The Debye formulation of focused fields has been systematically used to evaluate, for example, the point-spread function of an optical imaging system. According to this approximation, the focal wave field exhibits some symmetries 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-dimensional 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 pattern in the Debye approximation, it is possible to find a similar pattern but magnified and situated at another plane within the Kirchhoff theory. Moreover, we may evaluate this correspondence as the action of a virtual thin lens located at the focal plane and whose focus is situated at the axial point 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 the best-focus location.
© 2000 Optical Society of America
Original Manuscript: June 25, 1999
Revised Manuscript: March 20, 2000
Manuscript Accepted: March 20, 2000
Published: July 1, 2000
Carlos J. Zapata-Rodrı́guez, Pedro Andrés, Manuel Martı́nez-Corral, and Laura Muñoz-Escrivá, "Gaussian imaging transformation for the paraxial Debye formulation of the focal region in a low-Fresnel-number optical system," J. Opt. Soc. Am. A 17, 1185-1191 (2000)