In this paper, we describe electron trajectory by vector form and express aberra-tions in matrix form. According to the more general viewpoint, we discuss some general linear transformations of gaussian trajectory parameters. The third order aberrations, aberration coefficients and their partial differentials in an electron optical system undergone these transformations are derived. Therefore the general electron optical Fraunhofer conditions for correcting isotropic and anisotropic coma are obtained. It is shown that under this condition, astigmatism and field curvature have extreme or stationary values. According to these results, some concrete problems are discussed; for example, the effect of aperture position on aberrations in the fixed or scanned electron beam systems, and the effect of magnetic immersion on aberrations in the emission electron system (cathode lens).
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