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Laser characterization of ultrasonic wave propagation in random media John A. Scales and Alison E. Malcolm Physical Acoustics Laboratory and Center for Wave Phenomena, Department of Geophysics, Colorado School of Mines.

ABSTRACT

Lasers can be used to excite and detect ultrasonic waves in a wide variety of materials. This allows the measurement of absolute particle motion without the mechanical disturbances of contacting transducers. In an ultrasound transmission experiment, the wave-ﬁeld is usually accessible only on the boundaries of a sample. Using optical methods one can measure the surface wave-ﬁeld, in eﬀect, within the scattering region. Here we describe ultrasonic wave propagation in randomly heterogeneous rock samples. By scanning the surface of the sample we can directly visualize the complex dynamics of diﬀraction, multiple scattering, mode conversion and whispering gallery modes. We will show measurements on rock samples that have similar elastic moduli and intrinsic attenuation, but diﬀerent grain sizes, and hence, diﬀerent scattering strengths. The intensity data are well ﬁt by a radiative transfer model, and we use this fact to infer the scattering mean free path. 1

INTRODUCTION

Rocks are complicated heterogeneous materials with microstructures that scatter acoustic or seismic waves. At the laboratory (ultrasonic) scale these microstructures are primarily cracks and grain boundaries. The scattering of waves from these boundaries is generally considered noise in geophysical applications, but these scattered waveﬁelds may give insight into how the rock formed, the environment in which it formed, its state of stress, ﬂuid saturation, etc. Heterogeneous media appear homogeneous when probed with waves whose wavelength is large compared to the scale of heterogeneity. Long-wavelength measurements provide the bulk or average properties of the medium. Traditional acoustic measurement methods using contacting transducers as sources and detectors work well to measure bulk properties since these properties can be inferred from waves which travel directly from the source to the detector. As the wavelength decreases relative to the heterogeneity, scattering from the microstructure becomes important and the transducers themselves act as scatterers, disturbing the measured waveﬁelds. We avoid these problems by using lasers as both sources and detectors of ultrasonic waves. Using lasers also allows us to collect dense, high-ﬁdelity data sets relatively quickly without having to physically couple transducers to the sample. We can introduce the laser beams into hostile environments (such as vacuum

chambers and ovens) via optical windows or ﬁber, create line or point sources by focusing the source beam, scan the surface waveﬁeld by mechanically pointing the detector beam, and focus the detector beam on spots that are much smaller than the smallest contacting transducer. Imagine having an array of tens of thousands of micron-sized, massless accelerometers! The ability to collect dense data sets allows us to visualize the entire waveﬁeld as it travels through the rock. By simply watching the waveﬁeld move as a function of time it is possible to distinguish phenomena such as absorption from scattering. To highlight these diﬀerences we perform experiments on samples with diﬀerent scales of heterogeneity. We have also used three diﬀerent experimental setups, each to highlight diﬀerent aspects of wave propagation. Ultimately our goal is to elevate the multiple-scattering speckle to routine use as an important source of information. We start here with the more modest goals of demonstrating the experimental feasibility of our plan and showing how strongly multiply scattered energy can be used to make inferences about the microstructure of rocks.

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EXPERIMENTAL SETUP

In our laboratory we study ultrasonic wave propagation in heterogeneous media such as rocks and engineered composites. A pulsed Nd:YAG laser (1064 nm

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Figure 1. A rotational scan of the Elberton granite core. The source and detector laser are along a line that crosses the cylindrical sample at antipodes. The sample is rotated through 360 degrees. On the left is the complete scan with no gain applied. The horizontal events visible are Rayleigh waves. On the right is a zoom showing the ﬁrst 10-50 µs of the data. The ﬁrst event is a compressional wave propagating directly across the sample. The low frequency event at about 33 µs consists of two counter-rotating Rayleigh waves. The angle-dependent dephasing of these waves is caused by heterogeneities along the path. The horizontal event at 22 µs is a whispering gallery mode (WGM) which skims the surface of the sample propagating at the compressional wave velocity.

wavelength, 5 ns pulses, .3 J/pulse) is used to excite ultrasonic waves via thermoelastic expansion or ablation. The source radiation characteristics are diﬀerent in the thermoelastic and ablation regimes (Scruby & Drain, 1990). In the ablation regime, the explosion of the plasma at the surface creates a reactive force into the sample; in the thermoelastic regime the thermal expansion of the sample creates a dipole radiation pattern. This diﬀerence could be signiﬁcant in experiments which depend strictly on source/detector reciprocity such as coherent backscattering. These waves are detected using a scanning laser interferometer that measures the absolute particle velocity on the surface of the sample via the Doppler shift. (See (Scruby & Drain, 1990) for the general principles of laser vibrometry. ) The output of the interferometer is then digitized at 14-bit precision using a digital oscilloscope card attached to a PC. The lasers and the sample are positioned on an optical bench with vibration isolation. The dominant frequency of the measured waves is 1 MHz, a limit imposed by the detector electronics. (At GHz frequencies, femtosecond lasers have recently been used to visualize ultrasound propagation in crystalline material (Sugawara et al., 2002) by scanning the surface, while in (Gahagan et al., 2002), ultrafast interferometric microscopy is used to characterize laser-driven shock waves.) In the sort of granular rock samples that we use,

scattering and attenuation strongly limit the ability to propagate high-frequency ultrasound. We have collected data on three samples representing diﬀerent scales of heterogeneity: aluminum (homogeneous), Elberton granite (smaller grains) and Llano granite (larger grains). We use three diﬀerent experimental conﬁgurations to highlight diﬀerent aspects of wave propagation. The ﬁrst is a “rotational scan” made by shooting across a sample (source and detector beams focused on antipodes, in the middle of a cylindrical sample), which shows the strong surface waves. The second experiment is a “surface scan” in which data are collected on a dense grid on the round top surface of the same cylindrical samples, allowing us to visualize the full waveﬁeld. Finally, to extract the scattering mean free path of the strongly scattering Llano sample, we collected data from a rectangular region of a sample large enough to enable us to ignore the strong reﬂections from the boundaries visible in the surface scan experiment.

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SURFACE WAVE TRAINS

Figure 1 shows an example of a rotational scan in the medium-grained Elberton granite (grain size about 1 mm). In this sample the compressional wave speed (Vp ) shows a characteristic sinusoidal dependence on the ori-

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Figure 2. The average over angle of the time series shown in Figure 1.

entation of the sample relative to the source/detector line. This body-wave anisotropy is due to the alignment of microfractures in the rock (Douglass & Voight, 1969). Only even multiples of the angle appear in a Fourier series of Vp (θ); odd powers are precluded by reciprocity (Smith & Dahlen, 1973). In our conﬁguration, however, waves that propagate along the surface, such as Rayleigh waves and whispering gallery modes (WGM), are unaﬀected by the anisotropy. The source-generated signal that is incoherent with respect to angle is largely due to grain scattering. At long times, surface waves dominate the signal since they spread in only two dimensions. If we were to average all these time series over angle, the body waves and the grain-scattered waves would tend to cancel, while the surface waves would be enhanced. This mean, or coherent, time series is shown in Figure 2. These repetitive surface wave pulses are phase-locked since they originate from the same pulse. Such a train of phase-locked pulses gives rise to a harmonic comb of frequencies (e.g., (Teets et al., 1977), (Bellini et al., 1997)). The theory is explained in the appendix but the basic idea is that no matter how widely separated in the time-domain, the pulses interfere, giving rise to fringes (closely spaced spikes) in the power spectrum. These fringes provide a precisely spaced ruler that converges to a Dirac comb as the number of pulses goes to inﬁnity. Even though we recorded only six pulses, the enhancement of the spectral resolution is striking. To show this we tapered the coherent signal around the ﬁrst two pulses and then zero padded to the length of the original time series. The power spectra of the two cases (two pulses versus six) are shown in Figure 3. In order to compute the Rayleigh wave speed we picked the peaks of the power spectrum. The spacing between these peaks corresponds to the rate at which surface waves propagate around the sample. We interpolated (by zero padding) the data in Figure 2 to yield a spacing of 305 Hz and to give us a non-zero variance of the peak spacing as a function of frequency. The resulting average peak spacing is 16378 Hz, yielding a surface wave speed of 2.83 ± 0.05 mm/µs, for a 55 mm diameter sample. (The uncertainty was conservatively taken to correspond to the 305 Hz frequency bin size.) The periodicity in the frequency comb indicates that there is no

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dispersion (wave speed is independent of frequency) and hence the microfracturing is not limited to the depth of penetration of the higher surface wave modes.

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VARIATION IN SCATTERING STRENGTH

In the Elberton granite, the ratio of wavelength to grain size is about 3 for surface and shear waves and 6 for compressional waves. We can see the eﬀects of grain scattering in the coda (i.e., the energy after the ballistic arrivals), but the wave propagation is coherent for hundreds of wavelengths. We have also made measurements in the Llano granite, which has similar elastic moduli and intrinsic attenuation (based on resonance measurements), but much stronger scattering. The ratio of the wavelength to the grain size in this sample is close to 1. In order to compare in detail the scattering of ultrasonic waves for these two samples we performed surface scans on the ends of cylindrical samples. The source laser was focused to a point on one end of the cylinders 7 mm from the edge. The ends of the cylinders were scanned at a density of 9 points per square milimeter. Both granite cylinders are approximately 100 mm long. The Elberton granite sample is 55 mm in diameter and the Llano sample is 50 mm in diameter. Having such spatially dense sampling of the waveﬁeld allows for a variety of spatial-Fourier-domain ﬁltering procedures. For example, in (Campman & Van Wijk, 2002) it is shown how to separate the direct from the scattered ﬁeld in media where surface scatterers are embedded in

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Figure 4. Elberton granite: on the left is the early part of the rotation scan of the sample as previously discussed. The full scan shows up to six surface wave circumnavigations of the sample. On the right are snapshots of the waveﬁeld measured on one end of the cylinder. The source laser is focused on a point 7 mm from the edge of the cylinder. Although the surface wave ﬁelds (Rayleigh waves) dominate the ﬁgure, both compressional and shear body waves are visible as well as Rayleigh and whispering gallery waves propagating around the edge of the cylinder. In the snapshots, G refers to the so-called ghost wave, which is a reﬂection of the source from the side of the sample. R is a Rayleigh wave on the end of the sample, R is a Rayleigh wave propagating around the circular edge (R4, for instance, denotes the Rayleigh wave that has gone around the sample four times).

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Figure 5. Llano granite: in this sample, even though the elastic moduli and intrinsic attenuation are similar to those in the Elberton granite, the scattering is much stronger, since the (c) ratio of wavelength to grain-size is close to 1. This strong scattering manifests itself in the rapid spatial decoherence of the waveﬁeld, both in the rotational scan (left) and in the surface waveﬁeld snapshots (right). After about 30 µs, the waveﬁeld is completely decoherent, consisting entirely of multiple scattering speckle.

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a background matrix. Here we limit the discussion to the estimation of the scattering mean free path. In this experiment we are almost measuring the Green’s function. When tuned to the thermoelastic regime, the radiation pattern of the source laser is not quite reciprocal to the detector (Scruby & Drain, 1990). (In the multiple scattering coda, the Green’s function

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can be synthesized from the two-point correlation function of the speckle (Malcolm et al., 2003). This relies on the equipartitioning of the energy amongst the modes, which only occurs beyond the mean free time.) Figures 4 and 5 show the rotational scans as above, next to snapshots of the scanned waveﬁeld. (Mpeg movies of these data are available on our web site at

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Figure 6. Many more modes are visible in the aluminum sample, including whispering gallery modes associated with both shear waves (WS) and compressional waves (WP). In addition, due to the conductivity of the sample, an ablation-generated sound wave in air is visible as a slow outgoing circular wave front in the snapshots. For the rock samples, the source laser was tuned to the thermoelastic regime.

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http://acoustics.mines.edu. These movies are useful for understanding the complex dynamics of wave propagation, scattering, caustic generation, etc.) For comparison, we show in Figure 6 a surface scan made in aluminum. Given that the intrinsic attenuation of the Llano sample is about the same as that of the Elberton, it is visually apparent that the scattering mean free path s is much shorter in Llano than in Elberton. To quantify this we look at ﬁtting the data with a radiative transfer model.

RADIATIVE TRANSFER

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where c is the group or energy velocity, r is the distance from the source, H is the Heaviside function and I(r, t) is the total intensity. (This model assume no mode conversion or localization and that the scattering is isotropic.) At “early” times this corresponds to the ballistic propagation of energy, while at “late” times RT is equivalent to diﬀusion. Early and late are, of course, relative to the strength of the scattering and attenuation. Our cylindrical surface scans were not ideal for this analysis because of the inﬂuence of the boundary. In order to eliminate the eﬀects of the boundary we made a surface scan on a long block of Llano granite (40x80 mm cross-section and over 1 m in length). We focused the source laser onto a 21 mm line using a cylindrical lens and scanned the surface waveﬁeld in a region 40x40 mm in size. Use of a line source gave largely one-dimensional propagation and allowed for ensemble averaging normal to the propagation direction. In order to ﬁt these data to the RT equation we must ﬁrst estimate the group or energy velocity. This we do by performing a regression on the peak intensity as a function of source-to-detector distance. This regression is shown in Figure 7. In the Llano granite, because of the strong scattering, our data do not constrain the absorption, a , though they do give information about the scattering. Extracting s from the radiative transfer equation is diﬃcult, however, because at late times the dependence of RT on s becomes algebraic rather than exponential. Thus it is important to average many realizations to reduce the spread (random ﬂuctuations due to the particular realization of the scattering) of the data enough to constrain s . To do this, we note that at large times (i.e., well after the ballistic peak) RT is independent of the distance r,

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allowing us to average over r as well as over our ensemble of diﬀerent lines of detector positions. The ensemble average intensity is shown in Figure 8. We then ﬁt the result to RT, as shown in Figure 9, giving us a mean free path of about 8 ± 2 mm.

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CONCLUSIONS

Laser ultrasonics provides an ideal tool for studying the complex dynamics of wave propagation in random media. It is possible to scan the surface of objects hundreds

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Figure 9. A radiative transfer ﬁt to the late-time intensity for mean free scattering path of 8 mm. Analysis of the optimization problem for s gives an uncertainty of about ±2 mm.

of wavelengths or more in extent at a suﬃcient density (many grid points per wavelength) to allow for the visualization of the surface waveﬁeld. Using a pulsed IR laser as a source allows us to create sources that are focused on points, lines or other distributed shapes, while using a scanning laser interferometer as a detector is analogous to having an array of tens of thousands of massless accelerometers. In addition, the interferometer provides an absolute measure of particle motion. Our goal is to use these non-contacting methods to measure and exploit the multiple scattering speckle in random media such as rock. Here we have shown an analysis of measurements made on three samples of varying degrees of heterogeneity. For the most heterogeneous sample, a coarse-grained granite, the scattering is so strong that the wave ﬁeld rapidly decoheres into speckle. In this sample we are able to infer the scattering mean free path by ﬁtting a radiative transfer model to the ensemble averaged intensities. Thus we can exploit the multiple scattering to characterize a microscopic feature of the rock sample that would be diﬃcult to measure otherwise.

ACKNOWLEDGMENTS We acknowledge many useful discussions with our CSM colleagues Roel Snieder, Kasper van Wijk, Alex Grˆet, Matt Haney, Huub Douma, and Carlos Pacheco and with Bart van Tiggelen of the Universit´e Joseph Fourier in Grenoble. This work was partially supported by the National Science Foundation (EAR-0111804) and the US Army Research Oﬃce (DAAG55-98-1-0070). A. Malcolm was also partially supported by TotalFinaElf.

Bellini, M., Bartoli, A., & H¨ ansch, T.W. 1997. Opt. Lett., 22, 540. Campman, X., & Van Wijk, K. 2002. preprint. Douglass, P.M., & Voight, B. 1969. Anisotropy of granites: a reﬂection of microscopic fabric. Geotechnique, 19, 376. Gahagan, K. T., Moore, D. S., Funk, D. J., Reho, J. H., & Rabie, R. L. 2002. Ultrafast interferometric microscopy for laser-driven shock wave characterization. J. Appl. Phys., 92, 3679. Malcolm, A.E., Scales, J.A., & Van Tiggelen, B. 2003. Imaging with multiple-scattering speckle. preprint. Paasschens, J. C. 1997. Solution of the time-dependent Boltzmann equation. Phys. Rev. E, 56, 1135. Scruby, C.B., & Drain, L.E. 1990. Laser Ultrasonics: techniques and applications. London: Adam Hilger. Smith, M.L., & Dahlen, F.A. 1973. The azimuthal dependence of Love and Rayleigh wave propagation in a slightly anisotropic medium. JGR, 78, 3321–3333. Sugawara, Y., Wright, O. B., Matsuda, O., Takigahira, M., Y. Tanaka, S. Tamura, & Gusev, V. E. 2002. Watching ripples on crystals. Phys. Rev. Lett., 88, 185504. Teets, R., Eckstein, J.N., & H¨ ansch, T.W. 1977. Phys. Rev. Lett, 38, 760. Udem, T., & Fergusen, A.I. 2002. Achievements in optical frequency metrology. In: Figger, H., Meschede, D., & Zimmermann, C. (eds), Laser physics at the limits. Springer-Verlag. Zadler, B., Le Rousseau, J.H.L., Scales, J.A., & Smith, M.L. 2002. Resonant ultrasound spectroscopy: theory and application. preprint.

Appendix: Phase-locked pulse trains 2

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Consider a Gaussian pulse p(t) = e−(t−∆t) /2σ . The time shift, ∆t introduces a phase shift in the Fourier 2 2 transform: p(ω) = σe−iω∆t e−σ ω /2 . Hence, the power spectrum for two identical pulses separated by ∆t will contain an interference term. The envelope will still be 2 2 the Gaussian e−σ ω /2 , but there will be modulation with peaks of full-width at half-max (FWHM) π/∆t and separation 1/∆t. For n such pulses the power spectrum is a geometric series the summation of which is: Pn (ω) = 2σ 2 e−σ

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In the n−pulse case the spacing between the peaks of the power spectrum is still 1/∆t but the FWHM is 2π/n∆t. If ∆t is constant, then for large n, the pulse superposition provides a precise ruler in the frequency domain, allowing pulse trains to achieve bandwidth far below that of a single pulse (see (Udem & Fergusen, 2002) for a historical account of the development of the pulsed laser comb). In the limit that the number of pulses goes to inﬁnity, Equation 1 converges to a comb of Dirac delta functions with a Gaussian envelope. In a way this is not surprising, since standing waves (i.e., normal modes), which have line spectra, can be thought of as a super-

Wave propagation in random media position of a large number of traveling waves. In fact, in our experiment the comb is nothing but the fundamental surface wave mode plus all the higher overtones. For our cylindrical sample, the number of pulses that can be used depends on the quality factor, Q, of the sample. Once the frequency of the fundamental surface wave mode is known, we could perform a resonance experiment by exciting the sample near this frequency and detecting the signal with a lock-in ampliﬁer as in resonance ultrasound spectroscopy (Zadler et al., 2002). Still, the frequency comb gives us direct access to any frequency dependence (dispersion) of the surface wave speed, and hence the depth of penetration of the microfracturing.

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