# Electronic Phase Detection with sub-10 fs Timing Jitter for Terahertz Time-
Domain Spectroscopy Systems

Felix Paries  1,2 Oliver Boidol  1 Georg von Freymann  1,2 and Daniel Molter1
1Fraunhofer Institute for Industrial Mathematics ITWM, Fraunhofer-Platz 1,
67663 Kaiserslautern, GERMANY
2Department of Physics and Research Center OPTIMAS, Technische Universität
Kaiserslautern, 67663 Kaiserslautern, GERMANY
<EMAIL_ADDRESS>
###### Abstract

Terahertz time-domain spectroscopy systems based on resonator-internal
repetition-rate modulation, such as SLAPCOPS [12] and ECOPS [11], rely on
electronic phase detectors which are typically prone to exhibit both a non-
negligible random and systematic timing error. This limits the quality of the
recorded information significantly. Here, we present the results of our recent
attempt to reduce these errors in our own electronic phase detection systems.
A more than six-fold timing-jitter reduction from 59.0 fs to 8.6 fs led to a
significant increase in both exploitable terahertz bandwidth and signal-to-
noise ratio. Additionally, utilizing our interferometrically monitored delay
line as a calibration standard, the systematic error could be removed almost
entirely and thus, excellent resolution of spectral absorption lines be
accomplished. These improvements increased the accuracy of our multi-layer
thickness measurements based on electronic phase detection by more than a
factor of five, pushing the overall performance well into the sub-µm regime.

††journal: oe††articletype: Research Article

##  1 Introduction

Over the last decade, terahertz time-domain spectroscopy [1] has evolved into
an established technology used for a variety of industrial applications such
as multi-layer thickness measurements of automotive coatings [2, 3, 4],
monitoring of drying processes [5], or mail inspection [6]. The idea is to
resolve a sample’s terahertz-pulse response in the time domain making both
amplitude and phase information accessible which in turn presents the
opportunity to calculate the optical path length and the absorption. Since the
desired information is only available on a sub-picosecond timescale, a
sampling approach based on two ultrashort laser pulse trains is required: one
pulse train is used to generate a train of terahertz pulses by repeatedly
short-circuiting a semiconductor-based photoconductive switch to which a high
voltage is applied [7, 8, 9]. Simultaneously, a second laser pulse train is
used to sample the electric field of the generated terahertz pulse train by
repeatedly short-circuiting a second photoconductive switch while measuring
the induced electrical current. For this pump-probe-approach to work, the
phase difference between the two pulse trains must be measured very precisely.
Traditionally, this has been realized through the use of external delay lines
that can be controlled and monitored with sub-2 fs accuracy via
interferometric measurement techniques [10]. However, due to the inertia of
the delay-line’s mechanical components, the scan rates are usually limited to
below 50 Hz [10]. Therefore, these approaches are not able to cover
applications where experimental conditions alter on a millisecond time scale.
To access this time regime, SLAPCOPS [12] and ECOPS [11] systems utilizing
resonator-internal repetition-rate modulation have been developed. These
delay-line-free systems can achieve scan rates up to 1600 terahertz pulse
traces per second [11] but come with the disadvantage of having to measure the
phase difference electronically via fast photo detection which is prone to
exhibit both a non-negligible random and systematic timing error. This in turn
yields a quality decrease in the recorded terahertz spectra. For this reason,
an improvement of existing electronic phase detection systems is needed.

In this paper, we present our recent attempt to reduce random and systematic
errors in our own electronic phase detection system. As will be seen, the
reduction of random errors led to a significant increase in both exploitable
terahertz bandwidth and signal-to-noise ratio. Furthermore, by using an
interferometry-based phase-difference detection as a measurement standard, we
could calibrate our system and thus, remove almost any systematic error.
Consequently, the drawbacks of our electronic phase detection unit could be
reduced resulting in the recording of terahertz spectra that can compete with
those recorded by the use of our external delay lines. To demonstrate the
impact on practical applications, we show that these improvements yield a more
than five-fold accuracy enhancement of our multi-layer thickness measurements
based on electronic phase detection.

##  2 Experimental setup

During the experiments, we utilized the terahertz time-domain spectroscopy
setup schematically shown in Fig. 1. To exclude errors caused by the laser-
intrinsic timing jitter of a dual-laser setup, we used the output of a single
mode-locked ultrashort-pulsed Erbium-doped fiber laser with a repetition rate
of 100 MHz (ELMO, Menlo Systems GmbH) and split it into two equal pulse trains
via a standard fiber-coupled 50:50 splitter. Each pulse train is then
amplified by an Erbium-doped fiber amplifier (ELMA, Menlo Systems GmbH). One
pulse train is guided to a photoconductive terahertz transmitter, whereas the
other one passes an interferometrically monitored delay line before being
guided to a photoconductive terahertz receiver [13]. Additionally, two fiber-
coupled 80:20 splitters are used to send 20% of each branch to a separate
photodetector, respectively. This setup allows for a twofold measurement of
the pulse trains’ phase difference: via an interferometric approach
determining the delay-line position with a precision of 1.1 fs [10], and via
an electronic approach utilizing a fast photo detection of the pulse trains’
higher-harmonics’ difference signal in combination with a frequency mixer. In
this way, the accurate interferometric phase detection can serve as a
measurement standard allowing to compare the electronic phase detection
against. This enabled us to examine both the random and systematic errors of
the electronic phase detection as well as their implications on the resulting
terahertz spectra. As a consequence of these experiments, we adjusted our
electronic phase detection unit from a previously used analog I-Q phase
transmission to a new 16-bit digital phase transmission. This paved the way
for a significant reduction of random and systematic timing error which in
turn causes a significant terahertz performance enhancement.

Figure 1: a) Experimental setup: Using a fiber-coupled splitter (50:50), the
output of an mode-locked Erbium-doped fiber laser (ML EDFL) is split into two
equal parts which are amplified by an Erbium-doped fiber amplifier (EDFA),
respectively. One pulse train is guided to a photoconductive terahertz
transmitter (Tx), whereas the other one passes a mechanical delay line before
being guided to a photoconductive terahertz receiver (Rx). 20% of each pulse
train is outcoupled for photo detection (PD). The phase difference is measured
twice: via an interferometric and via an electronic approach. This allows for
a comparison between the interferometric phase detection (measurement
standard) and the electronic phase detection. b) Previous analog phase
transmission architecture vs. new 16-bit digital phase transmission
architecture. Switching from an analog to a digital phase transmission
eliminated the digital-to-analog-conversion jitter and opened doors for a
bandwidth-controlled jitter reduction.

##  3 Simulation

Random and systematic errors influence the quality of the recorded terahertz
spectra in different ways. In order to substantiate our understanding of these
influences, we conducted various numerical simulations. The simulation’s
design is shown in Fig. 2. Starting with data from the well-known HITRAN
database [14], we calculated the terahertz absorption spectra for moist air
under consideration of the terahertz optical path length, the relative
humidity, the air pressure, and the temperature. The Kramers-Kronig-relation
[15] is then used to transform the absorption coefficient into a complex
refractive index. Additionally, the transformation of an ultra-short laser
pulse to an ideal terahertz pulse is modelled by taking into account the laser
pulse duration and the carrier lifetime in the semiconductor of a
photoconductive switch. The Fourier transformation of the derived terahertz
pulse then yields an absorption-free terahertz spectrum. To impose the water
vapor’s absorption characteristics onto the absorption-free terahertz
spectrum, a convolution of the complex refractive index and the terahertz
spectrum is applied. This yields a simulated absorption spectrum which can be
back-transformed into the time domain via an inverse Fourier transformation to
obtain a terahertz pulse containing the absorption information. By applying a
distortion model to the time axis, it is now possible to simulate an erroneous
phase detection and examine the influence of both a time-axis’ random and
static error. As a last step, the newly created distorted pulse is Fourier-
transformed to obtain the desired distorted terahertz absorption spectrum.

Figure 2: Design of the conducted simulation. Water vapor absorption
information from the HITRAN database is imprinted onto a simulated terahertz
spectrum and back-transformed into the time domain. Dependent on the model,
either a random or a systematic distortion is applied to the time axis of the
terahertz pulse containing the absorption information. A final Fourier
transformation reveals the influence on the terahertz spectrum.

###  3.1 Simulation of random error

As the simulation shows (Fig. 3), a random time-axis error translates to a
flickering of the FFT-generated spectrum’s tail in the frequency domain. One
hundred single shots of simulated dry-air terahertz spectra demonstrate this
phenomenon for three different jitter settings (0 fs, 10 fs, and 60 fs) (top).
The more time-axis jitter is applied, the earlier a tail flickering becomes
notable and less usable terahertz bandwidth is available. This effect is
further quantified by calculating the signal-to-noise ratio (left) as well as
the standard deviation over frequency (right). As expected, an applied time-
axis jitter reduces the signal-to-noise ratio and thus the simulated noise
floor is already reached at a lower frequency. The standard deviation
visualizes the different jitter levels in the frequency domain.

Figure 3: Top: One hundred single shots of simulated dry-air terahertz spectra
for three different random time-axis error settings (0 fs, 10 fs, 60 fs). An
applied random time-axis error reduces the usable terahertz bandwidth. Left:
Signal-to-noise ratio. Jitter reduces the signal-to-noise ratio and thus, the
noise floor is reached at a lower frequency already. Right: Standard deviation
over frequency. The different jitter levels can be seen clearly.

###  3.2 Simulation of systematic error

In contrast to a random error, a systematic error does not reduce the usable
bandwidth, but rather distorts the terahertz spectrum. This becomes apparent
by taking a close look at the shape and position of the spectral absorption
lines (Fig. 4). Dependent on the local distortion, the spectral lines develop
an asymmetry and are shifted away from their actual position. As a result,
spectral lines that are close together interfere and cannot be resolved
anymore, rendering the absorption information useless.

Figure 4: Top: Deviation over phase difference. The simulated systematic error
is shown in purple. Bottom: The systematic error distorts the terahertz
spectrum which affects the shape and position of the spectral lines. This
leads to a pronounced asymmetry and a positional shift. Thus, spectral lines
start to interfere and cannot be resolved anymore. The spectra are offset for
better visibility.

##  4 Results

###  4.1 Investigation of random error

####  4.1.1 Jitter

The setup shown in Fig. 1 practically exhibits four different processes that
contribute to the overall random error of the system: optical-to-electronical
conversion, analog-to-digital conversion, digital-to-analog conversion, and
externally induced noise. We found that the digital-to-analog conversion at
the output of the used FPGA causes a significant, non-linear noise
contribution which yields a phase-dependent jitter (purple plot in Fig. 5). To
eliminate this error, we switched from an analog phase transmission to a
16-bit digital phase transmission. This did not only remove the digital-to-
analog conversion as a non-linear noise source, but also opened the door for
transmission-bandwidth control and thus a methodical jitter reduction. Figure
5 demonstrates the results of these achievements. Firstly, the change from an
analog to a digital phase transmission eliminated the phase dependence.
Secondly, dependent on the applications requirements, a bandwidth-jitter
operation point can be chosen freely. For the terahertz experiments presented
in this paper, we chose the 250 kHz-8 fs operation point, since the DAQ
receives the interferometrically determined phase information at 250 kHz.

Figure 5: Measured jitter in dependence on phase difference for analog and
16-bit digital phase transmission. The phase-difference dependence of the
analog phase transmission is clearly visible (purple). It is mainly caused by
a non-constant jitter characteristic of the digital-to-analog converters. A
16-bit digital phase transmission solves this problem resulting in a nearly
phase-difference independent jitter. By adjusting the transmission bandwidth,
the jitter can be reduced to the sub-10 fs regime (orange). Dependent on the
application’s requirements the optimal bandwidth-jitter operational point can
be chosen freely.

####  4.1.2 Terahertz dry-air absorption spectra

To investigate the influence of the jitter reduction on the terahertz
performance, one hundred single shots of dry-air terahertz absorption spectra
have been recorded for each phase detection setup (interferometric, analog,
and 16-bit digital). Figure 6 (top) displays the achieved terahertz
performance enhancement. In accordance with the simulation (Fig. 3), the newly
developed digital phase detection setup with sub-10 fs jitter exhibits reduced
flickering and increases the practically usable terahertz bandwidth by approx.
1 THz.

####  4.1.3 Signal-To-Noise

To further investigate the achieved improvement, the signal-to-noise ratio as
well as the standard deviation versus frequency were calculated. The results
are shown in the bottom half of Fig. 6 and are in qualitative agreement with
the simulation. Comparing the previously used analog transmitting phase
detection unit with the newly developed digital transmitting phase detection
unit, an increase of the signal-to-noise ratio by up to a factor of 7 (left
side) can be seen. This shifts the practically usable upper terahertz limit to
above 2.5 THz, which is sufficient for most technical applications. In
addition, the increase in signal-to-noise ratio translates directly into a
significant reduction in measurement times. A look at the plot of the standard
deviation versus frequency completes the pictures (right side).

Figure 6: Top: One hundred single shots of dry-air terahertz absorption
spectra. The newly developed phase detection unit utilizing a 16-bit digital
phase transmission (green) clearly outperforms our previously used phase
detection unit utilizing an analog phase transmission (purple): The
practically usable terahertz bandwidth is increased by approx. 1 THz. This
closes the performance gap to the technically superior interferometric
approach (orange) further. Left: signal-to-noise ratio calculated from the
recordings above. Compared to the previously used analog transmitting phase
detection unit (purple), the signal-to-noise ratio of the newly developed
digital transmitting phase detection unit (green) increases by up to a factor
of 7, resulting in a practically usable terahertz bandwidth of more than 2.5
THz. The signal-to-noise ratio of the interferometric phase detection unit
(orange) illustrates the potential for improvement. Right: standard deviation
versus frequency calculated from the same one hundred individual images. The
graph confirms the significant improvement from the previous analog to the
newly developed digital approach. All results qualitatively agree with the
simulation.

###  4.2 Investigation of systematic error

####  4.2.1 Calibration

Besides a random error causing a time-axis jitter, there is a systematic error
causing a static distortion of the time axis. As a result, the spectral
absorption lines in the FFT-derived terahertz spectra develop an asymmetry and
shift away from their actual position (see simulation). Using the phase
difference information acquired by the interferometric phase detection unit as
a measurement standard, the systematic error could be determined. The result
is shown in Fig. 7: The systematic error is significant and the absolute value
exceeds 600 fs at the largest deviation. However, since this error is static,
it can be stored in a look-up table which is then used to calibrate the
system. After calibration, the systematic error is removed and only a random
error remains.

####  4.2.2 Influence on spectral features

The systematic-error-induced distortion of the terahertz absorption spectra
becomes evident by taking a close look at the spectral absorption lines of
water vapor in air. Figure 7 shows the mean of 20 ambient-air terahertz
absorption spectra, each recorded with an integration time of 1 s, for the
newly developed digital transmitting phase detection unit without calibration
(purple) and with calibration (green) as well as for the interferometric
measurement standard (orange). The zoom on the right side exemplarily reveals
the improvement achieved by the calibration: contrary to the non-calibrated
system, the calibrated system is able to resolve the spectral absorption lines
and performs as good as the measurement standard, nearly without any
noticeable difference.

Figure 7: Top: Deviation over phase difference. The systematic error is
clearly visible and its absolute value exceeds 600 fs at the largest deviation
(purple). Using a look-up-table approach, the system can be calibrated. This
results in a nearly systematic-error-free system with only a small random
error left (green). Bottom: Mean of 20 ambient-air terahertz absorption
spectra recorded with 1 s integration time (offset for better visibility). At
first glance, the spectra on the left side look quite similar. However, a
closer look reveals the differences in accordance with the simulation: the
non-calibrated system is not able to resolve the spectral absorption lines
(purple). In contrast, the calibrated system (green) resolves the spectral
absorption lines identically to the interferometric measurement standard
(orange).

###  4.3 Multi-layer thickness measurement

Non-destructive, contactless, multi-layer thickness measurement is an
established industrial terahertz application. By knowing the refractive
indices of the layers under test, the thickness of each layer can be derived
from extended analysis of the reflected (or transmitted) terahertz signal even
for sub-wavelength thicknesses [2, 3, 4]. The higher the resolution of the
recorded terahertz time-domain signal, the higher the accuracy of the layer
thickness determination. Figure 8 a) exemplarily visualizes the results of
1000 consecutive measurements for a three-layer system dependent on the used
phase detection unit. Clearly, the reduced timing jitter and systematic error
of the newly developed digital transmitting electronic phase detection unit
yields a convincing accuracy enhancement in comparison to the previously used
analog transmitting electronic phase detection unit. Similar to the signal-to-
noise ratios displayed in Fig. 6, meeting the performance of the
interferometric phase detection approach is still a few innovations ahead.
However, as shown in Fig. 8 b), the standard deviation could be pushed well
into the sub-µm regime which is fairly sufficient for most industrial
applications.

Figure 8: a) Layer-thickness measurement results of a three-layer coating on
metal of 1000 consecutive measurements each. The newly developed digital
transmitting phase-detection unit outperforms the previously used unit clearly
(green vs. purple). The performance of the interferometric measurement
standard reveals the potential for further innovations (orange). b) Standard
deviation for the three phase-detection systems under test. The standard
deviation of the electronic phase-detection unit could be reduced by a factor
of 5.8, 6.8, and 6.4 for the first, second, and third layer, respectively,
pushing the overall performance well into the sub-µm regime (green vs.
purple).

##  5 Conclusions

We have presented the recent design wins of our electronic phase detection
unit development which include a more than six-fold reduction of timing jitter
reaching the sub-10 fs regime, as well as a calibration technique to remove
almost any systematic error. We achieved this by switching from an analog I-Q
phase transmission to a 16-bit digital phase transmission which allows for
application-specific transmission bandwidth control, and by utilizing our
interferometrically monitored delay line with an accuracy of about 1.1 fs as a
calibration standard. This led to a significant increase in exploitable
terahertz bandwidth and signal-to-noise ratio, as well as an distortion-free
resolution of spectral absorption lines. These results were substantiated by
an extensive simulation which is in excellent qualitative agreement with the
measurements. Finally, we demonstrated that our design wins yield a more than
five-fold accuracy improvement of our electronic-phase-detection-based multi-
layer thickness measurements. These improvements represent an important step
towards higher resolution at faster measurement times.

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