So far, I've discussed what sort of a field will provide horizontal beam focusing. There are two problems left. For one thing, in my original discussion of weak focusing (link), I showed that the field index n had to be less than one for horizontal weak focusing. Later on, the constraints had mysteriously tightened, and I stated that for weak focusing to occur, the field index had to satisfy 0\le n<1. Secondly, we've seen that horizontal focusing can occur without vertical focusing in a uniform magnetic field, so we need a new constraint on the field index to ensure vertical focusing. Well, in this case, two wrongs almost make a right, and the second of the above-mentioned problems explains the first. Here's how it works.
In order to stabilize the vertical structure of the beam, the magnetic field needs to provide a restoring force in the vertical direction, something along the lines of F_z=-cz. In order to produce that, the magnetic field needs a horizontal component: B_x=-c'z. Well, from this we know that \frac{\partial B_x}{\partial z}=-c'. One of Maxwell's famous equations tells us \vec\nabla\times \vec{B}=0, so we clearly see that
\frac{\partial B_x}{\partial z} = \frac{\partial B_z}{\partial x}=\frac{\partial B_z}{\partial r}=-c'
Observe! We have shown that for vertical focusing, we need \frac{\partial B_z}{\partial r} to be negative. From our definition of the field index (n=\frac{-\rho}{B_0}\frac{\partial B_z}{\partial x}), then, we see that for vertical focusing to occur, the field index must be positive. Voila!
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