1. Introduction

Terahertz (THz) waves, coming to the foreground in optics, refer to electromagnetic radiations with frequencies lying between 0.1-10 THz. A salient feature of THz waves is their powerful penetration into most dielectric materials, which has inspired widespread applications in recent years . In particular, THz frequencies correspond to wavelengths shorter than the microwave, thus in favor of higher imaging resolutions. On the other side, THz radiations prove to be non-ionizing, owing to their lower photon energies than those of the infrared. These unique properties make THz waves highly applicable for non-destructive detections in chemical analyzing [1,2], biomedical sensing [3,4], security screening [5,6], etc. However, there is a fact as mentioned by Md. Saiful Islam that most present terahertz systems are bulky, and they typically depend on the free-space transmission [7]. Unfortunately, the THz radiation is highly sensitive to the atmosphere on account of water vapor contents. The free-space transmission may experience various undesirable losses due to the coupling with atmospheric components, which significantly reduces the transmission efficiency . An effective remedy consists in the development of low-loss THz waveguides, serving as a protection to trap THz waves inside the waveguide and isolate them from ambient atmospheres, thereby ensuring a stable transmission. Hence, THz waveguides has attracted considerable attentions over the past decades.

In terms of the flexibility for transmission paths, preferred THz waveguides are primarily dielectric fibers. It is worth mentioning that the fiber for THz frequencies can be perceived as a scaled-up version of that for the common infrared spectrum [8]. Therefore, pertinent concepts and technologies can be directly borrowed from those infrared fibers, which has greatly promoted the development of THz fibers. Unfortunately, there still exists an unpleasant but unavoidable fact: except for air, no dielectric material can be found to exhibit a negligible absorption over the THz spectrum. In order to achieve low loss guidances, a basic principle to design THz fibers is allowing the guided mode to distribute its field in the internal air as much as possible. Mindful of such a principle, various fiber structures have been proposed, roughly falling into three categories of mechanism: the total internal reflection (TIR), photonic bandgap (PBG) and antiresonance reflection (ARR). The first category is typified by porous core fibers. To meet the TIR requirement, these fibers have to employ a more porous cladding [9–13], or directly expose the core to air [14–17], so as to ensure a positive fiber delta. On the contrary, the PBG based guidance can trap electromagnetic waves in a core lower than the photonic crystal (PC) cladding. Whereupon, some PBG THz fibers have been proposed to effectively reduce the transmission loss. Specifically, the structure with a honeycomb cladding has been numerically and experimentally studied, however exhibiting relatively narrow spectrums [18–21]. Although the square or hexagon airholes with rounded corners can generate broader PBGs [22], but they are practically impossible to be fabricated. Another kind of scheme able to deliver broad PBGs is the Bragg structure, which arranges alternative covers with two different materials to achieve a periodic layout in the radial direction. In theory, the air is the optimal choice as one of the two materials [23], but this is apparently unrealistic. Some expedients concern supporting structures [24,25], or directly use two different solid materials [26]. With regard to the ARR mechanism, THz fibers widely adopt the Kagome type of PC cladding [27,28], tube lattice [29–35], or a single tube [36,37], which have also demonstrated very low losses. Nevertheless, mode lines in ARR fibers are not smooth, so their group velocity dispersion (GVD) curves, which are related to the second derivative of mode lines, should behave in an oscillation manner, thus to the detriment of THz pulse transmissions. Hence, the GVD problem has been rarely discussed for ARR fibers.

In this paper, a PC architecture consisting of honeycombed tubes (HTs) is proposed. It is well known that the honeycomb lattice has been widely adopted to arrange airholes in PC fibers [18–21,38–42]. When tube elements are introduced, the periodic PC presents a unit cell different from those in airhole based PCs, so it should provide a distinctive PBG map, which will be confirmed in the following. Also well known is that dielectric tubes have been commonly used to construct ARR THz fibers. In particular, a fiber with triangularly latticed tubes has been reported to present four PBGs [29]. However, the author claimed these PBGs are too narrow to support their evidently broader guidances, so they ascribe it to the ARR effect. Instead, here we adopt a honeycomb lattice to arrange dielectric tubes and, surprisingly, a broader and less dispersive PBG has been achieved. Then a fiber structure is designed for the effective THz guidance, which exactly takes advantage of the PBG confinement. Some important performances are numerically examined with the aid of full-vector finite element method (FEM), including the mode effective index, mode area, operation band, transmission loss, chromatic dispersion, and bending induced loss. Furthermore, an optimized sample is worked out to deliver low transmission losses, nearly zero flat dispersions, and negligible bending induced losses over a broad spectrum. Finally, possible fabrication methods on the proposed fiber are discussed.

2. HTs based PBGs

In order to demonstrate the superiority of the HTs based PC, we compare it to another two existing designs over the same scale. One of them arranges airholes in a honeycomb lattice [19], here termed as honeycombed holes (HHs). The other stacks tubes in a triangular lattice [43], and thus can be referred to as the triangulated tubes (TTs). Figure 1(a) schematically displays cross-sectional layouts for the three PCs. For comparability, all the tubes in HTs and TTs share the same inner diameter $d$, as well as the same outer diameter $D$. Moreover, the inner and outer diameters are also used to size the airholes and pitches in HHs, respectively. The base dielectric material of these PCs is cyclic olefin copolymer with the trade name of Topas, which features a low absorption loss and a constant refractive index ${{n}_{\textrm {Topas}}}=1.5235$ over 0.1-2 THz. The PBG calculation is performed through the plane wave expansion (PWE) method, enlisting the support of a freely available calculation package, i. e., the MIT Photonic-Bands (MPB) Package. Over a unit cell for calculations, the resolution is set to be 128 for the mesh, which can ensure a sufficiently high accuracy. Notice the resolution is a parameter for the mesh control, and larger resolutions produce finer meshes, thus yielding more accurate results. For a further confirmation, another software COMSOL has been employed with its eigenfrequency solver applied for a unit cell. Extremely fine mesh with mapped and triangular elements are adopted to produce a total of 11642 elements, and the same results have been obtained.

 

Fig. 1. (a) The first and second PBGs for three PCs with $d/D=0.9$, which are zoned by solid and dashed lines respectively. All these PCs are composed of Topas (light-grey region) and air (black region), with their layouts sketched in circular insets. (b) PBGs for different values of $d/D$ in HTs based PCs.

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In Fig. 1(a), the first PBGs for three PCs with $d/D=0.9$ are zoned by solid lines, while the second PBGs by dashed lines. As can be seen from this figure that, over low index regions, PBGs of HTs and TTs are remarkably less dispersive than HHs, holding promise for broader guiding spectrums. It is worth saying that the HHs based PC has lower effective indices supported in the PBGs, which can thereby support lower guided modes, and thus may suffer less material losses due to a larger fraction of power distributed in air. However, they are much more dispersive. Since the guided mode in a fiber is relatively flatter, it will experience a narrower horizontal bandwidth for HHs. Another difference can be found that the second PBGs of tubes based PCs are apparently narrower than that of the PC with airholes. Furthermore, the first PBG of HTs completely covers that of the TTs, and has a much larger part below ${{n}_{\textrm {eff}}}=1.1$. To be more specific, first bandgap spans at ${n_{\textrm{eff}}}$ = 1.2 are $1.56 \times {10^8}$ - $2.29 \times {10^8}$, $8.12 \times {10^8}$ - $11.31 \times {10^8}$ and $8.73 \times {10^8}$ - $11.19 \times {10^8}$ for HHs, HTs and TTs, respectively. At ${n_{\textrm{eff}}}$ = 1.1, first bandgap spans for HHs, HTs and TTs turn out to be $1.36 \times {10^8}$ - $1.66 \times {10^8}$, $5.39 \times {10^8}$ - $7.33 \times {10^8}$ and $6.51 \times {10^8}$ - $6.82 \times {10^8}$, respectively. Remember the design principle for THz fibers suggests that the guided mode should distribute its field in air as much as possible, which corresponds to a low effective index. Whereupon, the HTs based PC is more suitable to support low loss guidances for THz waves over a broad spectrum. It is worth noting that the PBGs of HHs can penetrate below the air line ${{n}_{\textrm {eff}}}=1.\textrm {0}$, but they are too narrow to effectively support a hollow core guidance. Furthermore, the manufacture of HTs can make demands on comparatively less materials, thus being more economic.

In Fig. 1(b), we have shown the first PBGs for different values of $d/D$ in HTs based PCs. It can be seen that, with increasing the $d/D$, the PBG shifts towards lower index regions, as well as towards higher frequency bands, while appears less dispersive. Since the guided mode over a lower PBG can achieve lower effective indices accordingly, it should have more power distributed in air. Hence, lower absorption losses can be expected for a larger $d/D$. However from the fabrication point of view, it is impossible for one to arbitrarily increase the $d/D$, while tubes with $d/D$ = 0.9 have already been experimentally achieved, as reported in an existing result [43]. So we will choose this value to exemplify guiding performances for the fiber design in the following.

3. PBG THz fiber design

Given the HTs based PC framework, what follows is to design PBG fibers for the THz guidance. The principle consists in engineering such a defect core that allows guided modes to lie within PBGs. Seeing that the second PBG is too narrow to support an effective guidance, here only the first PBG is chosen for the fiber design. Notice the PBG lies above the air line, so we do not consider the hollow core guidance. Instead, smaller tubes are triangularly latticed in the fiber center, as shown in Fig. 2. It is worth mentioning that the TTs have been employed previously as a THz fiber cladding [43], however without the discussion of guiding mechanism. They have not studied the PBG for TTs and, in view of a solid core, the guiding effect should be ascribed to the TIR. For the present design, TTs in the core have inner and outer diameters denoted as ${{d}_{\textrm {c}}}$ and ${{D}_{\textrm {c}}}$, respectively. Nevertheless, if the TTs are directly arranged as a core in HTs, it is apparent that a region about the core has an average refractive index being the globally highest. As a result, it can support the undesirable TIR guidance. To suppress the TIR mode, the region about inner cladding tubes can be depressed by removing a certain amount of Topas, producing a defect region with a lower average refractive index. In fact, an initial try we have made is thinning the six inner cladding tubes by enlarging their inner diameters. Although the TIR effect can be suppressed, many surface modes have been observed with a heavy field distribution around the contact between inner and outer cladding tubes. Then we came up with another ingenious design, which directly cut away a segment of each inner cladding tube, producing broken tubes with a breach. The top right inset in Fig. 2 has exemplified such a broken tube, with its breach region surrounded by a closed blue curve. Moreover, the cut segment should be as much as possible, so as to suppress those surface modes that would distribute their fields around the tube contacts between the core and inner cladding. So the cut positions are selected exactly where the core tubes contact with cladding tubes. Significantly, in comparison to the initial try based on the thinning method, the cutting strategy takes less toll on the PC area, thus requiring less honeycomb layers to serve the PBG function. Here only two honeycomb layers are employed, which has proven enough for an effective confinement of THz waves. It is known that employing more layers can reduce confinement losses, however at the cost of increasing fabrication difficulties. More to the point, total losses for THz transmissions are predominately blamed on material absorptions, while the decreased confinement loss by adding more layers is comparatively trivial. On balance, two layers of honeycomb configurations are more appropriate for the present design.

 

Fig. 2. The schematic cross section of the proposed THz fiber, which has an HTs based PC cladding. Smaller tubes are triangularly latticed in the fiber center, and the core region is zoned by a dashed yellow circle. Broken tubes around the core serve as the inner cladding.

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We now move on to determine structural parameters for the designed fiber. The ratio of inner to outer diameter in cladding tubes is fixed as $d/D=0.9$. The outer diameter ${{D}_{\textrm {c}}}$ in core tubes can be geometrically calculated with the aid of a sketched triangle, formed with green lines in Fig. 2. Three side lengths of this triangle are $\sqrt {3}{{D}_{\textrm {c}}}$, $D$ and $0.5({{D}_{\textrm {c}}}+D)$, respectively. Based on the Cosine law, we have ${{(\sqrt {3}{{D}_{\textrm {c}}})}^{2}}+{{D}^{2}}-2\sqrt {3}{{D}_{\textrm {c}}}D\textrm{cos}{{30}^{\textrm {o}}}={{(0.5({{D}_{\textrm {c}}}+D))}^{2}}$, yielding ${{D}_{\textrm {c}}}/D=3/11$. Then the angle $\theta$ in the triangle can also be calculated by the Cosine law, which is $\theta ={{21.8}^{\textrm {o}}}$. Seeing that this angle is half the radian of a breach, the proportion of the cut segment is then worked out to be 12.1 $\%$ of an intact tube. By now, the only pending parameter is the inner diameter ${{d}_{\textrm {c}}}$ in core tubes, which can serve to control transmission characteristics, as will be discussed in the following.

To study the influence of inner diameters in core tubes over guiding characteristics, we use the full-vector FEM for mode calculations based on COMSOL. The fiber model is meshed with the maximum element size of 10 $\mu$m. Then for the frequency of 2 THz ($\lambda$ = 150 $\mu$m), the maximum element size related to wavelength is $\lambda$/15, which should be sufficient to guarantee the result accuracy over the frequency range of interest (< 2 THz). Figure 3 has shown the mode effective indices ${{n}_{\textrm {eff}}}$ for fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.82 to 0.88 in increments of 0.01. To illustrate that the guidance is achieved by the bandgap, we have additionally calculated the fundamental space filling mode (FSM), as shown by a dashed yellow curve, which provides the highest effective index in a PC cladding [44,45]. If a mode is guided above the FSM, it can be ascribed to the TIR mechanism. However, when the guided mode lies below the FSM, it no longer satisfies the TIR condition, then the PBG or ARR effect should be take into consideration. However, the ARR guidance involves a hollow core and supports guided modes below the air line [27–35]. Since the present guided modes lie below the FSM as well as above the air line, they should be identified as PBG guidances. It can be seen from these mode lines that, with increasing ${d_{\textrm{c}}}/{D_{\textrm{c}}}$, the mode effective index turns out to be lower, on account of an accordingly increased air-filling fraction in the core. Furthermore, the guiding spectrum shifts to lower frequencies, and becomes narrower in concert with the PBG evolution. It should be noted that fibers with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ larger than 0.88 have mode lines outside the PBG, falling into leaky states. On the other hand, the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.82$ supports an undesirable surface mode within the PBG. The bottom right inset has exemplified the intensity profile for this surface mode at the normalized frequency $Df=9\times {{10}^{8}}$, which clearly features a large proportion of field distribution in broken tubes. In contrast, the fundamental mode distributes its field predominately in the core air, which has been exemplified by the top left inset for a fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.85$, working at $Df=6\times {{10}^{8}}$. Notice that, except for the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.82$, only fundamental modes can be observed in the other six fibers. That is to say, fibers with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ from 0.83 to 0.88 can support single mode guidances.

 

Fig. 3. Effective indices ${{n}_{\textrm {eff}}}$ versus the normalized frequency $Df$ for fundamental modes in fibers with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ from 0.82 to 0.88 in increments of 0.01 (solid lines), as well as for a surface mode in the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.82$ (dashed red line). The bottom right inset displays the intensity profile for this surface mode at $Df=9\times {{10}^{8}}$, while the top left inset for a fundamental mode in the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.85$, working at $Df=6\times {{10}^{8}}$.

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Next, we investigate the fraction of power in core for fundamental modes of the fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.83 to 0.88 in increments of 0.01, and the results are shown in Fig. 4. It can be seen that, about the the PBG center, these fibers have their largest fraction of power distributed in core. Over frequencies approaching PBG edges, dramatic falls can be observed evidently, implying leaky occasions for these modes. Furthermore, decreasing the inner diameter raises the mode line, making its center further away from the PBG edges, and thus increases the maximum fraction of power in core. Especially for the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.83$, the maximum fraction of power in core has a value larger than 85 $\%$. From the confinement point of view, a smaller ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ contributes to a tighter bound of the mode. In the meanwhile, it however corresponds to a larger effective index, which indicates a heavier field distribution in solid material, hence inviting a higher absorption loss. Given the fact that the absorption loss generally outweighs the confinement loss for THz guidances, the fiber with a smaller ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ would suffer more total losses.

 

Fig. 4. The fraction of power in core versus the normalized frequency for fundamental modes in fibers with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ from 0.83 to 0.88 in increments of 0.01.

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Since the cutoff definition by PBG edges is unreasonable for a practically finite structure, the fraction of power in core has been previously used to predict the effective guiding spectrum for PBG fibers. For example, the value of 25 $\%$ is suggested to designate the cutoff, and the fraction lower than this value is deemed to be a leaky state [18], however falling short of theoretical bases. Not only that, there is generally not a clear demarcation between the core and cladding in PC fibers, so the fraction of power in core is in fact no more than an estimation. Instead, here we exploit the effective area ${{A}_{\textrm {eff}}}$ of a mode to define its effective guiding spectrum for the proposed THz fiber. The effective area can be worked out from the formula ${{A}_{\textrm {eff}}}={{\left ( \iint {|E{{|}^{2}}\textrm {d}s} \right )}^{2}}/\iint {|E{{|}^{4}}\textrm {d}s}$, which concerns global integrations, thus obviating the need for any demarcations. Figure 5(a) shows the normalized effective area ${{A}_{\textrm {eff}}}/{{D}^{2}}$ as a function of the normalized frequency $Df$ for fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.83 to 0.88 in increments of 0.01. It can be seen that, about the PBG center, these fibers exhibit a small mode effective area, corresponding to a higher fraction of power in core. Near PBG edges, rapid expansions of the effective area can be observed, especially on the lower PBG edge for fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ = 0.86; 0.87; 0.88. Such a rapid expansion can be understood analogous to the behavior of confinement loss. Near the cutoff, both of them will be subject to a sudden change, which should follow an exponential form. It stands to reason that the rapid expansion is closely related to the mode leakage, so we designate the frequency corresponding to the rapidest expansion as the cutoff point. To find out these points, the second derivative of ${{A}_{\textrm {eff}}}/{{D}^{2}}$ is worked out for these fiber modes, with results shown in Fig. 5(b). In this figure, each curve has two peaks marked by open circles, exactly denoting the rapidest expansions. Whereupon, the interval between the two peaks can be defined as the effective guiding spectrum.

 

Fig. 5. (a) The normalized effective area ${{A}_{\textrm {eff}}}/{{D}^{2}}$ for fundamental modes in the fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.83 to 0.88 in increments of 0.01; (b) The second derivatives of ${{A}_{\textrm {eff}}}/{{D}^{2}}$, with their peaks marked by open circles.

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Now we discuss the transmission loss for the proposed THz fiber, which is composed of the effective material loss ${{\alpha }_{\textrm {eff}}}$ and the confinement loss ${{\alpha }_{\textrm {conf}}}$. With regard to the conventional method, the two losses have been separately calculated, and then their algebraic sum results in the total loss. In more detail, the effective material loss is derived from the formula ${{\alpha }_{\textrm {eff}}}={{\alpha }_{\textrm {mat}}}\eta$, where ${{\alpha }_{\textrm {mat}}}$ denotes the absorption loss of bulk solid material and $\eta$ the fraction of power in this material. It should take note that the calculation of $\eta$ requires an additional post-processing step, concerning a double integration over the mode field. Instead, we use another more convenient way to acquire the total loss, in which the material loss is directly factored into mode calculations. First, the bulk material loss in units of 1/m can be deduced from ${{\alpha }_{\textrm {mat}}}=10\ln (10){{\alpha }_{\textrm {dB}}}$, where ${{\alpha }_{\textrm {dB}}}$ in units of dB/cm represents the loss dispersion of Topas and can be worked out through ${{\alpha }_{\textrm {dB}}}=-0.13+0.63f+{{f}^{2}}$, with $f$ measured in THz [19]. Notice the imaginary part of material refractive index $\operatorname {Im}({{n}_{\textrm {mat}}})$ is related to ${{\alpha }_{\textrm {mat}}}=2k\operatorname {Im}({{n}_{\textrm {mat}}})$, where $k$ denotes the propagation constant in vacuum. As a result, we can have $\operatorname {Im}({{n}_{\textrm {mat}}})=5\ln (10){{\alpha }_{\textrm {dB}}}/k$. Then the complex form of material refractive index is directly introduced to calculate the mode effective index. Together with a perfect matched layer imposed on the fiber boundary, the mode calculation can result in mode effective indices having an imaginary part $\textrm{Im}(n_{\textrm{eff}})$. In this way, the total loss ${{\alpha }_{\textrm {tot}}}$ can be easily worked out from ${\alpha _{\textrm{tot}}} = 20k{\textrm{Im}} (n_{\textrm{eff}})/\textrm{ln}(10)$[dB/m], dispensing with an additional post-processing step.

Based on the above proposed method, we have calculated total losses for fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.83 to 0.88 in increments of 0.01, and obtained results are shown in Fig. 6(a). For each value of ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$, the outer diameter $D$ in cladding tubes takes such a value that the effective guiding spectrum is centered at 1 THz. It is worth noting that the obtained results have been verified to be the same as those obtained by the conventional method. Figure 6(a) has shown that these fibers suffer a low total loss, all of which are lower than 0.8 dB/cm for the most part of guiding spectrum. To further provide an insight into the evolution of ${{\alpha }_{\textrm {tot}}}$ against the frequency $f$, Fig. 6(b) gives the confinement losses ${{\alpha }_{\textrm {conf}}}$ on a log scale, which are obtained just by the mode calculation exclusive of the imaginary part $\operatorname {Im}({{n}_{\textrm {mat}}})$. Then the effective material loss ${{\alpha }_{\textrm {eff}}}$ can be derived from ${{\alpha }_{\textrm {eff}}}={{\alpha }_{\textrm {tot}}}-{{\alpha }_{\textrm {conf}}}$, with its dispersion behavior shown in Fig. 6(c). It can be seen that, in the middle of the guiding spectrum, the total loss is closely related to the effective material loss, while the confinement loss offers a negligible contribution, especially for those fibers with a smaller ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$. Hence, when the frequency increases, the total loss goes up in parallel with the effective material loss, which is ascribed to a stronger power convergence on the lossy material as a response to a higher frequency. Moreover, the fiber with a larger ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ suffers fewer total losses, implying a heavier power distribution in the air. However, near the guiding edges, the confinement loss becomes non-negligible, and even outweighs the effective material loss at lower frequencies for the fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.88$ . As a result, the total loss for a larger ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ goes up to be the highest over frequencies near the cutoff.

 

Fig. 6. (a) Total loss ${\alpha _{\textrm{tot}}}$, (b) confinement loss ${\alpha _{\textrm{conf}}}$ and (c) effective material loss ${\alpha _{\textrm{eff}}}$ as a function of frequency for fibers with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ from 0.83 to 0.88 in increments of 0.01. For each fiber, the outer diameter $D$ in cladding tubes takes such a value as to allow the effective guiding spectrum centered at 1 THz.

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In order to provide an optimal design at 1 THz, we have plotted in Fig. 7 the total loss as a function of the structural parameters $D$ and ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$, in form of a colored map. In this figure, the color brightness represents the loss intensity, as displayed by a legend on the right side. For each ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$, the fiber has limited $D$ values, to which the limit is prescribed by the normalized guiding spectrum $Df$ with $f$ = 1 THz. Seeing that the dark-colored regions correspond to lower losses, it stands to reason that the fiber with a larger ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ is more favorable. Then the optimized parameters are figured out to be ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.88$ and $D$ = 523 $\mu$m, which can deliver the lowest total loss 0.3 dB/cm for $f$ = 1 THz. Furthermore, we have studied this optimized fiber in more detail, over its effective guiding spectrum of 0.81-1.14 THz. Obtained results are shown in Fig. 8, including the mode effective index ${n_{\textrm{eff}}}$, total loss ${\alpha _{\textrm{tot}}}$ and GVD ${\beta _2}$. It can be seen that the fiber presents low mode effective indices, indicative of a power distribution predominantly in the air. As a consequence, this fiber suffers low total losses, which are lower than 0.45 dB/cm for the most part of guiding spectrum. Especially at frequencies near 0.9 THz, the fiber exhibits a total loss close to 0.26 dB/cm. In terms of GVD, the second order of dispersion coefficient ${{\beta }_{\textrm {2}}}$ is introduced to evaluate the transmission performance for THz pulses. The dispersion coefficient can be calculated through the formula ${{\beta }_{2}}=\frac {1}{\pi c}\frac {\textrm{d}{{n}_{\textrm{eff}}}}{\textrm{d }f}+\frac {f}{2\pi c}\frac {{{\textrm{d}}^{2}}{{n}_{\textrm{eff}}}}{\textrm{d }{{f}^{2}}}[\textrm{Ps}^{2}/\textrm{cm}]$, with $f$ measured in THz here. Obtained results demonstrate that this fiber features a wide spectrum of nearly zero flat dispersion, with $|{\beta _2}| 1\;\textrm{ps}^2/\textrm{cm}$ over 0.81-1.12 THz, implying a good transmission quality for THz pulses.

 

Fig. 7. The total loss as a function of the structural parameters $D$ and ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$.

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Fig. 8. Transmission characteristics in terms of the mode effective index ${{n}_{\textrm {eff}}}$, total loss ${{\alpha }_{\textrm {tot}}}$ and GVD ${{\beta }_{2}}$, as functions of the frequency $f$ for the optimized fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.88$ and $D = 523\;\mu\textrm{m}$.

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To be more practical, we have examined the bending induced additional loss ${\alpha _{\textrm{bend}}}$ for the optimized fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ = 0.88 and $D = 523\;\mu\textrm{m}$, working at 1 THz. First, the confinement loss of a bent fiber can be worked out from an equivalent straight model with ${n_{\textrm{eq}}}(x,\;y) = {n_0}(x,\;y) \textrm{exp}(s/{R_{\textrm{b}}})$. Specifically, ${n_0}(x,\;y)$ is the cross sectional distribution of refractive index in an originally straight fiber, while ${n_{\textrm{eq}}}(x,\;y)$ represents the distribution of another straight model equivalent to the bent one with radius ${R_{\textrm{b}}}$. Here the $s$ denotes $x$ or $y$, which is subject to the bending direction. Then the bending induced additional loss can be deduced by a subtraction of the loss in the originally straight fiber from that in the bent one. Results of ${\alpha _{\textrm{bend}}}$ versus ${R_{\textrm{b}}}$ are sketched in Fig. 9. It can be found that, no matter the bending direction, the bending induced additional loss can be completely negligible (< ${10^{ - 11}}$ dB/cm) for ${R_{\textrm{b}}}$ > 1 cm. Such a low bending loss can be attributed to the small core and the porous structure. Given the fact that a realistic packaged THz fiber is hard to be bent with so small a radius (${R_{\textrm{b}}}$ = 1 cm), we can say that the present design is highly resistant to the bending state, thereby ensuring the flexibility and stability for THz transmissions.

 

Fig. 9. Bending induced additional loss ${\alpha _{\textrm{bend}}}$ versus the bending radius ${R_{\textrm{b}}}$ for the optimized fiber with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}$ = 0.88 and $D = 523\;\mu\textrm{m}$ at 1 THz.

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Overall, Salient features of the proposed PBG THz fiber can be summarized as follows. First, this fiber can support single mode guidance and is tolerant of bending states, which can be ascribed to the small core size , i. e., only several hundreds of micrometers in the diameter. Second, its mode line is smooth over the transmission spectrum, so is the GVD curve, which is however not the case for ARR fibers. Moreover, the dispersion has shown to be nearly zero flat. To be more specific, we have compared the performance of this fiber to some previous works, as shown in Table 1, concerning the core diameter ${\phi }$, operation bandwidth (BW), total loss ${\alpha _{\textrm{tot}}}$, bending induced additional loss ${\alpha _{\textrm{bend}}}$, GVD $|{\beta _2}|$, and mode content. Notice the data in this table are all from reported simulation results, so as to ensure the fairness. As for fibers in Ref. [27,30,36,37], broader transmission bands have been achieved, each of which can even exhibit a portion of band with lower losses than our proposed fiber. However, all of them have arranged a much larger core diameter, generally several thousands of micrometers, with a view to favoring the ARR guidance. Therefore, these fibers would support multi mode guidances, which has indeed demonstrated explicitly in Ref. [36], while others have no discussion about this problem. Moreover, these fibers should exhibit higher bending losses [30], in comparison to our proposed fiber. Reference [37] has discussed the bending issue, however without giving specific data. By another way, obvious increments in total losses can be observed from their simulated results, on account of the non-negligible bending induced losses. On the other hand, these fibers present serious oscillation phenomenons in mode lines as well as transmission losses. So the GVD has been rarely discussed. Although Ref. [37] has reported a smooth dispersion curve with |${\beta _2}| 1$ < p${\textrm{s}^2}$/cm, it is achieved when the fiber is placed in water. It is worth noting that Ref. [19] has also reported a PBG fiber with single mode guidances, however without the dispersion and bending discussions. Their simulated results have demonstrated total losses below 1.0 dB/cm over 0.78-1.02 THz, with the minimum loss of 0.7 dB/cm at 0.88 THz. In our design, the fiber with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ = 0.88 has total losses below 0.45 dB/cm over 0.81-1.12 THz, with the minimum loss of 0.26 dB/cm at 0.9 THz. As for 0.88 THz, the total loss is 0.32 dB/cm. In comparison, our fiber can gain 0.7-0.32 = 0.38 dB/cm for the transmitted power. So for a length of 30 cm, the loss in our fiber can be 11.4 dB lower at 0.88 THz, corresponding to a transmitted power being 10 times higher. Notice the fiber with ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ = 0.88 is intentionally designed to present the lowest loss at 1 THz. As for ${d_{\textrm{c}}}/{D_{\textrm{c}}}$ = 0.83, we have obtained total losses below 0.98 dB/cm over 0.69-1.31 THz, which completely covers the design of Ref. [19], and the loss at 0.88 THz (0.37 dB/cm) is also smaller. It is exactly the less dispersive quality of the HTs based PBG that contributes to the better performance.

Table 1. Performance comparisons for different fibers.^a

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4. Fabrication discussion

With regard to the fabrication, the draw and stack method can be adopted, which has proven effective in THz PC fibers [30,43]. In particular, cladding tubes of the proposed fiber have outer diameters close to those in [43], and both the two fiber have the same value of $d/D = 0.9$. The challenge for the present design consists in the core region. For a feasible manipulation, some assisting elements can be adopted during the stacking. The stacking elements contains intact tubes, broken tubes, core tubes and assisting rods, as shown in Fig. 10(a). To be more specific, a thick tube is first cut into two sections. One section is drawn to produce a thinner tube with parameters $D$ and $d$, and it is then transversely cut into pieces. The other section is first longitudinally cut into a broken tube with the breach angle being $2\theta = {43.6^\textrm{o}}$, which is then drawn under the same condition as the previous section, aiming to obtain a thinner broken tube with same parameters $D$ and $d$. It is then also transversely cut into pieces. As for the core, thirteen tubes are triangularly stacked and then drawn to achieve the desired tube parameters ${D_{\textrm{c}}}$ and ${d_{\textrm{c}}}$. During the drawing processes, air pressure can be applied to sustain the structure, as used in silica PC fibers. Besides, six thin assisting rods can be prepared also by the drawing and cutting process, which serve to assist the follow-up stacking operation. The rod diameter is geometrically calculated to be $0.2(D + d)$, which ensures that these rods are tangent to the cut sides of broken tubes as well as to the core tubes. In the following, the stacking operation can be performed. As shown in Fig. 10(b), all tubes and rods are first schematically stacked into a jacket. It is worth saying that the insertion of assisting elements (tubes indicated by red circles and rods by yellow circles) can make the stacking more convenient. Moreover, the rods can also aid the aliment of broken tubes. After the stacking, those assisting elements must be pulled out to form the final fiber structure. For the convenience of pulling operation, assisting elements longer than remaining elements should be considered during the previous cutting process, ensuring their prominence for pulling manipulations. It is worth noting that, since the proposed fiber has a self-supporting structure, no glue or fusing is necessary after the stacking [30].

 

Fig. 10. (a) Stacking elements including intact tubes, broken tubes, core tubes, and assisting rods; (b) Stacked profile with inclusions of assisting elements.

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Another fabrication process may be also possible, which has been widely used in silica PC fibers with a lattice of only several micrometers, i. e., the stack and draw method. In contrast, the THz fiber has a much larger size, so it should be much easier to manipulate. Significantly, [19] has already involved the drawing process for the fabrication of a Topas based THz fiber with HHs. The difference is they use the drilling method to make the preform, while ours can use the stacking method. In this method, thick tubes with two different sizes is first prepared respectively for the cladding and core. Besides, broken tubes with the same size as those in the cladding is also prepared. By the same token, assisting elements can be included to aid the stacking process. All the tubes and rods are stacked with a profile as shown in Fig. 10(b), and then these assisting elements are pulled out to obtain the preform. Finally, the preform was placed in a draw tower and heated to above the glass transition temperature of the polymer using an oven. When the preform is softened, it can be pulled into the fiber with the desired parameters. Again, during the drawing process, air pressure can be applied to sustain the structure.

The above mentioned fabrication methods are just preliminary schemes, while more technical details should be obtained in practical experiments. In fact, these two methods both involve the stacking and drawing processes, however with different operation sequences. To figure out which one is more feasible, practical experiments have to be carried out. It is worth saying that the second method may be more favorable in fabricating longer THz fibers. As a matter of fact, the THz fiber is still lying in the exploration stage and requires further developments in the material, structure and fabrication. With the development of fabrication, such as the popular 3D technique, it may be possible to fabricate our design with a sufficient precision in the future.

5. Conclusion

In this paper, we propose a PC structure consisting of HTs. Calculations based on the PWE as well as the FEM have been performed to suggest that, in comparison with HH and TT based PCs, the one proposed in this paper can exhibit a broader and less dispersive PBG. We believe such a structure can inspire widespread designs and applications for optical fibers and devices. Based on the proposed PC cladding, a PBG fiber concerning the THz transmission has been designed, and the full-vector FEM is used to examine its guiding performances. Resultant mode lines have verified the effectiveness of the PBG guidance. Then the fraction of power in core and the normalized mode effective area have been studied. In particular, the second derivative of effective area has been proposed to define the effective guiding spectrum, which would be more reasonable for PBG fibers. As for the total transmission loss, we directly take into account the imaginary part of material refractive index for mode calculations, obviating the need of an additional post-processing step as used in conventional methods. The obtained results demonstrate that the fiber can deliver low losses over a broad spectrum centered at 1 THz. Further, the total loss as a function of structural parameters has been thoroughly studied for the frequency of 1 THz, yielding an optimal design with ${{d}_{\textrm {c}}}/{{D}_{\textrm {c}}}=0.88$ and $D$ = 523 $\mu$m, which can achieve the lowest total loss of 0.3 dB/cm. Then the optimized design has been explored in more detail, and the results demonstrate that it suffers total losses below 0.45 dB/cm and provides nearly zero flat dispersions with $|{\beta _2}| 1\; \textrm{ps}^2/\textrm{cm}$ over 0.81-1.12 THz, indicative of good transmission characteristics for THz pulses. Besides, bending induced additional losses have been worked out, which have shown to be negligible (< ${10^{ - 11}}$ dB/cm) even for a bending radius of 1 cm, thus demonstrating a flexible quality. Finally, we have discussed possible fabrication methods for the proposed THz fiber.