Super oscillatory functions in Super resolution imaging

Abstract

This thesis explores the concept of achieving oscillations faster than the diffraction limit of light, something that seems implausible, however with the discovery and utilisation of super oscillation functions, one can achieve such goal. While many classes of functions display super-oscillating behavior, this paper will focus mostly on the class associated with the method of the harmonic series. Work in this paper, therefore, demonstrates that by careful manipulation of the coefficients αn of a simple harmonic series, a function having superoscillatory regions can be realized, thereby proving the practicality of such functions in fields such as high-resolution imaging. Second, the work opted for an envelope method alternative, which attempts to compute it. Optimization of the harmonic series method points at significant progress for applications which require focusing ability at sub-wavelength. The culmination of this research lies in the harmonic superoscillatory function that oscillates approximately 500x faster than the diffraction limit, hinting at a profound impact on imaging technologies. This thesis is a contribution to the theoretical framework of superoscillatory phenomena, and at the same time, it lays out the ground for possible future work aimed at increasing the robustness and, in fact, the practicality of these functions for their use in real-world applications.