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Stealth and Equiluminous Materials for Scattering Cancellation and Wave Diffusion

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Stealth and Equiluminous Materials for Scattering Cancellation and Wave Diffusion Stealth and equiluminous materials for scattering cancellation and wave diffusion S. Kuznetsova,1, ∗ J.-P. Groby,2 L.M. Garcia-Raffi,3 and V. Romero-García2, y 1Laboratoire d’Acoustique de l’Université du Mans (LAUM), UMR CNRS 6613, Institut d’Acoustique - Graduate School (IA-GS), CNRS, Le Mans Université, France 2Laboratoire d’Acoustique de l’Université du Mans, LAUM - UMR 6613 CNRS, Le Mans Université, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France 3Instituto de Matemática Pura y Applicada (IUMPA), Universitat Politècnica de València, Camino de vera s/n, 46022, Valencia, Spain. (Dated: September 3, 2020) We report a procedure to design 2-dimensional acoustic structures with prescribed scattering properties. The structures are designed from targeted properties in the reciprocal space so that their structure factors, i.e., their scattering patterns under the Born approximation, exactly follow the desired scattering properties for a set of wavelengths. The structures are made of a distribution of rigid circular cross-sectional cylinders embedded in air. We demonstrate the efficiency of the procedure by designing 2-dimensional stealth acoustic materials with broadband backscattering suppression independent of the angle of incidence and equiluminous acoustic materials exhibiting broadband scattering of equal intensity also independent of the angle of incidence. The scattering intensities are described in terms of both single and multiple scattering formalisms, showing excellent agreement with each other, thus validating the scattering properties of each material. I. INTRODUCTION stealth acoustic materials have been numerically and experimentally designed to provide stealthiness on 24 Scattering of waves by a many-body system is an in- demand robust to losses . A subclass of stealth mate- terdisciplinary topic of interest in several branches of sci- rials is given by the stealth hyperunifom materials for which transparency appears in a subset of wave vectors ence and technology ranging from statistical mechanics 10–14 or condensed matter to wave physics. When such a sys- around the origin . The relevance of hyperuniformity tem is excited by an incident wave, the incoming energy appeared in condensed matter physics when classical is both scattered and absorbed by the obstacle. This re- systems of particles interacting with certain soft long- sults in a scattering pattern that is highly dependent on ranged pair potentials could counterintuitively freeze the geometry and size of the scatterer distribution as well into hyperuniform states. In other words, these systems as on the frequency-dependent properties of the material were counter to the common expectation that liquids of the constituent scatterers. The manipulation of wave freeze into crystal structures with high symmetry. An scattering has long been a topic of discussion in various increasing interest was focused on stealth hyperuniform classical areas of physics including electromagnetism1, materials, or simply on hyperuniform materials, as they photonics2 and acoustics3, but in recent decades signifi- have been used to design networks with complete band cant attention has been paid to artificial structured me- gaps comparable in size to those of a photonic/phononic dia to control waves. Photonic4–6 or phononic7–9 crys- crystal, while at the same time maintain statistical tals, hyperuniform and stealth materials10–14 as well as isotropy, enabling waveguide geometries not possible 15–18 with photonic/phononic crystals as well as high-density metamaterials are just a few examples of many-body 12,25–29 systems to control the scattering of the incident wave. disordered transparent materials. Another 4–6 important class of disordered many-body systems are Ordered structures, such as photonic and 12 7–9,19 equiluminous materials , which scatter waves uniformly phononic crystals, exhibit multiple overlapping in all directions. Such omnidirectional diffusion could Bragg diffraction peaks and thus peculiar dispersion play an important role in improving room acoustics by relations that can serve as efficient tools for the control avoiding unwanted reflections3,30,31. of wave scattering. Metamaterials are complex struc- arXiv:2009.01068v1 [physics.app-ph] 2 Sep 2020 tures that can be tuned and reconfigured to control the Materials with targeted scattering properties are usu- scattering of the incident wave through the resonance ally designed by inverse methods, i.e., their structure of their constituent building blocks9,20,21. Another way parameters are extracted from the scattering data. Al- of manipulating wave scattering is offered by disordered though this approach relies on an ill-posed problem32,33, structures, in which the phase transition between the various material design tools based on targeting the scat- wave diffusion and localization regimes occurs due to tering properties of the structure have been implemented the interference of the waves scattered in the media22,23. in both wave physics and condensed matter. Inverse Among the disordered systems, stealth materials are approach34–36 consists in optimizing the inter-particle in- characterized by the stealthiness, i.e. the suppression of teractions (thus minimizing some energetic characteris- the single scattering of the incident radiation for a given tics) leading to self-assembling from a simpler condition. subset of wave vectors11,12. Recently, one dimensional Optimization methods operating in direct space rely 2 on zero-temperature and near-melting temperature tech- II. SCATTERING IN MANY-BODY SYSTEMS: nique to obtain lattice ground state configurations34,37–41 STRUCTURE FACTOR AND MULTIPLE and collective-coordinates technique for soft matter and SCATTERING THEORY disordered ground states42,43. Usual numerical meth- ods include black-box optimization benchmarking44, We are interested in the scattering of acoustic waves by probabilistic45,46 and genetic algorithms47,48 to name a structures made of a distribution of N rigid cylindrical few. A flat acoustic lens49,50 focusing sound at a pre- scatterers with circular cross-section of identical radius 51 defined point, a photonic-crystal-based structure per- Ri = R0 and located at positions ~ri with i = 1; :::; N. forming requested optical tasks, or a sonic demultiplex- These N scatterers are embedded in a square area Ω of ing device52 spatially separating several wavelengths were the direct space of side L. We assume weak scattering, designed using a genetic algorithm in conjunction with i.e., the amplitude of the scattered field is small compared the multiple scattering theory (MST)53,54 to optimize a to that of the incident field. Under this condition, we as- cluster of scatterers. A 2-dimensional low loss acous- sume that the Born approximation is satisfied. Strictly tic cloak for air-born sound has also been designed by speaking, Born approximation corresponds to the case means of genetic algorithm and simulated annealing55. in which the incident field to the i-th cylinder is only Nonlinear conjugate gradient algorithm has been used composed of the incident wave, i.e. no scattererd waves to optimize a graded porous medium composed of a pe- by the other scatterers impinges the i-th scatterer. For riodic arrangement of ordered unit cells to provide the the geometries considered in this work, the weak scatter- optimal acoustic reflection and transmission56. Recently ing approximation is thus valid for low filling fractions scattering suppression of electromagnetic waves for pre- and when the scatterer radii are small compared to the scribed wavelengths and directions has been achieved by wavelength (see AppendixA for more details). pre-assigning the scattering properties in the reciprocal This discrete system can be characterized by the fol- space and using generalized Hilbert transform57. lowing scalar function defined in the spatial (direct) do- main Ω as N X ρ(~r) = f(~r) ∗ δ(~r − ~ri); (1) i=1 where ∗ is the convolution operator, δ(~x) is the Dirac’s In this work, we design disordered 2-dimensional (2D) delta and f(~r) is the transparency of the scatterer, de- acoustic structures consisting of rigid circular cross- fined without loss of generality as sectional cylinders embedded in air. These structures are designed to present prescribed scattering properties 0 if j~rj > R ; f(~r) = 0 (2) when excited by a plane wave. We target the information 1 if j~rj ≤ R0: on the scattering pattern in the reciprocal space and use an optimization procedure, which optimizes the positions Under these assumptions, the amplitude of the scattered of scatterers to ensure the targeted scattering properties. wave is proportional to the spatial Fourier transform of A weak scattering approach is followed, which allows us ρ(~r), FT (G~ ), where G~ is a vector of the reciprocal space. to characterize the system by its structure factor. This This follows from the well known theory in optics that the factor turns out to be proportional to the scattered in- diffraction pattern of a structure is equal to the product tensity and only depends on the scatterer positions when of the diffraction pattern of the base element and that of they are identical. Therefore, the optimization proce- the array58. Through this work we assume a time har- dure finds the distribution of scatterers producing the monic dependence of the type e−{!t where ! the angular targeted structure factor values by fixing the scattering frequency. With this, we simply end with properties in the reciprocal space and as a consequence N X ~ the desired scattering properties. The polar scattering FT (G~ ) = f(G~ ) × e−{G~ri : (3) pattern of the optimized distribution of scatterers is first i=1 evaluated from the representation of the structure factor in the reciprocal space by using the von Laue formulation. Therefore, the scattered intensity is given by This scattering pattern is then evaluated independently N N X X ~ by the MST, which is a self-consistent method account- I(G~ ) = jf(G~ )j2 × e−{G(~ri−~rj ); (4) ing for all orders of scattering. Comparison of the results i=1 j=1 of the two methods allows us to validate the approxima- tion of weak scattering and consequently the results.
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