37 research outputs found
Yield--Optimized Superoscillations
Superoscillating signals are band--limited signals that oscillate in some
region faster their largest Fourier component. While such signals have many
scientific and technological applications, their actual use is hampered by the
fact that an overwhelming proportion of the energy goes into that part of the
signal, which is not superoscillating. In the present article we consider the
problem of optimization of such signals. The optimization that we describe here
is that of the superoscillation yield, the ratio of the energy in the
superoscillations to the total energy of the signal, given the range and
frequency of the superoscillations. The constrained optimization leads to a
generalized eigenvalue problem, which is solved numerically. It is noteworthy
that it is possible to increase further the superoscillation yield at the cost
of slightly deforming the oscillatory part of the signal, while keeping the
average frequency. We show, how this can be done gradually, which enables a
trade-off between the distortion and the yield. We show how to apply this
approach to non-trivial domains, and explain how to generalize this to higher
dimensions.Comment: 8 pages, 5 figure
Analysis of Superoscillatory Wave Functions
Surprisingly, differentiable functions are able to oscillate arbitrarily
faster than their highest Fourier component would suggest. The phenomenon is
called superoscillation. Recently, a practical method for calculating
superoscillatory functions was presented and it was shown that superoscillatory
quantum mechanical wave functions should exhibit a number of counter-intuitive
physical effects. Following up on this work, we here present more general
methods which allow the calculation of superoscillatory wave functions with
custom-designed physical properties. We give concrete examples and we prove
results about the limits to superoscillatory behavior. We also give a simple
and intuitive new explanation for the exponential computational cost of
superoscillations.Comment: 20 pages, several figure
Optimising superoscillatory spots for far-field super-resolution imaging
Optical superoscillatory imaging, allowing unlabelled far-field super-resolution, has in recent years become reality. Instruments have been built and their super-resolution imaging capabilities demonstrated. The question is no longer whether this can be done, but how well: what resolution is practically achievable? Numerous works have optimised various particular features of superoscillatory spots, but in order to probe the limits of superoscillatory imaging we need to simultaneously optimise all the important spot features: those that define the resolution of the system. We simultaneously optimise spot size and its intensity relative to the sidebands for various fields of view, giving a set of best compromises for use in different imaging scenarios. Our technique uses the circular prolate spheroidal wave functions as a basis set on the field of view, and the optimal combination of these, representing the optimal spot, is found using a multi-objective genetic algorithm. We then introduce a less computationally demanding approach suitable for real-time use in the laboratory which, crucially, allows independent control of spot size and field of view. Imaging simulations demonstrate the resolution achievable with these spots. We show a three-order-of-magnitude improvement in the efficiency of focusing to achieve the same resolution as previously reported results, or a 26 % increase in resolution for the same efficiency of focusing
Roadmap on Superoscillations
Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding—still being explored—recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments
Roadmap on superoscillations
Superoscillations are band-limited functions with the counterintuitive property that they can vary arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals. Modern studies originated in quantum theory, but there were anticipations in radar and optics. The mathematical understanding—still being explored—recognises that functions are extremely small where they superoscillate; this has implications for information theory. Applications to optical vortices, sub-wavelength microscopy and related areas of nanoscience are now moving from the theoretical and the demonstrative to the practical. This Roadmap surveys all these areas, providing background, current research, and anticipating future developments
Realising superoscillations: A review of mathematical tools and their application
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