Eigenfunction representation of the integrals of the Debye-Wolf diffraction formula

The Debye-Wolf electromagnetic diffraction formula is now routinely used to describe focusing by high numerical aperture optical systems. In this paper we obtain the eigenfunction representation of the integrals of the Debye-Wolf formula in terms of Bessel and circular prolate spheroidal functions. This result offers considerable analytical simplification to the Debye-Wolf formula and it could also be used as a mathematical basis for its inversion. In addition, we show that numerical evaluation of the Debye Wolf formula, based on the eigenfunction representation of its integrals, is faster and more efficient than direct numerical integration. Our work has applications in a large variety of areas, such as polarised light microscopy, point spread function engineering and micromachining.