The avoided crossing behavior in the interaction of propagating sound or light waves with resonant inclusions is analyzed using a simple model of an acoustic medium containing damped mass-spring oscillators, which is shown to be equivalent to the Lorentz oscillator model in the elementary dispersion theory in optics. Two classes of experimental situations dictating the choice in the analysis of the dispersion relation are identified. If the wavevector is regarded as the independent variable and frequency as a complex function of the wavevector, then the avoided crossing bifurcates at an exceptional point at a certain value of the parameter , where γ and β characterize the oscillator damping and interaction strength, respectively. This behavior is not observed if the wavevector is regarded as a complex function of frequency.
Studies of the interaction of classical waves such as light and sound with a medium containing resonant inclusions go back to the elementary electronic theory of the dispersion of light in which an atom or molecule is represented by the classical Lorentz oscillator.1–3 More recently, the propagation of sound in artificial materials with resonant inclusions attracted considerable interest.4–11 Such media came to be known as locally resonant metamaterials,4 even though theoretical investigations of the interaction of surface acoustic waves (SAWs) with arrays of surface oscillators12–14 had appeared well before the term was coined.15 A characteristic phenomenon encountered in locally resonant media is the hybridization of the propagating wave with the local oscillator mode and the formation of an avoided crossing band gap. In optics, such hybridized mode is known as a polariton. In acoustics, local resonance band gaps similar to polariton band gaps in optics have been observed in various settings in solid, liquid, and gas media.4,6–9,11
It is generally known that a local resonance does not always lead to an avoided crossing. For example, in optics, not every absorption line yields a polariton band gap. Based on experimentalist's intuition we can suggest the role of the interplay of losses and the interaction strength: the avoided crossing behavior will be observed when the linewidths are smaller than the mode splitting and not observed in the opposite case. It would be interesting to know how, exactly, the transition between the two regimes occurs. For example, in a study of local resonance bandgaps in colloidal assemblies,7 the mode splitting was smaller than the linewidth, yet the band gaps were detected by the deconvolution of broad spectra into spectral components. Experiments on the interaction of SAWs with contact resonances of a monolayer of microspheres16,17 found either an avoided crossing or merely an increased attenuation depending on the strength of the interaction and the width of the resonance. In a theoretical study of the interaction of SAWs with damped local oscillators,14 it was observed that the mode splitting only occurs above some critical concentration of the oscillators, but no further investigation beyond this observation was undertaken.
In this report, we investigate the behavior of the avoided crossing in a locally resonant medium with damping using a simple spring-mass model of resonant inclusions,5,18 which is shown to be analogous to the Lorentz oscillator model in optics. We will see that if the wavevector is regarded as the independent variable and the frequency as a complex function of the real wavevector, the avoided crossing bifurcates at a certain value of a parameter , where γ and β characterize the oscillator damping and interaction strength, respectively. The bifurcation exemplifies the occurrence of an “exceptional point,” a phenomenon encountered in the transition between weak and strong coupling regimes in systems governed by non-Hermitian operators.19–23 However, no such bifurcation is observed if the wavevector is regarded as a complex function of the real frequency. We will argue that the two different representations correspond to two distinctly different classes of experiments.
II. THE MODEL AND DISPERSION RELATION
We consider mass-spring oscillators dispersed in a solid matrix as shown in Fig. 1. An oscillator consists of the mass M attached by springs to a rigid weightless shell. The resonant frequency of the oscillator is low enough so that the corresponding acoustic wavelength in the host medium is much greater than the average distance between the oscillators. In this case, we can use an effective medium approach to investigate the behavior of the medium on a distance scale much larger than the distance between the oscillators. Let us consider one-dimensional longitudinal motion of the effective medium along the x axis. In the absence of the oscillators, the displacement component ux is described by the wave equation
where is the density of the medium and is the longitudinal elastic modulus, , where is the longitudinal acoustic velocity in the host medium. If the oscillators are present, the momentum conservation necessitates the addition of a term containing the acceleration of the central masses
where Ux is the central mass displacement of the oscillators and n is the average number of oscillators per unit volume. We assume that the volume fraction of the oscillators is small enough as to not affect the density and the modulus of the host medium. This equation should be complemented by the equation of motion of an oscillator
where K is the total spring constant (for oscillators shown in Fig. 1 it is twice the spring constant of each spring) and η is the total spring damping coefficient. Here, we made an effective medium approximation by equating the displacement of the oscillator shell with the effective medium displacement ux. We are looking for a solution in the form of a harmonic plane wave, . Equations (2) and (3) are then transformed into ordinary differential equations for the amplitudes
where r = (U-u) is the amplitude of the oscillator displacement relative to the shell, , , and . Equation (4) leads to the following dispersion relation:
This equation can also be derived using the concept of the “dynamic mass density”5,18 (see the Appendix). We assume that the damping is small, , and that , the latter assumption being consistent with the assumed small volume fraction of the oscillators. The smallness of β means that far from the resonance the oscillators have a small effect on the propagating waves. Consequently, we are mainly interested in the behavior of the system in the vicinity of the resonance frequency. In this case, the term in the numerator in Eq. (5) can be disregarded due to the smallness of γ; however, the same term in the denominator cannot be disregarded as it yields a large contribution close to the resonance. The dispersion equation then takes the form
which is entirely equivalent to the optical dispersion relation obtained with the Lorentz oscillator model.2,3 This equivalence is not coincidental; rather, it indicates that Eq. (6) describes the generic behavior of the wave propagation in media with resonant inclusions. (An analogy between acoustic metamaterials with mechanical oscillators and the Lorentz oscillator model has already been noted.24) Thus, the following analysis will be equally applicable to the propagation of light and sound in media with local resonances.
We set and to unity (which is equivalent to using units of and for the frequency and wavevector, respectively), which yields a simple equation
containing only two parameters, β and γ. It is instructive to rearrange the terms as follows:
which makes it plain that the dispersion equation describes the interaction of the propagating wave with the oscillator mode; the right-hand side of the equation is responsible for the interaction of the two modes.
III. INVESTIGATION OF THE DISPERSION RELATION
In the investigation of the dispersion relation, we face a choice: one can either treat the wavevector k as the independent variable and use Eq. (7) to find the complex frequency as a function of the real wavevector or one can regard the frequency as the independent variable and the wavevector as a complex function of the real frequency. The two cases are indistinguishable as long as both the wavevector and frequency are real. However, as we shall see shortly, the difference is quite significant when the solutions are complex. The two cases correspond to distinct experimental situations, which can be schematically described as “scattering” and “transmission” measurements. For example, in Brillouin scattering measurements,7 the scattering geometry defines a real wavevector. In the transient grating experiment,9 acoustic modes with a well-defined (and real) wavevector are impulsively excited in an unbounded medium and measured via diffraction (scattering) of the probe laser beam. In optics, an impulsive stimulated Raman scattering study of phonon-polaritons25 exemplifies measurements with a real wavevector and complex frequency. Another type of experiments involves measurements of transmission through a “slab” of material.4,6,8 In this class of measurements, very common in optics, light or sound is attenuated with distance; consequently, the wavevector is complex, while the frequency is real (which is especially apparent for a monochromatic source). Thus, the choice between the wavevector and frequency as the independent real variable depends on the experimental situation. In a general theoretical treatment, both cases should be considered.
It is interesting to observe that the difference between the two cases exists even when the damping is absent. Figure 2(a) shows the frequency as a function of the real wavevector for the case β = 0.1, γ = 0. A classic avoided crossing pattern with a narrow band gap can be seen. Figure 2(b) shows the dispersion relation for the same parameter values but now the wavevector is plotted as a function of frequency (the frequency is still plotted on the vertical axis to facilitate a comparison between the two cases). The two dispersion branches for which both frequency and wavevector are real are identical to the case (a); however, now an extra branch with an imaginary wavevector appears inside the bandgap, in contrast to (a), where there are no solutions inside the bandgaps. The experimental implications are not insignificant: in a scattering experiment with an imposed real wavevector, no excitations will be found inside the bandgap; in a transmission experiment, however, frequencies inside the bandgap will be attenuated but still transmitted.
Let us now consider the case of damped oscillators. Figure 3 shows real and imaginary parts of the frequency in the vicinity of the avoided crossing as functions of the wavevector26 for different values of γ, with the value of β set at 0.01. As the damping increases, the avoided crossing in the dispersion of Re(ω) narrows and eventually bifurcates at a threshold damping value of 0.2. At higher damping, no avoided crossing is observed: in place of hybridized modes, which change character in the avoided crossing region, we have a predominantly propagating acoustic mode and a predominantly oscillator mode which cross each other and interact only weakly. It is also instructive to observe the behavior of the imaginary part of the frequency. At low damping, Im(ω) curves for the two modes cross each other as expected: since damping is only present in the oscillators, the mode that starts as a predominantly propagating acoustic wave at low wavevectors has low damping, while the other mode has the damping similar to that of the uncoupled oscillator mode [in which case Im(ω) = γ/2]. When the modes switch their character in the avoided crossing region, their damping goes up or down, respectively. By contrast, past the bifurcation point the oscillator mode has high damping while the propagating mode has low damping, which increases in the vicinity of the crossing yielding a nearly Lorentzian profile. Thus past the bifurcation point, the oscillators merely cause an attenuation of the propagating acoustic wave in the vicinity of the resonance.
The two distinct regimes observed at high/low damping are encountered in many different settings in quantum and classical systems and are often referred to as “strong coupling” and “weak coupling.”27–29 The bifurcation point separating the two regimes is of particular interest: this is the so-called “exceptional point,” i.e., a branch point singularity in the spectrum of a system governed by a non-Hermitian operator.19,20 The exceptional point is different from a degeneracy in that the eigenvectors of the system coalesce at the exceptional point. The coalescence of the eigenvectors can be seen in Fig. 4, which shows real and imaginary parts of the amplitude u calculated from Eq. (4), with r set to unity. The occurrence of exceptional points in physical systems has been attracting increasing attention, predominantly in optics and electromagnetism,22,30,31 but also in acoustics.23,32 Two coupled damped oscillators present a prototypical system yielding an exceptional point.20 We observe that Eqs. (2) and (3), after a Fourier-transform over the x coordinate, can be transformed into a form describing two coupled harmonic oscillators.
It would be useful to obtain analytical expressions for the values of parameters at the exceptional point. In the vicinity of the crossing point of uncoupled modes k = ω = 1, we can further simplify Eq. (8) by replacing and by and , respectively, and setting ω to unity elsewhere, with the following result:
This is a quadratic equation for ω, which yields the solution
The coalescence of both real and imaginary parts of ω at the exceptional point requires the square root to be zero, which leads to the following result:
This agrees very well with the numerical results obtained for β = 0.01. Thus, the bifurcation occurs at a threshold value of γβ−1/2 equal to 2. Consequently, it can be observed either as the damping is increased or as the interaction strength is decreased (for example, by reducing the concentration of oscillators, in agreement with the observation made in Ref. 14).
Let us now consider what happens if the wavevector is treated as a complex function of the real frequency. The analysis of this case is very simple as the wavevector is explicitly given by Eq. (7). From this equation, it is obvious that the dispersion is represented by a single branch as can indeed be seen in Fig. 5. [As in Fig. 2(b), we keep the independent variable ω on the vertical axis.] The two branches separated by a band gap can only be seen at γ = 0. As soon as a non-zero damping is introduced, the real and imaginary branches seen in Fig. 2(b) merge into a single branch with a characteristic “wiggle” in the real part of the wavevector observed at low damping. This wiggle is indeed observed in transmission measurements.6 In the imaginary part of the wavevector, all we see is a Lorentzian absorption peak. (It is instructive to observe that the peak value of the absorption is higher when the oscillator damping is low.) Interestingly, at γ = 0.2, when the bifurcation takes place in the frequency vs wavevector representation, nothing at all happens in the wavevector vs frequency representation: the dispersion of the real part of the wavevector is already very close to a straight line with a slope of unity (i.e., the dispersion of the uncoupled propagating wave) and the presence of the oscillators is mainly manifested in the broad absorption peak.
We have identified two kinds of experimental situations leading to different behavior of locally resonant media in the transition between the strong coupling regime γβ−1/2 ≪ 1, in which the hybridization and avoided crossing between the propagating waves and the localized oscillator mode are observed and the weak coupling regime γβ−1/2 ≫ 1, in which the effect of the oscillators on the propagating mode is reduced to an absorption peak at the resonance. If the experiment imposes a well-defined wavevector, then the frequency should be treated as a complex function of the real wavevector. In this case, the locally resonant medium behaves similarly to a system of two coupled oscillators and the avoided crossing bifurcates at an exceptional point at γβ−1/2 = 2.
It may seem surprising that this striking behavior of the classic Lorentz dispersion model has not been described in the literature. A possible reason is that optical dispersion is typically analyzed in terms of the frequency dependence of the refractive index, which corresponds to the second class of experiments, very common in optics, in which the frequency is well-defined and the wavevector is regarded as a complex function of the real frequency. In this case, the avoided crossing is, in the mathematical sense, absent at any non-zero damping. However, experimentally observed behavior at γβ−1/2 ≪ 1 may still resemble an avoided crossing. For example, the wiggle in the dispersion of Re(k) at γ = 0.05 in Fig. 5 will look like an avoided crossing if the central part of the wiggle near the resonance is missing in the measurements due to the strong attenuation, as often happens in transmission experiments.16,33 Yet in this case there is no bifurcation of the avoided crossing and no distinct boundary between the strong and weak coupling regimes.
The author greatly appreciates illuminating discussions with Prasahnt Sivarajah and Bo Zhen. This work was supported as part of the Solid State Solar-Thermal Energy Conversion Center (S3TEC), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award No. DE-SC0001299/DEFG02-09ER46577.
APPENDIX: DERIVATION OF THE DISPERSION RELATION VIA THE EFFECTIVE DYNAMIC DENSITY
It has been previously shown15,18 that in the absence of damping the “effective mass” of a weightless rigid shell containing a mass-spring oscillator for the external time-harmonic excitation at the angular frequency ω is equal to
It is straightforward to extend the effective mass concept to the case with damping. If the harmonic force is applied to the shell, the displacement of the shell us will be given by
Thus for the outside observer, the effective mass of the oscillator will be given by
As noted in the main text, we assume that the volume fraction of the oscillators is very small and does not affect the density of the host medium. (Note that even if the mass of the oscillators per unit volume nM is small compared to ρ0, they may have a large effect on the effective density close to the resonant frequency).
With a plane wave ansatz , Eq. (1) is transformed into the following form:
Equation (7) is a quartic equation in ω and yields four solutions. Figure 3 (just as Fig. 2) only shows solutions with a positive real part of ω; the solutions with a negative real part of ω correspond to waves propagating in the opposite directions but are otherwise equivalent to the solutions shown in the figure.