# Phase Retrieval in Terahertz Time-Domain Measurements: a “how to” Tutorial

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## Abstract

Terahertz time-domain spectroscopy (THz-TDS) is in many ways a well-established, proven, and versatile spectroscopic technique that is frequently and routinely used in many laboratories. The basis of high-quality optical data on materials using THz-TDS is the correct extraction of the complex-valued dielectric properties (index of refraction, permittivity, or conductivity) from the recorded amplitude and phase of the involved THz signals. The focus of this paper is to discuss stable methods for finding the physically meaningful frequency-dependent optical phase from time-domain signals, thereby avoiding some of the unphysical solutions to the inversion problem that is the central part of THz-TDS analysis. The paper discusses problems associated with the positioning of the THz signal in the recorded time window, phase offsets due to noise in the experimental data, and phase correction in the case of strongly dispersive media such as transparent semiconductors in the frequency range below but close to the transverse optical phonons.

## Keywords

Terahertz time-domain spectroscopy Phase retrieval Data analysis## 1 Introduction

Terahertz time-domain spectroscopy (THz-TDS) is a well-established spectroscopic technique that since 1989 has been used in almost 6500 peer-reviewed scientific works with over 100,000 citations.^{1} The very first THz-TDS demonstration by van Exter, Fattinger, and Grischkowsky [1] already demonstrated the basic and unique feature of THz-TDS, namely that the coherent generation and detection of an ultrashort THz pulse can be applied for a highly versatile spectroscopy technique, where the frequency-resolved amplitude and phase of the detected signal are recovered directly and without any model-based assumptions about the signal. The extraction of dielectric properties (index of refraction and absorption coefficient) from THz-TDS measurements relies heavily on correct retrieval of the optical phase of the THz wave. The general review of THz-TDS measurement and analysis by Withayachumnankul and Naftaly [2] discusses the fundamentals of phase retrieval in THz-TDS. In this paper, I will discuss some common pitfalls associated with a correct and physically meaningful treatment of the phase in a THz-TDS measurement, including correct estimation of general phase offsets and identification of situations where a standard blind phase unwrapping leads to incorrect estimation of the refractive index of a sample. Such situations occur for instance in highly dispersive samples, such as transparent polar semiconductors at frequencies below the fundamental transverse optical (TO) phonon.

There is a tremendous variety of applications of THz-TDS systems. While commercial systems are becoming more and more accessible, the majority of systems in use are custom-built by the same persons that operate them for spectroscopic applications. There are no universally established measurement protocols that ensure traceable results, and a recent study from the National Physical Laboratory shows a surprisingly large variation in spectroscopic results obtained on the same samples characterized by an international range of THz spectroscopy groups [3]. That important study indicated that as the THz-TDS technique now matures, the community shares a large responsibility, and needs to work together to establish and share common standards and best practices for recording and treatment of THz-TDS data. This tutorial contributes to the understanding of the optical phase in THz-TDS measurements, and advises methods that avoid common pitfalls in data analysis, thereby adding to the literature on best practices in THz-TDS [4, 5, 6, 7, 8, 9, 10]. While some of the discussed issues are trivial and others are more intricate and difficult to identify, the ambiguities associated with phase retrieval can play a role in the variability seen in results from different laboratories on the same materials.

The phase-sensitive detection is often emphasized as a unique capability of THz-TDS, but it should be emphasized that this is also possible with dispersive Fourier Transform Spectroscopy (DFTS), using incoherent light sources [11] and within an almost identical mathematical framework as is used in THz-TDS. DFTS was first demonstrated by Chamberlain, Gibbs, and Gebbie in 1963 [12], who used this technique for the measurement of the refractive index of crystalline quartz between 20 and 55 cm^{−1} (0.6–1.65 THz). The seminal THz-TDS article from Grischkowsky et al. from 1990 on the optical properties of dielectrics and semiconductors [13] applied THz-TDS to samples that had already been characterized by DFTS. THz-TDS demonstrated superior spectroscopic data quality combined with a straightforward data analysis and an optical setup that laboratories with access to femtosecond laser technology could build with standard components, thereby establishing THz-TDS as the technique of choice for future optical characterization in the far infrared. Today, the main use of DFTS technology is in optical coherence tomography (OCT), which is currently expanding its useful spectral range from the traditional near-infrared to the mid-infrared [14], and shares many characteristics with THz time-domain imaging (THz-TDI) [15, 16]. Also, scattering-type scanning near-field microscopy (s-SNOM) has recently been combined with DFTS to enable phase-sensitive spectroscopy across the THz and mid-infrared with resolution of a few tens of nanometers [17], and recently, the same technique has been applied to nanoscale optical pump—mid-IR probe imaging with femtosecond time resolution [18]. Hence, retrieval of the optical phase in time-domain measurements is relevant for a very wide community in optics.

In a THz-TDS measurement, the raw data from the experiment are two time traces of the detected electric field, recorded under some reference condition (*E*_{ref}(*t*)) and after interaction (reflection or transmission) with the sample under investigation (*E*_{sam}(*t*)). Spectroscopy is performed by Fourier transformation of the two time traces, *E*_{ref}(*f*) and *E*_{sam}(*f*), where *f* is the frequency. The amplitude and phase of the ratio of the two signals in frequency space (the complex-valued transmission function) is compared to a model of the transmission through the sample. Inversion of this relation is then performed in order to determine the best estimate of the frequency-dependent optical properties (complex index of refraction, permittivity, or conductivity). In the general case, this inversion is a problem on its own that is outside the scope of this paper, and we refer to the existing literature on this, for instance the papers by Duvillaret et al. [4, 5], Pupeza et al. [6], Scheller et al. [7], and Kruger et al. [8]. In these and most other works, the THz beam is assumed a plane wave so that the sample is not altering the geometric propagation of the beam. However, if the THz beam is focused tightly onto the sample, significant errors in the optical parameter extraction can arise from the plane wave approximation since the finite thickness of the sample, compared to the Rayleigh range of the THz focus, may alter the subsequent propagation and focusing of the broadband THz signal onto the detector. This effect was noted by Kužel et al. [19], who demonstrated a method that compensates the effect.

*n*and absorption coefficient

*α*by the simple relation [20].

*c*is the speed of light in vacuum, and

*d*is the sample thickness. This expression is valid within the spectral range of the spectrometer, where the signal of both reference and sample spectra are above the noise floor of the system [10]. The optical constants are thus related to the transmission amplitude ∣

*T*∣ and phase difference

*Δϕ*as

Equations (1) and (2) are approximations to any experiment. The most important assumptions are as follows. The THz beam is treated as a plane wave at normal incidence (finite size of the THz beam is ignored). Alterations of the beam path due to the sample are ignored (i.e., the sample is thin compared to the Rayleigh range). Only the directly transmitted part of the signal is considered (no echoes in the sample due to multiple reflections). In more complete descriptions of the interaction between THz waves and a sample, Eqs. (1) and (2) are replaced with more complicated expressions that may or may not have closed-form analytical solutions. However, the substance of the discussion that follows remains valid also in such situations.

*d*, compared to that of transmission through the same distance of air, directly determines the refractive index of the sample, as described by Eqs. (1) and (2),

*r*= − 1), contains information about the phase of the reflection coefficient of the sample, and thus mainly gives information about the extinction coefficient of the material,

*π*;

*π*] range, and therefore, the pitfalls related to correct phase retrieval described here are not as relevant as in transmission spectroscopy.

## 2 THz-TDS Phase Retrieval

*A*exp(

*iϕ*) =

*a*+

*ib*, the phase is recovered as

*ϕ*= arctan(

*b*/

*a*). The arctangent function by definition returns angles only in the interval [−

*π*/2;

*π*/2] (red curve, Fig. 1). The four-quadrant arctangent function returns angles in the full [−

*π*;

*π*] range (orange curve, Fig. 1). In either case, the continuous optical phase increase described by Eq. (3) (blue curve, Fig. 1) is not fully recovered; the winding number in the complex plane is not maintained in the angle restoration. This leads to undesirable phase jumps that can, however, be corrected. This is typically done computationally in a loop where phase jumps between adjacent frequency points are tested; if larger than some threshold (typically ±

*π*), then all phase values above the given frequency are correspondingly offset by ∓2

*π*. This very general and well-known process is referred to as “phase unwrapping.”

*N*equidistant sample points with a step size

*Δt*and a total scan length

*T*=

*NΔt*. This results in a frequency resolution

*Δf*= 1/

*T*and a Nyquist frequency

*f*

_{max}= 1/(2

*Δt*). Figure 2 shows an example of a THz-TDS measurement on a sample of quartz glass, thickness

*d*= 1.067 mm. The data sets consist of

*N*= 346 time points, spaced equidistantly by Δ

*t*= 0.0781 ps. This data set will be used in the following discussion. Figure 2a shows the time-domain traces with indication of the arrival time of the absolute maximum of the signals. Figure 2b shows the amplitude of the Fourier transforms of the signals, and the gray area (also used on subsequent figures) indicates the high-frequency region where the signal approaches the noise floor of the experiment [10].

Phase unwrapping can, under standard THz-TDS conditions, easily correct the undesired “2*π* phase jumps.” However, the phase unwrapping works under the assumption that the regular phase increase between adjacent frequency points in the discrete Fourier transform of the THz signals is actually smaller than the threshold value for phase jump correction. Three common situations can challenge this assumption leading to more or less obvious mistakes in the evaluation of the refractive index.

*t*

_{0}(

*t*

_{0, ref},

*t*

_{0, sam}, respectively, in Fig. 2), then the phase of its Fourier transform will be approximately

*π*for

*t*

_{0}>

*T*/2. Hence, unwrapping of the phase with a threshold of

*π*will mistakenly overcorrect the phase curve, and lead to an apparent phase curve with negative slope. In that case, the unwrapped phase can be corrected by

The same correction can be done automatically by subtracting the overall phase of the signals (Eq. (5)) before unwrapping, and then adding the same phase after unwrapping. This reduced phase is best calculated by multiplying the complex-valued spectra by exp(−*iϕ*_{0, ref}) and exp(−*iϕ*_{0, sam}), respectively, before unwrapping.

In the following, the term “blind unwrapping” refers to the use of the algorithm in Fig. 3 without further considerations, whereas the term “informed unwrapping” will refer to an unwrapping procedure where the noise properties of the data and the details of the sample are considered in the unwrapping procedure.

*π*due to phase unwrapping on too noisy phases. The optical phase should extrapolate to zero at low frequencies, so if there is a phase offset of

*m*⋅ 2

*π*, then the extracted refractive index from a transmission measurement will diverge at low frequencies to

*ω*

^{−1}low-frequency divergence. The effect of such accidental phase offset is shown in Fig. 5a. Since the refractive index is needed to account for reflection losses, the extracted absorption coefficient is also influenced by such phase offsets, as shown in Fig. 5b. Since the general phase curve deviates only weakly from the general linear behavior in most situations, the phase offset can be identified and compensated in an automatic manner by fitting a straight line to the phase data in the region of highest signal-to-noise ratio and extrapolating the linear fit to zero frequency. The extrapolated phase should be close to zero, and otherwise the phase curve can be shifted by a suitable multiple of 2

*π*.

- (1)
Locate the temporal position of the maximum of the absolute value of the reference and sample THz pulses, labeled

*t*_{0, ref}and*t*_{0, sam}, respectively, and determine the temporal offset between the sample and reference time windows,*t*_{offset}. - (2)
Calculate the Fourier transforms of the reference and sample signals, \( {\tilde{E}}_{ref}\left(\omega \right) \) and \( {\tilde{E}}_{sam}\left(\omega \right) \).

- (3)
Calculate the reduced phase of the reference and sample signals,

- (4)
Perform a standard unwrap of the reduced phase differences,

- (5)
Check if there is a global phase offset (integer multiples of 2

*π*) by linear regression—fit a linear function*ϕ*(*ω*) =*Aω*+*B*to the central part of the phase curve (in the spectral range of highest dynamic range). Offset the phase difference by an accordingly,

*B*/2

*π*.

- (6)
Find the full phase difference between the sample and reference signals as

The third case of phase problems is less obvious than the two situations described above. If the sample is thick (in the sense that there is a large difference in arrival times of the reference and sample pulses) and the frequency resolution in the experiment is limited, then the condition of small phase increase between adjacent frequency points may not be met by the experimental conditions, and simple phase unwrapping will no longer work as intended.

## 3 Phase Unwrapping in Dispersive Media

*d*is, correspondingly,

*t*

_{offset}, then the phase of the sample signal is corrected according to this,

Based on the previous discussion about general phase unwrapping, the numerical problem is to unwrap the phase of the first term on the right-hand side (RHS) of Eq. (15). Since the second term (the offset between the two scans) adds to the phase difference between adjacent frequency points, it is most convenient to unwrap the phase difference without this delay term, and then add it after the phase jump correction.

*f*

_{k}=

*k*⋅

*Δf*,

*k*∈

*ℤ*, the phase step between successive frequency points is

*dn*/

*df*= 0), the above relation simplifies to

*T*, the finer the frequency resolution is, and the smaller the phase step between successive frequency points. In this simplified situation, the required scan length

*T*that ensures that the absolute value of the phase difference between successive frequency points is smaller than

*π*is

*nd*/

*c*after the first transmitted signal. This shows that while this truncation is acceptable in relation to the phase unwrapping, Eq. (19) gives the minimum time window required for correct unwrapping of the phase.

*dn*/

*df*≠ 0), then the situation is a bit more complicated and slightly less favorable. In this case, the condition of absolute value of the phase step less than

*π*between successive frequency points can be written as

Since the shape of the refractive index is typically not known in advance, this expression is difficult to use in the planning of an experiment. For this, Eq. (19) may still be a first, approximate guideline.

*Δϕ*

^{'}(

*f*

_{k}) =

*Δϕ*(

*f*

_{k}) −

*Δϕ*

_{0}(

*f*

_{k}). Now the time window limitation (Eq. (20)) is modified to

Thus, unwrapping can be performed on *Δϕ*^{'} and the offset *Δϕ*_{0} then added back to the unwrapped phase difference. For nondispersive samples (\( n(f)\approx \overline{n} \)), there are then no practical limitations on the shortest time window to be used, at least with respect to correct phase unwrapping.

However, even the modified Eq. (22) in the case of dispersive media shows a feature that can be a complication for THz-TDS measurements in two situations.

Firstly, TDS measurements with high bandwidth, for instance with experimental setups based on THz generation in two-color air plasmas driven by femtosecond laser pulses [21, 22, 23] and THz detection by air-biased coherent detection (ABCD) [24], can reach frequencies of 30 THz or more [25]. This is approximately an order of magnitude higher bandwidth than traditional THz-TDS, which is limited to a few THz upper frequency cutoff in practical measurements. The higher bandwidth in ABCD measurements requires a small time step *Δt*, and thus a large acquisition time. It may therefore be tempting to reduce the full scan length *T* in such measurements, compared to a low-bandwidth THz-TDS measurement with larger time step. In such situations, one should still keep Eqs. (22) and (23) in mind.

Secondly, samples with large dispersion will show a significant variation of *n*(*f*) so that it in general can deviate significantly from the average value of the refractive index determined from the arrival time of the pulses (Eq. (21)). Samples with sharp resonances will feature a fast-varying change in the refractive index at the resonance frequency. This can lead to a very large value of *δ*(*Δϕ*^{'}(*f*)) at the resonance frequency, and therefore the requirement of an unexpectedly large scan window. In practical terms, a sharp resonance leads to a long ringing after the main THz signal, and the requirement in Eq. (23) is equivalent to a recording of a sufficiently long time window that catches the full decay of the ringing signal (and therefore secures a significant frequency resolution to reproduce the phase variation across the resonance with sufficient fidelity). As can be seen from the discussion here, this time can easily exceed the time between subsequent round-trip echoes in the sample, and thus the selection of a suitable time window poses a fundamental problem. The manifestation of this resonance-related phase problem is a large phase jump across a resonant feature, giving rise to incorrect jump of the phase across that resonance.

The implications of Eq. (23) with respect to both of these situations will be discussed after the following example of high-bandwidth THz-TDS spectroscopic analysis.

## 4 High Bandwidth Measurements of the Refractive Index of Strongly Dispersive Materials

*ω*= 2

*πf*. For the 4H polytype, two terms contribute to the dielectric function, namely a very strong term (

*W*

_{TO}) that describes the TO phonon, and a weak term (

*W*

_{TA}) that describes a transverse acoustic (TA) phonon, which is infrared-active due to polytype-specific folding of the Brillouin zone [28]. The experiment and the results are described in detail in ref. [28]. Important to note here is that the spectroscopy was performed with large-aperture samples of SiC (10 cm diameter) placed in the parallel section of the THz beam in the spectrometer, in order to obey the plane wave approximation behind Eqs. (1) and (2) as closely as possible.

*Δϕ*(

*f*

_{k}) −

*Δϕ*(

*f*

_{k − 1}) (red curve, calculated from the blue curve in Fig. 8b) and that of the model phase data sampled with the same frequency resolution (blue curve, calculated from the green dotted curve in Fig. 8). The green dashed line in Fig. 9 indicates the

*π*threshold. It is clear that this threshold is crossed at 18 THz, and the unwrapping fails. Since the phase curve, from this point on, is offset by 2

*π*for every frequency point with respect to the previous one, the phase curve can be completely unwrapped by

*k*

_{0}is the frequency index at which the correction starts. The resulting phase jump curve is shown in orange in Fig. 9. With this additional optics-based unwrapping, the phase difference curve is now in full agreement with the model curve, as shown in Fig. 8b (red curve).

The correctly interpreted phase difference leads to a refractive index (Fig. 10a, red curve) in close agreement with the model (green dotted curve) for 4H SiC. In contrast, the original unwrapped phase difference leads to wrong behavior of the refractive index, with an apparent peak near 18 THz. Figure 10b shows the absorption coefficient, calculated by using the index of refraction estimated from the blind unwrapping (blue curve) and the informed unwrapping (red curve). In this specific example, the extracted absorption coefficient is not influenced significantly by the improved phase extraction, since the strong absorption dominates over reflection losses at the highest frequencies.

## 5 Conclusions

To summarize, this tutorial has discussed the most important pitfalls to consider when analyzing the phase of the optical signals in THz time-domain spectroscopy. Six steps, summarized by Eqs. (9)–(12), form a stable method for phase retrieval that will work in virtually all situations commonly encountered in THz-TDS. These steps can be implemented in a rather straightforward manner in data analysis software, and requires no user interaction. The seventh step (summarized by Eq. (25)), to be employed if the sample material is strongly dispersive, typically requires inspection and decisions by the user, but represents a documentable additional treatment of the phase in such situations, in order to extract as much consistent information from a given experiment as possible.

## Footnotes

- 1.
Web of Science January 2019, search term “Terahertz AND time-domain,” excluding conference proceedings.

## Notes

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