  An analytical description of arbitrary strongly aberrated axially symmetric
focusing is developed. This is done by matching the solution of geometrical
optics with a wave pattern which is universal for the underlying ray structure.
The corresponding canonical integral is the Bessoid integral, which is a three
dimensional generalization of the Pearcey integral that approximates the field
near an arbitrary two-dimensional cusp. We first develop the description for
scalar fields and then generalize it to the vector case. As a practical example
the formalism is applied to the focusing of light by transparent dielectric
spheres with a few wavelengths in diameter. The results demonstrate good
agreement with the Mie theory down to Mie parameters of about 30. Compact
analytical expressions are derived for the intensity on the axis and the
position of the diffraction focus both for the general case and for the
focusing by microspheres. The high intensity region is narrower than for an
ideal lens of the same aperture at the expense of longitudinal localization and
has a polarization dependent fine structure, which can be explained
quantitatively. The results are relevant for aerosol and colloid science where
natural light focusing occurs and can be used in laser micro- and
nano-processing of materials.
