Abstract
In this paper a mathematical model is given for the scattering of an incident wave from a surface covered with microscopic small Helmholtz resonators, which are cavities with small openings. More precisely, the surface is built upon a finite number of Helmholtz resonators in a unit cell and that unit cell is repeated periodically. To solve the scattering problem, the mathematical framework elaborated in Ammari et al. (2019) is used. The main result is an approximate formula for the scattered wave in terms of the lengths of the openings. Our framework provides analytic expressions for the scattering wave vector and angle and the phase-shift. It justifies the apparent absorption. Moreover, it shows that at specific lengths for the openings and a specific frequency there is an abrupt shift of the phase of the scattered wave due to the subwavelength resonances of the Helmholtz resonators. A numerically fast implementation is given to identify a region of those specific values of the openings and the frequencies.
Highlights
Surfaces covered by a microscopic structure can display unforeseen physical properties, which can be applied in the everyday life to have advantageous effects like anti-reflection coating and high-efficiency light absorbers, or enhancers
In this paper we have established in Theorem 3.1 that the formula U k(z) − U0k(z) = Is e−i k1z1 ei k2z2 describes the scattered field in the far-field in our geometry, up to a small error, which depends on the lengths of the openings of the Helmholtz resonators
Dε,δk is given through a simple formula consisting of all εi and δk
Summary
Surfaces covered by a microscopic structure can display unforeseen physical properties, which can be applied in the everyday life to have advantageous effects like anti-reflection coating and high-efficiency light absorbers, or enhancers. From the proof of Theorem 3.1, we can extract a formula for a numerical evaluation of the scattered wave in the near-field and there we observe other remarkable outcomes like an enhancement of the amplitude of the scattered field, which is reminiscent of the results in [5] It is worth emphasizing on one hand the connection between our results and those obtained for periodic arrays of narrow slits in the series of papers [14,15,16,17,18,19] and on the other hand, the fact that, as shown in [13,20], when the typical size of the periodic unit strip goes to zero and the openings of the Helmholtz resonators go to zero, the gradient metasurface can be approximated by an effective boundary condition. In the Appendices we provide proofs of some technical results used for the proof of Theorem 3.1
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