High order aberration calculations of a quadrupole-octupole corrector using a differential algebra method.

A Differential algebraic method is of an effective technique in computer numerical analysis. It implements conveniently differentiation up to arbitrary high order, based on the nonstandard analysis. In this paper, the differential algebra (DA) method has been employed to compute the high order aberrations up to the fifth order of electron lenses with quadrupole and octupole corrector whose electric/magnetic fields are in the forms of discrete arrays, for example, the files computed by FEM or FDM method. The quadrupole and octupole electro-magnetic fields of arbitrary point are obtained by local analytic expressions, and then field potentials are transformed into new forms which can be operated in the DA calculation. The program has been developed and tested as well. The geometric and chromatic aberrations coefficients up to fifth order of electron lenses with quadrupole and octupole corrector are calculated by the developed program.