Complex and catastrophic ion beam focusing with a very thin electrostatic square rainbow lens

The focusing properties of a very thin electrostatic square rainbow lens are investigated theoretically with a parallel proton beam of energy of 10 keV. The lens electrodes lie on a circle of diameter of d = 10 mm, and the electrode voltage is 1 kV. The calculations are performed using the momentum approximation. It is shown that the spatial and angular distributions of transmitted ions are fully determined by the rainbow effect, which emerges from the interference of the contributions of the electrodes to the ion differential transmission cross-section. This makes it a complex effect. It is also demonstrated that the shape of the rainbow line in the central region of the transverse position plane far from the lens can be very accurately reproduced using catastrophe theory. Thus, the effect is a catastrophic effect too. The transmitted ion beam far from the lens contains only a strong central sub-beam. Special attention is devoted to the characteristics of this beam, which might find use in materials science and technology. Graphic Abstract