Superoscillation is a physical phenomenon of the local oscillations of a band-limited signal that fluctuate faster than the fastest Fourier component of the signal. In recent years, superoscillation leads to a method of super-resolution imaging, named superoscillatory imaging, and plays an important role in many areas, such as remote sensing and biomedical research. This dissertation investigated a key lens element for achieving superoscillatory imaging. Then, a vector-superoscillatory field provided a solution to a major problem associated with superoscillatory imaging. Lastly, the partial coherence effect, specifically circular coherence, was studied for vortex beam propagation in free space and can be considered in the quality of superoscillatory imaging.
This dissertation work began by studying the existing methods for designing filters to create superoscillatory fields in the image plane. A design method by Smith and Gbur tailors a superoscillatory field in two dimensions, from which a filter is calculated with both an amplitude profile and a phase profile (a complex filter). Accordingly, the first study of this dissertation aimed to simplify a complex filter into a filter with only one profile: an amplitude profile or a phase profile, which make the filters fabrication-friendly. This study derived the mathematical formula for generating simplified filter profiles (leading to the same superoscillatory field by complex filters). A step-by-step example of creating such simplified filters was demonstrated by following this approach. Performance criteria of the designed filters were discussed, including but not limited to energy efficiency. The designed phase-only filter showed an energy efficiency same as that of the complex filter.
The second study of this dissertation provided a method to eliminate the sidelobes that are inevitable to superoscillatory fields and causes a problem to superoscillatory imaging. As light is a transverse electromagnetic wave, the orientation of scattering patterns of Rayleigh scatterers is polarization-dependent. Then, superoscillatory fields with two polarization states (referred to as vector superoscillatory fields) were created, so that the sidelobes can be avoided in the imaging process. This study proposed an imaging system with vector superoscillatory illumination. Super-resolved scattering images of Rayleigh scatterer patterns were simulated under a vector superoscillatory illumination, whose resolution surpassed those obtained from a conventional imaging system. A device was proposed for generating a vector-superoscillatory field.
Light sources with circular coherence have perfectly coherent points on any concentric rings of their transverse planes. In the third study, we investigated from their ability in carrying optical vortices in free space to the self-focusing effect. Circular coherence was imposed onto vortex beams (with spiral phase structures). The free-space propagation of circularly coherent vortex beams showed that optical vortices remained their positions on free-space propagation and the beams revealed a focal region. This study also provided a model for propagating rotationally symmetric beams using two-dimensional Hankel transform. The self-focusing effect of circular coherence can be considered on further reducing the spot size of a superoscillatory field.
These three studies together make the superoscillatory imaging technique have more potential to be implemented.