Analysis of third order quadrupole aberrations

The formulae of 3(rd) order quadrupole aberrations are derived with Lie-algebraic map. It is the field extension along the axis that dominates the aberrations. For fixed focal length, the aberrations are small if the quadrupole length is long enough or its aperture is large enough. When the length is comparable to the aperture, the quadrupole with soft edge carries relatively small aberrations.