Theoretical study on sixth-order geometrical aberration correction

Spherical aberration correctors using hexapole fields are widely used and are pivotal in atomic-resolution imaging. Although hexapole-field correctors increase the aberration-free angular range, the angular range is limited by higher-order aberrations, such as six-fold astigmatism or sixth-order three-lobe aberration. Here, we propose two types of spherical aberration correctors to compensate for geometrical aberrations up to the sixth order. The first is a four-hexapole corrector, while the second is a two-hexapole corrector, where each hexapole has a nonuniform magnetic field. The four-hexapole corrector can increase the aberration-free angle up to almost 100 mrad. The two-hexapole corrector with a nonuniform magnetic field has a smaller aberration-free angle than that of the four-hexapole corrector, but it is more compact. The dominant residual aberration in these correctors is seventh-order spherical aberration or chaplet aberration, which is seventh-order geometrical aberration.