Predictions of Rayleigh's diffraction theory for the effect of focal shift in high-aperture systems

In their work on diffraction [J. Opt. Soc. Am.51, 1050 1961], Osterberg and Smith have computed in an exact manner from the Rayleigh-Sommerfeld diffraction integral of the first kind the irradiance distribution along the axis of a converging spherical wave, and they found that in a scalar optical system of high relative aperture and finite value of Fresnel number, the central peak value of the axial irradiance may occur inside, at, or outside the geometrical focal point as the angular semiaperture of the system is less than, equal to, or greater than, respectively, a particular angle that falls near 70 degrees . These findings are now reexamined using a different assumption that takes into account diffraction at the edge of the aperture. Different results are obtained that agree well with the predictions of other theories of diffraction of light and give confidence to the common conclusions drawn by investigators of the effect of focal shift, that the point of the principal maximum of axial irradiance is not at the geometrical focus but shifted toward the aperture in systems of different relative aperture and finite value of Fresnel number.