Nonlinear propagation and control of acoustic waves in phononic superlattices
Noé Jiménez, Ahmed Mehrem, Rubén Picó, Lluís M. García-Raffi, Víctor J. Sánchez-Morcillo
aa r X i v : . [ n li n . PS ] M a y Nonlinear propagation and control of acoustic waves in phononic superlattices
No´e Jim´enez a , Ahmed Mehrem a , Rub´en Pic´o a , Llu´ıs M. Garc´ıa-Ra ffi b , V´ıctor J. S´anchez-Morcillo a a Instituto de Investigaci´on para la Gesti´on Integrada de Zonas Costeras, Universitat Polit`ecnica de Val`encia, Paranimf 1, 46730 Grao de Gandia,Spain b Instituto de Matem´atica Pura y Aplicada, Universitat Polit`ecnica de Val`encia, Cami de Vera s / n, 46022, Valencia, Spain Abstract
The propagation of intense acoustic waves in a one-dimensional phononic crystal is studied. The medium consists ina structured fluid, formed by a periodic array of fluid layers with alternating linear acoustic properties and quadraticnonlinearity coe ffi cient. The spacing between layers is of the order of the wavelength, therefore Bragg e ff ects suchas band-gaps appear. We show that the interplay between strong dispersion and nonlinearity leads to new scenariosof wave propagation. The classical waveform distortion process typical of intense acoustic waves in homogeneousmedia can be strongly altered when nonlinearly generated harmonics lie inside or close to band gaps. This allows thepossibility of engineer a medium in order to get a particular waveform. Examples of this include the design of mediawith e ff ective (e.g. cubic) nonlinearities, or extremely linear media (where distortion can be cancelled). The presentedideas open a way towards the control of acoustic wave propagation in nonlinear regime. Keywords:
Nonlinear; phononic; multilayer
1. Introduction
One of the most celebrated e ff ects of wave propagation in periodic media is the appearance of forbidden propaga-tion regions in the energy spectrum of electrons, or band-gaps. Most of the physics of semiconductors, and thereforemany electronic devices, are somehow based on this concept [1]. In the late 80’s, these ideas where extended byYablonovich and John [2] to light waves (electromagnetic waves in general) propagating in materials where the op-tical properties like the index of refraction were distributed periodically. These materials were named, by analogywith ordered atoms in crystalline matter, as photonic crystals. The typical scale of the periodicity is given by thewavelength. Actually, not only light but any wave propagating in a periodic medium may experience the same e ff ects,and acoustic waves are not an exception. Sound wave propagation in periodic media has become very popular in thelast 20 years in acoustics, after the introduction of the concept of sonic crystals [3]. Exploiting the analogies withother type of waves many interesting e ff ects, as the mentioned forbidden propagation bands (band-gaps), but alsofocalization, self-collimation, negative refraction, and many others have been proposed. We consider in this paperthe simplest case plane waves propagating in a 1D structure, formed by a periodic alternation of layers with di ff erentproperties. Depending on the context, such a structure has been named a multilayer, a superlattice (particularly in thecontext of semiconductors) or a 1D phononic crystal (this include more exotic structures, as the granular crystal orlattice [4]). The huge majority of the studies considered so far have assumed a low-amplitude (linear) regime, neglect-ing the nonlinear response of the medium. Intense wave propagation in nonlinear periodic media, and in particularthe case of sound waves, is almost unexplored. In this paper we present di ff erent examples of new phenomena relatedto sound wave propagation in 1D periodic media, where each of the layer has a nonlinear quadratic elastic response.Nonlinear acoustical e ff ects in such structure have been studied only in a few works. In [5] the harmonic genera-tion process is described in a fluid / fluid multilayered structure (water / glycerine), based in a nonlinear wave equation.Also, acoustic solitons in solid layered nonlinear media have been presented in [6]. More recently, the complementaryaction of nonlinearity and periodicity has been considered in [7], where an asymmetric propagation device (acoustic Email address: [email protected] (No´e Jim´enez)
Preprint submitted to Elsevier August 18, 2015 ( ω ) a c k (2 ω ) c Figure 1: Layered acoustic system with two di ff erent layers and second harmonic generation scheme. Here the lattice period is a = a + a . diode) was proposed. There, the nonlinearity and the periodicity act at di ff erent locations and its e ff ect is consideredseparately. The e ff ects discussed in this paper are the result of the interplay between nonlinearity and periodicity.Here we describe how the geometrical and acoustic parameters of the structure can be used to control the harmonicdistortion processes in a multilayer. The conditions required to selectively act on the nonlinearly generated spectrum,and therefore manipulate the waveform in the desired way, are obtained and discussed.The theory presented here has been developed for fluid-fluid (scalar) structures, however the main conclusions areextendable to fluid-solid or to solid-solid multilayers, if particular conditions are given. Also, the main conclusionsof this paper are independent on the regime of the waves (audible, ultrasound,...), and therefore on the size or scale ofthe structure. Specially interesting is the domain when ultrasound waves belong to the Terahertz regime, where theseideas may find a great potential. The progress in miniaturization and the technological development allows currently tocreate phononic multilayers at scales even in the nanometer range (each layer contains then a small number of atoms).This structures are usually made of semiconductors and are often used in particular applications as phononic mirrorsto form phonon nanocavities [9], or microcavities to obtain a strong optomechanical coupling [10] (for a revent survey,see [8]). In a remarkable recent achievement, acoustic amplification was realized in doped GaAs / AlAs superlattices,where a SASER (Sound Amplification by the Stimulated Acoustic phonon Radiation) was demonstrated, in a deviceincluding a superlattice gain medium and GaAs / AlAs SLs acoustic mirrors [11].The structure of the paper is as follows: In Sec. 2 we present the model for nonlinear propagation of acousticwaves in periodic media. The next Sec. 3 describes the process of harmonic generation in homogeneous media,and how it is modified by the presence of periodicity. In Sec. 4 the possibility of manipulating the spectrum of apropagating sound wave by tuning the parameters of the layered medium is discussed, showing examples of particularsituation, as the case of a cubic-e ff ective medium made out of quadratically nonlinear layers. Finally, Sec. 5 presentsthe conclusions.
2. The model
We consider a periodic medium made of an arrangement of homogeneous fluid layers of thickness a and a with di ff erent material properties. For the shake of simplicity only longitudinal waves under normal incidence areconsidered. A scheme of the medium is shown in Fig. 1.The propagation of small amplitude waves in an infinite periodic system is completely described by its dispersionrelation, often known as band structure, that for 1D systems as in Fig. 1 can be expressed analytically as [12]cos ( ka ) = cos ( k a ) cos ( k a ) − k k + k k ! sin ( k a ) sin ( k a ) (1)2 Re( ka ) / π ω a / π c c /c Im( ka ) ω a / π c c /c Figure 2: Dispersion relation of the two-layers system for layer proportion α = . ff erent c / c ratio. Left: real wavenumbers. Right:imaginary part of the complex wavenumber. also known as the Rytov formula, where k is the Bloch wave-number, a = a + a is the lattice period, and k i = ω/ c i is the local wavenumber, with c i the sound speed in the i layer. For a wave of frequency ω incident in a medium withknown acoustical c i and geometrical a i parameters, the above equation results in a band structure of propagating andnonpropagating (bandgap) regions, as shown in Fig. 2. Thus, using Eq. (1), we can estimate the e ff ect of periodicityon the di ff erent harmonics of the incident wave as they propagate through the multilayer, which is the main premiseof this work. The ratio between layer thickness can be defined as α = a / a , leading to a = (1 − α ) a .An example of dispersion relation plot is shown in Fig. 2 for normalized parameters a = . ff erent soundspeed ratios c / c . Increasing the impedance ratio between layers increases the reflected intensity in the trans-layerpropagation, while the transmitted energy of the multiple internal reflections diminishes. As can be seen, due to thesescattering processes band-gaps are progressively opened around the wavenumber k = n π/ a with n = , , ... . Thus,the bandwidth of these band-gaps also increases when the impedance ratio grows.On the other hand, its imaginary part increases in amplitude with c / c , leading to shorter evanescent propagationinside the band-gap for high sound speed contrast layers, while remains zero (no attenuation) in the propagationband. We recall that the system is conservative: the physical interpretation of the complex wavenumber is not energyabsorption, but back-reflection of the incident wave. Thus, at band-gap frequencies the waves penetrate only a shortdistance into the medium with a forward evanescent mode, and if the medium is perfectly periodic and lossless theenergy is back-reflected (it behaves as a mirror). The nonlinear propagation of sound in the acoustic inhomogeneous media, and in particular in multi-layeredmedia can be described by several models, with di ff erent levels of accuracy. Here, we use the equations of continuummechanics for ideal fluids with space dependent parameters. These are the continuity equation for mass conservation[13]: ∂ρ∂ t + ∇ · ( ρ v ) = . (2)and the equation of motion that follows from conservation of momentum ρ D v Dt + ∇ p = , (3)where ρ is the total density, v is the particle velocity vector over a Eulerian reference frame, p is the acoustic pressure, t is the time and D is the material derivative operator.For non homogeneous media, the ambient properties of the fluid in the absence of sound are space dependent,so the total density becomes ρ ( t , x ) = ρ ′ ( t , x ) + ρ ( x ), where ρ ( x ) is the spatially dependent ambient density and3 ′ ( t , x ) is the perturbation of the density or acoustic density, that is space and time dependent. Then, using the materialderivative, Eq. (3) becomes ρ ∂ v ∂ t + ∇ p = − ρ ′ ∂ v ∂ t − (cid:0) ρ ′ + ρ (cid:1) ( v · ∇ ) v , (4)In this equation, the first two terms in the left-hand-side account for linear acoustic propagation, where the termsin the right-hand-side introduce nonlinearity in the Eulerian reference frame through momentum advection processes.On the other hand, we can expand Eq. (2) for nonhomogeneous media as ∂ρ ′ ∂ t + ρ ∇ · v + v · ∇ ρ = − ρ ′ ∇ · v − v · ∇ ρ ′ . (5)Here, the first two terms on the left-hand-side account for linear acoustic propagation, the third, also linear, ac-counts for the magnitude of the changes in the ambient layer properties. Note this term is space dependent but onlychanges at the interface between adjacent layers. For density matched layers, ρ i = ρ i − , this terms vanishes. The termson the right-hand-side are nonlinear and accounts for mass advection.Finally, a fluid thermodynamic state equation p = p ( ρ, s ) is needed to close the system, with s the entropy. Thelocal nonlinear medium response relating density and pressure variations, retaining up to second order terms, can bewritten as p = c ρ ′ + B A c ρ ρ ′ , (6)where B / A ( x ) is the quadratic nonlinear parameter and c ( x ) is the sound speed, that can be also spatially dependent.In this system of equations, quadratic nonlinearity appears in the equation of motion (4) and in the continuityequation (5), in the momentum and mass advection terms respectively, and also in the equation of state, Eq. (6), re-lating pressure and density acoustic perturbations. We note that here we only take into account nonlinear processesthrough the layer’s bulk. The nonlinear e ff ects at the boundary between adjacent sheets are neglected. These nonlinearboundary e ff ects include cavitation processes, that in the case of fluids with very di ff erent compressibility can be veryimportant. In the case of solid layers, other local nonlinear e ff ects relative to boundaries, e.g. clapping phenomena be-tween surfaces, can lead to nonlinearities that are orders of magnitude in importance compared to the bulk cumulativenonlinearities. For moderate amplitudes, the system of Eqs. (4-6) can be simplified. For that aim, we use a perturbative methodwith same ordering scheme as in [14], where O ( ε ), O ( ε ) and O ( ε ) represents the terms of generic smallness param-eter ε . The derivation of a second-order nonlinear wave equation requires the substitution of the linearized acousticapproximations (first order) into second order terms of Eq. (4, 5). This substitution procedure will give third ordererrors, so the final nonlinear wave equation will be a second order approximation of the full constitutive relations.These equations can be combined to form a single nonlinear wave equation valid for nonhomogeneous media upto second order approximation ∇ p − c i ∂ p ∂ t − ρ ∇ ρ ∇ p = − βρ c ∂ p ∂ t − ∇ + c ∂ ∂ t L + O ( ε ) . (7)where we introduced the coe ffi cient of nonlinearity β = + B A that accounts for material and mass advection quadraticnonlinearities. It is worth noting here that the second-order Lagrangian density vanish for plane progressive waves dueto the first order relation p = uc ρ that leads to L =
0. In this case, Eq. (7) simplifies to the well-known Westerveltequation for inhomogeneous media ∇ p − c ∂ p ∂ t − ρ ∇ ρ ∇ p = − βρ c ∂ p ∂ t + O ( ε ) . (8)In general, the Lagrangian density term can be discarded based on the distinction of cumulative and local non-linear e ff ects. In this way, for progressive quasi-plane wave propagation in homogeneous media the nonlinear local4 σ p n / p ω/ Ω n = 1 σ p n / p ω/ Ω n = 2 n = 3 n = 4 Figure 3: Harmonic generation in the layered medium at low frequencies (numerical results), and its comparison with analytical expressions(Fubini) for an homogeneous medium. e ff ects become insignificant in comparison to the nonlinear cumulative e ff ects, where in most practicals situations,beyond a distance of only few wavelengths away from the source local nonlinear e ff ects can be neglected. However,local nonlinear e ff ects can become significant in other complex situations including standing-wave fields and finiteamplitude acoustic waveguides. Concerning the layered media, in this work we solve numerically the full constitutiverelations, and the e ff ect of the Lagrangian term is shown to be negligible under the conditions of our study.
3. Harmonic generation in layered media
We will study the response of the layered system for plane-harmonic wave excitation. Then, as sketched in Fig. 1,the source is placed at one boundary of the layered system, and the acoustic relevant magnitudes are calculatedalong space and time. As the wave propagates, cumulative nonlinear e ff ects generate harmonics of the fundamentalfrequency, ω , and due to the multiple scattering processes into the layers, local nonlinear e ff ects also distorts thewave. However, the high dispersion of the layered system have a strong impact on the nonlinear harmonic generation.Dispersion modify the resonance conditions between fundamental and second harmonic wave, and also for othernonlinearly generated higher frequencies. In this way, nonlinear energy transfer e ffi ciency from one component toanother is modified in a wide variety of configurations, leading to the possibility of engineering and controlling thenonlinear wave processes by tuning the dispersion relation.Depending of the frequency of the input wave, di ff erent scenarios can be observed, as reported in the followingsubsections. We start studying the propagation in the layered system for harmonic excitation in the very low frequency regime,where we assume that ka ≪ ω ( k ) curve is nearly constant. The dispersion of all the spectral components is negligible, and they all propagate atnearly the same velocity and are correspondingly phase-matched. Thus, in the absence of dispersion and attenuationprocess, the system of Eqs. (2-3) and (6) can be reduced for a harmonic-plane wave to a Burger’s evolution equationexpressed in traveling coordinates with e ff ective parameters, namely ˜ c , ˜ ρ and ˜ β . An analytic solution of this equationin terms of the n th-harmonics of the fundamental wave of frequency ω and initial amplitude p is known as the Fubinisolution, p ( σ, τ ) = p ∞ X n = n σ J n ( n σ ) sin ( n ωτ ) , (9)where J n is the Bessel function of order n , and σ = x / x s is the propagation coordinate, normalized to the shockformation distance, x s = / ˜ β ˜ ε k , with the e ff ective match number ˜ ε = u / ˜ c and the e ff ective wavenumber k = ω/ ˜ c ,that can be also found from Eq. (1). This celebrated solution is valid for σ < ka/ π ω / Ω k k k ∆ k Figure 4: Scheme of the phase miss-matching situation. The fundamental wave vector k at frequency ω generates a forced wave 2 k at frequency2 ω . The free wave that the system allows to propagate is k , located in the dispersion relation curve. Due to dispersion, k , k , thus there exist aphase mismatch, ∆ k between both waves and the generation is therefore asynchronous. Simulations were carried out using a full-wave constitutive relations solver. Thus, we shall define the normalizedreference frequency as Ω = π ˜ c / a (located in the first band-gap). The source frequency was set to ω = . Ω .Figure 3 shows the analytical and numerical solutions for the low frequency limit of the layered system, wherean excellent agreement is obtained between Fubini and numerical solutions in the pre-shock region, σ < ω . . Ω . For frequencies above the (idealized) homogeneous-Fubini regime, finite (weak and strong) dispersion e ff ects areobserved. The dispersive e ff ects of the layered system deeply a ff ects harmonic generation processes.As intense waves propagate through a quadratic nonlinear medium, their frequency components interact with eachother and new frequencies arise at combination frequencies, including higher harmonics. The cumulative energy trans-fer from the interacting waves to the harmonics is dependent on the resonance conditions ω ± ω = ω , k ± k = k .Note these conditions express the laws of conservation of energy ( ~ ω ) and momentum ( ~ k ) in the quantum descrip-tion for the disintegration and merging of quanta [13]. These conditions can be satisfied in a variety of situations.The most simple case is observed in nondispersive media and for collinear waves k i = ω i / c . In this situation theresonance conditions are fulfilled all over the spectra and a large number of harmonics interacts synchronously: whenthere exist in the system a free wave with velocity ω / | k | that matches the excited ( forced ) wave ω ± ω / | k ± k | ,the free wave is excited in a resonant way. The resonant interaction leads therefore to synchronous (phase matched),cumulative energy transfer from the initial wave to the secondary wave fields.In the case of an initial monochromatic wave, the main wave generates its second harmonic. The resonant condi-tions in this situation read 2 ω = ω , 2 k = k , that holds true for nondispersive collinear waves, leading to the simplerelation 2 k ( ω ) = k (2 ω ). However, in the case of dispersive media this condition is, in general, not fulfilled and the forced and free waves interact asynchronously. Figure 4 shows such situation for a layered media with a fundamentalwave in the first dispersion band.In order to study asynchronous second harmonic generation processes, we recall here for the lossless second-order wave equation Eq. (8), for one-dimensional propagation. This equation does not include dispersion by itself,dispersion arises from the solution of the linearized wave equation with the layered media boundary conditions, wherethe eigenvalue problem leads to the Rytov’s dispersion relation Eq. (1).6n the following, we apply a perturbation method to obtain an approximate solution for the second harmonic field.We expand the pressure field as sum of contributions of di ff erent orders, i.e. p = p (1) + ε p (2) + · · · , where ε is thesmallness perturbation parameter, which we identify with the acoustic Match number. Thus, p (1) is the first order(linear) solution of the problem and p (2) its the second order contribution. By substituting the expansion in the secondorder wave Eq. (8), assuming constant density and neglecting O ( ε ) terms we get a coupled set of equations thatcan be solved recursively. The solution of the first order equation corresponds to a monochromatic plane wave offrequency ω p (1) = p sin ( ω t − k x ) (10)where k = k ( ω ) is the wave vector associated with the primary frequency ω , and p is the excitation pressureamplitude. Substitution of the first order solution into the equation obtained at the next order in the expansion, leadsto an inhomogeneous equation for the second harmonic field: ∂ p (2) ∂ x − c ∂ p (2) ∂ t = − βω p ρ c sin (2 ω t − k x ) . (11)The general solution of the this equation is the sum of the solution of the homogeneous equation ( p = p , p (2) = p (2) h + p (2) f , where the corresponding waves for this two solutions are the free , and forced wavesrespectively. Such homogeneous and particular solutions are: p (2) h = p (2) h (cid:12)(cid:12)(cid:12) x = sin(2 ω t − k x ) , (12) p (2) f = A ( k + k )( k − k ) sin (2 ω t − k x ) , (13)where k = k (2 ω ) is the wavenumber of the free wave at second harmonic frequency, and the constant A = − βω p /ρ c . It is worth noting here that as long 2 k , k , the forced and free waves in dispersive media havedi ff erent phase speed, i. e. the forced and free waves are phase mismatched as can be seen in the argument of thesin function in Eq. (12-13). Imposing the boundary condition, that the second harmonic must be absent at x =
0, thesecond harmonic field can be expressed as p (2) = Ak ∆ k sin ∆ k x ! cos (cid:0) ω t − k ′ x (cid:1) , (14)where the e ff ective wave number is k ′ = ( k + k ) / ≈ k and the detuning parameter that describes the asynchronoussecond harmonic generation is defined as ∆ k = k − k = k (2 ω ) − k ( ω ) . (15)Equation (14) describes the well-known e ff ect in second harmonic generation in dispersive media, that is thebeatings in space of the second harmonic field when the resonant conditions are not fulfilled. Thus, as ∆ k increases,the beating spatial period and also its maximum amplitude decreases. The position of the maximum of the beating,also called the coherence length, can be related to the second-harmonic phase-mismatching frequency as x c = π | ∆ k | = π | k (2 ω ) − k ( ω ) | . (16)This length corresponds to the half of the spatial period of the beating, where the maximum of the field is located.It can be expressed also for other higher harmonics simply as x c ( n ) = π/ | ∆ k n | = π/ | k ( n ω ) − nk ( ω ) | .In the limiting case of ∆ k →
0, the second harmonic field is generated synchronously and accumulates withdistance, so a linear growth is predicted. In this case, phase matching conditions are fulfilled and the free wave is
1. We neglect the ambient density variations for the sake of simplicity. Dispersion arise also for sound speed variations, that are assumed to beimplicit in the boundary conditions. B B σ a) σ | p | / p B B σ b) σ | p | / p B B σ c) σ | p | / p Figure 5: Second harmonic evolution for x c / x s = (1 , / /
8) obtained using Eq. (14) (continuous line), numerically (white circles), nondis-persive linear law of growth (dotted line) and Bessel-Fubini nondispersive solution (dashed). σ | p | / p P ( ω ) P (2 ω ) a) σ | p | / p P ( ω ) P (2 ω ) b) ω/ Ω k a ∆ ka/ π Im( ka )c) Figure 6: Evolution of the second harmonic field propagating in bang-gap for second harmonic frequencies (a) just above band-gap 2 ω = . Ω ,(b) in the middle of the bandgap 2 ω = Ω . All results for a layered medium with α = / c / c = /
2. (c) Detuning of the second harmonic(continuous line) and imaginary part (dotted line) in function of the normalized frequency. excited synchronous to the forced wave . Note here that in the derivation of Eq. (14) only second order processes aretaken into account and therefore, only second harmonic is predicted. This leads to overestimate the second harmonicfield: as long no energy is transferred to third harmonic, second harmonic predicted by Eq. (14) in the absence ofdispersion grows indefinitely. The validity of this model can be explored expanding the Bessel functions of Fubiniseries near the source. A simple comparison between the full Fubini solution and linear second harmonic growth givesa reasonable approximation for distances σ < . p (2 ω ) < p / ff erent simulations in the dispersive regime of the layered media where the wave amplitudeand frequency has been selected to match x c / x s = , / /
8. The higher beating spatial period waves correspondsto lower frequencies. The analytical solution for the second harmonic matches the full-wave numerical solution.However, di ff erences can be observed in the second harmonic amplitude estimation for x c / x s = Waves with frequencies falling into the band-gap of the dispersion relation are evanescent due the non negligibleimaginary part of its complex wave number. Thus, its amplitude decays exponentially with distance. If the nonlinearlygenerated second harmonic falls into a band-gap, its amplitude does not decay but reaches a constant value [5]. Figure6 shows this case for two di ff erent frequencies. The constant amplitude value of the second harmonic wave dependson the imaginary part of the wave vector.This e ff ect can be understood in terms of the free and forced waves. If the second harmonic is evanescent (asfollows from the dispersion relation), the wave will not accumulate with distance. The fundamental wave is “pumping”8nergy to the second harmonic field at every point in space. Thus, the second harmonic field is generated locally andremains trapped inside the layered media. It reaches a constant level that depends on three main factors. In first place,the ”pumping” rate, characterized by the fundamental wave amplitude and medium nonlinearity, or more strictly theratio between the layer thickness and the shock distance a / x s . Secondly, it also strongly depends on the magnitude ofthe imaginary part of the complex wave number, i.e. the ratio between its characteristic exponential decay length andthe shock distance in a layer. The characteristic decay length of the evanescent propagation is always shorter when thesecond harmonic is in the middle of the band-gap, leading to a weaker second harmonic field in this frequency region,as seen in Fig. 6. Finally, it depends also on the detuning of real part of the wave number, where for the first band-gapis minimum at the center. The first factor can be isolated and studied separately. However, the two last factors arelinked through the specific dispersion relation of the medium.Figure 6(c) shows the detuning of the second harmonic and the imaginary part as a function of the frequency fora medium with α = / c / c = /
2, showing that at the middle of the band-gap these two factors have oppositee ff ects: detuning is null (phase matching) when evanescent decay is nearly maximized, and viceversa. However, themagnitude of the e ff ects can be very di ff erent. As the rate of the second harmonic generation (see the initial slopein Fig. 5) is independent on the detuning, and the evanescence implies that the wave decays after few layers, therenot exist a practical compensation of the e ff ects at the center of the band-gap. However, the situation is di ff erent forfrequencies around the limits of the band-gap, where the coherence length is of the order of the exponential decaycharacteristic length. Thus, for frequencies just above bad-gap and for amplitudes with shock distance comparableto the evanescent characteristic decay length, the beatings can be also observed, as shown in Fig. 6 (a). Then, iffrequency is increased the characteristic decay length becomes shorter than the shock wave distance and beatingscannot be observed, leading to to the characteristic constant second harmonic field shown in Fig. 6. When the fundamental frequency of the wave lies within the band-gap, small amplitude waves propagate evanes-cently. Essentially, the same applies to finite amplitude harmonic waves. In general, if the shock distance is largecompared to the characteristic decay length of the evanescent wave, the nonlinear e ff ects have no chance to accumu-late and harmonic amplitude is negligible. Since the characteristic exponential decay is about few lattice sites, thismeans that the initial amplitude necessary to achieve nonlinear e ff ects in this configuration is much higher than thosein the preceding sections. Figure 7 shows the evolution of the first and second harmonic waves for a fundamentalfrequency at the Bragg frequency, 2 ω = Ω , and with a frequency just above but into the band-gap, ω = . Ω for a layered media of α = / c / c = /
2. In the first case, the imaginary part of the wave vector is remarkablehigh and the waves decay fast after few lattice units. Due to this fast decay, the second harmonic interacts only overa short distance with the first, and its amplitude is very limited. After a few lattice units, the fundamental wave canbe treated as a small-amplitude evanescent-wave. The second harmonic, that also falls in bandgap (but in the secondband gap) also decays exponentially.On the other hand, if the fundamental frequency is set just above the band-gap, where the imaginary part ofthe wave-vector is smaller, the amplitude of the fundamental wave decays more slowly, penetrating deeper into thematerial. The interaction region with the second harmonic is larger, and nonlinear e ff ects result in a more e ffi cientgeneration of the second harmonic. Furthermore, as long the di ff erent (higher order) bandgaps in the layered mediacan have di ff erent bandwidth, in this configuration at ω = . Ω second harmonic does not fall inside a bandgap.Therefore, the generated second harmonic wave at the beginning of the lattice propagates through the medium essen-tially without amplitude change. Due to the evanescence of the fundamental wave, there is only forced wave at thebeginning of the medium. Therefore, although in this configuration waves are phase mismatched, beatings are notpresent: only the free wave propagates through the medium.
4. Nonlinear acoustic field management
In the preceding sections we have explored the fundamental behavior of nonlinear waves generated inside thelayered media. But also, medium parameters can be designed to provide specific conditions. The material parameterscan be tuned to get coherence at one frequency of interest, e.g. at one of the harmonics of the fundamental wave, or9 P ( ω )a) σ | p | / p P (2 ω )b) σ | p | / p Figure 7: (a) Evolution of the fundamental harmonic wave field with its fundamental frequency falling just above into band-gap, ω = . Ω ,(continuous line), and in the middle of the bad-gap, 2 ω = Ω (dotted line). (b) Corresponding second harmonic field, where for ω = Ω (dotted line) second harmonic frequency falls in the 2nd band band-gap while for ω = . Ω , (continuous line) lies into a propagating band. to get detuning or evanescent propagation at other specific harmonics. Using these mechanisms the layered mediumcan be used to provide a balance of the harmonic amplitudes, or to obtain specific nonlinear waveforms, providing acontrol of the nonlinear process inside the medium.In the design of a system for this purpose, the coherence length is a useful control parameter. For this aim, theanalytic Eq. (1) is used, which is shown to provide an excellent framework to tune the layered parameters to obtain thedesired balance between detuning, evanescent propagation, synchronous generation and, at the same time, it allowsto find those conditions for a specific phase / group speed. Figure 9(a,b) shows an example of a dispersion relation, thecoherence length for the second and third harmonic. The resulting harmonic amplitudes when phase matching of allharmonics is achieved is shown in Figure 8. This happens for a set of frequencies ω = (0 , . , . , ... ) / Ω . On theother hand, there also exist frequencies at which there exist coherence for the second but a non-negligible detuning isobserved for the third. The opposite e ff ect can be also obtained, where coherence is achieved for the third harmonicbut second harmonic presents strong dispersion. Finally, other interesting regions are those where second harmoniccomponent is almost phase matched and for the same frequency third harmonic falls into a band-gap.In the following subsections, we propose and analyze di ff erent configurations of the layered medium with specificbalance between detuning, evanescent propagation and synchronous generation. One can expect that second harmonic generation is maximized in homogeneous nondispersive media. However,in nondispersive media coherence is achieved not only at second harmonic frequency, but also in the higher spectralcomponents. As a result, energy is transferred from second harmonic field to higher spectral components and thereforesecond harmonic field does not grow indefinitely. Moreover, shock waves are formed and nonlinear absorption reduceswave intensity for σ > π/ σ | p / p | p ( ω ) p (2 ω ) p (3 ω ) Figure 8: Harmonic distribution for the frequency ω = . / Ω . Coherence is recovered for at least the lowest spectral components. Blackstocksolution (dotted lines). | p / p | p p p σ −1 0 1−1012 t f σ =0.5 | p / p | | p / p | | p / p || p / p | −1 0 1−1012 −1 0 1−1012 σ = 3.3 −1 0 1−1012 σ = 7.8 ωa / c π k a / π x c / a ωa / c π σ = 1 t f t f t f (a)(b) (c)(d) (e) (f) (g) Figure 9: (a) Dispersion relation for a layered media of c / c = .
33 and α = /
2. (b) Coherence length for second (red) and third (blue) harmonicsas a function of the fundamental frequency. Phase matched frequencies are those with x c → ∞ , while asynchronous generation is predicted for x c →
0. Frequencies at which the fundamental frequency is in band-gap are are marked in gray regions, while band-gap regions for second andthird harmonic are marked in dashed lines.(c) Harmonic distribution for ω = . Ω where a coherence is achieved for second harmonic whilethe frequency of the third harmonic falls into the bad-gap. (d-g) (continuous lines) Waveforms at di ff erent distances for ω = . Ω . At σ = . σ = . The dispersion of the layered system can be used to modify this situation by including phase mismatches that alterthe higher harmonic cascade processes, while maintaining coherence for the second harmonic. Figure 8 shows anexample of a dispersion relation where for ω = . Ω it can be observed that there exist a reasonable coherencefor the second harmonic ( x c / a ≈ σ < p | max ≈ . p , while in theexample of Fig. 9 a maximum second harmonic amplitude of p | max ≈ . p is predicted. As can be shown thedecreasing of the first harmonic follows the analytic nondispersive Blackstock solution for σ /
3. Thus, in thisregime all the energy of the first harmonic is being transferred to the second harmonic field. However, due to finitedetuning of the second harmonic a long spatial beating is produced, with normalized period 8 σ , and energy is returnedback to the first harmonic component.It is worth noting here that at distance σ ≈ t f t f t f t f | p / p | | p / p | | p / p | | p / p | σ =0.5 σ =2 σ =8 σ =12 σ ωa / c π k a / π x c / a ωa / c π (a) p p p (b) (c)(d) (e) (f) (g) | p / p | Figure 10: (a) Dispersion relation for a layered media of c / c = / α = .
3. (b) Coherence lengths for second (red) and third (blue)harmonics, (c) Harmonic distribution for ω = . / Ω where a coherence is achieved for second harmonic while the frequency of the thirdharmonic falls into the bad-gap. Bottom: (continuous lines) Waveforms at di ff erent distances for ω = . / Ω . At σ = . σ = . is observed at σ = .
3, as it can be appreciated in the waveforms of Fig. 9 the period doubling. Moreover, due tofinite detuning of the second harmonic the process is not cumulative for all distances and at σ = . σ = .
8, leading to a sinusoidal wave of di ff erent amplitude as can be observed in Fig. 9.The energy loss is mainly due to the artificial (numerical) viscosity necessary to nonlinear convergence[14]. For thesesimulations the total distance is 1200 lattice sites and therefore the e ff ects of attenuation are not negligible. However,the main nonlinear e ff ects related to strong lattice dispersion still appreciated. An analogous e ff ect has been alsostudied [13] where instead of dispersion, selective absorption at specific frequencies is used to modify and enhanceharmonic generation. In the first band ( ω < Ω ), coherence is always lower for the third harmonic than for the second. However, inthe superior bands the layered medium parameters can be tuned to obtain higher coherence for the third than for thesecond harmonic. Essentially we follow same ideas on the preceding section but for the third harmonic. In this case,the lattice is designed forcing the second harmonic to fall in bandgap. A the same time, perfect coherence can befound for the third harmonic at ω = . Ω . This situation is illustrated on Fig. 10 around ω = . Ω . In this case, thedispersion relation was obtained for a layered medium with parameters α = . c / c = / . p . As discussedin Sec.3.3, this constant field does not grow with distance and is related to the evanescent solution of the free waveand the local nonlinear “pumping”. On the other hand, due to the coherence of the third harmonic, all the energytransferred form second to third is accumulated with distance. Therefore, near the source the rate of energy transfer12rom second to third harmonic is constant. Thus, third harmonic start to grow almost linearly with distance, oppositeto quadratically in homogeneous media.Numerical simulations also show fourth and fifth harmonics grow (not shown in Fig. 10), but only fifth harmonicharmonic reach a remarkable amplitude, growing near the source almost quadratically with distance. Therefore, theentire system behaves as an artificially cubic-like nonlinear medium formed by quadratic nonlinear layers.The corresponding waveforms measured at σ = (0 . , , ,
12) are shown in Fig. 10(d-g). For σ = . ff ect, i.e. the steepening on the opposite side of the propagation direction, is characteristicof materials with negative parameter of nonlinearity. Therefore, the e ff ective nonlinear behavior observed by thesimulations in this conditions can be described as negative-cubic-like nonlinearity.
5. Conclusions