1. Introduction

During the past few decades, a rapid progress in THz opto-electronics is observed, being driven by novel opportunities in fundamental research and everyday life applications offered by THz technologies [1–7]. An increasing attention is paid to broadband coherent THz-wave generation and detection principles [8–11], such as those involving the single- and two-color laser plasma [12–23]. Among them, we particularly notice laser plasma formed by focusing of the two-color femtosecond pulses (at fundamental and second harmonic frequencies), that remains one of the most frequently-used laser-driven THz sources thanks to its ultra-broad emitted spectrum (it can span the frequency range as wide as $0.1$–$200$ THz [24]) and a relatively high THz peak fields [8]. Such THz-wave generation principles form a basis for novel methods of broadband THz spectroscopy, imaging, sensing and exposure, including the optical pump – THz probe experiments [25,26]. For long filaments, it is also possible to control the Carrier Envelope Phase (CEP) of the waveforms and, to some extent, the emitted THz spectrum using the interference effects [27–29].

Nevertheless, laser plasma-based THz emitters suffer from a number of disadvantages, considerably complicating their translation into practical fields. Along with technical difficulties associated with the use of cumbersome solid-state femtosecond lasers of a mJ-level pulse energy, quite complex angular distribution of the emitted THz radiation should be stressed, which is a function of the plasma excitation conditions, phase synchronism, sharpness of the laser radiation focusing, and length of filament [30–33]. Further manipulation and delivery of such complex-shaped THz beams to the object of interest in different THz applications appears to be a daunting task. This difficulty can be mitigated using some fiber/waveguide-based device for the THz-wave coupling and delivery, as well as for managing the THz-field spectrum, modal properties and angular distribution. In such a way, in Ref. [34], microstructured hollow-core polymer fiber was used to manipulate THz beam of a quantum cascade laser. This approach can be adapted for the laser plasma-based THz emitters, but it requires proper selection of the waveguide material, cross-section geometry, and underlying physical mechanism of electromagnetic waveguiding.

A number of THz waveguding principles, materials and related fabrication technologies have been developed during the past few decades, aimed at addressing challenging problems of THz waveguide and fiber optics [35,36]. Among them, we particularly notice the following solutions.

Most of the abovementioned THz waveguides are made of polymers, metals or polymer-metal structures with quite limited mechanical strength, thermal and radiation resistance, chemical inertness and biocompatibility. This reduces the range of such waveguides’ applications in THz sensing in harsh environments, under high mechanical load, thermal, radiation and chemical stress. Thus, they seem to be sub-optimal for operation with high-energy femtosecond laser pulses and intense THz fields that are usual for the THz-wave generation in laser plasma.

In order to mitigate this challenge, in our research, we considered sapphire shaped crystals grown by the Edge-defined Film-fed Growth (EFG) technique as a favorable materials platform for THz waveguide and fiber optics [57]. Sapphire waveguide and fiber optics possesses unique combination of physical properties owned by sapphire material (such as optical transparency in a wide spectral range spanning visible, infrared (partially) and THz ranges, high refractive index, mechanical and radiation strength, chemical inertness and biocompatibility) with advantages of the EFG technique (it allows to produce, directly from the Al$_2$O$_3$ melt, the sapphire shaped crystals with a complex cross-section geometry, the length of tens of centimeters, along with high surface and volumetric quality without any mechanical processing procedures, such as drilling, grinding, polishing, and others) [58,59]. In our earlier papers, we demonstrated that both ARROW and photonic crystal waveguiding mechanisms with small dispersion and propagation loss can be realized in the as-grown hollow-core multichannel-cladding sapphire shaped crystals with distinct geometries of cross section, with an ability to operate in aggressive environments [60,61]. Moreover, thanks to the high THz refractive index of sapphire, step-index sapphire fibers yield strong confinement of guided mode in a fiber core, which makes them ideal candidates for applications in the near-field THz imaging with the spatial resolution beyond the Abbe limit [45,62–64].

In this paper, we introduce hollow-core antiresonance THz waveguides for novel applications in THz optics – namely, for the THz radiation delivery from the two-color laser air plasma emitter to the object of interest, as well as for managing the spectrum and angular distribution of such a THz source. Slightly different waveguide geometries are considered in order to demonstrate flexibility of their design and fabrication. All of them are based on the EFG-grown sapphire tubes with different inner and outer diameters. Two of them are suspended in air, while another is coated by a polymer cladding. We carry out numerical simulations and experiments to uncover the optical performance of these waveguides. Numerical data reveal that they operate in the antiresonance mode, with effectively single guiding mode and small dispersion in a broad spectral range – i.e., the effective refractive index of this mode is $n_\mathrm {eff} \simeq 1.0$ in the $0.8$–$1.5$ THz range. Capabilities of broadband operation are then confirmed experimentally, by observing quite local (non-dispersed) THz pulses after the $\simeq 10$-cm-long waveguide fragments. For the considered waveguides, numerical simulations also predict small propagation loss, that overall decrease with frequency $\nu$, oscillate due to the ARROW mechanism, and can be as small as $\alpha \simeq 1$–$10$ dB/m in the $1.2$–$1.5$ THz range. Experimental data somewhat overestimate these numerical $\alpha$-values, but confirmed their oscillatory character, quite high optical performance of the developed waveguides, as well as their potential in broadband applications. These waveguides are then applied to couple and guide THz radiation from the two-color laser air plasma THz emitter (Fig., 1). We observed considerable narrowing of the THz radiation angular distribution after passing through the waveguides, as well as some changes in the THz spectrum shape. High optical performance of the developed waveguides and the aforementioned advanced physical properties of sapphire make them a favorable material and technological platform, that can boost the performance of laser plasma-based THz emitters and bring them closer to a variety of THz applications.

 

Fig. 1. Schematic of the THz-wave generation from the two-color laser air plasma, with its further coupling and guiding by a sapphire wavegude. Here, a scheme of the THz field angular distribution measurements at the far field zone, using a Golay cell on an angular translation stage, is shown.

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2. Numerical modeling of the sapphire waveguides

For the numerical simulations of the sapphire waveguides, the full-vectorial Finite-Difference Eigenmode (FDE) method [65] within the ANSYS Mode (Lumerical Mode Solutions) software was used, while accounting for both the THz optical properties of sapphire and various technological limitations posed by the EFG technique. Bulk refractive index and absorption coefficient of sapphire were taken from our earlier research work [63], while they are almost identical to those measured in Ref. [66]. The crystal c-axis was directed along the optical axis of a waveguide. Sapphire birefringence was accounted by appropriately defining its dielectric tensor, while no considerable impact of the crystal anisotropy on the guiding properties of sapphire shaped crystals were detected. In case of hollow-core waveguides, such a negligible role of anisotropy on their THz guiding properties can be attributed to the strong confinement of guided modes in a hollow core and, thus, to minimal interactions between these modes and the sapphire material. Similar effect was earlier reported in Refs. [57,60,61], where ARROW and photonic crystal hollow-core sapphire waveguides of different shape were analyzed. Also, it is worth mentioning that sapphire anisotropy plays an important role only when working with bulk (solid-core) sapphire waveguides and fibers [63,64]. Finally, dielectric properties of polymer coating, that is made of PolyTetraFluoroEthylene (PTFE / Teflon), were taken from Ref. [67].

As evident from Fig. 2, we considered three distinct waveguide cross-section geometries. The waveguides of Types I, II, and II are made of cylindrical sapphire tubes with the inner diameters of $4.5$, $4.2$, and $4.7$ mm, and the outer diameter of $5.5$, $6.0$, and $6.0$ mm, respectively. The waveguide of Type I is suspended in air (free space), with its refractive index $n_\mathrm {0} = 1.0$, while the waveguides of Types II and III are coated by the $\simeq 380$-$\mu$m-thick PTFE film. This coating is used to prevent de-coupling (from the waveguide) of the evanescent part of a guided mode, due to its interaction with any obstacle placed in contact with an outer waveguide surface. In Fig. 2 (b), we show numerical data on guiding properties of the Type I sapphire waveguide in form of the frequency-dependent effective refractive index $n_\mathrm {eff}$ and propagation loss coefficient $\alpha$ (defined by power) of the fundamental core-guided mode, while in Fig. 2 (c), we show the guided mode intensity distribution $I \left ( \mathbf {r} \right ) \propto \left | \mathbf {E} \left ( \mathbf {r} \right ) \right |^{2}$ in a waveguide cross-section at the frequency of $\nu = 1.0$ THz ($\mathbf {r}$ is a radius vector in the waveguide cross-section plane). In Figs. 2 (e),(f) and (h),(i), equal data sets are shown for the Type II and III waveguides.

 

Fig. 2. Numerical modeling of the antiresonance hollow-core sapphire THz waveguides. (a)–(c) Schematic of the waveguide cross-section geometry, effective refractive index $n_\mathrm {eff} \left ( \nu \right )$ and propagation loss $\alpha \left ( \nu \right )$ (by power) of the fundamental lowest-order core-guided mode, and guided mode intensity distribution $I \left ( \mathbf {r} \right )$ in a waveguide cross-section at $\nu = 1.0$ THz, respectively, for the Type I waveguide. (d)–(e) and (h)–(i) Similar data sets for the waveguides of Type II and III, respectively. In panels (b), (e), and (h), the propagation loss $\alpha$ is defined by the blue-to-red color bar, shown on the right from the panel (h), while in panels (c), (f), and (i), the guided mode intensity $I$ is defined by the black-to-yellow color bar, shown in the insert of panel (b).

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The observed spectral character of the $n_\mathrm {eff}$- and $\alpha$-functions are quite similar to that of the tube-based THz waveguides, that operate in the ARROW regime [49,50]. Namely, the THz wave is confined in a waveguide hollow core thanks to its Fabry-Perot reflections from the single- or multi-layer tube walls. The observed minima and maxima of $\alpha \left ( \nu \right )$-curves correspond to the constructive and destructive interference of reflected waves, respectively. Moreover, some modulations of $n_\mathrm {eff} \left ( \nu \right )$-curves and, thus, increased waveguide dispersion are observed around the maxima of $\alpha \left ( \nu \right )$-curves, where the waveguide Fabry-Perot reflectivity drops. Despite such a broken character of the guided mode, our numerical data predict high optical performance for all the considered waveguides with overall low dispersion and small-to-moderate loss. Namely, the loss are quite small $\alpha < 20$ dB/cm at high frequencies $\nu > 0.6$–$0.8$ THz, that are typical for the laser plasma-based THz emitters operation. All these waveguides can be applied to guide sub-ps-duration THz pulses over the considerable distances of tens of centimeter, while we might expect somewhat higher optical performance from the Type III waveguide (see Fig. 2 (d)). We also notice that polymer waveguide coating, that is aimed at preventing of the THz evanescent field de-coupling from the waveguide, does not considerably impact the waveguide optical properties.

In this work, we considered only the fundamental lower-order guided mode, which (among others) possesses the smallest dispersion and propagation losses due to the smallest mode area (in a fiber cross-section) and, thus, minimal mode – waveguide material interactions. Despite the higher-order core-guided modes can also be excited in such waveguides, we neglected them because of their higher dispersion and propagation loss, which are due to the larger mode area and stronger mode – waveguide material interactions, as evident, for example, from Ref. [50]. At the same time, these higher order core-guided modes can somehow affect the waveguide performance, especially at higher frequencies and larger waveguide diameter / radiation wavelength ratios. It is worth noting that the analyzed fundamental mode possesses a symmetric intensity distribution $I \left ( \mathbf {r} \right )$ in a waveguide core (see Figs. 2 (c),(f) and (i)) and, thus, it can be effectively excited by an axially symmetric laser plasma-based THz source, when it is centered at the waveguide optical axis in close proximity to the input waveguide end.

3. Fabrication of the sapphire waveguides

In Fig. 3, the antiresonant hollow-core sapphire THz waveguides fabrication using the EFG technique is shown. The crystals were grown in a 22 kHz induction-heated graphite susceptor/molybdenum crucible setup held within growth chamber [57]; see Fig. 3 (a). Verneuil crystals were used as a feed material for the growth process. The sapphire tube is grown from the melt Al$_2$O$_3$-film that is formed on the top of a molybdenum capillary die at the temperature as high as $2053^{\circ }$C and in the ambient high-purity Ar-atmosphere under the pressure of $1.1$–$1.3$ atm. Either single-point or tube-based c-axis sapphire seeds were used to initiate the crystal growth process. The pulling rate was in the range of $50$–$100$ mm/h. During the crystal growth, the Al$_2$O$_3$-melt rises to the top of the molybdenum die through the $0.25$-mm-thick ring capillary channel. The inner and outer sapphire tube diameters are mainly determined by the die design, with possible small variations caused by the surface tension of the melt in the meniscus region.

 

Fig. 3. Fabrication of the antiresonance hollow-core sapphire THz waveguides. (a) A scheme of the sapphire shaped crystal growth by the EFG technique. (b) An in situ photo of the crystallization zone during the crystal growth process. (c) An in situ photo of the as-grown sapphire tube atop of the heat zone during the crystal pulling. (d) A photo of thus fabricated $100$-mm-long sapphire waveguides, either without (Type I) or with (Types II and III) a PTFE coating, which is formed by the thermal polymer shrinkage.

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In order to improve the surface quality, control the crystal cross-section, and prevent the formation of defects (associated with the crystallization front overcooling), we used an automatic control system based on crystal weight sensors during the entire growth process [68–70]. This automatic weight control system relies on analysis and preventing of any discrepancies between the measured crystal weight and its desired a priori-calculated values. One of the most urgent problems in growth of the tubular crystals is a stabilization of the growth conditions at the die working edges and the related Al$_2$O$_3$-melt meniscus shape. Indeed, even small deviations in the Al$_2$O$_3$-melt level in a crucible lead to pronounce changes in the pressure of the melt meniscus and related temperature gradients in the crystallization zone [71]. Therefore, the meniscus height and, thus, the Al$_2$O$_3$-melt level in the crucible should be maintained in certain narrow ranges during the growth process. For this aim, the cruciable with a melt is handled in a vertical translation stage and can be displaced during the crystal growth, as shown in Fig. 3 (a). In addition, a high-resolution visible camera is used for monitoring the in situ tube diameters. A photo of the crystallization zone, obtained by such camera, is shown in Fig. 3 (b). Processing of such in situ images provides useful information about the crystallization zone state and the as-grown tube diameters. In Fig. 3 (c), we show an in situ photo of the as-grown tube, that contains a seeding part and rises above the heat zone.

The described experimental setup allowed us to fabricate sapphire tubes with the desired inner and outer diameters (Fig. 2.) and the lengths of about $50$ cm. These tubes were, then, cut into pieces with the length of $100$, and $150$ mm for further experimental characterization. Parts of the tubes were coated by a polymer film using a shrinking PTFE tube with the resultant coating thickness of $380$ $\mu$m, as measured by the standard contact micrometer. The representative examples of thus fabricated $100$-mm-long waveguides of Types I, II and III are shown in Fig. 3 (d).

4. Experimental characterization of the sapphire waveguides

Next, we proceed to the experimental study of the developed waveguide properties using the principles of THz pulsed spectroscopy, as well as to the combination of the waveguides with the two-color laser plasma-based THz emitter.

4.1 THz pulsed spectroscopy of the sapphire waveguides

First, the waveguides were characterized experimentally using the in-house transmission-mode THz pulsed spectrometer at BMSTU, which was designed specifically for the THz waveguide measurements, as detailed in Ref. [61]. This system uses LT-GaAs photoconductive antennas [3,5], as an emitter and a detector of the THz pulses, while these antennas are pumped and probed by the $\simeq 100$ fs laser pulses of TOPTICA FemtoFErb 780 laser. The spectrometer operates in an ambient laboratory environment – i.e. in a humid air. An important feature of this system is an ability to accommodate in the THz beam path the waveguides of different length ($l \leq 200$ mm). As described in Ref. [61], in the optical scheme of this THz pulsed spectrometer, one lens is used to focus the THz beam onto the input waveguide end, while another is used to collimate the THz beam, that leaves the waveguide. The first lens is rigidly fixed, while the second can be displaced manually along the THz beam axis direction, aimed at the THz field de-coupling from the output end of the waveguides with different lengths.

In Fig. 4, results of the THz pulsed spectroscopy of the developed waveguides are shown. Particularly, in Figs. 4 (a) and (b), we show a reference spectrum $\left | E \left ( \nu \right ) \right |$ and a reference waveform $E \left ( t \right )$, that represent the THz pulse propagation through the empty THz beam path, as well as the sample spectra and waveforms, after the THz pulse transmission through the $100$- and $150$-mm-long fragments of the Type I waveguide. From Fig. 4 (a), it is evident that the transmission spectra $\left | E \left ( \nu \right ) \right |$ drops with increasing waveguide length $l$ at all frequencies $\nu$, which results from the waveguide propagation loss $\alpha$, along with possible coupling and de-coupling losses. In turn, from Fig. 4 (b) we notice that after propagating through the waveguide, quite local wavelets still present in the sample waveforms $E \left ( t \right )$, being centered near the ballistic pulse of the reference waveform. For example, such a local THz pulse character in the THz waveforms can be compared with considerable THz pulse delay and broadening after propagation throught the individual $\sim 1$-cm-long step-index sapphire fiber suspended in air [45], or through the few-mm-long bundle of metal-coated sapphire fibers [63]. This qualitatively shows a small waveguide dispersion and justifies that the effective mode index of such waveguide is around unity ($n_\mathrm {eff} \simeq 1.0$), as predicted by our numerical simulations (see Fig. 2).

 

Fig. 4. THz pulsed spectroscopy of the antiresonance hollow-core sapphire THz waveguides. (a)–(c) Spectra $\left | E \left ( \nu \right ) \right |$ and related THz waveforms $E \left ( t \right )$, that correspond to THz-wave transmitted through the empty THz beam path (reference, $l=0$) and the THz beam path accommodating the Type I waveguide fragments with the lengths of $l = 100$ and $150$ mm, as well as estimated propagation loss $\alpha$ (by power), respectively. (d)–(f) and (g)–(i) Equal data sets for the waveguides of Types II and III, respectively. In (c),(f), and (i), the experimental data is overlapped with the numerical estimates of propagation loss of the fundamental lowest-order core-guided mode (see Fig. 2). For the experimental loss curves, the vertical error bars define the $\pm \sigma$ confidential interval of measurements, where $\sigma$ is a standard deviation. In (c), (f), and (i), the spectral noise level is $\simeq 5 \times 10^{-3}$.

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In order to estimate propagation loss $\alpha$, we applied an approach described in Ref. [60]. First, we normalized the sample spectra by a reference one to calculate the waveguide transmission spectra $\left | T \left ( \nu, l \right ) \right |$ (by field), as a function of the radiation frequency $\nu$ and the waveguide length $l$. Then, for simplicity, we neglected coupling and de-coupling losses, and fitted thus calculated waveguide transmission spectra by the following theoretical model, independently at each discrete frequency $\nu$ and waveguide length $l$

The observed experimental $\alpha$ estimates show moderate agreement with our numerical data. The experimental loss $\alpha$ somewhat overestimates the theoretical predictions. Experimental $\alpha$-curves also oscillate with frequency $\nu$, while only in several distinct frequency bands, that correspond to the $\alpha$-curve minima, the theoretically predicted small $\alpha$ is observed. The oscillations in the numerical and experimental $\alpha$-curves occur at a different characteristic period, while at certain frequency bands the measured $\alpha$ appears to be very high. The observed discrepancies between the numerical and experimental loss $\alpha$ can be attributed to a number of factors. Among them, we particularly notice the neglected coupling and decoupling losses, the existence of which will obviously lead to somewhat overestimated experimental propagation loss $\alpha$. In such hollow-core waveguides, the coupling/decoupling losses usually vary inhomogeneously with frequency, as well as depend on the excitation conditions and the guiding mode intensity distribution and polarization [52,57,61]. However, an impact of the coupling/decoupling losses on the waveguide performance is difficult to quantify at the moment, and more investigations are in order. Another factor is the neglected higher-order guided modes and related interference effects, that can lead to the observed $\alpha$-curve modulation. Finally, misalignments in the experimental setup should be notices as an important reason of the observed discrepancies, especially when measuring such long waveguide fragments with a reconfigurable THz beam path [61]. Nevertheless, in our experiments, we observed measurable values of the THz field amplitude either in time and frequency domains, which qualitatively justify overall efficient THz waveguidance in the developed structures.

4.2 Combining the sapphire waveguides with the two-color laser air plasma THz emitter

The $100$-mm-long pieces of the sapphire waveguides were then combined with the in-house two-color laser plasma emitter at GPI RAS, as schematically shown in Fig. 1. A key element of this setup is a Ti:sapphire laser with the central wavelength of $800$ nm, the pulse duration of $150$ fs, the pulse energy of $2.8$ mJ, the pulse repetition rate of $1$ kHz, and the output beam diameter of $12$ mm (at the $e^{-2}$ level). Part of the laser output was converted into second harmonic radiation using a $\beta$-BBO crystal of I-type with the dimensions of $10 \times 10 \times 0.1$ mm$^{3}$. To boost the THz-wave generation, the polarizations of fundamental and second harmonics were converted to the collinear state using a dual wavelength wave plate, while the delay between them was compensated by a calcite compensator plate. Finally, the two-color pulses were focused into an ambient air using an off-axis parabolic mirror with the focal length of $20$ cm. This results in a two-color laser air plasma excitation, with the length of the fluorescent plasma channel around $\simeq 8$ mm. During experiments, each sapphire waveguide was mounted towards our laser plasma-based THz emitter in the way that the plasma spark is centered on the optical axis of a waveguide and located just after the waveguide input end. Thus, the emitted THz waves are directly coupled to a sapphire waveguide.

4.2.1 Measuring the transmission spectra

First, we studied the effect of the THz radiation transmission through the waveguides on spectral parameters of the output radiation. For this aim, the THz spectra, emitted either without or with a THz waveguide in an optical scheme, were measured by a standard electro-optic sampling technique using a $\langle 1 1 0 \rangle$-cut ZnTe crystal with the dimensions of $3 \times 3 \times 1$ mm$^{3}$, a quarter-wave plate, a Wollaston prism, and two balanced photodiodes, as detailed in our earlier works [29,32].

Slightly different optical schemes were applied to collect the THz radiation, when measuring reference and sample spectra, that correspond to the laser plasma-based THz emitter without and with the sapphire waveguides, respectively.

Here, we should stress that such a discrepancy between the reference and sample data measurements does not allow us to compare them quantitatively (namely, to compare absolute values of the measured time- or frequency-domain THz signals). But we can still analyze qualitatively the shape of the emitted THz waveforms and spectra. In this way, in Fig. 5, the waveforms $E \left ( t \right )$ and spectra $E \left ( \nu \right )$ are normalized by their maximal values for convenience. We also notice some kind of pre-pulse appeared before the main pulse in the THz waveforms (i.e., the electric field is not flat, but looks bumped before the main pulse), which is especially pronounced for the waveguide measurements. These pre-pulses should not be attributed to any physical effect associated with THz waveguidance. They occurred due to some low-frequency oscillatory noise in the THz waveforms that is usual for the applied technique of THz pulse detection. We observed different levels of such noise in the reference and sample waveforms, which is due to the distinct setup arrangements, applied to collect these waveforms.

 

Fig. 5. Spectra and waveforms of the THz radiation emitted by the two-color laser air plasma, either without or with the sapphire waveguides in an optical scheme. (a),(b) Spectra $\left | E \left ( \nu \right ) \right |$ and waveforms $E \left ( t \right )$, respectively, that correspond to the THz pulses without a waveguide (reference) and with the $100$-mm-long waveguide of Type I (sample). (c),(d) and (e),(f) Equal data sets for the waveguides of Type II and III. In (a), (c), and (e), the spectral noise level is $\simeq 2 \times 10^{-2}$.

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For the three considered waveguides, the measured reference and sample spectra and waveforms are shown in Fig. 5. At low frequencies $\nu < 0.5$ THz, the THz spectra after the waveguides are more suppressed (Figs. 5 (a),(c), and (e)). In fact, this agrees well with our numerical simulations, which predicted high propagation loss $\alpha$ at low-frequency limit of the analyzed spectral range for all three wavegudies (see Fig. 2). Such effect leads to some shift of the THz spectrum maximum to higher frequencies after the THz pulse propagation through the waveguides. Moreover, from the measured THz waveforms we notice that the THz wavelets have a local character in both the reference (without a waveguide) and sample (with a $100$-mm-long waveguide) data (see Fig. 5 (b),(d), and (f)). This again highlights quite a small dispersion of THz pulses in our waveguides.

4.2.2 Measuring the far-field radiation patterns

Next, we studied the far-field radiation patterns formed by the laser plasma-based THz emitter, either without or with the developed waveguides. Schematic of such measurements is shown in Fig. 1. The angular THz beam power distributions were collected at the distance of $20$ cm from the plasma source (in a reference experiments), or the output waveguide end (in a sample experiment). The emitted laser radiation of Ti:sapphire laser was gated at the frequency of $20$ Hz, which resulted in the THz beam modulation. This allowed us to measure the angular distributions of the THz beam power, using a Golay (opto-acoustic) cell (Tydex GC-$1$P) with a $20$ Hz lock-in amplifier, since the Golay cell is capable of detecting only the modulated power flux of an optical beam. The Golay cell was mounted on a manual angular translation stage; the latter moves the THz detector in the horizontal direction $\theta$ with the angular step of $2.5^{\circ }$. To block the parasitic optical radiation, that co-propagates with the THz waves, we used a $1$-mm-thick PTFE-sheet handled normally towards the detector direction at the distance of $10$ cm from the laser plasma (in a reference experiment) or a waveguide output end (in a sample experiment). An additional iris metal diaphragm with the hole diameter of $5$ mm was installed in front of the Golay cell to ensure the equivalent angular resolution at all positions of the Golay cell, as well as suppress signal from the unwanted side light.

In Fig. 6, the detected angular distributions of the THz field intensity are shown for both the separately-operation laser plasma-based THz emitter and that equipped with the sapphire waveguides. The angular distribution formed by a separately-operating plasma source is typical for the considered conditions of the femtosecond laser pulses focusing (i.e., it features a near-axis plato, as reported in Ref. [31]), with the Full-Width at Half-Maximum of $\Delta \theta \simeq 26.56^{\circ }$. When the waveguides are introduced into the optical scheme, the far-field angular distributions are narrowed considerably (by a factor of $\simeq 4$) – i.e., down to the values of $\Delta \theta \simeq 9.40$, $8.22$, and $7.11^{\circ }$, for the waveguides of Type I, II and III, respectively. All waveguides demonstrate quite similar far-field distributions, when operating with such a broadband THz source, with some small deviations in the size of axial plato and diameter of the THz beam $\Delta \theta$.

 

Fig. 6. Angular distributions of the THz radiation emitted by the two-color laser air plasma, either without or with the sapphire waveguides in an optical scheme. (a) Experimental THz radiation distributions $I_\mathrm {C-S} \left ( \theta \right )$ at the far-field zone in the horizontal direction $\theta$ for the reference (without a waveguide, in blue) and sample (with a $100$-mm-long waveguide of Type I, in green) as compared with numerical predictions (in red). (b) 2D far-field distribution of the THz beam intensity $I_\mathrm {Far-Field} \left ( \theta, \phi \right )$, where $\theta$ and $\phi$ are the two orthogonal angular directions; this image is computed numerically using the vector diffraction integral transform within the ANSYS Mode software aided by Eq. (2). (c),(d) and (h),(i) Equal data sets for the waveguides of Types I and II. Table at the bottom summarizes the data on the far-field THz beam spot diameter.

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We also modeled numerically the two-dimensional (2D) far-field patterns formed at the shadow side of our waveguides, when they operate with such a broadband THz emitter. For this aim, we used the frequency-dependent spatial distributions of the lowest-order core-guided mode fields $\mathbf {E} \left ( \nu, \mathbf {r} \right )$, $\mathbf {H} \left ( \nu, \mathbf {r} \right )$ in the waveguide cross-section from our numerical analysis (they underlie the mode intensities in Figs. 2 (c),(f), and (i)) and the THz-wave spectra $\left | E \left ( \nu \right ) \right |$ measured at the shadow side of a waveguide in our experiments (see green curves in Figs. 5 (a),(c), and (e)). Using the vector diffraction integral transform within the ANSYS Mode software, we calculated the vector THz field distribution at the far-field zone $\mathbf {E}^{\mathrm {Far-field}} \left ( \nu, \mathbf {x} \right )$, where $\mathbf {x}$ is a radius-vector at the analyze far-field plane. Then, at each far-field point $\mathbf {x}$, we computed the detected THz beam power $I_\mathrm {Far-field} \left ( \mathbf {x} \right )$, by integrating (over the $\nu$) the weighted THz beam intensity

Then, we resorted from the radius vector $\mathbf {x}$ to the angular coordinates $\left ( \theta, \phi \right )$ at the far-field zone. The resultant 2D distributions are shown in Figs. 6 (b),(d), and (f), while their cross sections along the $\theta$-direction are compared with the experimental data in panels (a),(c), and (e).

On the one hand, we observe rather good agreement between our numerical and experimental estimates, which justifies the effectively single mode regime of our waveguides operation. Indeed, by considering only the THz field, that is carried by the lowest-order core-guided mode, we accurately described the observed far-field THz radiation patters. On the other hand, our findings highlight an ability to manage the shape of THz beams, emitted by the laser plasma-based THz source by their coupling to the guided mode of a hollow-core sapphire waveguide, as well as to deliver THz-wave to the object of interest using such a waveguide.

5. Discussions

Our numerical analysis (Fig. 2) and experiments (Fig. 4) revealed quite high optical performance of the developed sapphire-tube-based THz waveguides. Numerical data showed that these waveguides operate in the ARROW regime, with small dispersion and low-to-moderate loss in a broad spectral range. Namely, the effective refractive index of a guided mode is near unity $n_\mathrm {eff} \simeq 1.0$ in the frequency range of $0.8$–$1.5$ THz. Capabilities of broadband operation with small dispersion were then confirmed experimentally, since at the shadow side of the $\simeq 10$-mm-long waveguides quite local (non-dispersed) THz wavelets were observed. Moreover, numerical analysis of our waveguides predicted small THz-wave propagation loss $\alpha$, that overall decreases with frequency $\nu$, oscillates due to the ARROW guidance mechanism and can be as small as $\alpha \simeq 1$–$10$ dB/m (or $\alpha \simeq 4.3$–$43$ m$^{-1}$) at the high frequencies of $1.2$–$1.5$ THz. Using these estimates along with Eq. (1), one can easily calculate the distance $l$ over which the THz wave can be effectively guided by our structures. For example, by defining appropriate wavegudie transmission values (by power) of $\left | T \right |^{2} = 1/e$ (or $\simeq 37$%) and $\left | T \right |^{2} = 0.1$ (or $10$%), this distance appears to be as high as $l_\mathrm {1/e} \simeq 23.0$–$2.3$ cm and $l_\mathrm {0.1} \simeq 53.4$–$5.3$ cm, respectively, for the specified $\alpha$-values. Such loss seems to be realistic for a variety of applications, including endoscopic measurements and exposure of hardly-accessible biological tissues [2,7], or the discussed shaping of the THz emitter spectrum and angular diagram (Figs. 5 and 6) [34]. Despite our experiments somewhat overestimate the waveguide losses, they overall justify high efficiency of the optical elements.

Laser-plasma-based THz emitters can generally possess very high output power (up to $\sim 10^{-1}$–$10^{-2}$ W) and the bandwidth up to $\sim 10$–$100$ THz [24,72–74]. For such advanced THz emitters, the waveguide performance should be studied theoretically and experimentally in much broader frequency range, in order to objectively uncover their optical performance. In this study, we limited our numerical and experimental studies by the frequencies below $1.5$ THz due to a number of technical issues. Numerical analysis at higher frequencies requires much smaller discretization steps (in both time and spatial domains), thus, increasing complexity and duration of calculations. Experimental studies at higher frequencies also appear to be a daunting task due to the drop of the THz detectors sensitivity, including the photoconductive emitter in our THz pulsed spectrometer, and the electrooptical detector in our laser-plasma-based THz setup. Moreover, to excite laser plasma, quite long ($150$ fs) femtosecond pulses are applied, that also limits the THz radiation bandwidth in our experimental setup.

While it is almost impossible to ensure THz waveguidance in the total $10$–$100$ THz-range (with simultaneously small dispersion and loss) using a single optical element, one can still expect some THz waveguidance above $1.5$ THz from the analyzed structures. On the one hand, propagation loss of a waveguide should rise with frequency due to increasing THz wave absorption by sapphire [63]. At frequencies above $2.0$–$3.5$ THz, the sapphire tube walls will become almost non-transparent and, therefore, the waveguide will no longer operate in the ARROW mode. On the other hand, some THz waveguidance will still be observed at these frequencies and even higher thanks to the Fresnel THz-wave reflections from the inner wall of the high-refractive-index sapphire tube. Finally, we notice that, when the inner diameter of a tube will become much larger than the wavelength $d >> \lambda$, the waveguide should manifest strongly multimode waveguidance with the intermodal dispersion effect. In this case, the waveguide will lose its advanced optical performance.

High optical properties of thus developed waveguides allow them to compete with the advanced examples of polymer or metal-polymer THz waveguides [36,53,75–79], as well as with earlier-reported hollow-core structures based on sapphire shaped crystals of different geometry [57,60,61]; all these modern waveguides provide the propagation loss level of $\alpha \geq 1$ m$^{-1}$. High optical performance of our waveguides can be attributed to a favorable combination of the high refractive index and low absorption coefficient of sapphire at THz frequencies, strong confinement of guiding modes in the hollow waveguide core (it minimizes the THz wave–sapphire material interaction), as well as high quality of the EFG-grown shaped crystal. Finally, we notice that unique physical properties of sapphire enable advantage of sapphire THz waveguides over the polymer and polymer-metal ones, especially in sensing and exposure applications, that need optical elements with advanced mechanical, chemical, and radiation strength.

The EFG-grown sapphire shaped crystals form a favorable material platform for the THz waveguide optics, since they combine convenience and advanced performance of the EFG method with the relatively low material cost and modest fabrication time, required to achieve the desired cross-section geometry. The EFG technique provides rich opportunities for further optimization of the waveguide geometry, aimed at increasing or managing its optical performance. An interesting topic for further research is to study THz-wave generation in capillary channels inside the sapphire shaped crystals. The dimensions of such channels can be comparable and even smaller than the dimensions of laser plasma spark. Therefore, some interesting effects of sapphire shaped crystal on the waveform and spectrum of the emitted THz wave can be expected.

6. Conclusions

In this paper, we demonstrated capabilities of using the antiresonance hollow-core sapphire waveguides to deliver the THz radiation (over the considerable distances) from the two-color laser air plasma emitter to the object of THz measurements or exposure, as well as in managing the spectrum and angular distribution of a THz source. Our findings show that hollow-core sapphire waveguides can boost the performance of laser air plasma-based THz emitters and, thus, bring them closer to practical applications in different branches of THz science and technology.