The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

Abstract

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.