2.1.

Acoustic Theory

The wave energy of a SAW, and hence the wave motion, is concentrated within a penetration depth similar to its wavelength $\lambda$ , beyond which the wave amplitude quickly becomes negligible. ²³ This essentially means that the SAW velocity $c_R$ is governed by the elastic constants of the material depth it probes. The SAW penetration depth $z$ can be estimated from the relation:

where $f$ is the signal frequency.

In an isotropic homogeneous medium, $c_R$ is independent of the signal frequency as well as the propagation direction.²⁴ For a multilayer medium of different elastic properties, the wave propagation is influenced by all the layers it probes, such that $c_R$ is governed by a generalized dispersion equation.²⁵ To understand the dispersion effect, consider a broadband surface wave propagating in the $x_1$ direction on a sample consisting of a substrate coated with an upper layer that has a lower SAW velocity, as shown in Fig. 1 . Such a medium is of direct interest to our study, since a human tooth can be considered as a two-layer system within the scope of our intended investigation.

Fig. 1

Surface wave dispersion in a two-layer system.

A broadband SAW impulse can be regarded as the superposition of surface waves with different frequencies. Referring to Fig. 1, the higher frequency components have a shallow penetration depth and are more influenced by the surface layer, traveling with a relatively slower velocity, whereas the lower frequency components penetrate deeper and are influenced more by the substrate elastic parameters. Thus the phase velocity $c_R$ becomes a function of frequency and the SAW impulse is dispersed as it propagates.

2.2.

System Configuration

The schematic of the laser NDE system for SAW measurements is illustrated in Fig. 2 .

Fig. 2

Arrangement of the laser NDE system and the SAW dispersion measurement.

A solid-state quadrupled Nd:YAG laser, operating at $266\mathrm{nm}$ , was used to excite the surface wave. The source laser pulses, with $\sim 5\text{-}\mathrm{ns}$ duration, were directed through a rectangular mask and focused into a thin line by a cylindrical lens onto the sample surface (shown on the right of Fig. 2). The generated SAW propagates perpendicular to the line source and contains a wide frequency bandwidth enabling the signal to probe multiple layers in the material simultaneously, which is desired for the SAW dispersion measurement.

In our measurement setup, test objects were placed on a manual Z-stage for adjusting the separation between the lens and the sample, and hence the line-source width (defocusing the image). Reducing the line width can further increase the bandwidth of the generated SAW, but this inevitably raises the irradiating power density and may damage the specimen surface. During measurements, the laser pulse energy and the sample surface were constantly monitored to ensure no physical damage.

Detection of the ultrasound was performed by the OFI (shown on the left of Fig. 2) consisting of a $1550\text{-}\mathrm{nm}$ continuous-wave laser as the light source, a three-port optical fiber circulator, and a photoeceiver. These three components were joined using fiber connectors. The detected signal was averaged 32 times and displayed on a digital oscilloscope with a trigger signal from the Nd:YAG laser.

The tip of the middle port fiber of the circulator was cleaved to act as the probe head and placed on a micron-precision three-axis $\left(xyz\right)$ positioning stage, such that the probe-to-sample separation can be adjusted to optimize the returning signal strength and the spatial resolution, as well as make measurements at different locations. This positioning stage permits measurements between 1 and $12\mathrm{mm}$ from the line source, with $0.01\text{-}\mathrm{mm}$ precision. All measurements were made with a probe-to-sample separation of $5\mathrm{mm}$ or smaller.

The basic principle of this OFI is similar to a reference beam interferometer. The coherent laser light from the source $I_0$ was coupled into the circulator. As $I_0$ reaches the end of the middle port fiber (sensing tip), about 4% of the light will be internally (Fresnel) reflected at the silica/air interface and serves as the reference beam $I_1$ . The rest of the light is incident on the sample surface, and then reflection causes a small portion of it $I_2$ to be coupled back into the fiber and become the measurement beam. Throughout the rest of the system, $I_1$ and $I_2$ copropagate toward the third port, and this essentially common path environment means that the path difference of the two beams, and hence their relative phase, depends only on the separation between the fiber tip and the sample surface. In a typical ultrasonic measurement, the phase modulation is less than $\pi$ , which means that the elastic wave vibration amplitude is less than a quarter of the OFI source wavelength, and the interference intensity becomes a very good measure of the absolute acoustic waveform. However, the measured interference signal does not contain information about the absolute ultrasonic wave amplitude, since we did not have the means to know precisely the initial static phase difference (equivalent to controlling the probe-to-sample separation on a nanometre scale).

A more detailed discussion about the OFI can be found in Ref. 22.

2.3.

Determination of the Experimental Dispersion Spectrum

To determine the experimental dispersion curve, generated surface waves were measured at various locations along the epicentral axis of the line source. The recorded wave signals were digitally bandpass filtered $\left(1\text{to}50\mathrm{MHz}\right)$ to reduce the low and high frequency noise. Two signals, at locations $x_1$ and $x_2$ , were selected for dispersion analysis. They were cross-correlated to identify and exclude the noise signals, and then Fourier transformed. The phase difference between the ultrasonic signals $\phi \left(f\right)$ was determined from the phase angle of the Fourier spectrum, from which the frequency-dependent SAW velocity can be determined using:

2.4.

Validation Measurements on Common Metals

Prior to applying this system for evaluating human dental enamel, the capability of our system for measuring different types of surface waves (both nondispersed and dispersed) was investigated. In addition, the accuracy of the analysis technique for determining the experimental SAW dispersion spectrum must be tested by comparing with theoretical expectations. In this section we present results of evaluating different metal structures, and demonstrate the performance of the NDE technique. More extensive validation results of this system can be found in Ref. 22.

Homogeneous isotropic materials, namely aluminium and brass blocks, with known SAW velocities were measured [aluminium $\sim 2950{\mathrm{ms}}^{-1}$ , brass $\sim 1970{\mathrm{ms}}^{-1}$ (Ref. 27)]. Figure 3 shows the surface waves on the aluminium sample measured at various locations between 2 and $10\mathrm{mm}$ from the line source with $1\text{-}\mathrm{mm}$ step size. Note that the wave amplitudes are normalized, and that the $y$ axis is not the scale of the traces but rather it serves only as a label to indicate for each trace the relevant location from the line source. The absence of waveform broadening indicates that no dispersion occurs in this homogeneous medium.

Fig. 3

Surface waves on the aluminium sample measured at different distances from the source. (The traces are identically scaled arbitrary units; see text for details.)

Signals measured at 2 and $10\mathrm{mm}$ from the line source were chosen and analyzed as discussed earlier. Surface wave phase velocity as a function of frequency was calculated using Eq. 2 and is displayed in Fig. 4 . The same measurement was performed on the brass block, and the determined velocity is plotted also in Fig. 4. The signal-to-noise ratios for these sets of measurements were very high, such that the reliable frequency bandwidth extends to $30\mathrm{MHz}$ (demonstrated by the expected stable velocity value).

Fig. 4

Experimental SAW velocities for aluminium and brass samples.

These experimental results for both homogeneous block materials match very well with the theoretical values (less than 1% difference), and they demonstrate, as expected, that no dispersion occurred in these uniform media.

For the study of a dispersive two-layer system, a sample composed of a $\sim 40\text{-}\mu \mathrm{m}$ -thick nickel film sputtered onto a thick fused quartz glass substrate was used. The SAW velocities of the materials were known (nickel: $c_R$ $\sim 2700{\mathrm{ms}}^{-1}$ ; glass: $c_R$ $\sim 3400{\mathrm{ms}}^{-1}$ ). Figure 5 shows the surface waves measured along the propagation path up to $6\mathrm{mm}$ away from the line source with $1\text{-}\mathrm{mm}$ spacing. The distortion of the acoustic waveform is apparent, and in this case it is obvious that the low frequency components, more influenced by the glass substrate, travel faster.

Fig. 5

Surface waves on a two-layer system of Ni film on glass substrate measured at locations $1\text{to}6\mathrm{mm}$ away from the source. (Y-axis as per Fig. 3.)

Signals with the furthest separation were once again chosen and cross-correlated for the dispersion analysis. In this measurement the reliable frequency bandwidth spanned from $1\text{to}20\mathrm{MHz}$ . The experimental dispersion curve is plotted in Fig. 6 . To confirm the thickness of the nickel film, and hence the accuracy of the experimental result, the partial wave technique presented in Ref. 25 was used to calculate the theoretical SAW dispersion spectra using known parameters of the two substances with different film thicknesses (30, 40, and $50\mu \mathrm{m}$ ), also displayed in Fig. 6. Note that the local minimum of the theoretical dispersion curve shifted toward higher frequency as the thickness of the surface layer reduces. The experimental curve clearly fits best with the $40\text{-}\mu \mathrm{m}$ simulation. The actual thickness of the nickel film was then measured with an optical micrometer and found to be $42\pm 2\mu \mathrm{m}$ (it was not a perfectly uniform film).

Fig. 6

Experimental and theoretical dispersion curves for nickel film on glass substrate.

The prior measurements demonstrate that our optical NDE system can generate and detect different types of surface waves on various samples. Excellent matches were obtained in fitting the experimental dispersion curves with theoretical simulations. This demonstrates that the curvature and the trend of the experimental dispersion curve within the reliable bandwidth are accurate in terms of revealing the state of elasticity of the medium in which the SAW propagates, most importantly the elastic response as a function of depth. The main source of uncertainty in the current implementation of this technique will most likely be the error in measuring the signal separations from the OFI translation stage manual micrometer adjustment.

3.1.

Characteristics of Sound and Lesioned Enamel

Dental enamel envelops and protects the underlying dentin and is the hardest and the most mineralized substance of the human body. The unevenly distributed composition and the highly oriented microstructure of tooth enamel result in anisotropic and inhomogeneous mechanical and elastic properties.⁵ Kushibiki ²⁸ used line-focused-beam scanning acoustic microscopy (SAM) and measured the surface wave velocity as a function of propagation direction on the labial surface of extracted human incisors, and observed that the velocity values vary between about $3105\text{to}3155{\mathrm{ms}}^{-1}$ , with the maximum obtained in the direction parallel to the tooth axis (direction from crown to root). Peck, Rowe, and Briggs²⁹ repeated the investigation on the surface of human permanent molars using SAM and reported that the SAW velocity varied between about $3075\text{to}3142{\mathrm{ms}}^{-1}$ , with the maximum velocity also in the direction parallel to the tooth axis. The SAM technique has drawbacks in that the measuring steps can be tedious and lead to prolonged time requirements. Also, special polishing procedures for the specimens are a prerequisite that makes the technique impossible for clinical applications. In addition, it requires coupling fluid, whose physical conditions (e.g., temperature) need constant monitoring and maintaining.

An early caries lesion in enamel is observed clinically as a white opaque spot, and hence is referred to as a white spot lesion (WSL). ³⁰ Within the enamel WSL, the surface layer is porous but remains relatively intact and mineral rich; however, the subsurface area under the WSL (the body of the lesion) is low in mineral (10 to 70% of the sound enamel). The WSL enamel thus has lower hardness and elastic modulus than that of sound enamel. The WSL layer on top of sound enamel essentially simulates a two-layer system similar to the one depicted in Fig. 1, thus dispersion is expected to occur in broadband signals propagating through a WSL area.

3.2.

Sample Preparation and Artificial White Spot Lesion

Among the various teeth, it is advantageous to choose the incisor for initial studies because its front surface has the largest flat area of enamel. In addition, the enamel thickness is relatively constant $(\sim 1\mathrm{mm})$ over a wide region near the center of the incisor front surface. This is desirable because the measured SAW dispersion will not depend significantly on the uneven thickness of the enamel layer, but rather on the elastic modulus variation. However, it is very unusual for a natural lesion to occur on the front surface of an incisor, and thus artificial lesions were created for this study. Artificial WSLs, may not be identical with natural enamel WSLs, but are widely used for dental research³⁰ and accepted as sufficiently similar. Two recently extracted sound human incisors were selected for the measurements (labeled A and B). Thin layers from the backs of the teeth were removed with a diamond blade so that the sample could be mounted on the measurement holder (on the Z-stage).

For artificial WSL preparation, the center region of the front incisor surfaces was abraded and polished approximately to a depth of $\sim 100\mu \mathrm{m}$ below the natural surface with 400-grit abrasive papers. About $\sim 70\%$ of the front surface on both samples was polished, and the procedure was carefully performed such that very little of the enamel surface was removed so that the sample was still considered to be close to an in vivo condition. The samples were coated with a protective layer of nail varnish, while leaving windows of about $3\times 3\mathrm{mm}$ exposed in the polished regions, and then placed in prepared demineralization solution containing $2.2\text{-}\mathrm{mM}$ $\mathrm{Ca}{\left(\mathrm{N}{\mathrm{O}}_3\right)}_2$ , $2.2\text{-}\mathrm{mM}$ $\mathrm{K}{\mathrm{H}}_2\mathrm{P}{\mathrm{O}}_4$ , and $50\text{-}\mathrm{mM}$ acetic acid $(\mathrm{pH}=4.5)$ . ³¹ To produce lesions of different severity, sample A was soaked in the demineralization solution for five days, and sample B was treated for three days. After lesion treatment, the nail varnish was removed and the samples were rinsed with distilled water.

3.3.

Measurements and Results

Prior to actual measurement, a study was conducted to determine the laser absorption spectrum of human enamel,³² and found that UV light was more efficiently absorbed than near-IR light, hence $266\text{-}\mathrm{nm}$ emission from the Nd:YAG laser was chosen again to generate SAW. In addition, the laser damage threshold for enamel at $266\mathrm{nm}$ was experimentally determined. We used a tooth sample and irradiated it with a line source of laser pulse energy at $\sim 1\mathrm{mJ}$ , and focused the line source gradually until thermal damage (predominately ablation) started to occur on the enamel surface. The corresponding line-source dimension was measured to be $\sim 1.4\times \sim 0.02\mathrm{mm}$ , which gives a power density of $\sim 7.5\times {10}^8{\mathrm{Wcm}}^{-2}$ . For our measurements on human dental enamel, presented next, we operated at a power density value that is below this estimated threshold with a margin such that all our experiments are nondestructive.

The first measurement was performed on the healthy region of sample A using the same system described before. Figure 7a shows how the healthy region of sample A was measured. Due to the limit in the usable sound enamel area, the line source (illustrated as the line) was irradiated near the edge of the enamel surface and the SAW propagated in the direction horizontal to the tooth axis. The recording of the surface waves was made at several positions along the propagation path (illustrated as the dots). The measurement was repeated ten times and each time we shifted the sample position randomly by a small amount $(\pm 0.2\mathrm{mm})$ in the direction perpendicular to the wave propagation path, such that the final evaluation results were the averaged contribution from an area of rectangular shape with $\sim 0.4\mathrm{mm}$ width. Recorded wave signals from one of the ten sound enamel measurements are presented in Fig. 8a . The temporal shape of the surface waves appeared to be consistent during propagation.

Fig. 7

Illustrations showing the positions of the SAW generation and detection on (a) the healthy region and (b) the WSL region of sample A.

Fig. 8

Surface waves recorded at different locations from the line source on (a) sound and (b) WSL regions of sample A. (Y-axis as per Fig. 3)

For measurements on the WSL region, as illustrated in Fig. 7b, we irradiated the line source near the top edge of the tooth and allowed propagation across the lesion in the direction parallel to the tooth axis. Surface waves were recorded at various locations, especially near the interface between the sound enamel and the WSL. Once again ten measurements, each with small random position variations, were repeated. Recorded wave signals from one set of the WSL measurements are shown in Fig. 8b. We present the signals measured before entering the WSL ( $2\mathrm{mm}$ from source), near the center of the WSL ( $\sim 4\mathrm{mm}$ from source), and just after the WSL ( $\sim 5.7\mathrm{mm}$ from source). The change in the temporal wave profile clearly indicates that dispersion has taken place.

Two signals, with a separation of more than $2\mathrm{mm}$ , were selected from each measurement for the dispersion analysis. The dispersion analysis was performed as discussed before. The reliable bandwidth of both sound and lesioned enamel measurements were approximately $1\text{to}25\mathrm{MHz}$ .

Dispersion spectra of sound enamel and WSL measurements were calculated, and the ten-times-averaged values were used as the final results. These are presented in Fig. 9 with standard deviation error bars.

Fig. 9

Dispersion curves of the final averaged results from the sound region (blue) and WSL region (red) of sample A ( $5\text{-day}$ demineralization). (Color online only.)

Several distinctive characteristics can be seen from the sample A results. In both sound enamel and WSL, the SAW velocities begin at a low value and increase with frequency. This can be explained by the influence of dentin, which has a much lower velocity value than enamel,^{33, 34} on the surface wave propagation at low frequencies. For SAW with frequencies of $1\text{to}6\mathrm{MHz}$ , the corresponding probing depths are relatively deep, and the propagation structure for these low frequency components can be considered as a two-layer system (sound bulk enamel on top of dentin). Dispersion is thus expected from the significant influence of the lower elastic modulus dentin.

In the sound enamel, once the SAW penetration depth is below a certain value (at $\sim 6\mathrm{MHz}$ ) at which the influence of dentin becomes negligible, the surface wave propagates in and probes only the enamel as a one-layer system. The velocity remains relatively constant at $\sim 3150{\mathrm{ms}}^{-1}$ ; this value is very close to that previously reported.^{28, 29} In the WSL region, the velocity begins to drop after $6\mathrm{MHz}$ and reaches a minimum of $\sim 2840{\mathrm{ms}}^{-1}$ at $\sim 16\mathrm{MHz}$ . The WSL dispersion profile after $6\mathrm{MHz}$ is similar to the result shown in Fig. 8, and thus confirms that the SAW propagation was influenced by the less elastic WSL layer (of unknown thickness at this stage) on top of the healthy bulk enamel. The differences between the two dispersion spectra are significant and clearly discriminate the two types of enamel.

The enamel anisotropy, as discussed in Sec. 3.1, is insignificant for this study, as previous studies^{28, 29} demonstrated that the maximum SAW velocity on human enamel is obtained in the direction parallel to the tooth axis. This suggests that if the sound region measurements of the current study were also performed in the direction parallel to the tooth axis, a slightly higher velocity value would be obtained that could further discriminate the WSL and the sound enamel dispersion curves.

Sample B, with the three-day artificial lesion treatment, was measured in an identical manner to sample A discussed earlier. The measured waveforms and the reliable frequency bandwidth are similar to that of sample A and for brevity are not shown. The final averaged dispersion spectra, with standard deviation error bars, are shown in Fig. 10 .

Fig. 10

Dispersion curves of the final averaged results from the enamel sound region (blue) and WSL region (red) of sample B ( $3\text{-day}$ demineralization). (Color online only.)

Comparing the results of samples A and B, one can first see that the sound enamel results share very similar dispersion profiles and velocity values, which indicates that the elasticity, as a function of depth, is similar between the two sound enamels. The WSL dispersion curves of both samples also have similar profiles, but there are a few important differences. For sample A the WSL curve has a minimum value of $\sim 2850{\mathrm{ms}}^{-1}$ , while the WSL curve for sample B has a slightly higher local minimum of $\sim 2910{\mathrm{ms}}^{-1}$ . This suggests that as expected the WSL of sample A, which has been demineralized for $5\text{days}$ , is less elastic (a more severe lesion) than the sample B WSL that has been demineralized only for $3\text{days}$ . In addition, the frequency at which the local minimum exists can help to provide insight into the thickness of the artificial lesion (for example, see Fig. 6). In sample A, the local minimum is at $\sim 16\mathrm{MHz}$ , while in sample B the local minimum is at $17\text{to}18\mathrm{MHz}$ . This indicates that the sample A lesion is thicker than the sample B lesion (due to prolonged demineralization) and influences the SAW propagation from a deeper probing depth (lower signal frequency).These results are very promising in terms of matching physical expectations regarding the surface wave propagation in enamel of different elasticity.