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Abstract
To address a challenging problem of super-resolution terahertz (THz) endoscopy, in this paper, an antiresonant hollow-core waveguide was coupled with a sapphire solid immersion lens (SIL), aimed at subwavelength confinement of guided mode. The waveguide is formed by a polytetrafluoroethylene (PTFE)-coated sapphire tube, the geometry of which was optimized to ensure high optical performance. SIL was judiciously designed, fabricated of bulk sapphire crystal, and then mounted at the output waveguide end. Study of the field intensity distributions at the shadow side of the waveguide–SIL system revealed the focal spot diameter of ≃0.2λ at the wavelength of λ = 500 μm. It agrees with numerical predictions, overcomes the Abbe diffraction limit, and justifies super-resolution capabilities of our endoscope.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Despite THz technology has a potential in a variety of socially- and industrially-important fields [1] (e.g., medical diagnosis [2] nondestructive testing of materials [3], quality control in pharmaceutical [4] and food [5] industries, security task [6]) and high speed communication [7], such applications are hampered by a number of problems, among which we focus on the two major ones.
The first problem is a rareness and low efficiency of the hard waveguides and flexible fibers in the THz range [8]. In fact, the existing waveguides, which are based on different material platforms and guiding mechanisms (e.g., parallel-plate [9] or tube [10] metal waveguides, polymer antiresonant or photonic crystal waveguides [11,12], step-index bulk [13] or porous [14] fibers, plasmonic waveguides [15,16]), either do not allow guiding THz radiation over considerable distances (with appropriate dispersion and loss), or possess limited technological reliability, low thermal, radiation, and chemical resistance [17]. Some faces the problem of guided mode decoupling due to the waveguide interactions with external obstacles, while others feature large outer diameter. These factors complicate their real-life applications of THz waveguides, such as measurements of hardly-accessible objects in harsh biological or industrial environments [18]
To mitigate this challenge, in our earlier work, the edge-defined film-fed growth (EFG) technique [19] was used to produce sapphire hollow-core antiresonant and photonic-crystal waveguides, as well as step-index fibers and fiber bundles [20]. A combination of the unique physical properties of sapphire (high hardness and radiation strength, chemical inertness, high THz refractive index and low absorption) with technological reliability of the EFG technique (as grown delicate geometry of cross section with optical surface and volume quality) makes such sapphire shaped crystals a favorable material platform of the THz waveguide optics.
The second problem is a spatial resolution of the common lens- or mirror-based THz optics, which obeys the Abbe diffraction limit for the free space focusing ($\simeq 0.5 \lambda$) and usually constitutes a few hundreds of microns (or even a few millimeters) due to large THz wavelengths [21]. As an alternative, earlier, we developed the THz solid immersion (SI) microscopy, that ensures a compromise between the resolution, operation rate, and energy efficiency [22,23]. It allows to quantify the sample optical properties distribution over the focal plane with the resolution as high as $0.15 \lambda$, while also providing superior energy efficiency thanks to the absence of any near-field probe in an optical scheme.
In this paper, the two aforementioned principles are merged to develop the super-resolution THz endoscope (Fig. 1 (a)). It is based on a hard hollow-core waveguide, made of a PTFE-coated sapphire ($\alpha$-Al$_{2}$O$_{3}$) tube and exploiting the antiresonant guiding mechanism [24]. It also uses a judiciously-designed sapphire SIL, mounted at the output waveguide end. THz-field intensity distribution at the shadow side of the "waveguide–SIL" system is studied using continuous-wave imaging systems based on a backward-wave oscillator (as an emitter at $\lambda = 500$ $\mu$m; this wavelength corresponds to the maximal power of our THz source), a Golay cell (as a detector), and a scanning aperture with the diameter of $\lambda$ (applied to collect images; panel (b)). To quantify the size of the subwavelength focal spot we used a test object with abrupt changes in transmission (panel (c)) [20,25]. Our findings revealed that the THz endoscope provides the focal spot dimensions as small as $\simeq 0.2 \lambda$, which agrees with numerical predictions and justifies its super-resolution capabilities.
Fig. 1. (a) Schematic of the super-resolution THz endoscope based on the hollow-core antiresonant sapphire waveguide with the sapphire SIL handled at its output end. (b),(c) Two approaches to visualize the THz-field intensity at the shadow side of the waveguide–SIL system, that are based on a scanning diaphragm and a razor blade, respectively.
2. Results
In this study, we used a waveguide similar to those from Ref. [26]. From Figs. 2 (a),(b), we notice that it is made of the EFG-grown sapphire tube, with sapphire c-axis collinear to the optical axis, the inner and outer diameters of $6.3$ and $7.6$ mm, respectively. The tube is coated by a $0.4$-mm-think PTFE film. A $50$-mm-long waveguide piece was cut and polished from both ends for further experiments.
Fig. 2. Hollow-core antiresonant THz waveguide based on PTFE-coated sapphire tube. (a),(b) Photos of the waveguide and its cross-section. (c) Numerical FEM data on the effective refractive index $n_\mathrm {eff}$ and propagation loss $\alpha$ (by power) of the fundamental mode. (d) Mode intensity $I \propto \left | \mathbf {E} \right |^{2}$ in the waveguide cross section at $\lambda = 500$ $\mu$m.
In Fig. 2 (c), results of numerical waveguide modeling using the finite-difference eigenmode (FDE) method [27] within the ANSYS Mode software are shown in form of the effective refractive index $n_\mathrm {eff}$ and propagation loss $\alpha$ (by power) for the fundamental mode, that is the only effectively supported by our waveguide [26]. At the output wavelength of our backward-wave oscillator (BWO) ($\lambda = 500$ $\mu$m), the loss is $\alpha \simeq 2.6$ dB/m. As evident from Fig. 2 (d), mode intensity $I \propto \left | \mathbf {E} \right |^{2}$ ($\mathbf {E}$ is an electric field vector) decays monotonically from the optical axes to the cross-section periphery, revealing quite a large mode area ($>\lambda ^{2}$).
Sapphire SIL of a bullet-like configuration (Fig. 3 (a)) was optimized numerically using the finite-different time-domain (FDTD) method [28]. The unitary sapphire SIL with a total thickness $L$ comprises two components – a hemispherical lens with a radius $R$ and a sapphire window with a thickness $L-R$. For both elements, the sapphire c-axis is collinear to the optical axis. In our analysis, we assumed that the fundamental mode of our waveguide (Figs. 2 (c),(d)) has a negligible axial electric field $\mathbf {E}$, as well as an effective diameter larger than SIL. In this case, SIL can be modeled independently of a waveguide, by assuming its interactions with a plane TEM-polarized wave, that radiates SIL from the left (Fig. 3 (a)). To further reduce the computational complexity, we resorted to modeling of the cylindrical (2D) optical elements with a cross section equal to that of the axially-symmetrical (3D) ones, and considered the TM-polarized scattered field; in Ref. [29], such a 2D approximation provided quite relevant predictions for the 3D SIL systems.
Fig. 3. Numerical FDTD optimization of the sapphire SIL at $\lambda = 500$ $\mu$m. (a) Schematic of the bullet-shaped sapphire SIL with a radius $R$ and a total thickness $L$, that is comprised of a hemisphere and a window. (b) SIL resolution $\delta$ as a function of a radius $R$ and a normalized thickness $L/\left (2R\right )$. (c),(d) Field intensity $I \propto \left | \mathbf {E} \right |^{2}$ in the axial cross section of SIL with $R = 1$ mm and $L = 1.5$ mm, and focal spot at its shadow side (in free space), correspondingly. Red marker in (b) shows the $\delta = 0.152 \lambda$ resolution SIL, selected for further experiments.
In Fig. 3 (b), results of the bullet-like SIL optimization at $\lambda = 500$ $\mu$m are shown in form of its spatial resolution $\delta$ (i.e., the focal spot diameter at the full-width at half-maximum (FWHM) level) as a function of the lens radius $R \in \left (0.1, 5.0\right )$ mm and normalized thickness $L/\left ( 2R \right ) \in \left (0.2, 1.0 \right )$ mm. Based on the FDTD analysis, the SIL configuration with $R = 1$ mm and $L = 1.5$ mm was selected for the further experiments, since it features both the high resolution of $\delta = 0.152\lambda$ and the diameter of $2$ mm (small enough to be accommodated inside the waveguide; Fig. 2). It also can be fabricated by conventional mechanical processing of a bulk crystal. In Figs. 3 (c),(d), we show the FDTD data on the THz-field intensity distribution $I \propto \left | \mathbf {E} \right |^{2}$ in the axial cross section of SIL and the focal spot at the shadow of SIL, respectively. From our numerical analysis, we conclude that this SIL ensures subwavelength field confinement at a small distance behind the endoscope ($\ll \lambda$), where the focal plane almost coincides the back flat surface of SIL (Fig. 3 (a)).
The developed sapphire SIL was fabricated and coupled to the waveguide (Fig. 4 (a)). For this, the hemisphere was attached to the sapphire window using a polystyrene glue (Fig. 4 (b)), and then this assembly was mounted in a 3D printed polymer SIL–waveguide connector (Fig. 4 (c)). Such a detachable connector was used in this study for simplicity, while other options are also available: gluing the window to the waveguide end, or even obtaining the as-grown window directly in the EFG process, onto which the lens can be mounted.
Fig. 4. Continuous-wave ($\lambda = 500$ $\mu$m) imaging of the THz-field intensity $I \propto \left | \mathbf {E} \right |^{2}$ behind the waveguide, either without or with SIL. (a)–(c) Photos of the THz endoscope and separate SIL, either after gluing the hemisphere and window together and after mounting in a polymer holder). (d)–(f) Field intensity behind the waveguide without SIL, as well as its cross sections along OX and OY axes; all are probed by the scanning aperture. (g) Field intensity behind the waveguide–SIL system, probed by the scanning aperture. (h),(i) Cross sections (OX and OY) of field intensity behind the waveguide–SIL system ($I \left ( x \right )$ and $I \left ( y \right )$), probed by the razor blade, as well as the estimated focal spots ($d I \left ( x \right ) / d x$ and $d I \left ( y \right ) / dy$).
Using an original transmission-mode continuous-wave imaging system based on a BWO emitter at $\lambda = 500$ $\mu$m, the THz-field intensity $I \propto \left | \mathbf {E} \right |^{2}$ was first visualized at the empty output waveguide end (for details, see Ref. [20]). BWO is characterized by the slightly tunable output frequency (in the range of $100-200$ GHz), the linewidth of $10^{-5}\nu$, and the output power of $10^{-2}$ W. A $22$ Hz mechanical chopper is applied to modulate the THz homogeneous input beam, which further propagates throw the endoscope and demodulated on the detector. As a THz intensity detector, a Golay cell with sensitivity of $10^{-5}$ V W$^{-1}$, and the time response of $10^{-1}$ s is used. Finally, for intensity readout behind the endoscope the wide-aperture lens and a $500 \mu$m-diameter scanning diaphragm combination is applied. Both the diaphragm and the Golay cell is mounted on a 2D motorized translation stage, which yields raster-scan of the image plane with the positioning accuracy of $< 2\mu$m.
In Figs. 4 (d)–(f), intensity distribution over the empty output end (waveguide without SIL) is shown in form of a 2D image and its cross sections along horizontal (OX) and vertical (OY) directions, correspondingly. As predicted numerically (Fig. 2 (d)), for the empty waveguide, the guided-mode intensity drops monotonically from the optical axes to the cross-section periphery (panel (d)), while the FWHM mode dimensions are $\simeq 3$ times larger than the wavelength $\lambda$ (panels (e),(f)). These results justified guiding properties of the empty waveguide, but also showed that it is not capable of the super-resolution applications without SIL.
In Fig. 4 (g), the scanning-aperture-based imaging was applied to capture the THz-field intensity at the shadow side of the waveguide–SI system. From this THz image, we observed more local focal spot, which justifies that the guided mode was focused by SIL. Notice that peripheral part of the guided mode, which was not collected by SIL, was blocked by a non-transparent aperture, formed by the 3D printed SIL holder (it is made of conductive polymer) and featuring the diameter close to that of SIL (Fig. 4 (c)). However, thus captured THz image is limited by the scanning aperture diameter of $\lambda$, that is $\geq 5$ times larger than the expected subwavelength resolution of SIL.
To quantify the THz focal spot formed by SIL, we resort from scanning aperture-based imaging to the razor blade-resolution test [20,23]. In Figs. 4 (h),(i), the THz-field intensity profiles along OX and OY axes are collected at the shadow side of the waveguide–SIL system using a razor blade, from which the focal spot shape is estimated as first derivatives – $d I\left ( x \right ) / d x$ and $d I \left ( y \right ) / d y$. The focal spot has strongly subwavelength dimensions – i.e., $0.19 \lambda$ and $0.2 \lambda$ in the OX and OY directions, respectively, which overall agrees with the numerical FDTD predictions (Fig. 3 (d)). The observed discrepancy between the theory and experiment is smaller than $<25\%$. to the predictable differences between the numerically-modeled 2D and experimentally-studies 3D optical systems, the separate consideration of the waveguide and SIL in our numerical analysis, as well as the residual misalignments of the experimental setup. We also notice that geometry of the real (experimental) "SIL–window" system slightly differs from that considered in our numerical analysis. This is due to technological limitations, leading to the deviations of the waveguide and lens dimensions, or the presence of polystyrene glue between the sapphire lens and the output window. The observed discrepancy can also be attributed to the experimentally applied procedure of the resolution estimation via a razor-blade imaging, which is characterized by its own set of limitations and distortions. Nevertheless, the observed discrepancy is quite small, which allows us to claim the consistency of the theory and experiment and, thus, to justify superior performance of the developed THz microscopy.
Despite the observed $\simeq 15$-times resolution enhancement for the waveguide aided by SIL, the resultant energy efficiency of the waveguide–SIL system drops due to the Fresnel losses and beam attenuations in sapphire. Indeed, the integral beam power behind the waveguide drops, but we observe almost equal peak intensity in the small area around the optical axis, for the waveguide without (Figs. 4 (d)) and with (panel (g)) SIL. A signal-to-noise ratio (i.e., a ratio between the signal maximum and the noise level) for the waveguide–SIL system is $\simeq 65$ (panel (g)), which is $\simeq 1.6$ times smaller as compared to the empty one (panel (d)).
3. Discussions
Here, the THz endoscopic system based on the SI effect has been developed. In order to highlight prospective of the developed super-resolution THz endoscopy, let us compare it with the existing modalities of sub-wavelength-resolution THz imaging.
- • Although the options based on the traditional far-field THz imaging techniques, such as THz imaging holography [30], synthetic aperture [31], computational imaging [32] in combination with methods of image reconstruction [33], are capable of slightly sub-wavelength imaging resolution, they still obey the $0.5\lambda$ Abbe diffraction limit (for a free space focusing). Thus, the THz SI microscopy and endoscopy offers advancement over all these imaging modalities in terms of resolution.
- • THz imaging modalities based on a photonic jet phenomenon [34,35] offers slightly higher resolution – i.e., up to $10^{-1}\lambda$. However, they still remain far from practical applications, due to a problems of mesoscale dielectric particle handling in a focused beam, in front of an image object. Indeed, such a particle can interact with an object, resulting in image distortions. At the same time, combination of the THz waveguide with photonic jet effect seems a promising research direction, that forms some kind of alternative to the developed "waveguide–SIL" arrangement.
- • Finally, we mention different modalities of THz scanning-probe near-field optical microscopy, that can rely on the aperture-based [36,37] and tip-based [25,38,39] arrangements. For the aperture- and tip-based systems, the resolution can reach the impressive values of $10^{-1}-10^{-2}\lambda$ and $\sim 10^{-3}-10^{-4}\lambda$, respectively. However, the main common disadvantages of such systems are low optical throughout / energy efficiency (due to the presence of sub-wavelength probes in an optical scheme), possible interactions between the probe and an imaged object (they can distort image), as well as limited capabilities of endoscopic realization. Considering these disadvantages of the THz near-field microscopy methods, the THz SI microscopy and endoscopy seems to be more attractive for real life applications.
The developed SIL-aided THz endoscopic system has a potential in different THz applications, including sensing and exposure tasks. Slightly remote objects can be probed by such a THz endoscope, aimed at the non-destructive testing of materials at technological and exploitational stages [40,3], sensing in harsh environments [17,19], medical diagnosis and therapy of malignant and benign neoplasms with different nosologies and localizations [41]. However, a number of research and engineering problems should be solved before its implementation in these demanding areas. Resolution, energy efficiency, and spectral operation bandwidth of our endoscope can be enhanced by optimizing the waveguide and SIL geometry. For example, its energy efficiency can be boosted by both tuning the waveguide geometry (to minimize coupling / decoupling and propagation losses) and applying the anti-reflection SIL coatings or micro-patterns (to reduce Fresnel loss) [42]. For the THz sensing, also it seems to be important to combine the developed endoscope with the principles of continuous-wave or pulsed THz spectroscopy to remotely quantify optical properties of an analyte [22], while for the THz exposure, it has to be tested with intense THz fields [43].
4. Conclusions
In conclusion, in this paper, we developed the THz endoscope based on the hollow-core antiresonant sapphire waveguide coupled to the sapphire SIL. These elements were judiciously designed using the FDE and 2D-FDTD numerical simulations, aimed at ensuring high spatial resolution along with low beam power loss at the selected electromagnetic wavelength of $\lambda = 500$ $\mu$m. The bullet-shaped sapphire SIL was fixed at the output end of a sapphire hollow-core antiresonant waveguide using a sapphire window and a 3D printed polymer connector. Our numerical analysis and experimental studies revealed a formation of subwavelength focal spot at the shadow side of the "waveguide–SIL" system. The measured dimensions of such a beam spot are as small as $\simeq 0.19\lambda$ (in the direction perpendicularly to the electric field vector $\mathbf {E}$) and $\simeq 0.2\lambda$ (along $\mathbf {E}$). Our findings revealed super-resolution capabilities of the developed THz endoscopic optical system, which paves the ways for novel THz sensing and exposure applications.
Funding
Ministry of Science and Higher Education of the Russian Federation (075–15–2020–790).
Acknowledgements
The work of V.V.B. and S.V.G. was supported by the Ministry of Science and Higher Education of the Russian Federation, Project # $075$–$15$–$2020$–$790$ (THz-field intensity visualization).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon request.
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