Generalized vibration curves for apodization and super-resolution in far-field diffraction

Abstract: The Cornu spiral, widely used to analyze Fresnel diffraction, is a familiar example of a vibration curve, also known as an amplitude-phase curve or phasor diagram. A related construction appears in discussions of Fraunhofer diffraction, where the aperture function is uniform, making the curve's shape depend solely on phase. When the aperture function varies with position, however, both amplitude and phase determine the geometry. Here, vibration curves are extended to continuous, monotonic, symmetric, nonuniform aperture functions that produce either apodization (broadening of the central maximum with suppression of side maxima) or super-resolution (narrowing of the central maximum with enhancement of side maxima). These generalized vibration curves provide an intuitive visualization of how aperture structure shapes the far-field diffraction pattern. They clarify the mechanisms behind apodization and super-resolution and reveal how phasor rotation at the first diffraction zero provides a signature that characterizes the two effects. This unified perspective serves as a useful pedagogical tool for connecting Fourier transforms, optical diffraction, and vibration curves in the undergraduate and graduate curriculum.