Illustration of the Fresnel integral, with a modulating phase term, for two top-hat functions. The colour maps display the real part of the exponential term, , with the same colour scale as in Fig. 1. The alpha channel of the colour scale has been increased in regions where the spatial frequency of the exponent approaches the pixel size, so as to make transparent pixels that would otherwise be aliased. Phase terms that are constant with respect to x have been subtracted before display. These terms would not affect the final intensity of the image and removing them more clearly shows the line x′ = u(x) where . See text for further details.