Natural superoscillations in monochromatic waves inDdimensions

For monochromatic waves satisfying the Helmholtz equation with wavenumber k 0 , superoscillations correspond to local wavenumbers (magnitude of phase gradient) greater than k 0 . Large values of local wavenumber are associated with phase singularities. For isotropic random waves (superpositions of many nonevanescent waves) in D dimensions, we show that the probability that a point in the field is superoscillatory increases from 0.293 to 0.394 as D increases from 1 to infinity. The peculiar case D = 1 is examined in detail.