Asymmetric propagation using enhanced self-demodulation in a chirped phononic crystal

Asymmetric propagation of acoustic waves is theoretically reported in a chirped phononic crystal made of the combination of two different nonlinear solids. The dispersion of the system is spatially dependent and allows the rainbow trapping inside the structure. Nonlinearity is used to activate the self-demodulation effect, which is enhanced due to the particular dispersion characteristics of the system. The main feature of the device is that integrates the nonlinear and the filtering effects in a single propagating medium. The performed numerical study reveals an efficient generation of the demodulated wave, up to 15\% in terms of the pressure amplitude, as well as strong attenuation for undesired frequency components above the cut-off frequency. The obtained energy rectification ratio is in the order of $10^4$ for the whole range of amplitudes employed in this work, indicating the robustness of the asymmetry and non-reciprocity of the proposed device for a wide operational range.


I. INTRODUCTION
There is an increasing interest in non-reciprocal acoustic devices, those presenting an asymmetric propagation of waves due to the breaking of the time reversal invariance. In certain cases, such devices are able to transmit acoustic waves only in one direction. Inspired by their corresponding electrical counterparts, many interesting applications of this effect have been suggested, and the corresponding devices have been often called acoustic or phonon diodes, although there has been a little abuse with this denomination and must be used with care 1 . Phononics, the science of elastic/acoustic wave propagation in artificially periodic media, has been shown as an ideal platform for the design of acoustical nonreciprocal devices. Moreover, acoustic diodes have been proposed as one of the most promising applications of phononics 2 .
Once nonlinearity is considered in the wave propagation problem, it automatically introduces a plethora of rich wave phenomena in the system ? . It has been claimed that nonlinearity is needed to break reciprocity 1 . Nonlinear materials have the ability to change the spectrum of waves propagating in a medium, generating higher harmonics of a puretone input signal, or combination frequencies in the case of multi-frequency input. Original proposals of acoustic diodes exploiting nonlinear effects used highly nonlinear media, as artificial structures made of microbubbles 8 or granular chains 9 . In both cases, a nonlinear medium is located next to a periodic structure. When the wave passes first through the nonlinear medium, higher harmonics are generated, which can be blocked by the crystal if the bandgap is properly tuned. However, if we interchange input and output, the harmonics will be transmitted since the wave radiates the crystal in a propagating band and harmonics are generated later.
More recently, an acoustic diode has been proposed based on the self-demodulation effect 10 . It manifests due to the nonlinear generation of a low-frequency signal by a pulsed and high-frequency sound wave (or alternatively due to the nonlinear mixing of two highfrequency carrier waves) which produces a low-frequency wave corresponding to the difference frequency. Therefore, the proposal is based on the creation of lower, instead of higher frequencies as in Ref. 8, although the diode itself is structurally similar.
An interesting variation of conventional periodic crystals, which have been shown very useful for wave manipulation purposes, are the chirped (or graded) crystals, in which lat-tice constant is no more constant, but changes gradually along the propagation direction. Therefore, the dispersion properties change spatially along the structure and in some cases, they can be tailored to design a rainbow mirror 16,17 . In this case once the bandgap is reached, the wave is no more allowed to propagate and therefore is backward reflected with amplitude enhancement at different planes due to a progressive slowing down of the wave as it propagates along the structure. Many interesting wave-managing effects have been reported, as the mirage and superbending effect in bulk crystals 11 and waveguides 12 , or the sub-wavelength rainbow trapping [13][14][15] . Here we consider for the first time the propagation of sound or elastic waves in a chirped phononic crystal (a multilayered structure in the 1D case discussed here) in the nonlinear regime, i.e., we assume that the material forming the crystal (the layers in the 1D system considered here, as shown in Fig. 1) have itself a nonlinear response. Therefore, for a sufficiently high amplitude, nonlinearity and periodicity act together at any point in the material, differently from other proposals where the periodic and nonlinear parts of the devices are disjunct.
In order to introduce the main ingredients of our proposal, we consider a bi-cromatic input with two close frequencies, such that the bandgaps of each component are both located close to one boundary of the crystal. In the nonlinear regime, the propagation of the two frequency components creates a low frequency wave by the self-demodulation effect.
In addition, the dispersion introduced by the the chirped crystal, can be tuned to have bandgaps at the high frequency part of the wave field while the self-demodulated frequency lies in propagating regime, as schematically shown in Fig. 1. When the bi-cromatic wave impinges on the crystal from left to right, both components can propagate along the crystal before reflected, while the self-demodulated wave is generated and transmitted (see Fig. 1).
However, propagation from the opposite side is different. Incident waves find firstly the bandgaps, and they are reflected without activating the self-demodulation effect as they do not propagate, and therefore there is no transmittance at all. This asymmetric propagation is a direct consequence of the nonlinear behaviour of the system, that breaks the time reversal symmetry and provides a non-reciprocal device.
The article is organized as follows. Section II presents the chirped structure used for the investigation and describes the two configurations studied. The theoretical modeling based on a nonlinear wave equation and its numerical solution are described in Section III.
The linear properties of the system, as its dispersion relation, are described in Sec. IV A.
Section IV shows numerical results demonstrating the asymmetric propagation, and gives an estimation of the efficiency of the process. Finally, some concluding remarks are discussed in Sec. V. Reverse

II. CHIRPED PHONONIC CRYSTAL
The geometry of our proposed structure for an asymmetric propagation device is presented schematically in Fig. 1. The one-dimensional (1D) system is composed of two homogeneous slabs of material 1, coupled to a chirped phononic crystal (CPC) placed between of them. The CPC is composed of N layers of materials 2 and 3, placed alternatively at positions ranging from x = L/2 to x = 3L/2, where L represents the total length of the CPC. The spatial distribution of the chirped structure is given by the recurrence relation where i = 1, ..., N is the number of layer, a i is the thickness of the i -th layer, a 0 the first layer thickness, α the adimensional chirp parameter, and x i is the left boundary position of the i-th layer. The layer thickness increases (decreases) in the positive direction of the x-axis, hence the chirped adimensional parameter is negative, α < 0 (α > 0). The i-th layer lattice parameter is given by the sum of the thicknesses of materials 2, 3: The chirped structure length is given by L = N i=1 a i . We consider a CPC with the following values for the geometrical parameters: a 0 = 0.12 mm, α = −15.65, and N = 500, for a total length L ≈ 17, 6 cm. The structure presented in this work, is expected to be tested experimentally, therefore materials 1, 2, and 3, are selected on the basis of realistic materials as nickel-steel, aluminum and PMMA, respectively (nonlinear properties extracted

III. THEORETICAL MODEL
For the sake of simplicity, in this work we consider a purely 1D system. The source velocity vector points in the x-direction, leading only to the generation of longitudinal modes. Under this consideration, the nonlinear acoustic modes in the solid material can be described by a second order nonlinear wave equation. In analogy with nonlinear compressional waves in fluids, and up to second order of accuracy, we model the full-wave propagation using the Westervelt equation ? , which in inhomogeneous media is written as where p(x, t) is the acoustic pressure, and c 0 (x), ρ 0 (x) and β(x) are the space dependent sound speed, density and parameter of nonlinearity, respectively. Inside the multilayer structure these ambient properties are introduced as a space dependent function of alternating materials (c j , ρ j , β j , j = 2, 3) following Eq. (1).

IV. RESULTS
In this work we solve both the eigenvalue and the nonlinear propagation problems of the CPC previously defined. Although the CPC structure is not strictly periodic, if the chirp parameter α is sufficiently small, i.e., the layer thickness varies smoothly, the structure can be considered locally periodic, meaning that we can solve the eigenvalue problem at each position of the system considering that it behaves locally as an infinitely extended periodic crystal with local properties. Then, we can associate the spatial dispersion characteristics of the CPC, as demonstrated in Refs. 16 and 17.
In order to study the wave propagation inside the CPC, we will numerically solve the Eq. (2) by explicit finite differences using a computational grid of 200 elements per wavelength and a Courant-Friedrich-Levy number of 0.97. Two configurations are studied. First, in the forward configuration, "+", the incoming wave is generated at the left boundary (x = 0) and propagates in the positive direction of x -axis (see Fig. 1(a)). Second, for the reverse configuration, "−", the incoming wave is generated at the right boundary (x = 2L) and propagates in the negative direction of x -axis (see Fig. 1(b)). For both incidence directions the considered input signal is harmonic and composed of two slightly different frequencies, f 1 and f 2 , having an amplitude p 0 = p 0,1 + p 0,2 , where p 0,1 = p 0,2 = p 0 /2. We notice that for sufficiently small values of p 0 , the propagation is linear.

A. System characterization in linear regime
We begin describing the system behaviour in linear regime for the forward configuration.
The CPC is designed in order to fulfill three main conditions: (1) The two components of the input signal have been selected in such a way that their frequencies fall inside local bandgaps along their propagation through the CPC, (2) these local bandgaps are located nearly at the end of the structure, aiming the input wave propagates deep inside the structure to increase the interaction of both frequency components in the nonlinear regime, (3) the frequency of the self-demodulated wave is smaller than the cut-off frequency of the CPC. By cut-off frequency we mean the smallest frequency of the lower-edge of the spatially varying local bandgaps (see white lines in Fig. 2(a)). The local dispersion relations of every layer along the CPC are calculated analytically using the well-known Rytov formula considering the e ıωt harmonic dependence 18 cos(q i a i ) = cos(k 2 a i,2 ) cos(k 3 a i,3 ) − 1 2 where q i is the Bloch wavenumber corresponding to a periodic media made by a unit cell as the i-th layer, k j = ω/c j , where c j is the speed of sound in the layer of material j = 2, 3, and Z j = ρ j c j is the acoustic impedance. Figure 2 Fig. 2(a)). The pressure field in steady state for these frequencies is enhanced after its propagation along most part of the CPC, reaching the maximum value at different depths, as shown in Fig. 2(b). B. System characterization in nonlinear regime. Forward configuration.
The linear characterization shows how, for the selected input frequencies, the field is enhanced first and almost totally reflected later. We turn now our attention into the nonlinear self-demodulation process for the forward configuration. The pressure field distribution is shown in Fig. 3 for

C. Asymmetric propagation
The whole picture of the results obtained using the proposed asymmetric propagation device is shown in Fig. 4 for both forward (blue) and reverse (red) configurations, including all the relevant time signals (Figs. 4(a, c) for "+" direction, Figs. 4(d, f) for "−" direction), spectrum at the exit of the CPC (Figs. 4(b, e)) and the harmonic distribution of the acoustic field in space (Figs. 4(g, h)). The most relevant feature of the device is its asymmetric and non-reciprocal character, i.e., the ability to generate and transmit the difference-frequency component of the signal on one direction while not in the other. However, the device performance goes far beyond this main goal.
Firstly, the transmitted signal in the "+" direction, originated from the self-demodulation process, is efficiently generated, having a normalized pressure amplitude |p/p 0 | = 0.11 at x = 2L. In order to show that the origin of this efficient generation is related to the enhancement of the field inside the CPC, a secondary simulation is performed: the CPC is substituted by an homogeneous slab composed of material 2 (aluminum). Comparing the spectrum shown in Fig. 4(b), where the normalized intensity at x = 2L is shown in logarithmic scale, I + = 20 log(|p/p 0 |) , it is observed a nearly 20 dB reduction in the intensity of the demodulated wave in the homogeneous case (-19 dB using CPC versus -38 dB using the homogeneous medium).
Secondly, every frequency component above the cut-off frequency of the CPC is completely filtered out from the output signal in both configurations (see Figs. 4(b, e)), with intensities between -75 and -105 dB in "+" direction and -85 and -125 dB in "−" direction.
We notice that the spatial characteristics of the CPC offer greater options when selecting the input frequencies f 1 , f 2 , compared to a homogeneous slab of the same length. Fabry-Pérot resonances in the CPC are not equi-spaced in frequency, as velocity depends on it, f n = nc(f )/2L. Hence, the demodulated signal frequency can be modified just by selecting another pair of resonances whose difference lie below the cut-off frequency.
To quantify the asymmetry of the device, we make use of the energy rectification ratio 8-10 , which is defined here as σ = I(+)/I(−) in terms of the forward and reverse intensities, which are proportional to the acoustic energy. For the specific case shown in this Section, the obtained energy rectification ratio is σ = 3 × 10 4 , which is of the same order of other solutions proposed in the literature 8,9 .

D. Amplitude dependence
The amplitude dependence of the non-reciprocal device is calculated in terms of the input and output intensities, I in , I out , being I = p 2 /2ρ 1 c 1 . It is noted that due to the effective filtering performed by the CPC, the output signal, both in "+" and "−" directions, consists basically in the difference-frequency component, although in the latter case the amplitude of the wave is very small. Figure 5 In the previous Section we have seen that the enhancement of the acoustic field inside the chirped structure results in a more efficient self-demodulation process if compared to an homogeneous medium (see Figs. 4(b, e)). Here, the efficiency of the self-demodulation process in "+" direction in terms of the input pressure is also studied. Figure 5(b) shows that for every value of the input pressure the efficiency of the proposed device is considerably higher than the one obtained placing an homogeneous slab of material 1 instead of the CPC.
The amplitude dependence of the pressure shows different regimes. For small amplitude values of the input pressure, below 8 MPa, the output pressure increases linearly. As the amplitude increases the efficiency of the self-demodulation process is reduced as the energy of the input harmonics is transferred to higher harmonics. For the highest values, this energy transfer results in a slight reduction of the output pressure and as a consequence in a reduction of the efficiency.
In order to show the ranges of efficiency of our system, the dependence of the energy rectification ratio values on the input intensity are also studied and shown in Fig. 5(c). The obtained values of the rectification ratio are of the order of 10 4 for all the inputs used in this work. This means that the asymmetry and non-reciprocity character of the CMT structure is robust in terms of the intensities of the input signal, allowing a wide operational range.

V. CONCLUSIONS
A novel design of an asymmetric and non-reciprocal structure for acoustic propagation based on the nonlinear self-demodulation effect has been theoretically presented. The structure consist of a chirped phononic crystal made of nonlinear acoustic media. The numerical study performed here reveals that the structure plays a double role of frequency conversion and filtering. The enhancement of the field inside the CPC plays a key role to increase the efficiency in "+" direction compared to an homogeneous medium. The spatially varying dispersion, characteristic of chirped structures, allows to filter out the input frequency components and their higher harmonics, achieving strong attenuation values. This filter character of our structure has a direct consequence in the corresponding values of the rectification ratio, achieving a value of ≈ 10 4 for values of the input intensity that ranges between [200, 1300] W/cm 2 . The interplay between local periodicity and nonlinearity is revealed as a useful tool for the design of devices with asymmetric transmission properties.