Anderson localization, a phenomenon where waves are localized due to multiple scattering in a disordered medium, applies to both quantum and classical waves. While observed in quantum systems, demonstrating Anderson localization for classical waves has proven challenging. Soft matter offers a unique platform for observing this phenomenon due to the significant speed difference between fast and slow waves and the potential for strong scattering across wide frequency bands with minimal dissipation. This research aims to provide experimental evidence for Anderson localization of classical ultrasonic waves in a soft material, specifically a soft elastic medium doped with resonant encapsulated microbubbles (EMBs). The study's significance lies in its potential to advance our understanding of wave propagation in disordered media and to explore the possibility of broadband control of wave localization in soft matter, opening doors for various technological applications. The specific research question is whether the introduction of EMBs at specific concentrations and frequencies into a soft elastic medium will induce a transition into an Anderson-localized phase for ultrasonic waves, and if so, to characterize the properties of this localized phase.

Existing literature extensively explores Anderson localization in both quantum and classical systems. Studies on electronic systems have established the theoretical framework and experimental observations of the Anderson transition. However, observing Anderson localization in classical wave systems, particularly in three dimensions, has been more difficult. Previous research has demonstrated localization of light and ultrasound in various disordered media. Specifically, studies on the localization of ultrasound in three-dimensional elastic networks and the localization of light in strongly scattering media have provided valuable insights. However, these systems often suffer from limitations such as narrow bandwidths or difficulties in controlling the disorder strength. The use of soft matter, with its tunable properties and ability to introduce resonant scatterers like EMBs, provides a novel approach to investigate Anderson localization in classical wave systems more effectively. The paper draws upon these prior studies and leverages the unique properties of EMBs in soft matter to address the limitations of existing experimental approaches.

The study utilized a soft gel (Carbopol ETD 2050) doped with EMBs (Expancel), encased in a Uralite polymer shell for structural support. The EMBs, containing a mixture of isobutane and isopentane, were characterized using optical microscopy and scanning electron microscopy (SEM) to determine their size distribution and shell thickness. Normal incidence ultrasonic measurements were conducted in water using a 0.5 MHz piston-faced immersion transducer and a hydrophone. The sound level (SL), attenuation coefficient (α), and scattering mean free path (λs) were determined from the coherent part of the transmitted wavepacket using time-windowing. The experimental setup allowed for measuring both the coherent (average Green's function) and incoherent (higher-order Green's functions) components of the transmitted wave. To study late-time behavior and identify localization, a narrow impulse excitation was used, and the pressure was recorded across multiple speckles (spatial locations). The data was then analyzed to extract diffusion parameters and to fit the self-consistent theory (SCT) of localization, determining the localization length (ξ) and diffusion coefficient. Critical exponents were determined using power law fits to the mean free path near the mobility edge. This rigorous approach combining experimental measurements and theoretical modeling enabled precise characterization of the Anderson localization transition.

The study found significant reductions in sound level upon EMB doping, beginning near the anticipated minimum resonance frequency of the EMBs. A transition to a localized phase was observed, characterized by attenuation resonances exceeding 200 Np/m above 400 kHz. The transition was identified by attenuation peaks (mobility edges) that shifted to lower frequencies with increasing microbubble volume fraction (ϕ). At the lower-frequency mobility edge (ϕ = 1.2%, f = 527 kHz), the scattering mean free path (ls) decreased rapidly as the frequency approached the critical frequency (fc), obeying a power law with a critical exponent (γ) of approximately 1. Within the localized phase (between the mobility edges), ls/λ remained below 1.0, reaching values as low as 0.4. The critical density (ρc) for the phase transition was estimated. Late-time analysis of the transmitted intensity revealed a transition from diffusive behavior to localization, confirming the presence of an Anderson-localized phase. Fitting the late-time data to the self-consistent theory of localization provided estimates for the localization length. The localization length was found to decrease with increasing ϕ and was comparable to the wavelength within the localized phase. Strong attenuation of the incoherent wave at the mobility edge was observed and attributed to fluctuations in wave diffusivity. The study demonstrated broadband control of sound localization, observing localized phases as wide as 246 kHz.

The experimental findings provide strong evidence for Anderson localization of ultrasonic waves in a soft matter system with EMBs. The observation of a localized phase, marked by the anomalous decrease in mean free path at the mobility edge and confirmed by late-time intensity analysis and fitting to the SCT, directly addresses the research question. The results demonstrate the feasibility of achieving broadband control over localization in soft matter by tuning the concentration and resonance frequencies of the EMBs. The critical density found is in agreement with theoretical predictions for resonant point scatterers, highlighting the role of spatial dispersion in the coherent wavepacket. The observed strong attenuation at the mobility edge, due to fluctuations in wave diffusivity, provides a new experimental signature of the Anderson transition and differentiates these results from previously published results on sound localization in mesoglasses. This work significantly advances our understanding of classical wave localization and opens avenues for manipulating wave propagation in materials.

This study successfully demonstrated broadband control of sound localization and phase transitions in a soft matter system doped with resonant encapsulated microbubbles. The observation of a wide localized phase, the anomalous decrease in the mean free path at the mobility edge, and the confirmation of localization through late-time analysis provide substantial evidence for Anderson localization of ultrasonic waves. The study provides a valuable experimental comparison with existing theories, and suggests further investigations into the dynamics of wave propagation near the mobility edge and the influence of spatial dispersion. Future research could explore the use of this system for developing materials with tunable acoustic and elastic properties for diverse technological applications.

The study is limited by the finite size of the sample, which may affect the observation of localization. While the localization length was found to be comparable to the sample thickness in some cases, the finite sample size could restrict the measurement of closed transport loops, potentially influencing the measured localization length. The fabrication process of the samples could also introduce some variability, potentially affecting the reproducibility of the results. Although steps were taken to mitigate this variability, it is always a potential limitation in such experimental studies. The model used to characterize the EMBs may have limitations, and improvements to the EMB model may influence the precise quantitative interpretation of the results.