Fifth-order field aberration coefficients for an optical surface of rotational symmetry

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.