Temporal fluctuations in the phase of waves transmitted through a dynamic, strongly scattering, mesoscopic sample are investigated using ultrasonic waves, and compared with theoretical predictions based on circular Gaussian statistics. The fundamental role of phase in Diffusing Acoustic Wave Spectroscopy is revealed, and phase statistics are also shown to provide a sensitive and accurate way to probe scatterer motions at both short and long time scales.PACS numbers: 43.35.+d, 42.25.Dd, 43.60.Cg, 81.70.Cv For all waves, phase is irrefutably the most fundamental property. On macroscopic scales, however, phase is randomized by multiple scattering or obscured by decoherence. It is now generally accepted that a mesoscopic regime exists where wave phenomena persist on even hydrodynamic scales. Mesoscopic fluctuations can sometimes be long-range and non-Gaussian [1,2]. The universal conductance fluctuations are best known, originally discovered for electrons [3], and later also observed with visible light [4] and microwaves [5]. In the optics of soft condensed matter, the existence of dynamic mesoscopic fluctuations has led to a new technique called diffusing wave spectroscopy (DWS) [6]. In the acoustic counterpart, diffusing acoustic wave spectroscopy (DAWS) [7], the fluctuations of the scattered wave field are measured directly to probe the dynamics of disordered media. In seismology, the closely related technique of Coda Wave Interferometry [8] is extending the range of applications being studied.For acoustic, seismic or radio waves, the phase can be easily extracted. While many applications, including interferometric techniques such as InSAR [9], make use of phase for precise measurements, the phase of multiply scattered waves has often been neglected, since it is generally more challenging to extract useful information from phase in multiple-scattering systems. Mesoscopic studies have revealed the fundamental relation of phase to the screening of zeros of random fields [10], but most of the literature has focussed on quantities such as the probability distribution functions of intensity, transmission or conductance [1], and does not address the phase directly. Recent studies with microwaves [11], infrared light [12] and Terahertz radiation [13] have explored frequency correlations of the phase. In this Letter, we study time-dependent phase fluctuations of ultrasound in a dynamic, strongly scattering medium, and examine the statistics of both the wrapped and cumulative phase evolution. This combination of theory and experiment reveals a deeper insight into the mesoscopic physics of multiply scattered waves, and explicitly shows the relationship between the average phase evolution of a typical multiple scattering path -a crucial concept in D(A)WS modelling [6,7] -and the measured phase evolution of the transmitted waves. We also find that phase statistics can sometimes provide a more accurate method of measuring the dynamics than the field autocorrelation method that is used in D(A)WS. In our materials, the tempo...
We report the analysis of the statistics of the phase fluctuations in the coda of earthquakes recorded during a temporary experiment deployed at Pinyon Flats Observatory, California. The observed distributions of the first, second and third derivatives of the phase in the seismic coda exhibit universal power-law decays whose exponents agree accurately with circular Gaussian statistics. The correlation function of the spatial phase derivative is measured and used to estimate the mean free path of Rayleigh waves.PACS numbers: 46.65.+g, 91.30.Ab, 46.40.Cd In the short-period band (> 1 Hz) , ballistic arrivals of seismic waves are often masked by scattered waves due to small-scale heterogeneities in the lithosphere. The scattered elastic waves form the pronounced tail of seismograms known as the seismic coda [1,2]. Even when scattering is prominent, it is still possible to define the phase of the seismic record by introducing the complex analytic signal ψ(t, r) = A(t, r)e iφ(t,r) , with A the amplitude and φ the phase. In the past, many studies have focused on the modeling of the mean field intensity I(t) = A(t) 2 [see 3, for review]. The goal of the present paper is to study the statistics of the phase field in the coda. In the coda, the measured displacements result from the superposition of many partial waves which have propagated along different paths between the source and the receiver. Each path consists of a sequence of scattering events that affect the phase of the corresponding partial wave in a random way. For narrow-band signals, the phase field can therefore be written as φ(t, r) = ωt + δφ(t, r), where ω is the central frequency, and δφ denotes the random fluctuations. The trivial cyclic phase ωt cancels when a spatial phase difference is considered between two neighbouring points. Spatially resolved measurements are facilitated by dense arrays of seismometers that have been set up occasionally. We note that the phase of coda waves has not been given much attention so far. The advantage of phase is that it is not affected by the earthquake magnitude, and that it contains pure information on scattering, not blurred by absorption effects. For the statistical analysis of amplitude and phase fluctuations of direct arrivals, we refer the reader to e.g. Zheng and Wu [4].
In strongly scattering materials, multiple scattering tends to randomize the phase of transmitted or reflected waves and, as a result, the phase has often been overlooked. In this talk, the use of phase information to monitor the dynamics of multiply scattering media will be described and illustrated through measurements of the temporal fluctuations of ultrasonic waves transmitted through a time-varying mesoscopic sample. The probability distribution of the wrapped phase difference as a function of evolution time, as well as its variance, is measured and compared with theoretical predictions based on circular Gaussian (C1) statistics. Excellent agreement is found. A fundamental relationship between the variance in the phase of the transmitted waves and the fluctuations in the phase of individual scattering paths is predicted theoretically and verified experimentally. This relationship not only gives deeper insight into the physics of the phase of multiply scattered waves, but also provides a new way of probing the motion of the scatterers in the medium. To investigate dynamics on longer time scales, we also investigate the variance and correlations of the cumulative phase. This combination of wrapped and cumulative phase measurements allows both the short and long time dynamics to be probed with excellent sensitivity.
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