It is one of the critical parts of the simulation to select reasonably the time function of the Gaussian pulse. Therefore, Gaussian pulse light sources of g ₁ and g ₂ with the same energy are analyzed, whose pulse peak power density is P ₁ and P ₂, as shown in Figure 2. The pulse power spectrum density of the laser is considerably varied under the same space function as well as different time functions with the same upper and lower limit parameters. Figure 2A has pulse durations of approximately 60 ns, while (B) has a pulse duration of approximately 20 ns However, the maximum value of P ₁ is less than that of P ₂ . It can be seen that the laser’s power density increases as the time function narrows. As a result, selecting a time function should be based on personal experiment.

Different time functions induce changes in the temperature field, as seen in Figures 2C,D. The maximum temperature of the g ₁ function is greater than that of the g ₂ function at all detection points. The narrower the time function is, the less time there is for light energy to be converted into heat energy. This results in a lower temperature for the laser with a narrow time function at the same energy. Heat conduction causes a decrease in temperature away from the laser radiation point, which leads to differences in temperature at different detection points. The temperature field flattens out within 0.3 μs, which suggests that the temperature gradient is instantaneous. The time function g ₁ was employed in this simulation because of the higher temperature with a low power density.

After a pulse energy of 0.7 mJ, and a spot radius of 0.4 mm, the pulse width of the laser was adjusted from 1 to 100 ns to evaluate the impact on the laser excitation ultrasound waves. The temperatures of different positions detected by the probe in the transverse and longitudinal directions are shown in Figure 3. Different probes are spaced 2 μm apart in both directions. 533, 483, 456, 423, 403, 389, 378, and 351 K are the maximum temperatures produced by pulse lengths of 1–100 ns at (0, 3.5). The farther away from the (0, 3.5) point, as shown in Figure 3, the lower the temperatures are both transverse and longitudinal. We consider that the difference of the pulse width causes the peak power density to change, resulting in a different temperature acting instantaneously on the sample surface, leading to a change in the transient temperature field.

The displacements of pulse widths detected by the probe in different directions are shown in Figures 3C,D. Figure 3C indicates that a pulse width of 1 ns reduces the displacement from 22.6 nm at point (0, 3.5) to 16.8 nm at point (0.01, 3.5), while a pulse width of 100 ns reduces the displacement from 10.25 to 8.33 nm. This shift is more noticeable in the longitudinal direction; the displacement created by the laser with a pulse of 1 ns grows from 6.25 to 22.6 nm, whereas the displacement produced by the pulse of 100 ns increases from 1.05 to 4.15 nm, as shown in Figure 3D. The temperature field determines the stress field in the laser radiation region, and a larger temperature gradient results from a faster shift in the longitudinal temperature field; hence, longitudinal stress displacement is more variable. Due to the causal relationship between temperature and stress displacement, the overall trends of temperature and thermal stress wave displacements are similar. That is, the total displacement will decay away from the laser irradiation point. Therefore, it can be seen that the narrower the pulse width of the laser is, the higher efficiency of exciting the ultrasound.

Figure 4 shows the temperature field of the domain probe with varied spot radii after setting a pulse energy of 0.7 mJ and a pulse width of 10 ns The power density of the laser is directly affected by the size of the spot radius. The smaller the spot is, the higher the power density, which causes the temperature to increase. This trend is more noticeable in the smaller radius. In the case of lower than the damage threshold of the sample, the laser with a small spot radius should be selected as much as possible for exciting the ultrasound.

The displacement caused by thermal stress can intuitively reflect the ultrasonic wave’s amplitude. The laser’s incident point was detected to obtain displacement, the X component of which is illustrated in Figure 5A and the Y component in Figure 5B. The thermal expansion force generated by thermal energy interacts with the impeding force in the normal temperature area, resulting in a relatively complex value of the X displacement component. In Figure 5A, the X displacement after 1 μs is expanded, while the 0.05 mm displacement is masked for clarity. As heat energy is transmitted within the material, the X displacement becomes more noticeable. The spot radius is shown to be inversely linked with the X displacement. The longer the X displacement component exists, the smaller the radius is. In Figure 5B, the Y displacement corresponding to the spot radius of 0.3–1.0 mm is magnified for easy observation. The maximum Y displacements from 0.05 to 1.0 mm are 343.2, 142.4, 53.4, 28.4, 17.6, 11.9, 8.6, and 3.3 nm, showing a negative correlation. The smaller the spot is, the more energy is accumulated and the higher the power density. This will result in greater stress displacement in two directions. Usually, the data required in the actual detection is the Y displacement because it can be visualized to reflect the ultrasonic wave.

The wide spectrum and high amplitude of ultrasonic is one of the reasons why it is used to detect information. The influence of light spot radius on spectrum under the condition of same energy and same energy density is analyzed here. Figure 6A demonstrates that when the laser loading energy is constant, the smaller the spot, the greater the amplitude of the ultrasonic spectrum, and the information it can contain is easier to be extracted. Figure 6B shows that the spectrum amplitude of the ultrasonic wave will increase with increasing spot radius while the energy density of the laser remains constant. This is because the spatial function of the Gaussian pulse is inversely proportional to the square of the radius of the laser spot, and the size of the radius of the laser spot radiates with the same energy density without attenuation.

The propagation of ultrasonic waves and the mode transition that occurs following ultrasonic contact with the fracture are illustrated using a 70° crack model. Figure 7A shows different ultrasonic signals after laser irradiation for 2.0 μs The surface wave that directly reaches the crack is defined as the Rayleigh wave (RW), whose waveform is relatively apparent. At 3.1 μs, the ultrasonic waves reach the crack in Figure 7B, where the mode transition occurs. Some waves penetrate the edge of the crack and create a TRW, while other waves are reflected by it. A tiny percentage of waves are changed into mode transition shear or longitudinal waves after reflection; however, the vast majority of waves form crack-reflected Rayleigh wave (CRW). In Figure 7C, when the time reaches 3.75 μs, the TRW on the right side of the crack is reflected by the boundary, and its propagation direction reverses. After reflection, this waveform is called TRW-B. Figure 7D shows that the right-propagating reflection wave will also reflect through the right boundary.

If no special circumstances are specified, the default feature of the crack is that the angle is 70°, the depth is 0.3 mm, the width is 0.3 mm and Rayleigh waves have a wavelength of 890 μm. Different detection points are set to determine the angle of the crack. The Y displacement components obtained from these detection points are shown in Figure 8A. The Y displacement of these detection points from (2, 3.5) to (6, 3.5) is approximately −0.18 to 0.1 nm, while the Y displacement of the detection point (7, 3.5) is a maximum of 4.12 nm and a minimum of −4.7 nm, making it the largest of the detection locations. This phenomenon occurred at both acute and obtuse angles, with acute being the most noticeable. The tilt angle of the surface cracks in the numerical simulation is changed from 20° to 160° for different angle detection. The position (7, 3.5) was chosen as the detecting point to better examine the Y displacement components.

The Y displacements at different angles are shown in Figures 8B,C, which mainly reflects the variation in the RW waveform during this period of time. From 20° to 90°, there is a time delay in RW whose amplitude of Y displacement changes significantly as well. Part of the ultrasonic energy is blocked by the angled crack and continuously accumulates at the left end of the crack, which leads to the amplitude change as well as time delays of the RW. If the angle is greater than the right angle, the energy of the RW will no longer accumulate but will directly pass which is only reflected in the amplitude of the ultrasonic Y displacement. Therefore, the main change of the RW is in the displacement amplitude with angle changing from 90° to 160°.

It is a great method to fit correlation curves to solve the problem. The relationship between the angles and RW is obtained by fitting the difference between the peak and valley of RW. The relevant result is shown in Figure 8D. The curve from 20° to 80° is steeper, and the Y displacement difference between the peak and the valley drops from 0.714 to 0.0701 nm, as seen in the figure. The curve from 80° to 160° is flatter, whose Y displacement decreases from 0.714 to 0.0250 nm. The nonlinear curve fitting’s mean square error value (MSE) is 1.771810^–9. The fitting regression coefficient R² reaches 99.947%. The expression of the fitted curve is: Δ Y = 0.0248836 + 10.68 × exp ( θ − 24.68562 ) + 58462092 × exp ( θ − 1.02959 ) (9) where ΔY is the peak-to-valley displacement difference of the RW and θ is the degree of surface crack. It can be concluded that the higher the peak-valley difference of Rayleigh waves is, the smaller the surface tilt angle. Table 3 shows the data of some defect tilt angles and peak-to-peak values of direct Rayleigh waves under different crack depth/wavelength (D/λ) ratios. The trend is consistent in the case where the ratio of D/λ is less than 1.11. If the depth-wavelength ratio changes, then the peak-to-peak and Angle fitting expressions need to change as well. The quantitative correlation between the surface angle crack and Y displacement is obtained by fitting the curve, proving that it is a feasible analysis method.

By fixing the angle and width of the crack, the depth of the crack was increased from 0.1 to 0.6 mm. The identification of crack depth was discussed in the case that the detection point and excitation point are on the same side and opposite side. The change of crack depth will cause the change of D/λ ratio, so the depth detection will include a variety of ratios.

The detection point and the excitation point were (7, 3.5) and (0, 3.5), which constituted the excitation detection on the same side of the crack. The Y displacement component based on the detection point is shown in Figure 9A. The portion of the displacement from 2.40 to 2.58 μs was RW. The Y displacement from 4.5 to 5.5 μs indicates TRW-B. When TRW-B returns to the detection site, there is a time delay, and the Y displacement gradually decreases. It can be concluded that the deeper the depth is, the longer the propagation path along the crack. Most Rayleigh waves change into body waves, resulting in a decrease in the amplitude of TRW-B, which is more obvious with the increase of crack depth.

The maximum Y displacement of TRW-B and the peak-peak displacement of RW were fit to estimate the crack depth. The fitting result was shown in Figure 9B. The RW does not exhibit a monotonous change with increasing depth: it increases and then decreases slightly. The Y displacement decreases from −0.903 to −5.571 nm for TRW-B as the crack depth increases from 0.1 to 0.6 mm, which shows a negative correlation between displacement and crack depth. The expression of the TRW-B and RW fitting curve is: Y 1 = − 0.063264 + 0.185148 × exp ( − D 0.51517 ) (10-1) Y 2 = 0.1373 + 7.02 × D − 24.09 × D 2 + 35.7 × D 3 − 19.7 × D 4 (10-2) where Y ₁ is the maximum Y displacement of TRW-B, Y ₂ is the peak-peak displacement of RW, D is the depth of crack. The MSE of the fitting curves are 1.392 × 10^–12 (Y ₁ ) and 1.221 × 10^–11 (Y ₂ ), the fitting regression coefficient R² is 99.853% and 99.889%. The maximum error of TRW-B’s fitting data is 2.42% and that of RW is 0.38%. Therefore, TRW-B and RW is suitable for defect depth identification. And the higher the fitting correlation coefficient, the more accurate the fitting result.

To place the excitation and detection sites on both sides of the crack, point (8, 3.5) was chosen as the detection point. As the RW goes through the crack, the depth feature information will be included in the waveform. Detecting the Y displacement of TRW or TRW-B on the right side of the crack can provide the necessary information. Figure 10A shows the Y displacement components of the detection points at different times. TRW is the figure from 0.25 to 0.32 μs, and TRW-B is the figure from 0.39 to 0.45 μs TRW and TRW-B are two waveforms of transmitted waves that occur at distinct times. When stress propagates to the right boundary, the propagation direction is reversed, and the original negative stress becomes positive. The stress-related Y displacement is changed from a negative to a positive value, revealing a difference in displacement between the two waveforms.

These two waveforms were fitted into the same graph, as shown in Figure 10B. TRW’s displacement value increases from −0.1128 to −0.0328 nm when the crack grows from 0.1 to 0.6 mm, while TRW-B’s displacement value drops from 0.0826 to 0.0333 nm. The trend of these two waves is symmetrical, but there is a tiny difference in displacement because the stress is somewhat lost due to the resistance during propagation. Here are the two fitting expression: Y 3 = 0.0319678 + 0.0922522 × 0.0026 7 D (11-1) Y 4 = − 0.0340272 − 0.178835 × 0.000317 D (11-2) where Y ₃ is the extreme value of the Y displacement of TRW-B, and Y ₄ is Y displacement of TRW. The fitting curves were convergent, TRW-B with better fitting effect was introduced. Its MSE is 1.03764 × 10^–12, and the fitting regression coefficient R² is 99.591%. The maximum fitting error is 2.7% at a depth of 0.3 mm. TRW-B’s displacement is negatively associated with depth as it increases. The measured Y displacement and fitting the expression can yield the specific defect depth, demonstrating that the depth can be quantitatively detected by fitting the curve. TRW-B is therefore convenient for determining the quantitative depth of cracks on either the same or the opposite side. At the same time, it is also suitable for the D/λ range of 0.11 to 0.67 (depth of 0.1 to 0.6 mm).

The angle and depth of the crack were fixed, and the width was increased from 0.1 to 0.6 mm. The identification analysis of the width is still in the same and opposite cases. The ratio of defect depth to Rayleigh wavelength remained 0.34.

Point (6, 3.5) was taken as the detection point to keep the excitation and detection points on the crack’s left side. If the angle and depth of the crack remain unchanged, the structure impeding the propagation of RW will remain unchanged. With the increase in the crack width, the component of the RW converted to transmitted waves will be larger than the component converted to CRW. The Y displacement components at different times are shown in Figure 11A. The waveform at 3 μs is a direct CRW. The waveform at 5 μs is the right CRW, which is reflected by the boundary. To distinguish these two waveforms, they were called CRW-L and CRW-R, respectively. The Y displacement figure from 4.8 to 5.1 μs was magnified to make the relationship between CRW-R and width easier to see. Since the CRW-R wave will pass through the crack twice, the maximum Y displacement of the wave is selected for fitting, whose fitting result is shown in Figure 11B. The expression of the fitting straight line is: Y 5 = 0.0237416 − 0.0198169 × W (12) where Y ₅ is the Y displacement corresponding to CRW-R and W is the crack width. The fitting regression coefficient R² is 99.149%. The residual sum of squares is 5.0562 × 10^–13. When the crack width is 0.2 mm, the simulated value is 1.93936 × 10^–2 nm, and the fitted value is 1.977822 × 10^–2 nm, which is the farthest point from the fitted straight line with an error of 1.98%. CRW-R is linearly and negatively linked with crack width, according to the fit results. This negative correlation also exists at other inspection points on the left side of the defect.

The detection point was (8, 3.5) to keep excitation and detection points on both sides of the crack. The Y displacement components of (8, 3.5) at different times are shown in Figure 12A. The waveform of 2.5–3.0 μs in the figure corresponds to TRW, and the waveform of 4.0–4.5 μs corresponds to TRW-B.

The extreme displacement values of TRW and TRW-B are still selected for fitting, as shown in Figure 12B. The displacement of TRW increases from −0.0631 to −0.0369 nm as the crack width grows, indicating a positive correlation between displacement and width. TRW-B’s displacement falls from 0.0559 to 0.0413 nm, indicating a negative correlation. Y 6 = − 0.0740181 × exp ( − W 0.74738 ) + 0.00134864 (13-1) Y 7 = 0.0280565 × exp ( − W 0.032585 ) + 0.0354521 (13-2) where Y ₆ is the min Y displacement of the TRW and W is the width of the crack. Y ₇ is the TRW-B. TRW with better fitting effect was introduced here. The fitting curve’s fitting regression coefficient R² is 99.846%. MSE is 1.30148 × 10^–13. The maximum fitting error is 1.18% at a width of 0.4 mm. The essential crack width information can be obtained by detecting the TRW’s Y displacement extremes and substituting the calculation, which also proves that TRW can quantitatively identify the width while the D/λ is 0.34.