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, † 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 methods 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) domain 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 |~r| > R , f(~r) = 0 (2) when excited by a plane wave. We target the information 1 if |~r| ≤ 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−ıωt where ω 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−ıG~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~ ) = |f(G~ )|2 × e−ıG(~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 approximation of weak scattering and consequently the results. We where f(G~ ) is known as the atomic structure factor and apply the proposed approach to design and describe 2D only depends on the geometry of the scatterer as our scat- stealth and equiluminous materials showing broadband terers are considered rigid. Thus, the scattered intensity back-scattering suppression and broadband equally in- can be simply written as tense scattering respectively, independently of the angle of incidence. I(G~ ) = |f(G~ )|2NS(G~ ), (5) 3

G k G s k k s k 0 0

G G k k s k s k 0 0

FIG. 1. (Color online) Scattering by an array of N scatterers radiated by an incident plane wave characterized by the wavevector a ~k0. (a) and (f) represent a periodic and a random distribution of N = 64 cylinders with R0 = L/100 respectively. (b) and (g) show the representation of the structure factor S(G~ ) for the periodic and the random distribution respectively. (c) and (h) show the representation of the spatial Fourier transform |FT (G~ )| for the periodic and the random distribution respectively. In (b-c) and (g-h) the incident wavevector and the scattered wavevector are related through the Ewald circumference with the vectors of the reciprocal space. (d) and (i) show the polar distribution of the normalized scattered far ﬁeld intensity between θ = [90, 270] degrees for the periodic and the random distributions respectively. Black dotted (red dashed line) [continuous ~ ~ 2 f 2 blue line] represents the results obtained from the S(G) (|FT (G)| )[|Ps (θ, ω)| ]. (e) and (j) represents the scattering cross section of the periodic and random distributions respectively. a The coordinates of these distributions of points are provided in the Supplementary Material

f where Ps (θ, ω), at angle θ, reads as 2 N N N N 1 X X −ıG~ (~r −~r ) 1 X ıG~r~ f 2 X −ık|~r | cos (θ−θ ) X n i ınθ S(G~ ) = e i j = e j , (6) P (θ, ω) = e i ~ri (−i) A e , N N s k n i=1 j=1 j=1 i=1 n (8) is the structure factor. It should be noted here that the structure factor only depends on the position of the scat- with θ~ri , the azimuthal angle of the position vector of the ~ ~ i terers in Ω. Moreover, we notice that S(G) = S(−G) i-th cylinder ~ri and An, the scattering coefficients of the (see AppendixB for more details and additional demon- i-th cylinder calculated by MST. The scattered far-field f 2 strations). The structure factor is extensively used in intensity is thus proportional to I(θ, ω) ∝ |Ps (θ, ω)| . condensed matter or wave physics to describe the scat- The scattering cross section of the scatterer distribution tering of an incident wave by a given structure made of when excited by a plane wave eıkx is also computed by a distribution of scatterers. However, multiple scatter- applying the Optical Theorem (see AppendixA) via ing effects are neglected, although this approach has the benefit of allowing fast predictions. N ! f 4 X −ıkx X n i σ = −2Re(P (0, ω)) = − Re e ~ri (−i) A . The wave scattering by a distribution of scatterers can s k n effectively be more precisely described by the MST53,54. i=1 n The far-field expression of the scattered field provided (9) by the MST (see AppendixA for more details) when the structure is radiated by a plane wave with wave vector In order to illustrate the different results provided by ~k , is given by each method and the interpretation of the scattering in 0 the reciprocal space, the scattering properties of both a r k periodic and a random distribution of N = 64 rigid cylin- pf = P f (θ, ω) eıkr, r → ∞, (7) s s ı2πr ders of radii R0 = L/100 are analyzed (array of 8×8 rigid cylinders in the periodic case). The random distribution ~ where k = |k0| and the far-field scattered amplitude has been generated by choosing random positions of the 4 scatterers and avoiding overlapping between them. The the Bragg scattering is produced and that the results are periodic [random] distribution of scatterers is plotted in very close to those given by both S(G~ ) and |FT (G~ )|. For Fig.1(a) [1(f)]. 59. Figures1(b-c) [1(g-h)] respectively the random scatterer distribution, scattering along more depict S(G~ ) and |FT (G~ )| in the reciprocal space. In directions than in the periodic case is expected, because the periodic case, both S(G~ ) and FT (G~ ) consist of a more vectors are possible in the reciprocal space. Some periodic pattern of sinc-type functions. The maxima ap- directions predicted by both S(G~ ) and |FT (G~ )| are nev- pear due to the periodicity of the square distribution at ertheless missing when comparing the results with the √ √ 2π N 2π N 2 scattered far field as calculated with the MST. This is G~ = n , m with (n, m) ∈ Z . In the ran- L L due to the fact that scatterers are too close to each other dom case, the representations in the reciprocal space are in some area of the distribution for the weak scattering not periodic as shown in Fig.1(g-h). In both periodic approximation to be valid as shown in Fig.1(f). In and random cases, a hot spot in the center that repre- that case, the most realistic modeling is that provided sents the forward scattering is exhibited. The parity of by the MST. However, it should be noted that the main ~ ~ ~ ~ both S(G) = S(−G) and |FT (G)| = |FT (−G)| is clearly directions of scattering are captured by both S(G~ ) and visible. FT (G~ ). To interpret the scattering produced by these distri- To conclude this analysis concerning the scattering by butions, we first discuss how the scattering is directly a periodic and a random pattern of scatterers, Figs.1(e) interpreted in the reciprocal space using the von Laue and1(j) depict the scattering cross section as calculated formulation. Let us consider that the system is excited with the MST for the periodic and the random cases by an incident plane wave the wavevector of which is respectively. The Bragg interference produces a peak ~ k0 = k(~ex, 0), with ~ex the unitary vector along the x of scattering at the Bragg frequency fBragg = Nc/2L, ~ direction and k = |k0|. We choose k = πN/L in this where c is the speed of sound in the host fluid material, particular example to analyze the Bragg scattering in for the periodic distribution while the scattering cross ~ section almost continuously increases with frequency for the periodic case. This wavevector k0 is represented in Figs.1(b-c) [Figs.1(g-h)] for the periodic [random] case the random distribution. pointing one of the points of the reciprocal space. The von Laue formulation of the wave diffraction60 stipulates that the difference between the vector of the scattered III. MATERIAL DESIGN TOOL ~ wave, ks, and that of the incident wave, must be a vector ~ ~ of the reciprocal space, i.e., ks − k0 = G~ , for construc- Depending on the values of the structure factor in tive interference to occur. Assuming elastic scattering, the reciprocal space, different kinds of materials can be ~ ~ designed12. In general, the following constraint can be |ks| ≡ ks = |k0| = k = 2π/λ, only the vectors pointing imposed on the structure factor non zero values of S(G~ ) along the Ewald sphere60 can lead to scattered waves for 3D problems. This sphere of S(G~ ) = S for K ≤ |G~ | ≤ K , (10) ~ 0 1 2 radius k0 is centered at the origin of k0 in the reciprocal space. More precisely, all the possible scattered waves are where K1 and K2 are respectively the lower and the up- given by the Ewald sphere. The scattered wavevectors, per limits of an area of the reciprocal space for which ~ ks, are then given by the vector connecting the center of S(G~ ) has a constant value S0. Stealth materials present the sphere and the points along this sphere having a non zero structure factor, S0 = 0, for a general set of re- null value of S(G~ ). The scattering is finally activated ciprocal vectors. Hyperuniform materials12–14,25–28,61,62 ~ are specific type of stealth materials for which K = 0, along the direction given by these vectors ks. This dis- 1 cussion is valid for any dimension, the Ewald sphere is i.e., infinite-wavelength density fluctuations vanish up to ~ K2 = K. d-dimensional hyperuniform materials are in a circumference in 2D of radius k0 centred in k0, and is 1 KL d given by the limits of a segment of length 2k0 centred in addition characterized by χ = πd−1 2N , which repre- ~ sents the relative fraction of constrained degrees of free- k0 in 1D. dom for a fixed reciprocal-space exclusion-sphere of ra- ~ Following this procedure, the values of S(G) and dius K. Finally, equiluminous materials present a struc- ~ |FT (G)| along the Ewald circumference between ture factor that is other than zero, i.e., S0 6= 0 for speci- θ = [90, 270] degrees can be evaluated and, this po- fied wavevectors in the reciprocal space. lar distribution provides the back scattering produced In this section, we present a methodology to design by systems. Figure1(d) [Figure1(i)] shows the polar structured materials with prescribed scattering features. pattern of the scattered field by the periodic [random] In contrast to real-space methods, the desired scattering distribution. Both S(G~ ) and |FT (G~ )| present a strong characteristics are introduced directly in the reciprocal back scattering around 180o as expected from the Bragg space via the structure factor and an optimization proce- scattering. We then compare the results with the scat- dure is used to find, the scatter distributions that gives tered far field as calculated by the MST, i.e., when ac- rise to the targeted value of structure factor for a set of counting for all the scattering orders. We conclude that wavelengths. The cost function that is minimized during 5

µ G k s k 0

FIG. 2. (Color online) Structure factor and scattering produced by a Hyperuniform material made of N = 100 cylinders with a R0 = L/100. (a) represents the point distribution for the Hyperuniform material (with χ = 0.196). (b) shows the structure factor. (c) is a zoom the region of interest in (b). We plot the Ewald circumference corresponding to an incident wavevector ~k0. (d) represents the polar plot of the normalized scattered intensity calculated from the structure factor for the Hyperuniform √ 2π N (at k0 = ks = 0.35 L [white circumference in (c)] between θ = [90, 270] degrees in order to analyze the back scattering components. a The coordinates of this distribution of points are provided in the Supplementary Material

the optimization procedure is a function of the structure frequencies. This is√ the case of the Ewald circumfer- factor itself S(G~ ) [Eq. (6)]. For a given limit of wavevec- ~ 2π N ence for k0 = (0.15 L ~ex, 0) [white dashed line in Fig. tors, the structure factor must have a target value S0 for 2(c)]. In the opposite, strong back scattering occurs for √ all the wavevectors in domain Λ. The objective function ~ 2π N k0 = (0.35 L ~ex, 0). Figure2(d) shows the correspond- reads as ing polar diagrams of the normalized scattered intensities X φ(~r , . . . , ~r ) = S(G~ ) − S , (11) between θ = [90, 270] degrees as calculated with Eq. (5). 1 N 0 The intensity is normalized with respect to its maximum ~ √ G∈Λ ~ 2π N value for k0 = (0.35 ~ex, 0). Therefore, back scatter- L √ and is subjected to the following constrains to avoid over- ~ 2π N ing is clearly exhibited when k0 = (0.35 ~ex, 0) while lapping of scatterers of radius R , L √ 0 ~ 2π N no back scattering occurs when k0 = (0.15 L ~ex, 0). |~ri − ~rj| ≥ 2R0 ∀i 6= j. (12) Note that the asymmetry of the polar diagram in Fig. ~ 2(d) around the direction θ = 180 is a direct consequence We note that Eq. (11) is already norm L2 as S(G) is of the disorder of the scatterer distribution. already norm L2. The optimization algorithm looks for distribution of scatterers ~ri that minimizes the Eq. (11). Stealth, Hyperuniform, and Equiluminous materials or IV. RESULTS more generally any kind of materials with targeted properties in the reciprocal space can be designed. As an example, Figs.2(a-c) show a hyperuniform ma- In this section we show the results for diﬀerent mate- terial made of a distributions of N = 100 rigid cylin- rials with targeted scattering properties using the structure factor. We compare the results obtained from the ders with R0 = L/100 with χ = 0.196 designed by the present procedure.63 Figure2(b) represents the struc- structure factor, the spatial Fourier transform, and the ture factor of the corresponding scatterer distribution. MST. The scattering suppression area with S(G~ ) = 0 between √ K = 0 and K = 0.5 2π N is clearly visible. Fol- 1 2 L A. Broadband back-scattering suppression lowing the discussion on the interpretation of the scat- independent of the angle of incidence tering in the reciprocal space given in SectionII, Fig. 2(c) shows the Ewald circumferences for two diﬀerent √ We start by analyzing the properties of a stealth ma- ~ 2π N incident wavevectors k0 = (0.35 ~ex, 0) (white con- √ L terial presenting a scattering suppression area between ~ 2π N √ √ tinuous line) and k0 = (0.15 ~ex, 0) (white dashed 2π N 2π N L K1 = 0.75 L and K2 = 1.1 L . A distribution line). These two situations correspond respectively to of N = 64 identical cylindrical rigid scatterers with cases where the Ewald circumference is either partially or R0 = L/100 is considered. Figure3(a) shows the scat- completely within the scattering suppression area where terer distribution that minimizes Eq. (11) resulting from S(G~ ) = 0. In both cases, the points along the cir- the material design tool. Figure3(b) depicts the cor- cumference falling in the region S(G~ ) = 0 do not pro- responding structure factor showing the scattering sup- duce scattering. For this reason, hyperuniform materi- pression area between the imposed limits. Note that als suppress the scattering of incident radiation at low this particular shape of scattering suppression area im- 6

k s

k 0

k k k k | 0|=0.43 | 0|=0.43 | 0|=0.43 | 0|=0.43

k k k k | 0|=0.65 | 0|=0.65 | 0|=0.65 | 0|=0.65

FIG. 3. (Color online) Scattering properties of an stealth material for broadband and omnidirectional back scattering suppression. (a) Stealth distribution.a (b) Structure factor S(G~ ) of the stealth material. Ewald circumference distribution for two different incident directions of the same wave. Whitish area shows the√ scattering suppression√ for these two examples. 2π N 2π N Red circles in (b) represent the limits of scattering suppression, K1 = 0.75 L and K2 = 1.1 L . Green (yellow) arrow f represents the incident (scattered) plane wave. (c) Polar plot of the scattered field, |ps (θ, ω)| produced by this Stealth material obtained from the structure factor (dotted black line), the Fourier transform (dashed red line) and the MST (continuous blue √ 2π N f line) for k = 0.43 . (d) θ − |k| map of the scattered pressure field |ps (θ, ω)| obtained from MST for an incident wavevector √ L √ ~ 2π N ~ 2π N k0 = (0.43 L ~ex, 0). (e-h) Scattered pressure field |ps| for an incident wave with |k0| = 0.43 L (inside the scattering suppression region) along 0o, 270o, 180o, 90o of incidence respectively. (i-l) Normalized scattered pressure field |p | for an incident √ s ~ 2π N o o o o wave with |k0| = 0.65 L (outside the scattering suppression region) along 0 , 270 , 180 , 90 of incidence respectively. a The coordinates of this distribution of points are provided in the Supplementary Material

√ ~ 2π N plies that the back scattering is completely suppressed line) for k0 = (0.43 L , 0). The back scattering is al- for any frequency and for any incident wavevector in most suppressed as evidenced by the polar plot. Figure f this area. In Fig.3(b), two different Ewald circumfer- 3(d) represents the θ−|k| map of |ps (θ, ω)| obtained from ences are shown for two different incident directions at the MST. The scattering suppression is clearly in good the same frequency. The whitish areas represent those agreement with the results presented Fig.3(b). This of the suppressed scattered wavevectors at this partic- evidences the broadband back scattering suppression in- ular frequency. It should be noted here that fixing the dependent of the incident angle. frequency of an incident wave having scattered wavevec- Figures3(e-h) finally show the scattered pressure field √ tors in the suppression zone, the suppressed scattering 2π N |ps| for an incident plane wave with k0 = 0.43 L remains the same for any incident direction. ~ ~ for four different incident directions, k0 = (k0, 0), k0 = ~ ~ Figure3(c) shows the polar plot of the scattered field (0, k0), k0 = (−k0, 0), and k0 = (0, −k0) respectively f |ps (θ, ω)| by this stealth material as calculated with the (white arrow). The back scattering components are structure factor (dotted black line), the Fourier trans- strongly reduced for each angle of incidence. form (dashed red line) and the MST (continuous blue As predicted with the Ewald circumference in the 7

k k k k | 0|=0.55 | 0|=0.55 | 0|=0.55 | 0|=0.55

FIG. 4. (Color online) Scattering properties of an equiluminus material for broadband and omnidirectional diffusion. (a) Equiluminus distribution.a (b) Structure factor S(G~ ) of the Equiluminus material. White circles in (b) represent the limits √ √ 2π N 2π N of equally intense scattering area, K1 = 0.8 L and K2 = 1.2 L (with S0 = 1). (c) Polar plot of the scattered field, |pf (θ, ω)| produced by this Equiluminus material obtained from the structure factor (dotted black line), the Fourier transform s √ 2π N ( (dashed red line) and the MST (continuous blue line) for k = 0.55 . (d) θ − |k| map of the scattered pressure field |psθ, ω)| √ L 2π N obtained from MST for an incident wavevector ~k0 = (0.55 ~ex, 0). (e-h) Scattered pressure field |ps| for an incident wave √ L ~ 2π N o o o o with |k0| = 0.55 L (inside the equally intense scattering region) along 0 , 270 , 180 , 90 of incidence respectively. a The coordinates of this distribution of points are provided in the Supplementary Material

structure factor map, the forward component is the most minimizing Eq. (11) with S0 = 1. Figure4(b) shows the important. Although the values of the structure factor, corresponding structure factor showing the equally in- i.e., the scattered intensity, inside the suppression area tense scattering area between the imposed limits [white are not exactly zero, the independence of the back scat- circles in Fig.4(a)]. Similarly to the stealth material, tering with respect to the incidence angle is still remark- this particular shape of equally intense scattering area able. In order to prove the strong effect of this back scat- implies that the back scattering is equally distributed for tering suppression, we show the scattered pressure distri- any frequency and any incident direction (see Fig.3(b) bution for a frequency at which the scattered wavevectors for the Ewald circumference representation) in this area. fall outside the scattering suppression region. Figures3(i- ~ √ Nevertheless, the values of S(G) inside the equally in- 2π N tense scattering are not completely homogeneous due to l) show the scattered pressure field for k0 = 0.65 L at the same incident directions. Both the back scattering the discrete character of the proposed design contrary and the forward scattering are of equal importance. those of the stealth material. This finds translation in a quasi-equally intense scattering pattern. The values of S(G~ ) are however clearly smother inside the target area B. Broadband equally intense scattering than outside. independently of the angle of incidence Figure4(c) depicts the polar plot of the scattered field f |ps (θ, ω)| by this equiluminus material as calculated with Contrary to the stealth materials, we design an equilu- the structure factor (dotted black line), the Fourier transform (dashed red line) and the MST (continuous blue minous materials thanks to the proposed material design √ tool that produce broadband equally intense scattering 2π N line) for k = 0.55 L . Contrary to the stealth ma- independent of the angle of incidence. In this section, we terial, the back scattering is almost evenly distributed discuss an equiluminous material√ of equally intense scat-√ along the angles and quasi-equally intense as evidenced 2π N 2π N tering area between K1 = 0.8 L and K2 = 1.2 L . by the polar plot. Figure4(d) represents the θ − |k| map f For this particular case, and without loss of generality, we of |ps (θ, ω)| obtained from the MST. The quasi-equally consider a distribution of N = 100 rigid scatterers with intense scattering is in good agreement with the results R0 = L/100. Figure4(a) shows the scatterer distribution plotted in Fig.4(b). This evidences the broadband back 8

ıkx scattering behaviour of the structure independent of the of the form p0(~r) = e with a temporal dependence of incident angle. the type e−ıωt. The scattered wave by the i-cylinder can Figures4(e-h) show the scattered pressure field |p | be written as √ s 2π N X i ınθ for an incident wave with k0 = 0.55 for four dif- (~r−~ri) L ps(~r, ~ri) = AnHn(k|~r − ~ri|)e , (A1) ~ ~ ferent incident directions, k0 = (k0, 0), k0 = (0, k0), n ~ ~ k0 = (−k0, 0), and k0 = (0, −k0) respectively. The where Hn is the n-th order Hankel function of first type. back scattering components are angularly distributed i The total field incident to i-th cylinder pin(~r) is a su- with quasi-equal intensity for each angle of incidence. perposition of the direct contribution from the incident Although the values of the structure factor, i.e., the scat- wave p0(~r) and the scattererd waves from all the other tered intensity, inside the equally intense scattering area scatterers are not exactly constant, the quasi-equal intense back N X scattering independent of the angle of incidence is still pi (~r) = p (~r) + p (~r, ~r ). (A2) remarkable. in 0 s j j=1,j6=i This incident wave on the i-th cylinder can be expressed V. CONCLUSIONS as follows

i X i ınθ~r−~r pin(~r) = BnJn(k|~r − ~ri|)e i , (A3) Heterogeneous materials formed by a set of scatterers n embedded in a host material with tailored properties are where Jn is the n-th order Bessel function of first type. a useful tool for the control and manipulation of acous- We now express the scattered field by the i-th cylinder tic, electromagnetic and matter waves. In this work, we in the vicinity of the j-th cylinder. To do so, we use the present a methodology based on prescribing the scatter- Graff’s theorem: ing properties of the system in the reciprocal space, i.e., X j,i ınθ~r−~r prescribing its structure factor, to later obtain the spacial ps(~r, ~rj) = Cn Jn(k|~r − ~ri|)e i , (A4) distribution of scatterers with the corresponding scatter- n j ing properties. The developed methodology was applied ∀|~r − ~ri| ∈ [R , |~rj − ~ri| − Ri[, (A5) to construct stealth and equiluminous materials. The with scattered intensity was first obtained from the structure j,i X j ı(l−n)θ~ri−~rj factor based on their proportionality in the weak scat- Cn = Al Hl−n(k|~ri − ~rj|)e . (A6) tering approximation. The results were validated using l the multiple scattering approach that accounts for all The incident plane wave is then represented in the i-th the scattering orders. The scattered intensity patterns cylinder coordinate system, via obtained by these two approaches are in excellent agree- ıkx ıkxj ık|~r− ~rj | cos (θ~r−~r ) ment having similar angular distributions. First we have p0(~r) = e = e e j . (A7) designed a stealth system that exhibits broadband back- scattering suppression independently of the incidence di- At this stage, we use the Jacobi-Anger expansion to ex- ık|~r− ~rj | cos (θ~r−~r ) rections, having zero structure factor in the given fre- pand the term e j upon Bessel functions: quency range and as a consequence, a close to zero scat- ık|~r− ~rj | cos (θ ) X n ıθ e ~r−~rj = ı J (k|~r − ~r |)e ~r−~rj . (A8) tered intensity. Second, we have designed an equilumi- n j n nous system that provides broadband diffusion independently of the incident direction, having non zero constant Therefore, we end with structure factor in the desired range of frequencies. Al- X i ıθ p (~r) = S J (k|~r − ~r |)e ~r−~rj , (A9) though the scattered intensity is not exactly the same at 0 n n j n different scattering angles (we have a discrete distribution of scatterers), the scattering pattern is still quasi-intense where and is smoother inside the target frequency range than Si = ıneıkxj . (A10) outside it. The proposed methodology has proved itself n as a powerful tool to design and characterize disordered The factor eıkxj plays the role of a complex amplitude many-body systems with preassigned scattering proper- which depends on the horizontal projection of the posi- ties. tion of the j-th scatterer, xj. Now that we have expressed all the acoustic fields in- volved in the problem in the vicinity of the i-th cylinder, Appendix A: Multiple Scattering Theory we can obtain the following system of equations: N i i X X j ı(l−n)θ B = S + A H (k|~r − ~r |)e ~ri−~rj . We consider that the N cylinders of radius Ri are lo- n n l l−n i j j=1,j6=i l cated at ~ri with i = 1, ..., N to form the distribution in the x − y plane. The system is excited by a plane wave (A11) 9

At this stage, the Sn are known, but both Bn and Al are 0.04 unknown. The rigid boundary condition provides another equation relating them. At the interface of the i-th cylinder, we have 0.03

1 ∂pext = 0, (A12) ρ0 ∂r r=Ri 0.02 giving rise to H0 (kR ) i n i i i i 0.01 Bn = − 0 An ≡ ΓnAn, (A13) Jn(kRi) where the primes represent derivative. Therefore the am- 0 plitudes of the scattered and the incident ﬁelds on the i-th 0 1000 2000 3000 4000 i cylinder can be related by means of Γn. Finally the system of equations can be written as follows, FIG. 5. Scattering cross sections for cylinders with radius N [1/130, 1/120, 1/110, 1/100, 1/90, 1/80, 1/70, 1/60]. As the X X Γi Ai − Gj,i Aj = Si , (A14) radius increases the scattering cross section increases also. n n n,l l n The wide line corresponds to the scattering cross section of j=1,j6=i l the cylinders analyzed in this work. where

j,i ı(l−n)θ~ri−~rj Gn,l = Hl−n(k|~ri − ~rj|)e for i 6= j. (A15) with the far-ﬁeld scattered amplitude This system of equations is solved for every frequency by N 2 X −ık|~r | cos (θ−θ ) X n i ınθ S(θ, ω) = e i ~ri (−i) A e . truncating the inﬁnite sums. A good estimation for this k n truncation is i=1 n

1/3 (A21) l = n = ﬂoor kRmax + 4.05(kRmax) + 10, (A16) The scattering cross section is thus written as with Rmax = max(Ri). Once the system is solved, the N i 4 X −ıkx X n i coeﬃcients An are known and the total pressure can be σ = −2Re(S(0, ω)) = − Re e ~ri (−i) A . k n obtained from i=1 n N (A22) ınθ ır cos θ X X j (~r−~rj ) p(~r) = e + AnHn(k|~r − ~rj|)e . i=1 n (A17) 2. Weak scattering approximation

Strictly speaking the Born approximation implies that 1. Scattering cross section the interaction between the scatterers is negligible, in other words, the term ps(~r, ~rj) = 0 in Eq. (A2), i.e., From the previous equations, the expression of the the incident wave on the i-th cylinder is only the inci- scattering cross section of an array of scatterers can be dent wave without any contribution of the other scat- obtained. The scattered pressure field by a distribution terers in the structure. In this work, we will consider of scatterers can be written as that |ps(~r, ~rj)| << |p0(~r)| ∀j. For a single scatterer, the N scattering cross section, is defined as ınθ X X j (~r−~rj ) ps(~r) = AnHn(k|~r − ~rj|)e . (A18) Z I 1 2 dσ i=1 n σ = 2 |ps| ds = dΩ, (A23) |p0| dΩ In the far field, we have where the integral runs over a closed surface enclosing s the scatterer. This could be used to evaluate the inten- k n ık|~r| −ık| ~r | cos (θ−θ ) H (k|~r − ~r |) ' (−ı) e e j ~ri , n j ı2π|~r| sity of the scattered field by a single element. Figure5 shows the scattering cross section for scatterers with ra- (A19) dius [1/130, 1/120, 1/110, 1/100, 1/90, 1/80, 1/70, 1/60]. We have chosen a configuration for which the scattering considering that |~r − ~rj| ' |~r| − |~ri| cos (θ − θ~ri ). The far-field scattered pressure expression is also cross section is less than 0.015 in the range of frequencies analyzed in the work, meaning that the scattering is r k thus 1.5% of the incident wave, so the weak scattering pf = S(θ, ω) eıkr, r → ∞, (A20) s ı2πr approximation is valid. 10

Appendix B: Structure factor The structure factor, S(G~ ), is then deﬁned as this inten- PN 2 sity normalized by 1/ j=1 fj

N N 1 X X ~ S(G~ ) = f f e−ıG(~rj −~rk). (B2) N j k X j=1 k=1 Let us consider the scattering of an acoustic beam of f 2 wavelength λ by the distribution of N scatterers. We j j=1 assume that the scattering is weak, so that the amplitude of the incident beam is higher than the am- If all the scatterers are identical, then plitude of the scattering waves; absorption, refraction N N and higher order scattering can be neglected (kinematic X X ~ I(G~ ) = f 2 e−ıG(~rj −~rk), (B3) diffraction). The direction of any scattered wave is de- ~ ~ ~ j=1 k=1 fined by its scattering vector G~ = ks − k0, where ks ~ i i so and k0 = k0(cos θ , sin θ ) are the scattered and incident beam wavevectors with θi the incidence angle. For 2 N N N ~ ~ ~ 1 X X −ıG~ (~r −~r ) 1 X ıG~r~ elastic scattering, |ks| = |k0| = |k| = 2π/λ and then S(G~ ) = e j k = e j .(B4) ~ 4π N N G = |G| = λ sin(θ). The amplitude and phase of this j=1 k=1 j=1 scattered wave is the vectorial sum of the scattered waves N ~ P −ıG~ri ~ by all the scatterers Ψs(~q) = i=1 fie , with fi the Therefore, the structure factor S(G) is proportional to atomic structure factor. The scattered intensity reads as the intensity of scattered field by a configuration of N scatterers. It is worth noting here that the structure factor can be also related to the scattering cross section as follows

N N dσ X X ~ = f 2 e−ıG(~rj −~rk) = f 2NS(G~ ), (B5) dΩ j=1 k=1 where σ is the total cross-section and Ω is the solid angle. Note that the von Laue condition60,64 for the periodic ~ ~ ∗ ~ I(G) = Ψs(G).Ψs(G) systems implies that the constructive interferences will N N occur if the diﬀerence between the incident and reﬂected ~ ~ X −iG~rj X iG~rk wavevector is a vector of the reciprocal lattice. Therefore = fje × fke ~ |G~ | j=1 k=1 the Bragg scattering condition reads as |k| = 2 sin θ . In N N the 1D case (θ = π/2), the wavevectors are collinear and ~ X X −ıG(~rj −~rk) ~ = fjfke . (B1) then, |k| = |G~ |/2. j=1 k=1

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