Exact nonparaxial transmission of subwavelength detail using superoscillations

Abstract

Any object, described by a target function that need not be band-limited, can be sampled at any chosen set of points and then propagated without evanescent waves, using a function which is band-limited, so as to be imaged exactly (i.e. nonparaxially) at multiples of a given repetition distance. If the samples span a sub-wavelength region, the repeated images are superoscillatory. The number N of samples is equal to the repetition distance measured in wavelengths and also to the number of plane waves in the propagating field. If N ≫ 1, the waves form a quasi-continuum, and asymptotics enables an almost-explicit description of the superoscillations. But the matrix involved is ill-conditioned (many of its eigenvalues are very small), so this method of sub-wavelength imaging would be pathologically sensitive to noise, and the depth of focus is exponentially small. © 2013 IOP Publishing Ltd.