\section{Replacement of the RLA arcs by a compact FFAG lattice}

We developed a complete linear optics for a 4-pass muon dog-bone RLA
consisting of a single linac and two droplet-shaped return arcs at
each end of the linac.  
In order to reduce the 
number of required return arcs, we employ the  
non-scaling fixed-field alternating-gradient (NS-FFAG) arc lattice 
design, which allows transport of two 
consecutive passes with very different energies through the same string 
of arc magnets. 
As illustrated in figure \ref{fig:acc:rla:ffag_ab1}, each droplet arc
consists of a 60$^\circ$ outward bend, a 300$^\circ$ inward bend and
another 60$^\circ$ outward bend so that the net bend is 180$^\circ$. 
Such an arc geometry has the advantage that the outward and inward
bends are made up of similar cells and the geometry automatically
closes without the need for any additional straight sections, thus
making it simpler and more compact.
In the scheme presently under consideration, a 0.9~GeV/c muon beam is
injected at the middle of a linac with 0.6 GeV energy gain per
pass. 
Therefore, with a four-pass scheme, there are 1.2 and 2.4 
GeV/c passes through one arc and 1.8 and 3.0 GeV/c passes through the 
other.  In addition to accommodating 
the appropriate momenta, each arc must satisfy the following
requirements:
(a) for each momentum transported, the offset of the periodic orbit
must be zero at the entrance and the exit to the arc to ensure that
the beam goes through the center of the linac; (b) the arc must be
achromatic for each  momentum to guarantee matching to the linac; (c)
the arc must be mirror symmetric, so that $\mu^+$ and $\mu^-$ can pass
through the same lattice in opposite directions; (d) the arc must be
nearly isochronous for both energies to ensure the proper phasing with
the linac RF; and (e) the size of the orbit offsets, beta functions,
and dispersion should be small enough to keep the aperture size
resonable.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.85\textwidth]{06-Appendix/02-04-LinacRLA/ffag_ab1.eps}
  \end{center}
\caption{Layout and building-block-cell of a droplet-shaped return arc
  based on ns-FFAG optics.}
\label{fig:acc:rla:ffag_ab1}
\end{figure}

We used a ns-FFAG triplet magnet arrangement as the basic cell of the
arc design.  
The inward-bending triplet 
cell has an outward-bending combined-function magnet with positive gradient 
(horizontally focusing) at the 
center and two inward-bending magnets located on either side with equal 
negative gradients. The outward-bending triplet-cell has the same
structure but reversed dipole fields.
The cell symmetry ensures that their 
periodic solutions have $\alpha_x$ = $\alpha_y$ = 0 and $D^\prime_x$ = 0 
at the beginning and the 
end. Since perturbative-method codes 
are not suitable for large-momentum-offset cases, we studied the NS-FFAG 
optics using the Polymorphic 
Tracking Code (PTC) module of the MAD-X program \cite{Schmidt:2006sa}, which
allows symplectic integration through all elements with user control
over the precision (with the full or expanded Hamiltonian).

Since the two arc designs are very similar, we focus on the details 
of the optics of the 1.2\,GeV/c and the 2.4\,GeV/c arcs. 
We choose the lower 1.2 GeV/c momentum as the nominal momentum going
through the magnet centers. 
The constraint that the lower-energy periodic orbit must have zero
offset coming in and out of the cell is then automatically
satisfied. Besides, once the 1.2 GeV/c linear optics is adjusted using
quadrupole gradients, introduction of the sextupole and octupole
magnetic field components required for accommodating the 2.4~GeV/c
momentum does not change it. This decouples the 1.2 GeV/c linear
optics from the 2.4~GeV/c optics 
and ensures that, once the 
$D_x$ = 0 and $D^\prime_x$ = 0 
conditions are met for 1.2~GeV/c, they 
are not affected by the tuning 
of the 2.4~GeV/c optics.

The triplet is composed of 1 m long magnets separated by 20~cm 
gaps with each magnet's bending angle being 5$^\circ$. 
The quadrupole gradients are adjusted to make the cell 
achromatic. 
Figure \ref{fig:acc:rla:ffag_ab2}a shows the 1.2~GeV/c 
periodic orbit and dispersion, as well as the beta functions for 
the inward- outward-bending cells. In this case, 
the Bogacz-Lebedev 
$\beta_{11}$ and $\beta_{22}$
are equal to the usual horizontal 
and vertical beta-functions, respectively. 
The outward-bending cell has the same beta functions but its
dispersion has opposite sign. 
With a single triplet, one is not able to satisfy, at the same time,
the zero orbit-offset and achromatic conditions for 2.4~GeV/c optics. 
Therefore, three triplet cells are combined into a ``super-cell''. 
Sextupole and octupole field components are 
introduced in the middle magnets of the triplets while maintaining 
the super-cell's overall mirror symmetry. 
By tuning the sextupole components, we simultaneously satisfy the
conditions of the super-cell being achromatic and having zero incoming
and outgoing periodic-orbit offsets. 
The octupole components are adjusted, keeping their strengths the same
until transverse stability is reached in both dimensions. 
Figure \ref{fig:acc:rla:ffag_ab2}b shows the 2.4~GeV/c periodic orbit,
dispersion, and the beta functions for the inward- and outward-bending
super-cells. 
In the outward-bending super-cell, the dipole and sextupole fields are 
reversed. 
The beta functions in the outward-bending super-cell remain the same
while the periodic orbit and dispersion change sign. Since both the
inward and outward-bending super-cells are achromatic and have zero 
incoming and outgoing periodic orbit offsets, it is 
clear that the super-cells are automatically matched at both 
energies. 
Since the net bend of each super-cell is 
15$^\circ$, they can be combined easily to form the 60$^\circ$ outward
and 300$^\circ$ inward bends of the droplet arc.
\begin{figure}
  \begin{center}
    \includegraphics[width=0.85\textwidth]{06-Appendix/02-04-LinacRLA/ffag_ab2.eps}
  \end{center}
  \caption{
    Periodic orbit, dispersion, and beta functions for the
    inward-outward bending super-cell transition at 1.2\,GeV/c and
    2.4\,GeV/c, the two momenta of interest.
  }
  \label{fig:acc:rla:ffag_ab2}
\end{figure}

Finally, the linac optics needs to be matched to both arcs for 
all passes simultaneously by adjusting the strengths of the linac
fixed-field quadrupoles. 
The new solution was obtained by modifying the bi-sected linac profile
where the quadrupole strengths increase linearly from the linac's
center toward the ends. 
Investigation of the complete non-linear dynamics 
aiming to optimize the RLA's dynamic aperture is 
under study, as well as phase control to maintain RF synchronization.
