Terahertz time‐domain spectroscopy is a well‐established technique to study the far‐infrared electromagnetic response of materials. Measurements are broadband, fast, and performed at room temperature. Moreover, compact systems are nowadays commercially available, which can be operated by nonspecialist staff. Thanks to the determination of the amplitude and phase of the recorded signals, both refractive index and absorption coefficient of the sample material can be obtained. However, determining these electromagnetic parameters should be performed cautiously when samples are more or less transparent. In this chapter, we explain how to extract the material parameters from terahertz time‐domain data. We list the main sources of error, and their contribution to uncertainties. We give rules to select the most adapted technique for an optimized characterization, depending on the transparency of the samples, and address the case of samples with strong absorption peaks or exhibiting scattering.
- terahertz time‐domain spectroscopy
- transmission TDS
- reflection TDS
- attenuated total reflection
- extraction precision
- Kramers‐Kronig relations
Up to the end of the 1980s, the far‐infrared electromagnetic response of materials was mostly investigated thanks to Fourier transform infrared (FTIR) spectroscopy, which exhibits several advantages. During one scan, the recorded time‐equivalent waveform is built from information delivered by the entire spectrum, whereas other dispersive prism‐ or grating‐based spectrometers receive at any time only signal from a narrow band, i.e., a weaker signal with a smaller signal‐over‐noise ratio (SNR). Second, and oppositely to dispersive spectrometers, the resolution of FTIR instruments is not limited by the size of the source. These are respectively known as the multiplex and
In this chapter, we describe the principles of THz‐TDS and we explain how to extract the refractive index and the coefficient of absorption of materials from THz‐TDS data. We study the precision of this determination versus different error sources. We give rules to choose, depending on the samples under test, the most adapted THz‐TDS technique among transmission or reflection ones. We also treat the case of materials that exhibit strong absorption bands, and heterogeneous materials that scatter and/or diffract the THz beam.
2. Principles and basics of THz‐TDS
In THz‐TDS setups, a train of ultra‐short laser pulses excites a THz antenna, which converts each optical pulse into an electromagnetic (EM) burst and radiates it in free space. In other words, the carrier frequency of the optical pulse is rectified and only its envelope is saved. Because of a noninstantaneous response of the antenna, the conversion widens the EM pulse duration when compared to the optical one. Thanks to some THz optical system, the EM pulse is focused onto a receiving THz antenna, which is triggered by a delayed part of the laser beam. In the receiving antenna, a nonlinear process mixes both incoming EM pulse and laser pulse, giving rise to a signal integrated by the reading electronics, which is proportional to the convolution product of the laser pulse and the electrical field of the EM pulse. By varying the time delay between emission and detection, which corresponds to a time‐equivalent sampling, the temporal waveform of the convolution product is obtained. Two major features should be noted: (i) emitter and receiver are enlightened by the same pulsed laser beam, thus they are perfectly synchronously excited; and (ii) because it is triggered by ultra‐short laser pulses, the receiver records the EM pulse signal only when it is excited by the laser pulses (typically during a 1‐ps time slot): noise in the time interval between two consecutive laser pulses (typically 10 ns) is not recorded. This amazing 1 ps/10 ns = 104 ratio, associated with the perfect synchronization between emission and detection, together with the high stability of mode‐locked laser pulse comb, makes the dynamics of THz‐TDS extremely high, usually larger than 60 dB in power. The spectrum of the signal is calculated through a numerical Fourier transform of the convolution trace. The Fourier transform supplies a complex value, with a modulus and a phase. The phase is related, in the time domain, to the relative origin of the time delay between emission and detection while the modulus spectrum depends strongly on (i) the spectral efficiency of the emitting and receiving THz antennas, and (ii) the adjustment of the quasi‐optical THz‐TDS system. The most common THz antennas are photo‐conducting switches made from ultrafast semi‐insulating semiconductors, like low‐temperature grown GaAs (LTG‐GaAs). Basically, a microstrip line with a narrower gap at its center is deposited over the semiconductor substrate. In emission, the structure is DC biased and the gap becomes conductive when illuminated by a laser pulse. It behaves as a dipole whose moment varies promptly due to the photoconduction process. This dipole radiates in the far‐field region an EM signal proportional to the second time‐derivative of the moment variation, i.e., to the first derivative of the current (conduction and displacement) surge flowing through the gap. In such antennas, detection occurs through a complementary effect. The gap is biased by the incoming THz field that accelerates the free carriers synchronously generated by the triggering laser pulses. This current, proportional to the THz field, is read and time‐integrated by the electronics, usually a lock‐in amplifier and it writes:
THz emission can also be obtained either by illuminating the bare surface of an ultrafast semiconductor wafer, at which photo‐generated carriers are accelerated inside the wafer by surface fields or/and by the Dember effect , or by optical rectification in an electro‐optic (EO) crystal effect. Detection is also commonly performed by EO sampling. Characterization of a sample is mostly achieved in transmission by locating the sample in the THz beam, and recording the THz waveforms without (reference) and with the sample. Then, one Fourier transforms the waveforms, and the complex transmission coefficient of the sample is equal to the ratio of the signal and reference spectra:
The same procedure is performed in reflection, providing the experimental complex coefficient of reflection . In this case, the reference signal is the THz waveform reflected by a perfect mirror placed exactly at the same position as the sample to be tested. To avoid the difficulty of the exact positioning of the reference mirror, attenuated total reflection (ATR) scheme is preferred . ATR set‐ups are especially dedicated to characterize liquids or powders since (i) THz radiations are strongly absorbed by classical liquids like water avoiding any transmission measurement, and (ii) ATR scheme is sensitive enough to characterize dilutions whose solute concentration could be as small as 1%. In ATR experiment, the THz beam is reflected against the base of a prism whose index of refraction is higher than that of the studied substance to achieve total internal reflection of the THz beam. The reference is recorded without the material placed upon the prism base to finally obtain the measured complex total reflection coefficient .
The complex refractive index of the sample to be characterized is determined when the calculated transfer function (i.e., either , , or ), in which is the only adjustable variable, is equal to the experimental ones (, , or , respectively). When dealing with nonmagnetic materials, analytical expressions of the transfer functions are rather simple as long as the samples are slabs with parallel and flat faces. At the sample location, the incoming THz beam must be a plane wave. In practice, this is verified even with focused THz Gaussian beams, as far as the sample, placed at the focal point, is thinner than the Rayleigh length of the beam. Thus, analytical expressions of , , and are as follows:
with , and .
Solving Eq. (4) (or Eq. (5) for thin samples) is more difficult because of the oscillatory complex exponential term
To get rid of the oscillatory behavior, we proposed  to employ an error function that exhibits a monotonous shape with a single minimum that is quickly found with any numerical method:
with or . Figure 2 shows in transmission versus
3. Precision on the parameters determination
The material parameters (
3.1. Effect of a bad value of the sample thickness
A bad value of the thickness
Let us suppose, for the sake of sim and reflection are obtained by differentiating plification, that the Fabry‐Perot rebounds can be removed by a proper time‐windowing of the THz waveform. Thus, Eq. (4) of becomes:
A simple calculation leads to:
with . Usually, experiments are performed at normal incidence, for which Eq. (10) simplifies into:
Typically, an error of 1% on
3.2. Effect of an angular tilt
Using the same procedure, we investigate the influence of a bad orientation of the sample. The derivation is done versus the tilt angle Δ
A slight difference is obtained if the angular tilt is along the direction of the E‐field (TE) or perpendicular to it (TM). As shown in Figure 4, an angular tilt of 1° induces typically an error Δ
Even if the samples are perfectly well aligned relatively to the THz propagation axis, a noncollimated THz beam could lead to inaccuracies equivalent to those induced by angular tilt. Indeed, any converging Gaussian beam can be decomposed into plane waves arriving onto the sample under different incidence angles, from 0° (along the propagation axis) up to the diffraction angle whose value depends on the frequency. Referring to Figure 4, this could induce an error of about 1%. This unwanted effect gets even worse as THz converging Gaussian beam probes thick samples, because it defocuses the THz beam that impinges the receiver. Consequently, the detected signal can be respectively larger or weaker than expected, which leads to an over‐ or underestimation of the sample losses.
In reflection, a bad orientation (angular tilt Δ
By comparing expressions (12)—transmission—and (13)—reflection—it appears that the error Δ
3.3. Effect of a bad positioning of the reference mirror in reflection THz‐TDS
In the case of reflection THz‐TDS, the main geometrical error source is the misalignment of the sample as compared to the reference mirror. For the sake of simplicity, we suppose that the sample is thick enough to neglect the Fabry‐Perot rebounds. Let
This phase difference leads to an error :
Because of the
These rather large errors are induced by very small shifts, here , and get even larger at higher frequencies, according to Eq. (15). Therefore, in reflection THz‐TDS, a great attention must be paid to position the sample at the exact location of the reference mirror, or, if not possible, to either control, measure, or correct the induced phase difference. For that purpose, it exists several experimental [23–26] or numerical  solutions.
3.4. Effect of a noise with photo‐conducting THz antennas
Noise makes uncertain the measured values of the magnitude and the phase of the THz signals. Let us treat here only the case of photo‐conducting antennas. We call the noise power, i.e., the square of the variance of the THz signals. A first noise is generated by the emitter: shot noise due to the random arrival of the pump laser photons, fluctuation of the laser intensity, etc. Drift and mechanical vibrations of the optical delay line as well as random fluctuations of the laser beam direction add a noise‐equivalent contribution. It depends strongly on the equipment: Withayachumnankul et al.  have measured an amplitude variance of the order of 10-3 that was mostly due to delay‐line registration and mechanical drift. This noise is of the same order as defined previously and can be included in the emitter noise. The THz beam, together with its noise, is then reflected or transmitted by/through the sample toward the receiving antenna. At the receiver, two additional noises perturb the recorded signal, namely, the shot noise that is proportional to the recorded current (, Δ
where is the current recorded without sample in the set up (reference current), whereas is the current recorded with the sample. Taking into account the small value of the noise when compared to the signal (perturbation approach), we get:
Figure 6 shows the
Relations (17) and (18) indicate that the standard deviation is proportional to the detection bandwidth Δ
Assuming the noise is much smaller than the signal, one easily gets:
With a fully random phase noise, the related standard deviations of the modulus and phase of the measured signals are:
The validity of our analysis is experimentally verified as depicted in Figure 7. The standard deviations of the modulus
Finally, one should derive the experimental standard deviation from the actual measured signals. The
The standard deviation is calculated from Eq. (22):
The noise‐induced errors Δ
Examples of so‐determined errorare presented in Figure 8. For the Stycast samples, THz-TDS experiments performed in transmission leads to the amazing precision Δ
Figure 9 summarizes the influence for the different error sources in transmission THz‐TDS. The sample is a 1‐mm thick slab, with
The angular tilt is by far the largest source of error for the refractive index
4. Transmission or reflection THz‐TDS?
For opaque samples, it is compulsory to perform THz‐TDS measurements in reflection because no THz signal is transmitted. For low‐index transparent samples, transmission scheme is preferable, as the induced phase variation is integrated over the whole sample length, while the Fresnel phase change in reflection are weaker. For samples with a moderate absorption coefficient, the choice of the best experimental scheme is not obvious. However, the error study presented in the previous paragraph helps in selecting the optimized THz‐TDS characterization technique, i.e., either in reflection or in transmission. For the sake of simplicity, we address here only the case of normal incidence. We suppose that the samples are perfectly placed in the THz beam (no angular tilt) and that the sample thickness is perfectly known. We suppose also that the sample is thick enough to permit a record of only the directly reflected or transmitted THz pulse, by a proper time‐windowing of the other rebounds. Thus, the only source of inaccuracy is induced by intrinsic noises, given by relation (18), resulting on the errors expressed by Eqs. (24) and (25), where σx is expressed using Eq. (17). In reflection, Δ
The white curve indicates the limit . Above this limit, and thus transmission TDS is more precise than reflection TDS. This happens when absorption is rather small (). At higher absorption, reflection is preferable because the transmitted signal becomes weaker. The limit between both techniques is plotted in the (
5. Combined transmission and reflection THz‐TDS
In the far infrared, some materials exhibit strong absorption lines due to molecular resonances or/and due to collective excitations. In gases, narrow molecular resonances correspond to the excitation of mechanical vibrations of the whole molecule structure. In liquids and solid materials, the molecular resonances are coupled and broadened by thermal and density inhomogeneity at the molecular scale, resulting in wider absorption bands. In crystals, excitation of phonons leads also to a strong absorption of the THz waves. When characterizing such materials in transmission THz‐TDS, there could be no signal detected in transmission within the absorption bands. The phase is lost in these spectral regions, and thus extraction of the material parameters at higher frequencies is no more possible using common THz‐TDS extraction procedures as
The same procedure is repeated for each observed saturated resonant peak. Finally, is accurately obtained from corrected transmission in all regions of transparency, while in the absorption peaks, we set . An example is given in Figure 11 (right). An 890‐µm thick pellet of pure maltose has been characterized by THz‐TDS. In the vicinity of the absorption peaks at 1.15, 1.65, and 2.05 THz, transmission of the sample is below the noise level. In these spectral ranges, we save the value of extracted from corrected reflection data, while in the other regions of transparency, we retain the phase‐corrected transmission values . Absorption peaks as high as 250 cm
6. Combined transmission THz‐TDS and Kramers‐Kronig analysis
Reflection schemes are sometimes not available with commercial systems, preventing any phase correction procedure as detailed in the previous paragraph. Fortunately, phase jumps can be corrected from transmission measurements only. The causality of THz response of natural materials makes possible to calculate from with the Kramers‐Kronig (KK) relations. In fact, the phase jumps, due to saturation of transmission in the absorption peaks, impact mainly the extraction of and very slightly outside the absorption bands. Thus, the idea  here is to extract from TDS data in the transparency spectral regions and to perform a KK calculation to get . Missing values (in the absorption peak) induce an error on , which is quite small because it is spread over the spectrum thanks to the integral KK calculation. Comparing, in the transparency regions, determined by the KK transformation and the one extracted from THz‐TDS data permits to know and correct the phase jumps occurring at each resonance, if any. A last extraction, with the phase corrected transmission TDS data, leads to a very precise determination of between the absorption peaks. In fact, THz‐TDS data are obtained over a limited spectral range, while KK calculation should be completed from 0 to infinity. The resulting error is minimized by performing a singly subtractive Kramers‐Kronig (SSKK) transform :
7. Scattering effect
When dealing with chemical compounds, many samples are pellets fabricated by pressing a powder mixture of the material to be studied and of a transparent hosting matrix. This technique permits to characterize materials (named hereafter “inclusion” material) that are extremely absorbent in the THz range, or that cannot be manufactured as bulky or film samples, or that could be dangerous at a high concentration. The optical parameters (refractive index and absorption coefficient) of the inclusion material, diluted in the hosting powder, are retrieved through a model describing the electromagnetism response of the mixture. If this mixture can be considered as homogeneous, i.e., if powder grains and more generally inhomogeneity are smaller than the wavelength, there is no scattering: the electromagnetic response of the material can be described by effective medium theories. Among many theories, the most popular models are the Maxwell‐Garnett one (MG)  and the Bruggmann (BG) one : the inclusions are supposed to be spheroids or ellipsoids embedded in the hosting matrix. According to these models, the dielectric constant of the mixture respectively obeys:
Thus, in many heterogeneous samples, scattering occurs and depends on the relative size of the scattering particles as compared to the wavelength . If the particle size is smaller than about λ/20, scattering may be neglected while, if it is bigger than about 10 × λ, rays are geometrically deviated by powder grains. In between these limits, when particles are smaller than the wavelength, the scattering process is well described by the Rayleigh theory, while Mie scattering occurs for grain size comparable or bigger than the wavelength. Because of the typical size of the grains in common powders, Rayleigh scattering is almost negligible at THz frequencies and Mie scattering  model or even more complicated theories must be employed. As there is no analytical expression available to render the scattered amplitude, numerical codes are used. Nevertheless, Mie theory can be approximated at low scatterer concentration by simplified models, like the one proposed by Raman  to explain the Christiansen effect . In this case, the equivalent absorption is expressed as:
At different frequencies, the experimental data are well fitted by a linear curve for
We have described here the main features of THz‐TDS. This is a unique technique to quantitatively and precisely characterize the electromagnetic response of materials and devices over a broadband in the far‐infrared domain. However, because of limited space, we did not address in this chapter some additional possibilities of THz‐TDS in terms of material characterization:
Anisotropic materials: Their characterization by THz‐TDS is not tricky as most of THz antennas (photo‐conducting switches or electro‐optic crystals) are polarization‐sensitive, with a rather good rejection level (a few percent). Even cross‐polarization effects can be investigated by rotating the receiving antenna around the optical axis of the THz system. Moreover, THz‐TDS supplies the phase of the transmitted signals with respect to the involved THz beam polarization, from which the anisotropic parameters of a sample can be deduced.
Metamaterials: They are quite easy to manufacture for the THz range, as they require repetitive features whose size is smaller than the wavelength, i.e., in the range below a few tens of microns. This is achieved with MEMS or/and microelectronics technologies. Such THz metamaterials  show amazing properties, like negative refractive index (left‐handed property), chirality, and so on. THz‐TDS is especially well adapted to characterize these metamaterials because it delivers both amplitude and phase of the reflected or/and transmitted signal. Thus, for example, a negative phase, due to propagation in a left‐handed material, is clearly observed in the THz‐TDS experimental spectra.
Time‐resolved THz‐TDS : This technique can be applied if the sample under test is sensitive to light. In this case, a third part of the pulsed laser beam illuminates the sample with an adjustable delay as compared to the impinging THz pulse. The light‐induced modification of the sample properties, mostly by photo‐generation of free carriers, changes the transmission of the THz beam. By varying the optical‐THz time‐delay, the photo‐induced excitation in the material is time resolved. This method works very well to study the carrier dynamics in semiconductors and in superconductors, as far as the carrier lifetime is not too short as compared to the THz pulse duration. The main limitation of the technique is the rather long duration of the THz pulse, i.e., few hundreds of femtoseconds, as compared to very fast phenomena in matter. Investigating short events requires a proper deconvolution of the temporal records, which is not an easy task.
The list of applications of THz‐TDS is for sure quite long and giving it exhaustively is almost impossible. Moreover, the continuous progress of technology makes the technique really easy to use, especially with commercial systems, and numerous new scientific results are regularly published.
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