Cloaking of an acoustic sensor using scattering cancellation Free

Acoustic cloaking has been a topic of much interest in recent years. Different methods of cloaking have been proposed, including the use of coordinate transformations,^1–3 anomolous resonances,⁴ and scattering cancellation.^5–9 Although ideal transformation-based cloaks prevent any sound from entering the interior region, realizable cloaks of this type have thus far proven to be inherently imperfect, allowing a portion of the acoustic energy to penetrate the interior region. Hybrid configurations have also been proposed, which improve upon the imperfect transformation-based cloaks by introducing a scattering cancellation layer.¹⁰ 

One particular application of interest for acoustic cloaks is the creation of sensors that do not scatter the field that they measure. This concept, seminally introduced for electromagnetic waves a few years ago,¹¹ would enable one to detect sound without being detected, and allow for an ideal sensor which can measure an incident field without disrupting the field of interest. One such approach has been proposed using an imperfect transformation-based acoustic cloak, in conjunction with amplification¹² or utilizing an anti-cloak¹³ to restore the field within the interior (cloaked) region. However, such a cloak/anti-cloak configuration requires materials with strong anisotropy and double-negative effective fluid properties. While such extreme material properties have been proposed and demonstrated using acoustic metamaterials, they rely on resonant microstructures and are inherently narrow band and prone to appreciable losses.

Unlike the transformation-based cloaking of an object, scattering cancellation is obtained by eliminating only the scattering in the exterior fluid medium, without eliminating the field inside the object. Details about scattering cancellation can be found in previous work¹⁴ and in the supplementary material.¹⁵ One particular approach for electromagnetic (EM) waves has been achieved using plasmonic materials, and it is referred to as plasmonic cloaking.⁶ The non-resonant nature of the scattering cancellation achieved using plasmonic cloaking leads to a field in the interior which is closely related to the incident wave, and has been demonstrated to be an excellent approach for the cloaking of EM sensors.¹¹ In this Letter, an acoustic scattering cancellation cloak for a spherical acoustic sensor is examined, and the realization of such a cloak is explored. As a demonstration of the effectiveness of such an approach, we examine the behavior of a hollow radially polarized and anisotropic piezoelectric shell including mechanical losses in both uncloaked and cloaked configurations, shown in the inset of Fig. 1(a) using the material properties listed in the supplementary material.¹⁵ 

For underwater applications, a typical configuration for an acoustic sensor is achieved using a hollow spherical shell made of a piezoelectric ceramic, or piezoceramic, which converts mechanical force to electrical voltage and vice-versa. The coupling between mechanical and electrical fields are given by canonical equations for the strain vector, $ε→$⁠, and electric potential vector, $E→$⁠, in terms of the stress tensor, T, and electric displacement tensor, D,¹⁶ 

where the coefficients s_D, g and β_T are matrices which denote the elastic compliance at constant electric displacement, the piezoelectric strain coupling, and the dielectric impermeability at constant stress, respectively, and the superscript “t” denotes the matrix transpose. For a piezoelectric acoustic sensor, the appropriate electrical boundary condition is that of an open circuit, so the electric displacement is constant and the current flow is zero. In this case, the equations for the time-varying mechanical and electrical response of the piezoceramic decouple, and the mechanical stresses and strains in the piezoceramic can be treated separately¹⁷ and then used to calculate the voltage generated by the sensor by integrating the electric field over the shell thickness,

An approach to determine the necessary properties of an acoustic scattering cancellation cloak made up of two fluid layers has been developed for the case of isotropic elastic and fluid spheres.¹⁴ Due to the electrical polarization and crystalline microstructure of piezoelectric materials, anisotropic elastic properties are observed even when operating as an acoustic receiver. Therefore, application of the scattering cancellation approach to a piezoelectric shell requires the calculation of the field scattered by a coated spherically isotropic elastic shell.¹⁵ Such an approach enables efficient numerical solution techniques to be used for related problems, such as the scattering from spherically isotropic layers¹⁸ and active cancellation using a piezoelectric shell.¹⁷ To include a spherically isotropic elastic shell into the analysis used for the bilaminate acoustic scattering cancellation cloak, a revised expression for the scattering matrix of the core can be obtained,¹⁹ thus allowing for the determination of the necessary cloaking layer properties.

We will now examine the specific example illustrated in the inset of Fig. 1(a): An underwater acoustic sensor, consisting of a hollow piezoelectric shell made from a common type of piezoceramic, PZT-4. The properties of the two fluid cloaking layers are determined based on the method described in the supplementary material,¹⁵ which are applied here to a spherically isotropic shell of PZT-4 and summarized in Table I for a design frequency of kb = 1.25, which is typical of the operating range for such an acoustic sensor. As observed previously for isotropic elastic spheres,¹⁴ the outer layer (layer 1) is denser and stiffer than the surrounding water, whereas the inner layer (layer 2) is less dense and more compliant.

A useful metric to examine the scattering reduction achieved by the cloak is the scattering gain, defined as the ratio (in dB) of the total scattering cross-section, σ_sc, for a particular configuration to that of a rigid scatterer of the same size. The scattering gain for the cloaked and uncloaked configurations are shown in Fig. 1(a). At the design frequency kb = 1.25, the cloak is found to significantly reduce the scattering strength, achieving a reduction of nearly 50 dB relative to the uncloaked case. Even away from the design frequency, moderate reductions in the scattering gain are observed, particularly at lower frequencies. Thus, although designed to operate at a specific frequency, it is observed that the cloak achieves between 20 and 50 dB in scattering reduction for all frequencies at or below kb = 1.5 and, therefore, enables broadband scattering reduction.

Although scattering cancellation is achieved by reduction of the total scattering cross-section, in principle, we expect that the sensor is also able to extract a portion of the field induced in the core region. This topic has recently been explored in great detail for the scattering cancellation of EM waves by Fleury et al.,²⁰ and here, we apply these concepts to analyze the ultimate absorption and scattering properties of cloaked acoustic sensors by including mechanical losses in the piezoelectric layer. The total amount of energy intercepted by the object is characterized by its extinction cross-section, σ_ext, and the resulting absorption is given by the absorption cross-section, σ_abs = σ_sc − σ_ext. Fig. 1(b) shows the the ratio σ_abs/σ_sc for the cloaked and uncloaked configuration considered in Fig. 1(a), with the points of maximum and minimum scattering denoted by circles and an “x”, respectively. Following the work of Fleury et al.,²⁰ the results of Fig. 1(b) can be represented in a more illustrative manner by plotting σ_abs/σ_sc versus k²σ_abs, which is shown in Fig. 1(c). In this form, it can be observed that the point of minimum scattering of the cloaked configuration, corresponding to the design frequency of the cloak, reaches the maximum value in the absorption ratio, while the maximum scattering points correspond to the maximum in the abscissa, k²σ_abs, for each configuration. In other words, it is indeed possible to enhance the ratio σ_abs/σ_sc, but only at the price of reducing the overall absorption levels. These results are consistent with those found for EM cloaked sensors, even though the sensor considered here consists of a more complicated structure made of a spherically isotropic hollow shell at finite frequencies. This example, thus, illustrates the broad applicability of this approach for examining the absorption and scattering of acoustic waves.

In addition to the scattering and absorption, a useful means of comparing the effectiveness of the cloak is to examine the total acoustic pressure field, which is presented in Fig. 2 for the uncloaked and cloaked configurations at kb = 1.25. In Fig. 2(a), the disruption of the incident plane wave due to the presence of the sensor is clearly visible. With the cloak present, however, the incident plane wave passes undisturbed around the sensor, as seen from Fig. 2(b). Even with the cloak covering the sensor, a similar pressure field is observed within the piezoelectric shell.

To further examine the performance of the sensor, the relative voltage sensitivity measured in the piezoelectric shell as function of frequency is shown in Fig. 3. At or below the design frequency, there is only a slight variation in the sensitivity due to a shift in modal resonance of the piezoelectric shell, thus demonstrating that we are able to realize a significantly less detectable sensor over this range. An additional attribute of the cloaked sensor is demonstrated by the phase of the voltage sensitivity relative to the incident wave, which is shown in Fig. 3(b). For both configurations, the phase passes through 90 degrees at resonance (denoted by the circles). However, the cloaked piezoelectric shell phase is zero at and below design frequency of kb = 1.25, meaning that it is in-phase with the impinging wave. Utilization of this property can enable precise signal fidelity with the undisturbed incident wave and is also applicable for phased array sensing.

One of the primary challenges with the realization of cloaked systems is the feasibility of creating the composite materials and/or acoustic metamaterials to achieve the necessary effective cloaking layer properties. Compared with transformation-based cloaked sensor designs, the cloaking properties required to attain the performance described above can be much more readily realized due to the use of two effective isotropic fluids, each with positive density and bulk modulus on the same order of magnitude as the host fluid. However, the use of naturally occurring fluids to create cloaking layers with the particular properties specified in Table I is impractical, and therefore must be artificially engineered. Given these conditions, realization of layer properties can be achieved using phononic crystals, also known as sonic crystals, which provide broadband performance and have been extensively studied and used to create acoustic metamaterials for transformation cloaks.²¹ 

Sonic crystals consist of an arrangement or lattice of cylinders (2D) or spheres (3D). When the wavelength is much larger than the lattice spacing, the effective properties can be modeled using quasistatic effective medium homogenization techniques for a fluid host containing elastic inclusions. In terms of the compressibility, sonic crystals are similar to traditional composite structures, and the effective bulk modulus can be expressed as²² 

where κ and κ₀ are the bulk moduli of the inclusion and host, respectively, and $ϕ$ is the volume fraction of the inclusion. Unlike the static density of a simple composite elastic structure, the quasi-static effective density of a sonic crystal is due to the inertia arising from the reactive fluid motion in and around the inclusions, and for spherical inclusions can be expressed as²³ 

where ρ and ρ₀ are the densities of the inclusion and host, respectively.

To determine the necessary inclusion material properties and volume fraction, ρ_eff and κ_eff can be set equal to the respective property of the cloaking layer and then evaluated. Fig. 4 shows a graphical illustration of this process, as a function of $ϕ$⁠. The intersection of the necessary cloaking layer properties with the effective sonic crystal properties yields the appropriate volume fraction. From Fig. 4, one can observe that for layer 1 $ϕ=0.41$⁠, and for layer 2 $ϕ=0.51$⁠. With a host medium of water, the inclusions for layer 1 should be heavier and much stiffer than water, which could be achieved using a metal, such as aluminum. Conversely, for layer 2, the inclusions should be much less dense and much more compliant than water, which could be achieved using a material such as a low-density closed-cell foam. In both cases, these materials are readily available, and illustrate that the use of sonic crystals provide a realizable way to achieving a cloaked acoustic sensor.

In conclusion, in this Letter, a scattering cancellation cloak made of two fluid layers was examined for the case of an acoustic sensor consisting of a hollow piezoelectric shell with mechanical losses. The results for a representative case were considered, and the specific cloaking layers were presented. The performance of this cloaked sensor was explored in detail, and the relation of these results to the physical bounds for absorption and scattering were demonstrated. The effectiveness of the cloaking was examined, and found to give a 20–50 dB reduction in scattering strength relative to the uncloaked configuration over the typical range of operation for an acoustic sensor. The effectiveness of the sensor with and without the cloak was also considered, while the presence of the cloak was shown to eliminate the scattered pressure at the design frequency and enabled phase-matching with the incident wave. The feasibility of achieving the necessary properties was demonstrated using sonic crystals, which could be constructed using readily available inclusion materials such as aluminum and low-density closed-cell foam in the region around the sensor. Finally, while this paper has focused on cloaking of only the three lowest scattering modes, recent work has shown that it may be possible to enhance the ratio σ_abs/σ_sc while at the same time boosting the overall absorption of a sensor by optimally cloaking multiple scattering channels at the same time.²⁴ Our results may be generalized by considering additional cloaking layers, further enhancing the performance of the proposed acoustic sensor.

Work partially supported by the Spanish Ministerio de Economia y Competitividad (MINECO) under contract No. CSD2008-66 (the CONSOLIDER program) and by the Office of Naval Research (N00014-13-1-0216 and MURI Grant No. N00014-13-1-0631).