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Figure 3647a shows the simulated intensity distribution patterns of 200 keVelectron probes at 100 nm of B_{2} (secondorder axial coma), 100 nm of A_{2} (threefold axial astigmatism), 1 µm of S_{3} (axial star aberration of the 3rd order), and 2 µm of A_{3} (fourfold axial astigmatism).
Figure 3647a. Simulated intensity distribution patterns of 200 keVelectron probes: (a) B_{2} = 100 nm, (b) A_{2} = 100 nm, (c) S_{3} = 1 µm, and (d) A_{3} = 2 µm. In this simulation, the defocus C_{1} was set to 3 nm, the imaginary parts of the aberrations to zero, and the illumination semiangle to 30 mrad. [1]
Figure 3647b shows the schematic comparison of Zemlin (diffractogram)tableau characteristics for the axial aberrations up to third orders. Firstorder aberration (e.g. defocus and twofold astigmatism, A_{1}) shows the elliptical distortion even without electron beam tilting because the impact of firstorder aberrations does not depend on the tilt angle. For the aberrations of higher orders (n≥2), such as secondorder axial coma B_{2}, threefold astigmatism A_{2}, thirdorder spherical aberration C_{3} (>0), thirdorder star aberration S_{3} and fourfold astigmatism A_{3}, there is no elliptical distortion observable for the untilted case. In these cases, only at illumination tilts the characteristic distortion due to the aberrations becomes discernible. Note that the higherorder aberrations have equal symmetries to the ones in Figure 3647a as discussed in page3740.
Figure 3647b. Schematic representation of Zemlin (diffractogram)tableau characteristics for the axial aberrations up to third order.
After the corrections of lower order aberrations, the corrector normally introduces or leaves high order aberrations. For instance, the Ronchigram with a 54 mR aperture indicated in Figure 3647c shows the left 4^{th} order coefficient of order 0.5  2.0 mm limiting the probe shape and size after 3^{rd} order aberration correction by Nion quadrupoleoctupole corrector. The pattern is strongly 4fold symmetry with weak 2fold structure in the opposing corners. Note that this uniform 4fold symmetry was actually imposed by the octupoles in the corrector, so called parasitic aberrations.
Figure 3647c. Ronchigram obtained with 4^{th} order coefficients corrected to < 60 µm [2].
Note that an octupole always generates parasitic aberrations because it has cubic radial dependence together with fourfold azimuthal symmetry (fourfold astigmatism).
[1] Simulation of Rolf Erni.
[2] P. E. Batson, Control of Parasitic Aberrations in Multipole Corrector Optics, Microsc Microanal 14(Suppl 2), 2008.
