## Abstract

We demonstrate a reduction of THz material parameter estimation errors by a factor of 40, using a time-domain material-model-based extraction method, in comparison with the respective frequency-domain approach. Based on simulated and experimentally measured THz time-domain spectroscopy data with different levels of the signal-to-noise ratios (SNR), an improved robustness of the time-domain scheme with regard to noise is demonstrated. This approach thereby provides an additional tool for the extraction of material parameters, which is especially suited for the evaluation of spectral data of samples with strong resonant absorption features, even if the SNR at those features drops below 0 dB. The cause for the difference in performance between the frequency-domain and time-domain methods is identified and a formal description provided.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

In recent years, a large number of applications for THz systems have been demonstrated, ranging from 3D security and object imaging [1] to ultra-broadband device characterization [2], biosensing [3], and spectroscopic material analysis and recognition [4]. The advantages of THz radiation are its nonionizing nature, penetration capability through many dielectric materials (e.g., paper, plastics, cloth), and the presence of unique spectral characteristics of materials of interest, like explosives, drugs, pharmaceuticals, and chemicals [4–6]. It thereby provides a valuable tool for chemical compound analysis and process monitoring [7], sensing of gases [8], and security applications [4]. THz time-domain spectroscopy (THz-TDS) provides the most flexible access to this spectral information, as it covers a broad frequency range, exceeding those of electronic or tuneable continuous-wave laser systems [9]. In order to evaluate and compare the spectral characteristics of a substance, general material parameters, which are independent from the sample dimensions and the measurement system, have to be derived from the acquired measurement data, like the complex refractive index, the complex dielectric constant, and/or related material properties.

In the *classical approach*, these material
parameters are determined individually for every frequency point. Such
calculations are based on a frequency-space-based formulation of
propagation models, i.e., the mathematical representations of the measured
data in dependence of the sample material parameters. By comparing the
measured spectral data to the modeled data, the value of the material
parameters is extracted by numerical optimization, solving the posed
inverse problem. This optimization approach is necessary, as the
relationship between measured data and material data cannot be inverted
analytically. Such calculations are then repeated for every frequency of
the investigated spectrum in a point-by-point process, as shown in earlier
work [10,11]. Based on these material parameters, further
attributes of the sample, like relaxation-times, resonator strength, and
resonant frequencies or plasma frequencies can be derived. Given
sufficient *a priori* knowledge on a sample,
like the general material class and number of spectral features, these
material properties can be determined by assuming an analytic description
for the physical material model (e.g., Lorentz oscillator model) to the
previously extracted material parameters [6,12], which at the same
time provides a condensed representation of those parameters over the
frequency range. A schematic of such a standard frequency domain (FD)
processing approach is displayed in the left section of Fig. 1.

It has been demonstrated that the determination of material parameters is
notably improved by merging these two processing steps (i.e., propagation
model and physical material model) into one operation [13,14]. Since the noise-sensitive optimization of the individual
frequency-dependent material parameters [15] is replaced by the optimization of the frequency-invariant
global material properties of the physical models, the optimization
process operates synchronously on the complete spectral data set with
several hundred complex data points instead of on only two values
(amplitude and phase) for each individual frequency point. This
frequency-domain model-based analytic procedure (*FDm*) is schematically depicted in the central section of
Fig. 1.

In our analytic procedure, we make use of the fact that, as the sample and
its effect on the THz wave are described over the full spectral range by
the parametric material model, the use of inverse Fourier transforms to
derive time-domain model data becomes feasible, allowing the realization
of a time-domain minimization and parameter extraction procedure [16–18]. This provides an additional
time-domain alternative for the determination of THz material parameters,
which can be used as an extension of the frequency-domain-based
processing. While inherent limitations of generally applicable
frequency-domain methods can be circumvented, it is important to note that
this time-domain approach can only be used for application cases in which
some general *a priori* information on the
material are available to select the appropriate parametric material
model. This time-domain method (*FDm*)
depicted in the right part of Fig. 1 is therefore particularly appealing for quality control,
industrial analyses, or recognition tasks where the general analytic task
is predefined.

In this work, we present a detailed, comparative study of the performance of material-model-based approaches operating in the time or frequency domains, used for the determination of THz material parameters from measured THz data. The methods are applied to simulated data sets, including artificial noise, allowing the generation of system-independent, fully controllable testing scenarios with designable data characteristics to allow precise quantification of the quality of the extracted material data. This investigation is complemented by the demonstration and analysis of parameters extracted from measured, experimental data. The capabilities of the methods in both testing scenarios are compared quantitatively, and the cause for differences in their performances is formally determined.

## 2. FORMAL DESCRIPTIONS

#### A. Propagation Model

To determine the complex refractive index of a sample from measured data, its effect on the THz radiation has to be formally expressed as the samples’ refractive-index-dependent relative amplitude transmission coefficient ${T_{\rm{samp}}}$ [10]. In the frequency domain, this can be described by the ratio between the field transmitted through the sample ${E_{\rm{samp}}}$ and the corresponding air reference ${E_{\rm{air}}}$ measured without a sample. These are equivalent to their respective absolute transfer functions ${H_{\rm{samp}}}$ and ${H_{\rm{air}}}$ multiplied by the impinging electric field ${E_{\rm{inp}}}$:

In the following, the transmission through an optically thick, single slab sample at normal incidence is considered. As the internal reflections in the sample can be excluded for an optically thick sample by temporal windowing, this allows the formulation of the most elementary propagation model. Following the description in [10], the transfer function is provided in this case by

With ${E_{\rm{samp\_meas}}}(f)$ and ${E_{\rm{air\_meas}}}(f)$ being the Fourier transforms of the measured transients, the experimental transfer function ${T_{\rm{samp\_meas}}}(f)$ can be calculated. The complex refractive index can then be derived from the measured data, by adapting the parameters of the calculated transmission in Eq. (2) to the measured data ${T_{\rm{samp\_meas}}}(f)$ and solving the resulting equation:

A direct solution of Eq. (4) in the time domain in this form is not feasible, since this equation is defined for each frequency point individually by ${\eta _{\rm{samp}}}(f)$.

#### B. Material Models

As the information contained in THz-TDS data is not limited to the
samples’ complex dielectric material values, more relevant attributes,
like the properties of the underlying physical mechanisms that
determine the samples interaction with the THz radiation, can also be
derived. This is typically achieved by fitting a physical material
model, i.e., a mathematical description of those mechanisms, to the
complex refractive index data. Since the material models are
reasonably based on the desired physical material properties, values
like resonator strength, resonant frequencies, and relaxation times,
for instance, are directly available as part of the fitting result.
The selection of a suited material model is based on general *a priori* knowledge on the investigated material
type, including, for instance, the overall number and type of spectral
features.

As an example, a typical physical material model suited for many (nonmagnetic) electronically resonant materials in the THz range, like explosives, pharmaceutical substances, and chemicals, is the Lorentz oscillator model in Eq. (7) with multiple contributing oscillators [6,16]:

Considering that the propagation model as well as the physical material model are dependent on the complex refractive index, a combination of both models can easily be realized in one step. By substituting the refractive index in Eq. (4) with Eq. (7). This step not only replaces all individual frequency-dependent unknowns by frequency-invariant global quantities but also massively reduces their number, since the typically hundreds of refractive index values at individual frequencies are now represented by the material model with a few physically relevant material quantities. Utilizing this combined model, the physical material properties of a sample can directly be determined in one optimization step as elucidated for the FDm and TDm case in Fig. 1, matching the models’ complex output to the measured transmission data simultaneously over the complete spectral range. The corresponding material parameters are calculated by the combined model and do not require an additional extraction step as depicted for the FD case.

Since the transfer function can now be described for the complete investigated spectral range by the frequency-invariant material properties, an inverse Fourier transform can easily be applied, and the optimization procedure can alternatively be performed in the time domain by comparing the convolution product of the acquired reference transient and the calculated transfer function with the acquired sample transient. This convolution product is determined by calculating the transfer function based on the previously described combined model in the frequency domain, multiplying the result with the Fourier transform of the reference transient and performing the inverse Fourier transformation. With

#### C. Noise Propagation and Material Parameter Error

In order to determine and compare the performance of the time-domain (TDm) and frequency-domain (FDm) extraction procedures, the impact of noise and resulting errors on the processed data is investigated. Since measurements are performed in the time domain, providing the transient $E({t_k})$ in its discrete form $E(k\tau)$, the standard deviation ${\sigma _E}(k\tau)$ can directly be derived as a measure of noise error in this domain, with the sampling interval $\tau$ and the time index $k$. For the frequency-domain case, the propagation of this standard deviation after Fourier transformation has to be addressed in order to calculate the corresponding standard deviation for the signal magnitude ${\sigma _{| E |}}(f)$ and phase ${\sigma _\vartheta}(f)$. Following the formalism derived in [19,20], this dependency of the standard deviations can be described, for statistically independent time samples, as

The spectral signal-to-noise ratio can, in both cases, be defined as a ratio of the signal magnitude and standard deviation [21] as

Thereby, the SNR and measure of the resulting phase error at a given frequency can directly be determined from the ratio of the measured data and its standard deviation.The quality of the extracted parameters ${n_{\rm{samp}}}$ and ${\kappa _{\rm{samp}}}$ is evaluated by computing their root mean square error (RMSE), with respect to the exact reference values ${n_{\rm{ref}}}$ and ${\kappa _{\rm{ref}}}$ of the testing scenario. The RMSE for $N$ frequency points is defined as

This global measure of error is complemented by the investigation of the relative error of the parameters at every frequency point

and## 3. RESULTS AND DISCUSSION

#### A. Performance Evaluation by Simulated Data Sets

To evaluate the different methods for parameter extraction, two
categories of testing scenarios are used. The first applies the
methods to simulated noisy transient data sets and is presented in
Section 3.A. The second,
shown in Section 3.B,
processes actually measured data at varying experimental
signal-to-noise situations. All calculations (cf. Section 2) were implemented and performed
using MATLAB and the *lsqnonlin*
subroutine of its optimization toolbox.

The use of simulated data provides complete control over the pulse shape, the spectral characteristics of the sample, and the noise levels, independently from a particular measurement system and with full reproducibility. Since the material parameters are known by design, they can directly be quantitatively compared with the values extracted by the different, investigated methods. This provides an ideal direct measure of the errors of an analytic numerical approach and consequently its reliability for various SNR levels and sample properties.

The basis for the simulated data is a replica of an ideal THz transient, calculated from the derivative of a Gaussian function, similarly to the one shown in [22]. This transient is used as the air reference transient (Fig. 2). It has properties resembling those measured by the actual experimental system used later. The sample transient is obtained by convoluting this ideal air reference transient with the relative transfer function of the designed, simulated sample with four absorption resonances at 0.7, 1.25, 2.5, and 5.0 THz and a thickness of 900 µm. The corresponding calculated spectra are displayed in Fig. 3. In order to emulate the standard processing of measured data [10], typical preprocessing steps consisting of time-windowing the transient data, using a tapered cosine window, and applying a bandpass filter to the data in the frequency domain, according to the modeled and evaluated spectral range, are also performed. Normally distributed random numbers are used to generate the noise. White noise instead of field-amplitude-dependent noise [20,23] is used for the simulation, so that the effects of the noise can directly be interpreted as generally as possible, independently from a specific noise design. The amplitude noise is added to the simulated transients, which are scaled to different levels for the various investigated scenarios. These transients and their corresponding amplitude spectra are depicted in Figs. 4 and 5 along with the respective standard deviation, labeled according to their respective maximum SNR in the frequency domain.

Figure 6 shows the spectral SNR of the generated data sets, with identical sample properties and different input signal levels. The general reduction of the SNR at the absorptive feature positions at 1.25 and 2.5 THz is clearly visible as well as the overall decline toward higher frequencies. In the ${{\rm{SNR}}_{{\max}}} = 82 \;{\rm{dB}}$ case, the SNR of the second feature at 2.5 THz drops just below 0 dB. With the further decrease of SNR, an almost 500 GHz wide segment drops below 0 dB.

In the following, the FDm and TDm material parameter extraction approaches are applied to the simulated data, as described in Section 2. The resulting RMSE of the parameters determined by the time- and frequency-domain material model-based methods are depicted in Fig. 7. The comparison of the RMSEs shows a clear overall superior quality of the parameters extracted by the time-domain approach. At the low SNR scenarios below ${{\rm{SNR}}_{{\max}}} = 62 \;{\rm{dB}}$, the frequency-domain method fails completely at the parameter extraction over a broad range of frequencies, indicated by the hollow points and the RMSE saturation at very high levels. The time-domain concept, on the other hand, still reliably determines the material parameters despite the calculating parameters from data with sub-0 dB SNR levels at the strongest resonance. Furthermore, it maintains an at least 40 times lower RMSE for all investigated noise cases, compared with its frequency-domain counterpart, as evident from the parallel lower RMSE values. With the minimum SNR (${{\rm{SNR}}_{{\min}}}$) at a spectral feature position close to 0 dB, as is noticeable in Fig. 6 for the ${{\rm{SNR}}_{{\max}}} = 82 \;{\rm{dB}}$ case at 2.5 THz, the time-domain method resulted in an RMSE of $4 \times {10^{- 5}}$ compared with the RMSE $2 \times {10^{- 3}}$ of the frequency-domain concept. Especially if the SNR of several segments of the spectral features lies below 0 dB, the time-domain processing maintains an RMSE better than $1 \times {10^{- 3}}$ and is able to provide correct data over the complete spectral range.

Figures 8 and 9 show the relative errors ${\delta _n}$ and ${\delta _\kappa}$ of the extracted refractive indices $n$ and extinction coefficients $\kappa$ of both approaches for the input data of Fig. 6 in reference to the ideal material parameters. It is evident by comparing the absolute values of the relative errors that the time-domain-extracted data are in significantly better agreement with the original data for all scenarios, including scenarios with low SNR. For instance, for the refractive index errors in Fig. 8 for the ${{\rm{SNR}}_{{\max}}} = 92 \;{\rm{dB}}$ case, the TDm has errors below $2 \times {10^{- 5}}$ for all frequency positions, while the FDm has errors reaching $1 \times {10^{- 3}}$. Furthermore, the results of the frequency-domain method show large deviations at low SNR. This is evident from a more or less constant spectral shape for the TDm as a function of decreasing SNR (only the error amplitude increases), while the spectral behavior of the FDm completely changes and rises up to $7 \times {10^{- 2}}$, where the spectral feature at 2.5 THz first drops below 0 dB, for the reduced ${{\rm{SNR}}_{{\max}}} = 62 \;{\rm{dB}}$ to ${{\rm{SNR}}_{{\max}}} = 42 \;{\rm{dB}}$ cases. In this situation, where the SNR of wide segments of the spectral feature lies below 0 dB, the relative errors, especially of the refractive index, show a broadband error and an offset like behavior of the extracted data following the spectral feature at 2.5 THz. The time-domain approach, on the other hand, maintains in the respective cases a maximum relative error of only $5 \times {10^{- 5}}$ in the ${{\rm{SNR}}_{{\max}}} = 82 \;{\rm{dB}}$ case and $5 \times {10^{- 3}}$ in the ${{\rm{SNR}}_{{\max}}} = 42 \;{\rm{dB}}$ case, and no broadband offset errors are occurring.

Since both methods process the same input data, same number of data points, and use identical settings and preprocessing steps, it stands to reason that the variation in performance, especially in the low SNR cases, stems from the different implementation of the optimization procedures. In the following, we will demonstrate that this difference results mainly from the handling of phase data in the frequency-domain approach. The standard deviation of the phase data is calculated with Eq. (12) and depicted for different SNR scenarios in Fig. 6. The increase in the phase standard deviation (or the phase error) with a decrease of SNR, especially at the regions of the spectral features, is clearly visible and reaching values close to $\pi$ and above for ${{\rm{SNR}}_{{\max}}} \le 82 \;{\rm{dB}}$. As a consequence, the phase values at the absorption features are highly affected by these effects. The inset in Fig. 10 depicts the ideal phase change between consecutive frequency points in the noise-free case. At the 2.5 THz feature, the change is ${-}{1.6}\;{\rm{rad}}$. For the ${{\rm{SNR}}_{{\max}}} = 82 \;{\rm{dB}}$ case, the standard deviation of the phase data at that point rises close to 3 rad. As the phase change of the noisy phase data is the sum of the noise-free phase change and the phase error, it is easy to see that, in this case, the ${\pm}\pi$ ambiguity threshold is exceeded. Consequently, an unwrapping based on this data cannot be performed properly, resulting in the introduction of an erroneous phase jump. Since several neighboring points also reach a critical standard deviation, several jumps can occur at this feature position. The number of those jumps drastically increases with the reduction of SNR. As those unwrapping errors cannot be distinguished from phase changes induced by the sample optical thickness, their removal by a manual or supervised unwrapping is mathematically not possible. Such errors, then, do not only affect the data locally at the spectral features, but, as they induce a phase offset to all succeeding phase points (due to the dependency of the phase on preceding spectral phase values in the unwrapping procedure), this error leads to global errors at all frequencies above these erroneous spectral positions. This completely inhibits correct phase handling in the optimization procedure and is the origin of the broadband errors observed earlier. As the material-model-based method in the frequency domain operates simultaneously over the complete investigated range, all determined material parameters are made erroneous by this effect.

Since the time-domain method does not require any type of phase processing, as it directly performs the optimization step in the time domain, it is not affected by this limitation at all. Thereby, the effects of the rise of the error at the absorption features position remain spectrally localized and have only a limited degradatoriy effect on the overall extraction process. Broadband extraction of material properties thus becomes possible at limited SNR situations.

#### B. Performance Evaluation by Experimentally Measured Data Sets

The second section of the performance evaluation of the time- and
frequency-domain methods is their application to actually measured,
experimental transient data. Here, we investigate the data obtained
from alpha-lactose samples pressed into pellets of 13.4 mm diameter
and a thickness of 660 µm. The pellets were prepared without the use
of a matrix material [24]. The
measurements were performed with an in-house built TDS system using a
biased GaAs photoconductive switch emitter and an electro-optic (GaP)
differential detector in a lock-in measurement scheme, driven with a
commercial 20 fs pulsed Ti:Sapphire laser with a repetition rate of
85 MHz (Femtolasers INTEGRAL 20). The THz pulses are guided by a set
of off-axis parabolic mirrors. To emulate different SNR scenarios, the
system was operated at various optical pump-power levels, resulting in
a set of measurements with different THz field amplitudes. These
transients, their amplitude spectrum, and the respective standard
deviations are shown in Figs. 11 and 12. They are
labeled according to their respective maximum SNR in the frequency
domain. The calculations, as described in Section 2, are implemented and performed in the same fashion
as for the simulated data, again using the *lsqnonlin* subroutine of MATLAB for optimization. The number
of oscillators of the Lorentz model and their rough position is set
accordingly to three absorption features directly discernible in the
spectrum and a fourth feature visible as an absorption onset located
outside the investigated frequency range.

In an analogous fashion to the evaluation of the simulated data sets, Fig. 13 depicts the spectral SNR of the lactose sample measurements. The inset shows the resulting RMSE values. To exclude effects of varying sample preparation, measurement system, and data handling [25], with regard to published lactose data, the RMSE is calculated in reference to the time-domain extracted results of the highest SNR case. Similarly to the evaluation of simulated data, the time-domain-approach TDm outperforms the frequency-domain FDm at a moderate minimum SNR with a RMSE of $2.2 \times {10^{- 3}}$ versus $1.4 \times {10^{- 2}}$ also in the real measurement case. Even for the ${{\rm{SNR}}_{{\max}}} = 25 \;{\rm{dB}}$ case with the SNR at the feature position close to 0 dB, the time-domain method provides reliable parameters with an RMSE of $7 \times {10^{- 3}}$, while the frequency-domain RMSE is 20 times higher and thereby insufficient for a proper parameter determination. The slightly reduced difference in performance of the measurement case compared with the simulation case, where a difference of 40 was observed, can most likely be attributed to noise effects in the measurement that are not fully represented by the simulated noise.

The extracted refractive indices are plotted for comparison in Fig. 14. For all time-domain cases and the high SNR frequency-domain case, all curves of the refractive index and the extinction coefficient agree closely around the ideal values. It is directly discernible that, for the low SNR case determined in the frequency domain, the spectral feature at 1.3 THz is missing in the extracted data, and an overall offset error is manifest.

Following the same analysis of the simulated case, Fig. 15 shows the standard deviation of the measured phase data. The inset depicts the phase change between consecutive frequency points, which significantly peaks to a value of 2.9 rad at the sample feature position at 1.35 THz. A rise of the standard deviation at the absorption feature positions and an overall increase with decreasing SNR is again visible. For the ${{\rm{SNR}}_{{\max}}} = 38 \;{\rm{dB}}$ case, the ambiguity limit is approached due to the noise effects but not exceeded; therefore, the unwrapping can still be performed correctly, and an extraction including the feature in the frequency domain is possible. Still, the results are noticeably affected by the noise, resulting in a broadened extinction coefficient and a heightened change in the refractive index. For the low SNR case, the standard deviation at the feature position at 1.35 THz rises to 0.5 rad. Since this noise effect overlays with the 2.9 rad phase change caused by the samples’ properties, the overall phase change exceeds the ${\pm}\pi$ ambiguity limit, and the spectral feature is falsely removed from the phase data by the unwrapping procedure.

As a consequence of the false phase representation, the samples’ spectral properties cannot be correctly reconstructed by the material model, as it describes the full investigated spectral range simultaneously and is thereby globally affected and shifted, resulting in complete failure of the optimization of the calculated and measured phase data. This inhibits proper parameter extraction by the frequency-domain method. These findings are consistent with those of the simulated scenarios, confirming the sensitivity of the frequency-domain method to phase errors, due to the resulting failure in phase-unwrapping and the superior robustness achieved by the use of the time-domain approach.

## 4. SUMMARY

We have shown that, by use of the material-model-based time-domain
extraction method, material parameters can be extracted from THz transient
data with an approximate factor of 20–40 lower root-mean-square error
(RMSE), compared with the respective frequency-domain method. This was
demonstrated by applying the data analysis procedures to simulated
transient data and to experimentally measured data sets of alpha lactose
at varying SNR conditions, utilizing only general *a
priori* knowledge (visible from the spectra) on the number of
rough positions of resonances for the adequate material model selection.
The superiority of the time-domain concept is even more relevant in
scenarios where the SNR is close or even below 0 dB at the absorption
spectral feature position. While in such a case, time-domain methods still
allow correct parameter extraction over the full measurement bandwidth,
frequency-based approaches deliver erroneous results for all frequencies
if one spectral position reaches a low SNR level.

In the demonstrated examples, the time-domain method extracted material parameters with a 40 times lower RMSE from the simulated data, compared with the results of the frequency-domain approach. In the case of the measured data, a 20 times lower RMSE was achieved by the time-domain concept. This improvement of noise robustness, on the one hand, further expands the capabilities of the THz-TDS analysis of samples with particularly strong resonant absorption features. On the other hand, it offers the option to speed up analytic procedures and measurements, since experimental averaging could be reduced at the cost of decreased SNR, while still maintaining a reasonable parameter extraction quality.

The cause for the performance difference was investigated by analyzing the noise propagation from the time to frequency domains. In the simulated and measured scenarios, a drastic rise in the standard deviation of the phase data at the resonant absorption features positions could be observed. In the context of the necessity of phase unwrapping in the frequency-domain method, this has a strong impact. If the ${\pm}\pi$ ambiguity limits are exceeded due to phase errors, the correct operation of the unwrapping procedure is inhibited, leading to a broadband offset error of the phase over a wide spectral range, which prevents proper parameter extraction by the frequency-domain approach. As the unwrapping is not required in the time-domain optimization concept, it remains reliable, even if the SNR locally decreases at specific frequencies due to an absorption feature, without adding further processing steps or weighting procedures.

## Funding

Allianz Industrie Forschung (ZF4312305DF9).

## Acknowledgment

We gratefully acknowledge useful discussions and experimental support by G. Schulte, T. A. Pham Tran, and Dr. R. Bornemann.

## Disclosures

The authors declare no conflicts of interest.

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