Closed-form solutions of the two-dimensional homogeneous wave equation are presented that provide focal-region descriptions corresponding to a converging bundle of rays. The solutions do have evanescent wave content and can be described as a source–sink pair or particle–antiparticle pair, collocated in complex space, with the complex location being critical in the determination of beam shape and focal region size. The wave solutions are not plagued by singularities, have a finite energy, and have a limitation on how small the focal size can get, with a penalty for limiting small spot sizes in the form of impractically high associated reactive energy. The electric-field-defined spot-size limiting value is , which is about 38% of the Poynting-vector-defined minimum spot size and corresponds to a condition related to the maximum possible beam angle. A multiple set of solutions is introduced, and the elementary solutions are used to produce new solutions via superposition, resulting in fields with chiral character or with increased depth of focus. We do not claim generality, as the size of focal regions exhibited by the closed-form solutions has a lower bound and hence is not able to account for Pendry’s “ideal lens” scenario.
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