Abstract
We pose the problem of finding the conditions on a scalar homogeneous wave field in a given plane z = 0 so that the intensity patterns in the planes z and −z will be the same to within a rigid displacement of each other. If we call the wave field in the midplane f(x, y), then, for the case in which the displacement is a 180° rotation about the z axis, a sufficient condition for such intensity self-imaging is that the Fourier transform of f be real valued. For the case in which the displacement is a 180° rotation about an axis in the plane of the pattern itself, a sufficient condition is the conjugate symmetry of the Fourier transform of f about a line perpendicular to that particular axis. We show the relation of this kind of self-imaging to some previous results that concern the symmetry of spherical focused wave fields, and we derive a corresponding symmetry for cylindrical focused wave fields.
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