This thesis presents an investigation on imaging beyond the diffraction limit with superoscillatory electromagnetic waves. It begins with a survey on sub-diffraction imaging, which motivates the need for a new sub-diffraction imaging device which is free from drawbacks in devices which have heretofore been proposed. This work addresses this need through the design and demonstration of superoscillatory wave-based focusing and imaging devices. Through establishing a relationship between superoscillation and superdirectivity, a design procedure is formulated whereby antenna array theory is leveraged to design 1D and 2D superoscillatory waves. These waves are then physically synthesized or implemented as a filter in focusing and imaging systems at frequencies ranging from microwave to optical. Experimental characterization of these systems has led to successful demonstrations of sub-diffraction focusing and imaging, with working distances orders of magnitude farther than most other electromagnetic wave-based sub-diffraction imaging devices. In particular, the Optical Super-Microscope which achieves far-field sub-diffraction focusing has attractive merits to become a tool for general-purpose sub-diffraction optical microscopy. An investigation has also been conducted on superoscillatory waves in the time domain, which has resulted in the first reported demonstration of a superoscillatory temporal waveform, as well as a demonstrated improvement of radar range resolution beyond the Fourier transform limit.
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