Geometric focusing

A uniform magnetic field (field index 0, since the rate of change of the field is zero) provides a certain level of horizontal focus, in a phenomenon called geometric focusing.  From elementary E&M, we know that a particle in a uniform magnetic field with momentum perpendicular to the field lines will follow a perfectly circular path.  Let's examine the behavior of a nonconforming particle in the beam; call it Fred. If at some point it (he?) is in the ideal location, moving in the ideal direction, but has a lower momentum than a particle tracing out the ideal orbit (henceforth referred to as Ida), then the magnetic field will cause him to run around the ring in a smaller circle than Ida's trajectory. But after going all the way around, Fred ends up right back where he started, and while the beam may have defocused somewhat azimuthally (that is, the bunch is longer now, so it takes up a greater portion of the ring), it's once again focused horizontally. This is shown in the leftmost part of the figure below. Similarly, if Fred's momentum vector is pointed in a different direction than Ida's, he'll have a different trajectory, but that'll intersect Ida's twice, so we have geometric focusing.  Finally, just turning the previous example on its side, if Fred is slightly displaced relative to Ida, their trajectories once again meet twice in their trips around the ring. That's the premise of geometric focusing in accelerator physics; it's just a special case of horizontal weak focusing.

Some examples of geometric focusing. Black shows the ideal trajectory (Ida),
and red shows the trajectory of another particle (Fred) that differs slightly from
Ida in initial conditions (at the left of the image). In all cases, the trajectories
of Fred and Ida meet up at least once in each full revolution.

In (a), Fred starts in the same position as an ideal particle but with
lower magnitude momentum. In (b), he starts in the same position and with the same
magnitude momentum as an ideal particle, but pointed slightly outwards compared
to Ida's trajectory. And in (c), Fred has the same momentum as the ideal particle but
is slightly offset spatially.

There are a couple of problems with relying on geometric focusing, though.  For one thing, a very small deviation in the beginning can send Fred on a trajectory that is fairly far away from Ida's at certain points. In order for Fred to stay in the beam, he needs to not run into the walls, which creates a real headache for the accelerator designers. For another, a uniform magnetic field doesn't provide any vertical focusing effect; if Fred has even a tiny vertical component of his momentum, the magnetic field won't affect it, and Fred will end up moving higher and higher in the accelerator pipe until he runs into the material there and meets with an untimely end.