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Superoscillations in non-monochromatic naturally occurring waves

Sharif Alkhass

Published May 1, 2018• 0 citations
OpenAlex

Abstract

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.

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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.
A superoscillatory function is one that oscillates faster than its fastest Fourier component - A phenomenon created by subtle interference between Fourier components. In this thesis, the work of Professor Sir Michael Berry [1, 2] and Professor Sandu Popescu on superoscillations in free space [2] has been applied to the case of the quantum harmonic oscillator. Superoscillations in free space are seen to persist in time longer than expected and are also seen to reform periodically after times much greater than their disappearance - for the quantum harmonic oscillator superoscillations reform periodically faster. The time of disappearance, td, depends upon a larger set of variables, this is due to the added complexity of the quantum harmonic oscillator. The evolution of the wavepacket is investigated using an expansion in terms of eigenfunctions and in terms of the propagator using both an exact integration and a saddle point approximation. The creation and disappearance of superoscillations is shown to depend on the behaviour of the saddle points. The frequency of the quantum harmonic oscillator, ! is found to be an extra control parameter which dictates the strength and duration of the superoscillations.
Abstract. 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. 1 2 1.
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.
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations . Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
Abstract For complex scalar waves, a convenient measure of the local oscillations and (‘faster than Fourier’) superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions ψ( r ); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum components, i.e. K 2 r = − ∂ 2 ψ r / ∂ x 2 / ψ r , here called the ‘weak curvature’. Superoscillations correspond to places where K 2 lies outside the interval 0 ⩽ K 2 ⩽ 1. Two illustrations are given. First is an explicit family of real waves in dimension d = 2, with arbitrarily strong superoscillations; this could represent Neumann standing modes in a strip waveguide. Second is an exact calculation of the probability distribution of K 2 for Gaussian random real waves in d dimensions. This decays as 1 / K 2 2 , as a consequence of the codimension 1 of nodes (e.g. nodal lines for d = 2). The superoscillation probability varies from 0.3918 (d = 2) to 0.3041 (d = ∞).
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 ψ. The superoscillations persist for unexpectedly long times, but eventually disappear through the interaction of contributions to ψ 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 ψ 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.