1. Introduction

Terahertz (THz) technology has seen a growth in interest over the past few decades due to its application in the fields of communications [1–4], spectroscopy [5–8], and imaging [9]. Terahertz technology makes use of electromagnetic radiation over a spectrum of frequencies between 300 GHz and 30 THz. As THz technology has matured and novel applications of the THz spectrum have been proposed, the development of transmissive materials for elements such as lenses and windows has become imperative. This was initially achieved using crystalline materials such as quartz and high resistivity silicon (Si) [6], and these materials are still common today. As more materials were characterized in the THz spectrum, it was found that several polymers, including high-density polyethylene (HDPE) and cyclic olefin copolymer (COC), were also highly transmissive [7,10,11]. These materials have the added benefit of being low-cost and have a lower refractive index, reducing Fresnel losses. Both types of materials are adequate for standard THz optics such as lenses and windows; however, these materials are only available with specific material properties. If designs demand material properties, specifically refractive index, that do not correlate with one of these standard materials, an alternative is required.

Composites in the THz spectrum provide a means of designing materials with customizable properties. These THz composites are composed of two or more materials (called the host and inclusions) with significantly different material properties. Mixing the host and inclusions into a composite gives new material properties that are a combination of the individual components. There has been much work done on composite materials in the THz spectrum [12–16], with a focus typically on characterizing the refractive index of these materials with the appropriate effective medium theory [14,17,18]. Few studies focus on rigorously characterizing the extinction properties of THz composite materials, though some limited studies exist [16,17]. Characterizing, and subsequently minimizing, the extinction coefficient of THz composite materials is essential to designing composites that are transparent in the THz spectrum, which is a necessity for transmissive elements such as lenses.

In this work, a rigorous study of the loss mechanisms in THz composite materials is presented. Both HDPE and COC are used as low-loss host polymers. Either Si, silicon dioxide (SiO₂), or alumina (Al₂O₃), are used as inclusions. The inclusion materials all have significantly higher refractive indices than the two host materials, making them excellent candidates for THz composites. All these materials have been shown to have low loss in the THz spectrum, and so should produce low-loss composites. However, it is found that other considerations such as material crystallinity and charge carrier density greatly affect the loss in polymer composites.

2. Theory

The refraction and extinction properties of THz composites can be described by the Bruggeman model. While there are many different effective medium theories that have been explored to describe the refraction and extinction properties of composite materials (in any spectrum) [14,16–18], the Bruggeman model has been shown by ourselves [15] and others [12,14] to produce accurate results. The Bruggeman model can be cast as

The host and inclusion materials used in this work were chosen for their varying refractive indices and low extinction coefficients. Literature values for the materials are given in Table 1.

Table 1. Refractive index and extinction coefficients for the materials used in this work.

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There are three caveats with the numbers given in the table above. First, the refractive index and extinction coefficient of all materials are functions of THz frequency, while the values given are for 1 THz. Second, the numbers for inclusions all correspond to crystalline materials. Unfortunately, the inclusions for THz composites are typically microparticles that are not crystalline due to cost considerations or fabrication methods. The effects of crystallinity can be clearly seen by the differing extinction coefficients of quartz and fused silica, both pure SiO₂. Third, the extinction coefficient for the Si also assumes an intrinsic carrier density, with no extrinsic doping. The presence of any dopants in the Si microparticles will drastically increase the extinction coefficient due to free carrier absorption. The two additional loss mechanisms mentioned above will now be explored further.

The extinction behaviour of non-crystalline materials in the THz spectrum has been thoroughly investigated [8,16,19]. It has been well established that extinction in many polycrystalline and amorphous materials in the THz spectrum increases with frequency, via

In the equation above, k is a scaling constant, c is the speed of light, ħ is the reduced Planck constant, and β is an exponential constant. The constant β has been shown to be 2 for most semiconductor and oxide materials, though it is slightly less for polymers [19]. The increase in extinction with increasing frequency is typically attributed to the presence of disordered states in polycrystalline and amorphous media.

Free carrier absorption in semiconducting materials should also be considered when fabricating composite materials. The presence of free carriers in a material introduces additional dielectric susceptibility, ${\tilde{\chi }_\textrm{e}}(\omega )$ and ${\tilde{\chi }_\textrm{h}}(\omega )$, caused by both free electrons and holes, respectively. These dielectric susceptibilities affect both the refractive index and extinction coefficient of the material, though the change is more noticeable in extinction coefficient. The dielectric susceptibilities can be written as

3. Results

This section will show results for both the refractive index and extinction coefficient of composites fabricated using various combinations of host and inclusion materials. Each host material will be explored for multiple inclusion materials in each of the following subsections. All results were measured using THz time-domain spectroscopy, as was done in previous studies on THz composite materials [5–8].

3.1 COC composites

The first set of composite material results are based on COC as a host material. Fabricating COC composite materials is challenging since COC is a solid at room temperature, making it difficult to uniformly mix with solid microparticle inclusions. COC cannot be readily purchased as microparticles for dry mixing, while heating it to its melting temperature in an ambient environment results in denaturing of the polymer. Therefore, an alternative method of fabrication based on work by Johansen was used [20]. In this method, COC pellets are dissolved in a non-polar solvent (toluene) to create a liquid solution into which microparticles can be mixed. The solution is poured into sheets and heated to evaporate the toluene resulting in thin COC polymer composite sheets. The large amount of toluene required to dissolve the COC pellets prevents larger blocks of composite material from being formed due to the presence of voids that evolve during the evaporation of the toluene. Due to the complexity of this fabrication process, only three samples were made: one pure COC sample, one COC-SiO₂ composite, and one COC-Si composite. The average particle sizes of the SiO₂ and Si microparticles were both approximately 5 μm. The refractive index results for these composites are shown in Fig. 1. The COC-SiO₂ composite had a volumetric fraction of 34% and the COC-Si composite had a volumetric fraction of 21%. Thin horizontal lines are plotted in Fig. 1 corresponding to the theoretical refractive index based on the Bruggeman model with a depolarization factor of 1/8 along with the corresponding refractive index and extinction coefficient values given in Table 1. The depolarization factor was fitted to the experimental data. It should be noted that the refractive index solution to Eq. (1) is effectively independent of the chosen extinction coefficient as the extinction coefficient is over an order of magnitude smaller than the refractive index. The samples were all thin films with thicknesses of 1.5 mm for the pure COC, 0.94 mm for the SiO₂-COC composite, and 0.52 mm for the Si-COC composite.

 

Fig. 1. The refractive index of three COC composite materials is shown. The first material was pure COC, shown as the black dotted lines, the second sample was a COC-SiO₂ composite, shown as the solid red lines, and the third sample was a COC-Si sample, shown as the black dashed line. All samples were dissolved in toluene and poured into thin sheets. The thin horizontal black lines correspond to the refractive index predicted by the Bruggeman model.

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The results in Fig. 1 show the effect that introducing microparticles into the polymer has on refractive index. An average refractive index of n = 1.525 is seen for pure COC, which increases to n = 1.73 for the COC-SiO₂ composite and n = 2.0 for the COC-Si composite. These numbers mostly agree with the predicted values from the Bruggeman model. Unfortunately, the introduction of inclusions into the COC polymer increased the extinction coefficient of the composite samples, though these results were unreliable. In the case of the pure COC and COC-SiO₂ composites, the samples were too thin to produce significant enough loss to accurate measure the extinction coefficient. Conversely, the extreme loss present in the COC-Si composite yielded poor measurement results preventing the extinction coefficient from being measured accurately. This loss, which will be seen later to be due to free carrier absorption in the Si microparticles, is also the source of the noise in the refractive index of the COC-Si composites in Fig. 1. The losses for the SiO₂ and Si microparticles will be investigated further in Section 3.2 using HDPE as a host. In summary, it is possible to fabricate thin composite materials with variable refractive indices in the THz spectrum using COC as a host polymer. This is provided that SiO₂, or some other adequately low loss microparticle inclusion, is used. These materials can be used in THz windows, but thicker THz optics will require other composite materials with different fabrication processes.

3.2 HDPE composites

The second set of composite materials investigated in this work are based on HDPE as a host material. While still a solid at room temperature, HDPE is much simpler to mix with microparticle inclusions since it can be sourced as a fine powder. This allows the HDPE and microparticle inclusions to be dry-mixed, then hot pressed to create solid composite samples with uniform particle mixing. Hot pressing must be done at a high enough temperature that the HDPE particles fuse together, though not hot enough to melt the HDPE. Composite samples were fabricated by dry-mixing Si, SiO₂, or Al₂O₃ microparticles with powdered HDPE wax. The Si and SiO₂ microparticles were the same ones used with COC. The Al₂O₃ microparticles were spherical (N_d = 1/3) with an average particle diameter of 0.8 mm, while the powdered HDPE wax had an average particle diameter of 4 μm. These dry-mixtures were pressed in a cylindrical die at 83°C under 10 MPa of pressure for 30-60 minutes. The long pressing time was to allow heat to fully penetrate the sample, ensuring that all HDPE particles were fused. The effects of improper pressing will be investigated shortly.

To begin, results for HDPE-SiO₂ composites are shown. The first set of results in Figs. 2(a) and 2(b) show the refractive index (a) and extinction coefficient (b) for composites with volumetric fractions up to 20% at 1 THz. The solid black line in Fig. 2(a) denotes the Bruggeman model predictions based on a fitted depolarization factor of 1/8 along with the corresponding refractive index and extinction coefficient values given in Table 1. Similar to Fig. 1, the refractive index for these composite samples is constant between 500 GHz and 2.5 THz, so no spectra are plotted. However, the extinction coefficient of these samples was frequency dependent, so five representative extinction coefficient spectra are plotted in Fig. 2(c). The effects of poor sample pressing are clearly seen in the extinction coefficient results. All samples had thicknesses ranging from 2 mm to 5 mm.

 

Fig. 2. The refractive index (a), extinction coefficient at 1 THz (b), and extinction coefficient spectra of several representative samples (c) are shown here for HDPE-SiO₂ composites. The refractive index and extinction coefficients in (a) and (b) are shown as functions of volumetric fraction, while the spectra in (c) are shown as a function of THz frequency. In (a), the Bruggeman model is plotted as the solid black line. Spectra in (c) correspond to samples with volumetric fractions of 6.6% (green line), 10.4% (red line), 14.9% (solid black line), and 19.9% (blue line). The sample with a volumetric fraction of 14.9% was re-pressed and the extinction coefficient after re-pressing is shown in (b) as the hollow black square and in (c) as a dashed black line.

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The results for refractive index as a function of volumetric fraction in Fig. 2(a) show that the refractive index of these HDPE-SiO₂ composites can be very accurately predicted by the Bruggeman model with minimal error up to a volumetric fraction of 20%. The extinction coefficient of these composites, whose values at 1 THz are shown in Fig. 2(b) and whose spectra are shown in Fig. 2(c), are a different story. The extinction coefficients at 1 THz vary anywhere from a minimum value of approximately 1×10⁻³ up to 1×10⁻², showing no correlation with volumetric fraction. The minimum extinction coefficient corresponds to the extinction of the host and inclusions. This is evidenced by the results in Fig. 2(c), where the extinction coefficient spectra for correctly pressed samples is independent of volumetric fraction up to 20%. These spectra show a linear increase with frequency as is predicted in Eq. (2), and fitting these results gives β = 2 and k = 3.5×10⁴⁴ J^-1m^-2 for the SiO₂ microparticles. The scaling constant k measured here agrees with previous measurements for amorphous SiO₂ to within a factor of two [19]. This minor discrepancy could be due to slightly different sample crystallinities, or to the relatively limited frequency range over which the measurements in this work are carried out. Any extinction above the expected material extinction is assumed to be due to macroscale structural effects. This is suggested by the spectrum of the sample with a volumetric fraction of 14.9%, shown as the highest black marker in Fig. 2(b) and the solid black line in Fig. 2(c), which does not show the frequency dependence expected from non-crystalline materials nor free carrier absorption. To confirm the dependence on pressing parameters, this sample was re-pressed and its extinction coefficient was measured again. After re-pressing, the extinction coefficient for this sample dropped dramatically, indicated by the hollow square in Fig. 2(b) and the dashed line in Fig. 2(c), though some additional extinction still remains. Consequently, a proper manufacturing process is critical to the fabrication of low-loss composites in the THz spectrum when using this fabrication method. In summary, HDPE-SiO₂ composites are low-loss, but also have a narrow refractive index tuning range. Thus, these materials are useful for applications requiring low-loss materials with refractive indices near that of common polymers.

Next, results for HDPE-Al₂O₃ composites are shown. The primary advantage to using Al₂O₃ over SiO₂ is the fact that Al₂O₃ has a higher refractive index, which extends the refractive index tuning range. HDPE-Al₂O₃ composites were fabricated for volumetric fractions up to 25%. The refractive index and extinction coefficient results at 1 THz are shown in Figs. 3(a) and 3(b), respectively. Two sets of samples were fabricated: a set of thin samples with thicknesses of approximately 4 mm, and a set of thick samples with thicknesses of approximately 8 mm. The thin samples are shown as the solid black squares and the thick samples are shown as the hollow black squares. All samples were pressed with identical temperatures, times, and pressures. These results are compared to the Bruggeman model, shown as the dashed black line, with the corresponding refractive index given in Table 1 and extinction coefficient predicted by Eq. (2). Extinction coefficient spectra for three representative thin samples are shown in Fig. 3(c) along with fits using the Bruggeman model in Eq. (1) with the extinction from crystalline materials in Eq. (2) used for the Al₂O₃ extinction coefficient.

 

Fig. 3. The refractive index (a), extinction coefficient at 1 THz (b), and extinction coefficient spectra of several representative samples (c) are shown here for HDPE-Al₂O₃ composites. In (a) and (b), two sets of samples, thick and thin, were fabricated. The thick samples are denoted by the hollow squares and the thin samples are denoted by the solid squares. The refractive index and extinction coefficients in (a) and (b) are shown as functions of volumetric fraction, while the spectra in (c) are shown as a function of THz frequency. In (a) and (b), the Bruggeman model is plotted as the dashed line. In spectra in (c) correspond to samples with volumetric fractions of 3.5% (red), 9.4% (black), and 22.4% (blue). The experimental data, denoted by the markers, is fit with a Bruggeman model, denoted by the corresponding solid lines.

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The refractive index results in Fig. 3(a) show that measured refractive index for the HDPE-Al₂O₃ composites conform well with the Bruggeman model regardless of sample thickness. However, the extinction coefficient results for the same samples show two different correlations with volumetric fraction. Both sample sets show increasing extinction with volumetric fraction, though the extinction of the thick sample set is noticeably higher. Again, this is due to the thick samples being improperly pressed, with one thin sample also showing similar behaviour. The extinction coefficient spectra for thin samples with volumetric fractions of 3.5% (blue), 9.4% (red) and 22.4% (black) are shown in Fig. 3(c). The data points are experimental measurements, and the solid lines correspond to a theoretical model that uses the polycrystalline extinction coefficient from Eq. (2) in the Bruggeman model in Eq. (1). Note that Al₂O₃ must be polycrystalline as it cannot be fabricated in bulk as an amorphous material. The measured extinction coefficient spectra show good agreement with theoretical predictions when the Al₂O₃ microparticles are modelled using Eq. (2) with β = 2 and k = 1×10⁴⁶ J^-2m^-1. This extinction is also used to produce the Bruggeman curve seen in Fig. 3(b). To ensure that this extinction is due to the polycrystalline nature of the particles and not poor fabrication, these results can be compared to other Al₂O₃ composites in the literature. In particular, Headland et al. fabricated Al₂O₃ composites using a different host material and fabrication method [16]. Despite this, their data yields an extinction coefficient for Al₂O₃ microparticles of approximately 0.043 at 1 THz, which is similar to the extinction coefficient for the Al₂O₃ microparticles found in this work of 0.04 at 1 THz. This confirms that fabrication issues did not produce erroneous extinction in the thin HDPE-Al₂O₃ composites. In summary, HDPE-Al₂O₃ composites are capable of a wider refractive index tuning range than HDPE-SiO₂ composites. However, the polycrystalline nature of the Al₂O₃ particles results in higher losses, particularly for high volumetric fractions.

Finally, results for HDPE-Si composites are shown. Since the efficacy of the Bruggeman model to predict a composite's refractive index and extinction coefficient has already been established [15], these results are focused on the effects of free carrier absorption in the inclusions. Free carrier absorption is explored for only for Si as it is a semiconductor. The SiO₂ and Al₂O₃ microparticles in the previous results were insulating, so free carrier absorption will not be present. The Si microparticle inclusions used in the HDPE-Si composites are the same ones used in the COC-Si composites, where it was noted that these inclusions have extremely high extinction. Here, several samples with very low volumetric fractions (<1%) were fabricated to investigate the loss mechanisms in more detail without having the measured signal be entirely absorbed. The extinction spectrum for a representative sample with a volumetric fraction of 0.96% and thickness of 4.4 mm is shown in Fig. 4. The experimental data is denoted by red squares and a theoretical curve is denoted as the black line. The theoretical curve was plotted by solving Eq. (1) using the Drude model in Eq. (3) to predict the inclusion dielectric constant. The refractive index for Si in Eq. (3) was taken from Table 1 and the fitting parameters in Eq. (3) were charge carrier density and mobility. The host dielectric constant in Eq. (1) was taken from measured data for pure HDPE, which is why noise is present in the theoretical curve.

 

Fig. 4. The extinction coefficient spectrum of an HDPE-Si composite sample with a volumetric fraction of 0.96% is shown. Experimental data is shown as red squares; theoretical data generated using the Bruggeman model, with other experimental data for the HDPE dielectric constant and the Drude model for the Si microparticle dielectric constant, is shown as the black line.

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The results in Fig. 4 show how the extinction due to free carrier absorption in the Si microparticles dominates the extinction coefficient spectrum even at a volumetric fraction of less than 1%. Although the Si microparticles used here were the same ones used in the COC-Si composites, the depolarization factor of these Si particles was taken to be 1/6. This discrepancy is likely caused by error in the COC-Si composite measurement given how thin that sample was. Interestingly, the measured extinction coefficient of the HDPE-Si composite falls off for very low frequencies as opposed to rising infinitely as predicted by the Drude model. This effect is seen in both the experimental data and the theoretical curve when solving the Bruggeman model. Other samples fabricated with slightly higher volumetric fractions (<4%) showed similar behaviour, though their extinction coefficient spectra were more difficult to fit to the theoretical model due to poor measurement signal-to-noise ratio. Consequently, when fitting the Drude model to the experimental results, it was found that the particles are n-type with an extrinsic electron density of 4×10¹⁷ cm^–3 and a charge carrier mobility of 1100 cm²/Vs. These are reasonable values for a lightly-doped Si wafer that was likely ground into the particles. Finally, it is clear from the shape of the extinction coefficient curve that the losses in this sample are not due to material crystallinity as modelled in Eq. (2). In summary, materials with free carrier densities should be avoided when fabricating composite materials in the THz spectrum due to the extreme losses induced by free carrier absorption.

4. Conclusion

The extinction characteristics of several different types of composite materials in the THz spectrum are analyzed and experimentally validated in this work. It is found that there are many sources of extinction that may not be immediately apparent based on the materials in the composite. Factors such as material crystallinity, free carrier density, and improper fabrication procedures all increase material loss to varying degrees. Interestingly, none of these factors affect the composite material's refractive index, and so can only be seen in the extinction coefficient. These results give future designers of THz composites more options for materials, other than pure crystals, with the potential for customized refraction and minimal absorption.