1. Introduction

Terahertz (THz) radiation [1–3] has attracted increasing interest over the past several years due to its big potential for the applications such as imaging [4–6], spectroscopy [7–9], communications [10] and sensing [11]. So far, numbers of THz technologies have been developed to achieve THz sources, THz waveguides and other THz devices [12–16]. Among them, low-loss THz waveguides which are of great importance for the practical THz application have been considered as a big step towards the compact and robust THz systems. However, THz radiation is strongly absorbed in most kinds of materials, which gives a challenge to achieve low-loss THz waveguides. Fortunately, the propagation of THz wave in dry air is almost lossless which indicates a method to achieve some promising low-loss THz waveguides with large fractional power in the dry air. Several polymer fibers with complex cross sections (e.g. employing arrays of circular or rectangular air holes) have been proposed to reduce the loss for THz guiding [17–21]. Those research results indicate that the polymer THz fiber has risen up to be one of the most important candidates of THz waveguides.

Besides the polymer fibers with complex cross sections, for which complex and expensive fabrication system is usually needed, a simple solid polymer fiber (subwavelength-diameter plastic wire) for THz guiding has also been proposed [22], which has the ease of fabrication. In this paper, a kind of novel low-loss THz waveguide, polymer tube, is proposed. We comparatively investigate both the proposed polymer tube and the solid polymer fiber, which shows that the proposed polymer tube is with the better loss property than the solid polymer fiber and other advantages such as the better confinement, the ease of fabrication, large mode area and so on.

2. Structures and properties

Figure 1 shows the cross sections of (a) the proposed polymer tube and (b) the solid polymer fiber which we will investigate in this paper. The cross section of the polymer tube is a ring structure with an internal diameter (d) and an external diameter (D). The cross section of the solid polymer fiber is a circle structure with a diameter ($D\text{'}$). The heavy gray area in the cross sections show the polymer material which can possibly be one of the previously reported materials with low absorption in THz region (such as Polypropylene (PP), Polytetrafluoroethylene (PTFE), Polypropylene-high density (HDPE) and the cyclic olefin copolymer Topas) [21,23]. Since the refractive indexes of the most polymer materials for THz guiding are around 1.5 in the interested frequency region, to simplify analysis, in this paper the refractive index of the polymer material and the surrounding air is assumed to be 1.5 and 1, respectively. For the calculation in this paper, the typical parameters of the polymer tube and the solid polymer fiber are $D=300\mu m$, $d=250\mu m$ and $D\text{'}=165.8312\mu m$, which means the polymer tube and the solid polymer fiber have the same material area in the cross sections.

 

Fig. 1 (a) Polymer tube’s cross section with a ring structure and (b) solid polymer fiber’s cross section with solid cirlce structure.

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We employ a full-vector finite-element method (FEM) [24] and anisotropic perfectly matched layers to investigate the guided modes of the polymer tube and the solid polymer fiber. Figure 2(a) shows the effective indexes of the fundamental modes of the polymer tube (black curve) with the external/internal diameters ($D=300\mu m$/$d=250\mu m$) and the solid polymer fiber (red curve) with the diameter ($D\text{'}=165.8312\mu m$) as a function of the THz wave frequency. Figure 2(b) shows the mode area of the fundamental modes of the polymer tube (black curve) and the solid polymer fiber (red curve) as a function of the THz wave frequency. Inset shows the mode profiles of the power flow of the fundamental mode of the polymer tube for the THz wave frequencies of $\nu =0.6THz$,$\nu =0.75THz$, $\nu =1.0THz$, $\nu =1.2THz$, and $\nu =1.5THz$, respectively. In the most part of the calculated THz wave frequency region, the polymer tube has a much lower effective index than the solid polymer fiber. This can be understood due to the fact that the air core of the polymer tube ensures much larger part of the THz wave propagating in the air area, which essentially contributes the low-loss THz guiding of the polymer tube as discussed hereinafter. The polymer tube also has a larger mode area of the fundamental mode since part of the THz wave is well confined in the air core, which will be good for THz wave coupling from the free space to the polymer tube. For the THz wave frequency around or smaller than $0.5THz$, the effective indexes of both the polymer tube and the solid polymer fiber tends to be 1 and the mode area dramatically increases, which indicates the weak confinement of the THz wave.

 

Fig. 2 (a) Effective index of the fundamental modes of the polymer tube (black curve) and the solid polymer fiber (red curve) as a function of the THz wave frequency. (b) Mode area of the fundamental modes of the polymer tube (black curve) and the solid polymer fiber (red curve) as a function of the THz wave frequency. Inset shows the mode profiles of the power flow of the fundamental mode of the polymer tube for the THz wave frequencies of $\nu =0.6THz$, $\nu =0.75THz$, $\nu =1.0THz$, $\nu =1.2THz$, and $\nu =1.5THz$, respectively.,

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We also investigate the power fractions of the modal power in polymer material, air core and air cladding of the proposed polymer tube, which are defined as:

 

Fig. 3 (a) Power fractions of the modal power in polymer material [black (red) solid curves with hollow circles for polymer tube (solid polymer fiber)], air-cladding [black (red) dotted curves with hollow circles for polymer tube (solid polymer fiber)], and air-core (black dashed curve with hollow triangles for polymer tube) of the fundamental modes of the polymer tube and the solid polymer fiber. (b) Relative absorption loss of the solid polymer fiber (red curve with hollow circles) and the polymer tubes with the same external diameter of $D=300\mu m$and the different internal diameters of $d=225\mu m$ (blue curve with hollow circles), $d=250\mu m$ (black curve with hollow circles), and $d=275\mu m$ (green curve with hollow circles).

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The different power distributions of the modal power result in the different relative absorption losses of the polymer tube and the solid polymer fiber, which, according to a perturbation theory [25], can be expressed by

Figure 4(a) shows the mode profile of the power flow of the fundamental mode of the solid polymer fiber with the diameter ($D\text{'}=165.8312\mu m$) and the polymer tube with the same external diameter of $D=300\mu m$and the different internal diameters of $d=225\mu m$, $d=250\mu m$, and $d=275\mu m$ at the THz wave frequency of $\nu =1.0THz$. Figure 4(b) shows the normalized power flow distribution of the polymer tubes and the solid polymer fiber with the same parameters mentioned above in the radial direction. THz wave is well confined in the material area for the solid polymer fiber resulting in a large relative absorption loss and a small mode area. The polymer tube particularly with the internal diameters of $d=275\mu m$can achieve smaller relative absorption loss because large part of the THz wave is confined in the air core.

 

Fig. 4 (a) Mode profile of the power flow of the fundamental mode of the polymer tubes and the solid polymer fiber when the THz wave frequency is $\nu =1.0THz$. (b) Normalized power flow distribution of the polymer tubes and the solid polymer fiber in the radial direction.

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As an example, we show the experimentally measured (red solid curve) and theoretically calculated (black dashed curve) effective indexes of the fundamental modes of a PTFE tube with the external/internal diameters ($D=1200\mu m$/$d=600\mu m$) in Fig. 5 . Insets show (a) the reference THz signal and (b) the measured THz signal after the the PTFE tube in the time domain. The index of the bulk PTFE is about 1.44 in the calculated THz frequency region as reported in Ref [26]. Low index of the fundamental modes of the PTFE tube is observed in the low frequency region (less than $0.5THz$), which indicates low-loss property can be achieved when the operation wavelength is comparable to the external diameter of the polymer tube.

 

Fig. 5 Experimentally measured (red solid curve) and theoretically calculated (black dashed curve) effective indexes of the fundamental modes of the PTFE tube with the external/internal diameters ($D=1200\mu m$/$d=600\mu m$). Insets show (a) the reference THz signal and (b) the measured THz signal after the the PTFE tube in the time domain.

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3. Discussion and conclusion

In general, increasing the power fraction of the mode power in the air area of the polymer fiber is an effective approach to achieve low-loss THz waveguides, since the air is transparent to THz wave. A solid polymer fiber which is easy for fabrication is one of good candidates of the low-loss THz waveguides since large power fraction of the mode power in the air area can be achieved. However, this results in weak confinement of the guided THz wave and makes the guided THz wave susceptible to the surrounding environment since a large fraction of the guided power can be readily coupled into radiation modes. Furthermore, as a result of weak confinement, the guided modes within these polymer fibers suffer strong bend loss [17]. The proposed polymer tube which is as simple as the solid polymer fiber but offers an additional air part inside the tube, which is good to both reduce the relative absorption loss and to achieve better confinement (trapping power inside the tube). Moreover, the mode area of the polymer tube is larger than the polymer fiber with solid core of the same material area. Although there is still a part of mode power in the air cladding area (outside the polymer tube), the propose polymer tube can be used as a practical low-loss waveguide with better loss property and confinement property than the solid polymer fiber.

In conclusion, a polymer tube is proposed as a kind of novel low-loss THz waveguide. Both the polymer tube and a solid polymer fiber have been comparatively investigated. The effective indexes, mode area, power fraction, relative absorption loss and mode profile of the polymer tube and the solid polymer fiber have been presented in the paper. Simulation results show that the proposed polymer tube exhibits better loss property and confinement property than the solid polymer fiber. A PTFE tube for THz guiding has been experimentally demonstrated.