In this section, the experimental THz-TDS setup and sample are presented, followed by a compact overview of the scanning electron microscopy (SEM) measurement and analysis. The COMSOL simulation of the THz-TDS measurement is introduced. The results of the application of the algorithms on simulation and on experimental data are compared with the reference values.
3.1. Experimental Setup and Used Sample
The investigated sample was an Inconel 738 substrate with metallic bond coat and YSZ layer manufactured with electron beam physical vapor deposition. The sample represents the layer structure of a turbine blade and was provided by Siemens Technology (Munich, Germany). It has been investigated in a previous study [
16]. The sample comprises four steps (“6 mils”, “7 mils”, “9 mils” and “11 mils”) with different YSZ thicknesses ranging from nominal
mil to
mil or from
to
in SI units. (Mil
inch. The manufacturing parameters were specified in imperial units.) The exact manufacturing parameters are unknown. The sample is visible in
Figure 3.
The experimental data in this study were gathered with the TERA ASOPS THz-TDS system manufactured by Menlo Systems (Martinsried, Germany) combined with two TERA15-FC antennas as emitter and receiver from the same manufacturer. This system uses the Asynchronous Optical Sampling (ASOPS) technique utilizing two mode-locked lasers emitting femtosecond pulses at
with fixed repetition rate of
and tunable phase difference. One laser is used for excitation, while the other is used for detection. This technique does not require mechanical delay stages. The laser pulses are delivered to the antennas via optical fiber. Generation and detection of THz pulses is based on the principle of the superconductive switch (Auston switch [
5]). The system generates linearly polarized THz radiation with a bandwidth of
and a (THz) pulse energy of approx.
. Lenses (TPX35) focus the pulses on the target surface with a diameter of approx.
at full bandwidth.
The THz-TDS system was set up for reflection mode measurement with angled incidence at approx.
to the sample normal. The linearly polarized pulse had an angle relative to perpendicular polarization of approx.
. The setup is shown in
Figure 3. The measurement was performed in a laboratory environment without evacuation or dry air/nitrogen purging.
Following the previous comparison of pulsed thermography and THz-TDS measurements on the sample by Frisch et al. [
16] and within a second, forthcoming study by Frisch et al. [
23], a cross-section cut of the sample was prepared and analyzed with SEM. The SEM measurement data were extracted from [
23] and were used as the reference data for the validation of the presented adaption of the method proposed by Fukuchi et al. The SEM images were captured with the measurement software InTouchScope on the SEM JEOL JSM-6010 Plus (JEOL Ltd., Tokyo, Japan). Parameters were set at
acceleration voltage, and high-vacuum and backscattered electron images were recorded.
Figure 6 shows an example SEM scan.
3.2. SEM Image Analysis
The SEM images were analyzed and the thicknesses determined by visually averaging the air-TBC and TBC-BC interfaces and extracting the layer thickness with the SEM imaging software. Due to problems with charging of the sample during the SEM measurements, only one measurement could be performed for each of the four thickness steps. Since a statistical error cannot be determined, the uncertainty of the measurement was estimated as .
The porosity analysis was performed with the image processing software ImageJ. To determine the porosity of the samples, the images were segmented using the modified IsoData-Algorithm [
24]. For each sample, a polygon selection that covers a large part of the TBC area was traced. From this, the porosity was calculated as an area fraction. To determine the uncertainty of the measurement, for each SEM image, the porosity was determined in five arbitrary placed
squares separately. The standard deviation of these values is used as the uncertainty of the porosity measurement. The results are collected in
Table 1.
Watanabe et al. [
25] provide a thorough investigation of the dielectric properties of plasma-sprayed YSZ thermal barrier coating in the THz regime of 0.1–
for a range of porosity in the microstructure. In the study, they found a high transmittance of frequencies around
, falling to almost zero at
. We can confirm this frequency range for our experimental data (see
Figure 2). Watanabe et al. also provide measurements for the complex refractive index of YSZ layers in relation to the porosity ranging from bulk material (no porosity) to 25% porosity. A comparison between the real part of these results and the calculated effective refractive indices from our measurements is shown in
Figure 7. The time-of-flight (ToF) calculation method used to determine the refractive index for the TDS data is presented in
Appendix A.5. The uncertainty of the calculated refractive indices is determined via the propagation of uncertainty of Equation (
A26) with errors for
h (compare
Table 1) and the error for the time difference between pulses
estimated as
.
The values for the refractive index of YSZ measured in this study are 10–15% lower than those of [
25] and do not reproduce the expected inverse relationship in relation to the porosity. A possible reason could be a higher statistical spread of the porosity measurement via SEM images. For the following simulations, the averaged refractive index of
was used.
The extraction of the surface roughness of the air–TBC interface was performed with a simulated probe tip measurement. First, the TBC interface area for each sample was extracted and segmented with the IsoData-Algorithm. Then, small particles (
) were removed from the images. This was done to avoid a false surface detection in the following step. The resulting image was imported into MATLAB to calculate the surface height profile
. To mimic real surface roughness measurements with a scanning probe, for each lateral position
x, the height
of the first material pixel coming from the exterior towards the TBC-layer was registered. The width of the virtual tip was set to
, which is comparable to real measurement probes for the observed roughness range [
26]. To get the final surface profile
, the constant offset is subtracted
with
being the average of the surface profile. From this, the root mean square (RMS) surface roughness
is calculated.
The results for the surface roughness calculation are listed in
Table 1. As uncertainty of the measurement, the standard deviation of the four measurements of
is used. The roughness of the TBC-BC interface was not calculated in this study since this parameter is not relevant in the roughness correction presented in
Section 2.4.
Since the same manufacturing technique was employed for the different coating thickness steps, the surface roughness of the sample areas should also be comparable. To reduce the possible statistical spread of the SEM measurement, the average roughness of
is used in the roughness correction calculation for the measurement data in
Section 3.6.
3.3. COMSOL Simulation
To simulate THz-TDS, the experimental setup was recreated in COMSOL using the transient electromagnetic waves (ewt) interface with a 2D model. The geometry is shown in
Figure 8. The wave is excited at the left angled boundary (
) in the form of a prescribed electric field with both in-plane (parallel) and out-of-plane (perpendicular) polarization. The lenses focusing the beam in the experimental setup are not simulated. Instead, a plane wavefront with the approximate width of the focus spot of the experimental setup was used. The pulse shape is extracted from the first reflection of the thickest sample in the real measurement data. The extracted pulse is linearly windowed to zero at the edges to avoid discontinuities. The shape of the simulation was chosen to minimize the geometric size (simulation time) by ensuring that the center of sender and receiver point to the middle of the TBC interface, thereby maximizing the illumination of the interface. Two geometric domains are present: air on top and the TBC material below. Porosities in the coating were not modeled; instead, a bulk material with an averaged refractive index calculated in
Section 3.2 was used. This is essentially a simple effective medium approach. Rigorous effective medium models have been previously applied to YSZ coatings in [
27]. Since the proposed method only uses the real part of the refractive index, the modeled material parameters are real-valued. This means that absorption effects are not simulated. Furthermore, the modeled material parameters are not frequency-dependent. This is a reasonable simplification, since the real part of the refractive index of YSZ is fairly constant in the investigated frequency range [
27]. The outer boundaries are set as high absorption scattering with two exceptions: the sender, which has a scattering boundary condition without absorption, and the TBC–BC interface, which is set to perfect electric conductor. The width of the THz pulse is approx.
, which translates to an illumination projection size of about
. For meshing, a free triangular mesh with minimum element distance of
the minimal relevant wavelength (
) in the respective domain was chosen.
In this study, two types of simulations were executed. The first simulated the measurement with flat interfaces for both TBC and BC. Here, four different thicknesses were considered, equal to the SEM measurement of the thickness in
Section 3.2 and refractive indices according to
Appendix A.5. These simulations were used to verify the angle-correction method. The results are discussed in
Section 3.4. The second type of simulation used rough interfaces for TBC and BC with varying roughness to verify the correction presented in
Section 2.4. Here, the thickness and refractive index is kept constant (parameters as in flat simulation for “7 mils”). The rough interface was generated by adapting an algorithm presented in [
28], allowing the creation of random rough curves with specified RMS roughness
.
Table 2 shows an overview of the simulations and used parameters.
A comparison between real measurement data and simulated data of the same sample is shown in
Figure 9. The distances between reflections show good agreement for both simulations compared to the measurement data. Pulse amplitudes show some discrepancies, especially for the simulation without roughness. Here, lacking losses from absorption or interface roughness, the reflection at the second interface shows a higher amplitude than the reflection at the first interface. This is consistent with expectations, since the second interface allows no transmission. The simulation with roughness shows reduced amplitudes of consecutive reflections, stemming from the diversion of parts of the beam energy away from the receiving element. Here, a slight pulse widening is also visible, caused by stronger scattering for higher frequency components. The amplitudes for the measurement data show a stronger decay with only three reflections visible. This is mainly caused by the absorption in the coating, which was not simulated in this work. For the experimental data, the pulse widening is also stronger, which is a result of a strong frequency dependence of the imaginary part of the refractive index in the THz range for YSZ [
25].
3.4. Simulation: No Roughness
The results of the COMSOL model without roughness are shown in
Figure 10 and in
Table 3 for different states of polarization. The thicknesses are calculated without angle corrections for the refractive index calculation (but with correction for ToF) and with angle corrections for refractive index and ToF for three polarization states: parallel, perpendicular and linear polarization with a rotation of
relative to perpendicular polarization (see
Section 2.2). The latter signal was constructed from both the parallel and perpendicular data through superposition.
For all polarization states and thicknesses, the modified method shows good agreement with the reference values. The average error is 1.3%, compared to 10.6% for the unmodified method.
3.5. Simulation: With Roughness
The results for the four COMSOL simulations with rough interfaces are shown in
Figure 11 and
Table 4. Here, the unmodified method by Fukuchi et al. shows an average error of 15%. The highest deviation is observed for the case of parallel polarization and the lowest for perpendicular polarization. For the combined polarization case, the results depend on the difference between the roughnesses of the two interfaces.
The modified method without roughness correction shows an average error of 8%. As in the case for the unmodified method, the error is large for differing interface roughnesses.
The results for the modified algorithm with roughness correction show an error of under 1% on average with a maximum absolute error of . This shows the importance of the consideration of surface roughness in THz-TDS measurements.
The influence of the correction on the calculated refractive index is plotted in
Figure 12. Without correction, the roughness effects lead to increasing deviations in the calculated refractive index for higher frequencies and therefore smaller wavelengths. The roughness correction factor increases with the frequency and keeps the refractive index almost constant for a wider range.