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absolute value analytical angle anisotropic aperture astigmatism asymptotic axial chromatic aberration axial potential distribution azimuthal beam Boersch boundary calculated cardinal elements cathode charged particles chromatic aberration coefficient component consider constant coordinate current density curve cylinders defined by Eq deflection depends derivatives direction distance electrode potentials electron and ion electrostatic field electrostatic lenses energy spread expression field distributions figure of merit finite flux density distribution focal length focusing function given by Eq homogeneous image plane infinite magnification integration ion optics Laplace's equation limit linear magnetic field magnetic flux density magnetic lenses maximum method minimum multipole negative object-side focal obtain optical axis parameters paraxial approximation paraxial ray equation polynomial positive potential ratio principal planes properties quadrupole quantities radial radius refractive power relativistic respectively Section shown in Fig solution space space charge space-charge spherical aberration coefficient spline Substituting Eq surface Taylor series trajectory vector voltage ratio yields