Disguised assumptions and the conditions for validity of Debye integral representation of focused fields

In the paper of Wolf and Li [Opt. Commun.39, 205 (1981)OPCOB80030-401810.1016/0030-4018(81)90107-3], a sufficient condition was obtained under which the Debye integral representation (DIR) may be expected to give a good approximation to the solution of a boundary value problem that is generally taken to represent a field in the region of focus. After a long calculation of the solid angle of the circular cone of rays over which the Debye integral extends, the validity condition of DIR reduces to the requirement that the Fresnel number of the diffracting aperture, when viewed from the geometrical focus, is much greater compared to unity. The current study removes the long calculation of the cones of rays and replaces it with an analysis of Debye's disguised assumption relating to a replacement of the actual (unknown) field emerging from the exit pupil by the unperturbed incident field.