Tutorial: An introduction to terahertz time domain spectroscopy (THz-TDS) Free

Terahertz time-domain spectroscopy (THz-TDS) has become a valuable technique in the fields of chemistry,¹ materials sciences,² engineering,³ and medicine.^4–6 The range of applications is so large^7–9 that we could focus only on a few examples after introducing the basics.

The spectral range between 0.1 THz and 10 THz (3.3 cm $−1$–333 cm $−1$⁠) is an important region for low frequency dielectric relaxation and vibrational spectroscopy of liquids, such as water, methanol, ethanol, propanol, and others.^10,11 It has also been particularly important in the study of low frequency modes in molecular crystals. THz-TDS in combination with density functional theory (DFT)^12–18 can be used to study amino acids,^14,18–23 peptides,^24–26 drugs,²⁷ and explosives.^17,28,29 The latter two are particularly interesting due to the ability of THz radiation to penetrate through clothing³⁰ and packaging.³¹ There are also applications where THz-TDS is being optimized for the detection of bombs, drugs, and weapons.^32,33

In the field of materials science, THz-TDS is ideally suited to measure mobile charge carriers since they reflect and absorb THz radiation. THz-TDS has been used to measure conductivity,³⁴ topological insulators (TI),^35,36 and superconductors,^37,38 as well as phase transitions in these materials.³⁹ Furthermore, 2D materials like bismuth selenide (⁠ $Bi2Se3$⁠)⁴⁰ and graphene^41,42 have been studied. In the case of graphene, spatially resolved THz-TDS has been demonstrated as an effective tool to validate the homogeneity and uniformity of fabricated layers.⁴³ In addition, the interest in THz-TDS as a quality control technique is growing. For example, THz-TDS can be used to measure the drying process and final thickness of paint coatings, reducing costs and process times.^44–46 

The interest in THz-TDS from the medical community has been growing in recent years. Proof-of-concept experiments are able to detect skin cancer⁴⁷ and to monitor scar growth.⁴⁸ Furthermore, THz-TDS can also be used to detect yeast,⁴⁹ bacteria,^49,50 and viruses⁵¹ via THz-metamaterials.

Metamaterials comprise human-made, sub-wavelength structures. The local field enhancement in these structures significantly increases the sensitivity of THz-TDS and therefore allows the measurement of extremely small concentrations and new commercial applications.^52,53 THz wavelengths are on the order of $300μ$m, so achieving the sub-wavelength criterion is easily attainable with current lithographic techniques. In addition, the lack of classical optical components in this frequency range sparked the research of THz metamaterials. Metamaterial based filter (static^54,55 and dynamic⁵⁶) lenses,^57,58 beamsteerers,^59,60 and perfect absorbers⁶¹ have been characterized with THz-TDS.

The wide variety of applications has resulted in quite a number of different configurations of THz spectrometers.^62–64 We focus on general considerations and provide explanations and illustrations to provide an overview of the field of THz-TDS. That is, we explain THz-TDS in a general sense and how time domain measurements can provide spectral information. We also illustrate the method for extracting optical properties from the measured signal, as well as some potential pitfalls when acquiring THz-TDS spectra.

We first discuss the basic concepts of THz-TDS in Sec. II. In Sec. III, we present several techniques to generate coherent THz radiation, and in Sec. IV, we present how to detect THz. These sections are focused on photoconductive antennas, the most common THz-TDS emitter/detector. We define some figure-of-merit benchmarks for a THz-TDS system and discuss some common noise sources, which provide a starting point for the optimization of an existing system (Sec. V). We conclude this tutorial with a short overview of THz-TDS experiments related to materials science and spectroscopy in Sec. VI.

Spectroscopy refers to the energy, wavelength, or frequency of photons that pass through a sample. In the case of THz-TDS, the time-domain signal directly measures the transient electric field rather than its intensity.

The THz electric field at the detector is typically on the order of 10–100 V/cm⁶⁵ and has a time duration of a few picoseconds. Therefore, a fast and sensitive detection method of the electric field is required. Direct electrical detectors and circuits usually have rise times and fall times in the picosecond to nanosecond range and, therefore, do not have high enough time resolution. The way to reach sub-picosecond resolution is by using optical techniques wherein an ultrashort NIR optical pulse (typically shorter than 100 fs) is beamsplit along two paths to generate and detect the time-dependent THz field, as discussed below.

The measured signal corresponds to the THz field amplitude at a single point in time (⁠ $t1$⁠). The next step is to measure the signal at all time-points. This is achieved by delaying the read-out pulse relative to the THz pulse using a mechanical delay line. The output of the laser is split into two beams, as sketched in Fig. 1. One of the beams is used to generate THz radiation, and the other one is the read-out beam that detects it. The temporal delay is achieved by increasing the path length of one of the beams. The travel time of a laser pulse is $t=s/c$⁠, where $s$ is the path length and $c$ is the speed of light (this corresponds to 30 cm/ns or 300  $μ$m/ps). This simplifies the problem of femtosecond time resolution to one of micrometer spatial resolution. Precise micro positioning is achieved with computer-controlled positioning stages. In THz-TDS (and ultrafast spectroscopy in general), these stages are usually referred to as delay lines. The round trip delay time changes 6.6 fs for each micrometer of delay line displacement. The moving speed of the delay line defines the sampling speed in the time domain. It is, therefore, common to state the speed of the delay line in ps/s.

Water vapor has strong absorption features in the THz range that can interfere with measurements.^66,67 To minimize this absorption, the THz beam path is enclosed in a purge box (see Fig. 1) which is purged with dry air or nitrogen. Alternatively, the box can also be evacuated, but that requires a significantly higher level of design engineering.

The system sketched in Fig. 1 uses THz-TDS transmission geometry, which is the simplest one. However, it is also possible to measure the THz pulse reflected from the sample^68,69 or to use an attenuated total reflection geometry.⁷⁰ In addition, THz-TDS can also be used for the near-field detection of a sample.^57,71,72

The heart of any THz-TDS system is an ultrafast laser oscillator or amplifier. This laser emits optical pulses with femtosecond pulse durations which are transformed into a picosecond THz pulse. This is achieved by the THz emitter. There exists a large variety of THz emitters currently in use. We present a short overview of the most important emitters bearing in mind that the most appropriate emitter design for a particular application is not always trivial to identify. Therefore, in this overview, we focus on two figures-of-merit: the achievable THz-bandwidth and the suitability to different laser systems.

The different emitters, summarized in Table I, can be categorized in two different groups based on the underlying mechanism for THz generation. One way to generate THz radiation is to induce a short current pulse in a medium, which then emits THz radiation as described in detail below. The other commonly used method is via a nonlinear optical rectification process which is primarily used in amplifier based systems.

For the user, the most important concern is typically whether an emitter can be used with an existing laser system. To allow a rough estimation, we labeled the different mechanisms with either oscillator (Osci.) or amplifier (amp). For the purpose of this tutorial, a laser oscillator has a repetition rate between 40 MHz and 1 GHz, a pulse length of less than 150 fs, and a pulse energy below 10 nJ. In contrast, the amplifier has a repetition rate of 1-10 kHz, with a pulse energy of more than $10μ$J and also a pulse length of less than 150 fs. Emitters and detectors typically increase in bandwidth and efficiency with shorter laser pulses.

The short overview in Table I should help the reader navigate the large realm of THz emitters, but this list is not exhaustive. The actual bandwidth and performance of the emitter will depend on the experimental conditions. Here, we will focus solely on THz generation from photoconductive antennas (PCAs).

The net current is limited by the charge and energy stored in the capacitor. As soon as the charges in the gap are accelerated by the voltage, the capacitance energy (⁠ $W$⁠) is transferred into kinetic energy. The reduction of capacitance energy also results in a decrease in the local bias $Uloc$⁠, which in turn lowers the saturation velocity $vs∝Uloc$⁠.^65,104–106 The charge velocity is therefore larger than the saturation velocity which results in strong electron scattering and limits the photocurrent. Furthermore, the carrier lifetime in the semiconductor substrate limits the length of the current pulse. The precise calculation of both effects⁶⁵ is beyond the scope of this work, but if the materials and the geometry are chosen correctly, sub-picosecond current pulses $IPC(t)$ are produced in the PCA which then emit THz radiation.

The THz emission can be described as dipole radiation. The strong angular dependence of this radiation results in a strongly divergent THz beam. Usually, this radiation is collimated directly after the generation using a silicon lens. The refractive index of silicon is very similar to that of the PCA substrate, hence there are minimal reflection losses. Additionally, silicon has a large refractive index in the THz spectral range (⁠ $nSi≈3.42$¹⁰⁷) and is fairly easy to machine, which allows for a compact and small lens that can be mounted directly on the PCA chip. It is important to stress that the lens must be aligned precisely at the center of the emitter with micrometer precision. This can either be done with micrometer-precision screws in the research lab itself or the lens can be glued onto the chip, which is frequently offered in commercial PCAs.

The most commonly used semiconductor for the emitter is gallium arsenide (GaAs), either semi-insulating GaAs (SI-GaAs) or low-temperature grown GaAs (LT-GaAs). A less common one is radiation damaged silicon on sapphire (r-SOS). The carrier mobility in r-SOS is rather low (⁠ $μ≈1500cm2Vs$⁠)¹⁰⁶ compared to GaAs (⁠ $μ≈8500cm2Vs$⁠).¹⁰⁶ Additionally, the carrier lifetime in r-SOS (⁠ $τ≈$ 600 fs)¹⁰⁸ is longer than in LT-GaAs (⁠ $τ≈100$ fs).¹⁰⁹ Therefore, LT-GaAs is currently the material of choice for commercial emitters.

A drawback of GaAs is that it exhibits optical phonons at 8.5 THz¹⁰⁶ and, therefore, high frequency THz radiation is absorbed by the emitter before it can be routed to the spectrometer. This issue can be circumvented by using a thin GaAs layer of 1–2  $μ$m. However, this material is too fragile for practical applications and performing lithography on it is impossible. Therefore, this thin layer of LT-GaAs must be bonded to a back-substrate such as silicon, which has a similar refractive index and is THz transparent. THz pulses of up to 10 THz bandwidth have been demonstrated with this technique.^101,103

Similar to THz generation, there are several different approaches for THz detection. For THz-TDS, only coherent detection mechanisms are suitable, and we will not consider thermal detectors such as bolometers and Golay cells. As was the case for THz generation, we will present an overview of different techniques and then discuss detection via PCAs in detail.

After a broadband THz pulse is generated, a detector with a greater or equal bandwidth is must be used to collect the spectral information. Most detectors do not exhibit a flat gain profile and the best signal-to-noise ratio (SNR) is obtained when the spectral gain of the detector and the emitter bandwidths are closely matched. Table II provides a brief overview over some detectors and some references for further study.

THz detection via photoconductive switches can be performed with similar or even identical devices as used for THz generation. In contrast to generation, the metal gap fabricated on the semiconductor is not externally biased. The electrical bias is created by the electrical field of the THz electromagnetic pulse. The laser beam focused on the semiconductor gap generates free carriers, which increases the conductivity of the device. Thus, a current is produced that is directly proportional to THz electric field.

The response function of the detector depends on the laser pulse length and the carrier lifetime. In contrast to the generation, where the stored energy also limits the current-pulse length, only the carrier lifetime is important in detection. Shorter carrier lifetimes result in a sharper temporal response, which in turn leads to larger THz bandwidths. The carrier lifetime in LT-GaAs is typically about 100 to 300 fs.

Lock-in amplifiers are used to measure small signals within a noisy background. A lock-in amplifier is a system that selectively amplifies a signal at a given reference frequency $f$⁠. The noisy incoming signal is multiplied by a sine wave at the reference frequency and then sent through a low-pass filter to obtain the DC component, which is the signal of interest. The time constant of the low-pass filter is an extremely important parameter and is discussed below. For example, if a step function is passed through a low-pass filter, it will reach 95% of its final value after three time constants: $(1/e)3$⁠, and 99.3% after five time constants. Therefore, if one scans the delay line too fast, sharp features of the THz pulse will be broadened and attenuated. On the other hand, if one scans it too slow, there is no additional improvement.

The THz time-domain signal is modulated by switching the THz generation on and off at the desired reference frequency, which is referred to as “chopping.” This can be achieved by applying a square wave bias voltage between 0 V and $+10$ V or by mechanically modulating the excitation beam with a chopper wheel.

In general, electronic modulation is preferred when using PCAs. Electronic switching allows for tens of kHz modulation frequencies that are significantly higher than mechanical modulation frequencies of 0.1 to 4 kHz. These higher chopper frequencies are desirable because thermal noise (pink noise or 1/ $f$ noise) is proportional to 1/ $f$⁠, hence, higher lock-in frequencies result in lower noise. However, the highest usable modulation frequency is limited by the capacitance of the antenna as well as the frequency response of the cables, amplifiers, and other electronics. Therefore, it is recommended to empirically test which modulation frequency and method (mechanical vs. electronic) results in the best SNR.

In this section, we present suggestions for improving a THz-TDS system. However, we first define some figures-of-merit (FOMs) to benchmark a system. We will discuss all these FOMs by examples from the system used in our lab. For this section, the spectrometer was intentionally misaligned to provide substantial noise.

To estimate the usable bandwidth of the spectrometer, it is crucial to determine the SNR and the dynamic range (DNR). The noise floor is measured by placing a metal-block in the THz beam-path and measuring the signal using the same settings as during the actual measurement. The root-mean-square of the noise time trace is an excellent measure of the total noise. The amplitude of the time trace divided by this value is the first FOM for optimization. All frequency information of the THz pulse is contained in the time trace. For samples that are not strongly dispersive or absorbing, the amplitude of the time trace is a good approximation of the total THz power transmitted.

The Fourier transform of these measurements is used to determine the DNR, defined as ratio between signal intensity and noise intensity at a given frequency. In general, it is recommended to average both curves over a few GHz window to avoid spikes. These power spectra for our system are plotted in Fig. 4.

The maximum DNR at 0.9 THz is greater than 10 $6$ or 60 dB. At 2.5 THz, it is 25 dB. Therefore, measurements at frequencies below 2.5 THz are trustworthy. However, the user should keep in mind that the intensity DNR is only one of the two components of the complex-valued information from the measurement. Like the amplitude, the phase is also accompanied by noise. The origin of this noise is different. Amplitude noise is caused primarily by power fluctuations, while phase noise originates from temporal inaccuracy. Any imperfections in the delay line will increase the phase noise since the FFT assumes that the time-points are equally spaced.

To calculate the phase noise of a measurement, we perform the same measurement multiple times. The mean value of all iterations is defined as the baseline. The standard deviation of these iterations is a good approximation of the phase noise. The influence of the phase noise can be seen in Fig. 5, which shows the phase noise for the same measurement as in Fig. 4, and is surprisingly large.

After a rigorous characterization of the system, the best way to increase the SNR and DNR is by minimizing the noise, that is, it is typically easier to reduce noise rather than increase signal. There are many possible causes for noise in the system^123–125 and the remarks here are general in nature. We will not cover all potential noise sources for THz-TDS.

As stated previously, the ultrafast laser is at the heart of any THz-TDS system. Therefore, it is the largest source of noise. For example, if one completes his or her measurements for the day and returns the next morning to find that the noise has increased overnight, the best starting point for a noise investigation is the laser itself. Shot-to-shot fluctuations result in averaging over non-identical laser pulses. Since THz generation is usually nonlinear with respect to the laser power or pulse length, variations in laser intensity will, therefore, reduce the SNR due to a higher noise level. Furthermore, it is essential to ensure that the laser power is stable over the duration of a measurement, whether it is 5 min or 5 h.

Another source of noise is electrical noise. As discussed in Subsection IV C, the noise is typically lower for higher chopper frequencies (there are other types of noise, but 1/ $f$ noise dominates at the frequencies we use). However, there is a trade-off when the frequency response function of the antenna itself cannot support the high modulation frequency. Therefore, one must perform several SNR/DNR measurements with many different modulation frequencies to find the best one for the system. Naturally, one wants to avoid the AC line frequency and its harmonics, but there are often noise sources in the lab that are at different frequencies. Furthermore, it is possible that the bias voltage on the emitter somehow cross talks with the detector. Even minimal cross talk on the order of a femtoampere will result in a very high level of noise because the frequency of the cross talk is identical to the lock-in frequency. To track this down, it is advised to use a mechanical chopper to directly chop the THz beam instead of electrical modulation. This way, no electrical cross talk noise can occur. If the SNR is significantly better with mechanical chopping, the origin of the cross talk must be found.

As stated above, a particularly high phase noise is caused by temporal uncertainty. This can either be caused by the mechanical delay line, but air fluctuations and vibrations of the laser table are also potential culprits. To validate the performance of the delay stage, one should check the stability of beam pointing at a large distance. This can be done by eye, but more accurate results are achieved using a CCD camera to monitor the spot while moving the stage. This mechanical jitter can originate from the stage itself or from the optical mountings and posts on the stage. It is recommended to use highly stable full steel posts to avoid any bending or bouncing when moving the stage. If the jitter is not caused by the stage, air fluctuations need to be minimized.

Although it is tempting to evacuate the lab itself, this approach is unfeasible because it is difficult to adjust the optical mounts while wearing a space suit. Instead, any nitrogen inlets into the purge box should have a muffler to minimize both directional and turbulent air flow. In extreme cases, the optical beams must be enclosed in “light pipes” after the point of the beamsplitter.

The best technique to decrease the phase noise is to measure the difference signal between measurement and reference instead of making two separate measurements. This can be done by mounting the sample on a rapidly moving stage which brings it into and out of the beam path at tens to hundreds of Hz. This fast modulation can then be added to the chopper frequency, which allows the difference signal between sample and reference to be detected. This suppresses the noise and significantly increases the sensitivity.^126–128 

Even when all of these systematic sources of noise are identified and solved, noise will still remain in the system. Thus, there will always be scan-to-scan variability, which typically follows Gaussian statistics and will be proportional to $1N$⁠, where N is the number of scans averaged together. The only way to reduce this noise is to increase $N$⁠. In general, this increase can be accomplished either by measuring the full time trace multiple times (iterations) or by scanning slowly and integrating each time-point for a longer period of time. This integration is usually done at the lock-in amplifier by defining the time constant. As stated above, it is best to reduce the noise level itself to increase the SNR by that amount because signal averaging increases the SNR as a function of the square root of the time taken to make a measurement. For example, if the noise is reduced by a factor of five, then the SNR is increased by a factor of 5, but it would require an acquisition time of 25 times longer to achieve the same increase in SNR. Thus, if a typical scan takes 10 min to obtain, it will require 250 min (4 h 10 min) to increase the SNR by a factor of five. There are times when signal averaging is the only option, but one should strive to either increase the signal or decrease the noise before implementing it.

The lock-in amplifier averages the signal over a period of time which is defined by the time constant (⁠ $τ$⁠). This averaging has the advantage that some of the random noise is suppressed and the signal is smoother. If this were the only consideration, we would like this time constant to be as large as possible to give us the best SNR. However, while the delay stage is moving the signal is constantly changing because it is measuring different time-points of the THz signal. For example, in the case of a time constant of $τ=30$ ms and a high scan speed of 10 ps/s, every time-point is “averaged” with 0.9 to 1.5 ps of time-points after it (3 to 5 lock-in time constants).

It is important to choose the combination of lock-in time constant and scanning speed such that 3 to 5 time constants elapse in the time it takes the delay stage to move the distance corresponding to the sharpest feature in the THz pulse (and this could simply be a rising or falling edge rather than the actual pulse width). For example, if it is a typical value of 50 fs and the time constant is 30 ms, then 3 time constants will elapse over 50 fs of delay time when scanning at 0.55 ps/s, and five time constants will elapse when scanning at 0.33 ps/s.

The effect of the scan speed is illustrated in Fig. 6 by measuring the same pulse for the same total time, meaning that for faster scans more iterations were measured. The sampling rate was 128 Hz for a scan speed of $v=0.5$ ps/s, and 512 Hz for $v=5$ ps/s and $v=10$ ps/s, all with a 30 ms lock-in time constant.

Therefore, the THz spectra that would have been obtained with an infinitely short time constant are multiplied by a Gaussian window, centered around 0 THz, with a width of $1/σ$⁠. This windowing suppresses higher frequencies and, therefore, reduces the DNR. The Fourier transforms of the time traces in Fig. 6 are plotted in Fig. 7. The three spectra are normalized to the noise floor and were measured using the same total measurement time.

In addition to the scan speed and integration time constant, the total time range (in ps) must be considered. Technically, the THz pulse is infinitely long, however, its amplitude decays rapidly as a function of time. This decay time depends on the spectral characteristics of the sample and the THz pulse. Without significant dispersion or absorption, the spectral information in the first few picoseconds of the THz signal provides all of the information that is needed. However, if the sample material is strongly dispersive or shows a sharp absorption feature a longer scan is required.

The influence of a short time window is illustrated by the simulated spectrum of a metamaterial absorber, as shown in Fig. 8. The full 88 ps time trace of this simulation is Fourier transformed and plotted as black line in the graph which is obscured by the other lines. The FWHM of the resonance is 23 GHz. The same time trace was then truncated after 30, 20, and 10 ps. The resulting Fourier transforms are plotted in red, blue, and green, respectively. The 10 ps time trace has distinct ringing. Furthermore, the FWHM of the measured resonance broadens to 65 GHz and is slightly red shifted. The direction and strength of the shift depends on the experimental conditions. In this case, the shift is caused by the unequal distribution of the frequency components in experimental time traces.

In some cases, it is unavoidable to use a short time window to reduce the measurement time or to avoid reflections due to etalon effects. In the case of features due to etaloning, it is possible to not simply halt the data acquisition before the reflection feature appears, but rather to multiply the time traces with a Gaussian time window, centered around the temporal maximum of the THz-signal. The Fourier transform of a Gaussian window is again a Gaussian function, which will still broaden the spectral resonance, but not induce any ringing as is described above when a rectangular windowing function is used.

THz-TDS has been used to study molecules in the gas phase,^131–134 liquids and solutions,^{10,11,135,136} and the solid state.^{2,14,19–21,23–27,137–140} The goal of all of these applications is to measure the frequency-dependent complex refractive index of the sample. In some experiments, the user is more interested in the imaginary part of the refractive index (i.e., resonant absorption and conductivity). In other cases, the real part is more interesting (e.g., layer thickness measurements). A significant advantage of THz-TDS is that the real and imaginary part of the refractive index are measured simultaneously. While it is true that these two are related through the Kramers-Kronig relationship, spectral information of frequencies from DC to infinity is required. There are many optical techniques in which the Kramers-Kronig relation or other models are used, but the limitation of $0<ω<∞$ is always present.

We will illustrate how the refractive index of a sample can be calculated from experimental data. This illustration will start with the most general case and then present simplifications utilized with common experimental conditions. These examples are presented with a solid material as sample in mind, but they are easily generalized for liquids and gases.

These equations are identical to the results achieved by setting up an ABCD matrix¹³⁰ and calculating the transfer function. However, the ABCD matrix method is only valid if all reflections are measured and is hence less suited for THz-TDS, since one typically does not measure all reflections for thick layers.

In addition to the transmission through the sample, a reference scan without sample present must also be taken to ensure that the result is independent of the THz spectrometer being used. In the case of a thick substrate, such as a pressed pellet, referencing can be accomplished with a simple air measurement with no sample present. However, in the case of layered materials with a sample on a substrate, it is better to use transmission through the substrate alone without the sample material as a reference. The advantage of this referencing method is that the influence of the substrate thickness cancels out, and the coefficients $P1$ and $t01$ are not needed in the final calculations.

There is a large range of possible models, including Drude⁷⁶ and Drude-Smith^147,148 for conducting materials, the Debye-Model^10,136,149 for lossy materials and liquids, and loss free models such as Sellmeier¹⁵⁰ and Cauchy¹⁵¹. All these models have different assumptions and validation ranges, therefore, choosing one should be done with care.

As for any approximation, Equation (29) needs to be validated for the film thicknesses and materials used. While this formula works for several systems,^{34,38,126,152} it can also fail for seemingly similar experiments.^140,153

Determining the frequency-dependent absorption coefficient and refractive index of organic crystals such as amino acids, explosives, or drugs can be challenging. For example, amino acids are commonly received as polycrystalline powder. The powder itself is strongly scattering which significantly attenuates the transmitted THz signal. Furthermore, the absorption at the resonance frequency can reach values in the order of 500 cm $−1$⁠. The layer fabricated from this material must be homogeneous and less than $50μ$m thick to allow at least 10% transmission. Such a thin layer is challenging to handle and determining its thickness is even more challenging.

As an example, we explicitly demonstrate this procedure when measuring the spectrum of a polycrystalline molecular crystal (dl-norleucine) mixed with Teflon. THz-TDS is not only sensitive to the molecule itself but also to the crystal lattice of the material. Therefore, it is possible to identify the conformation, or polymorph, of a molecular crystal. These conformations are often temperature-dependent and, therefore, these experiments are performed in a cryostat in which the sample temperature is accurately controlled.

A mixed pellet containing the amino acids dl-norleucine in Teflon is mounted in the cryostat and cooled to 100 K. The THz-transient through this sample is measured and the results are plotted in part (a) of Fig. 10. Fig. 10 shows a reference pulse through air (black), a sample pulse though pure Teflon (blue) and a sample pulse through a Teflon pellet with dl-norleucine (red), at a temperature of 100 K. The time traces exhibit reflections at 22.5 (reference) and 27 ps (pellet), which are caused by the THz-detector. These reflections are suppressed by multiplying the time trace with a Gaussian window. The time traces are then Fourier-transformed, and Eqs. (33) and (32) are used to calculate the real and imaginary components of the complex frequency-dependent refractive indices, as is plotted in part (b) of Fig. 10. Note the correlation between real and imaginary part, a change in the real part (dispersion) results in a local maximum in the imaginary part (absorption) of the refractive index. The experimental results fulfill the Kramers-Kronig relation by definition, but are calculated without using Kramers-Kronig.

Terahertz time-domain spectroscopy (THz-TDS) is a valuable technique in the fields of chemistry, physics, electrical engineering, materials science, and medicine. This tutorial has presented the basic concepts and terminologies to someone without any prior experience. The core principle of THz-TDS is that the electric field is measured in the time-domain.

By adjusting the relative distance that two ultrafast laser pulses travel, their timing is changed and the THz field can be sampled with femtosecond resolution. The Fourier transform of these time traces yields the spectral information. We showed that scans with delay times that are too short results in spectral broadening of absorption features, and a scanning speed that is too high causes a reduction of the THz bandwidth.

A significant advantage of THz-TDS is that the THz spectrum is complex-valued, meaning it provides amplitude and phase information at every frequency point within the usable bandwidth. This information can be used to directly calculate the complex refractive index of a sample without the need of the Kramers-Kronig analysis or empirical models. This advantage was demonstrated with general examples, and then in more detail for mixed samples consisting of strongly absorbing materials diluted in a transparent host. The used matlab scripts can be downloaded free of charge from https://thz.yale.edu.

The authors acknowledge financial support by the National Science Foundation under Grant No. NSF CHE—CSDMA 1465085. We would like to thank Jacob Spies, Sarah Ostresh, Korina Straube, and Golo Storch for proof reading the manuscript and helpful suggestions.