Physical & Computational Acoustics

The propagation of ultrasound in solids is governed by the dynamic equations of elasticity and determined by boundary conditions (such as interfaces) and material properties.

We develop numeric and analytic models of wave propagation combined with laser-ultrasound (LUS) experiments to evaluate mechanic and geometric parameters of materials or parts. Examples are the determination of elastic moduli, grain size of polycrystalline materials, material damping or thickness of thin membranes.

Besides homogeneous samples, analyzing certain modes of wave propagation (e.g. surface acoustic waves or guided waves) allows the characterization of a layer on a substrate, or plate-like samples. The scale of such problems may range from mm thick steel plates down to micron thick coatings on Silicon wafers, for example.

If suitable samples are provided, our experimental equipment allows material characterization up to acoustic frequencies of 6 GHz.

We also utilize guided waves for the detection of cracks and delamination.

The basics for these applications and also fascinating physics are researched in publically funded projects. Our main research topics are:

    • Grain boundary scattering
    • Wave propagation in complex media, e.g. grain structures in metals and polycrystals
    • Grain boundary cracks
    • Zero-group-velocity (ZGV) and backward Lamb waves
    • Development of laser based ultrasound setups
    • Material damping

We attend international conferences and publish in renowned journals. Examples of our projects are introduced below.

Optical excitation and detection of zero group velocity Lamb waves

Our proposal deals with non-contact excitation and detection of a Lamb wave in a particular point of the dispersion relation. By using spatially and temporally modulated laser-ultrasound we aim to utilize the zero group velocity (ZGV) point of the first order symmetrical mode Lamb wave to characterize isotropic plates non-destructively.

Figure 1. (a) Lamb wave dispersion curves for a 1mm thick aluminum plate. The zero group velocity (ZGV) point of the S1 mode is marked with a black dot. The ZGV point of the S4 mode is also visible, however, the zero slope of this mode is less distinct. (b) Complex dispersion relation with a complex mode originating from the saddle point with ZGV of the real mode.

Lamb waves and ZGV points

It is widely acknowledged that certain wave modes within the Lamb wave dispersion relation segments with negative group velocity exist (Fig.1). The turning points of these curves, which exist between negative and positive slope and positive group velocity, are of significant interest as the group velocity becomes zero and at this point of the dispersion relation a strong and well-detectable resonance occurs. In contrast to thickness resonances in plates which occur with k=0, (where k denotes the wave number), this mode is associated with a finite wave number. The mode shape of the S1 mode at the ZGV point is shown in Fig. 2. The Poynting vector reveals that the power flow within the cross section cancels out this particular frequency leading to a resonance-like behavior. This can be explained by the fact that the propagation velocity of the energy of a wave packet is equal to the group velocity. Therefore, the introduced energy is unable to propagate away from the location of the generation, making them well detectable.

Figure 2. Mode shape of the S1 mode at the ZGV point.

In cooperation with the University of Colorado at Boulder we experimentally investigated the coupling of a laser pulse into ZGV modes of tungsten plates. In the experiment, we varied the diameter of a laser spot exciting a ZGV mode. The wavelength of a ZGV mode relates to the optimum spot diameter and can therefore be investigated in this way.

Figure 3. Experimental setup to measure dependence of coupling laser excitation into a ZGV mode on the diameter of the laser spot. The spot size was varied by translating the lens, which focused the laser on the sample surface. The acoustic waves were detected on the opposite side of the sample with a Michelson interferometer.

Frequency spectra of the time domain signals from simulations (Fig. 4; left); the ZGV points at 2.28, 4.00, 9.33 are apparent. The frequency domain response of the tungsten plate is shown as a function of the D/h ratio. The inset shows the amplitude of the Fourier transforms at the S1, A3, and A7 ZGV frequencies from the numerical simulations.

Results of the experimental investigations of the excitability of ZGV modes with constant pulse energy (Fig. 4; right): (b) Spectra for the different excitation spot diameters. (c) Variation of the spectra of the measured signals with increasing spot diameter. The vertical plane and the line within the plane mark the position of the S1 ZGV mode. (d) Variation of the amplitude of the S1 ZGV mode normalized to constant energy density for each pulse diameter. The maximum at D=h approx. 1.4 is apparent.

Location of the ZGV points

For the practical use of the ZGV points, their location in the k—w plane must be determined. ZGV points represent mathematically well-defined points within the dispersion relation as the wave number is non-zero and the group velocity vanishes.

The latter is given through the implicit derivative for the symmetric and antisymetric modes. The system of equations are given as:

The solution of the system of non-linear equations can be obtained numerically. Here, a Newton–Raphson iteration is used and the corresponding iteration process is given as:

The Poisson’s ratio of an isotropic plate can be determined with high accuracy by using two ZGV frequencies [4]. In addition, if the plate thickness is known then both elastic properties can be found. Although the basic idea is simple, the calculation of the locations of the ZGV points is more challenging. In our approach four non-linear equations are solved directly, yielding the unknown wave velocities cL and cT (or for example, Poisson’s ratio and one wave velocity).
The inverse problem is demonstrated with a tungsten plate with a thickness of 0.25 mm and an aluminum plate with a thickness of 0.125 mm.

Figure 5. Experimental signal and its Fourier transform for tungsten (a) and (b) and for aluminum (c) and (d) plates. The first two resonances indicated in the spectra are associated with symmetric and antisymmetric modes, respectively.

Spatially and temporally modulated Laser-Ultrasound (STeMoLUS)

We experimentally and theoretically pursue a unique combination of adjustable spatial modulation13 with freely tunable temporal modulation.8, 9 Our technique – spatial and temporal modulated laser-ultrasound (STeMoLUS) – combines the previous methods, allowing an arbitrary frequency resolution in the measurement without the necessity of using scanning optomechanics. Moreover, the dispersion relation is obtained directly from the maxima of the amplitude frequency scan, without the transformation techniques required for the evaluation of spatio-temporal scans.

Figure 6. Working principle of STeMoLUS: (a) Spatially modulated laser beams are achieved using SLM leading to excitation of interfering SAWs. The interference is constructive if the SAW’s wavelength, k, is equal to the periodicity, D. For clarity, the reverse propagating wave is omitted in the figure. (b) Excitation pattern and detection spot on the sample surface (c) Simulated interference amplitudes for excitation frequencies in the range of three interference maxima and different numbers of excitation lines.
Figure 7. (a)–(d) Comparison of experimentally (top) and theoretically (bottom) obtained interference plots: (a) and (b) non-dispersive SAWs in aluminum, (c) and (d) dispersive SAWs in molybdenum coated glass. The first interference maxima correspond to the dispersion relations. As the interference condition D=mk is fulfilled for arbitrary integers m, several interference maxima arise. (e) Experimentally obtained SAW velocities compared to a theoretical dispersion curve for the system in (c) and (d). (f) Experimental verification of constructive interference on the aluminum sample: the amplitude of the observed interference for D=54.5 mm remains constant using 2, 3, or 4 excitation lines whilst keeping the total energy on the sample constant. The magnitudes are normalized by the single line frequency response to remove the damping effects.

The working principle is shown in Fig. 6(a): by utilizing SLM, an intensity modulated laser beam with frequency is imaged onto the sample surface as a parallel line pattern. The surface displacement, produced by the superposition of SAWs emitted from the line array, is detected in a point outside of the array along a line perpendicular to the array as shown in Fig. 6(b).

We verified the Stemolus with a non-dispersive sample (uncoated aluminum) and with a dispersive system (glass substrate coated with a 3 mm thick molybdenum layer). In the measurement, the spatial periods in the excitation pattern were varied in steps of 6.05 mm. For each pattern, a frequency scan between 20 MHz and 150 MHz with a frequency step of 500 kHz was carried out. Subsequently, each frequency scan was filtered with a five point moving average filter. Figure 2(a) shows the recovered amplitudes for the measurement on aluminum. Since the interference condition is fulfilled for integer multiples of the SAW wavelengths, several maxima are obtained in the investigated frequency range, besides the distinct first maximum for k = D.

Numerical modeling of ultrasonic propagation and the generation of ultrasound by laser irradiation

The experimental application of laser-generated ultrasound in solids has been used extensively in experimental acoustics in the past decades because it enables contactless excitation of different elastic waves, such as bulk, surface, or guided waves. In search of better understanding and optimization of the generation mechanism, modeling by a thermoelastic source attracted significant attention from the initial stages of the experimental application. Thus, the generation process is defined by the coupled heat and wave equations, and various coupling terms are taken into account. Within laser ultrasonics the coupling of the heat and wave equations is due to thermal expansion leading to a semi-coupled problem.

In the generalized thermoelasticity, however, the thermal feedback of the propagating stress pulses is also included, hence, the problem becomes fully coupled and both thermal and elastic waves arise:

The spatial discretization of the two sets of equations is done with finite differences using staggered grids. Hence, the temperature, displacement, and stress components are discretized on different grids, which are shifted by a half-cell in one or both directions. The grids of the temperatures and normal stresses are identical whereas the grids of the displacements and shear stresses are shifted by a half grid dimension (Dx=2;Dy=2).

The spatially discretized equations could be written in the following matrix form:

Where M, C, K are the coefficient matrices. In the current case the wave equation is discretized by the conditionally stable, explicit Euler-method; therefore, the displacements in the (n+1)th time step are explicitly solved as a function of the displacements and stresses in previous time steps. Unconditionally stable, implicit integration techniques enable the use of the same temporal and spatial discretization and the heat equation is therefore integrated by the Wilson H method. During the irradiation of a metallic surface with a short laser pulse the electromagnetic waves penetrate a thin skin close to the surface (in the range of 10 nm), which is modeled as a surface flux.

Figure 8. Out-of plane surface displacements with Rayleigh and leaky-Rayleigh waves for aluminum (a) and for zinc (b). (c) Sections of the generated wave fields in the models for aluminum (left) and for zinc (right). In addition to the generated bulk longitudinal (L), quasi-longitudinal (QL), and shear waves (S), Rayleigh (R) and leaky-Rayleigh (LR) waves are also visible.

Numerical results represented by the vertical surface displacements are shown above. The generated Rayleigh waves are unipolar with no attenuation, as expected. The small, attenuating wave in front of the Rayleigh wave is assigned to the longitudinal bulk wave, or ’’surface skimming longitudinal wave’’. The magnitude of the displacements are shown in Fig. 7c at t = 4.48 ns (aluminum) and t = 5.8 ns (zinc) after excitation from a 1mm wide line source with 20 ps temporal duration. Besides the generated bulk longitudinal (L) and shear waves (S) also surface waves (R) are visible in aluminum. The longitudinal bulk wave shows an almost cylindrical wave front in the isotropic case with strongly varying pulse widths (and frequency content), as a consequence of the finite width of the excitation. The wave forms in zinc are different due to the anisotropy. In the transversely isotropic case in the investigated plane there exist three different wave types: quasi longitudinal (QL), quasi transverse (QT) and pure transverse (PT) modes.

Characterization of optical fibers for the detection of ultrasound

In this study we present a theoretical framework for calculating the acoustic response of optical fiber based ultrasound sensors. The acoustic response is evaluated for optical fibers with several layers of coating assuming a harmonic point source with arbitrary position and frequency. First, the fiber is acoustically modeled by a layered cylinder on which spherical waves are impinged. The scattering of the acoustic waves is calculated analytically and used to find the normal components of the strains on the fiber axis. Then, a strain-optic model calculates the phase shift experienced by the guided mode in the fiber owing to the induced strains.

Berer and Veres et al, J. Biophotonics (5), 518-528, 2012

Figure 9. Acoustic response of a bare optical fiber to an incident plane wave up to 100 MHz. The peaks correspond to cross-sectional resonances of the fiber. The framework is showcased for a silica fiber with two layers of coating for frequencies in the megahertz regime, commonly used in medical imaging applications. The theoretical results are compared to experimental data obtained with a sensing element based on a piphase-shifted fiber Bragg grating and with photoacoustically generated ultrasonic signals.
Figure 10. Acoustic response of a layered optical fiber to an incident spherical wave.

Figure 10 shows the response of the fiber compared to a set of experimental results where the source is shifted parallel to the fiber axis and the combined phase shifts along 15 and 30 degrees.

Veres et al, JASA 135(4), 1853-1862, 2014.

Figure 11. Problem statement: Spherical waves generated from an opto-acoustical source (R=50 mm) scatter from the optical fiber which is located at a distance of d from the source. The radii of the glass fiber and the coatings are 62.5, 110, and 130 mm, respectively. An FBG sensor in a distance of d is used to achieve the detection.

Waves in complex media

Industrial Applications

SHM - Structural Health Monitoring

For composite materials (or the whole lightweight construction) in particular, integrated quality monitoring (SHM - Structural Health Monitoring) allows us to take full advantage of the potentials of lightweight construction. A structure with permanently monitored integrity can, e.g., be realized by the integration of optical waveguides with FBG sensors. For such topics our experts in the field of Physical & Computational Acoustics can support you. 

Metal processing

Since Laser-Ultrasonics can be operated from a safe distance, this technology can be used in very early stages of metal processing at high temperatures. For example, the thickness of plates and the wall thickness of pipes can be measured in the still glowing state (even with only one-sided access). Also changes of the microstructure (e.g. grain growth), that are interesting in the rolling process, can be characterized with LUS.

Different materials (steel, aluminum, copper, semiconductors) - a wide range of issues (sheet thickness, defects, elastic properties, grain growth, anisotropy): the LUS technology combined with innovative data evaluation is highly versatile.

Characterization of thin-film structures

Laser-Ultrasonics allows a very efficient characterization of thin-film structures. High-frequency ultrasonic waves (SAW) are excited up to 1 GHz by laser pulses. The layer systems are characterized by analysing the dispersion behaviour of the waves. Ask our LUS experts!

Determination of layer thickness using surface acoustic waves (SAW)

Using Laser-Ultrasonics, the frequency dependent velocity of surface acoustic waves in a layer can be measured. The data can be evaluated to find the layer properties, for example its thickness. Our systems enable us to measure SAW dispersion curves up to 1 GHz, which allows us to characterize layers ranging from 500 nm up to several mm thickness. The illustrations show a core material (red) with an oscillating layer on top (blue) at increasing frequencies of a guided wave mode from case 1 to 4, whereas the penetration depth of the displacement field decreases until only the layer material influences the wave propagation (4). Thus, depending on the geometry and elastic properties of the layer material under investigation, the appropriate parameters of the measurement can be selected
The investigation of layered structures with higher complexity (e.g., multilayers or gradual layers) is possible by adapting the theoretical models for the calculation of dispersion curves.