Focusing properties of a very thin electrostatic quadrupole lens

The rainbow effect in transmission of a proton beam of energy of 10 keV through a very thin electrostatic quadrupole lens is considered theoretically. The incident ion velocity vectors are taken to be parallel to the lens axis. The distance between the positively or negatively charged lens electrodes is 10 mm and the electrode voltage modulus is 1 kV. The approach is based on the momentum approximation. The rainbow effect is studied via a detailed analysis of the ion differential transmission cross-section, which is determined by the Jacobian of the ion transmission angle components. This variable contains a term describing the ion deflection from the individual electrodes and a term describing the coupling of the electrodes that is seen by the ion. It has been found that the latter term is the source of the rainbow effect, which is, thus, a complex effect in ion beam dynamics. The final conclusion of the calculations is that the lens acts on the ion beam in a way to reshape it qualitatively into a beam containing four narrow sub-beams directed between the lens electrodes.