Spatial rainbows and catastrophes in transmission of protons through electrostatic hexapole lens

This work reports on the spatial rainbows occurring in transmission of 10 keV protons through hexapole lens. Positive potentials of the lens electrodes are set to be 0.9, 2, and 5 kV. The spatial rainbows and corresponding proton distributions are calculated by using an accurate analytical approximation of the numerically obtained lens potential. Further, for the positive potential of the lens electrode equal to 0.9 kV, it has been shown that application of catastrophe theory leads to a simple polynomial non-linear mapping, generating accurate spatial rainbows at the exit and in the drift space behind the lens.