Periodic sampling errors in terahertz time-domain measurements

Arno Rehn; David Jahn; Jan. C. Balzer; Martin Koch

doi:10.1364/OE.25.006712

1. Introduction

THz Time-Domain Spectroscopy (THz TDS) has become an excellent tool for spectroscopic analysis of a large variety of materials in the THz frequency range. In the range from 0.1 THz to 10 THz many interesting phenomena can directly be accessed, such as intra- and intermolecular vibrations [1, 2], transitions between rotational lines of gases [3] and carrier dynamics in semi-conductors [4,5] - to name only a few. For more than two decades, THz TDS has been used for numerous studies ranging from pure spectroscopy applications over pump-probe experiments to imaging setups in industrial environments [6, 7]. A lot of effort has been spent during the last years to improve data analysis of the spectroscopic information recorded with THz TDS [8–13]. Moreoever, for further development of new THz TDS systems increasing the SNR and bandwidth is mandatory. In both cases, data interpretation and system improvement, understanding the sources of error is of paramount importance. Comparability and reproducability of THz TDS data can only be achieved if all contributions of errors are understood and if systematic errors are eliminated. A wide variety of publications dedicated specifically to measurement uncertainties are available, even with special focus on THz TDS [14–17].

To date, however, systematic errors in the sampling process of waveforms have not yet been specifically targeted by any such publication. While several systematic and statistical errors from sample alignment to optical data extraction have been discussed in detail, the origin and impact of systematic errors in the sampling process are not yet understood.

In this paper, we have investigated the sources as well as the effects of systematic sampling errors. We show that the periodic sampling time error is potentially the most common type and also the most severe.

First, we discuss the different origins of the periodic sampling error. Two examples are given to illustrate the origin and impact of this type of error. In a second step, we model the systematic error for arbitrary waveforms f (t) and investigate the impact on the waveform’s spectrum. These observations are not restricted to the field of THz TDS and apply generally to optical sampling experiments.

On the basis of simulated and measured THz TDS data, we provide real-world examples and examine the influence on optical parameter extraction from THz measurements.

2. Problem assessment

Figure 1 shows a typical THz TDS setup. The sampling of the THz pulse is achieved by changing the time difference of arrival of the optical pulses at emitter and detector. Enumerating the ways in which the sampling of a waveform can be affected, four main points come to mind immediately:

The sampling clock is going too fast or too slow by a constant factor.
This type of error will simply stretch the time and frequency axes. It can usually be remedied by doing a calibration on a sample with known features. For this purpose, water absorption lines are often used as a quick check for the frequency calibration. Their frequencies are well known and they are naturally present in broadband measurements. Other methods might use CO absorption lines or exploit the etalon effect [18, 19].
The sampling clock rate is fluctuating randomly.
This effectively amounts to a statistical sampling error and thus just decreases the Signal to Noise ratio (SNR). A more thorough analysis was done by Jahn et al. with focus on THz TDS measurements [16].
Some samples are missed.
Missing samples can potentially create jumps and discontinuities in the data. As such, they mainly influence the high-frequency regime of the data. The missed samples are usually statistically distributed and consequently should only decrease the dynamic range (DR) of the upper frequency regime. Oversampling can easily dampen this effect. Hence, it will not be discussed further in this paper.
The sampling clock period is fluctuating periodically.
This type of error easily occurs when multiple, unsynchronised clocks are part of the data acquisition process or when the sampling device itself is imperfect. The resulting sampling error τ(t) is highly periodic and systematic. Thus, it often can’t be effectively eliminated by averaging many measurements. Because of its perseverance and potentially high impact on measurements, we will further investigate this error in the following sections of the paper.

Fig. 1 This figure shows a sketch of a typical THz TDS setup. The delay line changes the optical path lengths of the emitter and/or detector paths. The sampling of the THz pulse is thus controlled by the delay line position and trigger pulses are generated by the controller, acting as a clock. The actual time of acquisition is, however, determined by the lock-in amplifier’s clock.

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First, to demonstrate the relevance, the next paragraphs give two real-world examples where this type of sampling error has visibly affected the measurement. In the first example it is demonstrated that two unsynchronised clocks (the delay line trigger and the lock-in data acquisition) may yield a periodic sampling error. The second one is more straightforward: In this example, a periodic non-linearity of the movement of the delay line causes a periodic sampling error. This type of error is present in low-cost delay lines but may also become apparent in high precision measurements using highly accurate laboratory components.

2.1. Triggered acquisition of THz TDS data

THz TDS data is often acquired by delaying the laser pulse used for sampling with a motorised translation stage. For fast acquisition, the controller of this translation stage emits trigger pulses when the stage has travelled a specified distance. This signal triggers the connected lock-in amplifier to record a sample.

While not immediately obvious, this setup includes two unsynchronised clocks: First, there is the controller, which generates equidistant trigger pulses (assuming perfectly linear motion). Second, there is the internal clock of the lock-in amplifier with which it checks for the incoming trigger pulses. Since the frequency basis for both of these clocks is different, there will be sawtooth-like time difference between the incoming trigger signal and the actual recording of the sample.

In the case of the popular Stanford Research SR800 series lock-in amplifiers, the internal clock runs at 512 Hz. This creates a comparably large time window of about 2 ms in which any incoming trigger signal (of sufficient length) is mapped to the same sample. The temporal error for sample number n can be expressed as

σ_{LIA} (n) = n \cdot Δ t \mod \frac{v_{move}}{512 Hz},

where Δt denotes the spacing between two trigger pulses and v_move is the speed of the translation stage (both as seen from the stage’s controller). Figure 2 shows the resulting sampling time error τ(t) for three different movement speeds. The sampling error τ(t) is the deviation of the linear time axis and the sampled time axis. It follows a sawtooth like pattern and its amplitude increases for higher delay line speeds. The data is presented in units of the optical delay for easier comparison with usual THz TDS measurements.

Fig. 2 These graphs show the deviations τ(t) of the correct time axis to an unsynchronised second time axis for different movement speeds. The shape of the deviation τ(t) appears to be sawtooth-like. In the context of optical sampling, the error’s amplitude scales linearly with the movement speed of the translation stage and can lead to complex periodic structures.

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The effects of this periodic sampling error on the spectrum are shown in Fig. 3. For data acquisition a realistic movement speed of 3 ps/s has been used. The step width Δt is 12 fs. It can be observed that the signal in the spectrum seemingly never fully drops to the noise floor. Instead, even the very high frequency regime is dominated by “wobbles”. These artefacts of the sampling error appear so close to the actual THz spectrum to reasonably assume that the highest THz frequency components are influenced in some way as well.

Fig. 3 The effects of the unsynchronised clocks are evident: If each acquisition of a sample is triggered individually (green), the signal never fully drops to the noise floor. Instead, some kind of “wobbles” dominate most of the frequency range. It is very likely that the actual THz spectrum at around 1.5 to 2 THz is influenced by these artefacts, even though it is not apparent in which way. These effects do not occur when only the scan’s start is triggered and samples are acquired continuously afterwards (orange). Here, the sampling rate is solely determined by the lock-in amplifier’s internal clock.

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2.2. Delay line non-linearity

Since in most THz TDS measurements the delay line acts as one of the sampling devices, any non-linear movement that is not correctly tracked will directly result in a periodic and systematic sampling time error. Note that in contrast to the unsynchronised sampling clocks described above, the error is now independent of the delay line’s movement speed. Instead, the non-linearity is symptomatic of the hardware’s mechanics. Most high-accuracy, commercial translation stages do not exhibit such non-linear behaviour. However, this is not always the case for fast-scanning mechanical delay lines or those optimised for low cost [20].

For demonstration, we inspect the data from Probst et al. [20] more closely: In the mentioned publication, a THz Quasi-TDS system was built by using the sled from a CD drive as the delay line’s translation stage. Interferometric monitoring of the movement profile showed strong periodic deviations from linear movement. These deviations can be approximately described by a superposition of three sines:

σ_{CD} (t) \approx \sum_{i = 1}^{3} A_{i} \sin (2 π f_{i} t + ϕ_{i})

The acquired THz data’s frequency domain representation again shows “wobbly” artefacts that do not appear when a better translation stage is used (c.f. Fig. 4). The interferometric measurements in Fig. 4 a) shows that in case of the cheap delay line a strong periodic sampling error can be measured directly. The spectrum reveals similar patterns than those shown in Fig. 3. These spectral features are related to the periodic sampling error, which will be shown in the next section.

Fig. 4 Non-linear movement of the delay line effectively amount to periodic sampling time errors in optical sampling systems. a) shows the deviation from linear movement for two investigated delay lines. b) displays the resulting THz spectra. Due to the pronounced non-linearity of the re-purposed CD drive, spectral artefacts at around 1 THz appear.

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In the following, we want to illustrate that the periodic sampling error will eventually have an impact on the extracted optical parameters. Probst et al. [20] evaluated the birefringence of a liquid crystal polymer (LCP) sample. The sample was measured with both the Thorlabs translation stage and the CD drive. Average refractive indices were calculated from the time shift between two measurements as determined from their cross-correlation. The authors discovered that the birefringence as it has been extracted from the measurements performed with the CD drive distinctly depend on the used evaluation method (either directly or indirectly evaluated, see Table 1). On the other hand, the results obtained from the measurements with the Thorlabs delay line were reported to be consistent with themselves. This shows that the systematic error can have unforeseen influences on the optical parameter extraction.

Table 1. This table shows the birefringence Δn_LCP of an LCP sample as extracted by Probst et al. The periodic error exhibited by the CD drive creates inconsistent data, leading to results which depend on the used evaluation method.

View Table

3. Theoretical model

To understand the influence of systematic sampling errors, we first have to create a model of the measurement process and the deviation. In a first step we derive the incorrectly sampled time-domain trace f_s (t) and its spectrum ${\hat{f}}_{s} (ω)$ . The systematic sampling error is described by σ(t), which is the deviation from the linear time axis at each time position t.

In a second step, we restrict ourselves to periodic deviations σ(t) = A cos(νt + ϕ). More general periodic deviations can be written as a Fourier series f (t) = ∑_i A_i sin (ν_it + ϕ_i) making it sufficient to do the following calculation for the generic case of a sinusoidal deviation with a fixed angular error frequency ν and phase ϕ.

3.1. Ab initio calculations

Generic examination

Consider that we want to sample a function f(t) at different values of t. However, our method of sampling at the correct point in time is imperfect and a systematic sampling time error σ(t) is introduced. Thus, instead of f(t) we are actually sampling a function f_s(t):

f_{s} (t) = f (t + σ (t)) .

In order to remove σ from f’s argument, we expand f_s(t) into a Taylor series in point t:

f_{s} (t) = \sum_{n = 0}^{\infty} \frac{1}{n!} \frac{\partial^{n} f}{\partial t^{n}} σ^{n} (t) .

To understand the influence of σ on the spectrum, we need to do a Fourier transform into the frequency domain:

{\hat{f}}_{s} (ω) = \sum_{n = 0}^{\infty} \frac{1}{n!} ℱ_{t} [\frac{\partial^{n} f}{\partial t^{n}} σ^{n}] (ω) .

By applying the convolution theorem, we can individually transform f and σ:

{\hat{f}}_{s} (ω) = \sum_{n = 0}^{\infty} \frac{1}{{(2 π)}^{n} n!} {ℱ_{t} [\frac{\partial^{n} f}{\partial t^{n}}] * ℱ_{t} {[σ]}^{* n}} (ω)

= \sum_{n = 0}^{\infty} \frac{1}{{(2 π)}^{n} n!} ({(i ω)}^{n} \hat{f} (ω)) * {\hat{σ}}^{* n} (ω)

where x * y is the convolution of x with y and x^*ⁿ the convolution power, i.e. the n-fold convolution of x with itself.

Periodic deviation

We will now consider the case of a simple periodic deviation, which represents the basis for the systematic sampling errors discussed in the previous section. As explained in the introduction of this section, other periodic error functions can easily be constructed as a superposition of this fundamental case.

We define the sampling time error function

σ (t) = A \cos (v t + ϕ)

with amplitude A, angular frequency ν and phase ϕ. Its frequency domain representation is

\hat{σ} (ω) = A π [e^{i ϕ} δ (ω - ν) + e^{- i ϕ} δ (ω + ν)] .

The n-fold convolution of $\hat{σ}$ with itself is

{\hat{σ}}^{* n} (ω) = {(A π)}^{n} {[e^{i ϕ} δ (ω - ν) + e^{- i ϕ} δ (ω + ν)]}^{* n}

= {(A π)}^{n} \sum_{k = 0}^{n} (\begin{array}{l} n \\ k \end{array}) e^{i (n - 2 k) ϕ} δ (ω - (n - 2 k) ν)

\equiv {(A π)}^{n} \sum_{k = 0}^{n} (\begin{array}{l} n \\ k \end{array}) e^{i Φ_{k}^{n}} δ (Ω_{k}^{n} (ω)) .

This follows from applying the binomial theorem to the n-fold convolution and using the translation property of the Dirac delta. For simplicity, the definitions

Ω_{k}^{n} (ω) \equiv ω - (n - 2 k) ν,

Φ_{k}^{n} (ω) \equiv (n - 2 k) ϕ

were introduced.

Inserting (12) into (7), we obtain

{\hat{f}}_{s} (ω) = \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} (\begin{array}{l} n \\ k \end{array}) \frac{{(A π)}^{n}}{{(2 π)}^{n} n!} e^{i Φ_{k}^{n}} ({(i ω)}^{n} \hat{f} (ω)) * δ (Ω_{k}^{n} (ω))

= \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{{(i A Ω_{k}^{n} (ω))}^{n}}{2^{n} k! (n - k)!} e^{i Φ_{k}^{n}} \hat{f} (Ω_{k}^{n} (ω))

= \hat{f} (ω) + \sum_{n = 0}^{\infty} \sum_{k = 0}^{n} \frac{{(i A Ω_{k}^{n} (ω))}^{n}}{2^{n} k! (n - k)!} e^{i Φ_{k}^{n}} \hat{f} (Ω_{k}^{n} (ω)) .

As is apparent, the spectrum of the imperfectly sampled signal contains the original overlaid with shifted and scaled copies of itself (henceforth called “mirror spectra”).

Assuming a weak sampling error, we can drop all terms but the first in the above sum:

{\hat{f}}_{s} (ω) = \hat{f} (ω) + \frac{i A}{2} e^{i Φ_{0}^{1}} Ω_{0}^{1} \hat{f} (Ω_{0}^{1}) + \frac{i A}{2} e^{i Φ_{1}^{1}} Ω_{1}^{1} \hat{f} (Ω_{1}^{1})

= \hat{f} (ω) + \frac{i A}{2} [e^{i ϕ} (ω - ν) \hat{f} (ω - ν) + e^{- i ϕ} (ω + ν) \hat{f} (ω + ν)] .

In this first order approximation, the periodic sampling error thus creates a phase-shifted mirror image of the original spectrum $\hat{f} (ω)$ about the central frequency ν of the error.

An important conclusion from this result is that the error will not be cancelled out in the transfer function $H_{s} = {\hat{f}}_{s}^{samp} / {\hat{f}}_{s}^{ref}$ of a measurement. Since the reference and sample spectra are different, the appearing mirror spectra will also be different and hence they will not cancel in the transfer function H_s. More mathematically speaking, even though the sampling error is present in both acquired spectra ${\hat{f}}_{s}^{ref}$ and ${\hat{f}}_{s}^{samp}$ , the resulting transfer function reads

H_{s} (ω) = \frac{{\hat{f}}_{s}^{samp} (ω)}{{\hat{f}}_{s}^{ref} (ω)} = \frac{{\hat{f}}^{samp} (ω) + \sum ({\hat{f}}^{samp}, ω, ν, ϕ)}{{\hat{f}}^{ref} (ω) + \sum ({\hat{f}}^{ref}, ω, ν, ϕ)} .

The periodic sampling error’s additional summands can not be reduced from the fraction when reference and sample measurements are carried out. Hence, this error will also influence the calculated optical constants which are derived from the transfer function H(ω). We will quantify the effect of the periodic sampling error on the extracted optical constants in the following two sections.

3.2. Application to THz TDS spectra

We will now apply the derived formula to model THz data. Following Xu et al. [21], we model a THz pulse as

f (t) \propto \frac{t}{\sqrt{2 π τ^{2}}} \exp (- \frac{t^{2}}{2 τ^{2}})

in the time-domain. τ is a parameter defining the width of the pulse. The frequency domain representation is

\hat{f} (ω) \propto - i ω τ^{2} \exp (- \frac{1}{2} τ^{2} ω^{2}) .

In all following calculations, we chose τ = 150 fs.

Figure 5 shows the original THz spectrum and its variant under the influence of a periodic sampling error. The spectra are calculated using the equations (19), (17) and (22). As expected, we see that the approximation closely follows the exact calculation up to the first mirror spectrum. The exactly calculated incorrectly sampled spectrum contains additional mirror spectra at frequencies higher than ν, but at a significantly lower amplitude.

Fig. 5 Comparing the original spectrum with the spectrum under the influence of a periodic sampling error, “mirror spectra” appear at integer multiples of the error frequency ν. The first order approximation only describes the most intense mirror spectrum at ν.

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Compared to the unaffected spectrum, the periodic sampling error can potentially have a significant impact on the high frequency components (depending on the amplitude A and the angular frequency ν of the error). An example for this can be seen in Fig. 6. For illustration, the original spectrum contains three absorption lines. The sampling error is a single sine with amplitude A = 0.15 fs, frequency ν/(2π) = 3.3 THz and phase ϕ = 1.5 π. Even though the sampling deviation’s amplitude is very low (A = 0.15 fs) compared to typical time scales of THz measurements, the influence on the spectrum is distinct. Due to the mirror spectrum, a higher bandwidth is faked and absorption features are artificially created. The fake features are not even readily identified as such, since the transition from “real” spectrum to “fake” spectrum is smooth. The unwrapped phase is linear over the whole extended spectrum (except for the expected discontinuities at the absorption lines) and thus also conceals the presence of the error. The phase is not pictured for the sake of readability.

Fig. 6 Depending on the parameters, a periodic sampling error can have a significant impact on the spectrum. In this figure, the effects of a very weak error (with regard to it’s amplitude, compared to typical THz time scales) with a frequency within the bandwidth of the original THz spectrum are shown. The spectrum affected by the sampling error has a higher, but fake, bandwidth and mirrors some of the original absorption features. The error is not readily discernible as such and the fake spectral features might be taken for real.

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But as also can be seen from this example, if the error frequency ν is larger than double the pulse’s bandwidth, the error will not have any effect on the usable part of the spectrum.

For further study later on, it is useful to know the frequencies at which the mirror spectrum has the same power as the original spectrum (we will further call these the “equi-power lines”). For that, we solve

{| f (ω) |}^{2} = {| \frac{i A}{2} \exp (i ϕ) \cdot (ω - ν) \cdot f (ω - ν) |}^{2}

\overset{(22)}{\Rightarrow} ω = \frac{A}{2} \cdot {(ω - ν)}^{2} \cdot \exp (ω ν τ^{2}) \exp (- \frac{1}{2} ν^{2} τ^{2})

\Rightarrow \frac{2}{A} \exp (\frac{1}{2} ν^{2} τ^{2}) \equiv C = \frac{{(ω - ν)}^{2}}{ω} \exp (ω ν τ^{2})

for ω. This equation is not analytically solvable. We therefore apply some approximations:

ω ≪ ν: $\frac{{(ω - ν)}^{2}}{ω} \approx \frac{ω^{2}}{ν}$ $\Rightarrow ω \approx - \frac{1}{ν τ^{2}} W_{n} (- \frac{ν^{3} τ^{2}}{C})$
ω ≫ ν: $\frac{{(ω - ν)}^{2}}{ω} \approx ω$ $\Rightarrow ω \approx - \frac{1}{ν τ^{2}} W_{n} (C ν τ^{2})$

W_n denotes the Lambert W function, defined by z = W_n(ze^z). The function is multi-valued and hence has multiple branches n. Since we only want to look at the real-valued solutions for ω, we can restrict our attention to the branches n = −1 and n = 0 for ω ≪ ν and n = 0 only for ω ≫ ν [22].

3.3. Influence on the transfer function

To investigate the influence on extracted optical constants and spectral features, the transfer function of two model THz pulses at different temporal positions has been calculated for different periodic error frequencies. The shift is Δt = 0.83 ps, corresponding to the optical path length difference of an idealised, completely transparent sample with refractive index n = 1.5 and thickness d = 500 µm. The simulations were done by evaluating Eq. (3) with a periodic sampling error as described in Eq. (8). As before, the THz pulse shape is taken from [21].

The bandwidth of this simulated system is approximately 5.3 THz with a peak dynamic range (DR) of 100 dB (single shot). The DR is limited by artificially introducing white noise in the simulated time-domain trace. The sampling error’s amplitude was set to A = 1 fs.

Examining the transfer function of this idealised sample is an easy yet effective way to analyse the impact of the sampling error. With perfect sampling, its magnitude is flat and equals |H₀| = 4n/(n + 1)² due to reflection losses, where n denotes the refractive index of the sample. Any differences in magnitude are thus due to the error. Note that we are only considering a single pass of the THz pulse and neglect any Fabry-Pérot reflections. The other, equally important parameter of the transfer function is its phase. While we could look at all phase values individually, it is more useful to explore the difference of the refractive index to its nominal value. In a simple model, the refractive index is just a differently scaled measure of the phase shift introduced by the sample. Again with perfect sampling, this difference to the nominal value should be zero – any deviations can be attributed to the sampling error.

The magnitude of the transfer function and difference to the expected refractive index of n = 1.5 are shown in Fig. 7. The influence can be seen very well and is strongest in the high frequency regions. More specifically, the dominant influences are bounded by the equi-power lines towards the low-frequency regime. Strong influences in the region ν/(2π) > 10 THz are higher order effects which are not accounted for by the approximation in Eq. (19).

Fig. 7 Top left: Deviation of the the transfer function’s magnitude |H| from the proper transfer function’s magnitude |H₀| in logarithmic scale. Top right: The resulting refractive index error Δn. Bottom: Zoom of the refractive index error to the region of main interest. The THz frequency ω/(2π) and error frequency ν/(2π) are in the region 0 THz ≤ ω/(2π), ν/(2π) ≤ 4 THz. Errors in the magnitude of the transfer function as well as in the calculated refractive index appear dominantly on the high-frequency side of the equi-power line.

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As is evident, even seemingly small sampling errors can have a significant impact on low intensity spectral features. In the presented example, errors in the magnitude can exceed 8 dB and the refractive index can be off in up to the first decimal place.

Now one may argue that a peak DR of 100 dB is not usually achieved and that consequently, the sampling error will not be of any significant impact. While that might be true for single shot measurements, the viability of averaging many measurements to achieve a peak DR of 90 dB has already been demonstrated [23]. If the source of the periodic sampling error has no random components (e.g. if the delay line or associated electronics such as the encoder exhibit non-linear behaviour), then the averaging procedure will not dampen its impact – it is thus still relevant. In this case, examining the impact of the sampling error on an averaged measurement is equivalent to examining the impact on a single shot measurement with the same peak DR, as was done in this simulation.

It should be noted that in the previously discussed special case of unsynchronised delay line controller clock and lock-in amplifier sample clock, the phase between both clocks will be different and random with every measurement. Hence, averaging many measurements of this type will actually mitigate the periodic sampling error. In terms of the initially presented examples, this simulation only describes the case of delay line non-linearity.

But even considering a system with a lower dynamic range, Fig. 7 still provides important insights. Namely, only the dominant influences of the error are on the high-frequency side of the equi-power line. Looking more closely at the low frequency side (with regard to both signal frequency and sampling error frequency), the error of the refractive index is in the third decimal place:

\begin{array}{l} | Δ n_{\max} | \approx 0.003 for ν / (2 π) \in [0 THz, 3 THz], \\ ω / (2 π) \in [0.25 THz, 3 THz] . \end{array}

This calls the accuracy of some high precision material characterisations into question; in particular those in which the refractive index is stated with three decimal places [24–27]. Considering that these characterisations rarely present any margins of error, the findings here may provide a rough estimate for the uncertainty of the values.

For comparison, another error with a similar impact on refractive index extraction is the Gouy phase shift [28, 29]. However, this applies only to focused measurements. Notably, according to [28], the Gouy phase shift dominantly affects the low frequency regime of the spectrum, whereas the presented periodic sampling error is most visible at higher frequencies.

4. Conclusion

We have described two different sources of periodic sampling errors. First, we considererd non-linear movement of the delay line which originates from the periodic deviation from the set-point of the position. This error can be compensated if the error function is e.g. characterized by an interferometric measurement. Second, two unsynchronised clocks in the experimental setup can lead to a sawtooth-like deviation in the time-domain which, in turn, lead to a complex structure in the frequency domain.

A typical error source is a constantly moving delay line which triggers a lock-in amplifier. We have developed a theoretical model to describe this kind of periodic sampling error and utilise it to explain the influence on THz TDS spectra. For the case of sinusoidal errors, we have found that mirror spectra are created around the error’s frequency. As a consequence, a higher artificial bandwidth is generated and the spectrum in the evaluable frequency range is falsified. The investigated case serves as the basis for describing more complex error functions. According to the Fourier theorem, these can be constructed as superpositions of the sinusoidal error and consequently generate many overlapping mirror spectra.

An investigation of the transfer function has revealed that this type of error is significantly influencing the extracted refractive index over the complete bandwidth. Hence, it has a severe impact on THz TDS studies where small changes in the refractive index are investigated.

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Stage	Δn_LCP, direct	Δn_LCP, indirect
Thorlabs	0.129 ± 0.001	0.129 ± 0.001
CD drive	0.133 ± 0.001	0.137 ± 0.003

Periodic sampling errors in terahertz time-domain measurements

Abstract

1. Introduction

2. Problem assessment

2.1. Triggered acquisition of THz TDS data

2.2. Delay line non-linearity

3. Theoretical model

3.1. Ab initio calculations

Generic examination

Periodic deviation

3.2. Application to THz TDS spectra

3.3. Influence on the transfer function

4. Conclusion

References and links

Cited By

Figures (7)

Tables (1)

Equations (30)

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