Defect Detection in Carbon Fiber-Reinforced Plate by Imaging of Mechanical Nonlinearity-Induced Sideband Vibrations

Abstract

Laser Doppler scanning vibrometry is used for imaging spectral vibration components in a carbon fiber-reinforced composite plate that contains a sub-surface delamination defect caused by hammer impact. The images reveal sideband generation at the location of the defect, reflecting mechanical nonlinearity-induced mixing between a high amplitude, low-frequency vibration that modulates the stress–strain behavior near the defect and a low amplitude, high-frequency probe vibration. In this work, a multifrequency probe is used to tackle the problem that the mixing coefficients are, in practice, frequency dependent. Based on the measured sideband amplitudes, a study is presented on the expected feasibility of detecting defects by a full field imaging scheme based on a photorefractive interferometer that is configured as a vibrometer acting as a bandpass filter around a sideband frequency of interest.

1. Introduction

Composite materials occupy a significant share of structural elements in a number of critical applications: transportation, wind turbines, pressure vessels, and civil engineering structures, to name a few. To guarantee the safety and reliability of the composite structures, non-destructive testing (NDT) techniques, able to detect and size possible defects, are constantly being adapted to the needs of the manufacturers and operators. The major points of interest for the operators are high sensitivity to defects so that they can be detected in an early stage, and the possibility to inspect large surfaces in a rapid way, in order to reduce the non-operational time of the item under testing. Downtime reduction is particularly critical for commercial sectors such as commercial airlines because of the economic impact that it involves. Over the past decades, a number of techniques have been developed to satisfy the needs of the different industries. Several ultrasound-based techniques based on linear [1,2,3] and non-linear [4,5,6,7,8,9,10,11] effects, infrared imaging [12,13,14,15,16,17,18], interferometric techniques, both full field (speckle interferometry [19], shearography [20,21]), and scanning [8,22,23,24,25].

In this work, the defect type of interest is barely visible impact damage (BVID) in a carbon fiber-reinforced polymer (CFRP) plate. This type of damage occurs after one, or a series, of heavy impacts that leave little or no trace on the target surface but cause delaminations in the inner layers. When it comes to aircrafts, typical examples are tool drops, vehicle impact (airport facilities), bird strikes, and impact with stones during take-off or landing. After an impact, an unscathed surface does not guarantee that the component is perfectly intact: ply debonding, core damage (in the case of sandwich components), and fiber breakage may have happened in depth [26,27].

In spite of the great advantages of classical ultrasonics in terms of sensitivity to mechanical disturbances caused by defects and the flexibility of letting guided waves probe large surfaces [28], due to the often closed state of delamination defects that make them appear like intact material, the risk of such defects remaining undetected by classical ultrasonic techniques is rather high.

Non-linear acoustic (NLA) methods in which the induced vibrations open the delamination and thus reveal their presence have been successful at localizing those defects in a wide variety of samples (plexiglass, glass fiber, concrete, metals [29], and CFRP) [4,30]. Two approaches are possible: harmonic generation and cross-modulation. In harmonic generation, the defect non-linearity acts as a source of harmonics of the pump vibration. The sample is set into vibration with amplitudes spanning from several hundred nanometers to a few micrometers, depending on the sample type and the defect characteristics. The pump wave should induce sufficient strain variations on the defect to bring it into the non-linear section of its stress–strain curve. Figuratively speaking, the pump wave should modulate the defect between a “closed” (linear) state and an “open” (non-linear) state. As a detector, a piezoelectric transducer or an optical vibrometer can be used, the tradeoff between these two choices being that the PT has a larger sensitivity, while the laser vibrometer can be scanned over the surface of the sample and returns a spatial profile of the spectral density of the vibrations in the sample, thus allowing for defect localization. The second approach, cross modulation, requires a probing wave in addition to the pump wave. Usually, the probing wave has a higher frequency and a lower amplitude than the pump wave. As a consequence of the defect non-linearity, sidebands are generated at frequencies that are linear combinations of those of the pump and probing wave. The same schemes for detection for the harmonic generation case can be used here, with the same tradeoff.

Contrary to the successful application of scanning NLA methods, to the best of our knowledge, no camera-based full-field approaches exploiting cross-modulation for defect detection have been reported. Interestingly, in the first half of the 1990s, a full optical lock-in (or bandpass) camera based on a photorefractive interferometer was demonstrated [31,32,33,34,35,36] to be capable of real-time imaging of vibrations happening at a known frequency. The possibility of configuring a full-field photorefractive interferometer as an optical bandpass filter (PRI-BP) was also demonstrated for a scenario in which the sample was excited with multiple frequencies [37].

In Section 2 and Section 3, we first report on the characterization of a CFRP affected by BVID using scanning laser Doppler vibrometry (LDV), by exploiting the nonlinear mechanical behavior of the defect, which induces modulation of a probing vibration by a delamination opening pump vibration, thus causing sidebands near the probing vibration frequency. Given that the efficiency of sideband excitation has been observed to be highly frequency dependent [8], here a frequency comb excitation is used as a probing signal mapping a region of interest along the sample surface. A final map, resulting from the combination of the displacement maps at all sidebands, allows for locating the defect with precision comparable with infrared imaging.

The obtained sideband amplitude maps are then used for quantitative estimation of the magnitude of the defect-induced sideband vibration, which in turn is employed for determining the sensitivity requirements for full-field PRI imaging of those sidebands.

In Section 4, the possibilities and limitations of Michelson-based full-field interferometry, applied for sideband imaging, are shown, by means of simulations and by means of experimental characterization of a PRI-BP device in point detection and full-field mode. In point detection, light is collected by a photodetector and processed by a lock-in amplifier. The minimum detectable displacement, the bandpass filtering response, and the extent of unwanted masking effects caused by the nonlinear optical response of the used device are quantified experimentally. Lastly, a simulation-based evaluation of the feasibility of using point detection-based and full-field PRI-BP for the detection of defects exploiting sideband generation is presented.

2. Materials and Methods

When a composite material is delaminated, its stress–strain function becomes non-linear [4,38,39] at the delamination location. Generally speaking, this leads to phenomena such as higher harmonic generation and sideband generation [40,41]. Vibrational signatures of the nonlinearity provide an ideal indication for the defect location [8,9,20]. There are different types of responses that can give rise to frequency mixing, the most prominent being a quadratic response [42] (Figure 1a), hysteresis [43,44,45] (Figure 1b), and a polygonal chain or mechanical diode, due to the so-called clapping phenomena [46] (Figure 1c).

In the investigated case of a hammer-impacted CFRP, we aim at exploiting contact acoustic non-linearity, going along with clapping between interfaces at the two sides of impact-induced delamination, as a signal feature characteristic for the defect. Two plates of equal size and layup were manufactured. The plates had a dimension of 30 × 30 cm² and consisted of 8 plies stacked in a 0–90° configuration, meaning the orientation of the fibers in each ply was orthogonal with respect to the adjacent ones. The first four plies were stacked symmetrically to the last four. This was completed to balance the internal stresses, and thus avoid the occurrence of wrinkles and warping. The total thickness of the plates was 1.7 mm. The internal delamination was created by hitting the plate with a rubber hammer.

In general, a non-linear acoustic response can be revealed by injecting two mechanical sinusoidal waves in the sample, one “probe” wave at a high frequency, f_P, and small amplitude, which probes the response, and one “modulating” wave at a low-frequency, f_M, and high amplitude, which modulates that response at the nonlinearly responding defect. The injected waves typically result in standing wave patterns along the plate surface. By mapping the spectrum of the resulting total vibration along the plate, spatially resolved information is obtained. A graphical scheme of such an experiment is presented in Figure 2. In locations where nonlinearity is present, frequency mixing occurs as a result of

$$\epsilon (t)=C(\sigma (t))\sigma (t)=\left(C_0+\frac{\partial C}{\partial \sigma }\sigma (t)+\dots \right)\sigma (t)$$

(1)

with ε(t) the local strain resulting from the local stress σ(t), which also induces changes in the compliance C(t). In case of a sinusoidal pump and probe wave, this can be rewritten as

$$\begin{array}{cc}\hfill \epsilon (t)& =\left(C_0+\frac{\partial C}{\partial \sigma }\left({\sigma }_{P,0}\sin ({\omega }_{P}t)+{\sigma }_{M,0}\sin ({\omega }_{M}t)\right)+\frac{{\partial }^{2}C}{\partial {\sigma }^2}{\left({\sigma }_{P,0}\sin ({\omega }_{P}t)+\left({\sigma }_{M,0}\sin ({\omega }_{M}t)\right)\right)}^2\dots \right)\hfill \\ & \times \left({\sigma }_{P,0}\sin ({\omega }_{P}t)+{\sigma }_{M,0}\sin ({\omega }_{M}t)\right)\hfill \\ & =C_0{\sigma }_{M,0}\sin ({\omega }_{M}t)+\frac{\partial C}{\partial \sigma }{\sigma }_{M,0}{}^2\sin (2{\omega }_{M}t)+\frac{{\partial }^{2}C}{\partial {\sigma }^2}{\sigma }_{M,0}{}^3\sin (3{\omega }_{M}t)+\dots \hfill \\ & +C_0{\sigma }_{P,0}\sin ({\omega }_{P}t)+\frac{\partial C}{\partial \sigma }{\sigma }_{M,0}{\sigma }_{P,0}\sin (({\omega }_P+{\omega }_M)t)+\frac{\partial C}{\partial \sigma }{\sigma }_{M,0}{\sigma }_{P,0}\sin (({\omega }_P-{\omega }_M)t)+\dots \hfill \\ & +\frac{{\partial }^{2}C}{\partial {\sigma }^2}{\sigma }_{M,0}{}^2{\sigma }_{P,0}\sin (({\omega }_P+2{\omega }_M)t)+\frac{{\partial }^{2}C}{\partial {\sigma }^2}{\sigma }_{M,0}{}^2{\sigma }_{P,0}\sin (({\omega }_P-2{\omega }_M)t)+\dots \hfill \end{array}$$

(2)

In addition to harmonics of the modulating vibration, sidebands appear in the measured probe vibration spectrum around the probing wave’s frequency at f_P ± mf_M, with m an integer number (|m| ≥ 2 components are caused by self-modulation and harmonic generation by the pump vibration). Using a scanning vibrometer, it is possible to measure the spatial dependence of the vibration spectrum along the plate. When scanning over the sound region of the sample, the spectrum presents terms only relative to the modulating vibration (f_M) and to the probe (f_P) vibration. When scanning over the BVID region, the spectrum also contains the sideband frequency terms.

In industrial applications, choosing a modal frequency of the sample is not a major complication. Normally, the vibration modes of an engineering item are known for simulation and design purposes. If they are unknown, they can be estimated with a number of existing approaches (simulation, shearography, scanning LDV, using accelerometers). Choosing the proper frequency for the probing term poses a challenge instead. In NDT, the defect characteristics are considered unknown, so it is not possible to estimate the local defect resonance (LDR). For this reason, instead of using a pure tone, a frequency comb was chosen. Using comb excitation, it is possible to combine the spectral density maps of multiple sidebands and obtain a compound image that may offer more insight on the defect silhouette, compared to the spectral density map originating from a single sideband. Sweeping the frequency of the probing wave over a frequency range [47] would return a similar insight but at the cost of a longer acquisition time per point and a more complex data analysis.

In view of the latter, here, we used a combination of multiple probing waves, with harmonic frequencies pf_P and |p| = 1..12, resulting in a total vibrational content containing frequencies pf_P ± mf_M and thus sidebands around every peak in the comb spectrum of the probe:

$$\begin{array}{cc}\hfill \epsilon (t)& =C_0{\sigma }_{M,0}\sin ({\omega }_{M}t)+C_0{\sigma }_{p,0}\sin ({\omega }_{p}t)+{\displaystyle \sum _{m=2}^{+\infty }\frac{{\partial }^{m-1}C}{\partial {\sigma }^{m-1}}{\sigma }_{M,0}{}^m\sin (m{\omega }_{M}t)}\hfill \\ & +{\displaystyle \sum _{m=1}^{+\infty }{\displaystyle \sum _{p=1}^{+\infty }\frac{{\partial }^{m}C}{\partial {\sigma }^m}{\sigma }_{M,0}{}^m{\sigma }_{P,0}\sin ((p{\omega }_P+m{\omega }_M)t)}}\hfill \end{array}$$

(3)

Figure 3 shows a simulated example of such a scan in a plate with a nonlinearly acting region in the middle. Contrary to the intact parts of the plate (e.g., power spectral density (PSD) left panel), in the defect region (right panel), sidebands around the probe frequencies reveal the presence of nonlinear frequency mixing.

In the performed experiments, we used a laser Doppler vibrometer (Polytec (Polytech GmbH, Waldbronn, Germany) OFV-5000/sensor head OFV 505 (Roland Corporation, Hamamatsu, Japan)) in velocity mode (sensitivity 5 mm/s/V) for scanning, and a Roland (Roland Corporation, Hamamatsu, Japan) Octa-Capture sound card for signal acquisition (Figure 4). Probe waves were generated by sending a sum of (phase locked) sinusoidal signals, with frequencies f_p = pf_P with f_P = 4000 Hz (p = 1..12), and V_pp = 0.1 V for each individual sinusoidal component from the Roland Octa-Capture sound card through a home-made pre-amplifier to a 35 mm diameter PZT/brass piezoelectric transducer glued on the sample plate. The peak-to-peak amplitude of the probing comb was 20 V. Also, the sinusoidal modulating waves (frequency f_M = 530 Hz) were synthesized by using the Roland Octa-Capture sound card, but in view of efficiently generating nonlinear effects, the sound card output signal was amplified by an AA Lab Systems (A.A.Lab System Ltd., Wilmington, DE, USA) power amplifier to about 120V_pp, which excited a second, identical PZT/brass piezoelectric transducer glued on the plate. In the scanned region of interest, retroreflective tape was glued on the plate in order to improve the amount of collected backscattered probe light and thus improve the SNR of the LDV signals.

3. Results

Figure 5 shows maps of the amplitude of the probe frequency (f_P) component and its nearest neighbor sideband components (f_P ± f_M) for 3 of the probe frequencies. The color ranges are scaled according to the minimum and maximum amplitude in the map. For the lowest probing frequency of 4000 Hz, features of the standing wave of the probe (wavelength 75 mm, calculated on the basis of the frequency and the bending wave velocity for an estimated Young modulus of 70 GPa and Poisson ratio of 0.25) appear in the center frequency as well as in the sideband frequency maps. However, the amplitudes of the sidebands were found to be roughly 10 dB weaker than one of the center frequency components and no sign of a particular defect location is visible.

The situation for the 24,000 Hz probing frequency is different. As expected, here the maps reflect the occurrence of standing waves (estimated wavelength 3 mm), but the amplitude at the probing frequency and its two satellites are clearly different at the location where we expected the (hidden) hammer impact defect to be. The fact that the defect is not only revealed in an enhancement of the amplitude of the sideband components but also at the original probing frequency indicates that the 24,000 Hz probing frequency may be exciting a local defect resonance, caused by delamination-induced softening and possible mass–spring–mass behavior of poorly attached delaminated plies. The existence of the sidebands must be due to nonlinear mixing between that probing vibration and the modulating vibration.

The picture for the 48,000 Hz probing frequency is less clear than for 24,000 Hz. There is amplitude enhancement at the defect location, mainly for the two sidebands, but with a substantially lower contrast to the other parts. This is because the amplitude level of the sidebands at 48,000 Hz is close to the noise floor of the measurement. Additionally, the sidebands at 48,000 Hz do not highlight the whole defect region but only a part of it.

The different pictures obtained for these three probing frequencies indicate that the use of multiple frequencies is quite useful in typical cases where the defect resonance frequency is unknown. The combination of one of the comb frequencies matching a defect resonance and nonlinear response-induced sideband generation increases the probability of detection.

Figure 5 only shows selected spectral maps, illustrating best- and worst-case scenarios. The generation of sidebands has taken place consistently at every comb frequency component with consistent efficiency. The spectrum averaged around the defect location (Figure 6) shows this.

An interesting view of the saliency of the defect in LDV displacement amplitude scans is given in Figure 7. The top panel shows that at 24,000 Hz, the increased vibration amplitude at the defect clearly stands out from those at other locations, with a defect-to-background amplitude contrast of more than 30 dB. The defect-generated sideband amplitudes are about 10 dB smaller than the amplitude of the probe vibration at the defect, but still with a defect-to-background amplitude contrast of more than 20 dB. In addition to the peak at the defect around x = 3 cm, there is a small peak at x = 18 cm. This could be an artifact, but it cannot be excluded that it reveals a second delamination location, which exhibits nonlinearity and thus sideband generation, but which is not resonating at 24,000 Hz.

Since in real-life circumstances, it cannot be known a priori whether there is a defect, whether this defect exhibits a resonance and, if yes, at which frequency it occurs, the possibility of exciting the sample with a probe signal with a multi-frequency comb spectrum can be implemented by considering the root mean square (RMS) value of all the center comb components, and the same for the sidebands, as follows:

$$S_{center}(x,y)=\sqrt{\frac{1}{N_p}\sum _{p=1}^{N_p}{S}^2(x,y,p{f}_P)}$$

(4a)

$$S_{left}(x,y)=\sqrt{\frac{1}{N_p}\sum _{p=1}^{N_p}{S}^2(x,y,p{f}_P-f_M)}$$

(4b)

$$S_{right}(x,y)=\sqrt{\frac{1}{N_p}\sum _{p=1}^{N_p}{S}^2(x,y,p{f}_P+f_M)}$$

(4c)

The bottom panels of Figure 7 show the resulting cross-sections. For defect localization, it is not meaningful to distinguish between left and right sidebands and therefore they are combined in a sideband RMS figure obtained as:

$$S_{s\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{s}}\left(x,y\right)=\sqrt{S_{left}^2\left(x,y\right)+S_{right}^2\left(x,y\right)}$$

(5)

Figure 8 (top panel) shows the resulting map. Thanks to the fact that for some of the comb components, the center frequency and/or the sidebands are strongly excited at the defect, the defect location clearly stands out from the background. In spite of their lower amplitude, the defect-to-background contrast is larger for the sidebands than for the central component, indicating that nonlinear sideband generation may be a more adequate phenomenon to exploit for the detection of this kind of defects than counting on enhanced probe frequency amplitudes due to a defect resonance.

Figure 8 shows a comparison between the 2D maps of the RMS of the left and right sideband signals and a thermographic phase image. The RMS map (top) shows a sharp feature at the defect on a somewhat fluctuating background, and high RMS values towards the top left corner of the plate, which probably can be attributed to contact nonlinearities caused by improper clamping of the plate. The bottom map was obtained by pulsed photothermal flashlamp excitation (using 2 halogen lamps, 6 kJ per lamp, 2 ms pulse duration, lamp–sample distance approximately 1 m) and detection using a FLIR X8500SC at approximately 1 m from the sample, placed in between the halogen. The map shows the phase of the f = 0.3 Hz component obtained by a temporal Fourier transform of the signal for every pixel. The hammer-impacted region clearly shows up in the IR signal phase map, but it is substantially larger than in the RMS. Also, the contrast compared to intact parts of the plate is larger, mainly because the thermographic image is smoother. In Figure 9, the phase profile of the IR image, taken over a 4 cm strip running along the x-direction highlights the contrast between the defective region, marked by black dashed lines, and the adjacent region. Hence, compared to the sideband detection approach, the IR approach is more sensitive to hammer impact damage and somewhat less vulnerable to the detection of false positives. However, it should be mentioned that the front excitation—back detection IR thermography approach is intrinsically more sensitive to poor thermal contact between delaminated ply surfaces than front excitation—front detection IR thermography but that in practical circumstances the back side of plates is often not accessible, which is impeding the use of back detection. Also, the cross-section in the horizontal direction through the defect of the thermographic map shows that the phase exhibits quite some slowly varying values. In conclusion, the RMS image resulting from using a frequency comb as a probing wave proved to overcome the limitation of the traditional approach of using a single frequency probing wave and has a precision comparable to that of IR imaging.

4. Discussion

4.1. Feasibility of Cross-Modulation Detection Using PRI-BP

In the previous section, it was shown that when a CFRP plate containing a hammer impact-induced delamination defect is subject to the combination of a high-frequency probe wave with a comb-like spectrum and a strong low-frequency vibration, LDV scanning reveals modulation of the high-frequency amplitude by the low-frequency vibration, resulting in sidebands in the signal spectrum near the comb frequencies, with particularly high contrast at the defect location when a comb frequency matches the clapping resonance of the defect. The following approach is thus feasible for inspecting the integrity of CFRP components in the framework of NDT campaigns but unfortunately, LDV scanning is quite time-consuming. This indicates the translation of the approach to a full-field vibrometry scheme. Combining the detection of defect-induced sum and difference frequency signals with standard full-field vibrometry approaches such as speckle interferometry and shearography however stumbles over the intrinsic optical nonlinearity of these methods, which means that the mixing of probe and pump frequencies is not only caused by acoustic nonlinearity but also by optical nonlinearity. More specifically, the interference intensity I_tot (Watt/m²) of Michelson-type (full-field) interferometers is given by I_tot(δ) = I_ref + I_sample + (I_refI_sample)^1/2cos(2πδ/λ_optical + ϕ), with λ_optical (m) the wavelength of the used monochromatic light and δ (m) the motion induced change in optical path difference between the sample beam and the reference beam. The sensitivity of I_tot to δ is maximum starting from an optical phase angle difference Φ (rad) between the two beams of π/2, so that cos(2πδ/λ_optical + ϕ) = −sin(2πδ/λ_optical) but even in that case I_tot(δ) is not linear. Especially when displacement δ(t) contains probe contributions δ_probe(t) = δ_probe,0exp(iω_probet) and pump contributions δ_pump(t) = δ_pump,0exp(iω_pumpt) larger than λ_optical/4, this leads to frequency mixing with signal variations at sum and difference frequencies between f_probe and harmonics of f_pump=, f_probe ± nf_pump (with n the integer index of the pump harmonics).

The hurdles that Michelson-type interferometers encounter when applied in the framework of cross-modulation experiments are here illustrated by means of numerical analysis. Figure 10 illustrates three vibration patterns present in a sample plate: the pump, the probe, and a sidelobe. Their amplitude was chosen to resemble the previous LDV experiments. Round numbers have been chosen for the different parameters, without losing generality, with the intention of illustrating the phenomena of interest more clearly. Figure 11 shows a simulation of the different frequency components of the vibration pattern as would be measured by a full-field interferometer in the case of the probe and pump amplitudes used in the LDV experiment in the previous section, δ_probe,0 = 10 nm and δ_pump,0 = 3.3 μm, for a probe frequency f_probe = 24 kHz and a pump frequency f_pump = 0.53 kHz. Due to the nonlinearity of the sensitivity of the interferometer to sample displacements, mixing frequencies are present near all the antinodes across the whole sample of the standing wave pattern of the pump frequency vibration. The truly acoustic mixing frequency component, with an amplitude of 50 pm, which was simulated in the form of added frequency components δ_defect = δ_defect,0exp(i(ω_probe ± ω_pump)t) at the two sideband frequencies is totally masked by the optical nonlinearity-induced features.

In Xiong et al. [37], the use of a photorefractive interferometer in bandpass mode (PRI-BP) has been shown to have the potential to tackle the above problem. Photorefractive materials are a class of nonlinear optical materials whose refractive index is dependent on the local light intensity [49,50]. The light impinging on the crystal excites the electrons in the valence band which rearranges in a space charge distribution resembling the light interference pattern [51,52]. The resulting space charge field affects the refractive index of the crystal according to the Pockels effect. Photorefractive materials have therefore been extensively used as media for continuous hologram reading and writing [35,53]. Since the space charge field formation is not immediate, using photorefractive crystals it is possible to isolate and filter out light phase variations that are faster than the space charge field formation characteristic time (τ) [54]. This property has enabled several schemes of adaptive interferometry for ultrasound detection [55,56,57] in which the resulting signal coming from the sample could be immune to disturbances due to sample roughness and low-frequency environmental vibrations. By exploiting a heterodyne interferometer scheme and modulating the phase of the reference beam, it is possible to realize a full-optical bandpass filter (PRI-BP) (also referred to as full optical lock-in) [31,33]. In PRI-BP (layout in Figure 12), the lowpass characteristic of the photorefractive crystal that causes the diffraction of the reference beam when mixed with the probe beam, combined with the use of a Pockels cell that modulates the optical phase of the reference beam at a frequency near to the one of a vibration frequency of interest (in casu: one of the sideband frequencies), results in a full-field vibrometry device with high selectivity to vibrations at the phase modulation frequency, and strong rejection of other vibration frequencies.

The intensity of the diffracted reference beam in a PRI-BP is a function of phase modulations induced in the sample beam and in the reference beam and is given by [37].

$$I_s=K_{eff}\left[J_0^2({ϵ}_2)J_0^2({ϵ}_{pump,0})J_0^2({ϵ}_{probe,0})+4J_0({ϵ}_{defect,0})J_1({ϵ}_{defect,0})J_0({ϵ}_2)J_1({ϵ}_2)J_0^2({ϵ}_{pump,0})J_0^2({ϵ}_{probe,0}) \frac{cos(\mathsf{\Omega }t+\mathsf{\Phi })}{{(1+{\mathsf{\Omega }}^2{\tau }^2)}^{1/2}}+\dots \right]$$

(6a)

where the terms ${ϵ}_{defect,0}, {ϵ}_{pump,0}, {ϵ}_{probe,0}$ express the optical phase modulations induced by the defect, pump, and probe vibrations.

$${ϵ}_j=\frac{4\pi {\delta }_j}{{\mathsf{\lambda }}_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}}$$

(6b)

where j = defect, pump, probe. The term ${ϵ}_2$ (rad), indicates the optical phase modulation induced in the reference beam by the Pockels cell, tuning this value so that it maximizes the factor $J_0\left({ϵ}_2\right)J_1\left({ϵ}_2\right)$, maximizes in turn the signal modulation depth. $J_n\left(x\right)$ is a n-th order Bessel function of the first kind The term $J_0\left({ϵ}_{defect,0}\right)J_1\left({ϵ}_{defect,0}\right)\approx {ϵ}_{defect,0}$ when ${ϵ}_{defect,0}\ll {\lambda }_{\mathrm{o}\mathrm{p}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}$. The time-varying terms of interest having angular frequencies Ω (rad/s) that equal the difference in angular frequencies of the optical phase modulation ω_Pockels = 2πf_Pockels and of different signal components. Since f_Pockels can be chosen close to the acoustic mixing frequency of interest (e.g., f_probe + f_pump) and far from f_pump, the lowpass factor (1/(1 + (Ωτ)²)^1/2 that affects the amplitude of the time-varying part in the measured diffracted intensity is made large (Ωτ << 1) for the former component and small (Ωτ >> 1) for the latter component. In this way, the unwanted time-varying component at angular frequency Ω can be efficiently suppressed, because of its filtering characteristic, measured in Figure 13 (linear) and Figure 14 (logarithmic), and as shown in the simulated amplitude map in Figure 15, which was calculated for the same conditions as in Figure 11. The experimentally observed bandpass-type rejection of unwanted signal components for the PRI-BP used in this work is further illustrated in the top panels of Figure 13. A fit of the time constant τ (s) of the device yielded τ = 1.5 ± 0.5 ms, corresponding with a bandwidth of 106 ± 50 Hz for a central bandpass frequency of 18.5 kHz. The PRI optical bandpass has a rejection of 20 dB/decade. The PRI-BP is particularly suited to rejecting large unwanted background signals because, in addition to the bandpass attenuation term (1/(1 + (Ωτ)²)^1/2, its response in displacement is proportional to $J_0\left(ϵ\right)J_1\left(ϵ\right)$, which is linear only for $ϵ\ll \lambda /4$, while for large $ϵ$, $J_0\left(ϵ\right)J_1\left(ϵ\right)\approx {ϵ}^{-1}{\cos }^2\left(ϵ+\phi \right)$ [58], it is a decreasing function of the displacement.

4.2. Artefacts Caused by Strong Pump Frequency Component

Looking at Equation (6a), one complication of the PRI-BP interferometer is that, in spite of the rejection of time-varying signal components at unwanted frequencies, the measured amplitude of the component of interest is affected by other vibrational components: even if the amplitude Bessel function—factor of interest, J₀(ε_defect,0)J₁(ε_defect,0) is linear with the very small vibration of interest (Figure 17), δ_defect = δ_defect,0exp(i(ω_probe ± ω_pump)t, one of the other pre-factors, $J_0^2\left({\epsilon }_{pump,0}\right)$ varies strongly with the amplitude δ_pump,0(x,y) of the pump vibration. Figure 18 shows that the magnitude of the bandpass characteristic of the PRI-BP is strongly suppressed by the presence of a pump vibration. In this case, with amplitude δ_pump,0 = 0.38 μm, the masking factor corresponds to $J_0^2\left({\epsilon }_{pump,0}\right)=0.88$. The dependence of the masking factor on the amplitude of the pump, measured by scanning the amplitude of the pump vibration while measuring the light modulation happening at the downmixed frequency of interest, is illustrated by Figure 16. Even a very simple pump modal pattern, the fundamental mode of a rectangular plate, for example, causes a non-trivial masking pattern, which may lead to misinterpretation of the defect size. This is graphically illustrated in Figure 19.

Figure 16. The dependency of I_s on ${\delta }_2$, the pump amplitude, is illustrated in the swept network experiment of this figure with two vibrations excited in the sample: a “weak” vibration (57 kHz,44 nm) which was measured as a function of the amplitude of a second vibration (2.4 kHz, spanning 2.7–1015 nm). The frequency difference between the reference beam optical modulation and the weak vibration was kept fixed at 2 Hz. The amplitude of the downmixed light intensity modulation was then measured with the lock-in detector. The data (dots) are then fitted to the masking term J₀(${\delta }_2$).

Figure 16. The dependency of I_s on ${\delta }_2$, the pump amplitude, is illustrated in the swept network experiment of this figure with two vibrations excited in the sample: a “weak” vibration (57 kHz,44 nm) which was measured as a function of the amplitude of a second vibration (2.4 kHz, spanning 2.7–1015 nm). The frequency difference between the reference beam optical modulation and the weak vibration was kept fixed at 2 Hz. The amplitude of the downmixed light intensity modulation was then measured with the lock-in detector. The data (dots) are then fitted to the masking term J₀(${\delta }_2$).

The effect of this strongly position-dependent factor $J_0^2\left({\epsilon }_{pump,0}\right)$ on the measured signal pattern is illustrated in Figure 21. The masking factor increases non-monotonically as a function of the pump amplitude. The parts of the defect located close to the antinodes of the pump vibration pattern are dimmed by the masking effect by several orders of magnitude compared to the defect portions closer to the antinodes. A possible approach to mitigate the masking effect is to combine the data obtained using different pump amplitudes. The mask resulting from the average of the different masking patterns, $M$, can be calculated as:

$$M(x,y)=\frac{1}{N}\sum _{n=1}^N{J}_0^2\left({\mathsf{\alpha }}_n{\mathsf{\epsilon }}_{pump,0}(x,y)\right).$$

(7)

where ${\mathsf{\alpha }}_n$ is a series of real numbers used to scale the pump amplitude between two extremes of choice. Each of these individual masking patterns presents local maxima and local minima at different locations from each other. If a sufficient number of pump amplitudes are chosen, the ripples present in an individual masking pattern are smoothed and the non-monotonicity of the mask is circumvented. The resulting smooth mask is reported in Figure 20, where the masking pattern of 15 different pump vibration components of amplitude spanning between 2.5 and 6 μm were combined in an average.

Another issue caused by the masking pattern is the reduction in signal amplitude. To get around this issue, it is necessary to combine images resulting from different pump vibration patterns chosen either by sweeping the pump frequency or by selecting a number of different standing modes as pump vibrations. These choices should be evaluated experimentally since there is no guarantee that the sideband conversion efficiency is the same for every pump frequency.

Figure 20. Average of 15 masking patterns generated by 15 different pump amplitudes. The pump displacements spanned from 2.5 μm to 6 μm in steps of 250 nm. The pump displacement pattern is the same as in Figure 19. The advantage of combining multiple pump masks is that the resulting averaged mask is void of ripples. However, the signal amplitude coming from the regions where the pump amplitude is the largest will inevitably be dimmed.

Figure 20. Average of 15 masking patterns generated by 15 different pump amplitudes. The pump displacements spanned from 2.5 μm to 6 μm in steps of 250 nm. The pump displacement pattern is the same as in Figure 19. The advantage of combining multiple pump masks is that the resulting averaged mask is void of ripples. However, the signal amplitude coming from the regions where the pump amplitude is the largest will inevitably be dimmed.

Figure 21. Illustrated map of the sinusoidal PRI-BP signal component in the same conditions as in Figure 10, i.e., at a frequency of vibration of interest (23,470 Hz) of a plate subject to a superposition of a 530 Hz large (δ_pump = 3.3 μm amplitude) sinusoidal vibration with wavelength of about 1 unit and a 24 kHz small (δ_probe= 10 nm amplitude) sinusoidal vibration with wavelength of about 0.13 units across the whole surface. A small 23,470 kHz (δ_defect = 0.5 nm amplitude) sinusoidal vibration, mimicking a mechanical nonlinearity-induced vibration of interest, is superposed on the other two vibrations in the white rectangle. Thanks to the strong suppression of vibrational components other than the vibration of interest, the image clearly shows the presence of the vibration of interest in the rectangular, white dashed, “defect region”, and it is not affected by the presence of the two other vibrations. The data are normalized to the maximum value across the image.

Figure 21. Illustrated map of the sinusoidal PRI-BP signal component in the same conditions as in Figure 10, i.e., at a frequency of vibration of interest (23,470 Hz) of a plate subject to a superposition of a 530 Hz large (δ_pump = 3.3 μm amplitude) sinusoidal vibration with wavelength of about 1 unit and a 24 kHz small (δ_probe= 10 nm amplitude) sinusoidal vibration with wavelength of about 0.13 units across the whole surface. A small 23,470 kHz (δ_defect = 0.5 nm amplitude) sinusoidal vibration, mimicking a mechanical nonlinearity-induced vibration of interest, is superposed on the other two vibrations in the white rectangle. Thanks to the strong suppression of vibrational components other than the vibration of interest, the image clearly shows the presence of the vibration of interest in the rectangular, white dashed, “defect region”, and it is not affected by the presence of the two other vibrations. The data are normalized to the maximum value across the image.

4.3. Feasibility of PRI-BP Detection of Acoustic Cross Modulation in Terms of SNR

The previous sections have shown that it is feasible to detect, by using LDV scanning, defects by the mixed frequency components they induce between a sinusoidal pump vibration and sinusoidal probe vibration, and that full-field photorefractive imaging in bandpass mode allows for detecting small vibrations at a frequency of interest, while rejecting other vibrations, no matter their amplitude. Combining the concept of doing NDT via frequency mixing with PRI-BP detection gives the potential for full-field PRI-BP tuned on mixed frequency components to generate damage maps without time-consuming scanning. The challenge for this approach is that the mixed frequency components typically have a very small displacement amplitude.

In the experimental setting described in Section 2, the modulating displacements were of the order of 3.3 μm at the defect location, the probe vibration amplitude 250 pm, and the defect-induced sidelobe amplitude 50 pm.

Table 1. Overview of the minimum detectable displacement of the LDV setup and the PRI-BP setup in different configurations.

Minimum Detectable Displacement/Noise Level	pmW1/2/Hz1/2	pm/Hz1/2
PRI-BP point detection theoretical value	32 × 10−6	0.14@P = 50 nW
PRI-BP point detection experimental value	22 × 10−3	10@P = 10−5 nW/pixel
PRI-BP full-field experimental value	300 × 10−6	3000@P = 10−5 nW/pixel

gives an overview of the shot-noise limited minimum detectable displacement (MDP) of PRI-BP implemented in point (scanning) detection and full-field detection, expressed per unit of generated signal power $I_1$ and for our particular setup, in which part of the probe laser intensity that was sent to a circular brass sample was 9.6 mW/cm², the intensity of the light collected and focused by a commercial camera objective lens (positioned at 500 mm from the sample) was 3.06 mW/cm², measured at the PR crystal surface, and the intensity of the reference beam that was mixed with the sample in the PR crystal was 93 mW/cm². The power reaching the camera (FLIR Chameleon 3 (Teledyne FLIR LLC, Wilsonville, OR, USA), using a SONY IMX265 CMOS pixel array, 3.19 MPixels, pixel size 3.45 μm × 3.45 μm) (full-field mode) of the PRI-BP was about 10⁻⁵ nW per pixel (600 nW/cm²). For point detection, a photodetector Thorlabs DET36 A/M was used, and no lenses were employed to illuminate the sample and collect the light and the resulting signal power was 50 nW. The PRI-BP signal is comprised of a constant term and an oscillating term. The first and second terms are between square brackets of Equation (6a), respectively [33]. In an ideal setting, with perfect rejection of undesired light, the average shot noise is, therefore, a function of the constant term, and the SNR is the ratio of the electrical powers resulting from the oscillating term and the constant term.

Some of the values in Table 1 are striking. The experimentally observed noise value for PRI-BP point detection is almost three orders of magnitude larger than the value predicted considering optical shot noise on the anisotropically diffracted reference beam. Part of the reason is that the rejection of the transmitted sample beam by the polarizing beam splitter between the PR crystal and the detector is 10⁻³, so the intensity of the anisotropically diffracted reference light on the detector was 30% of the transmitted sample light. Using high extinction polarizer, with a rejection ratio of 10⁻⁵, an additional order of magnitude in the SNR can be gained. A factor to investigate is that the diffracted and the transmitted light displayed elliptical polarization. This goes in contrast with the experimental evidence reported in the literature and may suggest that the photorefractive process is not happening entirely according to the presented theory and the diffraction efficiency may be far from optimal. Possible reasons for this could be birefringence induced by stresses in the crystal [33] or suboptimal interaction length between the interfering beams. The crystal thickness may also need optimization. Additionally, electrical noise (e.g., read noise in the camera, dark noise) might have been underestimated.

The theoretically expected MDP of PRI-BP in scanning mode (0.14 pm/Hz^1/2) is of the order of the MDP detected by LDV and thus in principle more than small enough to detect defect-induced frequency mixing components of 50 pm amplitude as in Section 2. Also, the practically observed MDP of 10 pm (for a bandwidth of 1 Hz) would suffice.

As a consequence of the decrease in SNR with decreasing signal power, the experimentally observed MDP with full-field PRI-BP, evaluated per pixel, is much smaller than the one where all the light is focused to a single detector. For a bandwidth of 1 Hz, the noise level of 3 nm would be far too large to detect the 50 pm mixing frequency components in the experimental setting of Section 2. Taking into account that the LDV scan took T_measurement = 13 h for 85 × 290 pixels, corresponding with an overall bandwidth of 1/T_measurement = 21μHz. The extrapolated MDP of PRI-BP for characterizing a whole sample in the same measuring time would be 5 nm/(T_measurement)1/2 = 23 pm.

5. Conclusions

The presented results show that, although invisible for the naked eye, impact-induced delamination damage can be detected in a measurement scheme that exploits the mechanical nonlinearity-induced generation of mixed frequency components between a strong vibration that makes the defect open, and a combination of probing vibrations that together form a comb spectrum. Vibrometry performed by a scanning laser Doppler vibrometer system yielded maps that show a clear contrast between the defect location (mixing frequency component amplitude of the order of 50 pm) and intact regions of an impacted carbon fiber-reinforced plate. The probability of defect detection of the method is shown to be favored by employing a comb spectrum for the probe, by virtue of the higher chance of one of the probe frequencies matching the frequency of a local defect resonance.

While detecting a small vibration of interest in the presence of other, masking, large vibrations in a classical full-field interferometric scheme is cumbersome due to the high nonlinearity of interferometric detection when large vibrations are present, photorefractive interferometry in a bandpass configuration around the frequency of the small defect-induced vibration of interest was shown to mitigate the masking problem. Further work is needed to decrease the minimum experimentally detectable displacement amplitude from 3 nm/(Hz)^1/2 to 50 pm. One of the important steps to make is an enhancement of the polarization selection that favors the information containing part of the detected light (anisotropically diffracted part of the reference beam) compared to unwanted light (transmitted sample beam). Also, temporal coding of the reference beam is expected to improve the detection sensitivity. Also, in the case of PRI-BP, the same principle learned with LDV, that the defect silhouette is strongly frequency dependent, holds. In principle, only a few seconds of acquisition time are necessary to acquire a bandpass video. Therefore, future work should investigate the generation of an RMS figure exploiting different probing frequencies when working with PRI.