Focal shift effects in diffracted focused beams

We study the propagation characteristics of focused, apertured radially symmetric beams and show that the axial point at which the mean-square radius of the diffracted beam is a minimum tends to be displaced towards the aperture plane depending on the Fresnel number associated with the aperture and the Fresnel number associated with the radius of the incident beam. The magnitude of this effect can be determined by a simple and general formula in terms of the amplitude distribution of the incident beam across the aperture and of the two Fresnel numbers. We investigate the focal shift effects for the case of a focused, apertured Gaussian beam. The irradiance and the encircled-power distributions in the plane at which the mean-square radius of the diffracted beam is minimum are calculated and compared with those corresponding to a uniform beam.