Functionality of an Einzel lens An Einzel lens is a device for ion optics to focus a beam of particles. It consists of three ring electrodes, with the outer ones being on earth potential and the middle one being on high voltage (see picture).
Now, due to the arrangement of the electrodes, the whole system including its electrical field is symmetric around an axis going through the middle electrode. Considering this symmetry, I don't understand how the beam can be focused. From my point of view, an in-going beam should leave the system the same way it entered it, thus leading to no focusing. Can someone explain how (and why) the beam is indeed focused by this device?

Source of picture: wikipedia.org on Einzel lenses, 23.09.2020
 A: I am not an expert on ion optics. However, I think I could make sense of the diagram, as follows. The lines drawn between the inner and outer electrodes are electric field lines (as it says in the diagram). They cannot have been drawn carefully because they are not orthogonal to the conducting surfaces, but if we take it that they are approximately correct elsewhere then their direction is from the inner electrode (where $V > 0$) to the outer ones (where $V=0$).
Now let's follow the upper of two ion paths shown (the dashed line), treating a positive ion. It first travels straight, then it encounters an upwards-directed electric field so turns in the upwards direction on the diagram (i.e. outwards, away from the axis of symmetry). Next as it reaches the inner electrode it encounters downwards or inwards directed field, and therefore turns in that direction. Finally it encounters outward-directed field and turns again.
In order to act as a focusing lens, what we require is that the net amount of inward steering is more than zero. In order to see that the symmetry allows this, imagine that the incoming beam was not parallel but was emitted from a point source on the left. Then, assuming the net steer is not zero, there exists a symmetric scenario in which ions from a point on the left are brought to a point on the right. This is analogous to the $2f, 2f$ object, image distance case for a thin lens in optics. This being so, then a collimated input beam will be focused at $f$.
