Fraunhofer diffraction by a spiral slit

Abstract

A plane slit in the form of an Archimedean spiral with two arms has a Fraunhofer diffraction pattern consisting of a central Airy pattern on the scale appropriate to the circular aperture that bounds the spiral, surrounded by grating ring lobes composed of fringes that are themselves arcs of spirals. The diffraction pattern for a one-armed spiral involves phase variations that are not apparent in the laboratory Fraunhofer intensity pattern and are difficult to show but that become evident when the Hartley transform is used instead. Computation of an unfamiliar function, the half-order derivative of the sine function, which is essential to this discussion and which occurs in other systems with circular grating lobes, is discussed.