The approximate formulas for the principal ray parameters, such as directional cosines and heights of incidence, as well as for the paraxial sagittal quantities h_s and H¯_s have been expressed by paraxial quantities and Seidel aberrations to fifth-order accuracy. On the basis of these relations an expression for the sagittal radius of curvature r_s, (for a given y¯) has been obtained. These quantities are used to derive fifth-order field aberration coefficients for arbitrary surfaces of rotational symmetry by using the wave aberration formula for sagittal focus {M. Gaj, Opt. Spectrosk. 21, 373 (1966) [Opt. Spectrosc. 21, 209 (1966)]}. The resulting expression has four terms. The first one depends only on asphericity and tends to equal zero when the surface becomes spherical. The second is a disturbance term and disappears in the Seidel region. The third and fourth terms may be treated as a generalization of the Petzval curvature and of the Seidel astigmatism, respectively. The limits of the terms, when h¯ tends to zero, has been examined.

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