The ability to control and direct acoustic energy is essential for many engineering applications such as vibration and noise control, invisibility cloaking, acoustic sensing, energy harvesting, and phononic switching and rectification. The realization of acoustic regulators requires overcoming fundamental challenges inherent to the time-reversal nature of wave equations. Typically, this is achieved by utilizing either a parameter that is odd-symmetric under time-reversal or by introducing passive nonlinearities. The former approach is power consuming while the latter has two major deficiencies: it has high insertion losses and the outgoing signal is harvested in a different frequency than that of the incident wave due to harmonic generation. Here, a unique approach is adopted that exploits spatially distributed linear and nonlinear losses in a fork-shaped resonant metamaterials. This compact metamaterial design demonstrates asymmetric acoustic reflectance and transmittance, and acoustic switching. In contrast to previous studies, the non-Hermitian metamaterials exhibit asymmetric transport with high frequency purity of the outgoing signal.

Controlling wave propagation using metamaterials—composite materials with a specifically tailored impedance profile—has led to many unprecedented technologies in recent years. Photonic (Joannopoulos et al., 2008; Saleh and Teich, 1991) and phononic (Deymier, 2013) crystals, negative and modulated index materials (Eleftheriades and Balmain, 2005; Lakes, 2001; Nassar et al., 2017), cloaking (Werner and Kwon, 2014), and super-resolution systems (Zalevsky and Mendlovic, 2004) are testaments to the increasing ability to design and fabricate dielectric and mechanical impedance profiles of the underlying structures that can modify the propagation of light and sound to yield exotic wave phenomena. Prospects of utilizing loss or absorption as a critical ingredient in metamaterials is emerging as a new design paradigm (El-Ganainy et al., 2018; Hodaei et al., 2017; Konotop et al., 2016; Kottos and Aceves, 2016; Suchkov et al., 2016). Tailoring loss, together with a judicious design of the impedance profile, can lead to the realization of devices with unconventional properties and novel functionalities. The best-known example of such non-Hermitian methodologies is the creation of Parity-Time symmetric materials. These materials steer light (El-Ganainy et al., 2007; Makris et al., 2008; Musslimani et al., 2008) and sound (Fleury et al., 2015a; Shi et al., 2016; Zhu et al., 2014) propagation using loss and/or gain, in stark contrast to the previous approaches in photonic and phononic devices (Boechler et al., 2011; Liang et al., 2010) and metamaterials (Haberman and Guild, 2016; Lee et al., 2012; Milton et al., 2006) that primarily employ differences in the real part of the dielectric or mechanical impedance of their constituents or purposefully insert defect modes within a bandgap. Novel technologies that can emerge from the manipulation of loss and gain include shadow-free sensing (Fleury et al., 2015a), unidirectional transparency (Zhu et al., 2014), coherent perfect absorption (Merkel et al., 2015; Song et al., 2014), and asymmetric transmission (Popa and Cummer, 2014). Active elements are incorporated through piezoelectric and piezoacoustic effects in those systems to provide gain and loss. For instance, speakers (Fleury et al., 2015a; Shi et al., 2016) and nonlinear electronic circuits (Popa and Cummer, 2014) have been used to control the flow of sound. Nevertheless, such elements face challenges in terms of energy consumption and physical size. Passive conservative nonlinearities have also been used for achieving asymmetric transmission (Boechler et al., 2011; Lepri and Casati, 2011; Lepri and Pikovsky, 2014). In many of these conservative nonlinear systems, the output signal is harvested at a different frequency than the input signal.

Here, we demonstrate asymmetric acoustic reflection in a linear non-Hermitian metamaterial (LnH-MetaMater), and asymmetric acoustic reflection and transmission in a nonlinear non-Hermitian metamaterial (nLnH-MetaMater). In contrast to the majority of existing nonlinear acoustic rectifiers [see recent reviews, Fleury et al. (2015b), Maznev et al. (2013)], our design provides a compact subwavelength system with high frequency purity that utilizes the natural losses of the constituent materials without using any external power.

To realize asymmetric acoustic energy transport exploiting non-Hermiticity, we designed a metamaterial such that the two resonant components of the metamaterial have dissimilar imaginary parts of the complex resonance frequency while the real parts remain approximately equal. The metamaterial consists of two air-coupled tuning forks that resonate at a desired matched frequency: a metal fork made of aluminum (Al) (density ρ = 2700 kg m−3, Young's modulus E = 69 GPa, Poisson's ratio ν = 0.33) and a polymer fork made of polycarbonate (PC) (ρ = 1200 kg m−3, E = 3 GPa, ν = 0.4). We modify their resonance mode of interest to be at the same frequency by tailoring the geometry of each fork. We utilize both the principal and the sway modes of resonance to fabricate two distinct metamaterials for this study. For each metamaterial, we fabricate a pair of forks that are frequency matched at a specific mode, i.e., a pair of forks that are frequency matched at the principal mode, and another pair of forks that are frequency matched at the sway mode. At the matched frequency of resonance, the Al fork acts as the low-loss element while the PC fork acts as the high-loss element. An intriguing feature of this design approach is that it permits constituents with different material properties and geometries provided the resonators resonate at the same frequency. It allows designing metamaterials with dissimilar imaginary parts without affecting the ability to maintain similar real parts.

A tuning fork has two primary modes of resonance (Rossing, 1992): (i) the principal mode at which the tines of the forks move out-of-phase [Figs. 1(a) and 1(b)] and (ii) the sway mode at which the tines of the forks move in-phase [Figs. 1(c) and 1(d)]. We fabricated two forks from each material employing different dimensions to have their respective resonant frequencies matched at approximately 2175 Hz for principal mode (corresponds to a wavelength in air of 159 mm) and at approximately 1925 Hz for the sway mode (corresponds to a wavelength in air of 180 mm). The principal mode is less susceptible to the fork stem mounting boundary conditions as the oscillation of the tines occurs about a node at the top of the stem [node 1 in Figs. 1(a) and 1(b)]. The response of both the Al and PC forks at their principal mode is linear and independent of the excitation amplitude for the range of incident sound pressure amplitudes we use in this study [node 1 in Figs. 1(a) and 1(b); see Fig. S2 in the supplemental material1) while exciting the forks at the sway mode induces strong nonlinear behavior in the PC fork [Figs. 1(c) and 1(d); see Fig. S3 in the supplemental material1). We characterized the loss behavior of each fork by its full-width-at-half-maximum (FWHM) of the resonance. The measured loss (FWHM) at the principal mode is 5.60 ± 0.40 Hz for the Al fork, and 22.08 ± 0.84 Hz for the PC fork for all excitation amplitudes [Figs. 1(a) and 1(b)]. Our experiments also indicate that the transmittance and reflectance are not equal for Al and PC forks (see Fig. S2 in the supplemental material1).

FIG. 1.

(Color online) Resonant behavior of the tuning forks at their principal and sway modes. (a) and (b) Absorbance at the individual resonances of the Al (a) and PC (b) forks at their principal mode (∼2175 Hz); insets show the schematic illustration of the principal mode displacement of the respective tuning forks. (c) and (d) Absorbance at the individual resonances of the Al (c) and PC (d) fork at their sway mode (∼1925 Hz); insets show the schematic illustration of the sway mode displacement of the respective tuning forks. (e) Dependence of the peak frequencies (solid lines) and the losses (FWHM) (dashed lines) of the sway mode of resonance on the excitation amplitude (Pi). (f) Dependence of resonance frequency of the PC fork on its loss, γ (FWHM).

FIG. 1.

(Color online) Resonant behavior of the tuning forks at their principal and sway modes. (a) and (b) Absorbance at the individual resonances of the Al (a) and PC (b) forks at their principal mode (∼2175 Hz); insets show the schematic illustration of the principal mode displacement of the respective tuning forks. (c) and (d) Absorbance at the individual resonances of the Al (c) and PC (d) fork at their sway mode (∼1925 Hz); insets show the schematic illustration of the sway mode displacement of the respective tuning forks. (e) Dependence of the peak frequencies (solid lines) and the losses (FWHM) (dashed lines) of the sway mode of resonance on the excitation amplitude (Pi). (f) Dependence of resonance frequency of the PC fork on its loss, γ (FWHM).

Close modal

In contrast to the principal mode, the sway mode oscillation is about a node at the bottom of the stem and it strongly couples to the system boundary through the fork's stem—a mechanism that leads to a strongly nonlinear response [Figs. 1(c) and 1(d)] at the sway-mode resonance. The strength of this nonlinear response is typically determined by the material properties. Its nature—dissipative or conservative—can be quantified by a detailed analysis of the variations of the position of the resonance peak (for conservative) and of the FWHM of the resonance peak γ (for dissipative) as a function of the excitation amplitude. In the Al fork, for example, we find that the FWHM of its sway-mode resonance is minimally affected by the excitation amplitude [Fig. 1(e)]. We therefore consider the resonance peak of the Al fork [Fig. 1(e)] to be affected only by conservative nonlinearities present within the range of excitation amplitudes used in our experiments.

The PC fork, however, shows a strong increase of the FWHM of the sway mode as a function of the excitation amplitude, Pin [Fig. 1(e)], a clear indication that the PC fork experiences nonlinear losses. The observed shift in the resonance peak [Fig. 1(e)] is primarily a direct consequence of these nonlinear losses. This dependence of resonance peak frequency (ω) on the loss (γ) is modeled as ω=ω02αγ2, where ω0 is the linear resonance frequency of the PC fork, and α is a fitting parameter [Fig. 1(f)]. In the case of the PC fork, the losses are nonlinear and this is also reflected in the resonance shift and peak broadening. The agreement of the experimentally observed response with the above damped-oscillator model allows us to exclude the existence of any other strong conservative nonlinear mechanisms in the resonance of the PC fork.

Our LnH-MetaMater consists of an Al fork and a PC fork that have the same resonance frequency at their principal mode, and our nLnH-MetaMater consists of a different pair of Al and PC forks designed to have their resonance frequency matched at the sway mode. The forks are placed at a designated gap, G, thereby forming a resonance cavity between them. We devised a two-port acoustic testing setup [Åbom, 1991; ISO 10534-2:1998(E)] to experimentally investigate the acoustic transport in the LnH-MetaMater and nLnH-MetaMater (Fig. 2; see Fig. S1 in the supplemental material1). The forks are placed inside an impedance tube where the plane wave from a speaker is incident on the metamaterial in port-A. The incident wave is partially transmitted into port-B while the rest is absorbed and/or reflected back into port-A. Back reflections from the end of port-B are minimized by absorbent foam. Two pressure-field impedance-tube microphones in each port are used to measure the incident and reflected sound waves. We implemented a lock-in amplification system (Molerón et al., 2015) for phase-sensitive measurements of the waves (see experimental techniques section in the supplementary material1). From the measured pressure, we calculated the reflectance [R=(PrPr*/(PiPi*))] and transmittance [T=(PtPt*/(PiPi*))], where Pi is the incident pressure wave, Pr is the reflected pressure wave, Pt is the transmitted pressure wave, and * indicates the complex conjugate; see the Methods section in the supplementary material.1 We used two configurations of the metamaterial corresponding to left and right wave incidence on the sample as shown in Fig. 2 to investigate the directional wave propagation characteristics. The pair of tuning forks, the center-to-center distance between them, and the rest of the experimental setup remain the same between these two configurations. Since the structure incorporates losses, the sound flux is not conserved, and the absorbance A is defined as A1TR.

FIG. 2.

(Color online) Experimental apparatus used for measuring the asymmetric acoustic energy transport. The metamaterial is embedded in a two-port acoustic testing system for phase-sensitive measurement of the incidence, reflection, and transmission: (a) the wave incident from the left (incident first on Al fork) (inset shows a magnified view of the forks embedded in the waveguide); (b) the wave incident from the right (incident first on PC fork).

FIG. 2.

(Color online) Experimental apparatus used for measuring the asymmetric acoustic energy transport. The metamaterial is embedded in a two-port acoustic testing system for phase-sensitive measurement of the incidence, reflection, and transmission: (a) the wave incident from the left (incident first on Al fork) (inset shows a magnified view of the forks embedded in the waveguide); (b) the wave incident from the right (incident first on PC fork).

Close modal

To investigate the asymmetric wave propagation in the LnH-MetaMater, we plot the degree of asymmetry in reflectance, transmittance, and absorbance characterized by QR=(RPMRMP)/(RPM+RMP), QT=(TPMTMP)/(TPM+TMP), and QA=(APMAMP)/(APM+AMP) as functions of the frequency and the incident sound pressure level (SPL) (Pi) (Fig. 3). Here, the subscript MP of R, T, and A denotes that the wave is first incident on the metal (Al) fork (left incidence), and the subscript PM denotes the wave is first incident on the polymer (PC) fork (right incidence). The color density quantitatively describes the QR, QT, and QA in each plot. The two forks in the LnH-MetaMater have the same design frequency of ∼2175 Hz, but different losses at the principal mode of resonance [Figs. 1(a) and 1(b)]. The reflectance [Fig. 3(c)] exhibits directional asymmetry at the resonance frequency while the transmittance [Fig. 3(d)] remains nearly symmetric. The color on the positive scale of QR implies that the intensity of the reflectance corresponding to the right incidence is higher than that of the left incidence (also see Fig. S4 in the supplemental material1). The slightly negative constant value of the entire transmission asymmetry plot is an outcome of the small differences between the left and right configurations of the experimental setup.

FIG. 3.

(Color online) Experimental response of the LnH-MetaMater. (a) and (b) Illustrations of the asymmetric response of the LnH-MetaMater for left and right incidences. Asymmetry in (c), reflectance (QR), (d) transmittance (QT), and (e) absorbance (QA) as functions of frequency and incident sound pressure amplitude Pi for a gap between the forks, G = 157 mm (∼λ) [the color scale −0.3 to +0.3 on the right shows the range of values for all the asymmetries in (c)–(e)].

FIG. 3.

(Color online) Experimental response of the LnH-MetaMater. (a) and (b) Illustrations of the asymmetric response of the LnH-MetaMater for left and right incidences. Asymmetry in (c), reflectance (QR), (d) transmittance (QT), and (e) absorbance (QA) as functions of frequency and incident sound pressure amplitude Pi for a gap between the forks, G = 157 mm (∼λ) [the color scale −0.3 to +0.3 on the right shows the range of values for all the asymmetries in (c)–(e)].

Close modal

The asymmetric reflectance is a direct consequence of non-uniform losses present in the metamaterial and their influence at various reflection paths that contribute to the left and right total reflections. Incident sound waves from the right (PC) side will have a total reflection that can be written in the form of a geometric series RPM=rPC+tPCrAltPC+tPCrAlrPCrAltPC+=rPC+tPCrAl(1/(1rPCrAl))tPC, where tPC/Al and rPC/Al are the transmission and reflection amplitudes of the PC/Al forks, respectively. Similarly, the total reflection for incident waves encountering the metamaterial from the left (Al) side can be written as RMP=rAl+tAlrPCtAl+tAlrPCrAlrPCtAl+=rAl+tAlrPC(1/(1rPCrAl))tAl. The convergence of the geometric series is guaranteed since |tPC|2,|rPC|2,|tAl|2,|rAl|2<1 due to absorption [Fig. 1(a) and 1(b) and Fig. S2 in the supplemental material1]. Direct comparison of RMP with RPM indicates that these two expressions are not the same. Their difference is already evident at the first term RPM/MPrPC/Al, which represents wave-paths associated with direct reflections from the PC/Al fork.

At the principal-mode resonance frequency, we observe that QR > 0 [Fig. 3(c)] and QA < 0 [Fig. 3(e)] indicate that the reflection is higher and absorption is lower when the sound is incident from the right (PC fork) side. Since the PC fork is a strong-loss element, the incident waves from the right side get highly reflected immediately. This is a manifestation of an overdamping behavior (corresponding to a system with a strong imaginary part of the impedance; notice RPM in Fig. S4 in the supplemental material1). However, the waves incident on the LnH-MetaMater from the Al side (left incidence) are transmitted into the resonant cavity formed by the two forks and dwell in the cavity due to multiple scattering events between the two forks (see the various terms in the geometric series above). These multiple scattering events with the PC fork lead to an overall enhancement of the absorption and consequently decrease the overall reflection, thus resulting in asymmetric reflection (and absorption) by the LnH-MetaMater. We, therefore, have an asymmetric resonance-enhanced absorption due to the non-uniform losses in the metamaterial.

Although the reflectance and absorbance in the LnH-MetaMater are asymmetric, the transmittance is symmetric [Fig. 3(d)]. Similar considerations as those for the reflectance indicate that the transmission paths for both left and right incidences contain the exact same scattering events resulting in symmetric transmission. Additionally, it is evident from these color density plots that the response of the LnH-MetaMater does not show any dependence on the amplitude of the incident pressure wave, as expected for a linear system.

The nLnH-MetaMater, on the other hand, consists of an Al fork and a PC fork having closely matched frequencies (∼1925 Hz), but very dissimilar losses at the sway mode of resonance [Figs. 1(c) and 1(d)] with the PC fork exhibiting predominantly dissipative nonlinearities [Figs. 1(e) and 1(f)]. The nLnH-MetaMater exhibits left and right asymmetry in transmittance [Fig. 4(d)] in addition to asymmetric reflectance [Fig. 4(c)] and absorbance [Fig. 4(e)]. The transmission asymmetry is most pronounced in the regions near resonance, and is predominantly a result of the dissipative nonlinearity [i.e., nonlinear imaginary part of the resonance frequency; Figs. 1(d)–1(f)] present in the PC fork [the contrast ratio of transmission asymmetry (TMP/TPM), which reaches values above 5 within the range of amplitudes tested, can be seen in Fig. S7 in the supplemental material1]. Because we utilize dissipative nonlinearity, the outgoing signal has high frequency purity unlike in the use of conservative nonlinearities (Boechler et al., 2011; Lepri and Casati, 2011; Lepri and Pikovsky, 2014), i.e., there are no significant higher harmonics observed, for example, as seen in the frequency response of the nLnH-MetaMater for the incidence at 1903 and 1919 Hz [Figs. 4(f) and 4(g)]. These two incidence frequencies correspond to the highest asymmetry in transmission [Fig. 4(d)]. As shown in the frequency response measured in the empty impedance tube with no forks [Fig. 4(h)], the higher harmonics seen in Figs. 4(f) and 4(g) have been generated by the speaker.

FIG. 4.

(Color online) Experimental response of the nLnH-MetaMater. (a) and (b) Illustrations of the asymmetric response of nLnH-MetaMater for left and right incidences as a function of frequency and incident sound pressure amplitude Pi at the gap between the forks G = 157 mm (∼λ). Asymmetry in (c) reflectance (QR), (d) transmittance (QT), and (e) absorbance (QA) [the color scale -0.6 to +0.6 on the right shows the range of values for all the asymmetries in (c)–(e)]. Frequency response of the nLnH-MetaMater for the (f), left incidence, (g), right incidence at the incidence frequencies (1903 and 1919 Hz) corresponding to the two highest transmission asymmetries. (h) Frequency response in the empty impedance tube for the speaker excitation at 1903 and 1919 Hz.

FIG. 4.

(Color online) Experimental response of the nLnH-MetaMater. (a) and (b) Illustrations of the asymmetric response of nLnH-MetaMater for left and right incidences as a function of frequency and incident sound pressure amplitude Pi at the gap between the forks G = 157 mm (∼λ). Asymmetry in (c) reflectance (QR), (d) transmittance (QT), and (e) absorbance (QA) [the color scale -0.6 to +0.6 on the right shows the range of values for all the asymmetries in (c)–(e)]. Frequency response of the nLnH-MetaMater for the (f), left incidence, (g), right incidence at the incidence frequencies (1903 and 1919 Hz) corresponding to the two highest transmission asymmetries. (h) Frequency response in the empty impedance tube for the speaker excitation at 1903 and 1919 Hz.

Close modal

The transmission asymmetry can be understood qualitatively by using a geometric series consideration similar to that of the LnH-MetaMater. The terms that describe the sound-wave-paths for left (Al side) and right (PC side) incidences include “time-reversal” pairs such as tAltPC and tPCtAl, tAlrPCrAltPC, and tPCrAlrPCtAl, etc., which regardless of containing the same sequence of scattering events with PC and Al, but in reverse order, are no more equal. Specifically, the transmission tAl/PC and reflection rAl/PC terms are amplitude dependent. As a result, they are affected by the direction of the sound incidence in the metamaterial (air-coupled forks). For example, if the wave is incident on the PC side of the nLnH-MetaMater, it will strongly activate the nonlinear mechanisms of the PC fork. On the other hand, if the wave is incident on the Al side of the nLnH-MetaMater, only a smaller portion of the energy arrives at the PC fork to activate nonlinear mechanisms. Therefore, the transport characteristics, tPC and rPC, will be different for left and right incidences. The exact summation of such nonlinear (amplitude dependent due to incidence direction) series in the case of the nLnH-MetaMater is a difficult task. Nevertheless, one can get a quantitative understanding of the transmission asymmetry of the nLnH-MetaMater by modeling the system as a set of two coupled oscillators.

When the principal and sway modes are well-separated spectrally, the tuning fork system can be modeled as two coupled masses [Fig. 5(a)]. By adding a semi-infinite lead of identical coupled masses on each side, we are able to investigate the scattering properties of the system. In our model, the oscillator that describes the PC fork [green circle in Fig. 5(a)] has nonlinear losses while the other oscillator that describes the Al fork [red circle in Fig. 5(a)] is modeled as a linear lossless oscillator. The oscillators are coupled with a spring that has a coupling constant k, which models the distance between the forks (the shorter the distance the higher the coupling constant k). At its left and right, this system is coupled to two semi-infinite arrays of linear non-lossy oscillators [black circle in Fig. 5(a)] that propagate the sound in free space [Fig. 5(a)].

FIG. 5.

(Color online) Numerical model: (a) Illustration of the nLnH-MetaMater model (the top and bottom labels indicate the index, n, of each mass and the spring constant, vn, of each coupling, respectively); (b) Asymmetry in transmittance QT as a function of frequency and incident intensity assuming both lossy and real nonlinearity (here, aI=0.06 and aR=0.02); and (c) Asymmetry in transmittance QT as a function of frequency and incident intensity assuming only the lossy nonlinearity (same aI as in b, aR=0).

FIG. 5.

(Color online) Numerical model: (a) Illustration of the nLnH-MetaMater model (the top and bottom labels indicate the index, n, of each mass and the spring constant, vn, of each coupling, respectively); (b) Asymmetry in transmittance QT as a function of frequency and incident intensity assuming both lossy and real nonlinearity (here, aI=0.06 and aR=0.02); and (c) Asymmetry in transmittance QT as a function of frequency and incident intensity assuming only the lossy nonlinearity (same aI as in b, aR=0).

Close modal

The equation of motion for each of the masses shown in Fig. 5(a) is given by,

mnx¨n=vn(xnxn1)vn+1(xnxn+1)[μn+f̃n(xn)]ẋn+g̃n(xn)xn,
(1)

where mn is the nth mass, xn is the displacement of the nth mass, vn is the spring constant of the spring connecting the nth mass to the (n1)th mass, μn is the linear friction coefficient, and we have assumed that the masses are at a unit distance between one another at their equilibrium points. The arbitrary functions f̃n(xn) and g̃n(xn) describe the lossy and real nonlinearity of the system, respectively. Of these parameters, μn, f̃n, and g̃n are zero for n1,2. For the spring constants and masses, we have

vn={v,n=2v0,n2mn={m1,n=1m2,n=2m0,n1,2.
(2)

Dividing Eq. (1) by m0 results in

mnm0x¨n=κn(xnxn1)κn+1(xnxn+1)[γn+fn(xn)]ẋn+gn(xn)xn,
(3)

where κnvn/m0, γnμn/m0, fnf̃n/m0, and gng̃n/m0.

By assuming harmonic solutions of the form xn=aneiωt, we obtain

ω2mnm0xn=κn(xnxn1)κn+1(xnxn+1)+iω[γn+fn(xn)]xn+gn(xn)xn.
(4)

From Eq. (4) and using the lead values for κn, γn, fn, and gn, it can be shown that the leads—where we prepare the incident propagating waves from −∞ and measure the scattering outgoing propagating waves (to −∞ for reflected and to +∞ for transmitted)—can support propagating waves with a dispersion relation

ω=2κ0(1cosk),
(5)

where k is the associated wavenumber of the propagating wave with frequency ω.

The transport properties of the coupled nonlinear oscillators of Eqs. (2) and (4) can be further investigated using the backward transfer map

xn1=κn+1κnxn+1+(1+κn+1κnω2κnmnm0iωκn[γn+fn(xn)]gn(xn)κn)xn,
(6)

which super-imposed with appropriate scattering boundary conditions can allow for a unique solution of the nonlinear problem. We assume incident and transmitted plane waves of the form

xn={i0eikn+r0eiknn1t0eiknn2,
(7)

where i0, r0, and t0 are the complex amplitudes of the incident, reflected, and transmitted waves, respectively. Scattering boundary conditions [Eq. (7)] assume a left incident wave. For a right incident wave one needs to modify Eq. (7) accordingly.

We can obtain i0 and r0 in terms of x1 and x2 from Eq. (7),

i0=x1x0eikeikeik,r0=x1x0eikeikeik.
(8)

These relations, combined with Eqs. (5) and (6), allow us to obtain i0 and r0 in terms of t0 and k by recursively calculating the displacement of each of the masses, and in turn we can compute reflectance and transmittance for the system. This way of calculating the scattering properties of nonlinear systems by fixing the output scattering conditions is necessary because of the multi-stability often present in these problems (D'Ambroise et al., 2012; Hennig and Tsironis, 1999; Lepri and Casati, 2011).

We evaluate the transmittance T|t0/i0|2 and reflectance R|r0/i0|2 using the backward map approach. First, we perform the transformation Anxnei2k/t0. This allows us to simplify the form of the wave at the output. Assuming for simplicity that m0=1, Eq. (6) becomes,

An1=κn+1κnAn+1+(1+κn+1κnmnκnω2iωκn[γn+Fn(An)]Gn(An)κn)An,
(9)

where Fn(An)fn(xn(An)), Gn(An)gn(xn(An)). From Eq. (7), we have A3=eik, A2=1. By recursively applying Eq. (9), we obtain,

A1=δκ0κeikA0=κκ0+α(δκ0κeik)δ=1+κ0κm2κω2iωκ[γ2+F2(1)]G2(1)κα=1+κκ0m1κ0ω2iωκ0[γ1+F1(δκ0κeik)]G1(δκ0κeik)κ0,
(10)

which together with Eq. (8) allows us to obtain the transmittance and reflectance,

T=|t0i0|2=|eikeikκκ0(δκ0κeik)(αeik)|2R=|r0i0|2=|κκ0(δκ0κeik)(αeik)κκ0(δκ0κeik)(αeik)|2.
(11)

As we discussed previously, the above calculations assume left-incident scattering boundary conditions, see Eq. (7). To evaluate the transmittance and reflectance for a right-incident wave one needs to flip the subscripts of m, γ, F, and G that appear in the expressions for α and δ in Eq. (10), i.e., 12,21, which is equivalent to flipping the sample.

Equation (11) represents the most general solution of the non-linear scattering problem. To make a qualitative comparison with the experimental results we assume specific functions for the nonlinear functions F1,2 and G1,2 (associated with f1,2 and g1,2, respectively). From the experimental observations of the resonant behavior of the PC fork (resonant frequency and FWHM vs SPL), we obtain the functional form of g2 and f2 vs displacement, respectively, mapping displacement to sound pressure. The functions used are

f2(x2)=aI(log10(1+|x2|))2g2(x2)=aR(log10(1+|x2|))2F2(A2)=aI(log10(1+|tA2|))2G2(A2)=aR(log10(1+|tA2|))2,
(12)

where aI, aR are real numbers. Assuming that the nonlinearity of the Al fork and its linear losses are negligible, g1, f1, and, γ1 are equal to zero. We also adjust m1 so that resonators 1 and 2 have the same resonant frequency in spite of the losses in resonator 2.

We then obtain the transmittance from the left (TL) and right (TR) for a range of incident frequencies ω and transmitted wave amplitudes t0 using Eqs. (5) and (11). Once we have these, we compute the incident wave intensity for a given frequency and transmitted wave amplitude as |i0|2=|t0|2/T. We have observed multistability (i.e., multiple possible output waves for a single input) in our results for a range of parameters (i.e., |i0|2 corresponding to more than one value of T), mostly around small frequencies and relatively large values of |i0|2 when g2(x2) is large. However, multistabilities are suppressed for the parameter range that is presented below. To verify that the model reproduces the features of the system observed in experiments, we interpolate the results of TL/R(|i0|2,ω) so that we can obtain the degree of asymmetry QT=(TRTL)/(TR+TL). Obviously, QT can only be obtained for the parameter range where both TL/R are single-valued functions of the incident intensity.

In Figs. 5(b) and 5(c), we present our computed QT vs the frequency in the x axis, which is normalized by ω0=κ0/m1, and as a function of the incident signal intensity in the y axis, for two sets of parameters: (a) combination of lossy and real nonlinearities at the resonator modeling the PC fork and (b) lossy nonlinearity only. The rest of the parameters used are κ=1.35, κ0=1, m1=1.26, m2=1.20,γ1=0,γ2=0.5. In Fig. 5(b), we observe that as the incident intensity increases and the nonlinearity becomes significant, we transition from a region of symmetry in transmission to a region of asymmetry. In the asymmetric region, for small frequencies, QT takes small values close to zero. As the frequency increases, QT transitions to positive, then back to zero and finally it takes negative values. We also observe a shift to the left of the lobe of positive QT in the color-density plot with increasing incident intensity. We notice a very similar behavior in Fig. 5(c), where the real nonlinearity is set to be zero in our calculations, with a slight increase in the shifting of the positive QT lobe. This effect can be more dramatic for different nonlinear functions. This result suggests that the dominant effect that leads to asymmetric acoustic energy transmission and its switching behavior we observe in the experiment is the presence of lossy nonlinearity.

To understand the source of asymmetry in transmittance, we examine the dependence of TL/R on the nonlinearities. For simplicity, we will assume κ0=κ and m1m2=m. Starting with the expression for transmittance in Eq. (11) and using the appropriate expressions from Eq. (10), we obtain

TL=|2sink1η[ηiγ̃2iF̃2(1)G̃2(1)]|2TR=|2sink1η[ηiγ̃2iF̃2(η)G̃2(η)]|2,
(13)

where β2(m/κ)ω2, ηβeik, γ̃2γ2(ω/κ), F̃2(x)F2(x)(ω/κ), and G̃2(x)G2(x)/κ. The discrepancy between these expressions stems from the arguments of the nonlinear functions. This is true for both the real and imaginary nonlinear terms. If we assume no real nonlinearity [inspired by the fact that Figs. 5(b) and 5(c) are similar] and small values of x2 so that f2(x2)aI|x2|2 and F2(A2)aI|tA2|2, Eq. (13) can be further reduced to the form

TL=|AB+iΓ|2TR=|AB+i(Γ+Γ̃)|2,
(14)

with A2sink, B1η2, Γη[γ̃2+F̃2(1)], Γ̃ηF̃2(1)(|η|21). Equation (14) demonstrates how the asymmetry arises even in a simple case of lossy non-linearity.

From the numerical modeling, we have found that the transmittance of an incident sound wave from the right (PC) side of the nonlinear lossy oscillator is TPM=|A/(B+i(Γ+Γ̃))|2. Similarly, the transmittance of an incident wave from the left (Al) side of the linear lossless oscillator is TMP=|A/(B+iΓ)|2. The variables A and B contain “linear” information such as the frequency of the incident wave, the dispersion properties of the propagating medium, structural/geometric information of the resonant cavity. The variable Γ includes information about the linear and nonlinear losses associated with the lossy oscillator. Typically, Γ is an increasing function of the excitation amplitude and is responsible for the nonlinear resonant shift toward lower frequencies that we observe in the transmission spectrum of the nLnH-MetaMater (Fig. 5). The term that is responsible for the asymmetry in the transmittance is Γ̃, which is also an increasing function of the excitation amplitude. It incorporates nonlinear (i.e., amplitude-dependent) losses and directionality—it is zero for left-incident waves and different from zero for right-incident waves. In the latter case, it adds to the existing reciprocal losses Γ, thus “pushing” the resonant transmittance for the right-incident wave further to lower frequency values. In other words, TPM/MP(ω) gets their corresponding maximum values at different resonant frequencies ω=ωPMres<ωMPres, respectively. The separation between ωPMres,ωMPres becomes larger as the amplitude of the incident wave (and thus Γ̃) increases. Consequently, the transmittance asymmetry QT(ωPMres)>0 as the amplitude of the incident wave increases. This behavior further verifies our experimental results shown in Fig. 4.

We also find from the coupled oscillator model that the asymmetric transport can additionally be affected by the coupling strength between the oscillators. By changing the center-to-center gap, G [see Fig. S1(b) in the supplemental material1], between the two forks in the nLnH-MetaMater, we demonstrate the effect of coupling strength [see Figs. S5 and S6 in the supplemental material1). This suggests that the asymmetric energy transport in the nLnH-MetaMater can be controlled not only as a function of the incident amplitude, but also as a function of the gap between the constituent resonant elements. Interestingly, for a gap G = 38 mm, the reflection is completely suppressed (RMP = 0) for left incidence, leading to a unidirectional extraordinary absorber with absorbance greater than 80% [see Figs. S6(g)–S6(i) in the supplemental material1].

Transmission asymmetry is forbidden by the reciprocity theorem in the case of linear time-reversal symmetric systems, consistent with the symmetric transmittance in LnH-MetaMater that we discussed initially. In contrast, asymmetric acoustic transport can be realized in systems with conservative nonlinearities and in the absence of mirror symmetry (Boechler et al., 2011; Lepri and Casati, 2011; Lepri and Pikovsky, 2014; Liang et al., 2009). In such cases, however, the outgoing signals at the fundamental frequency are noisy because of high insertion losses due to a high impedance mismatch (Lepri and Casati, 2011; Liang et al., 2009, 2010; Maznev et al., 2013). Instead, one may utilize the existence of large asymmetric frequency conversion for efficiently transmitting power from one side to the other. The obvious drawback of this approach is that the harvested signal is at different frequencies than the input signal. Use of amplifiers can overcome insertion losses (Popa and Cummer, 2014), which, however, makes the structure bulky while consuming external power. Our compact (thickness ∼subwavelength) nLnH-MetaMater overcomes both of those challenges and provides asymmetric transmission in the same frequency as the incident wave, while consuming no power. Designing an nLnH-MetaMater using different lossy materials such as polymers, domain-switching ferroelectric materials, and carbon nanotube foams where the loss can be controlled actively will enable the creation of novel devices that can control the sound propagation in desired ways for various applications. Acoustic switches, diodes, and unidirectional perfect absorbers are some of the many potential devices that can be developed using nLnH-MetaMater.

Details of the experimental methods, sample fabrication, and additional data can be found in the supplementary material.1

We acknowledge the financial support from the Air Force Office of Scientific Research via a Multi-University Research Initiative (Contract No. FA9550-14-1-0037). R.T. additionally acknowledges the financial support from the University of Wisconsin-Madison. We also acknowledge David Smith, Michael Bonar, and Gary L. Woods of Rice University, and Miguel Moleron of ETH Zurich for their support during the acoustic testing apparatus development. The authors declare no competing financial interests. R.T. and R.M.B. contributed equally to this work.

1

See supplementary material at https://doi.org/10.1121/1.5114919 for experimental methods and additional data.

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Supplementary Material