The Ewald sphere construction for radiation, scattering, and diffraction

Abstract

In most electrodynamics textbooks, the directional gain of an antenna is calculated using analytical integration, and the resulting expression is plotted as an afterthought. From a student's perspective, the analysis may be difficult, mysterious, or unrevealing. In this paper, we show that the Ewald sphere construction, a powerful tool for predicting crystallographic diffraction patterns, can also be used to help students gain direct geometrical insight into antenna radiation patterns. The radiation pattern from a sinusoidally varying current distribution can be obtained intuitively by sketching the reciprocal-space current density and examining how it behaves on an “Ewald” sphere centered at the origin. Furthermore, the nodes of the radiation pattern can be determined quantitatively by locating the intersections of the Ewald sphere with the nodes of the reciprocal-space current density. We illustrate this procedure with several examples, in the context of quantum mechanics, acoustics (sound), and electrodynamics (light). We provide an alternative formulation using the reciprocal-space polarization and magnetization, which treats loop antennas and coil antennas as easily as linear antennas. We make the connection to the original Ewald construction for scattering. We also show how the Ewald construction applies to diffraction through a planar aperture, within the Kirchhoff approximation.