Four aspects of superoscillations

@article{Kempf2018FourAO,
  title={Four aspects of superoscillations},
  author={Achim Kempf},
  journal={Quantum Studies: Mathematics and Foundations},
  year={2018},
  volume={5},
  pages={477-484}
}
  • Achim Kempf
  • Published 14 February 2018
  • Mathematics, Physics
  • Quantum Studies: Mathematics and Foundations
A function f is said to possess superoscillations if, in a finite region, f oscillates faster than the shortest wavelength that occurs in the Fourier transform of f. I will discuss four aspects of superoscillations: (1) Superoscillations can be generated efficiently and stably through multiplication. (2) There is a win–win situation in the sense that even in circumstances where superoscillations cannot be used for superresolution, they can be useful for what may be called superabsorption, an… Expand
Evolution of Superoscillations in the Dirac Field
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The study of the evolution of superoscillations as initial datum of fieldExpand
Evolution of Superoscillations in the Klein-Gordon Field
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. There is nowadays a large literature on the evolution of superoscillations underExpand
Holomorphic functions, relativistic sum, Blaschke products and superoscillations
Superoscillating functions are band-limited functions that can oscillate faster than their fastest Fourier component. The notion of superoscillation is a particular case of that one of supershift. InExpand
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. ModernExpand
A unified approach to Schr\"odinger evolution of superoscillations and supershifts
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time dependent Schrödinger equation is an important andExpand
Evolution by Schrödinger equation of Aharonov–Berry superoscillations in centrifugal potential
In recent years, we have investigated the evolution of superoscillations under Schrödinger equation with non-singular potentials. In all those cases, we have shown that superoscillations persist inExpand
Schrödinger evolution of superoscillations with $δ$- and $δ'$-potentials
In this paper, we study the time persistence of superoscillations as the initial data of the time-dependent Schrödinger equation with δand δ′-potentials. It is shown that the sequence of solutionsExpand
How superoscillating tunneling waves can overcome the step potential
Abstract We consider the Cauchy problem for the Schrodinger equation with step potential with jump V 0 at the origin and whose initial datum is a superoscillatory function F n that is traveling fromExpand
Aharonov–Berry superoscillations in the radial harmonic oscillator potential
In this paper, we study the evolutions of Aharonov–Berry superoscillations under the radial harmonic oscillator potential. For this model, we know the Green function and, taking advantage of it, weExpand
Synthesis of Super-Oscillatory Point-Spread Functions with Taylor-Like Tapered Sidelobes for Advanced Optical Super-Resolution Imaging
Recently, the super-oscillation phenomenon has attracted attention because of its ability to super-resolve unlabelled objects in the far-field. Previous synthesis of super-oscillatory point-spreadExpand
...
1
2
...

References

SHOWING 1-10 OF 59 REFERENCES
Scaling Properties of Superoscillations and the Extension to Periodic Signals
Superoscillatory wave forms, i.e., waves that locally oscillate faster than their highest Fourier component, possess unusual properties that make them of great interest from quantum mechanics toExpand
Suppression of superoscillations by noise
Bandlimited functions can vary faster than their highest Fourier component. Such 'superoscillations' result from near-perfect destructive interference among the Fourier components and correspond toExpand
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 practicalExpand
Evolution of quantum superoscillations, and optical superresolution without evanescent waves
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.Expand
Superoscillations with Optimal Numerical Stability
TLDR
Time translation σ is introduced as a design parameter and an explicit closed formula is given for the condition number of the matrix of the problem, as a function of σ, which enables the best possible condition number to be determined. Expand
Driving Quantum Systems with Superoscillations
Superoscillations, i.e., the phenomenon that a bandlimited function can temporary oscillate faster than its highest Fourier component, are being much discussed for their potential forExpand
Superoscillations: Faster Than the Nyquist Rate
TLDR
This paper investigates the required dynamical range and energy (squared L2 norm) as a function of the superoscillating signals' frequency, number, and maximum derivative, and shows that the required energy grows exponentially with the number ofsuperoscillations, and polynomially with the reciprocal of the bandwidth. Expand
The mathematics 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 thatExpand
Unusual properties of superoscillating particles
It has been found that differentiable functions can locally oscillate on length scales that are much smaller than the smallest wavelength contained in their Fourier spectrum—a phenomenon calledExpand
Construction of Aharonov–Berry's superoscillations
A simple method is described for constructing functions that superoscillate at an arbitrarily chosen wavelength scale. Our method is based on the technique of oversampled signal reconstruction. ThisExpand
...
1
2
3
4
5
...