1. Introduction

The analysis of the propagation of elastic waves in solids is the fundamental principle behind all major schemes for non-destructive evaluation and structural health management systems [1–5]. One major approach to introduce ultrasonic elastic waves into materials and structures is the use of piezoceramic transducers which enable the excitation of small amplitude transient disturbances. The propagation characteristics of these waves, in particular propagation velocity and interaction with structural features such as cracks and delaminations, are used to determine material and structural characteristics such as elastic moduli and thicknesses and to locate and characterize (quantify) damages. Various types of sensors are used to measure both surface perturbations and propagating waves including piezoelectric transducers [6,7] (also with thermal excited was via laser pulses [8]), fibre optic sensors [9], polarization imaging [10], interferometric imaging [11,12], holographic imaging [13] and laser Doppler vibrometer (LDV) systems [14,15].

The key advantage of the laser measurement systems is in their non-contact nature: they do not act as artificial inhomogeneities affecting the properties of the structure under test. Simply, the sensor does not interfere in any way with the propagating elastic wave. While the currently available LDV systems (such as, for example, Polytec scanning vibrometers) offer high precision and sensitivity, they are invariably large and costly measurement systems still primarily used in laboratory environments, and typically only measure velocities or displacements at points on the surface of the structure and not inside the solid. The cost of such scanners can range from tens of thousands to hundreds of thousands of (US) dollars depending on the configuration, sensitivity (nm to pm), number of sensing points and built in scanning configurations. The aim of this article is to investigate the application of laser feedback interferometry (LFI) as a cost effective, non-contact, technique to measure and visualize the propagation of elastic waves inside solids.

LFI uses the laser, and the laser cavity itself, as both the reference and measurement path to create a highly sensitive interferometer [16–20]. The LFI systems have been implemented using a wide range of lasers and wavelengths, selecting appropriately spectral windows in which materials under test are transparent [21–27]. Previously this technique has been shown to visualize acoustic and pressure waves waves in air [28–31] and in turbid media [32], by measuring the effect the acoustic wave has on compressing the medium it travels and thus altering the index of refraction along the path of the laser beam. The LFI technique has also been previously used to measure static changes in refractive index of transparent solid materials [33], dynamic index changes in the wave-guides of electro-optic modulators [34], as well as small changes in relative refractive index between different materials [35–39]. Most importantly this technique has been demonstrated effectively with cheap (sub ten dollars a piece in bulk) widespread telecommunications lasers, [21] and with simple customized electronics could easily create a sub-hundred-dollar sensing system.

Thus, the fundamental question investigated in this study is whether the LFI can be used to measure and visualize very small, transient changes in refractive index caused by propagating elastic disturbances in a optically transparent solids. In this article, we demonstrate that an ultra-compact and simple implementation of an LFI is a powerful tool for accurate visualization of the elastic waves in solids, thus opening the pathway to a range of new opportunities in ultrasonic non-destructive testing and evaluation.

2. Principles of operation

The refractive index of a solid supporting the elastic wave is influenced by mechanical strain caused by the wave. In this work we use LFI in a double transmission arrangement. The laser beam is transmitted through a region of the material under test, of thickness $d$, in which the stress (and therefore the refractive index) change due to the presence of the elastic wave. The laser beam then exits the solid and is reflected from an external wall located behind the structure under test (typically covered by the retroreflective material), back through the material and into the laser sensor (See Fig. 1). The dynamic change in the optical path length brought about by the elastic wave can be interpreted as an equivalent mechanical displacement of the backing retro reflective surface [40–42]. It should be noted here that this equivalent displacement, caused by the change in refractive index of the solid, is much larger than the real mechanical change of the material thickness $\Delta d$ caused through the Poisson’s effect. Interpreting the change in refractive index as an equivalent displacement reduces the problem to that of displacement measurements, a problem commonly dealt with in the LFI literature [43–46].

 

Fig. 1. Schematic diagram of the LFI setup used for measurements of the acoustic field in the acrylic beam.(LD - Laser Diode, PD - Photodiode, TIA - Transimpedance Amplifier).

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The conventional application of the LDV systems for wave visualization is through measurement of nanometer-scale displacement of the surface. This calls for the use of small retro reflective targets affixed to the front (or the back) surfaces of the object under test. The LDV system then detects the small (nanometer) scale displacement of the surface of the structure caused by Poisson’s effect as the elastic wave propagates through the structure. LDV systems have also been used (in double transmission arrangement) to observe the change in index due to stress [47,48]. The direct replication of the LDV small surface displacement measurement (on the order of $\approx \lambda /100$) is possible using LFI, but requires complicated optics [49,50], plurality of optical paths or reference beams [51–53], or narrow line-width lasers [54,55].

The method we propose here represents a way of measuring dynamic changes in refractive index and visualizing elastic waves, while retaining the simplicity of the conventional LFI approach in a double transmission arrangement. The primary contributor to this index change is the linear change in refractive index with mechanical stress [56,57] and the change in the sample thickness is a second order effect.

A laser operating under optical feedback emits a power $P_{\rm {F}}$ that is calculated from the power emitted by the stand-alone laser $P_0$ as [28]:

The round-trip time in the external cavity is the sum of three terms: $\tau = \tau _0 + \tau _s + \delta \tau$, where $\tau _0 + \tau _s$ is the (constant) round trip time in the external cavity including that outside and inside the solid sample, and $\delta \tau$ is a time dependent part resulting from the change in the refractive index of $\delta n$ over the sample thickness $d$ [28]:

The system will observe a line integral of the acoustic pressure field along the axis of the laser beam varying with time $t$, resulting in an interferometric waveform at each spatial pixel [28,30]. By varying the spatial position of the laser perpendicular to the beam axis (in an $x$–$y$ plane as indicated in Fig. 1), a 2D array of interferometric signals captured.

3. Experimental setup

Figure 1 shows the LFI experimental setup. Guided elastic longitudinal waves propagating in an 10 $\times$ 10 $\times ~380~{\rm mm}^{3}$ acrylic beam were selected to demonstrate the potential to apply LFI to the quantitative measurement of small amplitude elastic waves in a solid. A 10 mm diameter, 2 mm thick piezoelectric transducer (Ferroperm Pz27) was adhesively bonded to one end of the acrylic beam in order to generate the waves. A 10 mm diameter, 10 mm thick brass backing mass was attached to the piezoceramic disc to increase the amplitude of the excitation pulse. The transducer was driven from a signal generator (33210A, Agilent Technologies Inc.) amplified using a high power amplifier (Model 7500, Krohn-Hite Corporation) resulting in an $\approx {80}$ V pulse waveform at a repetition rate of 1 kHz.

This repetition rate was selected to allow waves traveling in the rod to fully dissipate before the next pulse was sent. The pulse waveform consisted of 8 sinusoidal periods at 100 kHz fundamental frequency covered by a Hanning window. Based on fundamental theory of the propagation of longitudinal elastic waves in a solid beam the pulse with 100 kHz centre frequency the wavelength is estimated to be approximately 16 mm and propagates at an theoretical calculated velocity of approximately 1640 m/s. Because the wavelength is comparable to the cross-section dimension of the beam and the acrylic material is not purely elastic, it is expected that the 8 cycle pulse is substantially distorted during the propagation along the beam. Its amplitude will decrease due to frequency dependent viscous damping and the high frequency components travel slower than the low frequency components based on the dispersion curve of the fundamental longitudinal wave mode in a beam [1–4].

The laser diode (LD) used was an 850 nm distributed feedback laser (EYP-DFB-082-00150-1500-SOT02-000, Eagleyard DFB) which was collimated with an aspheric lens ($f = 8mm, NA = 0.5$ C-240, Thorlabs Inc.). A laser driver (LD205, Thorlabs Inc.) was used to operate the laser in continuous-wave mode at $\approx {100}$ mA (resulting in an output power of $\approx {40}$ mW). The interferometric LFI signal was monitored using the back facet photo-diode (PD), mounted inside the laser package, with a custom built trans-impedance amplifier (TIA) (10 kV$/$A) with additional voltage gain ($\times$ 300). The laser and lens assembly were mounted on an $x$–$y$ motorised stage (LSM050A and LMS450A, Zaber Technologies Inc.), with the beam propagating perpendicular (along $z$) to the plane of travel of the stage. The fixed reflective backing wall was an upright aluminum optical breadboard with a retro-reflector surface (Oralite Reflective Film 5700, ORAFOL Europe GmbH) and was placed at a distance 100 mm from the laser.

The acrylic beam was placed roughly midway between the laser ($z=$0 mm) and the retroreflective screen ($z=$100 mm) at a slight angle to minimize reflections back from the air-acrylic and acrylic-air interfaces. This effectively removes the displacement component of the LFI signal.

The signal from the PD is acquired over a 73 $\times$ 1601 pixel scan area in a 18 $\times$ 400 mm$^2$ area (0.25 mm pixel step size , $x=$-12 to 388 mm $y=$-9 to +9 mm). The piezoelectric excitation is located at $x=0$ mm and centred at $y=0$ mm. Each of these signals was acquired with a sample rate of 10 MS/s using a high speed 10-bit Oscilloscope (DSOS404A, Keysight Technologies) over 500 µs.

For comparison, the longitudinal wave pulses were also captured using a single point, commercial LDV system (Polytec OFV 303), at several points along the beam (where small 1 $\times$ 1 mm$^2$ squares of retro-reflective tape were placed on the beam). The LDV system measures the nanometer-scale transverse displacement on the surface of the beam caused by Poisson’s effect as the longitudinal wave travels along the beam.

4. Results & discussion

Figure 2(a) shows the spatial distribution of strain in the acrylic beam created by the elastic wave at $t=200$ µs (at this moment the pulse is approximately midway along the rod). Visualization 1 shows the time evolution of the pulse between $0 - 500$ µs. It shows the forward traveling pulse, its attenuation and broadening as it propagates along the rod, as well as the superposition of the incident and the reflected pulses at the distal end of the acrylic beam. From these it is possible to see that the wavelength is almost exactly the calculated value (16 mm) and the phase velocity is 1669 m/s (also very close to the value calculated above). Figures 2(b) and 2(c) show the amplitude and phase information for each waveforms at each scanned pixel point. These amplitude and phase values were calculated from the information in a 1D the fast Fourier transform (FFT) of each of the time domain signals at each pixel point with the values from the bin at 100 kHz.

 

Fig. 2. Results of the imaging of the acoustic wave in the acrylic beam. a) Time Snapshot at $t=200$ µs (see also Visualization 1), b) FFT Amplitude, c) FFT Phase .

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To elucidate fine details of the signal both the time-domain snapshot and Visualization 1 — the recorded time domain signal — were first squared and then displayed on a logarithmic scale (dB). This allowed extended dynamic range of the signal to be visualized showing additional interesting features of the signal. This also removes the issues related to the shifting quiescent point of the system — an interesting phenomenon explained in detail in the Appendix. The Appendix discusses how the location of the quiescent point of the interferometer on the (highly nonlinear) LFI transfer function determines the signal magnitude, and more critically its phase, and the effective way for mitigating its effects.

Visualization 1 shows the propagation of the pulse through the acrylic beam. One will notice that the beam is quite dispersive as the spacing between the peaks increases as the pulse travels along the rod. At the distal end of the rod one also observes the reflection of the pulse and the interference effects of the incident and the reflected waves.

Figure 3 compares the results of the LFI measurements against the Polytec LDV sensor signal. We would like to reiterate here that the LDV sensor is measuring the small transverse displacement of the acrylic beam’s side surface (the surface of the beam facing the LDV sensor). This lateral movement is caused by the propagating elastic longitudinal wave because of the Poisson’s effect (change in the beam thickness $d$), while the LFI implementation is measuring predominantly the change in refractive index of the material due to the strain caused by the elastic wave. This assumption was verified by observing no appreciable LFI signal from the retro-reflector spot affixed on the side surface of the acrylic beam used for the LDV measurements. Therefore, while the LDV is measuring the secondary effect of the wave propagation (lateral deformation of the beam), the LFI signal is proportional to the distribution of strain within the material caused by the wave.

 

Fig. 3. Comparison of signals acquired via LFI (blue lines) vs those acquired by LDV measurement (orange lines), at different $x$ positions along the acrylic beam.

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To explore the effects of the movement of the reflective wall behind the acrylic rod, we measured the LFI signal caused by its controlled displacement and confirmed that the LFI signals created by the acoustic waves are easily separable form the spurious signal caused by the drifting reflective back surface. The controlled displacement also revealed that the amplified LFI signal for a full interferometric fringe was 4 V (equivalent to $\lambda /(2 n_\textrm {ext})$ displacement), while the measured effects from the acoustic wave (see Fig. 3) were at least 1–3 orders of magnitude below this (maximum peak–peak values ranging from 0.3 – 0.005 V along the acrylic beam).

Figure 3 compares the LFI signal (blue line) and the LDV signal (orange line) — the match between the two is almost perfect. The LFI signals were acquired at 25 spatial pixels from the small 5x5 pixel ($1~\rm {mm}^{2}$) area along the centreline of the acrylic beam at each of the $x$ positions being compared, and the signal with the best quiescent point location corresponding to the minimum phase distortion was selected. The fact that the time domain signals from the LFI and the LDV systems are practically identical suggest that the LFI scheme proposed here is a viable alternative to the well established LDV technique for inspection of transparent materials. It may be worth noting that being "transparent" does not necessarily mean transparent in the visible spectral range. Optical absorption of materials depends strongly on the wavelength; as most materials have low-absorption windows spread throughput the electromagnetic spectrum these high transparency spectral regions can by utilized by proper selection of the laser. For example several plastic materials commonly used in manufacturing are transparent at terahertz frequencies [58]. This may allow for use of terahertz quantum cascade lasers in the LFI configuration [21] for visualizing elastic waves in materials which are not transparent in the visible or infra-red parts of the electromagnetic spectrum. In the visible and near-infrared low-cost in-plane [59] and vertical-cavity surface-emitting lasers [60] represent a viable solution. The use of monolithic vertical-cavity surface-emitting laser arrays [61,62] will allow for parallel acquisition form the plurality of points, and ultimately for real time imaging of non-periodic transients.

5. Conclusion

The use of ultrasonic elastic waves is a well established technique for non-destructive testing of materials and structures, in particular to explore the interaction of waves with structural features to detect and characterize defects such as cracks and delaminations. The advantage of the laser measurement systems for visualizing the distribution of elastic waves is in their non-contact nature and they do not interfere in any way with the interrogating elastic wave. In this article we introduced laser feedback interferometry (LFI) as a cost effective, alternative to a well established LDV technique and demonstrated its effectiveness on an example of a wave propagating through a prismatic acrylic rod. We have demonstrate that an ultra-compact and simple implementation of an LFI offers significant potential for accurate visualization of the elastic waves in solids, thus opening the pathway to a range of new opportunities in ultrasonic non-destructive testing and evaluation including the use of laser array based sensors for parallel acquisition and real time imaging of acoustical fields [66,67].

Appendix

Effects and limitations of sub-fringe measurements in LFI systems

Fig. 4 shows one period of the LFI signal obtained by displacing the target by $\lambda /(2 n_\textrm {ext})$ away from the laser, or for the equivalent change in the refractive index of the external cavity ($n_\textrm {ext}$) [63]. For large displacements, one obtains the periodic LFI signal with each fringe corresponding to the displacement of $\lambda /(2 n_\textrm {ext})$. The shape of the waveform is dependent on the feedback parameter $C$, and the line-width enhancement factor $\alpha$; the waveform in Fig. 4 was calculated for a laser in a feedback regime of $C=0.7$ ($\alpha =5$). For displacements much smaller than $\lambda /(2 n_\textrm {ext})$ (external cavity phase change much smaller than $2\pi$) the displacement can still be measured, but the signal morphology is entirely different: the LFI signal becomes proportional to the stimulus [64].

In order to obtain good linearity of the stimulus–response transfer function, one needs to ensure that the small displacements occur with the quiescent point at a quadrature point of the transfer function (points $B$ or $R$ in Fig. 4). quiescent point $R$ (on the positive slope) provides minimum signal distortion, where the quiescent point $B$ provides increased sensitivity, and a larger LFI signal for the same stimulus. Obviously, operating in the vicinity of the point $G$ yields virtually no LFI signal. Due to refractive index fluctuations in the external cavity, the quiescent point in the course of this experiment tends to randomly move between all three locations ($B$, $R$, and $G$), resulting in the noticeable and sudden "phase inversion" of the LFI signal and the accompanying change in amplitude as the quiescent point switches between from $R$ to $B$, with the brief loss of signal as the quiescent point transitions through $G$. This, initially puzzling, phase inversion and signal loss behavior, once understood can easily be dealt with and does not impede the operation of the measurement system in any way.

 

Fig. 4. effects on fringe starting position on small ($\ll \lambda /2$) equivalent displacements. Red - ideal case in center of the positive slope of the fringe. Blue - effect on a steep negative slope of the fringe. Green - effect when near one of the turning points of the fringe.

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Taking the square and then the logarithm of the LFI signal allowed for the visualization of the signal with the extended dynamic range (See Fig. 5), and markedly reduces the effect of the unstable quiescent point.

 

Fig. 5. Actual measured examples of effects on fringe starting position so small ($\ll \lambda /2$) equivalent displacements. Red - ideal case in center of the positive slope of the fringe. Blue - effect on a steep negative slope of the fringe. Green - effect when near one of the turning points of the fringe.

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Alternatively, a fringe stabilization system could be implemented if the stable quiescent point was required, as explained in [65].

The exemplar traces corresponding to LFI signals experimentally obtained under conditions which correspond to quiescent points $B$, $R$, and $G$, are shown in Fig. 5. These traces were acquired over a small 5$\times$5 pixel area ($1~\rm {mm}^{2}$) at a distance 10 mm away from the ultrasonic transducer.

The signal phase inversion and the signal loss phenomena are much more pronounced in weak ($C<1$) feedback regimes, and become almost imperceptible as the feedback increases ($C>{1}$). This can be illustrated using the Fig. 6 which depicts the LFI phase transfer functions for four different feedback conditions corresponding to feedback coefficients $C$ between 0.1 and 3.0. As the feedback increases from $C=0.1$ [Fig. 6(a)] to $C=3$ [Fig. 6(d)] the red region — the location of the quiescent point $R$ — becomes wider and the likelihood of the system operating in the regime with the positive linear transfer function (region $R$) increases. Blue region all but disappears. The remaining existence of the green region suggests that signal will occasionally be lost, but not phase inverted. This also explains why we did not observe this effect in our previous study [28].

 

Fig. 6. Effects of different feedback levels (C) on expected LFI signals for small ($\ll \lambda /2$) equivalent displacements. Red - the positive slope of the fringe. Blue - the negative slope of the fringe. Green - turning point of the fringe.

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In [28] the laser beam propagated only through air, and the retroreflector performs almost perfect coupling back into the laser cavity. In the current study, the acrylic beam located between the laser and the retro-reflector adds four air/acrylic interfaces (forward and return path) to the optical path with Fresnel losses associated with each interface. The acrylic beam itself does not have optically flat sides which steers the beam and decreases further the coupling efficiency with the corresponding decrease in $C$. Consequently the system operates closer to the weak feedback regime ($C<1$), leading to the frequent signal inversion and the occasional signal loss.

Funding

Australian Research Council (DP210103342).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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