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arXiv:1512.00109 · Journal of Physics A Mathematical and Theoretical

Scaling properties of superoscillations and the extension to periodic signals

Eugene Tang, Lovneesh Garg, Achim Kempf

Published Jul 8, 2016• 11 citations

Abstract

Abstract 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 to signal processing. However, the more pronounced the desired superoscillatory behavior is to be, the more difficult it becomes to produce, or even only calculate, such highly fine-tuned wave forms in practice. Here, we investigate how this sensitivity to preparation errors scales for a method for constructing superoscillatory functions which is optimal in the sense that it minimizes the energetic expense. We thereby also arrive at very accurate approximations of functions which are so highly superoscillatory that they cannot be calculated numerically. We then investigate to what extent the scaling and sensitivity results for superoscillatory functions on the real line extend to the experimentally important case of superoscillatory functions that are periodic.

DOI: 10.1088/1751-8113/49/33/335202

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It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.

It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible - a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.

Superoscillations, band-limited signals that locally oscillate faster than their highest Fourier component, have recently enabled superspectroscopy and super-sensing. Previous experiments [Phys. Rev. Lett. 131, 153803 (2023), https://doi.org/10.1103/PhysRevLett.131.153803 and APL Photonics 10, 086107 (2025), https://doi.org/10.1063/5.0271556] relied on combining four quasi-sinusoidal harmonics to generate temporal superoscillations. Here, we show that just two harmonics are sufficient to produce superoscillations of comparable quality, as quantified by the local frequency. By reanalyzing prior experimental data with two harmonics (0.5 and 0.6 THz) and performing new experiments with 0.9 and 1 THz harmonics, we demonstrate a twofold enhancement in local frequency relative to the highest frequency component -- matching the enhancement previously achieved with four harmonics. We provide a simple analytical explanation for the bichromatic superoscillations based on near-complete destructive interference, and identify an optimal frequency separation of approximately 10% that balances the superoscillatory frequency enhancement against signal amplitude. This simplification substantially lowers the experimental barrier to implementing superoscillation-based technology.

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