Axially symmetric focusing as a cuspoid diffraction catastrophe: Scalar and vector cases and comparison with the theory of Mie
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.