# Aberrations and their correction

Every rotational symmetric electromagnetic field acting as a lens on an electron beam has intrinsic lens aberrations. This fundamental theorem was already shown by Scherzer (Scherzer, Z. Phys. 101 (1936) 593 ).

## chromatic aberration

The term chromatic aberration was adopted from light optics. Due to the dispersion of glass lenses light rays with different colours are focused with different focal lengths. Just like a beam of light usually consists of different wavelengths the electrons coming from the electron source have slightly different energies. One can easily understand that faster electrons are less influenced by electromagnetic fields so that the position of the electron focus depends on the electron energy. Chromatic aberration is decreased by using a small energy width and a high acceleration voltage respectively. The chromatic aberration coefficient has about the same magnitude as the focal length for weak lenses and about 0.6 times the focal length for strong lenses (in strong lenses the specimen is immersed in the lens field).

## spherical aberration

Spherical aberration arises because the focus depends on the axial distance of the electron path. Physically this is a consequence of the fact, that lens fields have to satisfy the Laplace equation. The aberration coefficient is substantially of the same order of magnitude as the focal length. If the specimen is immersed in the lens field, the minimum value of the aberration coefficient is half of the focal length. It diminishes when stronger lenses are used.

## coma and astigmatism

Coma can be found if the electron beam is off-axial and/or oblique to the optical axis. Coma aberration can be avoided by proper adjustment of the beam.

Astigmatism results from construction errors. Beams in the sagittal and the meridional plane respectively can have different focal lengths. This can be compensated for by a so-called stigmator, the function of which can be understood by contemplating the optical analogue. Axial astigmatism can be simulated by adding a cylindrical lens to a rotational symmetric lens. In electron optics two quadrupoles mounted in an angle of 45° serve as a stigmator.

## correcting chromatic and spherical aberrations

Prof. Otto Scherzer always believed that the most promising way to get rid of lens aberrations was to introduce non-rotational symmetry in the electron path. Then one of the preconditions, that render aberrations unavoidable, would be abandoned - see also "Scherzer Theorem". From 1948 on he designed and built together with his co-workers a first electrostatic corrector, which consisted mainly of three octupole elements to compensate for the spherical aberration. But due to instabilities in the mechanical alignment the resolution of the microscope could not be improved. Nevertheless this corrector compensated for the spherical aberration of the objective lens.

In order to correct the spherical aberration one has to use a combination of magnetic quadrupole and octupole. The chromatic aberration can be compensated by the use of electric and magnetic quadrupoles.