An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces

Antonakakis, Tryfon; Guenneau, Sébastien; Skelton, E. A.

doi:10.1098/rspa.2013.0467

Cited by 13 publications

(27 citation statements)

References 57 publications

(107 reference statements)

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“…The excitation is a line source just outside the leftmost edge of interface and a very clear ZLM is excited that can be identified, on each ribbon, with the eigenstate from the ribbon. One feature that is also evident is the long-scale wave envelope with a wavelength that alters with frequency; this can be described via an effective medium approach [36] as applied to edge states [29].…”

Section: Edge States: Zero Line Modesmentioning

confidence: 99%

Topological beam-splitting in photonic crystals

Makwana¹,

Craster²,

Guenneau³

2019

Opt. Express

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We create a passive wave splitter, created purely by geometry, to engineer three-way beam splitting in electromagnetism in transverse electric and magnetic polarisation. We do so by considering arrangements of Indium Phosphide dielectric pillars in air, in particular we place several inclusions within a cell that is then extended periodically upon a square lattice. Hexagonal lattice structures are more commonly used in topological valleytronics but, as we discuss, three-way splitting is only possible using a square, or rectangular, lattice. To achieve splitting and transport around a sharp bend we use accidental, and not symmetry-induced, Dirac cones. Within each cell pillars are either arranged around a triangle or square; we demonstrate the mechanism of splitting and why it does not occur for one of the cases. The theory is developed and full scattering simulations demonstrate the effectiveness of the proposed designs.

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Section: Edge States: Zero Line Modesmentioning

confidence: 99%

Topological beam-splitting in photonic crystals

Makwana¹,

Craster²,

Guenneau³

2019

Opt. Express

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“…Research on metamaterials (employed to guide and control elastic waves for applications in microstructured devices [1][2][3][4][5][6][7] and earthquake resistant structures [8][9][10][11][12]) has focused a strong research effort to time-harmonic vibrations of periodic beam networks. These networks can be analyzed via Floquet-Bloch analysis for free vibrations of an infinite domain (which can be either 'exact', when performed with a symbolic computation program [13] or approximated, when solved numerically [14]), or using the f.e.…”

Section: Introductionmentioning

confidence: 99%

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Bordiga

Cabras

Bigoni

2019

International Journal of Solids and Structures

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In-plane wave propagation in a periodic rectangular grid beam structure, which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed both with a Floquet-Bloch exact formulation for free oscillations and with a numerical treatment (developed with PML absorbing boundary conditions) for forced vibrations (including Fourier representation and energy flux evaluations), induced by a concentrated force or moment. A complex interplay is observed between axial and flexural vibrations (not found in the common idealization of out-of-plane motion), giving rise to several forms of vibration localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These localizations are triggered by several factors, including rotational inertia and slenderness of the beams and the type of forcing source (concentrated force or moment). Although the considered grid of beams introduces an orthotropy in the mechanical response, a surprising 'isotropization' of the vibration is observed at special frequencies. Moreover, rotational inertia is shown to 'sharpen' degeneracies related to Dirac cones (which become more pronounced when the aspect ratio of the grid is increased), while the slenderness can be tuned to achieve a perfectly flat band in the dispersion diagram. The obtained results can be exploited in the realization of metamaterials designed to control wave propagation. and the solution techniques are well-known, many interesting features still remain to be explored. This exploration is provided in the present article, where an exact Floquet-Bloch analysis is performed and complemented with a numerical treatment of the forced vibrations induced by the application of a concentrated force or moment, including presentation of the Fourier transform and energy flow (treated in [26] for free vibrations). It is shown that (i.) aspect ratio of the grid, (ii.) slenderness and (iii.) rotational inertia of the beams decide the emergence of several forms of highly-localized waveforms, namely, 'channel propagation', 'X-', 'cross-', 'star-' shaped vibration modes. Moreover, these mechanical properties of the grid can be designed to obtain flat bands and degeneracies related to Dirac cones in the dispersion diagram and directional anisotropy or, surprisingly, dynamic 'isotropization', for which waves propagate in a square lattice with the polar symmetry characterizing propagation in an isotropic medium.The presented results open the way to the design of vibrating devices with engineered properties, to achieve control of elastic wave propagation.2 In-plane Floquet-Bloch waves in a rectangular grid of beams An infinite lattice of Rayleigh beams is considered, periodically arranged in a rectangular geometry as shown in Fig. 1a, together with the unit cell, Fig. 1b.

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“…In particular, the novel properties of the metamaterials are usually at frequencies beyond the low frequency, or the long-wavelength limit, that is the validity constraint of the conventional homogenization approaches [31]. There are efforts to extend the conventional homogenization to higher frequencies * Corresponding author: erwinstu@ust.hk † Corresponding author: sheng@ust.hk by a scheme denoted as "two-scale asymptotic" [32,33] in which a separate set of expansions (different from the longwavelength asymptotic) is applied near the standing-wave modes at the edges of the Brillouin zones (BZs) for periodic structures. The effective-medium characteristics in the pass bands can thereby be approached from two directions.…”

Section: Introductionmentioning

confidence: 99%

Homogenization scheme for acoustic metamaterials

Yang

et al. 2014

Phys. Rev. B

116

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We present a homogenization scheme for acoustic metamaterials that is based on reproducing the lowest orders of scattering amplitudes from a finite volume of metamaterials. This approach is noted to differ significantly from that of coherent potential approximation, which is based on adjusting the effective-medium parameters to minimize scatterings in the long-wavelength limit. With the aid of metamaterials' eigenstates, the effective parameters, such as mass density and elastic modulus can be obtained by matching the surface responses of a metamaterial's structural unit cell with a piece of homogenized material. From the Green's theorem applied to the exterior domain problem, matching the surface responses is noted to be the same as reproducing the scattering amplitudes. We verify our scheme by applying it to three different examples: a layered lattice, a two-dimensional hexagonal lattice, and a decorated-membrane system. It is shown that the predicted characteristics and wave fields agree almost exactly with numerical simulations and experiments and the scheme's validity is constrained by the number of dominant surface multipoles instead of the usual long-wavelength assumption. In particular, the validity extends to the full band in one dimension and to regimes near the boundaries of the Brillouin zone in two dimensions.

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An asymptotic theory for waves guided by diffraction gratings or along microstructured surfaces

Cited by 13 publications

References 57 publications

Topological beam-splitting in photonic crystals

Topological beam-splitting in photonic crystals

Free and forced wave propagation in a Rayleigh-beam grid: Flat bands, Dirac cones, and vibration localization vs isotropization

Homogenization scheme for acoustic metamaterials

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