Impact of Strong Scattering Resonances on Ballistic and Diffusive Wave Transport

The strong impact of scattering resonances on all the key transport parameters of classical waves in disordered media is demonstrated through ultrasonic experiments on monodisperse emulsions. Through accurate measurements of both ballistic and diffusive transport over a wide range of frequencies, we show that the group velocity is large near sharp resonances, whereas the energy velocity (as well as the diffusion coefficient) is significantly slowed down by resonant scattering delay. Excellent agreement between theory and experiment is found, elucidating the effects of resonant scattering on wave transport in both acoustics and optics.

Figure 1

Ballistic pulse transmitted through the emulsion (solid red curve) and a reference pulse through gel (dashed blue curve) for a propagation distance $z=2\mathrm{mm}$. The inset shows the normalized FFT magnitudes versus frequency.

Figure 2

(a) Measured extinction length ${\ell }_{\mathrm{ext}}$ ($\approx {\ell }_s$), (b) phase velocity $v_{\mathrm{ph}}$, and group velocity $v_{\mathrm{gr}}$ (circles and squares, respectively), compared to the phase velocity $v_0$ ($=1.48\mathrm{mm}/\mu \mathrm{s}$) in the pure water-based gel matrix (dotted line). Theoretical calculations are based on the independent scattering approximation (solid lines). Red arrows indicate the resonant frequencies of isolated fluorinated-oil droplets with $a=170\mu \mathrm{m}$. For this emulsion, the polydispersity was 3%.

Figure 3

(a) Transmitted unfiltered incoherent field for one realization of disorder (at $\rho =0\mathrm{mm}$ and $z=20\mathrm{mm}$), with the input signal shown in the inset. (b) Average energy density (normalized by the incident energy density $U_0$) for different distances $\rho$ after numerical filtering of transmitted codas around $f_0=2.7\mathrm{MHz}$. (c) The linear fit of the width squared of the diffuse halo between 50 and $500\mu \mathrm{s}$ (red line) provides an accurate measurement of the diffusion coefficient $D$. In (b) and (c), symbols denote experimental data and the black lines correspond to the theoretical predictions over all times up to $1000\mu \mathrm{s}$.

Figure 4

(a) Measured diffusion coefficient $D$ as a function of frequency (symbols) from the fits for $w^2(t)$ [cf. Fig. 3], compared with theoretical predictions from the independent scattering approximation (dashed lines). (b) Absorption time ${\tau }_a$ deduced from the fits for $U(\rho ,t)\propto \exp (-t/{\tau }_a)$ [cf. Fig. 3]. The experimental data shown here were obtained from measurements done for three different depths $z$ ($=15$, 20, and 25 mm) within the sample.

Figure 5

Comparison of the energy and group velocities from the experiments (circles and squares, respectively) and theory (dashed and solid lines). The sound speed in the pure gel matrix is represented by the dotted line.