Vertical focusing

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!