Derivation of conventional formula of the third-order aberration for off-axial optical system
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A formula of the third-order aberration based on the primary aberration expansion for an off-axial optical system is investigated. The point eikonal analysis and the coordinate system using the surface normal vector lead to a simple wavefront-based expression. Three-ray system employing the central, the principal and an arbitrary ray is examined for the derivation. The rays are forced to reach from the last surface to a virtual image point. Therefore, the contradiction of the optical path lengths brings the wavefront aberration. The extracted third-order aberrations are not strictly the same expressions as the conventional formula but show clearly the features of the five kinds of the primary aberrations along with the term originated by the off-axial layout. The relation between the second-order term and the expanded gaussian matrix for paraxial ray tracing and a term related to field curvature are discussed for off-axial systems.