Terahertz Multilayer Thickness Measurements: Comparison of Optoelectronic Time and Frequency Domain Systems [original]

Vol.:(0123456789)

https://doi.org/10.1007/s10762-021-00831-5

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Terahertz Multilayer Thickness Measurements: Comparison

ofOptoelectronic Time andFrequency Domain Systems

LarsLiebermeister1 · SimonNellen1 · RobertB.Kohlhaas1 ·

SebastianLauck1· MilanDeumer1 · SteffenBreuer1 · MartinSchell1,2·

BjörnGlobisch1,2

Received: 22 June 2021 / Accepted: 3 November 2021

Abstract

We compare a state-of-the-art terahertz (THz) time domain spectroscopy (TDS) sys-

tem and a novel optoelectronic frequency domain spectroscopy (FDS) system with

respect to their performance in layer thickness measurements. We use equal sample

sets, THz optics, and data evaluation methods for both spectrometers. On single-

layer and multi-layer dielectric samples, we found a standard deviation of thickness

measurements below 0.2 µm for TDS and below 0.5µm for FDS. This factor of

approx. two between the accuracy of both systems reproduces well for all samples.

Although the TDS system achieves higher accuracy, FDS systems can be a com-

petitive alternative for two reasons. First, the architecture of an FDS system is essen-

tially simpler, and thus the price can be much lower compared to TDS. Second, an

accuracy below 1µm is sufficient for many real-world applications. Thus, this work

may be a starting point for a comprehensive cross comparison of different terahertz

systems developed for specific industrial applications.

1 Introduction

Terahertz (THz) spectroscopy is an interesting sensing technology for many appli-

cations in material and structural analysis, compound identification, and testing

[1, 2]. The measurement is non-contact and non-destructive, can easily handle air-

material interfaces, and uses non-ionizing radiation. One of the key applications for

THz spectroscopy is the thickness measurements of paint and coating layers. Until

today, time domain spectroscopy (TDS) is almost exclusively used for this kind of

* Lars Liebermeister

[email protected]er.de

1 Fraunhofer Institute forTelecommunications, Heinrich Hertz Institute, Einsteinufer 37,

10587Berlin, Germany

2 Institut Für Festkörperphysik, Technische Universität Berlin, Hardenbergstraße 36, EW 5-1,

10623Berlin, Germany

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applications, as these systems are mature and commercially available. In addition,

THz TDS offers high acquisition speed and high THz bandwidth. The latter ena-

bles thickness measurements of sub-mm dielectric layers [3–5] and the detection of

defects in polymers, foams, and other non-conductive materials [6].

In contrast, frequency domain spectroscopy (FDS), which is based on the gen-

eration and coherent detection of continuous wave (cw) THz radiation, hitherto is

barely used for non-destructive testing (NDT) [7, 8]. The inherently low measure-

ment rate of most scientific FDS implementations and all commercial FDS prod-

ucts makes NDT with cw THz radiation highly inefficient. However, it was shown

already in 2014 that THz FDS can be used for multilayer thickness measurements

[9]. We have recently demonstrated a THz FDS system that allows acquisition rates

of 200 Hz, which is comparable to the speed of most commercial TDS systems.

With this FDS system, a PET film with a thickness of 23µm could be measured with

an accuracy better than 2%, which corresponds to an uncertainty below 0.5µm. This

demonstrates that FDS can compete with TDS for accurate thickness measurements

on thin dielectric layers [10]. Thereby, the main advantage of FDS compared to TDS

is the simplicity of its system architecture. FDS systems are all-fiber coupled, and

they do not require femtosecond optical pulses, moving optics, or complex phase

locking electronics. This is why many industrial applications may benefit from using

frequency domain spectroscopy instead of TDS. To date, there is no detailed com-

parison between the two methods in terms of measurement accuracy, measurement

time, and reproducibility.

In this paper, we present the first direct comparison of a state-of-the-art TDS and

optoelectronic FDS system for layer thickness measurements in reflection geometry.

Both TDS and FDS measurements were performed on the same samples, namely

PET films with thicknesses between 23 and 350 µm, a Si wafer with a thickness

of 350µm, and a ceramic coated with spray paint on both sides. Based on this, we

compare the ability of the two systems to achieve the same results in consecutive

measurements. Thereby, we analyze the effects of instrument noise, dynamic range,

and bandwidth of the respective system on thickness determination. As the central

figure of merit, we investigated the standard deviation of consecutive measurements

at the same position on the sample. We found that the standard deviation of both

TDS and FDS is always lower than 2%.

This paper is organized as follows: In Section2 we describe in detail the TDS

and FDS system used for the comparison. Section3 explains the experimental setup,

data evaluation, and thickness determination. The measurement results are presented

and discussed in Section4 before we summarize our results in Section5.

2 TDS andFDS System

The TDS measurements in this comparison were done with a commercial state-of-

the-art system, the TeraFlash pro system from TOPTICA Photonics AG [3]. A func-

tional diagram of its optical components is shown in Fig.1a. The system uses a fem-

tosecond fiber laser centered at 1560nm, which generates pulses with a duration of

100fs at a repetition rate of 100MHz. A voice-coil driven-optical delay allows for

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time-domain sampling with update rates up to 100Hz. An additional optical delay

line in the receiver arm of the spectrometer allows for compensating a wide range of

THz free space path lengths. Fiber-coupled InGaAs-based photoconductive modules

are used as emitter (Tx) and receiver (Rx), respectively. The photoconductive mate-

rial of the receiver is rhodium-doped InGaAs (InGaAs:Rh) [11, 12]. The Tx uses

iron-doped InGaAs (InGaAs:Fe) [13]. In this configuration, the spectral maximum

of the system is centered at 1THz, and the peak dynamic range reaches 70dB in

single shot (66ms measurement time) and exceeds 95dB for 1000 averages (see

Fig.2a). The total available bandwidth is more than 6THz and the spectral resolu-

tion can be as low as 1GHz. All measurements presented here were acquired with a

scan range of 70ps, corresponding to a spectral resolution of 14GHz.

FDS measurements were performed with a recently published optoelectronic

continuous-wave terahertz spectrometer [10]. The working principle of this sys-

tem is based on frequency-modulated continuous-wave (FMCW), which is an

Fig. 1 Schematics of the two optoelectronic THz spectrometers used in this paper. (a) The TDS system

compromises a pulsed fiber laser (fs-laser), a fast (voice-coil delay) and a slow (path length compensa-

tion) free-space optical delay, photoconductive emitter (Tx) and receiver (Rx), and a real-time control-

ler to drive the voice-coil and acquire the data. All components are fiber-coupled. (b) The FDS system

compromises a fixed-frequency (cw-laser) and a swept laser (Swept cw-laser), an erbium-doped fiber

amplifier (EDFA), a PIN-photodiode emitter (Tx), and a photoconductive receiver (Rx) as well as a

data acquisition unit (DAQ). The Tx- and Rx paths are indicated with yellow- and green-shaded arrows,

respectively. The different path length in the Tx and Rx arm of the spectrometer are a prerequisite for the

optoelectronic FMCW technique [10]

Fig. 2 Comparison of typical spectra recorded with the TDS (a) and FDS (b) systems. The dynamic

range is plotted against frequency. Both records use 1min of averaging. In contrast to the measurements

on the sample, these signals are recorded in a transmission setup consisting of two off-axis parabolic mir-

rors, which is filled with air of ambient humidity

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established technique used in purely electronic THz systems [14]. A schematic dia-

gram of the optoelectronic THz FDS system is shown in Fig. 1b. The spectrom-

eter uses two fiber-coupled, continuous-wave semiconductor lasers emitting in the

c-band (1530–1565nm). The frequency of one of these lasers is swept periodically,

while the emission frequency of the second laser stays fixed. The laser outputs are

spatially overlapped in a 3dB coupler, which generates an optical beat note. This

beat note is amplified by an EDFA before being converted into THz radiation via

photomixing in a waveguide-integrated PIN photodiode [15, 16]. The same beat sig-

nal is also used to drive the coherent detection in the photoconductive Rx [15]. Tx

and Rx are based on commercially available, fiber-coupled modules from TOPTICA

Photonics AG. Further details on the THz FDS system can be found in the litera-

ture [10]. The most significant and fundamental difference of this FDS system com-

pared to common setups with coherent detection of cw THz signals is the method

employed to obtain the phase. When no phase modulation is used, fringe-detection

or Hilbert-transform [17] are common methods to extract the phase information.

Alternatively, active phase modulation by a free-space optical delay line [18], a fiber

stretcher [19], or an optical phase modulator [20, 21] can be employed, which allows

for amplitude and phase determination with a quadrature lock-in detector. The FDS

system used here bases on an optoelectronic adaption of the FMCW technique [10],

which results in passive phase modulation. The tunable cw-laser is frequency swept

with more than 500THz/sec. In combination with a path length imbalance of 20cm

between the emitter and the receiver arm (indicated by shaded arrows), an inter-

mediate frequency of 500kHz is generated in the photomixing receiver, which can

be directly used for coherent detection with a software-based lock-in amplifier. This

quadrature lock-in detection allows to detect amplitude and phase as a function of

frequency. Note that this FDS system neither requires optomechanics nor free space

optics nor electro-optic phase modulation. The THz amplitude spectrum acquired

with the FDS system is depicted in Fig.2b. The spectral maximum is centered at

100GHz with a peak dynamic range exceeding 90dB for 4000averages. In a single

shot measurement, which is acquired in 14ms, the dynamic range measures 60dB

with a 2THz bandwidth. With averaging, the peak dynamic range and bandwidth

can reach 117dB and 4THz, respectively [10]. Figure2 compares the THz spectra

acquired with the TDS and the FDS system for a measurement time around 60s.

3 THz‑Setup andMeasurement Procedure

This section covers the experimental setup, the set of samples, the data preparation,

and the algorithm for thickness determination.

3.1 Experimental Setup

All measurements were carried out in reflection geometry, which is the most

industrially relevant setup. It requires only single-side access to the sample under

test, and reflection measurements can be performed independent of the substrate.

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A schematic diagram and photograph of the THz reflection head used for TDS

and FDS measurements are depicted in Fig.3. Since the mechanical dimensions

and beam profile of pulsed and cw emitter and receiver modules are almost identi-

cal, the same THz optical beam path is used for both systems. A 90° off-axis par-

abolic mirror with 2inch focal length collimates the THz beam of the emitter and

another 90°off-axis parabolic mirror with 4inch focal length focuses the beam

on the sample surface under an angle of 8°. The reflected THz beam is collected

and focused on the detector by an equal configuration of mirrors. The diameter

of all parabolic mirrors measures 1inch, which allows for constructing a compa-

rably compact reflection head. The resulting measurement spot is about 1mm in

diameter and the THz-beam is s-polarized with respect to the plane of incidence.

The beam path is oriented upwards (see Fig.3), which allows for placing the sam-

ples reproducibly in the THz focus. In order to suppress THz absorption from

water vapor, the reflection head is encapsulated (housing in Fig.3) and purged

with nitrogen (see photograph in Fig. 3)). Between background, reference, and

sample measurement, the alignment of the reflection head was kept unchanged.

The placement of the sample in the THz-beam path as well as the alignment

is a common cause of variations of the measurement result. This effect can even

dominate the uncertainty of thickness measurements in industrial applications.

However, these variations are mainly related to the geometry of the samples,

the sample holder, the design of the THz optics, and environmental conditions.

Therefore, it is not an effect caused by the THz system used. In our comparison,

we tried to minimize the influence of positioning errors and sample inhomogene-

ity by focusing our analysis on the reproducibility of consecutive measurements

taken on the same position on the sample.

Fig. 3 Schematic and photo-

graph of the reflection head used

for non-contact layer thickness

measurements. The reflection

head was identical for TDS and

FDS measurements. The diam-

eter of the parabolic mirrors

measures 1”. In this view, THz

transmitters and receivers and

their respective beam paths are

arranged one behind the other.

Both beam paths hit the sample

at the same spot and under an

angle of 8°. Tx and Rx have a

distance of 5mm. The housing

is purged with nitrogen to avoid

detrimental effects from water

vapor absorption

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3.2 Measurement Conditions

For the comparison presented in this paper, the FDS and TDS system were driven

under almost identical conditions. It is well known that bandwidth and dynamic

range increase significantly with a higher number of averages, i.e., longer meas-

urement time. In general, the central limit theorem states that the dynamic range

increases by 10dB per tenfold increase in acquisition time, assuming a random noise

background [10]. Therefore, the measurement time is a crucial parameter in the sys-

tem comparison, which can be adjusted by applying a different number of averages

for TDS and FDS. Figure4 compares the THz spectra acquired with the two sys-

tems in reflection geometry. Here, the terahertz reflection head shown in Fig.3 and a

metal reflector were used. The signal acquisition time was almost identical for both

systems. The TDS data was acquired by averaging two subsequent pulse traces with

total acquisition time of 127ms. The peak dynamic range was > 70dB at 1 THz

and the bandwidth exceeded 5THz. The FDS data was acquired with ten averages,

resulting in a total measurement time of 147ms. Under these conditions, the peak

dynamic range was > 50dB at 100GHz and the total bandwidth was 2THz. Hence,

the bandwidth of the TDS system is approximately twice as large as the bandwidth

of the FDS system. In contrast, the frequency resolution of the FDS system meas-

ures 1GHz, whereas the frequency resolution in TDS is 14GHz only. Before the

results are being discussed and compared in Section4, the procedure of data analy-

sis is described in the next paragraph.

3.3 Data Preparation

For a direct comparison of THz data acquired with the two different systems, the

data post-processing and thickness evaluation methods have to be identical for

both systems. Figure5 depicts this unified post-processing procedure. For TDS,

a fast Fourier transform (FFT) converts the measured pulse trace into a complex

spectrum that can be represented in the frequency domain either by amplitude

Fig. 4 Spectra recorded with the TDS (a) and FDS (b) system using the terahertz reflection head shown

in Fig.3 and a metal reflector. The spectra are recorded using the same optics, alignment procedure,

acquisition and averaging parameters, and environmental conditions as the sample measurement in this

work. The right spectrum is smoothed with a 10GHz window

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and phase or by a complex number. These spectra have the same data structure as

the spectra acquired with the FDS system. Hence, all following data processing

steps are identical for both data sets. First, the complex transfer function of the

sample is determined by dividing the spectra of the sample measurement by a ref-

erence spectrum recorded with a metal reflector (see Fig.4). Ideally, this transfer

function contains the spectral response of the sample only. Spectral features that

stem from the respective spectroscopic system and the setup are thereby elimi-

nated. Next, an inverse FFT (iFFT) transforms the transfer function into the time-

domain, and subsequently, frequency filters are applied to the time-domain signal

to remove noise and artefacts. In the resulting time trace, each dielectric interface

with an increase or decrease in the refractive index appears as a positive or nega-

tive peak, respectively (see Fig.6). Note that these processing steps correspond to

a deconvolution of the TDS pulse trace. The frequency-filters are of Butterworth-

type and applied forward–backward so that a flat phase response is ensured. Ini-

tially, the filter parameters are optimized manually to suppress background noise

and ringing while keeping the closest peaks well separated. The low-pass 3dB-

frequency was found to be critical for the thickness evaluation, and therefore a

value sweep was performed to find an optimal value (see Section4 for details).

After this optimization, the filter parameters were kept constant for all measure-

ments and identical for TDS and FDS.

Fig. 5 Preparation procedure for the TDS- and the FDS-data. A representative sketch of the data after

each step is shown at the bottom. The procedure corresponds to a deconvolution of the TDS pulse trace.

Abbreviations: FFT, fast Fourier transform; Ampl., amplitude; iFFT, inverse fast Fourier transform

Fig. 6 Thickness determination procedure

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3.4 Thickness Determination

In the processed time trace data, thick layers can be clearly identified by well-

separated peaks, which enable thickness determination via basic peak-finding

algorithms. However, this simple procedure fails for thin layers where individual

peaks overlap. Therefore, we use a model-based approach to thickness determina-

tion. Figure6 is a guide through the procedure:

1. Ideally sharp interfaces are assumed, which correspond to measurements with

infinite bandwidth and dynamic range. Under this assumption, each interface is

represented by a

𝛿

-peak with a scaling factor for its amplitude and its position in

the time domain. A model time-trace is generated from a series of

𝛿

-peaks with

initially guessed positions and amplitudes.

2. Frequency filters are applied to the model time-trace using the same coefficients

as they were applied to the measured data.

3. This filtered model is compared to the recorded transfer function in the time

domain. Steps 1 and 2 are repeated with varying position and amplitude of the ini-

tial

𝛿

-peaks. Thereby, the model is optimized to fit the recorded data. Commonly,

a least squares method is used for such an optimization processes. However, this

approach does not converge well for a time trace containing a few peaks only.

In the least squares method differences between model and data have the same

weight for all data points. In our case most of the data points of the time-trace are

close to zero and carry no information about the thickness. In order to increase

the weight of the peaks, we multiply the square of the model function with the

square of the difference between model and data. The result is then used as the

penalty function, which is to be minimized.

4. We determine the layer thickness from the positions of the

𝛿

-peaks of the opti-

mal model. The time difference between two adjacent peaks is multiplied by the

literature values of the refractive indices of the respective material.

Note that this procedure is similar to the frequently-used transfer matrix

method. However, it is essentially simpler since refractive index and absorption

coefficient of the material are not optimized. This simplification makes the pro-

cedure more robust against inaccurate alignment, surface roughness, and environ-

mental influences. A possible disadvantage of this method is the detailed knowl-

edge required about the sample under test, which includes the material of the

different layers, the number of interfaces, and the THz refractive indices. How-

ever, for most industrial applications, this information is either already available

or it can be determined in off-line characterization measurements.

The measurements with TDS and FDS were conducted using two reflection heads

of equal design; therefore the data was not necessarily recorded on precisely the

same spot of the sample. Inhomogeneities in material composition, refractive index,

and local thickness can lead to differences in the thickness values obtained with the

two systems. Hence, we choose the relative uncertainty of consecutive measure-

ments as the figure of merit for the comparison of TDS and FDS measurements.

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Each measurement was repeated 50 times on the same position without movement

of the sample. Afterward, we determined the average thickness value and the stand-

ard deviation. The latter is a measure for the repeatability of the thickness meas-

urement and indicated as error bars where applicable. In general, absolute accuracy

of a measurement is a matter of calibration, which corrects for systematic errors.

However, the feasibility of the technology depends on the capability to obtain the

same results in repeated measurements. Therefore, we use the standard deviation

of subsequent measurements as central figure of merit. This value is independent of

both, the precise calibration of the system and the knowledge of the exact material

parameters. Thereby, the systems can be compared directly without considering any

calibration procedures.

4 Results

In this paper, we measured the same set of single and multi-layer dielectric samples

with both the TDS and the FDS system. The single layer samples were PET foils

(polyethylene terephthalate) with nominal thicknesses of 23µm, 125µm, 250µm,

and 350µm and two Kapton foils (PI, Polyimide) with thicknesses of 50µm and

75µm. The multi-layer sample consists of a ceramic substrate with green spray paint

on both sides. All samples were placed on the sample holder shown in Fig.3 and

measured under nitrogen atmosphere. The thicknesses were determined by using lit-

erature values for the refractive indices, which are

nKapton =

1.87,

nPET =

1.75 [22],

npaint =1.5

, and

nAL2O3=3

[23]. For simplicity, this evaluation does not account

for frequency-dependent refractive indices. The resulting small systematic error

in the absolute thickness values affects both systems similarly. Note that the thick-

ness evaluation procedure does not evaluate the reflection intensity at the interfaces;

therefore, absorption coefficients are not included.

Figure7(a) shows the measured thickness as a function of the nominal thickness

for the six single layer samples. The diagonal, dashed gray line indicates perfect

Fig. 7 (a) Measured thickness as a function of nominal thickness for the set of single layer dielectric

foils. Blue (orange) symbols correspond to FDS (TDS) measurements. The dashed gray line is a guide to

the eye showing perfect agreement of nominal and measured thickness. (b) Standard deviation as a func-

tion of the nominal thickness for 50 consecutive measurements on the single layer PET and Kapton foils

shown in (a)

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agreement between nominal and measured thickness. Blue/orange symbols denote

TDS/FDS measurements, respectively. Above all, the measured values are very

close to the expected nominal thickness. The highest deviation can be observed for

the 350µm-PET foil. As both FDS and TDS measure a lower layer thickness than

expected, we attribute this discrepancy to a slightly thinner or not perfectly homo-

geneous sample. However, both measurement techniques deliver layer thicknesses

with small standard deviations in consecutive measurements (indicated as error

bars) for all single-layer samples. Hence, the thickness can be measured in a repro-

ducible way with both techniques.

As the error bars in Fig. 7 (a) are very small, Fig. 7 (b) compares the standard

deviations of all samples as a function of the nominal thickness. In addition, Table1

summarizes the results. The standard deviation of the thicknesses measured with TDS

ranges from 0.04µm to 0.20µm, whereas the standard deviations of the FDS measure-

ments are 0.10µm to-0.37µm. Hence, the accuracy of the TDS system is by a factor

of approx. two higher compared to the FDS measurements. However, all uncertainties

are in the sub-micron regime, which demonstrates that both systems are well suited

for precise layer thickness measurements. Even for the thinnest sample, the standard

deviation is not higher than 1% of the nominal sample thickness. In particular, this

shows that the single layer PET foil with 23µm in thickness is not yet below the reso-

lution limit of the FDS system. This underlines the quality of the signal acquired with

this system as well as the efficiency of the thickness evaluation procedure.

As mentioned before, the cutoff frequencies of the low- and high-pass filters,

which are applied to the data and the model function in the time domain, affect

the reproducibility of the thickness determination. Here, we demonstrate the effect

exemplarily for the low-pass filter. For the high-pass filter, the arguments are

analogous. If the cut-off frequency of the low-pass filter was too small, a signifi-

cant amount of the signal would be cut out. This would reduce the signal strength

and lead to overlapping peaks at thin interfaces, which increases the uncertainty of

the thickness determination. If the cut-off frequency of the low-pass filter was too

high, the noise in the time trace would increase, which again increases the standard

deviation. This effect can be observed in Fig.8, where the standard deviation of 10

Table 1 Layer thicknesses with absolute and relative standard deviations obtained with TDS and FDS

measurements on single layer PET and Kapton foils

TDS FDS

Material Nominal

thickness

(µm)

Thickness

(µm)

Std. Dev.

(µm)

Rel. Std.

Dev. (%)

Thickness

(µm)

Std. Dev.

(µm)

Rel. Std.

Dev. (%)

PET 23 27.97 0.17 0.61 26.50 0.27 1.02

Kapton 50 47.86 0.06 0.12 45.74 0.18 0.40

Kapton 75 70.75 0.04 0.06 70.31 0.10 0.14

PET 125 126.20 0.10 0.08 125.50 0.16 0.13

PET 250 252.00 0.20 0.08 253.20 0.37 0.15

PET 350 331.85 0.03 0.01 325.80 0.30 0.09

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thickness measurements on a 125µm PET foil is plotted as a function of the cutoff

frequency of the low-pass filter between 0.2THz and 3.5THz. Blue/orange symbols

correspond to TDS/FDS measurements, respectively. The open symbols denote the

values used for the results shown in Fig.7. The standard deviation for both systems

shows a broad minimum, spanning from 0.5THz to 3.0THz for TDS and from 1 to

2THz for FDS. Within these frequency ranges, the standard deviation is well below

0.2µm. Hence, the overall higher bandwidth and dynamic range of the TDS system

leads to a larger interval of possible cutoff frequencies. However, these results also

show that the cutoff frequency used in the previous single layer measurements (open

symbols) did not artificially reduce the accuracy of the TDS system.

Finally, we compared the TDS and FDS system in thickness measurements on a

tri-layer sample. The sample consists of a ceramic substrate (Al2O3) with a nomi-

nal thickness of 640µm and layers of green spray paint of different thickness on

the front and on the backside. The experimental setup and the data evaluation was

identical to the single layer samples. The filters used in the data evaluation were a

first order high-pass with a 3dB cut-off frequency at 0.22THz and a fourth order

low-pass with a 3dB cut-off frequency at 1.72THz. Figure9 shows the normalized

transfer function of the TDS (blue) and the FDS (orange) measurement. The ceramic

substrate and the two paint layers are highlighted by gray and green background

colors, respectively. The different material boundaries can be identified clearly by

their corresponding peaks in the transfer function. Note that differences between

TDS and FDS measurement are small and almost invisible in this figure. The calcu-

lated thicknesses and standard deviations of 50 consecutive measurements are sum-

marized in Table2.

Similar to the measurements on the single layer samples, the absolute values of

the layers differ slightly between the measurements with the two systems. This can be

explained by different measurement positions on the sample. More interestingly, the

standard deviation of subsequent measurements is as low as 0.2µm for the TDS and

0.5µm for the FDS system. Similar to the single layer samples, the standard deviation

of the FDS system is about twice as large as the standard deviation of TDS. Neverthe-

less, the uncertainties of the measurements are well below 1µm for both systems. Note

Fig. 8 Standard deviation of the

thickness measurements on a

125µm PET foil as a function

of the cutoff frequency of the

low-pass filter used in the data

preparation procedure (see

Fig.5). Orange (blue) symbols

correspond to FDS (TDS)

measurements, respectively.

Open symbols denote the cutoff

frequencies used for the results

shown in Fig.7

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that the standard deviation of the first and the second layer is essentially lower than

the value for the backside paint layer. This effect can be explained by the lower signal

strength from the backside, which is caused by THz absorption in the material as well

as reflection losses on previous interfaces (see Fig.9).

5 Summary andOutlook

In this paper, we compared thickness measurement results on single and multilayer

dielectric samples obtained with a commercially available THz TDS system and a

novel optoelectronic FDS system. The measurements were performed in reflection

geometry on the same sample set for TDS and FDS, respectively. The central figure

Fig. 9 Processed transfer function of the TDS (blue) and FDS (orange) measurement on a tri-layer sam-

ple in the time domain. The sample consists of a ceramic substrate with a nominal thickness of 640µm

and green spray paint on the front and the backside. The peaks in the transfer function indicate the inter-

faces between different layers. Green and grey shaded areas indicate paint and ceramic, respectively

Table 2 Layer thicknesses of a tri-layer sample determined with the TDS and the FDS system

TDS FDS

Material Thickness

(µm)

Std. Dev. (µm) Rel. Std.

Dev. (%)

Thickness

(µm)

Std. Dev. (µm) Rel. Std.

Dev. (%)

Spray paint

(front)

46.1 0.1 0.2 44.3 0.2 0.4

Ceramic sub-

strate

619.4 0.1 0.02 616.5 0.2 0.04

Spray paint

(back)

72.0 0.2 0.3 72.8 0.5 0.7

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of merit for the comparison was the standard deviation of 50 consecutive measure-

ments at the same position on the sample. This allows to minimize external error

sources like the inhomogeneity of the sample or the positioning accuracy of the

sample. Hence, this comparison focusses on inherent system properties like instru-

ment noise, dynamic range, and effective bandwidth. For five single layer PET and

Kapton foils with thicknesses between 23µm and 350µm, we found standard devi-

ations smaller than 0.4µm with FDS and below 0.2µm with TDS. Although the

TDS system features a bandwidth of more than 5THz and a peak dynamic range

of 70dB at 1THz, the accuracy of the thickness measurements was just a factor of

two higher than the accuracy of the FDS system, which has a 2THz bandwidth and

a 50dB peak dynamic range at 100GHz. In the thickness measurements on a tri-

layer sample, this tendency manifested itself in the same way: the accuracy of TDS

measurements was again just a factor of two higher than the accuracy of the FDS

measurement. This demonstrates that FDS systems with a comparably simple archi-

tecture can be a promising alternative to complex TDS systems for layer thickness

measurements. Additional effects such as sample positioning and alignment reduce

the accuracy and thereby decrease the advantage of the TDS system. In future work

we aim to extend this cross comparison by including different system architectures

like electronically controlled optical sampling (ECOPS), asynchronous optical sam-

pling (ASOPS), quasi TDS (QTDS) and THz cross-correlation spectroscopy with a

superluminescent diode. Additionally, more sophisticated algorithms and physical

modelling are planned to be applied to the data, which is likely to provide even more

accurate results. Furthermore, different approaches might turn out to be optimal for

a cw or a TDS system.

Authors’ Contributions Conceptualization, L.L.; validation, S.N, R.B.K, S.L., M.A, S.B. and B.G.; inves-

tigation, L.L.; data curation, L.L.; writing—original draft preparation, L.L. and B.G.; writing—review

and editing, M.S. and B.G.; supervision, M.S. and B.G.; funding acquisition, B.G. and L.L. All authors

have read and agreed to the published version of the manuscript.

Funding Open Access funding enabled and organized by Projekt DEAL. This research was funded in

part by Bundesministerium für Bildung und Forschung (BMBF), grant VIP + TeraLayerII, number

03VP07261.

Availability of Data and Material The data that support the findings of this study are available from the

authors on reasonable request.

Declarations

Conflict of Interest The authors declare no competing interests.

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,

which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as

you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-

mons licence, and indicate if changes were made. The images or other third party material in this article

are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the

material. If material is not included in the article’s Creative Commons licence and your intended use is

not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission

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directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen

ses/ by/4. 0/.

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