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Abstract
Terahertz (THz) near-field imaging is attracting a lot of attention for its potential applications in medical diagnosis and material characterization. However, the spatial resolution of the recorded THz image was mainly limited by the diffraction limit of the commonly used lens- and mirror-based THz optical systems. Alternatively, a solid immersion lens (SIL) can be a promising approach for achieving super-resolution imaging as it reduces the spot size of the focused THz beam by a factor of 1/n, where n is the refractive index (RI) of the lens material. In this work, we present the design and fabrication of hemispherical THz SIL using powder mixes of titanium dioxide (TiO2) and polypropylene (PP) whose RIs are ≈10 and ≈1.51, respectively, at 1.0 THz. In particular, we present two different lens fabrication strategies that are simple and cost-effective solutions. The first strategy uses pressing the TiO2 powder with a PP powder at the Vicat temperature of PP while controlling the concentration of TiO2 and the resultant lens porosity. The second design consists in pressing the TiO2 powder in a hollow hemisphere that is 3D printed using PP. The fabricated lenses are then characterized physically and optically, and their RIs are compared to the theoretical estimates using the Bruggeman model of the effective media. From the experimental measurements of the proposed SIL, a resolution limit as low as 0.2λ was achieved at 0.09 THz (λ ≈ 3.3 mm), which is comparable to the best resolutions reported in the literature.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, THz imaging has become one of the popular techniques in scientific and industrial applications mainly due to peculiarities of the THz-wave – matter interactions, as well as non-invasive and high-resolution imaging capabilities of THz waves when compared with infrared-visible and microwave counterparts, respectively [1]. Due to the rapid development of high-power THz sources and efficient detectors, the commercial availability of THz imaging systems has become a reality. However, there are many challenges yet to be addressed before increasing its widespread applications in many areas. In THz imaging, one of the main challenges is the physical limitation in the spatial resolution, which in free space is ${\approx} 0.5\lambda$ according to Abbe’s diffraction limit [2]. For example, this limit implies that a THz pulse centered around $0.1$ THz could not resolve features of the image smaller than $1.5$ mm. This particular limitation hinders the application of THz imaging in several areas such as spectroscopy [3], medicine [4,5], semi-conductor analysis [6], and security [7,8]. Therefore, the development of a novel method to increase the spatial resolution of the THz imaging is the need of the hour.
Near-field imaging is one of the promising approaches which was developed to resolve the smallest features of the image [9]. Generally, in an NF imaging system, the information is collected within a few wavelengths of the sample, before the negative effects of diffraction degrade the image resolution [10,11]. Solid immersion microscopy is an example of such a system. In this system, the focused incident beam propagates through a high RI lens – i.e., so-called, SIL, located in the near-field of the sample. While propagating in this lens, the beam is confined by the dielectric response of the high RI medium. Behind the lens, the evanescent field still contributes to the beam’s intensity, yielding a spot size smaller than the diffraction limit [12]. This system was first fabricated by Mansfield and Kino in 1990 when they reported the creation of a SIL microscope in the visible range [13]. Over time, the concept behind the SIL microscope was rationalized, and the hemispherical SIL became a useful microscopy component in imaging laboratories [14]. In general, the spot size of a focused beam can be estimated as
where $\delta$ is the spot size, $\lambda$ is the free space wavelength, $\textrm{NA}$ is the numerical aperture of the SIL, and $\theta$ is the half-angle of the cone of light focused by the lens. From Eq. (1), it follows that the RI of the SIL plays a significant role in reducing the spot size and improving the resolution.In the THz regime, the availability of high RI material makes the solid immersion microscopy system a promising solution to achieve higher resolution. As reported in the literature, materials with RI above $3$ give a significant improvement in the resolution of the THz imaging system [15]. The most widely used SILs are currently made of the high-resistivity float-zone silicon, with the THz RI as high as ${\approx} 3.5$, along with negligible dispersion and material absorption. This could be further improved by using the material having higher RI. By taking the advantage of longer THz wavelengths, many authors have reported metamaterials and metasurfaces with very high RI that will eventually be used in those systems [16,17]. However, it could be more practical to simply use natural elements and more classical fabrication techniques to design the SILs.
In this view, we aim at using TiO2 in the fabrication of SIL as it has an RI of ${\approx} 10$ in the THz range [18]. TiO2 is a ceramic that is widely used in the industry, mainly because of its opacity which makes it a perfect white pigment [19–21]. It is generally produced as a powder by sintering, although it can as well be grown as a crystal in its most general form of rutile [22]. Moreover, it is commercially available in powder form and is rather inexpensive. As it is generally described as inert and not highly hazardous material, however, when used as a nanoparticle, it can also become an irritant and its inhalation can be dangerous. Nevertheless, the high RI property of TiO2 can be of great interest in fabricating THz optics. For example, a SIL that is fabricated using pure TiO2 could theoretically increase the resolution by a factor of ${\approx} 10$. Therefore, the main objective of this work is to maximize the SIL RI by increasing the concentration of TiO2.
So far, we have shown that TiO2 could be an ideal material for the fabrication of a high RI SIL. However, a few fabrication challenges need to be addressed. TiO2 is generally difficult to shape when it is in powder form and it requires high temperature (the sintering temperature of TiO2 is between $400$ and $800$°C) and high pressure (${\approx} 1.5$ GPa) [23,24]. This requires very specific and expensive equipment for fabrication. TiO2 can also be grown as a rutile crystal cylinder by epitaxial growth, and then be cut and polished in the shape of a hemisphere. This whole process, although providing a monocrystalline, optically anisotropic, and ideally dense TiO2 structure, would also prove itself extremely long and expensive [25]. Alternatively, shaping of TiO2 structures using sol-gel techniques can be possible but demands specific equipment, and also it allows very limited concentration of TiO2 which further results in lower RI of the SIL [20,26]. Therefore, a relatively simple and inexpensive technique must be identified for the fabrication of TiO2-based THz SIL.
The paper is organized as follows. Firstly, we present two different approaches that we carried out in designing and optimizing the high-performance hemispherical SIL which is followed by the fabrication procedures. In both designs, we use PP as a secondary material to help shape the TiO2 in a hemispherical form. Among all polymers, PP is chosen because of its reasonably high RI ($1.51$ at $1.0$ THz) and low losses in the THz regime [27]. It is equally a resistant material whose low melting temperature ($160$°C) helps to shape and is already available in many forms for multiple applications. Secondly, we present the theoretical and experimental characterization of the proposed SILs. Since the lenses fabricated are composites of the two elements (TiO2 and PP), their measured RIs will be compared to the theoretical prediction of Bruggeman’s model [28]. Finally, by carrying out the imaging experiment, we show that the resolution of the proposed SILs can be reduced to as low as $0.2\lambda$ at $0.09$ THz which is well below Abbe’s diffraction limit.
2. Design and fabrication of hemispherical SIL
In this section, we present the two simple and repeatable approaches to fabricating SIL of TiO2 powder. The important parameters will be highlighted along with their limits and appropriate design conditions. The schematic of the hemispherical lens is shown in Fig. 1. We have chosen the diameter D of the hemispherical solid immersion lens as 25.4 mm (1 inch) as it is large enough for the collimation of THz beams generated by the standard pulsed THz imaging systems. Similarly, the chosen diameter is small enough to minimize the fabrication cost and allows convenient handling and manipulation. In this study, we used the TiO2 powder in the rutile form (Alfa Aesar), with a purity of 99.5% and a particles size of $1$–$2$ µm, and the PP powder Propyltex 325S (Micro Powders), with a particle size of $10$–$15$ µm.
2.1 Fabrication of SILs using a mechanical hot press (pressed SIL)
The first design presents the pressing of a heated mix of TiO2 and PP powders inside of a hemispherical mold. To help estimate the porosity once the SILs are fabricated, each mix is characterized by a weight concentration of TiO2 (i.e., ${m_{\textrm{TiO}_{2}}}/({{m_{\textrm{TiO}_{2}}} + {m_{\textrm{PP}}}} )$, where ${m_{\textrm{TiO}_{2}}}$ and ${m_{\textrm{PP}}}$ are the mass of the TiO2 and PP powders, respectively) and a mixture density
Table 1. Mass of TiO2 and PP powders mixed to produce the pressed SILs.
In what follows, we present the detailed fabrication procedure. The first step is the mixing of the powders since the homogeneity of the mixture is critical to the successful fabrication of pressed SILs. On one hand, as PP will effectively act as glue (supporting matrix for TiO2), it needs to be uniformly distributed within the mixture to keep the TiO2 particles in a solid structure. On the other hand, one of the assumptions of any effective medium theory is that the inclusions composing the material are uniformly distributed over its volume and, thus, yield a uniform effective RI.
The TiO2 and PP powders are therefore mixed, for $30$ min, in a high-energy ball mill (Fig. 2 (a)), which is a type of grinder consisting of a canister, that rotates at high speed and contains the powders and hard ceramic balls [29]. As the canister rotates, the ceramic balls generate several high-speed collisions with the particles (Figs. 2(b) and (c)), reducing the size of particles and destroying the agglomerations, and improving the homogeneity of the mixture [30]. This particle size reduction also helps minimize the voids between the TiO2 and PP particles. Next, the powders are mixed again, for $30$ min, using the V–1 Mini V Type Powder Mixer Machine (Fig. 2(d)), aimed at improving the mixture homogeneity. During the rotation cycle of a V-shape canister, the powders are continuously gathered to the bottom of the canister before being separated equally on the two sides (Figs. 2(e) and (f)). Considering different densities of the two powders, this step significantly improves the mixing efficiency.
Fig. 2. Improving the homogeneity of the TiO2 and PP powder mixtures. (a) Photo of the high-energy ball mill.
(b),(c) Schematics of the operation principle of the high-energy ball mill, where the arrows point the direction of the force. (d) Photo of the V-1 Mini V Type machine, where the red arrow shows the direction of rotation. (e),(f) Schematics of the operation principle of the V-1 Mini V Type machine, where the red circles show the areas where the particles are gathered and separated.
In the second step, the mixed powders are transferred to the specially designed mold and pressed mechanically under heat. The mold consists of three main parts made of aluminum (Fig. 3 (a)). The bottom plates are drilled with a cylindrical hole on top of a hemispherical hole, both with a diameter of $25.4$ mm ($1$ inch). An aluminum cylinder of the same dimensions as the cylindrical hole is used to apply pressure on the powder located inside of the hemispherical hole. The top plate was used to apply pressure evenly on the bottom two plates that contain the powder. The design of the mold with two bottom parts is intended to facilitate the removal of the SIL after the application of high pressure. Next, the hot press is heated to temperatures close to the Vicat temperature of PP ($150$°C) [31], at which the polymer softens and can be penetrated by other particles. The optimal temperature and pressure for mechanical stability of the resultant lenses are experimentally found at $135$°C and $2$ metric tons, respectively. The mold was pressed for $1$ h under this condition.
Fig. 3. An aluminum mold designed for the fabrication of hemispherical SIL by mechanical pressing of the mixture of TiO2 and PP powders. (a) Schematic. (b) Photo.
To estimate the RIs of the resultant SIL materials, another mold was fabricated that is similar to the one shown in Fig. 3 (a), and where the hemispherical hole was replaced with the cylindrical geometry, while similar processing conditions and materials concentrations were used during sample fabrication (Table 1). The shape of a cylinder sample makes it more practical to measure its RI and easier to press, as the resultant samples are less fragile than a hemisphere. The cylinder RIs were then measured using THz spectroscopy and used as an approximation to the RIs of the pressed SILs. Particularly, two to four cylinders with a $25.4$ mm diameter and $2$ to $3$ mm thickness are pressed for the same duration as the SIL while varying somewhat the porosity for every mixture (Table 2) by changing the amount of the material in the mold. Next, the mold was left to cool down to room temperature. Then, the cylinders and hemispheres are carefully taken out of the molds and polished with sandpaper of $120$, $400$, and $1000$ grits [32]. After polishing, their mass and dimensions are measured.
Table 2. Properties of the cylindrical samples fabricated by mechanical hot press technique for their THz spectroscopic characterization.
The volume of the air inclusion ${f_{\textrm{air}}}$ can be estimated by comparing the sample density ${\rho _{\textrm{sample}}} = {m_{\textrm{sample}}}/{V_{\textrm{sample}}}$ to the powder mix density from Table 1. In the case of an ideally dense lens, there would be no air inclusion, and the density of the sample would be the same as the density of the powders. The air inclusion (or porosity) is however unavoidable due to the limitations of the press and of the mold (aluminum is a rather soft metal that cannot sustain very high pressure). The porosity (i.e., the air filling factor by volume) is estimated as
while the derivation of this Eq. is given in the appendix. From thus estimated porosity ${f_{\textrm{air}}}$ it is possible to estimate the theoretical effective RI of the SIL using, for example, the Bruggeman model [33]. It is observed during the experiments that the lowest achievable porosity also depends on the concentration of TiO2. This is because the deformation capacity of the PP allows it to better fill the pores when it is in higher concentration. Thus, using higher concentrations of TiO2 (which leads to higher RI of the samples), generally results in higher sample porosities, and in samples that are more fragile and tend to break during removal from the mold. At $94.1\%$ TiO2, the lowest porosity achieved at which a lens is still mechanically stable is $41\%$. A slightly lower porosity is achieved in the cylindrical samples for a similar concentration. The properties of the fabricated cylinders and SILs are presented in Tables 2 and 3, respectively.Table 3. Properties of the SILs fabricated by the mechanical hot press technique for the THz imaging experiments.
The volume filling factor of the TiO2 and PP inclusions in each sample can be calculated simply through their mass in the three-component mixes and their density. Dividing the mass by the density gives the volume fraction of each component:
In Fig. 4, we show the fabricated SILs that are removed from the mold structure. Polishing must be carried out to improve the surface quality. It is noted that the grey stains on the $85.6\%$ TiO2 and $90.2\%$ TiO2 SILs are due to silicon oil used in the mold to simplify lens release. As fabricated hemispherical lenses feature imperfections on their round surface that are several hundred microns deep that appear during lens removal from the mold. At the same time, the flat surface of the lens is of very high quality with no visible scratches or non-uniformities. Note that for imaging applications, it is the quality of the flat surface that is most important and that requires surface flatness below the resolution limit of the SIL lens. As in this work we operate near 0.1 THz, according to Eq. (1), the planar surface flatness should be below $\frac{\lambda }{{2{n_{TiO_{2}}}}}\sim 150\; \mathrm{\mu} m$, which is readily satisfied. At the same time, the quality of the lens round surface is not as critical, with imperfections mostly causing enhanced scattering and not the reduction in SIL resolution.
Fig. 4. Photos of (a)–(c) the top / spherical surface and (d)–(f) the bottom / flat surface of the pressed SILs with the TiO2 volume fraction of $94.1\%$, $90.2\%$, and $85.6\%$, respectively.
2.2 Printed SIL
The second design is intended to create a SIL made of pure TiO2 powders (without PP inclusions). A thin hemispherical shell was 3D printed (Raise3D pro2) using PP filament, inside which the TiO2 powders are pressed manually. The thickness of the shell is $1.5$ mm, with PP deposited using $50$µm-thick layers and a $200$-µm-diameter nozzle. The high 3D printing precision is necessary to minimize the surface roughness of the hemispherical shell and the related THz-wave scattering effects, thus, sustaining the THz-beam quality reduction (Fig. 5).
Fig. 5. Comparison of the thin hemispherical shells of PP that are 3D printed using $100$ µm nozzle and layer heights of (a) $100$, (b) $80$, and (c) $30$ µm.
The thickness of $h = 1.5$ mm is chosen so that the shell is solid enough for the TiO2 powder to be compressed inside of it while being thin enough to minimize the shell absorption losses. At the same time, one has to be aware of the fact that the shell acts as a simple antireflection coating with the frequencies of the maximal transmission that are multiples of $\frac{c}{{4h{n_{PP}}}}\sim 33GHz$. This, in turn, leads to spectral intensity modulation of the reflected data, and one has to make sure to operate at the frequency of maximal transmission of the SIL plastic shell to enhance its imaging sensitivity. It is noted that 3D printing with PP is challenging as it is very sensitive to temperature gradients which affects strongly the model adherence to the build plate. To improve adherence of the 3D printed sample to the build plate, the plate temperature was set to 95°C. The filament extrusion temperature was $240$°C, and the extrusion speed was $15$ mm/s.
Next, the 3D printed hemispherical shell was filled with pure TiO2 powder and manually pressed to avoid the breakage of the shell (Figs. 6(a)–(e)). The porosity of the SIL fabricated using this approach can be estimated using Eqs. (3) and (4), thus resulting in ${f_{\textrm{air}}} = $ 52.7% and ${f_{\textrm{Ti}{\textrm{O}_2}}} = $ 47.3%, which is the highest fraction of TiO2 among all the fabricated lenses.
Fig. 6. Fabrication of the SIL using the pure TiO2 powder compacted in a 3D printed hemispherical shell. (a) The 3D printed shell in an aluminum mold. (b) The shell filled with pure TiO2 powder. (c) The aluminum cylinder is used to manually press the TiO2 powder. (d) The shell with compacted TiO2 powder. (e) The resultant SIL.
The geometry of SILs presented in this work can be further optimized to compensate for various aberrations and diffractive effects, which are of particular importance in the long-wavelength regime. Furthermore, the hemispherical SILs fabricated using two techniques presented in this paper have different surface qualities which can result in different scattering losses. While the pressed SILs feature somewhat smoother surfaces than the printed SILs, the surface roughness of the printed SILs can be further optimized by postprocessing of the printed shells such as polishing, annealing, etc.
3. Characterization of the composite TiO2–PP material fabricated via the hot pressing
In this work, we used two different THz spectroscopy systems (continuous wave (CW) and time-domain (TD)). Since the CW THz spectroscopy system has a higher spectral resolution (∼50 MHz) than the THz time-domain system (∼1 GHz), and as it offers a considerably higher spectral power density than TDS (higher signal to noise ratio), it is a system of choice when performing material characterization, such as complex RI. Particularly, the THz RIs of the pressed cylinders, with the filling factors listed in Table 3, were characterized experimentally using the THz CW spectroscopy system using the cut-back approach detailed in Refs. [34,35]. The main disadvantage of the THz CW system is the long acquisition time when broadband characterization is required. Therefore, for THz imaging and characterization of the SIL resolution (see section 4), we resorted to using the THz time-domain system due to its potential for fast broadband acquisition, as well as the possibility to operate both in the time and spectral domains. The description of the CW THz spectroscopy system is briefly discussed as follows. This setup is based on a pair of photomixers (as an emitter and a detector of THz waves) pumped by the two distributed feedback infrared fiber-coupled lasers with slightly different center wavelengths. During the THz measurements, the cylindrical samples are placed in the collimated THz beam. Their frequency-dependent complex RI:
where n is its real part, $\alpha $ is the THz-wave absorption coefficient (by power), ${c_0} \approx 3 \times {10^8}$ m/s is the speed of light in free space, $\nu $ is the electromagnetic-wave frequency, was retrieved from the measured data using the phase-sensitive cutback technique [34].In Figs. 7(a) and (b), we show the measured THz optical properties of the pressed cylinders made of the composite TiO2–PP material and featuring the different content of components and porosity. From thus measured data, we notice that the developed material platform provides an ability to manage, in wide limits, the THz refractive index, by changing the material composition. This can be very useful for the THz optics synthesis and optimization. For the considered range of the TiO2 concentrations and porosities, quite high RIs $n \in ({2.80,3.95} )$ are observed, which are equal to or even higher than those of such common high-refractive-index crystalline THz optical materials, as sapphire [36] and HRFZ-Si [37]. As expected, the observed RI increases with an increase in the TiO2 content ${f_{\textrm{TiO}_{2}}}$ and, oppositely, decreases with an increase in the porosity ${f_{\textrm{air}}}$. From Figs. 7(a) and (b), we also notice that our material has quite small dispersion of optical properties, as well as moderate THz-wave absorption $\alpha \in ({1-5} )$ cm–1 (by power) in the 0.1-0.25 THz spectral range. This makes the developed material suitable for applications at the lower THz range, as well as for the fabrication of relatively large-size free-space THz optical elements. All these justify the potential of the considered material platform in the superresolution THz imaging.
Fig. 7. THz optical properties of the composite TiO2–PP materials with the different TiO2 contents and porosities. (a) Measured RIs (solid line) and absorption coefficients (dotted line) of the pressed cylinders with the TiO2 content of 90.2% wt. and the different porosities. (b) Equal data set for the pressed cylinders with the different TiO2 content and porosity. (c) Theoretical estimates of the effective RIs using the Bruggeman model (at $0.1$ THz), which are compared with the measured RIs of the cylinders, and the results of imaging experiments that are discussed below.
To be able to predict the THz optical properties of the developed composite material platform, as a function of the volume fraction of TiO2 and PP, as well as the porosity, the Bruggeman effective medium model [33] is applied, considering the data from Table 3. In this way. assuming that the shape of the particle is spherical, the effective RI ${n_{\textrm{eff}}}$ is calculated by resolving the following Eq.
4. Characterization of the THz SILs made of the TiO2–PP composite
In what follows, we characterize the spatial resolution of thus fabricated SILs, when used in the pulsed THz microscopy setup operating on the principles of THz time-domain spectroscopy [39–42]. A schematic and a photo of the experimental setup are shown in Fig. 8. A femtosecond laser (Mai Tai) with an average power of ${\approx} 2.5$ W and a central wavelength of $800$ nm was used as the optical source to drive the photoconductive antenna (PCA) emitter and the PCA detector. The THz emitter is an interdigital array antenna (iPCA-21-05-1000-800-h from Batop Optoelectronics). The antenna array was mounted with a hemispherical HRFZ-Si lens and a TPX lens in the same housing to produce an almost collimated beam. A secondary TPX lens (L1) with a focal length of $151.5$. mm is added and positioned experimentally to better collimate the beam. The collimated beam from the lens L1 is then deviated vertically by a flat gold-coated circular mirror with a diameter of $50$ mm (M1) so that the sample (which should be located atop of the SIL) can be scanned in the horizontal plane while illuminated from the bottom. The THz beam passes through the beam splitter BS1 for the first time before L3, and a second time after being reflected on the sample. A thin silicon wafer of thickness ∼200 µm was used as the beam splitter (BS1). Similarly, L3 is a TPX THz lens from Batop optoelectronics with a diameter of $50$ mm and a focal length of $35$ mm whose NA is $0.54$. The theoretical spot size of lens L3 is estimated as $0.9259\lambda$. The THz lenses L1 and L2 are identical and are made of polytetrafluoroethylene (PTFE) with a focal length of $151.5$ mm and a diameter of $2$ inches. The detector is also a PCA (TERA8-1 from Menlo). It is not that the spot size of the SIL was measured in the NF of the flat surface.
Fig. 8. Pulsed superresolution THz SIL-based microscope. (a) Schematic of the SIL microscope. (b) Photo of the SIL microscope. (c) Photo of the razor blade and a SIL head block adapted for the knife-edge resolution test. (d) Photo of the 3D printed SIL head showing placement of different optics.
To characterize the THz beam produced by the emitter PCA, a first experiment was conducted by focusing the collimated beam directly onto the detector PCA (Fig. 9 (a)). For this arrangement, the detecting PCA is placed on a motorized stage in order to measure the electric field $E(t )$ at different position close to the focal point of the lens L2. We present the measured THz intensity ($I(t )\; \propto |{E{{(t )}^2}} |$) (Fig. 9 (b)) and its Fourier spectrum $|{I(\nu )} |$ (Fig. 8(c)), that has a bandwidth of ${\approx} 1.5$ THz. These curves are useful for quantitative analysis of the THz signal drop while resorting to reflection-mode SIL arrangement. Finally, from Fig. 9 (a), the PCA is moved in the focal plane of L2 to measure the distribution of the pulse intensity over the area of 20 mm x 20 mm. When plotting the maximal intensity of each pulse recorded, we notice that the THz field intensity is homogeneously distributed over the beam aperture, which is an important condition for further work with the SIL optics, in which both propagating and evanescence waves are excited by different parts of the beam aperture [43].
Fig. 9. Characterizing the collimated THz beam of the experimental setup. (a) Schematics of the THz beam path and results of visualizing the THz field intensity distribution over the collimated beam aperture. (b),(c) THz intensity $I(t )$ and its Fourier spectrum $|{I(\nu )} |$ (by field), respectively at the location of maximal intensity.
A second experiment was conducted with an optical scheme, in which a simple flat mirror is used instead of the focusing SIL (Fig. 10 (a)). Here, support is 3D printed in 4 parts to help align the optical component of the microscope (Fig. 7 (d)). A reference (without SIL) THz waveform $E(t )$ (Fig. 10 (b)) and its Fourier spectrum $|{E(\nu )} |$ (Fig. 10 (c)) were measured, while this mirror allows to redirect the reflected THz beam towards the detector. The measured signals highlight that the THz beam intensity decreases significantly as it passes through the beam splitter and reflects from mirrors. We also note some interference patterns in the Fourier spectrum (Fig. 10 (c)), which occur due to the internal reflection of the THz pulse inside the beam splitter. As compared to the initial ${\approx} 1.5$ THz bandwidth of the collimated beam spectrum (Fig. 9(c)), the spectral width for the experimental setup shown in Fig. 10 (a) reduces to ${\approx} 0.7$. THz (Fig. 10 (c)). Despite such a notable loss in the spectral width, it is still sufficient enough to carry out the THz imaging experiments. The complete THz microscope with the developed and fabricated SILs is therefore ultimately built.
Fig. 10. Characterizing the THz beam in an optical system, where a simple flat mirror is used instead of the focusing SIL. (a) Schematic of the THz beam path. (b),(c) THz waveform $E(t )$ and its Fourier spectrum $|{E(\nu )} |$ (by field), respectively.
Then, a knife-edge experiment was carried out, as shown in Fig. 11 (a)–(c), to uncover the spatial resolution of our microscope, when it operates either without a SIL (using the Batop optoelectronics’ TPX-D50-f35, a standard diffraction-limited lens with a diameter of 50 mm, a focal length of 35 mm, a RI of 1.45 at 1 THz and a NA of 0.54) or with different abovementioned SILs, including the pressed ones with $85.6\%$, $90.2\%$, and 94.1% TiO2 content (Table 3), and the printed SIL. Due to the chromatic aberration of L3, the focal plane will not be the same for all frequencies carried by the THz pulse. To take advantage of the higher power generated by the PCA at $0.1$ THz, the system is aligned to image the beam at the focal point of that frequency (Fig. 11 (d)). The measure of the probe is made by a knife-edge experiment. The measurement entails moving a razor blade in the focal plane of L3 (which coincides with the flat side of the SIL when present) and recording the reflected intensity with a spatial resolution much smaller than the beam spot size. As the edge of the razor blade moves toward the center of the beam, the reflected intensity will increase and define the intensity profile of the beam. The first-order derivative of this profile yields a gaussian-like function representing the increase in reflected intensity as a function of the axial coordinate of the beam. This function is considered to give a point spread function of the optical system, and, then, the spatial resolution is retrieved as the full width at half maximum (FWHM) of the Gaussian-like function (Fig. 11 (c)) [44].
Fig. 11. Measurements of the spatial resolution of the THz microscope, when it operates without SIL and with the different SILs. (a)–(c) Schematics of the resolution estimation based on the knife-edge measurement. (d) THz spectrum $|{E(\nu )} |$ that reveals the maximal spectral amplitude at ${\approx} 0.1$ THz. (e) Resolution test without SIL (only L3 lens). (f)–(h) Resolution tests for the pressed SILs with the $85.6\%$, $90.2\%$, and $94.1\%$ TiO2 contents, respectively. (i) Resolution test for the printed SIL.
To characterize the SILs, the displacement step of the knife-edge is set between $10$ and $25$ µm. At every step, $65$ to $100$ waveforms are recorded and averaged. Each pulse corresponds to $1753$. measures of the THz electric field taken by an acquisition card at $2500$ Hz with a delay line moving at $25$ mm/s, giving a scan duration of approximately $120$ ps. Although the SNR of the detected THz pulse was improved, the recorded intensity profile was still somewhat irregular and the first-order derivative (when used directly with a signal) results in the incorrect estimation of the FWHM. This irregularity of a signal is fundamental in SIL microscopy as the measurements are performed in the reflection mode. Indeed, in the reflection mode, when conducting knife-edge measurements, there is always a reflection from the SIL flat surface both with and without a knife, while for the intermediate positions of the knife small parasitic peaks in reflection are common due to various diffraction effects within SIL, as well as due to the complicated structure of the focal spot itself (as demonstrated in our prior works [Ref: 4,15,36,43 and 44]). Therefore, to estimate the FWHM of a focused spot, we first smoothened the THz intensity profile using a moving average algorithm which would replace each group of n points with thaverage intensity of the group (n varied between 3 and 8 depending on the profile). The first-order derivative of the THz intensity profile was further smoothened heuristically using the built-in function of MATLAB (smoothdata()). Finally, we estimated the FWHM from half the maximum value of the first-order derivative. The vertical lines are drawn at half the maximum value of the first-order derivative. In Fig. 11, we plotted the raw data of the measured intensity profile and their first-order derivative (see dotted lines). It must be noted that the smoothened profiles of the THz intensity and its derivative do not change the profile substantially. Therefore, we believe that the abovementioned procedure which was adapted from the references [15,44] is valid for estimating the FWHM of the focal spots from somewhat irregular data generated by the reflection mode SIL microscopy.
As evident from Fig. 11, the spot size of the bean focused by L3 at $0.094$ THz is $3.189$ mm, or $0.904\lambda$. This beam size corresponds to the diffraction limit of the system and will serve as a reference for the SILs. The same experiment is made for each of the SILs, and their results are summarized in Table 4. We can see from Fig. 11 that the presence of SIL dramatically increases the system resolution. The measured resolution is represented in Table 4 in form of the resolution enhancement factor – i.e., a ratio between the resolution of the L3 lens only and the SIL optical system. Such a resolution enhancement can be interpreted as a measure of the SIL RI ($n = \textrm{FWH}{\textrm{M}_{\mathrm{No\; SIL}}}/\textrm{FWH}{\textrm{M}_{\textrm{SIL}}}$), as follows from Eq. (1). These imaging-based RI estimates are compared with the RI model and experimental data from Fig. 7 (c), where overall reasonable aggrege is observed. In this experiment, the CW setup was used to characterize the RI of the pressed cylinders, since such a system has a better spectral resolution, making it more suited for spectroscopic applications. The pulsed microscope system was however built to take advantage of the faster acquisition time of such a system due to the simultaneous recollection of all the spectral information. The advantage of both systems were therefore used for their corresponding purpose to provide the measure of the SILs’ RI. Finally, we notice that the estimated minimal resolution of $0.20\lambda$, which is achieved using the printed SIL competes with the highest resolution of $0.15\lambda$ reported in the literature that uses HRFZ-Si [44].
Table 4. Measured resolution of the pressed and printed SILs at $0.09$ THz.
5. Conclusion
In conclusion, this paper reported on the fabrication of SILs based on two design strategies (mechanical hot press technique of TiO2/PP mixes, and 3D printed SIL shell with TiO2 powder compactification), that could be used to enhance the resolution of THz imaging well beyond the Abbe’s diffraction limit. The fabricated SILs are characterized and tested using a pulsed SIL THz microscope system. The improvement in resolution is quantified using the knife-edge measurements of the beam waist after focusing with and without SIL. The best resolution of $0.2\lambda$ at $0.1$ THz is achieved using a 3D printed SIL shell with compacted pure TiO2 powder, while the second-best resolution of $0.25$ is achieved using a pressed TiO2/PP SIL with 90.2 wt. % TiO2 concentration. Both results compete well with the highest resolution of $0.15\lambda$ reported today in a THz SIL system that uses expensive high resistivity Si materials. We also note that the effective RI of the demonstrated hemispherical SIL using polypropylene/TiO2 is only somewhat higher (∼4.0) than the RI of a Silicon (∼3.4). Therefore, the resolution of our SILs is only somewhat better than that of their Silicon counterparts. The main reason for that is that there is a considerable air filling factor of ∼30% (porosity) which is still present in the lens. We believe that by further optimizing the lens fabrication methodology one could further reduce the porosity, and augment the lens RI and resolution. It can be achieved, for example, by reducing the particle size of TiO2 and by using a hard mold that can withstand higher pressure. Furthermore, in terms of absorption losses, the Silicon lens performs better than our TiO2 lens. However, the main idea of the paper is to achieve a higher resolution than a Silicon lens by profiting from the TiO2 high RI, even at the expense of the lens losses. Finally, we also note that the cost of our TiO2-based lens is significantly lower (by a factor of ∼10) than that of a crystalline Silicon hemispherical lens of a similar diameter due to the difference in their fabrication techniques
Appendix – Calculation of the volume fraction of pores
Here, we show derivation behind Eq. (3) that allows experimental measurement of porosity of the pressed samples using the weights of the TiO2 and PP powders in the mix, as well as the total volume of a sample. In this development, ${V_{\textrm{total}}}$ and ${m_{total}}$ represent the volume and mass of the sample (cylinder or SIL), ${V_{\textrm{air}}}$ represents the volume of the air inclusions (or porosity), ${V_{\textrm{TiO}_{2}}}$ represents the combined volume of the TiO2 particles, ${V_{\textrm{PP}}}$ represents the combined volume of the PP particles
Funding
Canada Research Chairs (Ubiquitous Terahertz Photonics project).
Acknowledgements
The work of M. Skorobogatiy was supported by the Ubiquitous Terahertz Photonics Project of the Canada Research Chairs Program. We would also like to acknowledge the financial support of the Institut de l'Énergie Trottier for the project on Meta Windows.
Disclosures
The authors declare no conflict of interest. Q. Chapdelaine conducted process development for the pressed and 2D printed SIL fabrication, assembled the SIL microscopy system, performed materials characterization, and wrote the paper. K. Nallapan, Y. Cao, and H. Guerboukha have developed the original pulsed and CW imaging and spectroscopy systems and helped adapt them to conduct material characterization and SIL microscopy reported in this paper. K. Nallapan, co-supervised the project and contributed to writing the paper. N. Chernomyrdin and K. Zaytsev have advised on the layout and design of the pulsed SIL imaging system, provided some of the custom optics for the system, and contributed to the paper writing. Finally, M. Skorobogatiy was the author of the idea, supervised the project, and contributed to writing the paper.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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