*{1cm}
\begin{figure}[htbp]
   \centering
    \begin{tabular}{cc}
   \includegraphics*[width=75mm]{Advanced_diags/E203_high_comp_rho.png}  & 
   \includegraphics*[width=75mm]{Advanced_diags/E203_high_comp_profile.png} \\   
   \includegraphics*[width=75mm]{Advanced_diags/E203_med_comp_profile.png}  & 
   \includegraphics*[width=75mm]{Advanced_diags/E203_low_comp_profile.png} \\   
       \end{tabular}
   \caption{ Top left: The magnitude of the Fourrier Transform of the time profile of the bunch for a ``high compression'' run on the 9 April 2013 (see text in~\cite{E203prstab}). Other plots: bunch profile reconstructed from measurements at different times and with different bunch compression settings (top right: 9 April 2013, high compression; bottom left: 9 April 2013, medium compression; bottom right: 25 june 2013). Plots taken from~\cite{E203prstab} where more explanations can be found.}
   \label{fig:E_203_profiles}
\end{figure}

Background rejection is an important problem for radiative measurements. Because CSPR uses the dispersion induced by the grating it is less sensitive than other methods but still not immune from background noise. One method to measure the background is to expose a "blank" grating, that is a grating without teeth, to the beam and measure the signal produced. This signal is then considered to be the background.

Another solution could be to measure the two polarisation components of the signal: CSPR is known to be polarized whereas most backgrounds aren't.  We verified this with the E-203 apparatus~\cite{E203prstab} (see figure~\ref{fig:E203_polar}) and I worked further on that topic with a student, Solène Le Corre, who had taken part in the experiment~\cite{rapport_solene}.


\begin{figure}[htbp]
   \centering
\includegraphics*[height=60mm]{Advanced_diags/E203_polar.png} 
   \caption{The measured degree of polarisation of the grating signal as a function of observation angle. The solid line is the theoretically predicted polarisation of CSPR, as a function of wavelength (see text in~\cite{E203prstab} for details; plot taken from~\cite{E203prstab}).}
   \label{fig:E203_polar}
\end{figure}



Additional measurements of the polarization and the azimuthal distribution of CSPR have been done by the E-203 experiment and are being analyzed.


\subsection{Smith-Purcell radiation measurement at SOLEIL: SPESO}
\label{sec:SPESO}

Many questions about Coherent Smith-Purcell Radiation arose when we formed the E-203 collaboration and FACET being far from Europe, it was not the best suited place to make detailed comparison between theory and experimental results.

With colleagues from Synchrotron SOLEIL I therefore started another experiment called SPESO (Smith-Purcell Experiment at SOLEIL). SPESO is installed at the end of Helios, the SOLEIL linac. For simplicity reasons SPESO uses a fixed grating on a movable arm. Opposite to this window is a z-cut quartz window (transparent at the wavelengths at which the radiation is expected). In front of that window a set of 3 translation stages and 4 rotation stages allow to move detectors in 3D and mesure the intensity of Smith-Purcell radiation at different locations. 
The layout of the experiment is shown on figure~\ref{fig:SPESOlayout} and a photo is shown on figure~\ref{fig:SPESOphoto}. The experiment has been described in details at~\cite{delerue:in2p3-00862652}.

%\vspace*{2cm}
\begin{figure*}[tbp]
   \centering
\includegraphics*[width=85mm]{Advanced_diags/test_stand_step1_v5_cleaned.pdf} 
   \caption{The experimental setup of SPESO at the end of the SOLEIL Linac (image taken from~\cite{delerue:in2p3-00862652}).}
   \label{fig:SPESOlayout}
\end{figure*}

%\vspace*{2cm}
\begin{figure}[htbp]
   \centering
          \hspace*{-0.5cm}\begin{tabular}{cc}
 \includegraphics*[width=90mm,angle=90]{Advanced_diags/SPESO_layout.jpg} &
    \includegraphics*[width=0.45\textwidth]{Advanced_diags/MOPAB025f1.jpg}
       \end{tabular}
   \caption{The experimental setup of SPESO at the end of the SOLEIL Linac: left, as it was in 2015 (image taken from~\cite{delerue:in2p3-00862652}) and right with the polarizer installed in 2016 (image taken from~\cite{Delerue:IPAC17-SOLEIL}). On both images the electrons travels from left to right.
   }
   \label{fig:SPESOphoto}
\end{figure}

The first SPESO signal was difficult to obtain because the pulse length had been underestimated. After some searching we finally found some signal at wavelength much longer than expected (see figure~\ref{fig:SPESO_first_data}). These data allowed us to reconstruct the spectrum emitted by the bunch and its profile (see figure~\ref{fig:SPESO_signal_spectrum}). These results were reported in~\cite{Delerue:2016zgr} where detailed explanations of these figures are available. Discussion with persons who contributed to the commissioning of the linac showed that our measurement is in agreement with the measurements done during the linac commissioning.




\begin{figure}[htbp]
%   \vspace*{-.5\baselineskip}
   \centering
       \hspace*{-0.5cm}\begin{tabular}{cc}
   \includegraphics*[width=75mm]{Advanced_diags/MOPMB002f2_negated.png} &    \includegraphics*[width=75mm]{Advanced_diags/MOPMB002f5a.png}
       \end{tabular}
   \caption{Left: Example adapted from~\cite{Delerue:2016zgr} of the initial data recorded with the two SPESO detectors (Pink is the Q-band detector and blue is the Ka-band detector, see explanation in~\cite{Delerue:2016zgr}).  Right:  Example taken from~\cite{Delerue:2016zgr}  of raw signal collected with the the  Ka-band  detector while the detectors were moving vertically. The red line corresponds to the grating in the inserted position and the blue line to the grating in the retracted position (background). }
   \label{fig:SPESO_first_data}
%   \vspace*{-\baselineskip}
\end{figure}




\begin{figure}[htbp]
%   \vspace*{-.5\baselineskip}
   \centering
       \hspace*{-1.5cm}\begin{tabular}{cc}
   \includegraphics[height=45mm]{Advanced_diags/MOPMB002f6.eps} &
   \includegraphics[height=45mm]{Advanced_diags/MOPMB002f7.eps}
\end{tabular}   
   \caption{Images taken from~\cite{Delerue:2016zgr}. Left: Distribution of the signal measured as function of the observation angle. The red dots correspond to the Ka band detector and the blue dots to the Q-band detector. The solid lines correspond to data points at frequencies within the sensitivity range of the detector and the dashed lines to data points at frequencies outside that range. These data are compared to simulations for 4 different bunch FWHM. Right: Bunch profile recovered from the data presented on the left using the method described in~\cite{reco_paper}.}
   \label{fig:SPESO_signal_spectrum}
%   \vspace*{-\baselineskip}
\end{figure}

The SPESO experiment has been upgraded in 2016 to study the polarisation of the Smith-Purcell radiation. The result has been presented in~\cite{Delerue:IPAC17-SOLEIL} and is shown in figure~\ref{fig:SPRESO_polarization_2017}. This result is a surprise to us as we expected almost 100\% polarization but it could be compatible with what was observed at FACET (see figure~\ref{fig:E203_polar}). The measurement campaign on SPESO is continuing to increase the range of the measurement.


\begin{figure*}[!htb]
%   \vspace*{-.5\baselineskip}
   \centering
   \includegraphics*[width=70mm]{Advanced_diags/MOPAB025f6.pdf}
   \caption{Degree of polarisation as function of the longitudinal displacement (along the beam propagation axis). Image taken from~\cite{Delerue:IPAC17-SOLEIL}. }
   \label{fig:SPRESO_polarization_2017}
%   \vspace*{-\baselineskip}
\end{figure*}


\subsection{Coherent Smith-Purcell radiation measurement at CLIO.}
\label{sec:CLIO}

SPESO is meant to be a test bench to study the properties of Coherent Smith-Purcell Radiation. However to have this diagnostic accepted in control rooms, it is necessary to demonstrate that it can work as a control room tool and produce results quickly. This is the aim of the Smith-Purcell experiment at CLIO. CLIO (Acronym in french for Infrared Laser Center at Orsay - {\em Centre Laser Infrarouge à Orsay}) is a Free Electron Laser installed in Orsay. It has been described in~\cite{Ageron:1989eq,bourdon:in2p3-00020744,Glotin:1992aj}.

Before installing the experiment we studied the  dynamics of the beam and the predicted signal~\cite{delerue:in2p3-01322166}. Example of expected signal are shown on figure~\ref{fig:CLIO_sp-normalised}. To avoid searching the signal for too long in case the pulse length was very different from the expectations we decided to first measure CTR and made a comparison of expected signal level for CTR and CSPR~\cite{delerue:in2p3-01322182}. On figure~\ref{fig:CLIO_CTR_CSPRr} the single electron yield and the predicted spectrum at CLIO are compared.
Our conclusion was that using the CLIO parameters and with a beam-grating separation of \SI{3}{mm} we expect a signal (in the range 0.03-3 THz [ 0.1 - 10 mm]) of \SI{8.37e-7}{J} for CSPR and \SI{7.35e-08}{J} for CTR. This result was questioned by a few colleagues but finally accepted and confirmed by independent experimental verifications made at another facility by another group~\cite{PhysRevAccelBeams.20.024701}.


\begin{figure}[!htb]
  \centering
       \hspace*{-0.05cm}\begin{tabular}{cc}
    \includegraphics[width=0.45\linewidth]{Advanced_diags/MOPMB005f6.pdf} &
  \includegraphics[width=0.45\linewidth]{Advanced_diags/MOPMB005f7.pdf}
\end{tabular}   
  
  \caption{  Coherent Smith-Purcell spectrum as a function of the observation angle for different bunch profiles in CLIO. The grating used for these simulations has a pitch of 8~mm and a blaze angle of $30^o$. The left figure gives the energy distribution, the right figure is normalized so that the maximum amplitude of each line is 1. Image taken from~\cite{delerue:in2p3-01322166}.
}
  \label{fig:CLIO_sp-normalised}
  \end{figure}

\begin{figure}[htb]
  \centering
  \label{sey}
\end{figure}

\begin{figure}[!tbp]
  \centering
       \hspace*{-0.05cm}\begin{tabular}{cc}
  \includegraphics[height=55mm]{Advanced_diags/MOPMB003f1.pdf} &
  \includegraphics[height=55mm]{Advanced_diags/MOPMB003f9.pdf}
  \end{tabular}   

  \caption{Left: Single electron yield for transition radiation (TR) and Smith-Purcell radiation (SP). The screen diameter for transition radiation is \SI{40}{mm}. The SP  single electron yield is presented for different beam-grating separation (\SIlist{3;6;9}{mm}). The grating used here is   $40\times180$ \si{mm^2} with 8 mm pitch and $30^o$ blaze angle. The signal is measured as integrated with a \SI{50}{mm} diameter parabolic mirror located \SI{300}{mm} from the beam axis. Right:  CSPR and CTR energy density as function of wavelength with the same parameters than on the left figure. Both images are taken from~\cite{delerue:in2p3-01322166}.}
  \label{fig:CLIO_CTR_CSPRr}
\end{figure}

After a first phase during which we measured CTR to validate the detection chain the experiment has quickly been upgraded with a grating instead of the CTR screen. Unlike the E-203 experiment we use single quartz window to let the radiation out. We also decided to use Off Axis Parabolic mirrors (OAP) instead of Winston cones to focus the radiation of the detectors. Photos of the experiment are shown on figure~\ref{fig:CLIO_expstp} and the experiment has been described in details in~\cite{Delerue:IPAC17-CLIO}.

\begin{figure}
  \centering
       \hspace*{-0.05cm}\begin{tabular}{cc}
  \includegraphics[height=55mm]{Advanced_diags/20170203_193939_resized.jpg} &
  \includegraphics[height=55mm]{Advanced_diags/MOPAB026f1.jpg}
    \end{tabular}   

  \caption{The experimental setup for CSPR measurements at CLIO: a set of pyrodectors with off axis parabolic mirrors is placed equidistantly with \ang{7} separation around the window and the experimental chamber with the grating inside (right image taken from~\cite{Delerue:IPAC17-CLIO}). The two images have been taken at different dates and have a different number of detectors.}
  \label{fig:CLIO_expstp}
\end{figure}

\begin{figure}[!htb]
  \centering
       \hspace*{-0.05cm}\begin{tabular}{cc}
  \includegraphics[height=55mm]{Advanced_diags/MOPAB026f3.pdf} &
  \includegraphics[height=55mm]{Advanced_diags/MOPAB026f6.pdf} 
     \end{tabular}
  \caption{Signal amplitude as function of the grating position with respect to the center of the beam for three different detectors angles for a long range (left) and a short range near the beam (right). On the right the data (circles) is compared to a fit (solid line). Images taken form~\cite{Delerue:IPAC17-CLIO}. On the left image the peak between \SI{20}{mm} and \SI{30}{mm} is still under investigations.}
  \label{fig:CLIO_gscan}
\end{figure}


After the publication of~\cite{Delerue:IPAC17-CLIO} further work was done and has been reported in~\cite{rapport_vitalii_CLIO_2017}. The figure~\ref{fig:CLIO_form_factor} shows form factors that were measured on CLIO and figure~\ref{fig:CLIO_profile} shows the reconstructed profile.

\begin{figure}[!htb]
  \centering
       \hspace*{-0.05cm}\begin{tabular}{cc}
  \includegraphics[height=55mm]{Advanced_diags/FF.eps} &
  \includegraphics[height=55mm]{Advanced_diags/FF6.eps} 
     \end{tabular}
  \caption{Form factor measured at CLIO as reported in~\cite{rapport_vitalii_CLIO_2017}. Left: Variation of the form factor when the beam grating separation varies. Right: Variation of the form factor when the buncher phase varies.
    }
  \label{fig:CLIO_form_factor}
\end{figure}

\begin{figure}[!htb]
  \centering
  \includegraphics[width=80mm]{Advanced_diags/Profile1.eps}
  \caption{Bunche profile reconstructed on CLIO for different buncher phase as reported in~\cite{rapport_vitalii_CLIO_2017}.
    }
  \label{fig:CLIO_profile}
\end{figure}

During the recent runs we took data with two different grating pitches. As the data were taken on different days (at the moment changing the grating requires breaking the vacuum) it is difficult to put them together but the comparisons we did looks promising.

The experiment will be upgraded in a few months so that we will be able to change the grating without breaking the vacuum. We also hope to be able to test strategies were 3 gratings will be exposed at the same time (at different azimuthal angles) to be able to do single shot measurements a scheme showing a transverse view of such layout is shown on figure~\ref{fig:multiple_gratings}.


\begin{figure}[!htb]
  \centering
  \includegraphics[width=50mm]{Advanced_diags/multiple_gratings.png}
  \caption{Proposed scheme for single show measurement: in the transverse plane three different grating, \ang{60} apart are used to measure three different wavelength ranges from the beam.
    }
  \label{fig:multiple_gratings}
\end{figure}


\subsection{Outlook: Application to laser-driven plasma accelerators, ERLs and the ESS}

Coherent Smith-Purcell Radiation has the potential to become a single shot diagnostic to measure longitudinal profile. 

One of the initial motivation for this diagnostic is to be able to measure the longitudinal profile of beams coming out of plasma accelerators. The very short pulses produced by such accelerator make them a very interesting topic to study and the ESCULAP facility at LAL (see section~\ref{sec:ESCULAP}) will offer good testing opportunities. One important question that CSPR should allow to settle on a plasma acceleration experiment is wether the electron pulse has been produced alone or if it accompanied by satellites in the neighboring buckets.

However on the way other applications have appeared. The European Spallation Facility will have ps-bunches in its high energy and their length will have to be measured non destructively. CSPR is well suited for such measure as it is non intercepting and some preliminary studies have been performed~\cite{barros:in2p3-01020770}. Another location where the non intercepting nature of CSPR combined to the moderate cost of the associated detector can be an advantage is Energy Recovery Linacs (ERL). In an ERL longitudinal deformation of the electron bunch can be amplified at each pass leading to beam disruption. A monitor capable of monitoring the bunch profile at several locations without destroying it  is therefore important. This can be done by a CSPR monitor.


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