1. Introduction

The terahertz (THz) region of the electromagnetic spectrum is typically considered to occupy 300 GHz to 10 THz (1 mm to 30 µm), bridging the gap between millimeter waves and the infrared. There are unique properties of THz radiation that make it a potentially powerful technique in security [1–6], pharmaceutical [7–9], non-destructive testing [3,10], and medicine [11]. First interest of THz radiation is its modest attenuation when passing through common physical barriers such as clothes, woods, papers, and plastics [1]. Secondly, many chemical products exhibit characteristic spectral absorption which can be used to distinguish them. Furthermore, as THz waves are non-ionising and their wavelengths are sub-millimetric, they have the potential to combine safe-to-use high-resolution imaging and identification through spectroscopy.

THz imaging and spectroscopy are now widely used by many research groups and have begun supporting homeland security for detection of illegal material. The identification of chemical products using THz spectroscopy requires a reliable reference database. Nevertheless a large variety of explosives [3,5,6], drugs [4], and biological products [12] have been indexed in the THz range, the absorption spectra of common materials show different features depending if the sample is granular or ground in fine powers [1,13]. Moreover, the absorption spectra of common materials such as fabrics have been characterized using THz time-domain spectroscopy and presented different spectral features. For example, cotton’s fabric exhibited sometime an absorption peak at 2.5 THz [1], thick cotton’s fabric shows attenuation peaks at 0.54 THz, 0.83 THz, and 1.0 THz [14], while others measurements presented a continuous increase of the spectral absorption for the same frequency range [6,15]. Similar variations occurred also for other fabrics like wool, polyester and nylon [1,14–16]. These discrepancies demonstrate the difficulties to obtain reliable identification of chemical products since some of these common barrier materials can have an attenuation peak corresponding to the main absorption peak of usual explosive like RDX. In order to better understand these discrepancies, we demonstrate that the distortion of the spatial coherence of the THz pulse reflected or transmitted through a non-uniform sample, like fabrics or granular products, can result into the overestimation of the electromagnetic absorption, the generation of false peaks and even to the disappearance of characteristics peaks into the absorption spectrum of the material studied.

2. Numerical model

To clarify these discrepancies between the different absorption spectra of common materials, we developed a simple model based on a standard THz time–domain spectroscopy (THz TDS) system [17]. This model relies on a broadband THz pulse propagating through the sample to be analyzed, and the sample is imaged on the detector by using focusing optics. Thereafter, the electric field distribution of the transmitted THz pulse is measured in time through a pump-probe scheme. The spectrum of the transmitted THz pulse is obtained by performing the Fourier transform of the average electric field detected. In this model, the initial electric field (E_I) of the THz pulse is expressed in the temporal domain by E_I(x, y, z, t)=A(x, y, z, t)exp(ikz−iω ₀ t) where A is the envelope of the electric field, ω ₀ is the central angular frequency, t is the time, k is the wavevector, z corresponds to the propagation axis, and the transverse coordinates are given by x and y. The sample to be analyzed is modeled by a non-uniform thickness d(x,y) media, and the detection in a THz-TDS system is done over the collected cross-section of the THz beam. Therefore, we integrated the electric field of the THz pulse over the transverse coordinates x and y of the beam. Finally, the spectrum of the transmitted THz pulse is obtained by taking the Fourier transform of the integrated transmitted electric field and is given by:

where α(ω) is the sample spectral absorption, c is the speed of light, OPL(ω,x,y)=d(x,y)[n(ω)-1] is the optical path length, and n(ω) is the index of refraction of the sample in the THz band [17]. There is no analytic solution for Eq. (1) when the distributions of the sample’s thickness d(x,y) or refractive index n(ω) (and consequently OPL(ω,x,y)) are random. Therefore, numerical integrations have to be carried out.

3. Results

3.1 Teflon samples

We performed a first series of experiments to validate the effect of the spatial coherence distortion by using Teflon samples. A THz-TDS system based on photoconductive antennas for the THz source and detection was used. The spectral bandwidth of the generated THz pulse spanned from 0.1 THz to 3 THz. For this first series of experiment, we used three samples of Teflon having different distribution of thickness, namely a Teflon foil, a soft Teflon felt, and a rough Teflon felt. The distribution of the sample’s thickness was measured with a digital caliper on which a rod of 500 µm diameter was mounted.

The Teflon foil had a uniform thickness of 800 µm. The measured absorption spectrum is shown in Fig. 1(a) and is close to zero (dotted line) over the spectral range of the THz-TDS setup used. The second sample, the soft Teflon felt, had a mean thickness of 560 µm and a standard deviation of ~80 µm. The corresponding standard deviation of the OPL(σ_OPL) for the THz pulse transmitted through the soft Teflon felt was therefore around 80µm(n_Teflon − n_air)≅35µm. In the case of the rough Teflon felt, the standard deviation of sample’s thickness was ~300 µm (3.8 times larger than for the soft Teflon felt) and the corresponding σ_OPL was equal to 300µm(n_Teflon − n_air)≅140µm. Figure 1(a) shows the measured attenuation spectra for the soft and the rough Teflon felts. For the soft Teflon felt (Fig. 1(a), dashed line), we measured a continuous increase of the attenuation for higher frequency up to the limit of detection of the THz-TDS system. For the rough Teflon felt, we clearly observed three attenuation peaks positioned at 0.83 THz, 1.72 THz, and 2.46 THz. It is important to remember here that the Teflon absorption (α(ω)) is negligible and these attenuation peaks are the result of the interference on the THz-TDS’s detector [17]. These interference features were caused by the non-uniformity of the sample’s thickness. Thus, when the THz pulse passed through the sample, a random phase (or OPL) distributed across the section of the beam was added to the local electric field. From the statistical point of view, the standard deviation corresponds to the most representative fluctuation of a random phenomenon. If on the detector, the standard deviation of the OPL (σ_OPL) satisfies (2m+1/2)σ_OPL=2π c/ω where m is an integer, the frequency ω would mostly interfere destructively.

 

Fig. 1. (a) Experimental results of the spectral attenuation for an uniform Teflon foil of 800 µm thick (dotted line), for a Teflon felt having a measured OPL standard deviation of ~35 µm (dashed line), and for a Teflon felt having a measured OPL standard deviation of ~140 µm (solid line). (b) Numerical simulations for the spectral attenuation of a THz pulse passing through a Teflon felt with an OPL standard deviation of 42 µm (dash line), and 144 µm (solid line).

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Figure 1(b) presents the simulated attenuation spectra using the Eq. (1) for the THz pulse passing through the soft and the rough Teflon felt. We did the numerical integration of the Eq. (1) by assuming α(ω)=0 and a constant index of refraction [18] n(ω)=1.44. We optimized the fit between the measured and the simulated attenuation spectra by varying σ_OPL in the simulations. By using an σ_OPL=42 µm for the simulations (which is close to the experimental value of σ_OPL=35 µm), we retrieved a similar continuous increase of the attenuation up to 36 cm⁻¹ at 2.3 THz for the soft Teflon felt. The first peak of attenuation was calculated to be at 3.4 THz, which is beyond the working spectrum of our THz-TDS system. What is observable on Fig. 1 for the soft Teflon felt (dashed line) is therefore only the “wing” of the attenuation peak. For the simulated rough Teflon felt, we used an σ_OPL=144 µm for the simulations (experimental value is σ_OPL=140 µm) in order to match the three attenuation peaks positioned at 0.84 THz, 1.68 THz, and 2.52 THz. We should note here that variations in scattering and diffraction by transverse inhomogeneous edge features may also induce frequency dependent destructive interference [19–21]. Considering these additional effects to our model would certainly contribute to refine our numerical simulations. However, the phase distortion is mainly caused by the fluctuation of the OPL and the simulated attenuation spectra are in qualitative agreement with the experimental results, and this, even by neglecting the chromatic aberration of the THz beam, and the Teflon felt’s scattering.

3.2 Fabric samples

These previous results obtained with the Teflon felts clearly show that non-homogenous samples induced phase distortion resulting into frequency dependent destructive interference on the THz-TDS detector. If THz-TDS is to be used for detecting hidden illegal substances, it is important to know how these barriers modify the detected THz spectrum. As mentioned in the introduction, many barrier materials such as fabrics exhibited different spectral absorption, while the composition of these fabrics was the same [1,14,15]. Since most of the barrier materials are non-homogenous, absorption spectral patterns caused by interference on the TDS detector should be observed. Moreover, absorption spectra for fabrics having different wire diameters should also be different. In order to confirm these expectations, Fig. 2(a) shows the measured attenuation for a single layer of polyester fabric having small wire diameter (500 deniers) and large wire diameter (1000 deniers). Clearly, the attenuation spectra of polyester are different depending of the wire diameter used for the fabric. This difference is simply due to the larger variation of the thickness (or OPL) for the 1000 deniers polyester fabric, and as observed with the Teflon, the destructive interference occurring on the THz detector is shifted towards the lower frequencies [17] as compared to the 500 deniers sample. Using the same model as with the Teflon, we performed numerical simulation in Fig. 2(b) of the THz pulse passing through the 500 and 1000 deniers polyester samples. Since the electromagnetic absorption of the polyester might be weak (see below), we performed the numerical integration of the Eq. (1) by assuming α(ω)=0. The index of refraction for the polyester was estimated to be 1.55 from the THz-TDS measurement. Figure 2(c) shows a sketch of the fabric structure. Over a single unit cell in Fig. 2(c), the thickness of the 500 deniers fabric fluctuated from 0 µm (hole in the unit cell) up to 250 µm (crossing point of two wires), while for the 1000 deniers sample, the thickness would vary from 0 µm up to 350 µm. By using these parameters, we retrieved again qualitative agreement between the measurements (Fig. 2(a)) and the numerical simulations (Fig. 2(b)). This points out that the recorded attenuation spectra in Fig. 2(a) are mainly caused by the structure of the fabric, and not its electromagnetic absorption.

 

Fig. 2. (a) Experimental results of the spectral attenuation for a polyester sample of 500 and 1000 deniers. (b) Numerical simulations for the spectral attenuation of a THz pulse passing through a polyester fabric of 500 and 1000 deniers. (c) Sketch of the fabric structure.

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To further verify that the spectral features observed in Fig. 2(a) are due to the structure of the fabric and not its electromagnetic absorption, we measured the absorption spectrum of a ground polyester fabric. The results are presented in Fig. 3. Here, we compared the measured attenuation before and after grinding a 500 deniers polyester fabric. For the ground sample, the thickness was more homogenous than the original polyester fabric, and thus, we observed a decrease by a factor of 4 for the measured attenuation.

 

Fig. 3. (a) Experimental results of the spectral attenuation for a polyester fabric of 500 deniers and a ground polyester sample. Optical images of (b) the 500 deniers polyester fabric and (c) the ground polyester sample.

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3.3 alpha-lactose samples

Up to now, we presented the effect of coherence distortion on THz-TDS by using samples with negligible or very low electromagnetic absorption. In the following section, we present the impact of the coherence distortion on chemical product having narrow electromagnetic absorption bands and well referenced spectral signature in the THz range. First samples used are α-lactose monohydrate powders with different grain size. Figure 4 compares the measured attenuation spectra for two samples. The first sample (black curve) was made of α-lactose powder having a diameter lower than 5 µm. The powder was compressed into a Teflon capsule forming a sample with a uniform thickness of 500 µm. The second sample (red curve) was made of α-lactose flakes having a diameter varying between 355 µm to 1 mm obtained using a sieve. The powder of this sample was not compressed, but was spread uniformly as much as possible inside the Teflon capsule, and the variation of the sample thickness was limited by the size of the α-lactose flakes. In Figure 4, for the 5-µm α-lactose sample, we can clearly observe the well-referenced absorption peak at 0.54, 1.20, 1.38 and 1.82 THz. On the order hand, for the α-lactose flakes, the two absorptions peaks at 1.20 and 1.38 THz completely disappeared and a new broadband peak centered around 1.0 THz appeared. This broadband peak is again likely due to the destructive interference on the THz-TDS detector. However, we could not perform numerical simulations for this sample since we could not measure precisely the OPL distribution. It is also worth mentioning that this broadband peak does not originate from the scattering of the powder. In fact, the scattering cross section is proportional to ω ⁴, implying that higher frequencies should be scattered more strongly, which is not the case in Fig. 4.

 

Fig. 4. Experimental results of the spectral attenuation for α-lactose monohydrate samples having powder diameter lower than 5 µm (black line), and flake size between 355 µm to 1mm (red line).

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3.4 RDX samples

Finally, a last experimental example of the coherence distortion effect is shown in Fig. 5 for RDX, a nitramine explosive. We compared the absorption spectrum of two RDX samples having different grain sizes. The first sample was a class I RDX having grain diameter inferior to 5 µm and the second sample was a class III RDX for which the grain size varied between 200 to 300 µm. Clearly, the attenuation spectra in Fig. 5 for the two RDX samples are dissimilar. For the 5-µm RDX powder, we can see the well referenced absorption peak positioned at 0.81, 1.05, 1.38, 1.53 and 1.98 THz. However, for the large diameter RDX powder, the phase distortion altered the attenuation spectrum and we cannot retrieve the same spectral signature than for the 5-µm RDX powder. The phase distortion from the 200–300 µm RDX seems to shift the main RDX absorption peak from 0.81 THz to ~0.70 THz, but in reality, the observed attenuation peak at ~0.70 THz is the combination of the electromagnetic absorption peak at 0.81 THz from the RDX and a destructive interference peak positioned around 0.65 THz. In fact, the size of the crystal or grain powder does not shift the position of the electromagnetic absorption peak, it is only its superposition with a destructive interference peak that might give an apparent spectral shift.

 

Fig. 5. Experimental results of the spectral attenuation for RDX samples having powder diameter lower than 5 µm (black line), and between 200 µm to 300 µm (red line).

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4. Conclusion

In summary, these results clearly demonstrated that the distortion of the spatial coherence of a THz pulse can generate artifacts with a standard THz-TDS system [17]. We presented some examples of the effects of coherence distortion on the measured attenuation spectra of chemical products having well referenced absorption peaks in the THz range. Evidently when the coherence distortion of the THz beam increased, the background of the attenuation spectrum increased too. Also, for sample having a non-uniform thickness, false absorption peaks appeared, some attenuation peaks of the sample apparently shifted in frequency, or even worse, some absorption peaks disappeared.

Up to now, the measurement of the THz absorption spectrum has been done on many materials by using a THz-TDS. Generally, the analyzed sample was prepared with precaution in order to have a uniform thickness and a smooth surface. However, there are many samples having non-uniform thickness (or OPL) such as clothes, woods or granular samples which have also been studied extensively with a THz-TDS. With such kind of samples, the measured electromagnetic absorption might have been overestimated due to the interference occurring on the THz-TDS’s detector, and could explain the discrepancies between the measured absorption spectra for samples having an OPL fluctuating more than the wavelength of the THz beam. It is important to note that explosive samples having uniform thickness and covered with a single layer of fabric could be identified [6] with THz-TDS, however, such identification become harder as the number of covering cloth layers increase [1].

The main challenge here is to apply the THz-TDS to the field where most of the samples or targets to analyse are non-uniform and non-homogenous. Consequently, the phase distortion of the THz pulse induced artifact on the measured absorption spectrum. Evidently, the use of detector’s matrix instead of a single detector would minimize the integration of the distortion over the complete cross-section of the THz beam, and thus, minimize the destructive interference [22]. However, many research groups are now working on different type of THz sources and detectors. THz-TDS is a very powerful technique for the analysis in lab environment of well prepared and homogenous samples [1–11]. On the other hand, there are new tunable continuous wave THz source that are more and more efficient. Such tunable THz sources when combined with a sensitive broadband detector would have great advantage in the field since such detection scheme would be less sensitive to the phase distortion.