Multi-Turn Current Coil Magnetic Lens

My last couple of posts on magnetic lenses here & here left me mostly satisfied, but still curious about the electron beam trajectory.  From the research I did on the topic I thought that the beam should spiral around the axis, but equally I can make a case for it not doing so.  Electrons tend to move in spirals around magnetic field lines, and that is evident in the the high field simulation as the spiral tightens and moves in towards the axis as the field lines bunch up and pass through the coil.

I thought that I may see some of sign of the beam spiralling around the axis if I had a larger region where the field was more uniform.  To test this hypothesis, I did another simulation where I replaced the single coil with twenty, all carrying 5 percent of the single coil simulation current.  The results however were essentially the same.  The graphs below are from the simulation output.  The Octave code for the simulation is located here.

Cross section of the field through the coil axis

Cross section of the field near the centre of the coil

In the two graphs above it can be seen that the field is more uniform throughout the length of the coil.  The second diagram also shows that there is almost no radial component to the field in the centre of the coil.

Electrons spiralling along through the coil

Electron trajectory through the coil

I have selected a high current for the simulations to exaggerate any effects that may be present and once again I got the same result, the electrons spiral along "kissing" the axis and moving away from it again.  In reality, most of these simulations are unrealistic and can't be tested, and without some sort of physical experimentation I can't confirm my results, but the model seems sound.  The results from basic simulations with a constant field agreed with theory, and there aren't any obvious instabilities.

If anyone has any experience with a similar model, either a simulation or a physical device, I would like to hear from you.  What type of trajectory should I be expecting to see?  Are these simulations close to reality?

I found these simulations quite satisfying really, the way a simple equation like the Lorentz force equation, combined with some initial parameters, produces such a complex result.  At least I have a better idea of how electrons move in magnetic fields, that and I got to sharpen my MATLAB skills again.  The code isn't optimised but it does the job.