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 = ∞).
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 = ∞).
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.
Abstract Bohmian mechanics is a deterministic account of quantum phenomena in terms of particle trajectories. Guided by a quantum force, the gradient of a quantum potential, trajectories bend to account for quantum interference. The quantum potential is expressible in terms of weak values, complex numbers that characterize quantum systems for which both the initial and final states are predetermined. In Berry and Shukla (2010 J. Phys. A: Math. Theor. 43 354024), the typical probabilities of weak values were calculated. In this paper, expressing the quantum potential in terms of weak values, we find the typical probability distribution of the quantum potential for randomly selected states. The latter allows the calculation of the typical probability of semiclassical superoscillation, a generalization of superoscillations (functions that oscillate faster than their fastest Fourier component) to functions that are not band-limited.
Superoscillations, namely regions where a band-limited function f (x) varies faster than the fastest of its Fourier components k, generate the illusion that the Fourier content is ?supershifted? so as to lie outside the spectrum of the function. The relation between supershifts and superoscillations, central to the quantum weak measurements scheme, is explored in terms of two different representations of the local Fourier transform in the ?phase space? (x, k). The Wigner function W(x, k), regarded as a function of k for fixed x, inherits the band-limited property of f (x). Neverthless, its local k average can lie outside the spectrum because W, although real, posesses negative values. The local Wigner average of k equals the local wavenumber at x (local weak value of momentum), defined as the phase variation k loc(x)?=?? x arg f (x). By contrast, the Husimi function H(x, k), i.e. the windowed Fourier transform with window width L, corresponding to squeezing of the coherent state associated with (x, k) (and representing the pointer wavefunction after a weak measurement), is positive-definite. But it is not band-limited, and the local Husimi average of k equals k loc if L is small enough. These properties are illustrated numerically with two superoscillatory functions.
Abstract Superoscillating functions, i.e. functions that locally oscillate at a rate faster than their highest Fourier component, are of interest for applications from fundamental physics to engineering. Here, we develop a new method which allows one to construct superoscillations of arbitrarily high frequency and arbitrarily long duration in a computationally efficient way. We also present a method for constructing non-singular Schrödinger potentials whose ground state is a superoscillating wave function.
The concept of superoscillations is extended beyond bandlimited functions, to include monochromatic waves in space-varying media, such as wavefunctions representing quantum particles in non-constant potentials. 'Semiclassical superoscillations' are defined as places where the phase gradient (local wavevector, or weak momentum) exceeds the classically allowed momentum. The phenomenon is essentially quantum (or, in optics, essentially beyond geometrical optics), and can arise in three different ways: from interference, evanescence (e.g. barrier penetration), and WKB corrections. Illustrations are given, including superpositions of few and many states in a linear potential, angular-momentum circular harmonic-oscillator states, eigenstates of degenerate harmonic-oscillator modes, and above-barrier scattering.
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.