1. Introduction

Intensive research has been conducted to develop fibers and waveguides to transmit infrared radiation for applications including medical treatments, sensing, and spectroscopy etc. Dielectric-coated hollow metallic fibers (HMFs) have been extensively studied as low-loss transmission waveguides for lasers in the visible and infrared (including terahertz) wavelength regions [1–4]. For the simplest HMF without a dielectric coating, the fundamental transverse-magnetic (TM) mode suffers from a loss which is several orders higher than that of the fundamental transverse-electric (TE) mode. The high loss experienced by the TM modes is generally regarded as the main limiting factor that prevents the long-distance transmission of light in such bare HMFs. Introducing a dielectric coating helps to significantly reduce the transmission loss in this kind of waveguides [1–4]. However, in such a dielectric-coated HMF, the loss of the TM mode heavily depends on the thickness of the dielectric coating as a result of a change in the resonance condition of the light in the dielectric layer [1]. It follows that for a particular dielectric-coated HMF, the propagation loss is dependent on the frequency of the guided light. Yan and Mortensen recently have proposed a HMF for guiding infrared light by incorporating a metal-wire based metamaterial coating in replacement of the dielectric coating [5]. Their proposed metamaterial fiber is able to guide the TM-polarized light with a relatively low loss, which is insensitive to the coating thickness or the working frequency. The broadband operation is due to the fact that the TM light is evanescent in the metamaterial coating.

In Ref. [5], it is found that a very thin metamaterial coating with even one layer of metallic wires can significantly reduce the propagation loss of the TM₀₁ mode. Inspired by the finding, in this study we propose a HMF with longitudinally corrugated inner surface with a corrugation period much less than the operation wavelength. The corrugated inner metal surface can be regarded as a thin layer of metamaterial in close analogy to the thin metal-wire metamaterial deployed in [5]. We theoretically confirm that the introduction of a corrugated metal surface can similarly reduce the propagation of the TM₀₁ mode. Compared with the metamaterial fiber discussed in Ref. [5], our proposed fiber structure avoids the isolated micrometer-scaled metal wires; therefore, it is much more practical from the fabrication point of view.

2. Fiber structures

The simplest cross-section of our proposed HMF with a corrugated inner surface is schematically shown in Fig. 1(a) . In practical realizations, the spacing between corrugation teeth can be filled with some dielectric material; the corresponding schematic diagram is shown in Fig. 1(b). Compared with the metamaterial fiber design in Ref. [5], where the metamaterial is formed by inserting metal wires in a dielectric material, the structures shown in Figs. 1(a) and 1(b) are much easier to realize [6]. Notice that strict periodicity in surface corrugation is not required as long as the average corrugation period is kept much smaller than the free-space wavelength. However for ease of analysis, in this work we focus on fibers with periodic corrugations. Referring to Fig. 1(b), Λ denotes the period of the corrugation, D denotes the depth of the corrugation (or the thickness of the corrugation layer) and d denotes the width of the corrugation teeth. The metal filling fraction for the corrugation layer is defined as f _m = d/Λ . In Fig. 1(c), we also illustrate the traditional IR fiber which is basically a HMF with a dielectric coating. We will compare our fiber design with the traditional design in the following investigations.

 

Fig. 1 (a) The cross-section of a HMF with a corrugated inner surface. (b) The cross-section of a corrugated HMF with dielectric fillings. (c) The cross-section of a conventional HMF with a dielectric coating. The golden region is metal and the light blue regions represent some dielectric material.

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We focus on the wavelength of CO₂ lasers, i.e. 10.6µm. This wavelength is useful for thermal sensing, medical surgeries and general material cuttings. The metal is chosen as silver, which has a permittivity ε = −2951 + 1654i [7] at the mentioned wavelength. The filling dielectric could be Zinc Selenide (ZnSe) or Arsenic Selenide (As₂Se₃) with a refractive index of ~2.5.

3. Numerical results and analysis

HMFs are most commonly used for high-power IR light delivery applications. In such applications, a large core is desired to accommodate a high beam power. Correspondingly, a large number of modes can be guided in such a hollow-core fiber. Despite the large number of modes, if a fiber is not severely bent, its guiding characteristics is in general determined by a few lowest-order modes. HE_11, TE₀₁, and TM₀₁ modes are the particular low-order modes one should pay attention to. Among these modes, TM₀₁ mode is the most lossy one; TE₀₁ mode is the least lossy one; and the HE₁₁ mode has a loss between those of the TE₀₁ and TM₀₁ modes [8]. In practice, the HE₁₁ mode is recognized as the most useful mode for delivering a laser beam owing to the fact that both its profile and the polarization match quite well with those of a common laser beam. However from the theoretical perspective, in the case of HMFs one should really concentrate on engineering the TM₀₁ mode. For a conventional HMF without a dielectric coating or surface corrugation, the propagation loss of the TM₀₁ mode can be orders of magnitude higher than that of the TE₀₁ mode. Reducing the propagation loss of the TM₀₁ mode will naturally reduce the loss of the HE₁₁ mode, since the HE₁₁ mode consists both TE and TM field components. A reduction in the HE₁₁ mode loss as a consequence of a reduction in the TM₀₁ mode loss was numerically observed also in [5]. In this paper we will therefore solely discuss the loss characteristics of the TM₀₁ mode and investigate the effects of the structural parameters on the loss. From another perspective, our emphasis on the guidance characteristics of the TM₀₁ mode is also motivated by the valuable properties of such a radially-polarized beam, such as its ability to be focused into a tighter sport size [9], not to mention its distinct polarization state. Lasers directly generating such a beam have been devised [10]. Hence studies on waveguides optimized for guiding TM light are both important and urgent.

We employ the commercial COMSOL Multiphysics package based on the finite element method to investigate the guided mode properties of the proposed structures. The core diameter is chosen at 700µm and a corrugation depth of 1.0 µm is used. The filling dielectric has an index n _f = 2.5. Figure 2 shows the guided TM₀₁ mode properties including its loss and effective mode index (n _eff) as a function of the metal filling fraction f _m, examined at a number of corrugation periods. Notice that when f _m = 1 the corrugated HMF becomes the traditional HMF without a dielectric coating; in the case f _m = 0 the fiber becomes a traditional dielectric-coated HMF. In general, we can see that a low metal filling fraction will lead to a relatively low-loss propagation of the TM₀₁ mode. At f _m = 0.1 for Λ = 2.16 µm, the mode loss is about 50 times smaller compared to the fiber at f _m = 1 (i.e. a traditional HMF without coating). The effect of the corrugation period Λ is also manifested in Fig. 2. One can see that at a fixed f _m, the TM₀₁ mode loss is reduced when Λ increases from 0.54 µm to 8.64µm. However when Λ increases to 17.28µm the TM₀₁ mode loss gets higher, with a loss peak arising at f _m≈0.25. When the corrugation period becomes large enough, resonant modes are supported by the dielectric filling, which in turn cause the high loss of the TM₀₁ mode. In Fig. 2, one should expect that the simulated loss and n _eff curves should unite at f _m = 0 since they all correspond to an identical traditional dielectric-coating HMF. In the same figure, we have superimposed the loss and n _eff curves for the TM₀₁ mode guided in the same corrugated HMF but without dielectric fillings. In this case we have only examined the case of Λ = 2.16µm; other Λ values are also examined but without significant effect on the loss. By comparing the loss curves for HMFs with and without dielectric fillings, we conclude that a dielectric filling can significantly reduce the propagation loss of the TM₀₁ mode.

 

Fig. 2 (a) Propagation loss and (b) n _eff of the TM₀₁ mode as a function of the metal filling fraction at various corrugation periods. R = 350µm, D = 1 µm, n _f = 2.5.

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The above analysis has revealed that the refractive index of the filling dielectric plays an important role on the propagation loss of the TM₀₁ mode in the proposed fiber. Naturally we would like to know how the TM₀₁ mode loss depends on the refractive index of the dielectric filling. Our calculation result is shown in Fig. 3 . First we set the corrugation depth at D = 0.5μm. It is observed from Fig. 3 that the TM₀₁ mode loss decreases drastically when the refractive index of the filling material increases from 1 (without filling). When the refractive index increases further from 2.5 to 5, the TM₀₁ mode loss changes little. This study explains the necessity of a dielectric filling, and tells that preferably one should use a filling with index no less than 2.5 (therefore there exists a large degree of freedom in choosing the filling material). However, care should be taken to avoid deploying simultaneously a large filling refractive index and a large corrugation depth. Indeed, from Fig. 3 when we increase the corrugation depth to 1µm or 2µm, some loss peaks arise at large refractive index values of the filling material. The high losses are again due to the resonant modes supported by the filling materials with a relatively large transverse area. Since in this work we concentrate on a filling material with an index of 2.5, it is appropriate to use a corrugation depth of 2µm, provided other parameters are fixed as in the caption of Fig. 3. This choice of corrugation depth is also advantageous for maintaining a low propagation loss for the TE₀₁ mode, which will only be confirmed in Fig. 4 .

 

Fig. 3 Propagation loss of the TM₀₁ mode as a function of the refractive index of the filling material. R = 350µm, f _m = 0.1, Λ = 2.16µm.

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Fig. 4 Propagation losses of the TM₀₁ and TE₀₁ modes as the coating thickness varies for both the conventional HMF and the corrugated HMF (R = 350µm, f _m = 0.1, Λ = 2.16µm, and n _f = 2.5).

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The propagation losses of the TE₀₁ and TM₀₁ modes in our proposed HMF as functions of the core diameter are shown in Fig. 5 . Refer to the figure caption for the fiber’s material and geometrical parameters. For comparison, we also superimpose the TE₀₁ and TM₀₁ mode losses for a traditional HMF without a dielectric coating in the same figure. It can be seen that both losses decrease with the core size. By comparing the loss curves for the two types of fibers, we can see that the TM₀₁ mode loss can be easily reduced by 50~100 times at a relatively large core size with the corrugated fiber structure. The TE₀₁ mode loss of the proposed HMF is slightly increased compared to that of the traditional HMF; however the loss value is still at a very small value (~0.1dB/m), which should not deteriorate light propagation in the fiber in a noticeable manner. By using a curve fitting procedure we find that for the corrugated HMF, its TE₀₁ mode has a loss value that follows the 1/R ³ dependence; while the loss of its TM₀₁ mode decreases with respect to R at even a greater speed.

 

Fig. 5 Propagation loss of the TE₀₁ and TM₀₁ modes as a function of core diameter for both the proposed HMF and the traditional HMF. f _m = 0.1, Λ = 2.16µm, D = 2.0µm, n _f = 2.5.

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The traditional technology of using a dielectric coating on top of the metal cladding works well for reducing the propagation loss of the TM₀₁ mode. However, the propagation loss is quite sensitive to the thickness of the dielectric coating. To confirm this, we carried out a simulation for a conventional HMF with a core diameter of 700µm and a coating refractive index of 2.5. Figure 4 shows the propagation losses of the TM₀₁ and TE₀₁ modes when the coating thickness of the traditional HMF increases. The simulation reveals perodic loss variations for both the TM₀₁ and TE₀₁ modes. Moreover, the loss peaks for the TM₀₁ and TE₀₁ modes “discord” with each other by half of a period, which suggests that one can only have a narrow low-transmission spectral window for a fixed fiber structure for guiding a hybrid mode. In the same figure, we also show the losses for the TM₀₁ and TE₀₁ modes in our proposed corrugated HMF as functions of the corrugation layer thickness (or corrugation depth). The following parameters are chosen in the simulation: R = 350µm , f _m = 0.1, Λ = 2.16µm, and n _f = 2.5. It is noticed that our proposed fiber design does not suffer from a periodic variation in the TM₀₁ mode loss. Therefore the proposed HMF can serve as an efficient waveguide for delivering TM-polarized IR light with a broadband spectrum. We have however noticed that in the proposed HMF, the loss of the TE₀₁ mode does change periodically with respect to the thickness of the corrugation layer. Nevertheless, due to the fact that the TM₀₁ mode is insensitive to the coating thickness, the achievable low-loss spectral region can be twice as large as that of the conventional HMF for guiding a hybrid mode.

4. Conclusions

In conclusion, we have proposed a type of practical hollow-core metallic fibers with corrugated inner surface for delivering broadband IR light with relatively low loss. The effects of the metal fraction, the corrugation period and the corrugation depth to especially the TM₀₁ mode propagation loss have been numerically investigated. Our simulation results show that the TM₀₁ mode loss in the proposed HMF can be reduced easily by about 50 times compared to that in a traditional HMF without a dielectric coating. Compared to a traditional HMF with a dielectric coating, our proposed fiber has similar minimum loss levels for both the TM₀₁ and TE₀₁ modes. The key advantage of our proposed HMF fiber is that the propagation loss of the TM₀₁ mode is insensitive to the corrugation layer width, which can potentially double the low-loss spectral window for a fixed fiber geometry for guiding a hybrid mode compared to the conventional HMF with a dielectric coating.