Changeset 736 in ETALON


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Timestamp:
Sep 1, 2017, 5:58:20 PM (7 years ago)
Author:
delerue
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Progress on the HDR

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papers/2016_HDR_ND/Advanced_diags
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137 added
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  • papers/2016_HDR_ND/Advanced_diags/smithpurcell.tex

    r640 r736  
    44\chapter{Bunch length measurement}
    55
    6 \section{Bunch length measurement techniques}
    7 
    8 The length of a particle bunch is rather difficult to measure at lepton accelerators where this length is typically much shorter than what can be measured by fast detectors. 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.
     6The length of a particle bunch is rather difficult to measure at lepton accelerators where this length is typically less than a millimeter,
     7corresponding 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.
     8Coherence 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.
    99
    1010In 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.
    … …  
    1212When 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 methods.
    1313
    14 I have worked on this technique for several years.
     14I have worked on one of these techniques using Coherent Smith-Purcell Radiation for several years.
     15
     16
     17\section{Bunch length  and bunch profile measurement techniques}
     18
     19Several other techniques allow for bunch length measurements and I will first review them.
     20
     21\subsection{Techniques based on current transformers}
     22
     23Bunch 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.
     24The working principle and an example of current transformer are shown on figure~\ref{fig:current_transformer}.
     25
     26When 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).
     27
     28
     29\begin{figure}[htbp]
     30\begin{tabular}{cc}
     31\includegraphics[height=3.6cm]{Advanced_diags/torus_CT.jpg} &
     32\includegraphics[height=3.8cm]{Advanced_diags/ICT.png}
     33\end{tabular}
     34\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}.}
     35\label{fig:current_transformer}
     36\end{figure}
    1537
    1638
    1739\subsection{Techniques based on longitudinal to transverse exchange}
    1840
    19 
    20 
    21 
    22 - streak
    23 
    24 
    25 
    26 - 3 phases
    27 
    28 \subsection{Measures based on transition radiation}
    29 
    30 EOptique
    31 CTR
    32 
    33 \subsection{Measures based on Coherent Smith-Purcell Radiation}
    34 
    35 CSPR
    36 
    37 Resolution max.
    38 
    39 \subsection{Single shot Bunch length measurement}
     41For 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.
     42
     43A {\bf Streak Camera} is a device available commercially that can achieve this fate~: in a streak 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.
     44After the electrodes the electron beam hit 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.
     45
     46A {\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.
     47
     48{\bf Streak camera} are often used in rings to measure the bunch length using the synchrotron radiation emitted\cite{Thomas:2006db,Labat:dipac2007}.
     49
     50This 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.
     51
     52\begin{figure}[htbp]
     53\center
     54\includegraphics[height=3.6cm]{Advanced_diags/TCAV}
     55\includegraphics[height=4.2cm]{Advanced_diags/TCAV_photo.png}
     56\caption{Left: The principle of the Transverse deflecting cavity used at SLAC to measure bunches longitudinal profiles (Image taken from~\url{https://portal.slac.stanford.edu/sites/ard_public/facet/facilities/Pages/TCAV.aspx}). Right: A photo of the FACET transverse deflecting cavity (image taken from~\cite{PhysRevSTAB.17.102801}). }
     57\label{fig:TCAV}
     58\end{figure}
     59
     60
     61A 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.
     62
     63\subsection{Measures based on radiation emitted by the beam}
     64
     65Instead 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.
     66
     67In 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.
     68The 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.
     69 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.
     70This 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}).
     71This technique is illustrated on figure~\ref{fig:electro_optic}.
     72
     73\begin{figure}[htbp]
     74\center
     75%\includegraphics[height=3.6cm]{Fig-1-Layout-of-Electro-optic-sampling.png}
     76\includegraphics[width=10cm]{Advanced_diags/urn-cambridge_org-id-binary-alt-20160717172124-83101-mediumThumb-S2095471915000092_fig3g.jpg}
     77\caption{Principle of bunch length measurements using electro-optic sampling  (taken from \cite{tomizawa_sato_ogawa_togawa_tanaka_hara_yabashi_tanaka_ishikawa_togashi}}
     78\label{fig:electro_optic}
     79\end{figure}
     80
     81
     82
     83The 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 two photons, then the two photons will contribute and the emission probability is increased. When $N$ photons contribute the probability increases as $N(N-1)$.
     84
     85It is possible to define the ``form factor'' (${\cal F}$) of the radiation emitted by an electron bunch as:
     86
     87\begin{equation}
     88{\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
     89 \left| \int_{-\infty}^{+\infty}  S(z) \exp{\left( i\frac{2\pi \nu}{c}z \right) } dz  \right|
     90 \label{eq:form_factor}
     91\end{equation}
     92
     93where $\nu$ is the photons' frequency, $S(x)$, $S(y)$ and $S(z)$ are the profile distribution along $x$, $y$ and $z$ respectively.
     94And the total radiation intensity $I_{tot}$ emitted for a given radiative phenomena at a given frequency will be:
     95
     96\begin{equation}
     97I_{tot}(\nu) = I_1(\nu) \left( N + N(N-1) {\cal F}(\nu) \right)
     98\label{eq:coherent}
     99\end{equation}
     100
     101where $I_1$ 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.
     102
     103{\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.
     104The 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}).
     105
     106\begin{figure}[htbp]
     107\center
     108\begin{tabular}{cc}
     109\includegraphics[width=7.cm]{Advanced_diags/CTR_SLAC.jpg} &
     110\includegraphics[width=7.cm]{Advanced_diags/Experimental-layout-for-extraction-top-and-detection-bottom-of-THz-radiation.jpg}
     111\end{tabular}
     112\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}). }
     113\label{fig:CTR_setups}
     114\end{figure}
     115
     116
     117
     118\begin{figure}[htbp]
     119\center
     120\begin{tabular}{cc}
     121\includegraphics[width=6.5cm]{Advanced_diags/1-s2_0-S0168900211020791-gr3.jpg} &
     122\includegraphics[width=6.5cm]{Advanced_diags/1-s2_0-S0168900211020791-gr4.jpg}
     123\end{tabular}
     124\caption{Principle XX of the Coherent Transition Radiation single shot setup at DESY (schematic and picture). }
     125\label{fig:CTR_setups_DESY}
     126\end{figure}
     127
     128The SLAC group has published their ability to reconstruct \SI{3}{fs} to \SI{60}{fs}-long bunches.
     129
     130
     131Another 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}.
     132
     133\subsection{Comparison of longitudinal bunch measurement techniques}
     134
     135The different longitudinal bunch measurement techniques are summarized in tables~\ref{tab:long_meas_properties} and~\ref{tab:long_meas_capabilities}.
     136
     137
     138From theses tables it appears that there is a wide range of cost and complexity among longitudinal bunch length measurements.
     139The 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}.}.
     140This 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.
     141
     142The 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.
     143
     144\newgeometry{width=200mm,height=280mm,top=5mm,bottom=10mm}
     145\pagestyle{empty}
     146\begin{landscape}
     147\hspace{-1cm}
     148\begin{table}[htbp]
     149\begin{center}
     150\begin{tabular}{l|c|c|c|c||}
     151Name & Main  technology  & Best resolution & Cost  & Availability status \\
     152\hline
     153\hline
     154Current transformer &  Coil & Limited by electronics &  $\mathcal{O}$ 10k\texteuro  & Commercially available\\
     155& & $\mathcal{O}$ 100 \si{ps} & & Commonly used  \\
     156\hline
     157Streak camera& Photocathode, High Voltage  & Limited by high voltage frequency  &  $\mathcal{O}$ 300k\texteuro  & \small Camera commercially available\\
     158 with radiator  & & $\mathcal{O}$  \si{ps} &  & Setup: R\&D\\
     159\hline
     160RF deflector & High power RF + cavity &  $\mathcal{O}$ 10 \si{fs} &  $\mathcal{O}$ 500k\texteuro - 2M\texteuro   & R\&D product distributed\\
     161& C-band or X-band RF & & (energy dependent)  & by one manufacturer \\
     162\hline
     1633 phases & RF cavities &  $\mathcal{O}$  \si{ps} & Use linac & Commonly used  \\
     164& & & infrastructure &  \\
     165\hline
     166Electro-optic sampling & Laser &   $\mathcal{O}$ 50 \si{fs} &   Laser: $\mathcal{O}$ 200k\texteuro & Commonly used  \\
     167 &  Birefringent crystals  & (better res. in optics) &  Other:  $\mathcal{O}$ 50k\texteuro  & in optics \\
     168\hline
     169Coherent Transition  & Optics  &  $\mathcal{O}$ 10 \si{fs} & $\mathcal{O}$ 50k\texteuro  & In use at \\
     170Radiation & THz filtering & & &  several facilities \\
     171\hline
     172Coherent Smith-Purcell & Diff. gratings & $\mathcal{O}$ 50 \si{fs}  & $\mathcal{O}$ 50k\texteuro &  R\&D \\
     173 Radiation &  THz filtering &  & & \\
     174\hline
     175\hline
     176\end{tabular}
     177\caption{Comparaison of the properties of the different techniques of longitudinal bunch length and profile measurement}
     178\label{tab:long_meas_properties}
     179\end{center}
     180\end{table}
     181
     182\begin{table}[htbp]
     183\begin{center}
     184\begin{tabular}{l|c|c||}
     185Name  & Single shot  capability & Beam destruction \\
     186\hline
     187\hline
     188Current transformer  & Yes & No \\
     189\hline
     190Streak camera with radiator & Yes  & Radiator dependent \\
     191& & Synchrotron radiation: No; Screen: Yes \\
     192\hline
     193RF deflector & Partially & Yes \\
     194& The phase of the measurement must be checked & \\
     195\hline
     1963 phases &  No  &  Yes (beam off-energy)\\
     197\hline
     198Electro-optic sampling  & Yes & No \\
     199\hline
     200Coherent Transition Radiation & Interferometer: No   \small Multpl. Gratings: Yes & Yes \\
     201\hline
     202Coherent Smith-Purcell Radiation & Yes  & No \\
     203\hline
     204\hline
     205\end{tabular}
     206\caption{Comparaison of the capabilities of the different techniques of longitudinal bunch length and profile measurement}
     207\label{tab:long_meas_capabilities}
     208\end{center}
     209\end{table}
     210
     211\end{landscape}
     212\restoregeometry
     213\pagestyle{plain}
     214
     215
     216\section{Smith-Purcell Radiation}
     217\label{sec:CSPR}
     218
     219\subsection{First observation of Smith-Purcell Radiation}
     220
     221The 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}.
     222
     223\begin{figure}[htbp]
     224\center
     225\begin{tabular}{cc}
     226\includegraphics[width=6.5cm]{Advanced_diags/smith_purcell_first_image} &
     227\includegraphics[width=6.5cm]{Advanced_diags/grating_radiation.pdf}
     228\end{tabular}
     229\caption{Left: Image taken from~\cite{SP1953} showing the observation of radiation by Smith and Purcell in 1953. Right: Schematic of Smith-Purcell radiation. }
     230\label{fig:first_SPR}
     231\end{figure}
     232
     233The first observation of Smith-Purcell radiation from relativistic electrons was reported only much later in 1992~\cite{PhysRevLett.69.1761} and
     234the first observation of Coherent Smith-Purcell Radiation~\cite{PhysRevE.51.R5212} came  in 1995.
     235
     236XXX coherent / relativistic images
     237
     238XXX Van den berg P. M. van den Berg, J. Opt. Soc. Am. 63, 1588 ~1973!.
     239
     240\subsection{Interpretation of Smith-Purcell Radiation}
     241
     242The phenomena was further studied and several interpretations were proposed:
     243\begin{itemize}
     244\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).
     245\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).
     246\end{itemize}
     247
     248
     249\begin{figure}[htbp]
     250\center
     251\begin{tabular}{cc}
     252\includegraphics[width=6.5cm]{Advanced_diags/grating_dipole_radiation.pdf} &
     253\includegraphics[width=6.5cm]{Advanced_diags/grating_EM_diffraction.pdf}
     254\end{tabular}
     255\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).}
     256\label{fig:SPR_interpretation}
     257\end{figure}
     258
     259Although 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 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}.
     260
     261
     262\begin{figure}[htbp]
     263\center
     264\begin{tabular}{cc}
     265\includegraphics[width=7.5cm]{Advanced_diags/MOPMB004f2.png} &
     266\includegraphics[width=7.5cm]{Advanced_diags/MOPMB004f3.png}
     267\end{tabular}
     268\caption{Comparison of the single electron yield of different Smith-Purcell Radiation models (as published in~\cite{malovytsia:in2p3-01322176}):
     269the 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.}
     270\label{fig:SPR_comparison}
     271\end{figure}
     272
     273
     274
     275
     276\subsection{Application of Smith-Purcell Radiation}
     277
     278Smith-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}.
     279As 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}.
     280
     281The use of CSPR as a bunch longitudinal profile diagnostic was proposed soon after the first observation of CSPR and it was even 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}.
     282
     283
     284
     285\subsection{Theoretical aspects of Smith-Purcell Radiation}
     286
     287
     288The 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.
     289
     290\begin{figure}[htbp]
     291\center
     292\includegraphics[width=7.5cm]{Advanced_diags/grating_radiation_coordinates.eps}
     293\caption{Coordinates system used in this chapter. The direction $z$ comes out of the figure plane as is the azimuthal angle $\phi$.}
     294\label{fig:SP_coordinates}
     295\end{figure}
     296
     297A 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.
     298
     299\begin{figure}[htbp]
     300\center
     301\includegraphics[width=7.5cm]{Advanced_diags/GratingSurface.png}
     302\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}.}
     303\label{fig:grating_orders}
     304\end{figure}
     305
     306The coordinate system used in this chapter is shown on figure~\ref{fig:SP_coordinates}.
     307
     308
     309The 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
     310
     311\begin{eqnarray}
     312\lambda & = & \frac{l}{n} \left( \frac{1}{\beta} - \cos \theta \right)
     313\label{eq:SP_wavelength}
     314\end{eqnarray}
     315where $n$ 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.
     316This 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.
     317
     318The  intensity of radiation emitted by a single electron (single electron yield), per unit solid angle ($\Omega$) and per frequency ($\omega$) is given by
     319\begin{eqnarray}
     320\frac{d^2I_1}{d\omega d\Omega} & = & \frac{e^2 \omega^2 l^2}{4 \pi^2 c^3} R^2 \exp{\frac{-2 x_0 }{\lambda_e}}
     321\end{eqnarray}
     322where $e$ is the electron charge, $x_0$ 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.
     323
     324The evanescent wavelength is given by
     325
     326\begin{eqnarray}
     327\lambda_e = \lambda \frac{\beta \gamma}{2 \pi \sqrt{1 + \left(  \beta \gamma \sin \theta \sin \phi   \right)?2 }}
     328\end{eqnarray}
     329where $\gamma$ 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).
     330
     331
     332As 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:
     333\begin{eqnarray}
     334\frac{d^2I}{d\omega d\Omega} & = & \frac{d^2I_1}{d\omega d\Omega} \left[ N + N(N-1) \mathcal{F}(\omega) \right]
     335\label{eq:SP_coherent}
     336\end{eqnarray}
     337where $\mathcal{F}(\omega)$ is the form factor introduced in equation~\ref{eq:form_factor}.
     338
     339One 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.
     340
     341\begin{figure}[htbp]
     342\center
     343\includegraphics[width=7.5cm]{Advanced_diags/SP_signal_different_gratings.eps}
     344\caption{Radiation profiles observed for the same pulse but for different grating pitches.}
     345\label{fig:grating_effect}
     346\end{figure}
     347
     348
     349
     350
     351\subsubsection{Improving the beam-coupling calculation}
     352
     353
     354The factor $R^2$ 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.
     355
     356
     357The 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}:
     358\begin{eqnarray}
     359\lambda = \frac{2 l \sin \theta}{n}
     360\end{eqnarray}
     361
     362This has led me to study with a student~\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}.
     363
     364
     365It 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}.
     366
     367\subsection{Profile recovery}
     368
     369As 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.
     370
     371We 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.
     372
     373The 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.
     374
     375To recover the phase of a form factor, it must first be written in the following form:
     376\begin{eqnarray}
     377log(\mathcal{F}(\omega))=log(\rho(\omega))+i\Theta(\omega)
     378\end{eqnarray}
     379where $\rho(\omega))$ is the amplitude and $\Theta(\omega)$ the phase associated to this Form factor. $\rho(\omega))$ is thus the result of the measurement and  $\Theta(\omega)$  the information that needs to be recovered.
     380
     381The Hilbert transform gives then the relation
     382\begin{equation}
     383\Theta(\omega_0)  =  -\frac{1}{\pi} \textit{P}\int^{+ \infty}_{- \infty}\frac{ln(\rho(\omega))}{\omega_0-\omega}d\omega.
     384\end{equation}
     385
     386and the Kramers Kronig relation gives:
     387\begin{equation}
     388\Theta(\omega_0)  =  \frac{2\omega_0}{\pi} \textit{P}\int^{+ \infty}_{0}\frac{ln(\rho(\omega) )}{\omega_0^2-\omega^2}d\omega
     389\end{equation}
     390
     391In most cases the value given by these two methods will be very close.
     392
     393Before 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.
     394
     395To 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:
     396
     397\begin{equation}
     398\Delta_{FWXM} = \left| \frac{FWXM_{\mbox{orig}} - FWXM_{\mbox{reco}}}{FWXM_{\mbox{orig}} }\right|
     399%\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|
     400\end{equation}
     401
     402
     403Figure~\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}.
     404
     405
     406\begin{figure}[htbp]
     407 \centering
     408 \begin{tabular}{cc}
     409  \includegraphics*[width=65mm]{Advanced_diags/plots_11541.eps} &  \includegraphics*[width=65mm]{Advanced_diags/plots_11658.eps}\\
     410     \includegraphics*[width=65mm]{Advanced_diags/plots_12231.eps} &    \includegraphics*[width=65mm]{Advanced_diags/plots_12667.eps}
     411\end{tabular}
     412  \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.}
     413   \label{fig:good_bad_profiles}
     414\end{figure}
     415
     416
     417
     418\begin{figure}[htbp]
     419 \centering
     420  \includegraphics*[width=70mm]{Advanced_diags/plots_61.eps}
     421   \includegraphics*[width=70mm]{Advanced_diags/plots_62.eps}
     422  \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}. }
     423   \label{fig:sampling_frequency}
     424\end{figure}
     425
     426
     427
     428\begin{figure}[htbp]
     429 \centering
     430  \includegraphics*[width=70mm]{Advanced_diags/plot1510_31.eps}
     431    \includegraphics*[width=70mm]{Advanced_diags/plot1510_32.eps}
     432  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
     433  \caption{Comparison of different samplings methods using the $\chi^2$ criterion (left) and $\Delta_{FWHM}$ (right), as presented in~\cite{reco_paper}.}
     434   \label{fig:detectors_positionning_comparison}
     435\end{figure}
     436
     437
     438\begin{figure}[htbp]
     439 \centering
     440  \includegraphics*[width=70mm]{Advanced_diags/plots_41.eps}
     441    \includegraphics*[width=70mm]{Advanced_diags/plots_42.eps}
     442  %\includegraphics*[width=70mm]{newFig/lin27e203line203.eps}
     443  \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} .}
     444   \label{fig:detectors_positionning}
     445\end{figure}
     446
     447
     448\begin{figure}[htbp]
     449 \centering
     450  \includegraphics*[width=70mm]{Advanced_diags/plot2210_181.eps}
     451  \includegraphics*[width=70mm]{Advanced_diags/plot2210_182.eps}
     452  \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\%.}
     453   \label{fig:reco_noise}
     454\end{figure}
     455
     456
     457
    40458
    41459\section{Experimental study of Coherent Smith-Purcell radiation}
     460
     461One of the remaining difficulties preventing the adoption of Smith-Purcell radiation as a longitudinal profile is that
     462
     463
    42464\subsection{Smith-Purcell radiation measurement at FACET}
    43465\subsection{Smith-Purcell radiation measurement at SOLEIL}
    44466\subsection{Smith-Purcell radiation measurement at CLIO}
    45 \subsection{Outlook: Application to laser-driven plasma accelerators}
     467\label{sec:CLIO}.
     468
     469
     470
     471
     472Image CLIO 3 diff profiles
     473
     474
     475\subsection{Outlook: Application to laser-driven plasma accelerators and ERLs}
    46476beam grating sep.
    47477stability
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