Evolution of quantum superoscillations and optical superresolution without evanescent waves

A superoscillatory function-that is, a band-limited function f (x) oscillating faster than its fastest Fourier component-is taken to be the initial state of a freely-evolving quantum wavefunction psi. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to psi with complex momenta that are exponentially disparate in magnitude; this is established by applying the asymptotics of integrals, supported by numerics. f (x) can alternatively be regarded as the wave generated by a diffraction grating, propagating paraxially and without evanescence as psi in the space beyond. The persistence of superoscillations is then interpreted as the propagation of sub-wavelength structure farther into the field than the more familiar evanescent waves.