three-fold astigmatism

The three-fold astigmatism, one of the second-order axial geometrical aberrations, is a parasitic aberration exhibiting three-fold symmetry. (Note that it is called a third-order aberration in terms of the wave aberration.) The aberration is corrected using magnetic fields produced by hexapoles.
A widely used two-stage hexapole Cs corrector corrects the spherical aberration of the objective lens using a combination effect in the three-fold astigmatism field. In the Cs correction system, the first hexapole simultaneously produces a large three-fold astigmatism and a negative spherical aberration. Then, the second hexapole placed below the first hexapole produces a three-fold astigmatism having an opposite sense to the former and a negative spherical aberration. The three-fold astigmatism produced in the first stage is canceled by that with an opposite sense produced in the second stage. The positive spherical aberration of the objective lens is corrected by the negative spherical aberration produced in the two stages.

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