IMAGE: Combined Effects of Aberrations and Coulomb Interactions

The Image software package computes effects of aberrations and discrete Coulomb interactions in electron and ion beams, in a unified way, by direct ray-tracing.  The software is applicable to a very wide range of systems, including combinations of round lenses, deflectors and stigmators, multipole systems, cathode imaging systems and electron mirrors.  Accurate direct ray tracing eliminates the need to use conventional aberration theory, and the method is therefore applicable to systems with aberrations of any order.  The software handles fields with an arbitrary combination of multipole components, and can compute the effects of asymmetry errors, for use in tolerancing calculations.  Post-processing facilities are included, for plotting point spread functions (with systematic or random initial conditions) and through-focal series of edge blur diagrams.

To analyze a system containing any combination of optical elements (e.g. lenses, deflectors, stigmators, dynamic focus coils, etc.), the axial field functions of each element are first computed

using any suitable field analysis software (e.g. SOFEM for lenses, deflectors and stigmators, or EO-3D and MO-3D for multipole elements).  The axial field function of each optical element is then fitted with a set of orthogonal analytic functions.  After fitting the axial fields by exact analytic functions, the off-axis fields at any point are obtained very accurately by radial power series expansions.  The software can compute these expansions to the 11th power of the off-axis distance (r11), for all multipole field components up to dodecapole (12-pole) fields.  Thus the software can handle the aberrations of all types of optical element likely to be encountered in practice.  The analytic representation of the fields for each optical element means they are all exact solutions of Laplace’s equation, suitable for use in direct ray-tracing.

The paths of electrons or ions are traced through these analytic fields by direct ray-tracing, using a fifth-order Runge-Kutta formula, with adaptive step size for automatic error control.  This ray-trace algorithm typically gives an accuracy of better than 1 picometer over a column length of 1 meter.

In the ray tracing, a whole ensemble of particles can be traced down the column simultaneously.  This makes it possible to include the Coulomb fields between the N particles in the ensemble, on each time step.  A subset of the IMAGE software can be supplied, which allows the analysis of the combined effects of aberrations and Coulomb interactions, in imaging systems containing lenses only.

If you are interested in the IMAGE package, please contact us at info@electronoptica.com.

Spot diagrams of geometrical aberrations (left), for a system with a deflected sub-field, and combined effect of aberrations and discrete Coulomb interactions (right).