# Einzel Lens

Many modern ion optic elements utilize time-varying electric fields to focus ion beams.  Moreover, they utilize collisional cooling to further focus the ion beam, working better in moderate vacuum than they do in high vacuum.  While these elements can be quite powerful, they require somewhat complex control electronics to generate multiple phases of RF and often DC gradients.  There are simpler focusing elements that utilize electrostatic fields to focus ion beams.  Among these, the most common is the so-called "Einzel Lens".

## Basic Geometry

An Einzel lens is comprised of three metal cylinders/electrodes arranged coaxially in a row.  The first and third electrodes have the same voltage applied to them, while the second electrode has a voltage that is "uphill" to ions of the polarity of interest.  For example, if one wants to focus positive ions, then the second electrode will have a voltage that is more positive than the first and second electrodes.

## Focusing Mechanism

An Einzel lens works via a semi-delicate balance of ion kinetic energy and the potential difference between the central electrode and the outer electrodes.  As an ion is exiting the first electrode, the uphill electric field slows its axial kinetic energy and also begins to push the ion outward radially.  However, as the ion climbs the uphill potential and slows down as it enters the second electrode, the electric field begins to push the ion inward radially.  The inverse process happens as the ion exits the second electrode.  It feels an inward radial force as it's exiting the second electrode and starts to fall down a "downhill" electric field.  As the ion is entering the third electrode it slows down and feels an outward radial force.  So the overall process is this:

1. Exiting first electrode: ion decelerates axially and is pushed outward radially
2. Entering second electrode: ion continues to decelerate axially (albeit at a slower rate) and is pulled inward radially
3. Exiting second electrode: ion accelerates axially while still being pulled inward radially
4. Entering third electrode: ion continues to accelerate axially (albeit at a slower rate) and is pushed outward radially