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\begin{document}
\title{Beam Dynamics of the CLIC Decelerator\thanks{Work supported by the Research Council of Norway and the Commission of the European Communities under the 6th Framework Programme �Structuring the European Research Area�, contract number RIDS-011899.}}

\author{E. Adli\thanks{Erik.Adli@cern.ch, University of Oslo and CERN}, D. Schulte and I. Syratchev,  CERN, Geneva, Switzerland}

\maketitle

\begin{abstract}
\small{
The CLIC Drive Beam decelerator will extract X-band RF-power from a 100 A Drive Beam.  The focussing and alignment systems must ensure transport of particles of all energies through the decelerator sectors, ensuring minimal losses.  This paper summarizes the current status of the beam dynamics studies, including the effect of wake fields, the focusing strategy, the alignment strategy and resulting tolerances.
}
\end{abstract}

\section{Introduction}

The CLIC Drive Beam Decelerator is responsible for producing the RF-power for the main linacs accelerating structures, operating at an X-band frequency of 11.9942 GHz.    

The RF-power will be generated by a 100 A electron Drive Bream being decelerated by a number of high-impedance Power Extraction and Transfer Structures (PETS).  In order to ensure a stable and uniform power production it is very important that the Drive Beam is transported with a minimum of losses.

This paper is a review of the the current status of beam dynamic studies of the CLIC decelerator, using CLIC 2008 baseline design parameters \cite{tecker_clic_note_2008} (some listed in Table \ref{tab:xband08_parameters}).  

\begin{table}[hbt]
\begin{center}
\caption{\small{Decelerator lattice and beam parameters.}}
\begin{tabular}{| l |c |r | c|}
\hline
\textbf{Decelerator parameters} & \textbf{Symbol} & \textbf{Value}  & \textbf{Unit}  \\ 
\hline
Maximum sector length  & $L$ & 1053 & m \\
FODO cell length  & $L_{FODO}$ & 2.01 & m \\
Initial energy & $E$ & 2.4 & GeV \\
Initial average current & $I_0$ & 101 & A \\
Final max. energy spread & $S$ & 90.0 & \% \\
Frequency  & $f_L$  & 12.00 & GHz \\
\hline
\end{tabular}
\label{tab:xband08_parameters}
\end{center}
\end{table}

\vspace {-10.0mm}

\section{Lattice layout}

Each decelerator sector consists of a FODO lattice, with up to two PETS inserted between each  quadrupole, as shown in Figure  \ref{fig:xband08_lattice}.   We require the maximum extracted energy to be the same for each sector. Because the fill pattern of accelerator structure varies, the length of the decelerator sector also varies.  As baseline we simulate the longest sector (about 1050 m).

\begin{figure}[htb]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig1.eps}
\caption{\small{Decelerator lattice}}
\label{fig:xband08_lattice}
\end{figure}

\section{Simulation methods and metrics}

The beam dynamics simulations have been performed with the tracking code PLACET \cite{PLACET}, and more detail about the simulation set-up can be found in \cite{adli_failure_modes_epac2008}.

As simulation criterion for minimum-loss transport we require the entire 3-sigma beam envelope, denoted $r$, defined as ${r\equiv\max\sqrt{(|x_{i}|+3\sigma_{x,i})^{2}+(|y_{i}|+3\sigma_{y,i})^{2}}}$, to be within half the available aperture, $a_0=11.5$mm, in order to have a margin for unmodelled effects like higher order wake fields. 


\section{Focusing strategy}

The decelerator focusing strategy is to keep the phase-advance per cell constant at $\mu_{FODO}\approx90^{\circ}$  for the \emph{lowest} energy particles.  For a perfect beam and machine, higher energy particles will then be contained in the envelope of the lowest energy particles.

In the baseline design the required quadrupole gradient starts at 81 T/m (field of 0.9 T close to the poles) at the beginning of the lattice, changing in steps of down to 0.05 T/m (corresponding to deceleration due to one PETS) along the decelerator lattice.  For the current baseline neither tune-up nor orbit correction strategies require that the lattice magnet strength be re-scaled.  Therefore, permanent magnets might be an option for the lattice.  However, the current and energy acceptance is only a few percent \cite{adli_failure_modes_epac2008}, meaning that a permanent magnet solution will not be flexible wrt. small changes in drive-beam "working-point".

For cost reasons it is of interest to see if we can accept a coarser quadrupole strength granularity.  Preliminary studies with a stair-case decrease in field strength show, however, that even a $\sim$1\% quadrupole strength granularity leads to an $\sim$10\% increase in the beam envelope.


\section{PETS transverse wakes}

The PETS dipole wake can at worst significantly distort the drive beam and cause heavy losses.  Part of the PETS design process was therefore to bring the effect of the dipole wake down to an acceptable level.  This work was performed as an iterative process, as outlined in Figure \ref{fig:xband08_pets_optimization}, where PETS design was improved until the effect of the wakes on the envelope growth was deemed acceptable.


\begin{figure}[tb]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig2.eps}
\caption{\small{PETS dipole wake optimization}}
\label{fig:xband08_pets_optimization}
\end{figure}


A variety of tracking simulations, taking into account misalignments, energy spread, adiabatic undamping, as well as beam jitter and offset, were performed in order to investigate the envelope growth and eventual instabilities driven by the transverse modes.  The outcome of this study - a PETS design were the dipole wake amplification is deemed sufficienty damped  - was a set of  nine dipole modes, used as baseline for further the baseline beam dynamics studies\cite{syratchev_adli_rf_power}.  Figure \ref{fig:xband08_dipole_amplification} shows the amplification of each dipole mode when driven by beam jitter at the mode frequency, for different Q-values.  We see that for nominal Q-values, as predicted by the current PETS design, the amplification of a single mode is at worst a factor two.

\begin{figure}[htb]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig3.eps}
\caption{\small{Amplification of PETS dipole modes}}
\label{fig:xband08_dipole_amplification}
\end{figure}

The dipole modes are used to simulate both multi-bunch effects and single-bunch effects.  A comparison with the K. Bane formula \cite{bane_short_range} for the short-range wake shows that extracted wake functions indeed corresponds very well with up to 4 $\sigma_z$, as shown in Figure \ref{fig:xband08_k_bane} (even though the PETS is outside the usual area of usage for this formula).   In Figure \ref{fig:xband08_dipole_amplification}  the $Q=Q_0$ graph corresponds to single-bunch effects only, and we see that the effect single-bunch effects are quite negligible.

\begin{figure}[htb]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig4.eps}
\caption{\small{Single-bunch wake compared to K. Bane formula}}
\label{fig:xband08_k_bane}
\end{figure}

Finally, we note that even if the current PETS design shows acceptable beam dynamics performance for nominal Q-values, we note that a scaling of the Q-factors by two leads to unacceptable increase in beam envelope (due to high-Q modes that don't die out, but grow along the beam).

\section{Beam Based Orbit Correction}

\subsection{The effect of quadrupole kicks}

In order to underline the need for beam based orbit correction we first discuss in some details the drivers of the beam envelope.  We have observed the previous section that the beam envelope amplification due to transverse wake fields is suppressed due to the PETS design.  The driver of the beam envelope is now quadrupole kicks due to quadrupole misalignments.  We estimate that static alignment of the machine at best can give a resulting rms offset of $\sigma_{quad}\approx$ 20 $\mu$m.  We can calculate the statistical average of a particle having a FODO phase-advance of $\mu_{FODO}$ by summing kicks, yielding an average centroid offset of : 

$<r>\approx2\frac{\sigma_{quad}}{\cos(\mu_{FODO}/2)}\sqrt{2N_{FODO}}\sqrt{\frac{(1-\frac{1}{2}S)}{1-S}}$

yielding 4 mm for the lowest energy particle.  This corresponds to incoherent build-up of the kicks.   However, for a given machine some particles will sample the quadrupoles with some coherence, and thus build up kicks faster.  This is illustrated in Figure \ref{fig:xband08_quadrupole_kicks} (transverse wakes are not included in these simulations).

\begin{figure}[htp]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig5.eps}
\caption{\small{The effect of quadrupole kicks}}
\label{fig:xband08_quadrupole_kicks}
\end{figure}


In the case above it is the lowest energy particles samples the quadrupole with some coherence and drives the envelope, but this will change from one random machine to another.  In fact, because the transient head of the drive beam pulse includes particles of a larger energy range, it is probable that the transient will drive the beam envelope. 

\subsection{Steering strategies}

In \cite{adli_beam_based_alignment_epac08} it is shown that after simple 1-to-1 steering the bean envelope growth is still significant, but that Dispersion-Free Steering can efficiently suppress the beam envelope growth due to quadrupole kicks, assuming the BPM resolution is about 2 $\mu$m.  Details about these steering schemes can also be found.  Figure \ref{fig:xband08_envelopes} show the envelopes without correction, after 1-to-1 correction and after Dispersion-Free Steering.  \cite{adli_decelerator_ctf3_steering} describes tests of steering algorithms for the decelerator in the CLIC Test Facility 3.

\begin{figure}[htb]
\centering
\includegraphics*[width=75mm]{ADLI_DECELERATOR_fig6.eps}
\caption{\small{Beam envelopes after steering}}
\label{fig:xband08_envelopes}
\end{figure}


\vspace {-5.0mm}



\section{Decelerator CTF3 test-benches}

\subsection{Test Beam Line (TBL)}

The main test-bench for the CLIC Decelerator will be the Test Beam Line (TBL), due for completion towards the end of 2010.  In the TBL it is foreseen to decelerate a combined CTF3 beam, of up to 30A and 150 MeV, extracting ca. 60\% of the total energy, using 16 PETS of 0.8 m.

Simulations predict that the beam envelope amplification due to transverse wakes is slightly larger in the TBL than in the decelerator.  Large over-estimations in Q-factors in the PETS design should therefore be observable in the TBL, e.g. in the form of losses due to high-Q modes leading to instabilities.

Because of it small numbers of quadrupoles in the TBL (16) versus the decelerator ($>$ 1000) we do not expect advanced beam-based alignment to be necessary.   However, simulations show that Dispersion-Free Steering will still be at least \emph{as effective} as 1-to-1, and the TBL could therefore be a potential test-bed for the Dispersion-Free Steering techniques suggested for the decelerator.

Finally, the TBL final energy will be a factor four lower than for CLIC, and the beam envelope for a perfect machine and beam (resulting from adiabatic undamping alone) is predicted to be almost 8 mm -  more than 2/3 of the PETS aperture - potentially leading to strong higher-order mode excitation (we remind that for the CLIC Decelerator we aim to keep the beam within 1/2 of the PETS aperture along the whole lattice).  A 3-sigma transport through the TBL therefore looks like a difficult task, but if we come close to achieving it, it will be a great step towards proving feasibility of the CLIC decelerator

\subsection{Two-Beam Test Stand}

Starting operation this year the Two-Beam Test Stand (TBTS) will decelerate the beam through a single PETS of 1 m, using recirculation of the extracted field to increase power production and deceleration for a given beam current.  In addition to proving RF-power extraction and transfer, the TBTS will provide the first tests for the decelerator scheme, allowing e.g to correlate simulations measurements of beam current, power production and measured deceleration in a dispersive region.  

\section{Conclusions}

In order to ensure minimum-loss beam transport in a robust way through the CLIC Decelerator, the following items have been studied and optimized :

\begin{Itemize}
\item Tightly focusing FODO lattice with smooth decrease of quadrupole gradient
\item Optimization of the PETS design in order to minimize dangerous modes of the transverse wakes
\item Advanced beam-based orbit correction to transport particles of all energies
\end{Itemize}

Simulations show that the a beam-envelope very close to minimum can be achieved with the current design. Test-facilities are underway to prove the feasibility of the above items.


\section{Acknowledgements}

Helpful discussions with S. Doebert, R. Zennaro and V. Ziemann are gratefully acknowledged.

\begin{thebibliography}{9}   % Use for  1-9  references
%\begin{thebibliography}{99} % Use for 10-99 references
\small{

\bibitem{tecker_clic_note_2008}
CLIC parameters 2008, F. Tecker ed., \emph{CERN CLIC-Note 764} (2008)

\vspace {-1.0mm}

\bibitem{PLACET}
https://savannah.cern.ch/projects/placet

\vspace {-1.0mm}

\bibitem{adli_failure_modes_epac2008}
E. Adli et al., A Study of Failure Modes in the CLIC decelerator, \emph{Proceeding of EPAC 2008} (2008)

\vspace {-1.0mm}

\bibitem{syratchev_adli_rf_power}
I.Syratchev et al., High RF Power Production for CLIC,  \emph{CERN CLIC-Note 720} and \emph{PAC 2007} (2007)

\vspace {-1.0mm}

\bibitem{adli_beam_based_alignment_epac08}
E. Adli and D. Schulte, Beam-Based Alignment for the CLIC Decelerator, \emph{Proceeding of EPAC 2008} (2008)

\vspace {-1.0mm}


\bibitem{bane_short_range}
K. Bane, Short-range dipole wake �elds in accelerating structures for the NLC, SLAC-PUB-9663 (2003)


\vspace {-1.0mm}


\bibitem{adli_decelerator_ctf3_steering}
E. Adli et al., Status of an Automatic Beam Steering for the CLIC Test Facility 3, \emph{Proceeding of Linac 2008} (2008)

\vspace {-1.0mm}


}
\end{thebibliography}

\end{document}



