Fractional Gouy phase

In general, the total Gouy phase shift has the form n pi, where n need not be an integer. As a result of the Fourier transforming property of a lens, the Gouy phase is found to be related to the types of discontinuities at the upper or lower range of the pupil function Q(c) resulting from the asymptotic order of the Fourier transform. The sign of the Gouy phase is also related to the slope of the pupil function. The oscillations of the Gouy phase shift arise from the strength of the nondominant discontinuity.