This invention relates generally to techniques for correcting aberrations in particle beams and, more particularly, to a technique for correcting high-order aberrations, such as spherical aberration.  In optical systems, the conventional analysis of image formation and beam focusing is based on "first-order" theory, otherwise known as Gaussian optics or paraxial theory.  The basic assumption in first-order theory is that the light rays being traced through various optical elements are close to and parallel with the optical axis of the elements. 
When there is a significant departure from this assumption, the optical images formed by the system contain aberrations referred to as high-order aberrations.  One of these is called spherical aberration, in which the effect on a focused image, for example, will depend on the cube of the radial distance of the ray from the optical axis.  In the case of a convex lens focusing a collimated light beam, the effect of spherical aberration is to focus the outer rays of the beam closer to the lens than the inner rays close to the axis.  In effect, the focal point is blurred into a line along the axis.  Other high-order aberrations are caused by effects dependent on the fifth power of the radius, or of other odd-numbered powers. 
These optical aberrations have counterparts in the field of particle beams.  Spherical aberration in particle beam systems results principally from the end fields of the lenses employed, that is from magnetic fields associated with the ends of focusing magnetic coils, which are usually quadrupoles.  There also appears to be an r.sup.3 -dependent space charge effect resulting from mutual repulsion of the like charges of beam particles.  Optical systems can be corrected for spherical aberration, at least to some degree, by the use of aspherical lens surfaces, but this option is not available to the designer of particle beam systems. 
The angular divergence produced by spherical aberration from a point source at the focal point of a lens of focal length F and radius R is given by: ##EQU1## where r is the beam radius at the output of the lens and k is a dimensionless constant for any particular case. 
It has been theorized by D. Scherzer that these aberrations cannot be corrected without introducing either currents or electrical charges into the region occupied by the beam.  This is known as Scherzer's Theorem (Z. Phys.  101, p. 593 (1936)).  Although the validity of Scherzer's Theorem is currently in doubt, there is still a need for some way to correct spherical and other high-order aberrations in charged particle beams.  The present invention addresses this need.