geometrical aberration

In geometrical optics where electron trajectories are described as motions of charged particles in an electromagnetic field, the deviation of the real imaging point of an electron from the ideal imaging point with no aberrations is called “geometrical aberration.” Optical properties are expressed by a power-series polynomial of r (distance of an electron beam from the optical axis) and α (angle between the beam and the optical axis), in which a beam emitted from one point on the object plane is mapped to a point on the image plane. If only the first order terms of the expression is taken, the polynomial expresses ideal imaging with no aberrations (Gauss imaging). If the higher order terms than the second order are taken into account, the deviation of the real imaging point from the ideal imaging point appears.
The order of the geometrical aberration is determined by the sum of the order of α (angle between the beam and the optical axis) and that of r (distance of an electron beam from the optical axis). It is conviniently used for describing the order of the aberration. On the other hand, the wave aberration is useful because the order of the wave aberration is closely related to the symmetry of the aberration. The order of the wave aberration is expressed by adding 1 (one) to the order of the geometrical aberration. For examples, the two-fold astigmatism is a first-order aberration in terms of the geometrical aberration but is the second-order aberration in terms of the wave aberration, and that the three-fold astigmatism is a second order in the geometrical aberration but a third order in the wave aberration.

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