Superoscillations in non-monochromatic naturally occurring waves

It is well known that one might describe a wave or a function in terms of it's Fourier components, therefore a function would be labelled as superoscillatory if it oscillates faster than the fastest Fourier component. This phenomena is known to be caused by the fine destructive interference between the Fourier components of a wave. There is a vast amount of literature on this special phenomena, however the main focus of this thesis will be studying and analysing the works of Sir Michael Berry, Sandu Popescu, Mark Dennis, and J.Lindberg [1][2][3][4][5], focusing mainly on naturally occurring superoscillations through the superposition of a large number of waves. In particular, investigating the effects of changing the band-width of the waves on the fractional superoscillatory area of the resultant wave as well as the effects of varying the number of waves (Fourier components) superimposing. In the papers by Mark Dennis et al [5] [4], an in depth description is given for superoscillations in speckle patterns, through the derivation of a joint probability density function of both intensity and phase gradient. Through this, a superoscillating fraction of 1 5 is found for a disk spectrum in addition an expression is found for an annular spectrum. The main work conducted for this thesis builds on this through a range of computational methods, specifically investigating the fraction of superoscillations obtained from the superposition of two dimensional non-monochromatic waves. An interesting result found shows that the fractional area which exhibit superoscillations is given to be much smaller than 1 5 for an annular spectrum as described in [5]. The calculations conducted here describes the effect of changing the bandwidth; in which the wavenumber of the individual waves are chosen from, has on the fraction of superoscillations. In addition the effect of varying the lowest wavenumber in the spectrum, as well as the number of sources, is provided. The study of all the papers mentioned earlier involves a vast amount of statistical analysis, ranging from Gaussian and speckle statistics to probability theory.