An Analytical Method of Vector Diffraction for Focusing Optical Systems with Seidel Aberrations II: Astigmatism and Coma

In this paper, a method developed by the author is used to study the image formation in various aplanatic systems of high Fresnel numbers and circular pupils with Seidel (fourth order) aberrations, particularly the astigmatic and comatic aberrations. The theory developed here is based on Richards and Wolf's formulation of the vector diffraction theory of focusing systems. The electric and magnetic vectors are derived to reflect the asymmetry of the surface of the wavefront with respect to the optical axis. The unit normal to the surface of an aberrated wavefront is expressed as the sum of two vectors. One vector is in the direction of the unit normal to the ideal Gaussian sphere, while the other, after a suitable normalization, characterizes the unit normal's deviation from the unit normal to the ideal Gaussian sphere. The components of the electric and magnetic vectors are then expressed in integral form. In the end, the integrals are evaluated in terms of the products of Gegenbauer polynomials of the first kind and spherical Bessel functions; both of these functions are evaluated by summing finite series. We present results for systems with astigmatic and comatic aberrations and provide information about the location of the diffraction foci and Strehl intensity. We also show that in the limit, the theory presented here provides solutions to aberration-free systems and systems with the Seidel aberrations of the first kind, and in cases when the numerical aperture is small, the results are in agreement with the scalar theory for large Fresnel numbers.