Acoustic cloaking in two dimensions: a feasible approach

Daniel Torrent and José Sánchez-Dehesa¹

Wave Phenomena Group, Department of Electronic Engineering, Polytechnic University of Valencia, C/ Camino de vera s.n., E-46022 Valencia, Spain

¹ Author to whom any correspondence should be addressed.

E-mail: jsdehesa@upvnet.upv.es

Received 13 March 2008
Published 13 June 2008

Contents

• 1. Introduction
• 2. The acoustic cloak: a proposal
• 3. Building the layers of the cloak: a feasible approach
• 4. Summary
• Acknowledgments
• References

1. Introduction

In a recent work by Cummer and Schurig [1], it was predicted that acoustic cloaking is possible in a two-dimensional (2D) geometry by means of a cloak made of an acoustic material having a strong mass anisotropy not existing in nature. This result in acoustics follows an analogous result in electromagnetism where different groups demonstrated that strong anisotropy of the corresponding electromagnetic (EM) parameters was also required [2]–[5]. However, while the EM cloaking has been experimentally demonstrated by using a specially designed metamaterial [6], its acoustic counterpart has not yet been demonstrated. Moreover, the demonstration for the acoustic cloaking is still waiting for some proposal of engineered material (metamaterial) that accomplishes the requirements on mass anisotropy predicted in [1]. In this regard, the work by Milton et al  [7] describes conceptually how the mass anisotropy could be possible by spring-loaded masses. Besides, in a recent advancement in the physical realization of metamaterial with mass density anisotropy these authors have demonstrated that such an uncommon property can be made possible by using non-symmetric lattices of solid cylinders [8].

In this work, we present an acoustic cloak that could be physically realizable. In brief, the proposed cloak is based on a multilayered structure consisting of two layers with the same thickness and made up of two different acoustic isotropic metamaterials. These metamaterials are built with sonic crystals (i.e. periodic arrays of sonic scatterers) based on two types of elastic cylinders that have to accomplish certain requirements on their mass density and effective sound speed. Numerical experiments based on multiple scattering method are presented to support the exact performance of the proposed cloak.

The paper is organized as follows. First, in section 2, we review the solution in [1] and report our approach to get the acoustic cloaking. Numerical experiments demonstrating the performance and properties of the proposed cloak are also presented and discussed. Section 3 describes the recipe to build the metamaterials needed to fabricate the multilayered cloak making it physically feasible. Finally, the work is summarized in section 4.

2. The acoustic cloak: a proposal

The solution reported by Cummer and Schurig for the acoustic cloaking in [1] requires a fluid material with an anisotropic density and a scalar bulk modulus. Moreover, these parameters must be dependent on the radial distance to the hidden object. The predicted functional form is

where R₁ and R₂ are the inner and outer radii of the cloaking shell, B is the bulk modulus of the shell, ρ_r and ρ_θ are the components of the diagonal mass density tensor, and the quantities with subscript b are those of the surrounding background that is a fluid or a gas.

Materials with such uncommon properties do not exist in nature and, therefore, some engineered material should be introduced to accomplish them. In a recent paper [8], these authors have shown that, in the low frequency limit, arrangements of cylinders in non-symmetric lattices lead to acoustic metamaterials with anisotropic mass density and scalar bulk modulus, as required by equations (1). The lattices considered in that work are single-cylinder lattices and, as a consequence, when the mass density of the cylinder is larger (smaller) than that of the background, the effective mass density tensor is always larger (smaller) than that of the background. This is an important drawback, because in conditions (1) one component of the mass density tensor is the reciprocal of the other and consequently the radial (angular) component of the mass tensor is always larger (smaller) than that of the background. Therefore, a material having certain mass density in the radial direction and its reciprocal along the tangential direction cannot be engineered by using the theory developed in [8]. However, we suggest below a path to get the actual realization of such a property by using periodic structures.

We arrive at the solution proposed here by exploring the possibility of building anisotropic materials based on sonic crystals with two types of material cylinders, following a combination of two approaches previously introduced by these authors [8, 9]. Unfortunately, the practical realization of conditions (1) was impossible to achieve because of the limitation imposed by the close packing condition of the lattice. Therefore, in a natural way, we conclude that a multilayered structure made of two materials overcomes such problem and give a solution to accomplish the required conditions. It is interesting to note that an approach similar to this was also proposed to get EM cloaking [10]. However, while the EM cloak only verifies a reduced set of conditions imposed for EM cloaking, the one reported here exactly matches the conditions for acoustic cloaking.

Let us consider a cloaking shell consisting of a multilayered structure that is made of alternating layers of materials of types 1 and 2. For any periodic system the bulk modulus (in the homogenization limit) does not depend on the type (isotropic or anisotropic) of lattice. In fact, the effective bulk modulus (at large wavelength), B*, can be determined by simply doing a volume average of its reciprocal. Thus, for a 1D multilayered system of materials 1 and 2, it was shown that [11]

where B₁ (B₂) is the bulk modulus of material 1 (2) and d₁ (d₂) is the length of layer 1 (2).

To obtain the tensor associated with the effective speed of sound, we need to calculate first the dispersion relation K(ω) of the system, i.e. the wavenumber as a function of the frequency. This calculation is very simple if the procedure explained in textbooks like [12] is followed,

where

for i = 1, 2.

The effective speed of sound is defined (in the low frequency limit) as the ratio between the angular frequency ω and the wavenumber K. This ratio can be obtained by making a power expansion of the trigonometric functions up to the second power of their arguments. It is easy to show that, c_⊥ and c_||, the diagonal components of the speed tensor for the propagation along the perpendicular and parallel directions, are

where ρ_⊥ and ρ_|| are given by

and represent the diagonal components of the effective dynamical mass density tensor.

Similar expressions were also reported by Schoenberg and Sen [11]. So the components of the reciprocal mass density tensor for d₁ = d₂ = d/2 and ρ₂/ρ_b = ρ_b/ρ₁ are

This structure satisfies the conditions (1a) and (1b); i.e. ρ_⊥/ρ_b = ρ_b/ρ_||. Moreover, the component of the mass density tensor along the perpendicular direction is always larger than one, as also required by condition (1a).

Now, we have to determine the material properties of media 1 and 2. If material 1 is selected as the high density material, from (7), the dependence of ρ₁ as a function of r is

The bulk modulus of both materials must also depend on the radial distance. To accomplish this condition, one possibility is assuming that both materials have the same speed of sound c₁. Therefore, the dependence of this quantity as a function of r can be obtained by inserting B₁ = ρ₁c₁² and B₂ = ρ₂c₁² into equations (1) and (2)

Equations (8) and (9) define the properties of medium 1, whereas those for medium 2 are derived from them as explained above

The proposed cloak is shown schematically in figure 1, where the 1D structure is transformed into a circular-shaped shell that it is expected to cloak a rigid core placed in its interior.

To check the functionality of the multilayered cloak, we have performed multiple scattering simulations by using the method developed in [14, 15]. Maps of the acoustic pressure at a time instant are represented by the real part of the complex amplitude p and are shown in figure 2 for the case of a rigid core of radius R₁ that is placed inside a multilayered shell of radius R₂ = 2R₁. The full structure is submitted to an acoustic field of wavelength λ = R₁/2. The performance of two different shells are depicted in figure 2, where the left panel corresponds to a shell made of 50 layers and the right panel to one composed of 200 layers, where each layer of thickness d is composed of two alternate layers of thickness of materials 1 and 2, respectively, i.e. d = d₁ + d₂ = d/2 + d/2. The cloaking effect is evident in both the representations, but that corresponding to 200 layers can be considered to be near perfect. These results can be compared with the case of the rigid cylinder with no cloak that is represented in the left panel of figure 3, where the incident wave is strongly scattered by the cylinder. On the other hand, in the right panel of figure 3 is depicted the acoustic cloaking by an extremely thin cloak, its thickness being two orders of magnitude smaller than the hidden cylinder, but it also is made of 200 layers. This result is very promising, because it indicates the possibility of building cloaks as thin as the available technology allows.

Now, it is also interesting to analyze the cloaking effect as a function of the number of layers employed in the fabrication of the cloak. The resulting behavior is important in order to simplify the fabrication of the cloaking shell as much as possible. We have studied the backscattered field as a typical parameter characterizing the cloak's performance and it is represented (in a logarithmic scale) in figure 4 as a function of the frequency for different numbers of layers. It is remarkable that in figure 4 only 50 layers are able to reduce by more than one order of magnitude (for a wide range of frequencies) the backscattered field in comparison with that for the corresponding naked rigid cylinder. Other interesting cases like penetrable and void regions are not reported here, but we expect results analogous to those that have already been published [13].

3. Building the layers of the cloak: a feasible approach

It has been shown above that the acoustic cloak can be exactly realized by using a set of N layers, each one made up of one isotropic fluidlike material of type 1 and another of type 2, their acoustic parameters being described by conditions (8), (9) and (10). This section is devoted to show that the requested dynamical properties of the fluidlike materials 1 and 2 can be actually realized by using lattices of solid cylinders embedded in the background where the sound propagates, i.e. by using sonic crystals. Particularly, we have shown that sonic crystals are a class of metamaterials that dynamically behave (in the homogenization limit) as true fluidlike materials whose properties can be tailored with practically no limitation. For example, the acoustic parameters of the homogenized sonic crystal made up of only one type of component basically depend on the filling fraction of the lattice [17, 18]. More recently, we have reported [9] that sonic crystals made up of two material cylinders increase the possibilities of metamaterial design. Here, we have used the last approach to design the acoustic properties needed for the layers of the cloaking shell.

For instance, let us assume that metamaterials 1 and 2 are going to be made up of two types of cylinders with acoustic parameters: (ρ_1α, c_1α) and (ρ_1β, c_1β) for metamaterial 1, and (ρ_2α, c_2α) and (ρ_2β, c_2β) for metamaterial 2. The parameters of metamaterials 1 and 2 can be tailored with the filling fractions f_1α, f_1β, f_2α and f_2β of the components involved in their fabrication. Thus,

where η_i = (ρ_i–ρ_b)/(ρ_i + ρ_b).

It is important to point out that the expressions (12) and (14) are only valid for small and moderate filling fractions. For large filling fractions (near the close-packing condition), the multiple scattering interaction between cylinders must be included [16] and calculations should be performed by using the exact expressions already reported in previous works [9, 17, 18].

In figure 5, we have represented the so-called phase diagram [9] of the metamaterial that can be obtained by using composites made of two component sonic crystals. This diagram has been obtained by considering cylinders made of fully elastic materials characterized by their parameters of density, ρ, longitudinal velocity, c_ℓ, and transverse velocity, c_t. Table 1 reports the parameters of the materials considered, where represents the associated fluidlike velocity [9]. Lines of the same color enclose the area in which a metamaterial with parameters (ρ*, c*) are available by just changing the filling fraction of the materials employed in the composite. For instance, the area enclosed by the blue lines defines the range of parameter that can be tailored using materials 1α and 1β in the composition of metamaterial 1. Within this area, the blue straight line represents the variation in (ρ*, c*) needed by material 1 in order to accomplish the cloaking by the multilayered structure. In other words, the conditions for material 1 given by (8) and (9) can be fully accomplished by using materials 1α and 1β in table 1. For the case of material 2, the red straight line in figure 5 plots the range of variation requested for cloaking. Let us stress that the set of elastic materials in table 1, which were selected to get the condition of acoustic cloaking, is not unique and another combination of materials could achieve a similar performance. In fact, in order to make the cloak real, a more appropriate set should be chosen to match the availability of materials in nature.

Table 1. Acoustic parameters of the materials forming the composites. The third column reports the effective velocity of the elastic cylinder [9]; , where cℓ and ct are the longitudinal and transverse velocities, respectively.

4. Summary

It has been shown that acoustic cloaking shells are possible by means of multilayered structures made with two types of acoustic isotropic metamaterials, whose acoustic parameters should change as a function of the distance from the layer to the center of the shell. The practical realization of the required radial dependence can be achieved by using homogenized 2D sonic crystals having two full elastic cylinders per unit cell. The parameters of the sonic crystal needed to get the cloaking effect can be obtained through a phase diagram analysis of the selected materials used in the cylinders. An important advantage of our proposed structures in comparison with its EM counterpart to get electromagnetic cloaking is their performance in a wide range of wavelengths. Besides, our proposed metamaterials are very robust against positional and/or structural disordering [17, 18].

We hope that this proposal motivates future experimental work demonstrating its performance. Finally, we should point out that the present proposal can be extended to the recently reported case of acoustic cloaking in three dimensions [19, 20], a result that was implicitly treated in [21]. By using an analogous development, we can foresee that the corresponding shell could be made of multilayers of two different isotropic metamaterials, which might consist of spheres of different solid materials.

Acknowledgments

This work has been partially supported by the Spanish MEC (project ref TEC2006-6723). DT acknowledges a grant paid by MEC. Both authors thank Liang-Wu Cai for useful discussions. JS-D also acknowledges the kind hospitality of Drs Shimamura and Garcia Villora at the National Institute of Materials Science in Tsukuba (Japan), where this paper was finalized.

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