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Cai Chen, Yuan Yin, Kan Wei-Wei, Yang Jing, Zou Xin-Ye. Improving the performance of acoustic invisibility with multilayer structure based on scattering analysis. Chinese Physics B, 2016, 25(12): 124302  
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Improving the performance of acoustic invisibility with multilayer structure based on scattering analysis
Cai Chen1, Yuan Yin2, Kan Wei-Wei3, Yang Jing1, Zou Xin-Ye1, †,
Key Laboratory of Modern Acoustics, MOE, and Institute of Acoustics, Department of Physics, Nanjing University, Nanjing 210093, China
The School of Mathematics and Physics, Jiangsu University of Technology, Changzhou 213001, China
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

 

† Corresponding author. E-mail: xyzou@nju.edu.cn

Project supported by the National Natural Science Foundation of China (Grant Nos. 11274168, 11374157, 11174138, 11174139, 11222442, and 81127901) and the National Basic Research Program of China (Grant Nos. 2010CB327803 and 2012CB921504).

Abstract
Abstract

In this paper, acoustic scattering from the system comprised of a cloaked object and the multilayer cloak with only one single pair of isotropic media is analyzed with a recursive numerical method. The designed acoustic parameters of the isotropic cloak media are assumed to be single-negative, and the resulting cloak can reduce acoustic scattering from an acoustic sensor while allowing it to receive external information. Several factors that may influence the performance of the cloak, including the number of layers and the acoustic dissipation of the medium are fully analyzed. Furthermore, the possibility of achieving acoustic invisibility with positive acoustic parameters is proposed by searching the optimum value in the parameter space and minimizing the scattering cross-section.

PACS: 43.20.–f;43.35.+d;62.90.+k
Keyword:metamaterials;acoustic cloaking;scattering;invisibility
1. Introduction

In 2006, Pendry et al. proposed the concept of transformation optics, and used this coordinate transformation technique to design an invisibility cloak for electromagnetic waves.[1] Ever since them, the coordinate transformation technique has attracted worldwide attention. Such a method was extended to the field of acoustics by Cummer and Schurig in 2006.[2] Since then there has been growing interest in acoustic invisibility[3–9] due to the possibility of hiding objects from detecting the acoustic signals, which may lead to applications in a variety of practical situations such as noninvasive acoustic sensors.

Recent research suggested that it is possible to “cancel” the cloaked object as well as to share information from its surroundings by using the complementary objects made of a double-negative medium (DNM).[10] While DNM has been proposed and demonstrated by using acoustic metamaterials,[11–15] the corresponding unit cells are usually related to a resonate effect and consequently are only effective in narrow band. Besides, the resonance-induced dissipation will inevitably harm the performance of the resulting device. Therefore the invisibility cloak based on DNM would be less practical for acoustic waves. Zhu et al. proposed the superlens cloaking by using single-negative medium (SNM) in 2011.[16] Compared with the DNM device, the invisibility cloak based on SNM would generally be easy to fabricate and effective to work in broader bandwidth.[17] Although the superlens cloaking by Zhu et al. is proved to be effective in reducing acoustic scattering from an acoustic sensor, such a device is composed of inhomogeneous anisotropic media which require, even in a reducible way, a large number of isotropic media with different parameters, and thus bringing the difficulty in fabricating it. Later, Zhu et al.[18] revised their folding transformation scheme and simplified the cloaking method, in which only three types of SNMs are needed. Xu et al.[19] further simplified the superlens fabrication process, and in their scheme, only one single pair of isotropic media is required for reducing the acoustic scattering.

In this paper, a recursive numerical method of calculating the acoustic scattering from the structure with a homogeneous isotropic multilayer shell is given in detail, and based on this method the SNM superlens cloaking is fully analyzed. The influences of several factors that may hinder the performance of the cloak, including the number of layers, acoustic dissipation of the media, etc., are investigated. According to the results above, we further investigate the possibility of achieving a positive parameter invisibility device, by searching, in the parameter space, for the optimum value that can render the scattering cross-section minimum.

The rest of the present article is organized as follows. In Section 2, we first introduce the theory of superlens cloaking and then give the theoretical formulation for scattering analyses with the recursive numerical method. In Section 3, we present in detail the numerical results for the cloaking performance in various situations, analyze the factors that may influence the invisibility effect, and give an example of an invisibility cloak based on scattering cancellation. Finally, we draw some conclusions from the present studies in Section 4.

2. Theoretical methods
2.1. Theory of superlens cloaking

For simplicity, we restrict the problem to a two-dimensional (2D) case. A sketch of the folding space transformation is shown in Figs. 1(a) and 1(b). During space transformation, the space A + C is transformed into the space A′ + C, and space B′ filled with complementary media is used to keep the continuity in the acoustic field. The unprimed and primed coordinate are used to represent the virtual space and the physical space, respectively. For the mapping A–A′ and C–C, the transforming equations can be written as r = f(r′), θ = θ′, with f(r) = (a′/a)r, (r < a), and f(r) = r, (r > b), respectively. According to transformation acoustic theory, the bulk modulus and inverse mass density tensor in the transformed space are given as κ′(x′) = κ (x)det(H) and ρ′−1(x′) = Hρ−1(x)HT/det(H), respectively. Here, H is the Jacobian transformation tensor with the components as in which hxi and are the scale factors in virtual space and physical space, respectively. If the virtual space is occupied by a homogeneous isotropic medium with bulk modulus κ0 and mass density ρ0, then the material parameters in medium A′ become κ′ = (a′/a)2κ and ρ′ = ρ0I.

Fig. 1. Sketches of the folding space mapping: (a) the virtual space consists of the bulk of the background medium; (b) the real space consists of three parts: circular area A′, annular shell B′, and the domain C outside. B′ is further divided by N bilayers with 2N rings. The mapping relationship is shown as A ↔ A′,

Now we focus on the annular shell B′ which is divided into N bilayers with the same thickness, and the two rings in each bilayer can be obtained through the folding transformation to meet the spatial continuity requirement. Thus the thickness of each bilayer can be obtained as

The n-th bilayer is at the position dn−1 ≤ r′ < dn, dn = a′ + ns (n = 1,2,...,N). In order to avoid anisotropy, the radial section of the two rings is selected as

where δn is the position of the dividing line between the two small rings, and As a result, and The relationship between the material parameters in and is given as

When N is large enough, the thickness of each bilayer tends to be 0, so

Since the dividing line is in the middle of the bilayer, the thickness values of 2N rings are the same. Meanwhile, when the thickness of each bilayer tends to be 0, the radial coordinates of all points in each small ring are almost equal, that is r′ ≈ δ, so equation (3) is rewritten as

The proposed structure can thus be realized with a pair of complementary SNMs, in which the parameters are independent of those of the background medium C or the medium A′.

2.2. Scattering analysis

In order to present the recursive procedure clearly, we begin with the case of only one single cylindrical scattering object. The analysis focuses on acoustic waves propagating along the cross section of a homogeneous infinite cylinder in a host medium. Cylindrical coordinates (r,θ,z) are used throughout the analysis, and the material is assumed to be isotropic. The wave vector is in the plane (r − θ), and the wave-front extends infinitely along the z direction. We assume that the external incident acoustic wave can be described as and the only scattering object is marked as number 1. So the corresponding incident, scattered and transmitted waves can be expanded into and where k0 and k1 are the wave vectors for the host medium and the scattering object, respectively; Jn is the Bessel function; is the Hankel function of the first kind; and are undetermined coefficients. For this scattering problem, and The radius of the infinite cylinder is a. According to the interface continuity condition, we can obtain

where ρ0 and ρ1 are the density of the host medium and the scattering object, respectively. Then, we can obtain

and it can also be expressed in the matrix form

where R1 is the scattering matrix and T1 the transmission matrix. It can be seen that the scattering matrix and the transmission matrix are diagonal matrices because of the symmetrical scattering object. According to the asymptotic expression for the Hankel function for large arguments, the far-field scattering pressure can be written as[20]

where pscr is the far-field scattering pressure, and the scattering directional factor f(θ) = |∑nβn e in(θ −π/2)|. Therefore, the total scattering cross section of the scattering object is

We show part of the multilayer structure in the first quadrant in Fig. 2. Layers and boundaries are numbered consecutively inward. It is assumed that there are N layers with the host as layer 0 and the innermost core as layer N. The acoustic impedance of layer j is zj = ρjcj, and the corresponding wave number and radius are kj and aj, respectively.

Fig. 2. First-quadrant cross section of the multilayer structure.

The acoustic field can be regarded as a superposition of the standing wave (corresponding to the Bessel function) and the scattered wave outward (corresponding to the Hankel function of the first kind). For instance, in the host medium, it is expressed as

In the innermost layer N, the standing wave field can be obtained by

In all other layers from layer1 to layer N – 1, the acoustic field is

where Sj,n and Fj,n are the coefficients for standing wave field and scattered wave field, j = 1, …, N − 1.

2.3. Recursive approach

If we already know the incident wave, that is, the value of α0n is given, then there are two general approaches to determining the acoustic scattering field. One is to list all the continuity conditions at all the interfaces, and solve the undetermined coefficient at one time;[5] the other is the recursive approach,[21] which is employed in this paper. The general idea of this recursive method is similar to the transfer matrix method.[22,23] Consider that the acoustic wave in layer j − 1 impinges onto layer j (where j = 1,..., N), the incident, reflected and transmitted waves can be written as and respectively, where the superscript i signifies that the direction of the incident wave is inward, and the subscripts j − 1 indicate that the incident wave is in medium j − 1. The coefficients of the reflected and transmitted coefficients, respectively, are

where zj and zj−1 are the acoustic impedances of their corresponding layers.

On the other hand, when the wave travels from layer j towards layer j − 1 (where j = 1,...,N), the incident, reflected and transmitted waves can be written as and respectively, where superscript o signifies that the direction of the incident wave is outward, the subscript j indicates the incident wave is in medium j. We can obtain coefficients of reflected and transmitted waves by solving boundary equations:

When only the innermost layer exists, there are two media in the space, where the scattering object is medium N with radius aN and the host material is medium N − 1. When the incident wave is given as the reflected and transmitted waves are

The total reflection coefficient in layer N is represented by which equals the reflection coefficient of layer N since there is only one layer. Then the outermost host medium is changed into medium N − 2, and the scattering object of medium N with radius aN is wrapped by layer N − 1 with outer radius aN−1, thus an effective dual-layer structure can be obtained. The acoustic wave will be reflected back and forth in layer N − 1, and we can obtain the reflection and transmission coefficients from Eqs. (14) and (15).

Supposing that the incident wave has the form of The wave reflected from boundary N − 1 into medium N − 2 is

Thus the total reflection coefficient in layer N − 1 can be written as The multiple scattering factor M in layer N − 1 is and the transmitted wave in medium N can be written as

The standing wave field in medium N − 1 can be written as The outgoing scattering field in medium N − 1 can be written as Comparing Eq. (17) with Eq. (18), we can see that the incident wave in layer N changes from to which is the standing wave field in layer N − 1 by adding a new shell.

According to the method above, the recursion moves outward in the descending direction of j until the outermost host medium becomes an actual host medium, and then we can obtain the whole distribution of acoustic wave field of the multilayered scattering structure. The details of this solution procedure are as follows.

First, from the innermost layer, the multiple scattering factor Mj,n and total reflection coefficient Rj,n can be obtained as

The recursion moves outward in the descending direction of j, where RN,n is defined by The total reflection coefficient of the multilayer structure is

Then we calculate the coefficient of the wave in the intermediate layer of the scattering object, and the coefficient of the standing wave in layer j is

The coefficient of the outgoing wave propagating from the core in layer j is

Finally, we can obtain the transmission coefficient of the innermost layer as

For the cylindrical multilayered structure under investigation in this paper, the acoustic field calculation by other methods may lead to singular values when the material parameters of adjacent layers are of substantial difference and the number of the layers becomes large. The proposed method in this paper can solve such problems more efficiently and accurately. Furthermore, the total scattering cross section of the structure is easy to obtain with this method, providing convenience in evaluating the performance of the invisibility cloak and the possibility of a new design approach.

3. Numerical results and discussion

Here, the capability of the multilayered shell structure for reducing the acoustic scattering from a cylindrical sensor system in the 2D problem is demonstrated by using the method above. The background medium is chosen to be water with ρ0 = 998 kg/m3 and κ0 = 2.19 GPa. The material parameters of the SNM are ρα = ρ0, κα = − κ0, and ρβ = − ρ0, κβ = κ0. The whole sensor system is regarded as being a scattering object with radius a′ = 1 cm, in which the effective mass density and the bulk modulus are ρs = ρ0 and κs = (4/9)κ0, respectively. The ratio between the outer radius b and the inner radius a′ of the annular shell is Figures 3(c) and 3(d) show the normalized acoustic pressure distributions of a bare sensor system illuminated by a plane wave propagating from the left side and a cylindrical wave generated by an infinite line source located at (–3.75 cm, 0 cm), respectively. The wavelengths of the plane wave and the cylindrical wave are chosen to be 0.5 cm. It can be seen that the two acoustic pressure wave fields are strongly disturbed, which leads to a shadow region behind the sensor system and considerable backward reflections. For comparison, figures 3(a) and 3(b) show the result for the sensor system wrapped in a 10-SNM layer shell. It can be found that the disturbance caused by the presence of the sensor system wrapped in the proposed structure is negligible. The incident wave can penetrate into the sensor region without changing the wavefront shape.

Fig. 3. Normalized acoustic pressure fields for the plane wave source [(a) and (c)] and for cylindrical wave source [(b) and (d)]. The scattering object is protected by the 10-layer shell ((a) and (b)), and unprotected ((c) and (d)). The unit a.u. is short for arb. units.

In order to quantitatively analyze the concealment effect of the proposed structure on the acoustic sensor, and compare the performances of the cloaking shells with different numbers of layers, the variation of total scattering cross-section with the number of layers increasing (up to N = 100) is computed, and the results are shown in Fig. 4. As we can see, when the scattering object is wrapped with only two layers (a pair of SNMs), the total scattering cross section is larger than that in the case without the cloak. The reason is that the thickness of each layer of SNM is only half the wavelength and the layer thickness does not meet the subwavelength requirement. However when the number of layers is more than four (two pairs of SNMs), the total scattering cross section of the structure is reduced by one order of magnitude. Meanwhile, the total scattering cross section decreases monotonically as the number of layers of the multilayer structure further increases.

Fig. 4. Relationship between the total scattering cross section of the structure and the number of layers of the multilayer structure.

Now we consider the case where the equivalent density of the scattering object (bare sensor) is not equal to that of the background medium. According to transformation acoustic theory, the phase matching and impedance matching are established at the same time only when the equivalent density of the scatter is equal to that of the surrounding environment. However, as is well known, the phase matching is more important in those structures. Therefore, on the premise of guaranteeing the phase matching, the impedance matching can be sacrificed to some extent to adapt to the unequal density between the scatter and background medium. So we multiply both the density ρs and bulk modulus κs by a robust factor ξ. It can be seen from Fig. 5 that the multilayer structure composed of complementary medium can still reduce the scattering by one order of magnitude when 0.5 ≤ ξ ≤ 2.

Fig. 5. Relationships between the total scattering cross section and the robustness factors for the cases of only a bare scatter (dotted curve) and the 10-layer shell scatter (solid curve).

Further, we calculate the acoustic field distribution when ξ values are 0.5 and 2 as shown in Fig. 6. As expected, for both cases, the scattered field is still suppressed by the multilayered structure, which agrees well with the result shown in Fig. 5. Figures 6(a) and 6(c) each show the normalized acoustic pressure distribution of a bare sensor system illuminated by the plane wave propagating. The wavelength is 0.5 cm. It can be seen that the two acoustic pressure wave fields are strongly disturbed. A comparison of the results in Fig. 6(b) with those in Fig. 6(d) shows the result for the reduced cloak shell, although neither of the wave fronts in Figs. 6(b) and 6(d) is as flat as that of the plane wave.

Fig. 6. Acoustic pressure field distributions when the robust factor ξ values are respectively 0.5 ((a) and (b)) and 2 ((c) and (d)) for the cases of only scatter ((a) and (c)), and a scatter wrapped in a 10-SNM layer shell ((b) and (d)).

According to the previous derivation process, it can be found that the selection of single negative material has no bearing on the parameters of the background medium and the scatter. Here, we investigate the scatterings of the multilayer structure under different material parameters of SNM. We consider two cases: ρα = −3ρ0, κα = 2κ0, ρβ = 3ρ0, κβ = − 2κ0, and ρα = − 0.5ρ0, κα = 0.3κ0, ρβ = − 0.5ρ0, κβ = − 0.3κ0. The corresponding acoustic field distributions are shown in Fig. 7.

As we can see from Fig. 7, the scattering becomes strong when none of the absolute values of the material parameters of SNM is equal to its counterpart of the background medium. But most of the scatterings can be suppressed by the multilayer structure, and these scatterings can be eliminated by increasing the number of layers.

Fig. 7. Acoustic field distributions when none of the absolute values of the material parameters of SNM is equal to its counterpart of the background medium. In panels (a) and (b): ρα = − ρβ = − 3ρ0 and κα = − κβ = 2κ0; in panels (c) and (d): ρα = − ρβ = − 0.5ρ0 and κα = − κβ = 0.3κ0.

Since single negative material achieved by metamaterial structure inevitably produces dissipation, it is necessary to study the scattering of the multilayer structure with dissipation in SNM. So we add a positive imaginary part to the mass density to describe the dissipation in SNM. Then the mass density used in calculation is ρα = (−1 + id)ρ0, ρβ = (1 + id)ρ0, where d is the dissipation factor. We calculate the variation of the total scattering cross section when the dissipation coefficient changes, and the results are shown in Fig. 8. It can be found that the multilayer structure composed of complementary media can still reduce the scattering by an order of magnitude when the dissipation reaches up to 10%.

Fig. 8. Relation between the total scattering cross section ratio and the dissipation factor, where σ0 represents the total scattering cross section of the scatter, σd denotes the total scattering cross section of the multilayer structure consisting of the scatter wrapped in a single negative lossy medium.

The corresponding acoustic pressure distributions are shown in Fig. 9 when the dissipation factor values are 5% and 10% respectively. The backscattering is almost zero when dissipation exists, but there exist some scatterings on the other side of the structure. The amplitude of scattering increases as dissipation increases.

Fig. 9. Acoustic pressure field distributions when dissipation exists in the multilayer structure consisting of SNM. The dissipation factors are 5% ((a) and (b)) and 10% ((c) and (d)) respectively.

The previous cloaking devices are designed according to the superlens cloaking theory, and analyzed with the scattering method. In the following we will show the possibility of the invisibility cloak design with the scattering method. As shown above, the scattering cross section is easy to obtain with this scattering method, by searching, in the realizable acoustic parameter space, for the optimum value of the cloaking shell that can make the sensor undetectable to the signals, and the scattering-cancellation-type cloak can be obtained. It is assumed that the sensor can be homogenized as a soft effective medium with acoustic parameters of 0.13ρ0, 0.15κ0, with radius of 0.5 cm. For the case without the cloak, the scattering field of this sensor excited by a plane wave source or cylindrical wave source with a wave length of 1 cm are shown in Figs. 10(a) and 10(b) respectively. The acoustic back scattering and shadow area can be clearly observed.

For simplicity, we only use two layers with the same thickness, i.e., 0.2 cm for cloaking, in which case the superlens cloaking method does not work effectively as shown in Fig. 3. The parameter space is assumed to be 0.05ρ0 ≤ ρ ≤ 3ρ0, 0.05κ0 ≤ κ ≤ 3κ0. The optimized parameter values are ρα = 0.5ρ0, κα = 0.38κ0, ρβ = 0.13ρ0, and κβ = 0.18κ0. The scattering cross section with these cloak layers decreases down to approximately 2% of the value for the case without the cloak. To verify the performance of this cloak, the acoustic fields for the sensor with the cloak are calculated and shown in Figs. 10(c) and 10(d). It can be seen that the scattering from the sensor and the corresponding shadow area are both cancelled, with the field for plane wave source or cylindrical wave source in free space recovered.

Fig. 10. Normalized acoustic pressure fields for the plane wave source ((a) and (c)) and cylindrical wave source ((b) and (d)). The scattering sensor is protected by the 2-layer shell ((c) and (d)), and unprotected shell ((a) and (b)) separately.
4. Conclusions

In this paper, the recursive numerical method is employed to study the cylindrical multilayered system and reduces the scattering from an acoustic sensor while allowing it to receive external information. The acoustic pressure field is obviously disturbed by the scattering of the bare sensor system, and the scattering can be cancelled when the sensor system is wrapped in a 10-SNM layer shell. We calculate the acoustic pressure fields in various cases. First, we analyze the scattering characteristics of the cloaking shell with different numbers of bilayers, the total scattering cross section of the structure decreases as the number of bilayers increases. Then we consider the case where the equivalent density of the bare sensor is not equal to the background medium. As expected, the scattered field can still be suppressed by the multilayered shell in this case. Further, we investigate the effect of the SNM parameter on the scattering field. When none of the absolute values of the material parameters of SNM is strictly equal to its counterpart of the background medium, the scattering of the acoustic pressure field emerges, but the scattering can still be suppressed by the multilayer shell. Finally we find that the backscattering nearly approaches to zero when acoustic dissipation induced by SNM is not very strong and some scatterings happen on the other side of the structure. The amplitude of scattering wave increases as the dissipation factor increases. Last but not least, the recursive method is efficient in calculating the scattering cross section and used to design a scattering-cancellation-type invisibility cloak.

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