## Abstract

In many industrial fields, like automotive and painting industry, the thickness of thin layers is a crucial parameter for quality control. Hence, the demand for thickness measurement techniques continuously grows. In particular, non-destructive and contact-free terahertz techniques access a wide range of thickness determination applications. However, terahertz time-domain spectroscopy based systems perform the measurement in a sampling manner, requiring fixed distances between measurement head and sample. In harsh industrial environments vibrations of sample and measurement head distort the time-base and decrease measurement accuracy. We present an interferometer-based vibration correction for terahertz time-domain measurements, able to reduce thickness distortion by one order of magnitude for vibrations with frequencies up to 100 Hz and amplitudes up to 100 µm. We further verify the experimental results by numerical calculations and find very good agreement.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Thickness measurement techniques for thin layers in the micrometer range are in high demand in many industrial and research fields. Especially, in automotive and aerospace industry thin coatings between 5 µm and up to 200 µm are applied to a variety of substrates, e.g., metals, ceramics, plastics, papers, rubbers, and composite materials. Terahertz technology [1,2] offers a suitable solution for thickness determination of thin dielectric coatings [3–5] on all types of substrates. This non-destructive measurement technology is able to investigate non-conductive layers in multilayer systems [6–8]. In contrast to conventional tactile methods, the contact-less terahertz approach determines the thickness of cured as well as wet coatings with short measurement and analysis times of less than one second in total [9–11]. Owing to the low photon energy in the terahertz spectral range, this technology is especially interesting for industrial quality control, as radiation protection measures are not required.

Terahertz thickness measurement techniques based on terahertz time domain spectroscopy [12] reach a high accuracy of better than 1 µm, a high reproducibility and a large measurable thickness range of more than 200 µm for a single layer. To measure amplitude and phase of the terahertz signals, a sampling approach is used, inherently relying on fixed and stable distances between all optical components including the sample during the measurement. Obviously, in most industrial environments vibrations cannot be avoided, challenging the proper use of terahertz time domain spectroscopy system for high accuracy measurements. While accurate position information inside the terahertz time domain spectrometer has recently been realized by us [13], vibration between sample and measurement head present a real challenge. Even if samples are free of vibration, common measurement cycles for robot mounted measurement heads do not allow for position stabilization of the measurement head before the next measurement position has to be addressed. Hence, without vibration control, terahertz technology might not be able to realize its full potential.

Here, we investigate in detail the influence of vibrations between sample and measurement head and present a high-speed interferometer-based vibration detection with a numerical correction of the measured terahertz signals. We design an experiment, which allows to apply arbitrary vibrations onto the system to identify critical vibration frequencies. We find that measurement errors are especially pronounced, if vibration frequency and driving frequency of the time-delay coincide. Our interferometric correction reliably eliminates the influence of vibrations and increases the accuracy by at least one order of magnitude.

## 2. Layer thickness measurement using terahertz time-domain spectroscopy

Let us briefly recall the well known concept of terahertz time-domain-spectroscopy based thickness determination, schematically depicted in Fig. 1. A femtosecond laser pulse is split via a beam splitter into two parts. One part is guided to the terahertz emitter antenna, the other part to the detector antenna. In our case the terahertz antennas are LT-GaAs photoconductive switches [14]. The photoconductive antennas and the out-coupling and collecting mirrors for the terahertz radiation are placed in an external measurement head, connected with 5 m long umbilicals with all electrical and optical connections to the 19″ supply unit. Inside this unit in the detector arm, a delay-line is placed to generate delayed pulses to the detector to sample the detected terahertz signals (in our experiment a fast 20 Hz shaking mirror (shaker)). Precise timing between emission and detection of the terahertz signal as well as a fixed travel distance for the terahertz pulses between emitter and detector are required to allow deterministic reproduction of the full signal along these lines.

Measuring thickness with terahertz pulses relies on partial reflections of the incoming light at each interface between two materials. Of course, materials have to have sufficiently different refractive indices and should be transparent enough to allow the light to sample the whole stack of layers under investigation. Considering a single incident wave with the field *E(t)*, the intensity of the reflected wave is described by *I*_{r} (*t*) = *R*·*I*(*t)* with *R* being the reflectance of the interface, 0·< *R* < 1. The reflectance *R* can be calculated from the amplitude reflection coefficient $R={r}_{1,2}^{2}$, which is derived from the Fresnel equations under perpendicular incidence:

*n*

_{1}and

*n*

_{2}represent the complex refractive indices of the involved materials. The fraction of transmitted radiation with wavelength

*λ*

_{0}and amplitude

*E*

_{0}is damped with an exponential decay while traveling through the material with thickness

*d*and extinction coefficient

*κ*. Using a metal substrate the transmitted part is almost completely reflected by the substrate with complex refractive index $N=n+i\kappa =\sqrt{\u03f5}=\sqrt{{\u03f5}_{1}+i{\u03f5}_{2}}$ and ${\u03f5}_{1}\ll 0$, ${\u03f5}_{2}<{\u03f5}_{1}$ and therefore,

*R*≈ 1. In the case of a single-layer sample two primary peaks are received with a time difference of $\Delta t=2\frac{dn}{c}$, which is the travel-time through the material. For a well-known refractive index the thickness of the layer can be easily determined by ∆

*t*. For thin layers more sophisticated approaches have to be used to determine the layer thicknesses from the time-domain signal [4,9], as the reflected pulses from the different interfaces strongly overlap and multiple reflections have to be taken into account.

## 3. Spectral influence of vibrations and correction-approach

While vibrations in principle might affect all relevant distances in the setup, rigid construction of the TDS unit allows to exlude vibrations inside the spectrometer. Additional control of the shaker’s position takes care of any remaining movement errors [13]. Hence, we will focus on vibrations influencing the distance between measurement head and sample, which principially are hardly to avoid for a robot-mounted measurement system even if the sample itself is in rest. In this case the travel-time between emitting the terahertz pulse and receiving the material response in reflection is not constant. Instead, the difference in distance introduced by the movement of the sample translates directly to a delayed arrival of the response pulse. The vibration either stretches or compresses the measured sampling signal locally, depending on the movement of the sample, see Fig. 2. The red curve displays the reflected pulse for a resting sample and the blue and green curves describe the reflected pulse for the other two cases: movement in and in opposite direction of the emitted terahertz pulse. The resulting change of the pulse shape due to the sample movement is clearly visible. This effect is solely due to the sampling principle and is not to be confused with the Doppler effect.

In order to reproducibly introduce vibrations commonly found in real industrial environments, the samples to be measured are mounted on a piezo-driven sample holder (see Fig. 3). Along these lines we are able to induce vibrations with adjustable amplitudes of up to 100 µm and frequencies of up to 100 Hz. The piezo module is mounted on a micrometer manual translation stage for adjusting the sample in the correct working distance to the terahertz transceiver head. The working distance of the transceiver depends on the used mirrors in the terahertz path and is set to 100 mm. All investigated samples were automatically measured with amplitudes from 0 µm to 100 µm and 0 Hz to 100 Hz in 1 µm and 1 Hz steps, respectively.

As a first example we study the influence of vibrations onto the water absorption lines recorded with just a single metal plate as reflecting sample. The results are depicted in Fig. 4(a) for 99 Hz vibration frequency with a maximum amplitude of 100 µm and after 1 s integration time. The spectrum of the vibrating sample (red) shows strongly distorted absorption lines compared to the signal of the resting sample (black) for both cases, single-shot measurement (top) and 40 times averaged data (bottom). With lower vibration frequencies and amplitudes, the strength of the distortion decreases (not shown).

In order to correct the influence of the vibration, we measure the vibration induced sample displacement using a high-speed displacement measuring interferometer (DMI) with sub-micrometer resolution [15]. The two mirrors required in a conventional interferometer are replaced in our fiber-based interferometer by the interface between fiber and air and the object itself. As two wave interference is desired, even objects with low reflectance act as mirror. For convenience, the wave reflected from the object is called sample wave while the wave which is immediately reflected by the end of the fibre is referred to as reference wave, as it does not travel to the object. The schematic setup of the interferometer is shown in Fig. 5. When the object moves along the optical axis, the optical path length for the sample wave changes, resulting in a phase shift between reference and sample wave. As the sample wave passes this distance twice, a change of the resonator length by λ/2 shifts the phases by a full period. Therefore, counting the occurring interference maxima while moving the object determines the displacement with a resolution of *λ*/2. The birefringent crystal adds polarization information to the interference patterns to discern the direction of movement.

The DMI is placed inside the terahertz supply unit while the output of the interferometer is guided via an optical fiber to the measurement head and coupled out by a collimator, normal to the sample surface. The DMI works at 1530 nm with an optical power of below 1 mW and provides a measurement bandwidth of 10 MHz. The sample vibrations are detected with a sampling rate of 200 kS/s and a resolution of 1 pm. The obtained positioning information is subsequently used for correcting the time axis of the measurement. As the obtained data describe the spatial displacement they can be easily transferred to a temporal displacement *t*_{vib}. Adding this data to the time axis from the shaking mirror *t*shaker the resulting time axis is non-equidistant: *t*_{dist} = *t*_{shaker} + *t*_{vib}. Starting with this non-equidistant time axis a cubic spline interpolation is performed to obtain an equidistant time axis again: *t*_{dist} ↦ *t*_{linear}. After this process, the resulting terahertz pulses do not contain the distortions due to the vibrations anymore.

The successful correction can be seen in Fig. 4(b). All distortions introduced by the vibration are corrected and all water absorption lines can faithfully be reproduced, reaching almost perfect agreement between the data taken with the resting sample (see inset for a magnification).

## 4. Vibration correction for thickness determination

To evaluate the vibration correction, we investigate different samples: two single-layer samples with identical coating but different thicknesses and a common automotive four-layer sample with e-coating, filler, base-coating and top-coating. Furthermore, the samples are measured without any averaging (measurement time of 25 ms) as well as with 1 s integration time to demonstrate the impact of the integration time on the vibration sensitivity. Table 1 summarizes the specifications of the samples.

For all samples, vibration frequency and amplitude are nominally modulated between 0 Hz and 100 Hz and 0 µm and 100 µm, respectively. To evaluate the influence of the vibrations, we plot the discrepancy between the measured thickness of the resting sample and the vibrating sample. These deviations are plotted as 2D color-code figures. Note that the piezo stage with the mounted sample is not able to actually induce vibrations with the full nominal amplitude range of 100 µm. Therefore, the edges of the parameter space are warped for the samples (see Fig. 6), as we plot the data for the actual vibration amplitude of the sample, received from the interferometric measurement.

We start our discussion with measurements on single-layer sample Single01 with a nominal thickness of *d*_{1} = 23.2 µm. Fig. 6 depicts the measured deviations for single-spectrum measurements (top row) and for an integration time of 1 s (bottom row). For single-spectrum measurements the deviations increase from less than 0.1 µm to up to 4 µm with increasing amplitude and frequency (see Fig. 6(a)). A prominent deviation is observed for multiples of 20 Hz vibrational frequencies. As the shaker moves with 20 Hz, a sample vibration of 20 Hz results in a fixed phase relation between the shaker movement and the sample vibration for the full measurement period.

The general deviations are reduced for increased averaging (see Fig. 6(c)) by roughly a factor of 2, except the feature at multiples of 20 Hz: although the results are averaged over a multitude of single measurements, all single measurements experience the same distortion because of the fixed phase relation, which counteracts the otherwise positive impact of the averaging process. Even with averaging the deviations are above 1 µm.

Applying the vibration correction, deviations are reduced to the noise level of a resting single-layer sample for the single-spectrum measurements (see Fig. 6(b)). Averaging the measurements, the noise is further reduced (0.03 µm), resulting in correct thickness determination even though vibrations are present (see Fig. 6(d)).

In the following, we will only discuss averaged measurements. For the thicker single-layer sample Single02 we find without vibration correction and disregarding the feature due to the frequency match deviations less than 0.3 µm for the complete parameter space (see Fig. 7(a)). Applying the vibration correction, again, all deviations are perfectly eliminated (Fig. 7(b)). The much better performance even for the uncorrected case can easily be understood: while for the thinner sample, several peaks from multiple reflections are measured, for the thicker sample just few reflections are recorded. With less reflection peaks contributing to the analysis, the influence of a distorted time-base is obviously reduced for thicker samples. One might assume now that reducing the scan-range for thinner samples might improve the data quality. This is definitely not the case, as for thinner samples reflections peaks are not well separated any longer and the additional information of the multiple reflection peaks is essential for correct thickness determination. As multiple reflections dramatically increase for multi-layer samples of thin layers as commonly found in automotive industry, terahertz thickness determination under the influence of vibration will be rendered quite useless as can be seen in Fig. 7(c). We evaluate the thickness deviation for each of the four layers and only plot the maximal deviation out of the four calculated ones. This demonstrates the much stronger influence of vibration onto thickness determination. Already for an amplitude of 50 µm and a frequency of 50 Hz the distortion is about 4 µm and increases to more than 10 µm for the full amplitude (note the truncated color scale). In particular, layer 1 (e-coating) and layer 2 (filler) are the most sensitive layers for this sample. This is illustrated in Fig. 8, in which the layer number with the maximum distortion is shown, used to generate the deviation plot in Fig. 7.

The vibration correction reduces the full-range deviation of the dominant layers 1 and 2 from about 3.0 µm to only 0.27 µm (not shown) and the maximal deviation from 10.1 µm to 0.8 µm. Without vibration correction, the results are useless for quality control. The higher deviations of the four-layer sample compared to the single-layer samples result from the complexity of the layer system leading to multiple overlapping reflections and, hence, drastically increased sensitivity to vibrations. Even the resonant feature at 60 Hz and 100 Hz is still visible after correction. Furthermore, with and without correction, increasing distortion values for constant frequencies and increasing amplitudes are observed. This observation can be explained in the following way: for each amplitude sweep, the distortion is calculated in relation to the thickness results of the resting sample with zero amplitude for each sweep. In case of varying temperature and humidity during the sweep, the different conditions may result in slightly different thickness values, especially for complex four-layer samples. To overcome this limitation in the real application, the environment conditions have to be constant or it is necessary to capture references more often.

Overall, the thickness determination with the terahertz time-domain spectrometer can be significantly improved with the presented vibration correction method for all investigated samples. In Table 2 the full-range deviations for the uncorrected as well as the corrected case are summarized. The deviation can be reduced by one order of magnitude for all samples, except the thick single-layer sample with an already very low deviation in the uncorrected mode.

## 5. Numerical calculations and vibration correction of measured data

In order to evaluate the experimental results, the experimental setup including terahertz pulse generation, water absorption, vibration, and noise is numerically calculated.

The generation of a terahertz pulse is based on the Drude model and can be described by the carrier density *n* (*t*)

*G*∝

*I*being the carrier generation rate proportional to the laser intensity

*I*and

*τ*

_{c}being the carrier trapping time.

After generating a numerical ideal terahertz pulse the environmental influences, in particular caused by the water absorption in air, are added. Detailed information about the absorption characteristics of water can be found in the freely available HITRAN (acronym: High-resolution transmission) database [16]. The database provides the spectral position *ν*_{0} of water absorption lines and the corresponding absorption intensity *S* as well as the pressure dependent parameters for air broadening *γ*_{air}, self broadening *γ*_{self}, and the pressure shift correction parameter *δ*_{air}. For every transition line, a Lorentzian profile is assumed to describe the absorption as

*γ*corresponds to the half width at half maximum (HWHM) of the Lorentzian profile, including both broadening coefficients, and ${\rho}_{{H}_{2}O}$ is the density of water molecules in air [17].

After adding the water absorption, the material response to the impinging terahertz pulse is determined based on the transfer matrix method of the layered system including the material parameters (refractive index and extinction coefficient) as well as the thickness of each layer [4]. The sample vibration is assumed to be a sinusoidal movement with constant frequency and amplitude as well as a random phase. To consider this vibration, the sinusoidal function ${t}_{\text{vib}}={A}_{\text{vib}}\xb7\text{sin}(2\pi {\nu}_{\text{vib}}t+\phi )$ describing the vibration is added to the linear time axis *t*shaker resulting in a non-equidistant time axis: *t*_{dist} = *t*_{shaker} + *t*_{vib}. A cubic spline interpolation of the terahertz pulse is calculated based on the new time axis: *t*_{linear} ↦ *t*_{dist}. Then, the interpolated terahertz pulse contains the information of the vibration with a new equidistant time axis. Finally, white noise is added to the terahertz pulse in time domain to achieve a dynamic range comparable to measured signals.

For many applications, especially for thickness determination an averaging is provided to improve the dynamic range and the bandwidth of the resulting terahertz pulses. In order to consider the averaging process, a well-defined number of pulses with different noise and phase are averaged after alignment of the pulse maxima. For a measurement time of 1 s in total 40 pulses are averaged and the dynamic range is improved by 16 dB compared to a non-averaged signal. In Fig. 9(a) the calculated thickness errors of the thin single-layer sample are shown and demonstrate a high qualitative agreement with the measurement results in Fig. 6(c). In addition, the results for the four-layer sample (Fig. 9(b)) indicate comparable qualitative results with the experimental data (see Fig. 7(c)). The overall behavior is confirmed by all calculations. Furthermore, the resonant features at multiples of 20 Hz shaker frequency are also present in the calculated data. In order to reduce the effort to experimentally measure the vibration influences of different coatings, the numerical tool can be used to obtain a forecast.

## 6. Conclusions

In conclusion, we demonstrated a vibration correction method for terahertz time-domain spectrometers. The presented results indicate the need of vibration correction for high precision thickness determination with terahertz time-domain spectroscopy systems. The investigated influences of vibrating samples to the spectral properties, illustrated with the water vapor lines, clearly show also the need of vibration correction for spectroscopy applications in industrial environments. With interferometric displacement measurement of the sample to be measured, the negative impact of vibrations can be completely eliminated. Hence, the thickness determination of multilayer systems, applied in many industrial fields, can indeed be realized with high accuracy in industrial environments. In order to prove the presented vibration correction single- and four-layer samples with induced vibrations of up to 100 µm and 100 Hz were investigated. We are able to reduce the thickness errors from more than 10 µm to the standard deviation of a resting sample for a four-layer system, opening the way for industrial application of terahertz thickness measurements.

## Funding

BMBF via Eurostars project INSPECT (E! 10823).

## Acknowledgments

The authors thank the German BMBF for funding via the Eurostars project INSPECT (E! 10823).

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