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 want to utilize the zero group velocity (ZGV) point of the first order symmetrical mode Lamb wave to characterize isotropic plates non-destructively.

It is well–known that for certain wave modes within the Lamb wave dispersion relation segments with negative group velocity exist (Fig.1). The turning points of these curves between negative and positive slope and positive group velocity are of significant interest as here the group velocity becomes zero and at this point of the dispersion relation a strong, 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 for 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; hence, the introduced energy cannot propagate away from the location of the generation, making them well detectable.

Earlier works have shown that laser-ultrasonic techniques are well suited to investigate these points due to their non-contact nature and the excited broad frequency content. Previous approaches made use of pulse-echo measurements evaluating only the frequencies of the thickness and ZGV resonance points. Although, the ZGV points are additional resonance frequencies to the thickness resonances, surprisingly, these measurement techniques do not deliver additional information, such as the material properties and plate thickness, compared to conventional pulse-echo measurements. This can be explained by the fact, that these works considered only temporal frequencies. We propose a new approach using frequency domain measurements - in both spatial and temporal frequency domains by which an arbitrary point of the dispersion relation can be evaluated. Measuring the ZGV resonance with both temporal (fZGV) and spatial (kZGV) frequencies allows to evaluate additional information compared to conventional pulse-echo measurements. Recent developments in laser-based ultrasound with the use of spatially and temporally modulated laser-sources allow direct frequency domain measurements, hence, the evaluation of both temporal (fZGV) and spatial (kZGV) frequencies become possible. The gained combined information of three frequencies fS1, fZGV, kZGV allows the evaluation of three unknowns cL, cS, h.

The working principle of the setup is shown in Fig.4 for SAWs: by utilizing a spatial light modulator (SLM), an intensity modulated laser beam with frequency f 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 Figs.4(a, b). The amplitude for a given number of lines N at frequency f for periodicity D (assuming infinite narrow line sources) yields:

For the cases ?=m?(f), (m=1,2,3,…), where ?(f)=2p/k(f) is the wavelength of the SAW, the contributions of all N lines add up to a maximum of the resulting amplitude. Figure 4(c) shows a simulated amplitude vs. frequency curve for a bulk material and a fixed excitation periodicity ?. This curve shows the interference of N plane waves from the N sources: the maximum repeats for integer multiples of f0 and side lobes arise for N >2. It can also be seen that the width of the peaks decreases with the number of excitation lines. The first maximum of the function A(?,f) at f0 corresponds to a point in the dispersion relation of the investigated system, given through the frequency-wavelength relationship. Therefore, our method allows a direct measurement of the dispersion relation, by scanning the frequencies f at different spatial periods ?. In the experiments, an excitation pattern with spatial periodicity ? is applied and the temporal frequency is scanned, while detecting the SAW amplitudes. This experimental technique will be extended in the current project to Lamb waves to investigate plates and plate-like structures.

In a 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, the diameter of a laser spot exciting a ZGV mode was varied. The wavelength of a ZGV mode relates to the optimum spot diameter and can therefore be investigated in this way.