\subsection{RFOFO ring cooler}

A ring cooler has been designed using an  RFOFO lattice of alternating solenoids. The design  employs a single cell for both transverse cooling and emittance exchange. It uses solenoids for focusing, giving large angular and momentum acceptances. The cell includes dispersion, acceleration, and energy loss in a single thick hydrogen wedge. Figure \ref{ring} shows the layout of the cooling ring drawn to scale.  The RFOFO lattice was chosen because, unlike in the SFOFO case used in Study 2, all cells are strictly identical, and the presence of an integer betatron resonance within the momentum acceptance is eliminated.


\begin{figure}[htb!]
%\vskip.2in
%\hskip-.4in
%\psset{unit=.8in}
%\input{ring3.fig}
\includegraphics*{rfofo_ring3.ps}
%\vskip-1.3in
\caption{Layout of RFOFO cooling ring}
\label{ring}
\end{figure}

The basic ring is made up of 12 identical 2.75-m long cells. In the figure, this symmetry is broken for injection and extraction, but the magnetic fields in this insertion are nearly identical to those in the rest of the ring.
Figure \ref{cells} shows a detailed view of three cells of the lattice.

\begin{figure}[tbh]
%\vskip.1in
%\psset{unit=.3in}
%\hskip.3in
%\input{h2reg1.fig}
\includegraphics*{rfofo_h2reg1.ps}
\vskip-.7in
\caption{Three cells of the RFOFO lattice}
\label{cells}
\end{figure}

The longitudinal field on-axis has an approximately sinusoidal dependence on position. The actual coils to generate the axial fields, in the presence of the bending fields, would have to be slightly different from those used in the simulation, but since the 3D fields used are consistent with Maxwell's equations, there is no question but that suitable coil positions can be found. 
The lattice transmits particles in the momentum band from 150 to 250 MeV/c. The average momentum for a small emittance beam varied from 191 to 201 MeV/c across each cell of the lattice.
The minimum value of the beta function at the central momentum is 40 cm.


Dispersion is provided by applying an approximately 0.125 T transverse bending field generated by alternately tilting the vertical plane of the solenoids by 1.5 degrees.
There is no attempt to control the field index $n$ (where $B\propto r^n$).
So the focusing in x and y are not identical.

It is found that the acceptance is reduced as the bending field is increased. We thus use a wedge with maximum possible angle (giving zero thickness on one side), and the least bending field consistent with adequate emittance exchange. 
The dispersion at the absorber of -8 cm in a direction 30 degrees from the y axis, 
The dispersion at the center of the rf is of the opposite sign, and also mostly in the y direction. Its direction is Larmor rotated by the axial fields.

The liquid hydrogen wedge has a central thickness of 28.6 cm and a total wedge angle of 76.93 degrees and is rotated 30 degrees from the vertical. No absorber windows are included in this simulation. 
The RF cavities had a frequency of 201.25 MHz and a gradient of 16 MV/m. No RF windows were included. 

The ICOOL simulations shown do not include the injection/extraction insertion, and use axial and transverse magnetic fields generated by a truncated Fourier decomposition of the fields from a straight solenoid lattice. The RF is represented as fields in perfect pillbox cavities. The input tracks are taken from a Study 2 simulation, using distributions from just before transverse cooling. The use of Study 2 simulated distributions is intended to allow a more realistic estimate of the ring's performance. 
No attempt was made to match the ring dispersion or slight differences in the transverse beta functions.

Figure \ref{all} shows the transmission, transverse emittance (in x, y), longitudinal emittance, 6 dimensional emittance, and a merit factor $M$ vs. length in the ring. $M$ is given by:

$$M~=~{\epsilon_6(initial) \over \epsilon_6(final)}~\times~{\rm Transmission}$$

\begin{figure}[tbh]
%\vskip-.4in
%\psset{unit=.45in}
%\hskip.5in
%%\input{fs33uk.fig}
{\large
%\input{all.fig}
\includegraphics*{rfofo_all.ps}
}
%\vskip.5in
\caption{Transmission, normalized transverse emittance, normalized longitudinal emittance, normalized 6-dimensional emittance, and the merit factor, as a function of distance.}
\label{all}
\end{figure}

Initially, the x emittance falls more rapidly than the y. This is expected because it is the y emittance that is exchanged with the longitudinal emittance, but the Larmor rotations soon mix the x and y emittances bringing them to a common value.

 After a distance of 400 m (\app 12 turns), the 6 dimensional emittance has fallen by a factor of 290, with a transmission of 44 \% (61\% without decay). The merit factor is 130. The same factor for the Study 2 cooling lattice, with no windows, is 13. With realistic windows and the injection/extraction insertion added, the merit factor will be much less than 130, but is likely to remain far better than the study 2 example.

This ring could not be used, as is, to replace the Study 2 cooling channel because the bunch train in this case is far too long to fit in the ring. But spiral 3D cooling channel could be used
and an even greater performance gain could be expected if the spiral were also tapered.

This approach seems very attractive, but it is still far from fully realistic, and much work needs to be done.