Time-Driven Superoscillations with Negative Refraction

M. Dubois, E. Bossy, S. Enoch, S. Guenneau, G. Lerosey, and P. Sebbah

Phys. Rev. Lett. 114, 013902 – Published 6 January 2015

Abstract

The flat-lens concept based on negative refraction proposed by Veselago in 1968 has been mostly investigated in the monochromatic regime. It was recently recognized that time development of the superlensing effect discovered in 2000 by Pendry is yet to be assessed and may spring surprises: Time-dependent illumination could improve the spatial resolution of the focusing. We investigate dynamics of flexural wave focusing by a 45°-tilted square lattice of circular holes drilled in a duralumin plate. Time-resolved experiments reveal that the focused image shrinks with time below the diffraction limit, with a lateral resolution increasing from $0.8 λ$ to $0.35 λ$ , whereas focusing under harmonic excitation remains diffraction limited. Modal analysis reveals the role in pulse reconstruction of radiating lens resonances, which repeatedly self-synchronize at the focal spot to shape a superoscillating field.

Received 25 February 2014

DOI:https://doi.org/10.1103/PhysRevLett.114.013902

Authors & Affiliations

M. Dubois¹, E. Bossy¹, S. Enoch², S. Guenneau², G. Lerosey¹, and P. Sebbah^1,3,*

¹Institut Langevin, ESPCI ParisTech CNRS UMR7587, 1 rue Jussieu, 75238 Paris cedex 05, France
²Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille, France
³Department of Physics, The Jack and Pearl Resnick Institute for Advanced Technology, Bar-Ilan University, Ramat-Gan 5290002, Israel

^*patrick.sebbah@espci.fr

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Vol. 114, Iss. 1 — 9 January 2015

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Images

Figure 1
Snapshots of the field intensity of vertical-displacement velocity measured at the surface of the plate at $t = 800 μ s$ (a), (c) and $t = 1150 μ s$ (b), (d). (a), (b) Experiment. (c), (d) Numerical simulations using 3D elastic FDTD with absorbing boundary conditions at the plate’s edges. The color scale outside the crystal is normalized at each time step to the maximum intensity on the right side of the lens. The color scale has been reduced 6 times inside the crystal to avoid image saturation. Blue dot: Source position.
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Figure 2
Time dependence of the lateral resolution in units of $λ$ ( $λ = 47.1 mm$ at 9 kHz). Open circles: Experimental data; Dashed red line: Numerical simulations. Rayleigh diffraction limit at $λ = 47.1 mm$ is marked by the horizontal line at $0.5 λ$ . Actually, the Rayleigh limit is ill-defined as the pulse covers a range of wavelengths. The double vertical arrow in Fig. 2 materializes the actual dependence on wavelength of the Rayleigh limit, which extends from $0.45 λ$ to $0.57 λ$ , for a Gaussian pulse at 9 kHz with standard deviation $σ = 2.1 kHz$ . The spot size goes well below the smallest Rayleigh limit. Vertical arrows: time steps shown in Fig 1. Lower inset: picture of the plate. Upper inset: larger time scale (numerical simulations). Solid magenta line: Monochromatic excitation. Dashed red line: Pulse excitation.
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Figure 3
(a) Snapshot of the reciprocal space at time $t_{focus} = 1176 μ s$ . $Γ$ , X, M are the symmetry points of the square lattice and $k_{x}$ and $k_{y}$ the reciprocal array axes. Dashed black circle: 9 kHz-isofrequency isotropic contour in “free space” (plate outside the platonic lens). Solid line: diamond-shaped 9 kHz-isofrequency contour within the platonic crystal. The white rectangle delimits the area of free-space propagative waves. The bright spots correspond to the modes of the flat lens selected spatially and spectrally by the point source. (b) Spatial amplitude distribution of the modes in the flat lens, obtained by inverse Fourier transform of the bright spots.
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Figure 4
Lateral profiles in the focal plane at three different time steps. Blue circles: Experimental data; red solid line: Coherent superposition of the four propagative components radiated by modes $P_{1}$ to $P_{4}$ of Fig. 3. Black dots: Rayleigh diffraction limit (sinc function). Histograms show the contribution of propagative components at the focal spot ( $P_{1}$ to $P_{4}$ from left to right). Red circles: Superoscillating function given by the sum of four weighted cosine functions (see text).
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