\section{Concept}
\label{DandPR:concept}
%\centerline{Juan C. Gallardo, Harold  Kirk}

%\centerline{\it Brookhaven National Laboratory}
%\subsection{Introduction}

The function of the phase rotation section of the Neutrino
Factory is to reduce the energy spread of the collected
muon beam to a manageable level that will allow reasonable
throughput to subsequent system components.  We see in Fig.~\ref{CPR:fig40}
the consequences of allowing the beam to drift without
the application of properly phased acceleration.  In Fig.~\ref{CPR:fig41}
we see the longitudinal structure of the beam after the
application of our proposed three-stage phase rotation system.

For a Neutrino Factory the requirements on the longitudinal
phase space are quite different than those for a muon collider.  
Unlike the muon beam for a muon collider we can permit the captured muon 
beam to grow in its longitudinal
dimensions and then put it through a buncher system that will permit the 
cooling section downstream to be operated
with rf cavities having relatively high frequency.
\begin{figure}[!bht]
\begin{center}
\centerline{\includegraphics*[width=3in,angle=90]{../template/report/ps-and-eps/PH-R-drift.eps}}
\caption{Beam longitudinal profile without phase rotation.}
\label{CPR:fig40}
\end{center}
\end{figure}

\begin{figure}[!tbh]
\begin{center}
\centerline{\includegraphics*[width=3in,angle=90]{../template/report/ps-and-eps/PH-R-ph_rot.eps}}
\caption{Beam longitudinal profile with non-distorting phase rotation.}
\label{CPR:fig41}
\end{center}
\end{figure}

If the process is done with a single drift and single induction linac,  
relativistic effects cause a distortion of the rotated bunch such that the initially 
high energy particles end up with a larger energy spread than the initially low energy 
ones. The use of at least two induction linacs, with a drift between them, allows this 
distortion to be greatly reduced.%\cite{nondistort}. 

It is natural for both these induction linacs to be bipolar, with
initial deceleration and later acceleration. For technical reasons,
such bipolar voltage pulses have been avoided in the baseline
design. In the case of the first linac, a hydrogen absorber is placed 
immediately after it to reduce the beam energy, allowing the first
linac to be unipolar. This absorber also reduces the emittance, and is
thus referred to as a ``minicooler.'' The absorber is in two parts,
with a field reversal between them to avoid the generation of
angular momentum.

To avoid a bipolar second linac, it has been replaced with two linacs:
the first unipolar decelerating, the second unipolar acceleration (A
slightly less conventional bipolar second linac solution is discussed
in Section~\ref{alternative-phase-rotation}).