Proof of concept of a frequency-preserving and time-invariant metamaterial-based nonlinear acoustic diode

Acoustic filters and metamaterials have become essential components for elastic wave control in applications ranging from ultrasonics to noise abatement. Other devices have been designed in this field, emulating their electromagnetic counterparts. One such case is an acoustic diode or rectifier, which enables one-way wave transmission by breaking the wave equation-related reciprocity. Its achievement, however, has proved to be rather problematic, and current realizations display a number of shortcomings in terms of simplicity and versatility. Here, we present the design, fabrication and characterization of a device able to work as an acoustic diode, a switch and a transistor-like apparatus, exploiting symmetry-breaking nonlinear effects like harmonic generation and wave mixing, and the filtering capabilities of metamaterials. This device presents several advantages compared with previous acoustic diode realizations, including versatility, time invariance, frequency preserving characteristics and switchability. We numerically evaluate its efficiency and demonstrate its feasibility in a preliminary experimental realization. This work may provide new opportunities for the practical realization of structural components with one-way wave propagation properties.


I. NUMERICAL MODELS
We first provide a detailed geometrical description of the filtering barriers FBi (with i ∈ [1, 2, 3]) presented in Fig. 1 of the main text, along with their dynamical behaviour in terms of band gaps (BGs) and pass bands.  Table I and the mechanical properties given in the text.  Table I and are chosen so as to provide proper filtering characteristics in specific frequency ranges (see Fig. 1 of the main text and

II. NUMERICAL RESULTS
In this section, dispersion diagrams for the regions described in the previous section are A. Dispersion diagrams of the A1, A2 and A3 regions Figure SM3 presents the band diagram for the unit cells composing regions A1, A2 and A3 in terms of reduced wavevector k * = k x ·ai/π with ai given in Table I. It is possible to see how the region A1 (Fig. SM3a) provides filtering lower frequencies (f < f 1 ) whereas regions A2 and A3 (Fig. SM3b,c) filter f > f 1 (BGs are highlighted by light pink rectangles).
B. Dispersion diagrams of the FB2 Figure SM4 presents the band diagram for FB2 in terms of reduced wavevector k * = k x · A/π with A given in Table II Taking for simplicity a quadratic nonlinearity, as stated in the main text, the generation of the frequency mixing in the sidebands is proportional to the product of the amplitudes of the two mixed frequency sources (A 1 and A 2 for sources S 1 and S 2 , respectively), through the nonlinear parameter (β 1 ). Then, we expect the amplitude of the sub-harmonic, f 0 , to be Similarly, for the generation of the harmonics of f 0 in the second nonlinear zone (with nonlinear parameter β 2 ), we expect the amplitude of the output to be From Eq. SM2 it is evident that the efficiency, e, which can be defined as the ratio between the squared input and output amplitudes, can be approximately estimated as:

FIG. SM5: Spectral content of the wavefield during Left to Righ (a-d) and Right to Left
To verify this hypothesis, we performed a specific parametric study of a 1-D device working at f 1 = 1.2 MHz (so that f 2 = 1.8 MHz and f 0 = 0.6 MHz). The principal propagating medium (the matrix) is an alluminium bar (ρ 1 = 2700 kg/m 3 , E 1 = 70 GPa), 180 mm in length. The phononic crystal is realized by alternating layers of alluminium and a second material with reduced modulus and density (ρ 2 = 200 kg/m 3 , E 2 = 7 GPa, e.g. wood). The three filtering barriers are designed by varying the characteristics of the unit cell (size and filling factor) as reported in Table III. FB2, in this case, is realized as a sequence of two phononic crystals (FB2 A and FB2 B ), in order to maximize the width of the filter and to cut frequencies above f 0 .
To verify Eq. SM2 and estimate the intensity of the nonlinear parameters and proportionality between the two nonlinear zones, in order to obtain the desirable efficiency in this 1-D model, we performed a series of simulations varying the quadratic nonlinear parameters of the two nonlinear zones (β 1 and β 2 ). As shown in Fig. SM6, simulations highlight the linear and quadratic dependence of the output vs input amplitudes on the nonlinear parameters.  FFT of the signal obtained with a propagation experiment.

V. EXPERIMENTAL RESULTS
To verify the functionality of the experimental set-up, we study here the spectral content of the signal generated by the transducer S1 (Fig.SM9a)