⇦Chinese Physics Letters, 2018, Vol. 35, No. 2, Article code 024301 Transparent and Ultra-lightweight Design for Ultra-Broadband Asymmetric Transmission of Airborne Sound * Jie Hu(胡洁)1,2**, Bin Liang(梁彬)1, Xiao-Jun Qiu(邱小军)3 Affiliations 1Key Laboratory of Modern Acoustics (MOE), Institute of Acoustics, Nanjing University, Nanjing 210023 2College of Information Science and Technology, Nanjing Forestry University, Nanjing 210000 3Centre for Audio, Acoustics and Vibration, Faculty of Engineering and IT, University of Technology Sydney, Sydney, Australia Received 27 October 2017 *Supported by the National Natural Science Foundation of China under Grant No 11634006.
**Corresponding author. Email: hujie@njfu.edu.cn Citation Text: Hu J, Liang B and Qiu X J 2018 Chin. Phys. Lett. 35 024301 Abstract Acoustic one-way manipulations have recently attracted significant attention due to the deep implications in many diverse fields such as biomedical imaging and treatment. However, the previous mechanisms of asymmetric manipulation of airborne sound need to use elaborate heavyweight structures and only work in certain frequency ranges. We propose a mechanism for designing an ultra-lightweight and optically transparent structure with asymmetric transmission property for normally incident plane waves. Instead of fabricating solids into complicated artificial structures with limited bandwidth and heavy weight, we simply use xenon to fill a spatial region of asymmetric shape which allows the incident plane wave to pass along one direction while reflecting the reversed wave regardless of frequency. We demonstrate both analytically and numerically its effectiveness of producing highly-asymmetric transmission within an ultra-broad band. Our design offers new possibility for the design of one-way devices and may have far-reaching impact on various scenarios such as noise control. DOI:10.1088/0256-307X/35/2/024301 PACS:43.20.Bi © 2018 Chinese Physics Society Article Text Recent emergence of the acoustic diode, as the acoustics counterpart of the electrical diode, gives rise to one-way manipulation of acoustic waves that are conventionally believed to always transmit symmetrically in a given path. During the past few years, asymmetric transmission of acoustic waves has attracted considerable attention due to the potential of such special kinds of devices to lead to revolutionary changes in various acoustic applications. Compared with the original design of acoustic one-way devices that only have low performance as a result of using nonlinearity to break through the limit of reciprocity,[1,2] many schemes have been proposed with substantial improvements in performance in linear regime. Thanks to the development of different mechanisms for achieving asymmetric transmission with linear structures such as those using asymmetric mode conversion, these designs are endowed with remarkable enhancement in terms of efficiency, bandwidth, thinness, etc.,[3-6] or with some special functionalities such as keeping the acoustic path open to light and heat.[7] However, the design and fabrication of acoustic metamaterials usually involve many difficulties due to their subwavelength nature, and the introduction of solids for building metamaterial units unavoidably increases the weight dramatically in comparison with air as the background medium.[8-15] Furthermore, the working bandwidths of the metamaterial-based one-way devices are still limited even for those composed of non-resonant units because they cannot work at high-frequency regime where the effective medium approximation fails.[16-22] Considering the broad spectrum of practical acoustic signals, it is highly desired to unveil a new mechanism for realizing asymmetric acoustic transmission without being restricted by the bandwidth. In this work, we make an attempt to address this issue by designing a structure capable of allowing the transmission of airborne sound only along one particular direction with theoretically unlimited bandwidth. The mechanism is that, instead of relying on artificial materials with limited bandwidth and complicated configuration, we simply use a natural material with innate dispersionless property to construct a polygon-shape model that has interfaces with different angles for the incident wave coming from the two opposite sides, regardless of the driving frequency. By judiciously designing the structural parameters of this structure, it becomes possible that the normally incident wave will be subject to total reflection only in one direction but will be guided to transmit through the system along a controlled trajectory as the incident direction is reversed. Furthermore, this material as inclusion can be chosen as another gas with high refractive indexes and low mass density, ensuring ultra-lightweight and transparent features of the resulting devices. We use analytical analysis as well as numerical simulations to demonstrate our mechanism. Firstly, we explain the mechanism underlying our scheme for producing ultra-broadband asymmetric transmission with an ultra-lightweight and optically transparent structure. For simplicity without losing generality, we consider a two-dimensional case in the current study. Figure 1 schematically illustrates the proposed structure that is simply formed by filling a polygonal region with a specific kind of gas with higher refractive index than that of the background medium (air), which is chosen as xenon here. The sound velocity and the normalized refractive index of air are $c_{\rm A}=340$ m/s and $n_{\rm A}=1$, respectively, and the acoustical parameters of xenon are $c_{\rm X}=169$ m/s and $n_{\rm X}=2.03$, respectively. The designed structure is situated inside a straight waveguide with hard boundaries on both sides for a better guidance of the incident plane wave. The two different kinds of gases, i.e., air and xenon, are separated by inserting a lightweight thermoplastic film, between which there is sufficiently thin space so that one could treat them as acoustically transparent layers. The shape of the whole structure is a convex polygon, with the interior angle of the leftmost vertex being $\alpha$ and the right side being a plane vertical to the waveguide boundary. The structural parameters of the designed model are $h=0.3$ m, $l=1.0$ m, $d=0.09$ m and $w=0.24$ m, respectively, with the parameters of $h$, $l$, $d$ and $w$ referring to the height of the structure, the length of the structure, the length of the upper and lower polygon sides and the maximum width of the whole polygon, respectively. This polygonal structure is designed for having an asymmetric transmission property for plane acoustic waves normally incident from the two opposite sides respectively, as will be demonstrated both theoretically and numerically in what follows. The mechanism of our proposed structure can be understood as follows: the acoustic wave will reflect and refract when it encounters an interface between two media (denoted as 1 and 2) with contrasting acoustic impedance, which is governed by the well-known Snell law $$\begin{align} \frac{\sin \theta _{\rm i} }{\sin \theta _{\rm t} }=\frac{c_1 }{c_2 }=\frac{n_2 }{n_1 },~~ \tag {1} \end{align} $$ where $\theta _{\rm i}$ and $\theta _{\rm t}$ refer to the incident and the transmitted angles, respectively, $c_p$ and $n_p$ are the sound velocity and refractive index, respectively, with the subscripts $p=1$, 2 referring to media air and xenon, respectively. As media 1 and 2 are chosen as air and xenon, it can be expected that the propagating wave will behave differently. When the acoustic wave propagates from air to xenon, the wave is entering a medium with higher refractive index; hence there is refraction regardless of the incident angle $\theta _{\rm i}$. The angle of refraction is smaller than $\theta _{\rm i}$ and can be calculated by $\theta _{\rm t} =\sin ^{-1}(\sin(\theta _{\rm i})/2.03)$. In contrast, the acoustic waves transmitting from xenon to air are supposed to meet a medium with lower refractive index, and will definitely be blocked due to the total reflection occurring as the incident angle exceeds the critical angle $\theta _{\rm c} =\sin ^{-1}(n_{\rm 1}/n_{\rm 2})=29.5^{\circ}$. In the current study we choose the vertex angle at the left side as a right angle (viz., $\alpha=90^{\circ}$). It is worth mentioning that, however, our scheme is universal and can yield different propagation directions for the transmitted wave by tailoring the value of $\alpha$, which should be limited in the range from roughly 75$^{\circ}$ to 121$^{\circ}$ for ensuring the incident angles in 'positive' direction (PD) and 'negative' direction (ND) cases to be smaller and larger than the critical angle respectively (viz., $\theta _{\rm i2} < \theta _{\rm c}$ and $\theta _{\rm i3} >\theta _{\rm c}$).
cpl-35-2-024301-fig1.png
Fig. 1. The schematic diagram of our proposed structure with asymmetric transmission property of airborne sound, which has a polygonal shape with a vertex on the left side and a plane on the right side for producing controlled refraction and total reflection for plane waves coming from the two opposite sides, respectively.
Our proposed scheme is based on the rational exploitation of this simple yet effective mechanism. For the specific configuration shown in Fig. 1, it would be reasonable to define the incident directions of waves coming from the left and right sides as PD and ND, respectively. When the incident plane wave comes from the left, as schematically illustrated in Fig. 1(b), one has $\theta _{\rm i1} =45^{\circ}$; and it can be readily predicted by Snell's law that the transmitted angle is $\theta _{\rm t1} =20.4^{\circ}$. Considering that $\theta _{\rm t1} +\theta _{\rm i2} =45^{\circ}$, the incident angle of the wave that propagates within the xenon filling the polygon and then impinges on the interface at the right side will become $\theta _{\rm i2} =24.6^{\circ}$, leading to a subsequent refraction angle of $\theta _{\rm t2} =57.7^{\circ}$. This suggests that the normal incident plane wave coming from the left side is allowed to pass through the whole polygonal region, and then leaves it along a predicted direction, as shown by the trajectories marked by blue arrows in Fig. 1. However, the plane wave incident from the opposite direction will impinge on the rightmost side normally and is permitted to penetrate into the xenon filling the polygonal region without changing in the propagation direction. In such a case, the incident angle on the left sides (denoted as $\theta _{\rm i3}$) will be $45^{\circ}$, which is apparently much larger than the critical angle derived above. It is therefore expected that the acoustic waves will undergo a total reflection and cannot pass, as shown schematically by the red dotted arrows in Fig. 1. As a result, the designed devices with a polygonal shape will allow only the transmission of acoustic waves coming along the PD while reflecting virtually all the incident acoustic energy propagating along the ND, leading to a substantially asymmetric transmission property. Notice that in our design we use a natural material with innately dispersionless acoustical parameters, which offers a high refractive index absolutely independent of the driving frequency in stark contrast to the previous artificial materials that have to depend on acoustic resonances to enhance the effective refractive index and will unavoidably lose their extraordinary acoustical properties when the frequency deviates from the eigen-frequency of individual units.[8] Furthermore, the introduced gas has nearly the same mass density as air and is optically transparent, thus there is no need for elaborate solid structures that increase the weight and opaqueness of the resulting device. As a consequence, our designed device can be expected to bear the advantages of ultra-broad bandwidth, ultra-light weight, total transparency and simple and low-cost fabrication. We have carried out a series of numerical simulations to evaluate the performance of our designed devices, by employing the finite-element method based on COMSOL multiphysics. Typical numerical results are shown in the following. The spatial distributions of acoustic pressures are calculated for the waves transmitting along two opposite directions at three specific frequencies, i.e., 1500 Hz, 10 kHz, 20 kHz, and are plotted in Fig. 2.
cpl-35-2-024301-fig2.png
Fig. 2. Spatial distribution of the acoustic pressure generated by the incident plane waves propagating along PD ((a), (c), (e)) and ND ((b), (d), (f)) for three particular frequencies: 1500 Hz ((a), (b)), 10 kHz ((c), (d)), and 20 kHz ((e), (f)), respectively.
To better understand the asymmetric transmission along the two different directions, here the transmission is defined as the ratio between the intensities of the transmitted wave and incident wave. It is observed that the transmission property of the proposed structure becomes nearly symmetric in the quasi-static limit. As the frequency of incident wave is extremely low, the typical dimensions of the polygonal region, e.g., the height and the maximum width that are chosen as 0.3 m and 0.24 m respectively as we have mentioned above, will become comparable with the operating wavelength (such as 0.23 m at 1500 Hz), leading to strong diffraction effects at the left part of the devices which will not yield the desired total reflection since the propagating wave is not governed by geometric acoustics in such cases. As a proof, the transmissions are numerically calculated to be 0.85 and 0.69 for PD and ND at 1500 Hz, showing that most of the incident wave energy is permitted to pass along either direction. However, when the frequency is sufficiently high, the designed structure produces a highly asymmetric transmission for the incident plane wave traveling along the two opposite directions, as proven by the numerical results shown in Fig. 3. In the ND, the transmitted waves are blocked as expected by the occurrence of nearly total reflection at the interface between xenon and air on the left side, despite it being allowed to penetrate the rightmost interface as a result of normal incidence, as illustrated by Figs. 2(d) and 2(f). Contrarily, the plane wave propagating along the PD is able to pass through the whole polygonal region and leaves it after being split into two parts. In particular, although the upper and lower parts of the wavefront of the transmitted wave overlap and lead to interference pattern in the output region of the waveguide, it can be clearly observed that these two parts of wavefront are traveling along the predesigned directions which agree quite well with the aforementioned analytical predictions obtained on the basis of Snell's law. In the PD, the transmission efficiency turns out to be very high (0.92 at both 1 kHz and 20 kHz), in comparison with the vanishing transmission in the ND (0.04 and 0.01 for these two particular frequencies, respectively). The imperfection in the one-way performance (viz., non-unity and non-zero transmissions in PD and ND) is mainly due to the wave interaction with the leftmost vertex, as evidenced by the disturbance in the acoustic fields near this point shown in the numerical results in Figs. 2(c)–2(f). Furthermore, we need to emphasize that such an asymmetric transmission should be independent of the driving frequency due to the fact that our scheme is based on a direct manipulation of the propagation trajectory for the incident wave as we discussed above. This is evidenced by the pressure fields displayed in Fig. 2, which shows that the propagation trajectories of the transmitted waves of different frequencies are always exactly the same as the results depicted in Fig. 1(c) despite the difference in their wavelengths.
cpl-35-2-024301-fig3.png
Fig. 3. The frequency dependence of the transmissions along the two opposite directions.
To better evaluate the ultra-broadband functionality of the proposed device, we have calculated and plotted the frequency dependence of the transmissions of left- and right-going waves in Fig. 3, which clearly shows how the acoustic energy transmitted along the two opposite directions varies as a function of the driving frequency. In Fig. 3, the blue and the red lines indicate the transmission along the PD and ND, respectively. In the quasi-static limit, the blue and red lines almost overlap, suggesting that the transmission is approximately symmetric when the frequency is extremely low, owing to the reason discussed above. With the gradual increase of the frequency, the designed polygonal structure becomes larger in terms of the wavelength, giving rise to the phenomenon of asymmetric acoustic transmission we have predicted on the basis of geometric acoustics. It can be observed that a highly asymmetric transmission persists throughout the spectrum except for the frequency region below approximately 1500 Hz. What deserves mentioning here is the essential difference between the ultra-broadband functionality we achieved with innately dispersionless natural materials and that possessed by the previous metamaterial-based designs for which even the non-resonant artificial structure will still become invalid when the frequency is so high that the effective medium approximation fails. In our design, both the transmissions along the two opposite directions become more stable as the frequency exceeds 6000 Hz ($>$0.9 for PD and $ < $0.1 for ND, respectively, except for some narrow transmission dips that may be caused by the reflection of hard boundaries) where geometric acoustics approximation holds quite well, and there is in principle no upper limit for such a working band.
cpl-35-2-024301-fig4.png
Fig. 4. Simulated contrast ratio as a function of frequency.
For the purpose of giving a quantitative estimation of the performance of asymmetric transmission, we further introduce a parameter of contrast ratio, defined as $r_{\rm c}=|T_{\rm PD}-T_{\rm ND}|/|T_{\rm PD} +T_{\rm ND}|$, where $T_{\rm PD}$ and $T_{\rm ND}$ are the transmissions along PD and ND, respectively, and the corresponding numerical results are displayed in Fig. 4. The more the parameter approaches 1, the more asymmetric the transmission is. The numerical results show that the value of $r_{\rm c}$ is extremely low in the low frequency regime below approximately 1100 Hz, where the diffraction effect dominates, corresponding to almost identical transmission coefficients for waves transmitting along the two opposite directions, which is consistent with the results in Fig. 3. The transmission becomes more asymmetric as the frequency gradually increases, and a near-unity $r_{\rm c} $ is observed despite slight fluctuations within a ultra-broad band beginning from approximately 6000 Hz, suggesting a high transmission efficiency along the PD while a virtually blocked transmission along the reversed direction. In summary, we have designed an ultralightweight and optically transparent structure with asymmetric transmission property. A spatial region of asymmetric shape filled with xenon is applied to permit the incident plane wave transmitting in the PD while blocking almost all waves that are incident in the ND in broadband. The asymmetric transmission behaves well at high frequency especially when the frequency is above 6000 Hz, while in the quasistatic limit, the transmission in PD is quite close to that of ND because of the size of triangle comparable with wavelength. We verify the effectiveness of the proposed design for achieving a highly asymmetric transmission of airborne sound in an ultrabroad frequency range via numerical simulations. The realization of high efficiency and broadband asymmetric acoustic manipulation by simple exploitation of naturally available materials opens up the possibility for the design of novel functional devices and may have great application in diverse fields such as noise control. References An acoustic rectifierAcoustic Diode: Rectification of Acoustic Energy Flux in One-Dimensional SystemsOne-way mode transmission in one-dimensional phononic crystal platesUnidirectional acoustic transmission based on source pattern reconstructionBroadband asymmetric acoustic transmission in a gradient-index structureUnidirectional acoustic transmission through a prism with near-zero refractive indexBroadband unidirectional transmission of sound in unblocked channelExtreme Acoustic Metamaterial by Coiling Up SpaceSubwavelength acoustic metamaterial with tunable acoustic absorptionA sound future for acoustic metamaterialsFocusing Ultrasound with an Acoustic Metamaterial NetworkDouble-negative acoustic metamaterialLocally Resonant Sonic MaterialsAcoustic MetamaterialsMembrane-Type Acoustic Metamaterial with Negative Dynamic MassSound and heat revolutions in phononicsSound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic CirculatorAsymmetric acoustic gratings P T -Symmetric AcousticsExperimental demonstration of anomalous Floquet topological insulator for soundLow-loss and broadband anomalous Floquet topological insulator for airborne soundNon-reciprocal and highly nonlinear active acoustic metamaterials
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