Extreme Acoustic Metamaterial by Coiling Up Space
Acoustic metamaterials are engineered materials that exhibit unusual constitutive parameters which are usually constructed from local resonating subwavelength structures [1–9]. Like their optical counterparts, these materials give rise to many novel phenomena including subwavelength focusing [10–16], extraordinary transmission [17–19], and, very recently, invisibility cloaking [20–25]. Alternatively, we can also use phononic crystals to demonstrate similar phenomena. Negative refraction, subwavelength imaging, and gradient index focusing have been demonstrated using this approach [26–28]. Such an approach can avoid local resonance and loss, while pushing the working frequencies to the diffraction regime in which a rigorous effective medium becomes ill defined. Here, we show that by coiling up space, a phononic crystal can be constructed with band foldings at very low frequencies so that it can actually have a valid effective medium description with extreme constitutive parameters. For airborne sound, this would normally require materials such as silicone rubber, which has a very high refractive index but unavoidably high sound attenuation. In our approach, we show that such a high refractive index can be achieved by coiling up space using curled channels of subwavelength cross section. The propagating phase along these curled channels can be arbitrarily delayed in order to mimic a high refractive index. Then, a metamaterial can be constructed from the high index material without any local resonance in order to generate a large dispersive spectrum of constitutive parameters, ranging from negative refractive indices, density near zero to very positive indices. Negative refraction associated with double negativity and tunneling through a density-near-zero metamaterial in two dimensions are also numerically demonstrated. Such a principle of coiling up space can be made isotropic and it can be easily applied to three dimensions.
First, we aim to show how space can be coiled up. In acoustics, waves can propagate within the perforations of subwavelength cross sections in the absence of a cutoff frequency. This concept has been used to generate a large anisotropy [5–9] and subwavelength imaging devices that employ such anisotropy in a fixed direction [12–14]. As an acoustic wave is simply a scalar field, these perforations can be further coiled up, whereas the waves can still propagate freely in the curled space. Figure 1(a) shows a schematic diagram of our metamaterial-repeating units. Identical thin hard solid plates (e.g., brass) of thickness and length are inserted into a background fluid (e.g., air) in a way that forms a network of fluid channels of width . The red dashed lines outline the primitive unit cell of lattice constant . For example, waves now propagate along a zigzag path, shown in blue arrows, instead of a straight line from point to and the space is effectively coiled up by a factor of approximately 4.2 in total path length. The structure is then simplified to that shown in Fig. 1(b) of the same size. The “X”-shaped region now represents the fluid channels that have a higher index ( is the relative refractive index normalized to the original background fluid and is approximately the same factor, namely, 4.2), while the rest of the regions represent the hard solid plates. By using such a high refractive index with materials of negligible loss in constructing the metamaterial, the corresponding band foldings should occur at low frequencies.
A simple model to describe the band structure of the metamaterial can be obtained approximately if we assume the channels in Fig. 1(b) are narrower than a wavelength. By applying the Floquet-Bloch theory, the dispersion relation is expressed as
where and , respectively, represent the elapsed phase of a Bloch wave in the and directions for the unit cell, or is the wave number of the acoustic waves in the background fluid. From Eq. (1), the dispersion relation can be solved. Different branches of the cosine function represent band foldings. Because the metamaterial coils up the space with the same factor in both the and the directions, the equifrequency contours (EFCs) should be very close to a circle near the point ( ). This generates an isotropic refractive index for the metamaterial. The normalized frequencies for the different branches intersecting at can be found as integral multiples of . The position of the bands can thus be tuned by , or the path length of the acoustic waves in the coiled space. A longer path length is equivalent to a higher relative refractive index . This results in band foldings in the band structure at low enough frequencies at which an effective medium description with both effective density and bulk modulus is still valid near the point.
Although Eq. (1) describes the qualitative picture of the band structure of the metamaterial, we can numerically obtain the band structure with all the microstructures in a full-wave simulation (COMSOL Multiphysics). The results are shown in Fig. 2(a), where the direction is defined along in Fig. 1(a). Except for the small gaps opening at the and points (shown by insets a1 to a3) and small frequency shift because of the finite width of the fluid channels, the whole band structure is very well approximated by Eq. (1) (gray dash-dotted lines). At lower frequencies, the channel width is much smaller than the wavelength, and thus the two band structures coincide with each other, e.g., the first band. The slopes around the point in both the and directions are almost the same at the first, third, and fifth bands owing to band folding. This indicates isotropic indices and thereby also that the EFCs of the three bands [ from 0 to 0.04, from 0.18 to 0.218, and from 0.22 to 0.26] shown in Figs. 2(b) to 2(d) are almost circular with variations in radius within 5%. The different relative indices can then be extracted from the sizes of the EFCs comparing to the sound lines shown by black dashed lines in Fig. 2(a). A large relative index is found for the first band at low frequencies, which is very near to the value 5.9 given by [Eq. (1)]. At the third and fifth bands, negative indices from 0 to approximately and positive indices smaller than 1 can be obtained, respectively. We note that there is a flat band around at the edge of the band gap in the effective medium region. This band is transverse in nature. For example, by examining the pressure field profiles (results not shown) for the modes along the or direction in the simplified view Fig. 1(b), they are actually odd modes regarding the mirror plane containing the vector. Thus these modes cannot be excited by incident plane waves of longitudinal nature propagating along the same vector in this picture. However, it is not completely deaf  when we go to the microstructure view in Fig. 1(a) because the structure has undergone a mirror-symmetry breaking regarding the incident waves. Here, we only focus on the bands with the longitudinal modes in the effective medium regime.
Indeed, all the above bands of the circular EFCs only span to a normalized frequency 0.26 at which the corresponding wavelength in the background fluid is still at least a few times larger than the lattice constant. The metamaterial can thus be described using an effective medium in more detail. By calculating the complex reflection and transmission coefficients of the unit in Fig. 1(a), we can now resolve the relative effective index and impedance for the above bands . Owing to the lack of local resonances (those associated with the resonating subwavelength units in conventional metamaterials) in our approach, the material absorption losses are not amplified and we have neglected these material absorption losses in our model as a simplification. We note that there may still be some material losses in practice. The results are shown in Fig. 3(a) with values of agreeing with the band structure in Fig. 2(a). The effective density and bulk modulus relative to the background fluid are then obtained by and , which are plotted in Fig. 3(b). In this work, all the quantities , , , and are defined normalized to the background fluid. We emphasize that only a refractive index but not both constitutive parameters for a phononic crystal can be defined in the diffraction regime instead of the effective medium regime we are working with here. At the long wavelength limit, and are simply constants that also agree with the geometric considerations: where is the filling ratio and relative density where is previously found from the zigzag path length. Our approach of coiling up space is effective at achieving a high refractive index which is rare in nature. In the frequency range from 0.18 to 0.26, changes from negative to positive and crosses zero at , which is the lower edge of the band gap, while also changes from negative to positive in a similar way and crosses zero at , which is the upper edge of the band gap. Below the band gap, we thus have a frequency region of simultaneously negative and to give . Both and are dispersive in the frequency range. Contrary to the conventional approaches in overlapping two different kinds of resonances to create double negativity, we only need to coil up the space to give a large enough .
Next, we demonstrate two applications using the extreme constitutive parameters given by our approach of coiling up space. First, an acoustic prism that has simultaneously negative density and negative bulk modulus can be constructed using the same structures shown in Fig. 1(a). A corner of the prism is shown in Fig. 4(a) with the inset of a closer view of the microstructures near the edge of the 45° inclination. A Gaussian beam of width with unit amplitude impinges on the prism from the bottom at a chosen normalized frequency in the background fluid. The relative effective index is at the frequency and the beam undergoes negative refraction and exits the prism with amplitude around 0.4 due to impedance mismatch in a horizontal direction in the simulation. Some reflection also occurs at the bottom of the prism (with values outside to 1 due to interference). For comparison, we also simulated the same prism using an effective medium description where the corresponding relative density and bulk modulus ( ) are extracted from Fig. 3(b). The result is shown in Fig. 4(b) with the same Gaussian beam as the input. The output beam direction, amplitude, and phase distribution are well comparable with those in Fig. 4(a), indicating our effective medium that has both negative density and negative bulk modulus faithfully represents the metamaterial.
As another example, our structure has a density near to zero at a very low frequency. This is the analog of near zero in electromagnetism and can support a novel wave tunneling phenomenon within a waveguide . Figure 5(a) shows the pressure field for such a rectangular waveguide in the acoustic version with a plane wave impinging from the left and a hard solid plate inserted in the middle. Because the solid blocks more than half of the width of the waveguide, the plane wave is scattered severely. In Fig. 5(b), the scatterer is now enclosed by our metamaterial. In both simulations, we chose a normalized frequency which is a little bit smaller than the frequency of the lower edge of the band gap where the relative effective density is zero. The small relative effective density together with the large relative bulk modulus implies the occurrence of tunneling. From Fig. 5(b), the plane wave can now squeeze through the narrow channel and exit the waveguide as a plane wave that has high transmission again.
The presented analysis and numerical results of achieving isotropic extreme constitutive parameters are based on the physical picture of coiling up the space the same ways in both the and directions. If we tune the path length in one direction [i.e., CA in Fig. 1(a)] to become longer, anisotropic refractive indices at the first band can be obtained. We can tune the anisotropy by controlling the ratio of the total paths lengths in the two directions. Moreover, if we coiled up the space in the third direction as well, a three-dimensional metamaterial that has extreme indices can be designed similarly. In conclusion, we have proposed a new way of generating a wide dispersive spectrum of acoustic extreme constitutive parameters through coiling up space. By coiling up space, we can obtain high refractive indices in different directions and obtain band foldings at low frequencies at which both the effective density and bulk modulus can be well defined. Such an approach avoids the usage of local resonances to achieve double negativity, density near zero, and very positive refractive indices. We believe our approach in coiling up space will be useful for designing transformation acoustical devices and other acoustic devices requiring extreme constitutive parameters.
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