A focusing system for monochromatic beams of charged particles, with magnetic quadrupole fields having common planes of optical symmetry, is treated theoretically. After specifying certain linear focusing properties for such systems, the calculus of variations is applied to obtain systems with minimum aperture aberrations. First order variational theory is applied to obtain the Euler equations that define extremal systems. This derivation gives four second order, coupled, ordinary differential equations. An iterative technique is presented for solving these Euler equations. Second order variational theory shows that the extrema of the aperture aberrations are local minima with respect to other systems having the same specified linear focusing properties. A digital computer was used to calculate a variety of systems with minimum aperture aberrations. Sample calculations are given for systems with magnifications of minus;1 and minus;2. General studies are presented for high magnification systems where the lenses are confined to a region near the object. The best orthomorphic systems have four lenses while the best anamorphotic systems have three. Several methods for regulating the quadrupole gradient strength are studied. It is found that optimum systems have aberration coefficients about one fifth as large as the best systems previously studied. It appears that quadrupoles alone are not as good as round lenses unless the quadrupole bore radius is less than 1sol;14 of the length of the shortest quadrupole.
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