Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system

The Debye formulation of focused fields has been systematically used to eva luate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmet ries about the geometrical focus. However, certain discrepancies arise when the Fresnel number, as viewed from focus, is close to unity. In that case, we should use the Kirchhoff formulation to evaluate accurately the three-d imensional amplitude distribution of the field in the focal region. We make some important remarks regarding both diffraction theories. In the end we demonstrate that, in the paraxial regime, given a defocused transverse patt ern in the Debye approximation, it is possible to find a similar pattern bu t magnified and situated at another plane within the Kirchhoff theory. More over, we may evaluate this correspondence as the action of a virtual thin l ens located at the focal plane and whose focus is situated at the axial poi nt of the aperture plane. As a result, we give a geometrical interpretation of the focal-shift effect and present a brief comment on the problem of th e best-focus location. (C) 2000 Optical Society of America [S0740-3232(00)0 0807-3] OCIS codes: 050.1960, 110.1220, 180.6900.