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All issues Volume 21 / No 2 (2025) J. Eur. Opt. Society-Rapid Publ., 21 2 (2025) 39 Full HTML
Open Access
Issue
J. Eur. Opt. Society-Rapid Publ.
Volume 21, Number 2, 2025
Article Number 39
Number of page(s) 6
DOI https://doi.org/10.1051/jeos/2025034
Published online 03 September 2025

© The Author(s), published by EDP Sciences, 2025

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Ultrasound is used in various ways in many methods from the fields of condition monitoring [1], structural health monitoring [2], non-destructive testing [3–5] or even process monitoring [6]. Ultrasound in the form of guided ultrasonic waves (GUWs) [7] can be used, for example, to detect component defects in metal [8] and fiber-reinforced composite plate structures [9] such as delaminations [10] or to test adhesive bonds [11]. If the method requires knowledge of the ultrasonic wave field or rather the mechanical displacement fields on the surface of the component, the current gold standard for the measurement is Laser Doppler Vibrometry (LDV) [12]. LDV is a point measurement method and usually operated point-by-point in scanning mode [13] for the measurement of two-dimensional wave fields, which results in a quite time-consuming measurement compared to full-field apporaches, capturing the complete domain at once [14]. LDV is therefore rather unsuitable for measuring two-dimensional ultrasonic wave fields in quasi-real time [15]. Examples of measurement methods that work in real time or quasi-real time are holography [15], shearography [16, 17] and photoelastic coating techniques [18]. In [16] and [18], the non-stationary wave fields were generated by transient excitation, whereas in [17] wave fields were also generated by continuous excitation. A disadvantage of the photoelastic coating techniques is that a corresponding photoelastic coating must be present on the technical surface. In the case of of shearography, a disadvantage is that the optimum shearing vector selection requires a previous knowledge of the acoustic wave vector [15]. This means that when measuring two-dimensional, non-reproducible, non-recurring, transient ultrasonic wave fields propagating in different directions, holographic techniques are advantageous over shearography. In [15], TV holography was used for the single-shot measurement of Lamb waves; however, the method requires a complex setup and is bound to the use of an optical imaging system with lenses, that has to be adapted for every slight variation of field of view (FOV) or distance, making the setup inflexible and bulky.

In this publication, we present for the first time the use of lensless digital holography for the single-shot detection of freely propagating two-dimensional transient ultrasonic wave fields on technical surfaces. Using the ability of digital holography to work without complex experimental setup or the use of a conventional imaging system, such as a lens, this method provides high resolution images of the object surface. Because of this, the FOV and the resolution can be adjusted without a change in the optics but by just changing the distances in the setup. This way, the setup is very versatile and easy to adapt for different measurement situations, which is especially convenient for process monitoring. Moreover, the shape of the GUWs can be captured without any additional demands on the sample, the frequency range or the type of excitation (i.e. continuous or pulsed). The only crucial requirement of stroboscopic coherent illumination is met by employing a pulse laser with a pulse length of 6 ns. Furthermore, with the measurement of the full wave field of the GUW, the technique allows to quantitatively determine the frequency and amplitude of the GUW. For this, we propose the evaluation of the structure function (SF) [19] for the measured height distributions, presenting a significant benefit in the field of GUW detection. We compare our results with those obtained from a simulation.

2 Experimental setup and methods

2.1 Holographic measurements of GUW-induced surface deformation

By means of digital holography, the whole complex wave field, i.e. phase and amplitude, is measured by coding the phase into the interference pattern between an undistorted reference wave r ( x ⃗ ) $ r(\vec{x})$ and an object wave u ( x ⃗ ) $ u(\vec{x})$ scattered by the object. The intensity of the interference pattern is given by I ( x ⃗ ) = | u ( x ⃗ ) + r ( x ⃗ ) | 2 = | u ( x ⃗ ) | 2 + | r ( x ⃗ ) | 2 + u ( x ⃗ ) ⋅ r * ( x ⃗ ) + u * ( x ⃗ ) ⋅ r ( x ⃗ ) . $$ I(\vec{x})=|u(\vec{x})+r(\vec{x}){|}^2=|u(\vec{x}){|}^2+|r(\vec{x}){|}^2+u(\vec{x})\cdot {r}^{\mathrm{*}}(\vec{x})+{u}^{\mathrm{*}}(\vec{x})\cdot r(\vec{x}). $$(1)

Several methods exist to separate the coherence function Γ ( x ⃗ ) = u ( x ⃗ ) ⋅ r * ( x ⃗ ) $ \mathrm{\Gamma }(\vec{x})=u(\vec{x})\cdot {r}^{\mathrm{*}}(\vec{x})$ from the intensity, with * denoting the complex conjugate. In our work, we make use of the so called spatial carrier method, introduced by Takeda et al. [20], which is based on tilting the reference wave, thereby modulating the object wave with a spatial carrier frequency. With the reference wave r ( x ⃗ ) $ r(\vec{x})$ known, this facilitates the determination of the full complex amplitude of the object wave by means of u ( x ⃗ ) = Γ ( x ⃗ ) ⋅ r ( x ⃗ ) / | r ( x ⃗ ) | 2 $ u(\vec{x})=\mathrm{\Gamma }(\vec{x})\cdot r(\vec{x})/|r(\vec{x}){|}^2$.

The object wave is commonly recorded across a plane parallel to the object under inspection. Since the entire wave field is known, we employ propagation methods from scalar diffraction theory, such as the Fresnel propagation formula [21], to calculate the wave field across the object plane u O ( x ' ⃗ ) $ {u}_{\mathrm{O}}(\overrightarrow{x\prime})$ by u O ( x ' ⃗ ) = i λ   z exp ( - i 2   λ z ) exp [ - i   λ   z ( | x ' ⃗ | 2 ) ] × F - 1 { u ( x ⃗ ) exp [ - i π λ   z ( | x ⃗ | 2 ) ] } ( x ' ⃗ λ   z ) $$ \begin{array}{c} {u}_{\mathrm{O}}(\overrightarrow{x^{\prime}})=\frac{\mathrm{i}}{\lambda z}\mathrm{exp}\left(-\mathrm{i}\frac{2\pi}{\lambda}z\right)\mathrm{exp}\left[-\mathrm{i}\frac{\pi}{\lambda z}\left(|\overrightarrow{x^{\prime}}|^2\right)\right] \\ \times \mathcal{F}^{-1}\left\{u(\vec{x})\mathrm{exp}\left[-\mathrm{i}\frac{\pi}{\lambda z}\left(|\vec{x}|^2\right)\right]\right\}\left(\frac{\overrightarrow{x^{\prime}}}{\lambda z}\right) \end{array} $$(2)where F - 1 $ {\mathcal{F}}^{-1}$ denotes the inverse Fourier transformation, z is the distance between the camera plane and the object plane and λ is the wavelength of the light used for illuminating the sample.

The wave field in the object plane provides us with the image I ( x ' ⃗ = | u O ( x ' ⃗ ) | 2 $ I(\overrightarrow{x^{\prime}})=|{u}_{\mathrm{O}}(\overrightarrow{x^{\prime}})|^2$ of the object. Since the propagation in equation (2) is a purely numerical process, we do not need any additional imaging device, such as a lens objective. Moreover, the phase distribution ϕ ( x ' ⃗ ) = arg { u O ( x ' ⃗ ) } $ \phi (\overrightarrow{x^{\prime}})=\mathrm{arg}\{u_{\mathrm{O}}(\overrightarrow{x^{\prime}})\}$ depends on the optical path between the light source and the camera and therefore also on the exact shape of the surface, with precision down to the single digit nanometer range. Guided ultrasonic waves can now be visualized by recording two different holograms: one with active piezoelectric excitation and one without any excitation. The phase difference Δ ϕ ( x ' ⃗ ) $ \Delta \phi (\overrightarrow{x^{\prime}})$ between the two measured object wave fields then yields the deformation, which is mainly caused by the GUW on the surface. Thereby, the wavelength and the amplitude of the GUW can be determined from the height distribution.

In order to measure the fast-travelling GUW, an INNOLAS SpitLight 600 pulsed laser was used with a pulse width of 6 ns to avoid motion artefacts. Thus, the pulse rate of max. 10 Hz constituted the limiting factor for the sample frequency. The experimental setup is shown in Figure 1. The mirror positioned at the edge of the object provides the reference wave, while the illuminated part of the object constitutes the object wave. Both wave fields then interfere and propagate towards the CMOS sensor, where the hologram is measured. For the measurement of the GUW, the sample plate was excited continuously with a piezo actuator attached on the back of the sample with a sine voltage wave with a defined frequency.

thumbnail Fig. 1

A digital holography setup with pulsed illumination is used to capture GUWs traveling on the object surface, which are excited by a piezo actuator on the back of the sample. The illumination is adapted to the FOV by a lens and an aperture, the neutral density (ND) filter is used to control the brightness. Part of the light is then reflected by a mirror to provide the reference wave r ( x ⃗ ) $ r(\vec{x})$, where the mirror is slightly tilted to perform spatial phase shifting. The rest of the light is scattered at the object surface and constitutes the object wave u ( x ⃗ ) $ u(\vec{x})$ . The interference of r ( x ⃗ ) $ r(\vec{x})$ and u ( x ⃗ ) $ u(\vec{x})$ is then measured with a CMOS sensor and numerically propagated in order to obtain information about the out-of-plane displacement caused by the GUW.

Large-scale defects in compound materials can lead to a change in the wavelength of the GUW due to a change of the effective thickness [3]. To demonstrate that our proposed method measures the correct wavelengths and thereby detects changes in the wavelength, we use two aluminum plates: sample 1 measuring 32.0 cm × 9.4 cm × 2.6 cm and sample 2 measuring 30.0 cm × 11.7 cm × 5.0 cm. For the excitation of GUWs, frequencies of fExc = [60, 300] kHz in steps of 20 kHz were applied to the piezo actuator, with a peak-to-peak amplitude of 60 Vp–p. The piezo actuator was a DuraAct patch from PI Ceramic GmbH with a length of 17 mm and a width of 13 mm. The thickness of the piezo actuator was 200 μm and its diameter was 10 mm.

2.2 Characterization of the out-of-plane displacement fields of the GUW with the structure function

By measuring the object wave field with digital holography, the phase of the light reflected from the surface is measured. Encoded in this phase distribution is the height of the object. For the characterization of the measured phase distributions, respectively the height distribution derived therefrom, the SF [19] can be used. The SF provides a statistical analysis of correlations on the surface by calculating finite differences and averaging these for all possible surface points. In the one-dimensional case, it is defined as S ( s ) = 1 N - s ∑ i = 1 N - s ( z i - z i + s ) 2 $$ S(s)=\frac{1}{N-s}\sum_{i=1}^{N-s} {\left({z}_i-{z}_{i+s}\right)}^2 $$(3)with N being the size of the array. The formula describes the mean squared height difference for value pairs (zi, zi+s) as a function of the separation s, which represents the relative distance between the points under consideration. In general, the SF reproduces periodicities on the surface [22]. For a 2D data input, the SF can be evaluated multidirectionally by taking into account all point pairs separated by vectors s ⃗ i $ {\vec{s}}_i$ with a length of | s ⃗ i | = s $ |{\vec{s}}_i|=s$ thereby averaging over all possible directions. This enables a one-dimensional statistical representation SF(s) of the surface profile. In previous work it has been demonstrated, how the evaluation of the SF over a surface can be used to determine the wavelength of periodic profiles on a surface [22, 23].

As the GUWs on the surface share the same frequency and SF(s) is evaluated multidirectionally, i.e. independent of the propagation direction, the SF can be applied here for the characterization of the GUWs deforming the surface. By definition of a periodic function, the separation s = s0 corresponding to the wavelength of the periodic structure leads to minimal differences and therefore minimal values in the SF. Thereby, the wavelength of the GUW can be determined from the first local minimum of the SF when applied to the measured height map.

2.3 Simulation of the wavelength of the GUW

The measured plate thicknesses and the excitation frequency range used were input parameters for the simulation of the wavelength of the fundamental antisymmetric mode of the GUW using the Dispersion Calculator v3.0 software [24] to validate the experimental results. The material parameters implemented in the software for the aluminum alloy 6061 [25] were used for the simulation. From the simulation, only the wavelength λsim of the fundamental antisymmetric mode is used as a function of the frequency fExc. The reason for this is that the fundamental antisymmetric mode has the highest out-of-plane displacement in the investigated frequency range.

3 Results for aluminum plates with different thickness

After the subsequent recording of the two holograms, the holograms were spectrally filtered and numerically reconstructed using the Fresnel approximation, resulting in a pixel pitch of Δξ = (52 ± 4) μm in the reconstructed image. Then, the phase difference was calculated and translated into height values. In Figure 2 some exemplary height distributions can be seen, featuring visible periodic deformations on the surfaces of sample 1 and 2 at different excitation frequencies fExc.

thumbnail Fig. 2

Using lensless digital holography with pulsed illumination, the deformation on aluminum surfaces caused by GUWs is measured at different excitation frequencies being applied to the piezo actuator. The excitation frequencies applied here were fExc = 60 kHz and fExc = 140 kHz for aluminum plates of thickness a), c) d = 5.0 cm and b), d) d = 2.6 cm.

In Figure 3a, we see another example for a digital holographic measurement of the deformation on sample 1 caused by GUWs at fExc = 200 kHz. Due to the superposition of different waves, caused by the continuous excitation and reflections from the edges, there is no clear propagation direction of the GUWs. Nevertheless, a periodicity can be seen. To characterize the GUWs, the structure function is then applied to the whole surface. As seen in Figure 3b, the periodicity is reproduced with the SF. In this case, the measured wavelength λm is determined to λm = 10.7 ± 0.8 mm from the position of the first local minimum. For this statistical analysis, all surface points are taken into account.

thumbnail Fig. 3

a) Digital holographic image of GUW on the surface of an aluminum sample of 32.0 cm × 9.4 cm × 2.6 cm resulting from a piezo excitation with fExc = 200 kHz. b) By statistical analysis of the measured height distribution with the SF, the wavelength of the GUW is given by the position of the first local minimum at s0 = 10.7 ± 0.8 mm. Overall, the multidirectional SF makes a periodicity visible, despite the superposition of various waves on the surface.

This evaluation was carried out for both samples at all different fExc to determine λm over a broader spectrum. The resulting λm are shown in Figure 4a. Here, we can see that the change in the thickness of the material is represented in a shift to shorter wavelengths in case of the thinner sample 1. Therefore, the change in the wavelength can be characterized in this case. As expected, the results also feature an exponential decrease in λsim towards higher fExc. In general, there is also an agreement with the results from the simulation for the wavelength of the asymmetric 0-mode (A0 mode), as seen in Figure 4b. This successfully qualifies our measurement method for the measurement and characterization for GUWs. The results for the thinner sample show a systematic deviation from the simulated results, in that λm is shifted slightly upwards.

thumbnail Fig. 4

a) GUWs on aluminum plates with different thickness d1 = 2.6 mm (red) and d2 = 5.0 mm (black) are excited at different frequencies fExc = [60, 300] kHz in steps of 20 kHz. By means of digital holography, the surface deformation is then measured and evaluated with the structure function to determine the wavelength λm of the GUW, which is plotted against fExc ·⋅d. The thinner plate shows shorter wavelengths. b) The measured wavelengths are plotted against the resulting wavelengths from the simulation for the A0 mode λsim. The blue line marks the ideal result for the case of λm = λsim, corresponding to a slope of m = 1. The experimental results are in agreement with the simulation results for both samples. The corresponding slopes are m1 = 0.95 and m2 = 1.08. This successfully qualifies our measurement method for the measurement and characterization for GUWs.

4 Conclusion

In this work, we have presented a full-field measurement technique for the single-shot observation of guided ultrasonic waves (GUWs). Based on a lensless digital holography setup, we can provide real-time deformation measurements featuring the height distribution caused by the GUW. For the characterization of the surface data, we used the multidirectional statistical evaluation provided by the structure function. This offers the determination of the wavelength of the GUW, where the resolution is only limited by the pixel size, and enables the comparison between two aluminum plates of different thickness based on the application of GUWs. The resulting wavelengths allow for a distinction between the samples and are consistent with simulated results, therefore qualifying the measurement method for the measurement and characterization of GUWs.

Funding

Large parts of this work were funded by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) under the project “Strukturfunktion 2” (project number 288038549). The authors also acknowledge the financial support of the research work on this article within the Research Unit 3022 “Ultrasonic Monitoring of Fibre Metal Laminates Using Integrated Sensors” (Project number: 418311604) by the DFG.

Conflicts of interest

There are no conflicts of interest.

Data availability statement

The underlying data is not publicly available at the time but can be provided by the authors upon reasonable request.

Author contribution statement

Conceptualization, O. Focke and C. Falldorf; Specimen preparation, C. Polle and M. Koerdt; Experimental measurements, B. Gutierrez and C. Falldorf; Evaluation and processing of experimental data, B. Gutierrez; Assessment of results, B. Maack and C. Polle; Simulation, C. Polle and M. Koerdt; Visualization, B. Gutierrez; Writing – Original Draft Preparation, B. Gutierrez, C. Falldorf, M. Koerdt and B. Maack; Supervision, R. B. Bergmann and D. May; All authors discussed the results and contributed to the final manuscript.

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All Figures

thumbnail Fig. 1

A digital holography setup with pulsed illumination is used to capture GUWs traveling on the object surface, which are excited by a piezo actuator on the back of the sample. The illumination is adapted to the FOV by a lens and an aperture, the neutral density (ND) filter is used to control the brightness. Part of the light is then reflected by a mirror to provide the reference wave r ( x ⃗ ) $ r(\vec{x})$, where the mirror is slightly tilted to perform spatial phase shifting. The rest of the light is scattered at the object surface and constitutes the object wave u ( x ⃗ ) $ u(\vec{x})$ . The interference of r ( x ⃗ ) $ r(\vec{x})$ and u ( x ⃗ ) $ u(\vec{x})$ is then measured with a CMOS sensor and numerically propagated in order to obtain information about the out-of-plane displacement caused by the GUW.
In the text
thumbnail Fig. 2

Using lensless digital holography with pulsed illumination, the deformation on aluminum surfaces caused by GUWs is measured at different excitation frequencies being applied to the piezo actuator. The excitation frequencies applied here were fExc = 60 kHz and fExc = 140 kHz for aluminum plates of thickness a), c) d = 5.0 cm and b), d) d = 2.6 cm.
In the text
thumbnail Fig. 3

a) Digital holographic image of GUW on the surface of an aluminum sample of 32.0 cm × 9.4 cm × 2.6 cm resulting from a piezo excitation with fExc = 200 kHz. b) By statistical analysis of the measured height distribution with the SF, the wavelength of the GUW is given by the position of the first local minimum at s0 = 10.7 ± 0.8 mm. Overall, the multidirectional SF makes a periodicity visible, despite the superposition of various waves on the surface.
In the text
thumbnail Fig. 4

a) GUWs on aluminum plates with different thickness d1 = 2.6 mm (red) and d2 = 5.0 mm (black) are excited at different frequencies fExc = [60, 300] kHz in steps of 20 kHz. By means of digital holography, the surface deformation is then measured and evaluated with the structure function to determine the wavelength λm of the GUW, which is plotted against fExc ·⋅d. The thinner plate shows shorter wavelengths. b) The measured wavelengths are plotted against the resulting wavelengths from the simulation for the A0 mode λsim. The blue line marks the ideal result for the case of λm = λsim, corresponding to a slope of m = 1. The experimental results are in agreement with the simulation results for both samples. The corresponding slopes are m1 = 0.95 and m2 = 1.08. This successfully qualifies our measurement method for the measurement and characterization for GUWs.
In the text

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