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͑ a ͒ The normalized ͑ to the geometrical cross section ͒ scattering cross section vs frequency for a plane sound wave scattered by a silica ( d ϭ 250 nm) sphere. The l denotes the spherical harmonic associated with the approximate eigenmodes appearing as peaks in the scattering cross section. Insets: Normalized energy density distributions for f ϭ 6 GHz ͑ 1 ͒ and for f ϭ 14.39 GHz ͑ 2 ͒ . At the former frequency the scattering cross section is low, while the latter corresponds to a peak in the cross section. ͑ b ͒ and ͑ c ͒ The partial energy density distribution for f ϭ 14.39 GHz and f ϭ 11.23 GHz, respectively, keeping only the contribution of the spherical harmonic responsible for the peak.
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Rich phonon spectra were observed experimentally by Brillouin spectroscopy in liquid, glassy, and crystalline state of colloidal systems of low and high elastic constant contrast. The nature of these phonons was elucidated by theoretical calculations of the single sphere scattering cross section, the energy density distribution, the light scatterin...
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... are distinct differences between the silica and PMMA colloidal suspensions. The silica suspensions are characterized by large mismatch in the elastic constant and density with the host medium. For comparable sizes the eigenmodes of silica spheres have higher frequencies and are well localized inside the spheres with weak leakage see Fig. 8 into the solvent as compared with the single PMMA sphere modes which are strongly coupled to the longitudinal c l k solvent phonon. The phonon spectrum of a silica colloidal glass or crystal should therefore be distinctly differ- ent from that of the lower volume fraction, liquid suspension. Figure 6a displays the Brillouin spectrum of ...
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... Fig. 8a we plot the residual scattering cross section divided by the geometrical cross section of a plane sound wave propagating in the solvent cyclohexane/decalin and im- pinging upon a silica sphere of radius r s 125 nm. Note that the peaks at 2.3, 11.23, 14.39, and 16.75 GHz correspond to approximate sphere eigenmodes associated with the ...
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... further support the identification of the resonances peaks in the scattering cross section as the approximate sphere eigenmodes, we have calculated the energy density associated with the wave given by Eqs. 5a, 6, and 7 for various values of the frequency. In inset 1 of Fig. 8a, we show the energy density distribution including the incident wave and averaged over the angle variables as a function of the distance from the center of the silica particle for f 6 GHz i.e., a frequency value not associated with any peak in the cross-section plot. We see that the energy is concentrated in the solvent with a minute ...
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... the scatter- ing cross section versus the reduced distance, r/r s , for the peak frequencies at 14.39 and at 11.23 GHz, respectively. In the case of the sharp peak at 14.39 GHz, the energy is concentrated mainly inside the particle with a minute leakage in the solvent demonstrating thus that we have an almost exact eigenmode. As we can see from Fig. 8c the broader the peaks the larger the leakage of energy in the solvent, in agreement with the discussion in Sec. V A. Note that the total cross section, , can be expressed as a sum l0 l of partial cross sections, l , each associated with a particular ...
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... E l (r) are the partial density energies of each mode l. Examples of energy density distribution are shown in the insets of Fig. 8a and of partial energy densities for l1 and l2 in Figs. 8b and 8c, ...
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... E l (r) are the partial density energies of each mode l. Examples of energy density distribution are shown in the insets of Fig. 8a and of partial energy densities for l1 and l2 in Figs. 8b and 8c, ...
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... we mentioned in Sec. V A, the global eigenmodes of the entire colloidal system can be thought of as a linear com- bination of the acoustic mode (c 0 k) and the various ap- proximate eigenmodes associated with each one of the nu- merous colloidal particles. There are cases see, e.g., inset 1 in Fig. 8a where the global mode is essentially the pure acoustic mode with minute hybridization. There are also cases where the hybridization involves mainly eigenmodes of the same l from individual spherical particle with small and to a first approximation negligible contributions from other l modes or the acoustic mode; the resulting global ...
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... flatness of these modes in the q space suggests that they are localized mostly inside each colloidal particle. In order to check the localization of these modes, we consider the computed energy density distributions for some phonon frequencies in Figs. 8b and 8c. One can observe that the energy is well concentrated inside the sphere with a small fraction leaked into the solvent; for the sharper peak at f 14.39 GHz the localization is better. However, this mode has not been detected experimentally probably because of his weak intensity. These localized modes interact weakly with similar modes ...
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Citations
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... 46,55 Additional related research on the subject of hybridization band gaps and negative index metamaterials have explored this attenuation. [61][62][63][64][65][66] It is worth noting that the full width at half maximum (FWHM) for the various resonances of the soda can array, visible as the peaks in Fig. 2, are on the order of 1 Hz. In room acoustics theory, the FWHM for a room mode is inversely proportional to the reverberation time, RT60, of the room as FWHM ¼ 2.2/RT60, which means that the effective reverberation time of the modes of the soda can array is about 2.2 s. ...
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... Green and magenta points indicate spacings a where one of the two wave modes highlighted in panel (a) are found to be destabilizing. These modes are further illustrated in Figure 5. (e) The black curve indicates the dependence on lattice spacing of the local wave amplitude h 0 beneath each droplet in the base lattice, normalized by the drop radius R. The orange curve denotes the corresponding product of impact phases S 0 C 0 that influences the lattice stability through ϕ in Equation (10). Figure 3. ...
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... The negatively charged silica colloids (Ludox LS, 12 nm, 30wt % ) and the positively charged silica colloids (Ludox Cl, 12 nm, 30wt % ) have been purchased from Sigma-Aldrich company. The neutral silica colloids feature a diameter of 250 nm (10wt % ) and 500 nm (5wt % ), both purchased from Applied Physics Inc. Si NPs have been chosen for this study because they were thoroughly studied in the past, and, owing to their transparency, also considered in several Brillouin scattering experiments on aqueous suspensions [34][35][36] . Furthermore, Si, as well as gold NPs have been extensively characterized due to their critical applicative interest. ...
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... One potential source of uncertainty is the evaluation of the angle α, but the angle can be calibrated with high precision. Here, we fitted α value using three reference materials of known refractive index and Brillouin shift; other methods to determine the angle of incidence of the α-geometry beam can be implemented [51]. The uncertainty on the refractive index can be expressed as a linear function of the shift measurement precision [52]. ...
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... One potential source of uncertainty is the evaluation of the angle α, but the angle can be calibrated with high precision. Here, we fitted α value using three reference materials of known refractive index and Brillouin shift; other methods to determine the angle of incidence of the α-geometry beam can be implemented [51]. The uncertainty on the refractive index can be expressed as a linear function of the shift measurement precision [52]. ...
... Various methods can be used to characterize the PnCs including piezoelectric transduction, Brillouin scattering, laser excitation, interferometry, and lithography-based methods [62][63][64]. There are two techniques generally employed to investigate the dispersion relation of periodic phononic structures, which are Brillouin light scattering (BLS) and the pump-probe technique with ultrashort laser pulses [65,66]. ...
... In terms of effective parameters, the existence of the band gap is associated to the negativeness of the effective compressibility which results in an imaginary propagating velocity, or equivalently evanescent waves. We have to mention here that in the acoustic community this type of band gap has been observed before the emergence of metamaterials and in the early 1990s researchers referred to them as "hybridization band gap" [58][59][60][61][62][63][64]. Hybridization is a very generic term originating from solid state physics that refers to the coupling between two states resulting on the existence of two hybrid states [65]. ...
... This seriously restrains the range of applications, especially in the low frequency regimes where the wavelength is large. Contrary to Bragg interferences based band gap, the hybridization one, which originates from the destructive interferences between the resonant response and the incident wave, is robust to a spatial disorder: breaking the translational periodicity of the medium does not close the band gap [60,63,81]. As a consequence we cannot use the trick of breaking the periodicity of the medium as performed in phononic crystals in order to create a defect state. ...
Photonic or phononic crystals and metamaterials, due to their very different typical spatial scales—wavelength and deep subwavelength—and underlying physical mechanisms—Bragg interferences or local resonances—, are often considered to be very different composite media. As such, while the former are commonly used to manipulate and control waves at the scale of the unit cell, i.e., wavelength, the latter are usually considered for their effective properties. Yet we have shown in the last few years that under some approximations, metamaterials can be used as photonic or phononic crystals, with the great advantage that they are much more compact. In this review, we will concentrate on metamaterials made out of soda cans, that is, Helmholtz resonators of deep subwavelength dimensions. We will first show that their properties can be understood, likewise phononic crystals, as resulting from interferences only, through multiple scattering effects and Fano interferences. Then, we will demonstrate that below the resonance frequency of its unit cell, a soda can metamaterial supports a band of subwavelength varying modes, which can be excited coherently using time reversal, in order to beat the diffraction limit from the far field. Above this frequency, the metamaterial supports a band gap, which we will use to demonstrate cavities and waveguides, very similar to those obtained in phononic crystals, albeit of deep subwavelength dimensions. We will finally show that multiple scattering can be taken advantage of in these metamaterials, by correctly structuring them. This allows to turn a metamaterial with a single negative effective property into a negative index metamaterial, which refracts waves negatively, hence acting as a superlens.
