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The present document is the Deliverable 5.1 Experimental set up of the Dynamo project (https://dynamo-project.eu/), funded by the European Commission’s European Innovation Council and SMEs Executive Agency (EISMEA), under its Horizon Europe Research and Innovation programme. The objective of Work Package 5 (WP5) is to develop imaging applications for the samples designed by our project partners AGH, IC, and UJI, which were fabricated by CSIC and characterized by CNRS. This work package represents the final step in the process described in this project. To develop these applications, we need to excite the phononic modes previously characterized at CNRS. Due to the complex nature of experiments in this domain, we cannot expect the spatial distributions of the field to be identical to those measured by CNRS, although the frequencies at which these are excited will be identical or very close. For a proper development of applications, it is desirable then that the experiments be as similar as possible. For this reason we proposed to implement the setup with the same elements, especially the laser source. Therefore, the first task of WP5 was to replicate the picosecond ultrasonic setup of CNRS so that we can later develop applications based on the optimal phononic architectures. Despite some delays in acquiring the necessary equipment, we have finally completed the setup, and it is working correctly. We can now excite acoustic waves in the range of a tenth of a GHz and accurately resolve these waves over time. In this document, it isexplainedthe different components of the setup, including relevant parameters of the electronic equipment, and provide a list of immediate improvements to optimize the measurements, particularly regarding the spatial scanning of the acoustic fields. However, it's important to note that, like any homemade experimental setup, technical improvements will continuously be implemented throughout the project's duration.

The present document is the Deliverable 1.1.T-matrix of spheres and pillars of theDynamo project(https://dynamo-project.eu/)funded by the European Commission’s European Innovation Council and SMEs Executive Agency (EISMEA), under its Horizon Europe Research and Innovation programme. In this document it isdiscussed the possible approaches to speed up the computation of the eigen modes of phononic architectures. The initial proposal in the project was the use of multiple scattering theory, thus the first task to do is the development of expressions and algorithms for the computation of the T-matrix and Green’s functions of different resonators (spheres or cylinders) under certain scenarios (semi-infinite plane, free standing plate, layer over a substrate, etc.). However, it isfound this approach extremely complex for scatterers with a finite cross section, like holes or cylinders, and we are exploring other possibilities. Due to the large number of equations involved in the development of the theory, in this document it isadded just a summary of the methods and our conclusions, and the equations are formulated in the attached document. It must be pointed out that the objective of this first task was to setup the numerical problem, and the effective solutions of the different geometries is the second task of the Work Package 1. An extended version of this deliverableis available.The aim of this document is to develop analytical models for analysis of wave scattering from an arrangement of reflectors - for both surface and guided elastic waves. Within the developed tools and methods, we intend to include multiple scattering-based and also alternative methods for describing scatterers’ interactions with the wave field. &nbsp

With the growing interest in the field of artificial materials, more advanced and sophisticated functionalities are required from phononic crystals and acoustic metamaterials. This implies a high computational effort and cost, and still the efficiency of the designs may be not sufficient. With the help of third-wave artificial intelligence technologies, the design schemes of these materials are undergoing a new revolution. As an important branch of artificial intelligence, machine learning paves the way to new technological innovations by stimulating the exploration of structural design. Machine learning provides a powerful means of achieving an efficient and accurate design process by exploring nonlinear physical patterns in high-dimensional space, based on data sets of candidate structures. Many advanced machine learning algorithms, such as deep neural networks, unsupervised manifold clustering, reinforcement learning and so forth, have been widely and deeply investigated for structural design. In this review, we summarize the recent works on the combination of phononic metamaterials and machine learning. We provide an overview of machine learning on structural design. Then discuss machine learning driven on-demand design of phononic metamaterials for acoustic and elastic waves functions, topological phases and atomic-scale phonon properties. Finally, we summarize the current state of the art and provide a prospective of the future development directions.

This broad review summarizes recent advances and "hot"research topics in nanophononics and elastic, acoustic, and mechanical metamaterials based on results presented by the authors at the EUROMECH 610 Colloquium held on April 25-27, 2022 in Benicássim, Spain. The key goal of the colloquium was to highlight important developments in these areas, particularly new results that emerged during the last two years. This work thus presents a "snapshot"of the state-of-the-art of different nanophononics- and metamaterial-related topics rather than a historical view on these subjects, in contrast to a conventional review article. The introduction of basic definitions for each topic is followed by an outline of design strategies for the media under consideration, recently developed analysis and implementation techniques, and discussions of current challenges and promising applications. This review, while not comprehensive, will be helpful especially for early-career researchers, among others, as it offers a broad view of the current state-of-the-art and highlights some unique and flourishing research in the mentioned fields, providing insight into multiple exciting research directions., Nanophotonics, 12 (4), ISSN:2192-8614

The present document is the Deliverable 6.2 Data Management Plan (DMP)of the Dynamo project (https://dynamo-project.eu/), funded by the European Commission’s European Innovation Council and SMEs Executive Agency (EISMEA), under its Horizon Europe Research and Innovation programme. The Dynamo DMP follows the Horizon Europe Data Management Plan template. The Data Management Plan describes the data management life cycle for the data to be collected, processed and/or generated. The DMP aims at defining the management strategy of data generated during the project with the purpose to making research data FAIR: findable, accessible, interoperable, and re-usable. The DMP covers: (i) data handling during and after the project, (ii) what types and formats of data will be generated/collected, (iii) what methodologies and standards will be applied, (iv) whether the data be shared or made open-access and (v) how data will be curated and preserved. Dynamo is committed to give open access to data generated unless it goes against the beneficiary’s legitimate interests or it is contrary to any other constraints, such as data protection rules, privacy, confidentiality, trade secrets, Union competitive interests, security rules, intellectual property rights or would be against other obligations under the Grant Agreement. The DMP should be updated, when necessary, through the lifecycle of the Dynamo. Whenever the procedures are updated, all partners will be duly informed about the changes made with respect to the previous version. Further updating of the DMP will be in month 30 with the D.6.6. Data Management Plan (updated).

hermoacoustic amplifiers are analyzed in the framework of nonreciprocal Willis coupling. The closed form expressions of the effective properties are derived, showing that an applied temperature gradient causes the appearance of a nonreciprocal Willis coupling. Even and nonreciprocal Willis couplings are exhibited already in the first-order Taylor expansion of the solution and are of equal modulus but opposite sign, thus suggesting that the even Willis coupling is a reaction to the nonreciprocity introduced by the temperature gradients. These Willis couplings cause a coalescence point in the k space, which deviates from Re(k) = 0 (with k the wave number) and is thus a zero-group-velocity point, as well as the opening of an amplification gap at low frequency. Effective parameters and scattering properties are found in excellent agreement with experimental results. This article paves the way to further control the acoustic waves at very low frequencies with nonreciprocal systems.

Ponència presentada en el 15 Fifteenth International Congress on Artificial Materials for Novel Wave Phenomena, Metamaterials 2021 Localization of waves is studied by means of multiple scattering theory in a (2D) arrangement configuration. Studied structures belong to the so called twisted lattices. Results found show that, for a given frequency, there are localized modes for a discrete set of rotation angles, analogous to the recently studied angles for 2D materials such as graphene. The properties of these clusters whenever the rotation angle approaches a commensurable angle or a really small one are related to the dipolar interaction between scatterers. While the presented results are valid for any type of wave, flexural waves in thin elastic plates have been numerically studied and successfully explained in terms of the interaction between pairs of scatterers. The studied structures are shown as promising candidates for the design of tunable wave-trapping devices.

Ponencia presentada en Fifteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials), 20-24 Sept. 2021, New York, USA In this work, we discuss the interaction between Helmholtz resonators, using the finite element method. We show that the coupling between two spheres gives rise to two resonant modes at low frequencies. One is symmetric and corresponds to a mono-polar breathing mode, as currently observed. The second one is anti-symmetric and behaves as a dipole. We show that the first one generates an isotropic wave in the far pressure field while the second one gives rise to a directional propagative wave. The two modes are discussed as a function of the geometrical and physical parameters. When considering the thermo-viscous properties of air for a specific set of parameters, one finds that the behavior of the dipolar mode can be changed from a diffuser to an absorber. By managing double Helmholtz resonators, we aim to create a multifunctional meta-surface operating at two different frequencies.

We present a multiple scattering analysis of robust interface states for flexural waves in thin elastic plates. We show that finite clusters of linear arrays of scatterers built on a quasi-periodic arrangement support bounded modes in the two-dimensional space of the plate. The spectrum of these modes plotted against the modulation defining the quasi-periodicity has the shape of a Hofstadter butterfly, which as suggested by previous works might support topologically protected modes. Some interface states appear inside the gaps of the butterfly, which are enhanced when one linear cluster is merged with its mirror reflected version. The robustness of these modes is verified by numerical experiments in which different degrees of disorder are introduced in the scatterers, showing that neither the frequency nor the shape of the modes is altered. Since the modes are at the interface between two one-dimensional arrays of scatterers deposited on a two-dimensional space, these modes are not fully surrounded by bulk gaped materials so that they are more suitable for their excitation by propagating waves. The generality of these results goes beyond flexural waves since similar results are expected for acoustic or electromagnetic waves. D.T. acknowledges financial support from the “Ramón y Cajal” Fellowship under Grant No. RYC-2016-21188 and the Ministry of Science, Innovation and Universities through Project No. RTI2018-093921-A-C42. M.M.-S. acknowledges financial support from the FPU program under Grant No. FPU18/02725.

Homogenization theories provide models that simplify the constitutive description of heterogeneous media while retaining their macroscopic features. These theories have shown how the governing fields can be macroscopically coupled, even if they are microscopically independent. A prominent example is the Willis theory which predicted the strain–momentum coupling in elastodynamic metamaterials. Recently, a theory that is based on the Green’s function method predicted analogous electro–momentum coupling in piezoelectric metamaterials. Here, we develop a simpler scheme for fibrous piezoelectric composites undergoing antiplane shear waves. We employ a source-driven approach that delivers a unique set of effective properties for arbitrary frequency–wavevector pairs. We numerically show how the resultant homogenized model recovers exactly the dispersion of free waves in the composite. We also compute the effective properties in the long-wavelength limit and off the dispersion curves, and show that the resultant model satisfy causality, reciprocity and energy conservation. By contrast, we show how equivalent models that neglect the electromomentum coupling violate these physical laws.

The mechanical analog to the topological insulators brings anomalous elastic wave properties which diversifies classic wave functions for potential broad applications. To obtain topological mechanical wave states with good quality at desired frequency ranges, it needs repetitive trials of different geometric parameters with traditional forward designs. In this work, we develop an inverse design of topological edge states for flexural wave using machine learning method which is promising for instantaneous design. Nonlinear mapping function from input targets to output desired parameters are adopted in artificial neural networks where the data sets for training are generated by the plane wave expansion method. Topological edge states are then realized and compared for different bandgap width conditions with such inverse designs, proving that wide bandgap can promote the confinement of the topological edge states. Finally, direction selective propagations with sharp turns are further demonstrated as anomalous wave behaviors. The machine learning inverse design of topological states for flexural wave provides an efficient way to design practical devices with targeted needs for potential applications such as signal processing, sensing and energy harvesting.

As building blocks of acoustic metamaterials, resonant scatterers have demonstrated their ability to modulate the effective fluid parameters, which subsequently possess extreme properties such as negative bulk modulus or negative mass density. Promising applications have been shown such as extraordinary absorption, focusing, and abnormal refraction for instance. However, acoustic waves can be further controlled in Willis materials by harnessing the coupling parameters. In this work, we derive the closed forms of the effective parameters from the transfer matrix in three asymmetric and reciprocal one-dimensional resonant configurations and exhibit the differences in terms of coupling coefficients. The way in which Willis coupling occurs in spatially asymmetric unit cells is highlighted. In addition, the analysis shows the absence of odd Willis coupling for reciprocal configurations. These effective parameters are validated against experimental and numerical results in the three configurations. This article paves the way of a novel physical understanding and engineering use of Willis acoustic materials.

International audience; The asymmetric propagation of mechanical energy across interfaces is a challenging problem with a wide range of applications. In this work, we present a novel structure presenting the asymmetric propagation of elastic waves in thin plates in a broadband range. The structure consists of a combination of symmetrically and asymmetrically distributed pillars, so that the former decouple the different Lamb modes and the latter mix all of them. We show that a combination in tandem with these two structures can realize an efficient broadband asymmetric propagation at the subwavelength range and achieve a transmission difference larger than 200 dB between forward and backward directions. The proposed pillared meta-plate brings a new way for subwavelength and broadband wave manipulation in the fields of wave isolation, sensing and communication, among others.

The closed form expressions of the effective properties in periodic fluid laminates are derived thanks to the Padé approximation of the transfer matrix. A second-order Taylor expansion of the transfer matrix elements exhibits Willis coupling. This coupling is the sum of a local term and a nonlocal term. The nonlocal term arises from the apparent bulk modulus in quasi one-dimensional problems. The nonlocality directly impacts the governing equations modeling the acoustic wave propagation in these Willis materials, which then involve convolution products in space. As an example, a two-orthotropic porous material laminate is considered. The theoretically derived effective properties and scattering coefficients are found in excellent agreement with those numerically calculated. The Willis coupling widens the frequency range of validity and accuracy of the effective properties and thus of the calculated scattering coefficients when compared to classical homogenization results for which the Willis coupling is absent. This widening mostly relies on the effect of Willis coupling on the impedance of the fluid laminate. The effective properties are finally derived for a general laminate.

[EN] Topological phases have recently been realized in bosonic systems. The associated boundary modes between regions of distinct topology have been used to demonstrate robust waveguiding, protected from defects by the topology of the surrounding bulk. A related type of topologically protected state that is not propagating but is bound to a defect has not been demonstrated to date in a bosonic setting. Here we demonstrate numerically and experimentally that an acoustic mode can be topologically bound to a vortex fabricated in a two-dimensional, Kekul¿e-distorted triangular acoustic lattice. Such lattice realizes an acoustic analog of the Jackiw-Rossi mechanism that topologically binds a bound state in a p-wave superconductor vortex. The acoustic bound state is thus a bosonic analog of a Majorana bound state, where the two valleys replace particle and hole components. We numerically show that it is topologically protected against arbitrary symmetry-preserving local perturbations, and remains pinned to the Dirac frequency of the unperturbed lattice regardless of parameter variations. We demonstrate our prediction experimentally by 3D printing the vortex pattern in a plastic matrix and measuring the spectrum of the acoustic response of the device. Despite viscothermal losses, the measured topological resonance remains robust, with its frequency closely matching our simulations., J. C. acknowledges support from the European Research Council (ERC) through the Starting Grant No. 714577 PHONOMETA and from the MINECO through a Ramon y Cajal grant (No. RYC-2015-17156). J. S.-D. acknowledges support from the Ministerio de Economia y Competitividad of the Spanish Government and the European Union "Fondo Europeo de Desarrollo Regional (FEDER)" through Project No. TEC2014-53088-C3-1-R. P. S.-J. acknowledges support from MINECO/FEDER under Grant No. FIS2015-65706-P. D. T. acknowledges financial support through the Ramon y Cajal fellowship under Grant No. RYC-2016-21188 and to the Ministry of Science, Innovation and Universities through Project No. RTI2018-093921-A-C42.

Topological phases of sound enable unconventional confinement of acoustic energy at the corners in higher-order topological insulators. These unique states which go beyond the conventional bulk-boundary correspondence have recently been extended to non-Hermitian wave physics comprising finite crystal structures including loss and gain units. We use a multiple scattering theory to calculate these topologically trapped complex states that agree very well to finite element predictions. Moreover, our semi-numerical tool allows us to compute the spectral dependence of corner states in the presence of defects, illustrating the limits of the topological resilience of these confined non-Hermitian acoustic states. J.C. acknowledges the support from the European Research Council (ERC) through the Starting Grant 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (Grant No. RYC-2015-17156). D.T. acknowledges the support of MINECO through a Ramón y Cajal grant (Grant No. RYC-2016-21188) and of the Ministry of Science, Innovation and Universities trough project number RTI2018-093921-A-C42. Publicado

[EN] We report the finding of the analogous valley Hall effect in phononic systems arising from mirror symmetry breaking, in addition to spatial inversion symmetry breaking. We study topological phases of plates and spring-mass models in kagome and modified kagome arrangements. By breaking the inversion symmetry it is well known that a defined valley Chern number arises. We also show that effectively, breaking the mirror symmetry leads to the same topological invariant. Based on the bulk-edge correspondence principle, protected edge states appear at interfaces between two lattices with different valley Chern numbers. By means of a plane wave expansion method and the multiple scattering theory for periodic and finite systems, respectively, we computed the Berry curvature, the band inversion, mode shapes, and edge modes in plate systems. We also find that appropriate multipoint excitations in finite system gives rise to propagating waves along a one-sided path only., N.L and J.V.A. acknowledges financial support from MINECO grant FIS2015-64886-C5-5-P. N.L. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through The María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377), and also hospitality from the Universitat Jaume I in Castellon where part of this work was done. D.T. acknowledges financial support through the “Ramón y Cajal” fellowship under grant number RYC-2016-21188. P.S.-J. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness through Grant No. FIS2015-65706-P (MINECO/FEDER). J.C. acknowledges the support from the European Research Council (ERC) through the Starting Grant No. 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (Grant No. RYC-2015-17156).

We show that engineered diffraction gratings can considerably simplify the design and performance of acoustic devices. Acoustic reflecting gratings based on rectangular cavities drilled on an acoustically rigid surface are designed in such a way that all the incident energy is channeled towards the diffracted mode traveling in the oposite direction of the incident field (retroreflection effect), and this effect is used to cloak an object placed over a rigid surface. Axisymmetric gratings consisting in rigid surfaces with just one groove per unit cell are used to design acoustic thin acoustic carpet cloaks. Finally, full wave numerical simulations are performed and a conical carpet cloak is experimentally tested, showing an excellent scattering cancellation effect.

We present an extensive formulation of the inverse grating problem for flexural waves in which the energy of each diffracted mode is selected and the grating configuration is then obtained by our solving a linear system of equations. The grating is designed as a lineal periodic repetition of a unit cell comprising a cluster of resonators attached at points whose physical properties are directly derived by inversion of a given matrix. Although both active and passive attachments can be required in the most-general case, it is possible to find configurations with only passive (i.e., damped) solutions. This inverse design approach is an alternative to the design of metasurfaces for flexural waves, overcoming the limitations of gradient phase metasurfaces, which require a continuous variation of the surface’s impedance. When the grating is designed in such a way that all the energy is channeled into a single diffracted mode, it behaves as an anomalous refractor or reflector. The negative refractor is analyzed in depth, and it is shown that with only three scatterers per unit cell is it possible to build such a device with unitary efficiency.

Phononic crystals and acoustic metamaterials are periodic structures whose effective properties can be tailored at will to achieve extreme control on wave propagation. Their refractive index is obtained from the homogenization of the infinite periodic system, but it is possible to locally change the properties of a finite crystal in such a way that it results in an effective gradient of the refractive index. In such case the propagation of waves can be accurately described by means of ray theory, and different refractive devices can be designed in the framework of wave propagation in inhomogeneous media. In this paper we review the different devices that have been studied for the control of both bulk and guided acoustic waves based on graded phononic crystals.

The discovery of topologically nontrivial electronic systems has opened a new age in condensed matter research. From topological insulators to topological superconductors and Weyl semimetals, it is now understood that some of the most remarkable and robust phases in electronic systems (e.g., quantum Hall or anomalous quantum Hall) are the result of topological protection. These powerful ideas have recently begun to be explored also in bosonic systems. Topologically protected acoustic, mechanical, and optical edge states have been demonstrated in a number of systems that recreate the requisite topological conditions. Such states that propagate without backscattering could find important applications in communications and energy technologies. Here, a topologically bound mechanical state, a different class of nonpropagating protected state that cannot be destroyed by local perturbations, is demonstrated. It is in particular a mechanical analogue of the well‐known Majorana bound states (MBSs) of electronic topological superconductor systems. The topological binding is implemented by creating a Kekulé distortion vortex on a 2D mechanical honeycomb superlattice that can be mapped to a magnetic flux vortex in a topological superconductor. C.-W.C. and N.L. contributed equally to this work. J.C. acknowledges the support from the European Research Council (ERC) through the Starting Grant No. 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (Grant No. RYC-2015-17156). J.Y. gratefully acknowledges the support from the NSF (CAREER1553202 and EFRI-1741685). P.S.-J. acknowledges support from MINECO/FEDER through Grant No. FIS2015-65706-P. N.L and J.V.A. acknowledge financial support from MINECO grant FIS2015-64886-C5-5-P. D.T. acknowledges financial support through the “Ramón y Cajal” fellowship under grant number RYC-2016-21188. Publicado

Invisibility or unhearability cloaks have been made possible by using metamaterials enabling light or sound to flow around obstacle without the trace of reflections or shadows. Metamaterials are known for being flexible building units that can mimic a host of unusual and extreme material responses, which are essential when engineering artificial material properties to realize a coordinate transforming cloak. Bending and stretching the coordinate grid in space require stringent material parameters; therefore, small inaccuracies and inevitable material losses become sources for unwanted scattering that are decremental to the desired effect. These obstacles further limit the possibility of achieving a robust concealment of sizeable objects from either radar or sonar detection. By using an elaborate arrangement of gain and lossy acoustic media respecting parity-time symmetry, we built a one-way unhearability cloak able to hide objects seven times larger than the acoustic wavelength. Generally speaking, our approach has no limits in terms of working frequency, shape, or size, specifically though we demonstrate how, in principle, an object of the size of a human can be hidden from audible sound. This work was supported by the National Key R&D Program of China (Grant No. 2017YFA0303700), the National Natural Science Foundation of China (Grant No. 11634006) and the Innovation Special Zone of National Defense Science and Technology. Daniel TorrentThis work was supported by the National Key R&D Program of China (Grant No. 2017YFA0303700), the National Natural Science Foundation of China (Grant No. 11634006) and the Innovation Special Zone of National Defense Science and Technology. Daniel Torrent acknowledges the support from the MINECO through a Ramón y Cajal grant (Grant no. RYC-2016-21188). Johan Christensen acknowledges the support from the European Research Council (ERC) through the Starting Grant no. 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (Grant no. RYC-2015-17156). acknowledges the support from the MINECO through a Ramón y Cajal grant (Grant no. RYC-2016-21188). Johan Christensen acknowledges the support from the European Research Council (ERC) through the Starting Grant no. 714577 PHONOMETA and from the MINECO through a Ramón y Cajal grant (Grant no. RYC-2015-17156).

By developing and applying a homogenization scheme for elastodynamics, Willis discovered that the momentum of composite materials is macroscopically coupled with their strain through a constitutive tensor. This now-termed Willis tensor not only enlarges the design space of metamaterials, but is also necessary for obtaining a meaningful effective description that respects basic physical laws. In this talk, I will show how additional tensors of Willis type emerge by generalizing the homogenization theory of Willis to thermoelastic-, piezomagnetic- or piezoelectric media. I will provide examples for the latter case that exhibit an electromomentum coupling. I will show that this coupling is necessary for describing the effective properties of asymmetric piezoelectric media using a homogenized description that respects reciprocity and energy conservation. Finally, I will demonstrate how this coupling can be used to realize a device that actively control the phase of elastic waves.

Raspberry-like magnetic nanoclusters are synthesized and subsequently self-assembled to form a bulk metamaterial exhibiting strong isotropic optical magnetism in visible light. The magnetic response of the nanoclusters (metamolecules) and of the final assembled material are measured by independent optical experiments. The validity of the effective permeability parameter is probed by spectroscopic ellipsometry at variable incidence. Numerical simulations confirm the measurements.

In recent years, the advances in topological insulator in the fields of condensed matter have been extended to classical wave systems such as acoustic and elastic waves. However, the quantitative robustness study of topological states which is indispensable in practical realization is rarely reported. In this work, we proposed topologically protected edge states with zigzag, bridge and armchair interfaces in a new twisted phononic plate. The robustness of non-trivial band gap in bulk structure is clearly presented versus twisted angles, revealing a threshold of 5 degrees which is the key fundamental information for the robustness of topological edge states. We further defined a localized displacement ratio as an efficient parameter to characterize edge states. Due to the different orientation of the three interfaces, zigzag and bridge edge states show higher quantitative robustness in their localized displacement ratio. A map of robustness as a function of both frequency and twisted angle highlights the better performance of the topological zigzag edge state. Robustness is evaluated for twisted angle and for all possible types of interfaces for the first time, which benefits for the design and fabrication of solid functional devices with great potential applications.

We present an efficient method for the design of anomalous reflectors for acoustic waves. The approach is based on the fact that the anomalous reflector is actually a diffraction grating in which the amplitude of all the modes is negligible except for the one traveling towards the desired direction. A supercell of drilled cavities in an acoustically rigid surface is proposed as the basic unit cell, and analytical expressions for an inverse diffraction problem are derived. It is found that the the number of cavities required for the realization of an anomalous reflector is equal to the number of diffracted modes to cancel, and this number depends on the relationship between the incident and reflected angles. Then, the “retroreflection” effect is obtained by just one cavity per unit cell; also, with only two cavities it is possible to change the reflection angle of a normally incident wave, and five cavities are enough to design a general retroreflector changing the incident and reflected angles at oblique incidence. Finally, the concept of Snell’s law violation is extended not only to the incident and reflected angles, but also to the plane in which it happens, and a device based on a single cavity in a square lattice is designed in such a way that the reflection plane is rotated π/4 with respect to the plane of incidence. Numerical simulations are performed to support the predictions of the analytical expressions, and an excellent agreement is found.