Example phase patterns with histogram of the distribution (a–c) obtained by filtering an initial random phase map and their corresponding power spectra (e–g) for a uniform intensity for the disked-shape in the image plane. The filter widths used are w = 1 pixels (a, d), w = 4 pixels (b, d), w = 10 pixels (c, f). The oversampling factor is s = 4. The filter gain is ajusted so the vortex probability is 10−4: G0 ≈ 1, 2 (a–d), G0 ≈ 0 9 (b–e), G0 ≈ 58 (c–f). In (g) we show the distibution of the intensity across the pupil, averaged over all directions.