Vector diffraction in paraboloidal mirrors with Seidel aberrations. I: Spherical aberration, curvature of field aberration and distortion

Abstract

We study the image formation in paraboloidal reflectors with Seidel (fourth order) aberrations, particularly the spherical aberration, curvature of field aberration and the distortion aberration. The theory developed here is based on Richards' and Wolf's formulation of the vector diffraction theory of focusing systems. The expressions for the electric and magnetic vectors are derived to reflect the axially symmetric wave aberrations and it is established that these expressions are similar to the expressions for aberration-free parabolic reflectors except for the presence of the factor, exp(iκΦ), provided that these expressions are formulated in terms of the component, in the direction of the optical axis, of the unit normal to the wave front at the exit pupil. Here Φ is the aberration function. It is shown that if wave fronts emerging from the exit pupil suffer from the spherical aberration, the images formed at the respective diffraction foci are just as sharp as the images formed at the Gaussian focus by an aberration-free parabolic reflector, but with a slight reduction in the intensity. The similarity also extends to the state of polarization of the focused light. In the case of the curvature of field aberration, and in contrast with the scalar theory, we show that the Strehl intensity in the plane of the diffraction is less than unity. We also show that in the case of the spherical aberration, the intensity at the diffraction focus increases with increase in the numerical aperture, whereas for the curvature of field aberrations it decreases with the increase in the numerical aperture. For the distortion aberration, the intensity is the same as the Gaussian intensity and the generated image is merely shifted perpendicular to the optical axis.