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Abstract
Recently, many research groups worldwide have reported on the THz properties of liquids. Often these parameters, i.e., refractive index and absorption coefficient, are determined using liquids in cuvettes and terahertz time-domain spectroscopy. Here, we discuss the measurement process and determine how repeatable such measurements and the data extraction are using rapeseed oil as a sample. We address system stability, cuvette positioning, cuvette cleaning and cuvette assembly as sources affecting the repeatability. The results show that system stability and cuvette assembly are the most prominent factors limiting the repeatability of the THz measurements. These findings suggest that a single cuvette with precise positioning and thorough cleaning of the cuvette delivers the best discrimination among different liquid samples. Furthermore, when using a single cuvette and measurement systems of similar stability, the repeatability calculated based on several consecutive measurements is a good estimate to tell whether samples can be discriminated.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Terahertz (THz) time-domain spectroscopy (TDS) has become a standard method not only to study ultrafast processes in matter [1–6] but also to determine material properties between 100 GHz and a few THz [7–13]. Not only the hardware of THz systems made tremendous progress during the past 15 years [14–16] with THz systems reaching over 90 dB of dynamic range [17,18] but also the algorithms for parameter extraction from the experimentally obtained data became more sophisticated over the years [19–22]. Yet, one important question is how accurately and precisely material parameters can be determined. This question can be addressed in terms of measurement uncertainty, reproducibility and repeatability. So far, several studies have addressed the metrological analysis of THz time-domain spectrometers [23–27]. Here we assess measurement repeatability for the material extraction of liquids in the THz range.
1.1 Measurement uncertainty
The measurement uncertainty is a quantitative measure of how reliable a measurement is. Therefore, it is the basis for comparing measurement results and judging their meaning, e.g. for interoperability of components, for fair trade, for judging product specifications or for ensuring safety measures. A measurement result consists of two components, namely the best estimate for the measured quantity and the measurement uncertainty. The measurement uncertainty specifies the half width of the interval, in which the (still unknown) true value lies with a specific probability. Assuming a Gaussian distribution of measurements in general, it is referred to as standard measurement uncertainty (k = 1), if the probability corresponds to the 1σ value of ≈ 68%. Often, an expanded measurement uncertainty is specified by multiplying the standard measurement uncertainty with an extension factor of k, if a higher level of confidence is required (e.g. k = 2 for the 2σ value of ≈ 95%). The quantification of the measurement uncertainty is described in the Guide to the Expression of Uncertainty in Measurement [28]. It involves a detailed analysis of the measurement process, e.g. by a model equation that relates the measurement result to a number of input quantities with known measurement uncertainties. The measurement uncertainties of the input quantities can be specified based on statistical analysis (type A – the standard measurement uncertainty corresponds to the standard deviation of the mean of a data set) or other knowledge about the measurement process (type B – the standard measurement uncertainty corresponds to the square root of the variance of a probability density function that can be assumed). Typical type B contributions are specifications from product data sheets or from calibration certificates. A calibration certificate specifies an expanded measurement uncertainty (k = 2) assuming a Gaussian probability density distribution of the calibrated quantity. By propagating the measurement uncertainties of the input quantities through the model of the measurement process (e.g. by Gaussian error propagation using partial derivatives of the model equation if the model can be linearized or by using Monte-Carlo analysis) the measurement uncertainty of the output quantity can be determined. In case the input quantities have different probability density functions but there are many of them and the measurement uncertainties are of comparable magnitude, the central limit theorem of statistics leads to a Gaussian probability density distribution for the output quantity. By establishing an unbroken chain of calibrations with known measurement uncertainty between a specific measurement and the representation of the SI units at the National Metrology Institutes (e.g. NIST in the US, NPL in the UK, PTB in Germany, KRISS in Korea, and many others), metrological traceability can be established [29]. This is the basis for reliable measurement results.
In case of dielectric material parameter measurements of liquids using cuvettes in THz time-domain spectrometers, a full assessment of the measurement uncertainty is rather elaborate. Figure 1 depicts the measurement process that maps the dielectric properties of an oil sample to the best estimate for the dielectric properties together with its measurement uncertainty. As the measurement result is extracted from transmission measurements of a cuvette (and reference measurements without cuvette), not only these contribute to the overall measurement uncertainty but also the measurements that were used to characterize and model the cuvette before. Therefore, a multitude of unknown systematic errors and statistically occurring errors are conceivable. Unknown systematical errors might arise from insufficient modeling of the cuvette (e.g. neglecting higher order reflections, surface scattering and beam properties), truncating, windowing or filtering of measurements, misalignment, remaining humidity in the nitrogen atmosphere and/or thermal drift. Additional statistically varying contributions could arise from noise (e.g. laser noise and electronic detector noise) and electronic interferences and acoustical vibrations. Earlier efforts to evaluate the measurement uncertainty have shown that it depends strongly on spectrometer type and the properties of the sample [30]. For instance, the sample thickness has to be chosen carefully considering the spectrometer capabilities and the refractive index, absorption coefficient and homogeneity of the sample. An international comparison on dielectric material measurements has shown that many research groups have problems with reliable data extraction and underestimate the measurement uncertainty of their measurement results [31].
A comprehensive analysis of the overall measurement uncertainty of THz TDS on liquids in cuvettes is beyond the scope of this paper. Instead, we analyze how stable the measurement result is with regard to repeating measurements. This provides a lower limit for the measurement uncertainty, only, as it does not account for certain unknown systematic errors. However, this repeatability analysis is still useful as it indicates, which differences in dielectric properties of samples can be resolved from measurements.
1.2 Reproducibility and repeatability
Measurement precision, in contrast to measurement uncertainty which includes accuracy, is defined as closeness of quantity values repeatedly measured on the same or similar object under specified measurement conditions [32]. It is usually expressed by measures of imprecision (e.g. standard deviation). Furthermore, the precision determined in this way, can be understood as repeatability or as reproducibility, based on the measurement conditions. Repeatability is the precision for a set of measurements where every measurement is done on the same location, with the same operators and on the same objects using the same measuring system and procedure [32]. In case of reproducibility, the conditions differ more substantially. The conditions include different locations, different operators and different measuring systems.
In this paper, we address measurement precision of the dielectric parameters extracted from the performed TDS measurements in terms of repeatability, as all the measurements discussed here were performed by the same operators, on the same location, using the same measuring system. We present repeatability as an interval equal to two times the standard deviation (2·1σ) centered on the mean value and it can be understood as an interval within which the measurements are repeatable. We refer to it as the repeatability interval and it is important when comparing values to each other, which were measured and determined in the same way using the same system [33].
When studying dielectric materials, mostly one material layer is considered and the data extraction is straightforward [7,34]. Liquids, however, are typically measured inside a cuvette with two windows. Hence, this configuration represents a three-layer system. The dielectric THz properties of all three layers cannot be determined by a single sample and reference measurement. Therefore, the THz properties of the two windows must be determined for each window separately in advance. Only then the cuvette is assembled and filled with a liquid. The only layer with unknown properties to be determined by the THz TDS measurement is then the liquid layer (see [35] for details).
It has been shown that this approach can be employed to detect small levels of water contamination in engine oil [36] or to distinguish gasoline engine oils of different viscosities [37]. However, the differences in the dielectric parameters due to water contamination or different viscosity are very small. Therefore, the measuring procedure has to be highly repeatable. Hence, the question arises with what repeatability such measurements can be carried out. Between the measurements of the different liquids the cuvette needs to be cleaned, refilled and positioned into the beam path. It could also occur that the entire cuvette needs to be disassembled and reassembled.
In this paper, we experimentally investigate how these procedures affect the repeatability of the measurements. At first, we investigate whether small, random variations in the angular alignment when inserting the cuvettes in the spectrometer affect the measurement repeatability. Secondly, we investigate the effect of emptying, cleaning and refilling the cuvettes; with the same liquid but a fresh portion of it. Of course, this also involved repositioning the cuvettes. Finally, all these measurements were performed using five independently assembled and characterized cuvettes. By comparing the extracted material parameters from different cuvettes, we assess the effect of cuvette property variation and assembly on the repeatability of material parameter extraction.
All in all, with these measurements we address four experimental parts affecting the repeatability of the parameter extraction: (3.1) system stability, (3.2) positioning of the cuvettes, (3.3) cleaning of the cuvettes, and (3.4) cuvette assembly and characterization. Next, we will present the experimental details followed by the results of each of these experimental parts, followed by the conclusions.
2. Experiment
We used a commercial rapeseed oil as a sample liquid to test the effects of different experimental parts on the repeatability of the complex refractive index extraction. The oil was measured in cuvettes with sample volume of approximately 4 mL in a free-space THz TDS system as it is explained in this section.
2.1 Cuvettes employed
We constructed five cuvettes as schematically depicted in Fig. 2. Each cuvette is composed out of two low-absorbing fused silica glass windows and a metal holder. First, to characterize each cuvette, the transmission of all individual glass windows was measured. With these results, the complex refractive index and the thickness for each window were determined using the total variation algorithm for single-layer samples [21]. Afterwards, the windows were glued to the metal holders forming rectangular cuboids with approximate dimensions: 25.4 mm × 27.4 mm × 5.9 mm as shown in Fig. 2. Finally, the transmission through the empty cuvettes was measured and, in this way, the width of the liquid reservoir was determined using the total variation algorithm for multilayer samples [35]. The assembled and characterized cuvettes were placed on a caddy, which was mounted on a motorized stage within the experimental setup used.
Fig. 2. Schematic drawing of a cuvette consisting of two fused silica glass windows glued on a metal holder.
2.2 Experimental setup
A custom-built THz TDS system (Fig. 3) was used to measure the complex refractive index of the rapeseed oil. The key parts of the system are the frequency-doubled erbium-doped femtosecond fiber laser emitting at 780 nm, a pair of GaAs-based photoconductive antennas, and a mechanical delay line. A bias voltage applied to the emitter antenna was modulated and a lock-in technique was used to measure the THz electric field detected by the receiver. The generated THz radiation was first collimated by an off-axis parabolic mirror and after passing through a sample it was focused on the receiver antenna by a second off-axis parabolic mirror. All measurements in this study were performed under nitrogen atmosphere using the same procedure, which is described in the next subsection.
Fig. 3. Experimental setup composed of a pair of photoconductive antennas, a delay line and a femtosecond fiber laser emitting at 780 nm. The laser beam forms the optical path as depicted by the red lines. The blue region represents the THz radiation that impinges one of the five cuvettes.
2.3 Measurement procedure
The five cuvettes mounted on the caddy were filled with the sample oil and placed on the motorized stage in the THz TDS system. Each cuvette was positioned into the beam path by moving the computer-controlled translation stage after taking a reference measurement (no cuvette in the THz beam path). As a result, from each pair of measurements, it was possible to retrieve the complex refractive index of the oil, having the cuvette already characterized. This was repeated five times for each cuvette, resulting in 25 reference and 25 sample measurements, which we refer to as a measuring sequence. In total, ten measuring sequences were performed.
Between the first five measuring sequences the caddy with cuvettes was just taken out of the TDS system and placed back in (no cleaning). This can introduce small angular misalignments affecting the repeatability of the measurements. After these first five measuring sequences, the cuvettes were reassembled and newly characterized. Additionally, between the last five measuring sequences, the caddy was taken out of the system and the cuvettes were emptied, cleaned, and just then, measured empty. Then the cuvettes were refilled with a fresh portion of the same oil to observe the effect of cleaning on the repeatability. We cleaned the cuvettes using different solvents, cotton swabs and lens whips to get rid of any oil stains on the cuvette windows.
3. Results and discussion
3.1 System stability
The parameter extraction results for the first measuring sequence are shown in Fig. 4. The lines represent the frequency dependent mean of the refractive index and absorption coefficient for individual cuvettes. The frequency dependent error bars correspond to the sample standard deviation (1σ) and span the frequency dependent repeatability interval of 2·1σ centered at the mean. The standard deviation is defined as
Fig. 4. The effect of system stability on the repeatability. Average refractive index and absorption coefficient for all five cuvettes independently for a single measuring sequence. The error bars correspond to ± 1σ of five parameter extractions.
The sample’s refractive index decreases with the frequency from 1.516 at 0.3 THz to 1.498 at 1.2 THz and the absorption coefficient increases with the frequency from 2 cm-1 at 0.3 THz to approximately 11 cm-1 at 1.2 THz. The repeatability interval (2·1σ) increases at higher frequencies, indicating worse repeatability. This was expected, as the signal-to-noise ratio of the THz TDS system drops and sample’s absorption increases for higher frequencies.
For the sake of simplicity, from here on, we only show the results at 0.8 THz, as no significant frequency dependency affecting the repeatability between the four experimental parts was observed. The measured repeatability values at 0.8 THz for the refractive index and the absorption coefficient were (2.6, 2.8, 6.5, 6.0 and 3.0)${\times} {10^{ - 4}}\; $RIU (refractive index unit) and (0.12, 0.07, 0.11, 0.18, and 0.05) cm-1 for cuvettes one to five, respectively. These values of the repeatability interval correspond to the system stability (e.g. laser power fluctuations as discussed earlier). If we further average these five values, we get the average repeatability interval for the measuring sequence due to the system stability, which is done in section 3.4.
3.2 Positioning of the cuvettes
Between the first five measuring sequences, the caddy with cuvettes was removed and put back in place. This way, the effect of possible misalignments in the positioning of the cuvettes was evaluated. This effect was evaluated for each cuvette individually, by calculating standard deviation and the mean based on 25 measurements per cuvette (five measurements per single measuring sequence times the five measuring sequences). Furthermore, repeatability due to the system stability during a single measuring sequence was calculated based on the five measurements per cuvette. By averaging these repeatability values from the five measuring sequences, the average repeatability due to the system stability was determined independently for each cuvette. Refractive index and absorption coefficient of oil with corresponding repeatability intervals for individual cuvettes are shown in Fig. 5.
Fig. 5. Refractive index a) and absorption coefficient b) of oil with corresponding repeatability intervals due to the system stability (black) and system stability with cuvette positioning (green) for individual cuvettes. In addition, average values for all five cuvettes are shown as well as total repeatability interval corresponding to standard deviation of all 125 measurements. The columns correspond to the repeatability interval (vertical axis on the left) and the dots with error bars corresponds to the values of refractive index or absorption coefficient (vertical axis on the right). Note that the error bars around the dots correspond to the repeatability interval values.
For all cuvettes, the repeatability interval increased at least slightly when taking cuvette positioning into account compared to the repeatability interval due to the system stability only. On average for all five cuvettes, the repeatability interval increased by 0.4×10−4 RIU and 0.01 cm-1 for refractive index and absorption coefficient, respectively. Additionally, the total repeatability interval was calculated as the standard deviation of all 125 measurements comprising this experimental part. As such, it takes system stability, cuvette positioning and cuvette assembly into account. It increased by 1.7×10−4 RIU and 0.06 cm-1 for refractive index and absorption coefficient, respectively, compared to the average repeatability due to the system stability. These results indicate that cuvette positioning additionally limits the repeatability of the dielectric parameter extraction, however less than the cuvette assembly.
3.3 Cleaning of the cuvettes
Between the last five measuring sequences, the cuvettes were emptied, cleaned and the transmission of the empty cuvettes was measured. Then, we refilled the cuvettes with a fresh portion of the same oil and measured the transmission again. In this way, the effect of cleaning on the repeatability interval was assessed. In the same way as for positioning of the cuvettes, standard deviation and the mean of 25 measurements per cuvette (five measurements per single measuring sequence times the five measuring sequences) were calculated. Furthermore, repeatability due to the system stability during a single measuring sequence was calculated based on the five measurements per cuvette. By averaging these repeatability values from the five measuring sequences, the average repeatability due to the system stability was determined independently for each cuvette. Refractive index and absorption coefficient of oil with corresponding repeatability intervals for individual cuvettes are shown in Fig. 6.
Fig. 6. Refractive index a) and absorption coefficient b) of oil with corresponding repeatability intervals due to the system stability (black) and system stability with cleaning and cuvette positioning (green) for individual cuvettes. In addition, average values for all five cuvettes are shown as well as total repeatability interval corresponding to standard deviation of all 125 measurements. The columns correspond to the repeatability interval (vertical axis on the left) and the dots with error bars corresponds to the values of refractive index or absorption coefficient (vertical axis on the right). Note that the error bars around the dots correspond to the repeatability interval values.
On the average, the repeatability interval for all cuvettes increased when taking cuvette cleaning with positioning into account compared to the average repeatability interval due to system stability only. The increase was 0.3×10−4 RIU and 0.02 cm-1 for refractive index and absorption coefficient, respectively. However, for cuvette No. 1 the repeatability interval for refractive index even decreased. This suggests that the increase observed on average is of low significance. Additionally, total repeatability interval was calculated as the standard deviation of all 125 measurements comprising this experimental part. As such it takes system stability, cuvette cleaning with positioning, and cuvette assembly into account. It increased by 1.1×10−4 RIU and 0.15 cm-1 for refractive index and absorption coefficient, respectively, compared to the average repeatability due the system stability. Comparing the two increments of the repeatability interval, the results indicate that cuvette cleaning and cuvette positioning affect (3.2) the repeatability of measurements in a similar extent, however less than the cuvette assembly.
Since empty cuvettes were measured between each of the last five measuring sequences, the refractive index and absorption coefficient of empty cuvettes with corresponding repeatability interval were also computed. These are illustrated in Fig. 7. The values were computed in the same way as for cuvettes filled with oil. The extracted values per cuvette closely fit the expected value for refractive index and absorption coefficient for the empty cuvette, however, there is some deviation which can be attributed to the systematic uncertainties. This deviation is on the scale of the repeatability intervals indicating that cleaning of the cuvettes did not substantially affect the measurement repeatability.
Fig. 7. Refractive index a) and absorption coefficient b) of empty cuvettes with corresponding repeatability intervals due to the system stability (black) and system stability with cleaning and cuvette positioning (green) for individual cuvettes. In addition, average values for all five cuvettes are shown as well as total repeatability interval corresponding to standard deviation of all 125 measurements. The columns correspond to the repeatability interval (vertical axis on the left) and the dots with error bars corresponds to the values of refractive index or absorption coefficient (vertical axis on the right). Note that the error bars around the dots correspond to the repeatability interval values. The dashed lines present the expected value of parameter extraction for empty cuvettes (i.e. nitrogen).
3.4 Cuvette property variation and assembly
In order to take cuvette property variation and assembly into account, the repeatability interval was calculated as two times the standard deviation (2·1σ) for all 25 measurements for each measuring sequence. This value is associated with the system stability as well as the cuvette assembly and their characterization, since five different cuvettes were employed per measurement sequence. Furthermore, repeatability due to the system stability only was calculated for each measuring sequence as described in section (3.1). The results are shown in Fig. 8.
Fig. 8. Refractive index a) and absorption coefficient b) of oil with corresponding repeatability intervals due to the system stability (black) and system stability with cuvette assembly (green) for ten measuring sequences. In addition, average values for all ten measuring sequences are shown as well as total repeatability interval corresponding to the standard deviation of all 250 measurements. The columns correspond to the repeatability interval (vertical axis on the left) and the dots with error bars corresponds to the values of refractive index or absorption coefficient (vertical axis on the right). Note that the error bars around the dots correspond to the repeatability interval values.
The determined absorption coefficient of oil did not substantially vary between the measuring sequences, whereas for refractive index a consistent increase can be observed after the fifth measuring sequence. We assign this increase to the fact, that the different cuvettes were reassembled and newly characterized between the fifth and sixth measuring sequences. The repeatability interval due to system stability varied among different measuring sequences. Further, the repeatability interval taking system stability and cuvette variation into account varied as well but was always larger than the repeatability interval solely due to the system stability. The increase in the repeatability interval due to cuvette variation was on average 1.4×10−4 RIU and 0.10 cm-1 for refractive index and absorption coefficient, respectively. These results indicate that cuvette assembly affects the repeatability of measurements more substantially than cuvette cleaning and positioning.
4. Conclusions
In addition to the system stability, three further potential causes which affect the repeatability of complex refractive index extraction were investigated: possible angular misalignment of the cuvettes during positioning, cleaning of the cuvettes, and cuvette assembly and characterization. For the cuvettes and the spectroscopic system employed, we observed that system stability and cuvette variation are the main sources hampering the repeatability. Precisely inserting the cuvettes into the spectrometer (possible angular misalignment) and careful cleaning of the cuvettes hamper the repeatability additionally, however, to a smaller extent than the system stability and the cuvette variation and assembly. The repeatability interval in case of the investigated sample and our THz system was limited to the fourth decimal digit for the refractive index and to the first decimal digit for the absorption coefficient. Note that for strongly absorbing liquids (e.g. water) a significantly worse repeatability (i.e. a larger repeatability interval) may be expected.
Finally, our results suggest that it is sufficient to consider only the system stability to determine whether samples can be discriminated when using a single cuvette. However, we highly recommend other researchers to evaluate the repeatability for their system and cuvettes when measuring liquids in transmission with a THz TDS system in a similar way as presented here. Especially when measuring in a focus of the THz beam or employing cuvettes of significantly different thickness [38]. A further uncertainty analysis considering further systematic errors is mandatory, when exact absolute values for the dielectric properties are needed. It should be performed to be able to compare results between different research groups [38,39].
Funding
Ministério da Ciência, Tecnologia, Inovações e Comunicações (No 01250.075413/2018-04).
Disclosures
The authors declare no conflicts of interest.
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