Realising superoscillations: A review of mathematical tools and their application

Abstract

Superoscillations are making a growing impact on an ever-increasing number of real-world applications, as early theoretical analysis has evolved into wide experimental realisation. This is particularly true in optics: the first application area to have extensively embraced superoscillations, with much recent growth. This review provides a tool for anyone planning to expand the boundaries in an application where superoscillations have already been used, or to apply superoscillations to a new application. By reviewing the mathematical methods for constructing superoscillations, including their considerations and capabilities, we lay out the options for anyone wanting to construct a device that uses superoscillations. Superoscillations have inherent trade-offs: as the size of spot reduces, its relative intensity decreases as high-energy sidebands appear. Different methods provide solutions for optimising different aspects of these trade-offs, to suit different purposes. Despite numerous technological ways of realising superoscillations, the mathematical methods can be categorised into three approaches: direct design of superoscillatory functions, design of pupil filters and design of superoscillatory lenses. This categorisation, based on mathematical methods, is used to highlight the transferability of methods between applications. It also highlights areas for future theoretical development to enable the scientific and technological boundaries to be pushed even further in real-world applications.

Superoscillations: Realisation of quantum weak values

“…where [−a, a] is the superoscillatory region of the function bounded by one full period length 2π. By specifying M interpolation points then a superoscillatory function with M −1 arbitrarily fast oscillations can be controlled as the one in (14), only now the function is specified to be periodic. Applying such constraints to a periodic function, a truncated cosine Fourier series with N + 1 terms is taken in [16], the problem reduces to an eigenvalue problem with M linear equations and N + 1 unknowns, solved numerically to find the optimal energy ratio.…”

“…Therefore, ( 17) is, effectively, a finite-energy (it is normalised) band-limited signal (in momentum). This allows, again, to construct a superoscillatory function by specifying points for the wavefunction to pass through by the same method that resulted in (14). In which case, the wavefunction will contain sub-wavelength oscillations in a narrow region of very small amplitude compared to the large sidebands outside of it.…”

“…Because of this highly sensitive behaviour, quantifying superoscillations becomes important for their optical use. To do so, various authors have investigated properties like the size of the central spot, field of view (distance from central spot to first sideband) or a measure of energy efficiency, with a brief summary of results reproduced in [14].…”

“…The exponential relationship is quite prohibitive in terms of potential practical uses for superoscillatory functions, but the polynomial one poses smaller challenges. For optics, as pointed out in [14], if the focused energy decreased exponentially as the spot shrank, then vastly higher laser powers would be needed to realise small spots.…”

Superoscillations Made Super Simple

“…The term superoscillation was first coined by Michael Berry [27,28], in reference to work by Aharonov, Bergman, and Lebowitz [29]. Today superoscillations constitute a rapidly developing field of study in both mathematics [30,31] and physics [32,33].…”