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\begin{document}
\title{Focusing to nm beam sizes}
\author{R.~Tom\'as}
\maketitle
 
%1)      Issue: describing the required system and/or parameter and comparing with present state of the art when available.
%2)      R&D: describe R&D program envisaged or presently set-up to address the above issue
%3)      Achievements: Describe the results already achieved or aimed at first half of 2011
%4)      Outlook: describe further R&D envisaged in the near future (2012-13) and expected performances.

%\begin{abstract}
%.
%\end{abstract}


\section{Issue}
Future linear colliders require Final Focus Systems (FFS) of unprecedented demagnification
factors. This causes large chromatic aberrations that need to be compensated with
strong sextupoles, which in turn generate geometric aberrations that require special
arrangements of these sextupoles. 
The traditional design of such FFS consisted of two sections.
The first section hosted the sextupoles compensating the chromatic aberrations
generated by the second section containing the Final Doublet (FD). 
A more compact design was presented in~\cite{pan} featuring a local chromaticity
correction by placing the sextupoles at the source of the aberration, i.e., at the FD.
The CLIC Final Focus System (FFS) is based on this local chromaticity
correction scheme. However, residual aberrations of higher order were still limiting the
performance of the CLIC FFS with target vertical IP beam size in the nm level.
It was required to add extra non-linear elements, like octupoles and decapoles, 
to  better cancel residual aberrations~\cite{CLICnote}. Table~\ref{tabsy} 
reports the vertical IP beam size and the vertical chromaticity of various
projects, being CLIC  the most challenging.
%
\begin{table} 
\begin{center}
\small
\begin{tabular}{|l|l|r|r|}\hline
Project & Status   & $\sigma_y^*$ [nm]& $\xi_y$ [$10^4$]\\\hline
FFTB    & Measured & 70      &  1 \\\hline
{ATF2}    & Commissioning   & 37 & 1.9\\\hline
{ATF2 ultra-low $\beta$} & Proposed & 20 & 7.6\\\hline
{ILC}     & Design   & 6 & 1.5 \\\hline
ILC low power     & Alternative  & 4 & 3\\\hline
{CLIC } & Design   & 1  & 6.3\\\hline
\end{tabular}
\end{center}\vspace{-0.cm}
\caption{Vertical IP beam sizes and vertical chromaticities for different projects.
Chromaticity is computed from the elements of the transfer matrices as 
$(T_{346}R_{33}-T_{336}R_{34})/\sqrt{\beta^*_y}$.
}\label{tabsy}
\end{table}

In order to provide collisions with an acceptable background level it is mandatory to 
collimate the beam halo that would otherwise directly irradiate the detector or impact
machine components. This is achieved within the collimation section preceding the FFS.
Figure~\ref{bds} shows the optics functions through the CLIC Beam Delivery System (BDS) 
consisting of diagnostics, collimation and FFS sections.
%
\begin{figure}
\centering
\includegraphics[width=4.7cm, angle=-90]{BDS.eps}\vspace{-0.cm}
\caption{Optics functions along the CLIC BDS.}
\label{bds}
\end{figure}


\section{R\&D}
Achieving nm beam sizes with the current FFS designs is mostly challenged by the static 
and the dynamic imperfections of a realistic beam line, mainly misalignments and field errors.
Figure~\ref{initsigma} shows the probability distribution of the CLIC vertical IP beam 
size assuming realistic beam line imperfections. The most likely value is about 1~$\mu$m,
three orders of magnitude larger than the 1~nm from the ideal design.
%
\begin{figure}
\centering
\includegraphics[width=4.7cm, angle=-90]{/afs/cern.ch/eng/sl/lintrack/CLIC-DFS/bds_dfs_ff/RES_HV25fail/hist.sigma.init.HV.eps}\vspace{-0.cm}
\caption{Probability distribution of the CLIC vertical IP beam size assuming realistic beam line imperfections.}
\label{initsigma}
\end{figure}
% 
The static beam line imperfections must be compensated by using beam-based alignment and tuning techniques.
The dynamic errors must be mitigated via the use of feedbacks.
The validation of the entire process to reach the nm beam size
is done both via computer simulations and experiments.



The experimental verification
of this type of FFS is presently being investigated in the KEK ATF2 facility~\cite{ATF2}.
ATF2 contains a scaled-down version of the ILC FFS with a vertical IP beam size of about 37~nm. 
However the CLIC FFS has about 4 times more chromaticity than ILC or ATF2. An ATF2 R\&D proposal has been 
made~\cite{lowbeta,lowbeta09} to reduce the ATF2 IP vertical beta function by a factor of 4. 
This proposal has a two fold motivation, reduce the IP vertical size as close as possible to ILC and
CLIC values, see Table~\ref{tabsy}, and prove the CLIC chromaticity levels.


The ultra-low $\beta^*$ proposal for ATF2 will also serve to investigate the difficulty 
of tuning the FFS for different IP beam sizes. This might allow extrapolations
to the smaller beam sizes of ILC and CLIC. Simulations show that tuning difficulty increases 
for smaller IP beam sizes~\cite{lowbeta09}. ATF2 is presently under commissioning and 
it is expected that first experimental data
from ATF2 at relevant IP beam sizes will come before mid 2011.







\section{Achievements}

Montecarlo simulations of the FFS tuning are performed to estimate the probability
of success in achieving nominal parameters. The variables used during the tuning
are the transverse displacements and the strengths of the FFS quadrupoles and sextupoles. The figure
of merit to optimize is the luminosity. After a maximum of 18000 iterations 80\%
of the seeds reach 80\% of the design luminosity, see~Fig.~\ref{tuni}. This
result is encouraging but it is not fully satisfactory. It is desired that 90\%
of the seeds reach 90\% of the luminosity. 
%
\begin{figure}
\centering
\includegraphics[width=4.7cm, angle=-90]{hist.sigma.end.HV.eps}\vspace{-0.cm}
\caption{Probability distribution of the CLIC luminosity after tuning for a maximum
of 18000 iterations.}
\label{tuni}
\end{figure}
%
To this aim a large effort has been started during 2010 in which collaborators 
from different laboratories are sharing and comparing different alignment, tuning
and feedback algorithms~\cite{inte}. A first very important result was the combination
of the IP feedback and the beam-based alignment techniques to estimate the machine performance
at the steady state. A 59\% of the design luminosity in the energy peak is achieved
without having assumed any special stabilization of the final doublet.

\begin{figure}
\centering
\includegraphics[width=5.2cm, angle=-90]{FeedBack.lumi.eps}\vspace{-0.cm}
\caption{Performance of the IP feedback combined with beam-based alignment techniques.}
\label{IP}
\end{figure}




Due to the nanometric IP beam size CLIC faces the challenge of the sub-nanometer
stabilization of the last FFS quadrupole (QD0). In order to loose less than 2\%
luminosity the vertical jitter of QD0 has to be below 0.15~nm (for frequencies above 4~Hz) 
with the extra
complication that QD0 is embedded in the detector at 3.5~m from the IP.
There are very promising experimental results showing stabilization to
these levels via active ground isolation and structure resonance rejection techniques
in a laboratory environment, see Fig.~\ref{qd0sta} taken from~\cite{andrea2}.
The CLIC stabilization working group conducts the research in order
to find solutions in the detector environment.


\begin{figure}
\centering
\includegraphics[width=9.cm, angle=0]{int_rms_poutre_asserv_ss_quad_ss_title.eps}\vspace{-0.cm}
\caption{Demonstration of stabilization to the sub-nanometer level via
ground isolation and structure resonance rejection in a quiet 
environment~\cite{andrea2}.}
\label{qd0sta}
\end{figure}


A way to considerably ease the stabilization requirement and
difficulty would be to move QD0 out of the detector, thus allowing to support
it on the ground~\cite{Andrei}. 
Initially this required increasing L$^*$ to 8~m
and a consequent reduction of the design luminosity
by 28\%~\cite{rtomas}. However, recent developments of the experiment
allowed for a reduction of the end cups by 2~m. A new lattice has recently
been developed with an L$^*$=6m that would feature only a 16\% peak luminosity
loss with respect to the current design (L$^*$=3.5~m). Table~\ref{tablecilla}
displays all the existing CLIC FFS lattices with their luminosity performance.
It is clear from the table that longer L$^*$ implies lower luminosity.
The new lattice with L$^*$=6m has proved to tune in a similar success ratio
as the current design (80\% of the seeds reaching 80\% of the luminosity). 
Therefore this lattice becomes a very attractive
alternative in the case that the QD0 stabilization becomes impossible.    





\begin{table}[h]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
L$^*$ & {Total luminosity}  & {Peak luminosity} \\
$[m]$   &       $[10^{34} cm^{-2}s^-1{}]$       &       $[10^{34} cm^{-2}s^-1{}]$       \\\hline
%Requirements    &       5.9     &       2.0+20\%        \\\hline
3.5             &       6.9     &       2.5             \\\hline
4.3             &       6.4     &       2.4             \\\hline
6               &       5.0     &       2.1             \\\hline
8               &       4.0     &       1.7             \\\hline
\end{tabular}
\end{center}
\caption{Total and Peak luminosities for different lattices with increasing L$^*$.}\label{tablecilla}
\end{table}



%\section{Outlook}



\begin{thebibliography}{20} 
%
\bibitem{pan} P.~Raimondi et al, Phys. Rev. Lett. {\bf 86}, 3779-3782.\vspace{-0.1cm}
%
\bibitem{CLICnote} R.~Tom\'as, CERN-OPEN-2006-023 and Phys. Rev. ST Accel. Beams {\bf 9}, 081001 (2006).\vspace{-0.1cm}
%
\bibitem{CLICover}  R.~Tom\'as for the CLIC study team, ``Overview of the Compact Linear Collider'', Phys. Rev. ST Accel. Beams {\bf13}, 014801 (2010).
%
\bibitem{ATF2} ``ATF2 Proposal'', KEK Report 2005-2, SLAC-R-796, CERN-AB-2006-004, DESY 06-001, JAI-2006-001 and ILC-Asia-2005-26.

%
\bibitem{lowbeta} D.~Angal-Kalinin et al, ``Exploring ultra-low $\beta^*$ values in ATF2 -R\&D Programme proposal'',
CARE/ELAN Document-2008-002.\\\vspace{-0.1cm}
\href{http://esgard.lal.in2p3.fr/Project/Activities/Current/Networking/N2/ELAN/Documents/2008/ELAN-Document-2008-002.pdf}
     {\tiny \color{blue} \underline{http://esgard.lal.in2p3.fr/Project/Activities/Current/Networking/N2/ELAN/Documents/2008}}\vspace{-0.1cm}
%
\bibitem{lowbeta09}  D.~Angal-Kalinin et al, ``ATF2 ultra-low IP betas proposal'', these proceedings.\vspace{-0.1cm}
%
\bibitem{andrea2} B. Bolzon et al, "Study of vibrations and stabilization at the subnanometre scale for CLIC Final Doublets", Nanobeam 2008, LAPP-TECH-2008-05.\vspace{-0.1cm}
%
\bibitem{Andrei} A.~Seryi, "Near IR FF design including FD and longer L* issues", presentation in the CLIC08 workshop, October 2008, CERN.
%
\bibitem{rtomas} R.~Tom\'as et al, ``Status of the CLIC BDS'', Proceedings of Particle Accelerator Conference 09, Vancouver, Canada.
%
\bibitem{inte} B. Dalena et al, ``CLIC BDS tuning, alignment and feedbacks integrated simulations'' to be presented in IPAC 2010.
\end{thebibliography}
\end{document}
