Focusing properties of a square electrostatic rainbow lens doublet

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This is a study on the properties of a square electrostatic rainbow lens doublet. The said optical element consists of two square electrostatic rainbow lenses with the second lens axially rotated for 45 degrees with respect to the first one. The propagation of a proton beam with a kinetic energy of 10 keV through the doublet is in the focus of our analysis. The potential of the electrodes of both lenses is 2 kV. The electrostatic potential and the electric field components of the lens doublet are calculated using a 3-D computer code based on the method of moments. Spatial and angular distributions of protons propagating through the lens doublet, as well as the parameters defining beam quality, are investigated. As in the case of the single square electrostatic rainbow lens, the evolution of these distributions is determined by the evolution of corresponding rainbow lines, generated by the use of the theory of crystal rainbows. Our study shows that a beam core in the shape of a cusped square is formed by the spatial rainbow line that appears first. This rainbow line occurs during proton propagation through the first lens. The beam core retains the cusped square shape during the propagation through the second lens. The electrostatic field of the second lens causes the appearance of an additional spatial rainbow line, which encompasses the beam core and defines the outer border of the beam. This rainbow line constitutes the main difference between the cases of the lens doublet and a single lens.