1. Introduction

Hybrid optical fibers, sometimes called multi-material fibers, incorporate novel materials such as semiconductors and metals and enable the integration of sophisticated photonic and electrical functionality directly into the optical fibers [1, 2], thus enhancing the performance of fibers beyond light guidance. If the included structures have (at least in one dimension) feature sizes much smaller than the operating wavelength, an effective medium, or metamaterial (MM) is formed, which can possess vastly different optical properties to that of the individual constituent materials [3]. In the case of dense arrays of subwavelength-size metallic wires [4], fiber-based MMs can form indefinite metamaterials (iMMs) [5], i.e., effective anisotropic materials in which the tangential and longitudinal components of their permittivity tensor do not possess the same signs. Currently, metal nano- and micro-wires are routinely integrated into optical fibers (e.g., gold or gold-based alloys in silica [6,7], or indium in polymer [8]) either by direct-drawing [9,10] or post-processing [11,12]. By tuning the sizes of these subwavelength features, it is possible to reach optical properties beyond that of conventional step index fibers. In recent years, a number of groups have investigated fiber-based MMs [8,10,13–16], which can exhibit either negative electric permittivity with sub-wavelength metallic wires [8,10] or negative magnetic permeability via slotted-cylinder metallic resonators [17–19] with practical applications ranging from sub-wavelength terahertz imaging [20] to low-loss waveguiding in the infrared [14].

The dispersion equations and guidance conditions of hollow (air) core fibers with iMM cladding have recently been derived [21,22], showing that sub-wavelength guidance within the THz and GHz domain is possible [23]. The analysis considered core sizes comparable to (and smaller than) the operation wavelength neglecting material losses, which, however, strongly influence the modal attenuation and discrimination [24]. No explicit comparison to realistic structures containing discrete sub-wavelength units (or âǍIJmeta-atomsâǍİ), such as wires and split resonators, was presented.

In this contribution, we investigate the guidance of light inside of hollow-core iMM cladding fibers with large air cores (several hundred micrometers âǍŞ similarly to commercially available hollow-core bandgap fibers [25]) for low-loss guidance in the far-infrared (λ = 10.6µm, where metals have substantially lower losses than in the visible) due to the important applications in material processing and medicine [25–27]. Starting from the formalism presented in [21], and following a straightforward Scattering-parameter (or S-Parameter) retrieval procedure [28], we show that such fibers possess loss discrimination properties not found in hollow core waveguides with conventional isotropic claddings [e.g., glass-tube (i.e., capillary) or metal-tube waveguides]. Specifically, we find that the geometry can be tailored such that the linearly polarized HE₁₁ mode possesses the lowest loss, in contrast to hollow metal or dielectric tubes, where the TE₀₁ mode dominates (see for example [29] and references therein). Results obtained by solving the anisotropic Eigenvalue equation with retrieved parameters are in agreement with finite element simulations considering the full structure. Additionally, we show that it is possible to design an iMM cladding fiber where the radially polarized mode possesses the lowest loss, a situation which is not commonly found in other types of fiber structures. These results thus pave the way for the design of metamaterial hollow-core fibers with novel guidance properties, particularly at mid-IR wavelengths.

2. Device operation

The device under consideration is defined by an air-core fiber of diameter D = 2R (R: radius) with a iMM cladding formed by an array of sub-wavelength gold wires (d: wire diameter, a: centre-to-centre distance between two wires) in a silica background [Fig. 1(a)]. In all calculations, we take the separation between the centre of the wire and the inner edge of the silica wall to be a/2, which is always much smaller than the wavelength considered here (a < λ/5,λ = 10.6µm). Unless otherwise stated, we consider three layers of radially aligned wires, where a is the azimuthal separation between wires in the first layer, and also the center-to-center separation in the radial direction. The cladding is thus a concentrically arranged iMM which can be well-approximated as possessing an anisotropic diagonal permittivity and permeability tensors $\overline{\overline{\epsilon }}={\epsilon }_0\left\{{\epsilon }_r,{\epsilon }_{\varphi },{\epsilon }_z\right\}$ and $\overline{\overline{\mu }}={\mu }_0\left\{{\mu }_r,{\mu }_{\varphi },{\mu }_z\right\}$, respectively [Fig. 1(b)], with the the radial and azimuthal tensor components being identical (ε_r = ε_ϕ = ε_t and µ_r = µ_ϕ = µ_t).

 

Fig. 1 (a) Concept of the proposed iMM hollow-core fiber. Left: air-core fiber with diameter D = 2R with densely packed sub-wavelength gold wires embedded in silica defining the iMM cladding. Inset: square unit cell (diameter d, pitch a) used for the parameter retrieval. Right: equivalent metamaterial hollow-core fiber with permittivity tensors as defined in the text. (b) Sketch of the parameter retrieval procedure and the corresponding unit cell. Permittivity and permeability tensors are retrieved from the transmitted and reflected fields at the input/output ports boundaries using FE simulations. The unit cell is located in air and surrounded by periodic boundary conditions.

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2.1. Modal equations

A detailed derivation of the modal dispersion equations and guidance conditions of iMM cladding hollow-core fibers is found in [21], here we present the most important results relevant for this contribution. The Eigenvalue equation to be solved is given by

2.2. Metamaterial cladding parameter retrieval

The dielectric properties of the iMM cladding are investigated using a parameter retrieval procedure described in [28] [Fig. 1(b)]. Using the constituent material properties at λ = 10.6µm (gold: ε_gold = −3866.3 + 1711.9i [30], silica: ε_silica = 4.6861 + 0.23483i [31]) we retrieve the complex permittivity and permeability tensors of the cladding from the S-parameters [28] as a function of wire diameter d and inter-wire spacing a, using a 2D finite element (FE) solver (COMSOL). Figure 2 shows the real and imaginary parts of ε_z and ε_t, obtained by considering the electric field polarized parallel (ε_z) and perpendicular (ε_t) to the gold wires, respectively. The permittivity tensor is highly anisotropic [4], in particular ℜe(ε_z) varies over two orders of magnitude. Furthermore, ε_z and ε_t suggest a predominantly metallic- (ℜe(ε_z) < 0) and dielectric- (ℜe(ε_t) > 0) character respectively, with exceptions at low- and high- filling fractions, where ε_z and ε_t reveal a mixed dielectric and metallic character, respectively. Compared to $\overline{\overline{\epsilon }}$, the relative magnetic permeability tensor $\overline{\overline{\mu }}$ only varies comparatively weakly in this parameter range (|ℜe(µ_t)| = 1.24 − 2.23, |ℜe(µ_z)| = 0.1 − 1.0, whereas |ℜe(ε_t)| = 0.01 170, |ℜe(ε_z)| = 0.003 − 1576) and thus is not shown. To exemplify the various guidance properties, we choose three combinations of a and d [marked by the colored points in Fig. 2 (MM1, MM2 and MM3)] leading to the permittivity and permeability values summarized in Tab. 1.

 

Fig. 2 Pitch/diameter dependence of the different components of the retrieved permittivity tensors (gold wires, silica background, λ = 10.6µm) (a) ℜe(ε_z), (b) ℜe(ε_t), (c) ℑm(ε_z), (d) ℑm(ε_t). Coloured points in (a)–(b) number the example iMM cladding parameters discussed in the text and shown in Tab. 1.

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Table 1. Summary of metamaterial parameters.

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2.3. Modal discrimination

We now investigate the properties of the three lowest order modes of the hollow-core MM-clad fibers for various geometric parameters. We consider fibers with large core diameters (D > 100µm, corresponding to several tens of wavelengths), which are inherently multi-mode and have modal losses in the range of dB/m, which is sufficient for many practical applications. We consider only the most important fundamental- and higher-order modes, namely the HE₁₁, TM₀₁ and TE₀₁ modes, which are typically relevant in real-world applications, and analyze how their attenuation varies for fibers possessing different iMM claddings. It should be noted that the modes in such large core diameter hollow-core fibers have comparable ℜe(n_eff), slightly smaller than unity. Consequently, we will present only the corresponding ℑm(n_eff), which determines modal attenuation and thus relative mode discrimination.

We first present the modal attenuation of (isotropic) hollow tubes of the individual constituent materials, as a function of D. In the case of a hollow gold tube [Fig. 3(a)], the lowest-loss mode is the TE₀₁ mode, with the other two modes having ℑm(n_eff) orders of magnitude larger, as a result of the small field penetration of the TE₀₁ mode into the metal [24, 32]. Even in the case of a silica tube [Fig. 3(b)], the TE₀₁ is the lowest-loss mode, but its loss is comparable to that of the HE₁₁ mode, in contrast to the metal tube situation. In both cases, the TM₀₁ is the mode with the highest loss. If we then consider a iMM cladding with large metal filling fraction, (MM1, ℜe(ε_z) < 0 and ℜe(ε_t) < 0, see Fig. 2), we find similar properties. In this case, both axial and transverse permittivity have metallic-like behaviour, and the relative modal discrimination behaviour is comparable to that of a metal tube, whereby the TE₀₁ has the lowest loss. However, comparing Figs. 3(a), 3(b) and 3(c), we see that in the latter case the HE₁₁ and TM₀₁ modes have lower loss, and the attenuation properties of the TE₀₁ MM1 mode lies somewhere between its purely isotropic counterparts. Note that the numerical solutions of Eqs. (1)–(3) using the retrieved parameters of Tab. 1 [Figs. 2(c) and 2(d), solid lines] are in good agreement with the FE simulations using the discrete wire structure [Figs. 2(c) and 2(d), circles], confirming the viability of using retrieved metamaterial parameters for iMM cladding hollow core fiber designs.

 

Fig. 3 Modal attenation [ℑm(n_eff) and loss] vs. air core diameter for the three lowest-order modes in hollow-core fibers possessing claddings formed by (a) gold, (b) silica, (c) MM1 and (d) MM2 as defined in Tab. 1. Solid lines: numerical solutions of Eqs. (1)–(3). Circles: results obtain by FE modeling. Solid lines in (c), (d) use the permittivity tensors shown in Tab. 1. Also shown are the normalized axial Poynting vectors for the three modes with claddings formed by (e) MM2, D = 600µm and (f) MM3, D = 20µm (numbers below the repective row indicate the scale bar). Note that in (f) the TM₀₁ mode has the lowest loss. Arrows indicate the direction and magnitude of the electric field at a fixed point of time.

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In the case of an iMM cladding, (MM2, ℜe(ε_z) < 0 and ℜe(ε_r) > 0, see Fig. 2), the situation is radically different [Fig. 3(d)]. Most remarkably, the HE₁₁ mode now possesses the lowest loss [red points and curve in Fig. 3(d)], whereas the TE₀₁ mode reveals the largest loss. This can be qualitatively understood from the larger propagation constant of the HE₁₁ mode compared to all other modes, leading to a more grazing incidence (i.e., smaller incidence angle) onto the iMM cladding assuming a ray-type picture. One particularly unusual aspect is that in the case of the iMM hollow-core fiber [MM2, Fig. 3(b)] the HE₁₁ and TM₀₁ modes both possess ℑm(n_eff) values which are lower than in hollow tubes composed of the individual constituent materials [Figs. 3(a) and 3(b)]. Also at this point FE calculations (circles) are in agreement with calculations using Eqs. (1)–(3) (solid lines), confirming the feasibility of using retrieved parameters as a design pathway for hollow core iMM fibers. For completeness, Fig. 3(e) shows the axial Poynting vector profiles of the modes under consideration for a MM2 clad fiber, D = 600µm. Note that for D = 600µm at least 99.999% of the power is within the air core for all modes considered, making this hollow-core fiber geometry well-suited for high-power applications.

To fully illustrate the discrimination flexibility of iMM-HCFs, we investigate whether a material and geometry combination exists that enables the radially polarized (TM₀₁) mode to possess the lowest loss of all modes. A preliminary parameter search shows that this occurs for a fiber with D = 20µm and a MM3 cladding (d = 0.1µm, a = 0.2µm, see Fig. 2). In this configuration [Fig. 3(f), one-wire iMM layer], the TM₀₁ mode has significantly lower loss than the other supported modes, which confirms that our structure uniquely allows to adjust the modal discrimination.

2.4. Geometry dependence

As a further analysis, we now consider how changes in wire diameters affect the modal attenuation properties of the fibers presented thus far. For this purpose, we consider the largest wire separation examined (a = 2µm), and vary the wire diameter d, for a fixed core diameter D [Fig. 4(a)]. The modal behaviour transitions from a low-loss dielectric-type HE₁₁ towards a low-loss TE₀₁ mode [light green and blue regions in Fig. 4(a)], both for FE calculations with discrete wires [circles in Fig. 4(a)] and retrieved parameter model [solid lines in Fig. 4(a)]. Notice in particular the signs of the retrieved axial- and transverse-permitivitties change, as indicated by the different colored regimes in Fig. 4(a). It is important to note that within the iMM domain, the TE₀₁ mode has the highest loss, which indicates that the iMM cladding allows to adjust the modal loss by balancing between the penetration of the electromagnetic field of the different modes with the imaginary parts of the iMM tensor components. Interestingly, the HE₁₁ is the lowest loss mode even when all elements of the permittivity tensor are positive, which is contrast to what was observed for the case of a simple silica capillary presented earlier [see Fig. 3(b)]. To understand this further, we consider an entirely hypothetical case, yielding a qualitative explanation of the observed behavior. Assuming no magnetic reponse $\overline{\overline{\mu }}\sim {\mu }_0\overline{\overline{I}}$, we consider the case where ε_t = ε_silica and investigate the modal loss variation when changing ℜe(ε_z), keeping a constant ℑm(ε_z) = 1i. We observe that while the HE₁₁ mode possesses the lowest loss for ℜe(ε_z) ≪ 0, a transition occurs in the vicinity of ℜe(ε_z) ~ ℜe(ε_silica), causing its loss to increase above that of the TE₀₁ mode – notice that the TE₀₁ modal properties are not dependent on ε_z, resulting from the vanishing longitudinal electric field component of TE-modes in general [Eq.(2)]. Considering ε_z = ε_silica and varying ℜe(ε_r) for a constant ℑm(ε_r) = 1i, we see that the HE₁₁ and TM₀₁ modes only slightly depend on ℜe(ε_r), and that the TE₀₁ mode has the lowest loss across most of the parameter space, except for a region close to ℜe(ε_r) ~ ℜe(ε_silica).

 

Fig. 4 (a) Modal attenuation [ℑm(n_eff) and loss] vs. wire diameter d for a constant air-core diameter (D = 600µm) and pitch (a = 2µm, solid lines: FE simulations, dashed line anisotropic dispersion model). (b) Hypothetical calculations to investigate the modal transition behavior (D = 600µm, $\overline{\overline{\mu }}{~}_0\overline{\overline{I}}$) Top: ℑm(n_eff) vs. ℜe(ε_z) [ℑm(ε_z) = 1i], ε_r = ε_silica. Bottom: ℑm(n_eff) vs. ℜe(ε_r) [ℑm(ε_r) = 1i], ε_z = ε_silica.

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3. Conclusion

In conclusion, we have shown that the loss properties of indefinite metamaterial-cladding hollow-core fibers can differ significantly from those composed of conventional claddings and that the highly anisotropic cladding allows to tune modal discrimination such that either the HE₁₁, TE₀₁ or TM₀₁ mode has the lowest loss of all modes. By incorporating material attenuation into the anisotropic modal equations via complex optical parameters obtained from simple S-parameter retrieval, we reproduced the loss properties of such fibers as predicted by full finite element simulations, also elucidating the role of anisotropy. The unusual properties of the metamaterial cladding hollow core fibers emerge as a result of the fine balance between the field penetrations into the metal, and intrinsic material loss. These additional degrees of engineering freedom allows to uncover new loss regimes, for example regimes in which the TM₀₁ mode is the lowest loss mode. These results pave the way for the design of novel hollow-core fibers with guidance and tuning properties which are especially important for applications demanding cylindrical polarization states, for example laser cutting and materials processing [33,34], sharp focusing [35], particle trapping [36], particle acceleration [37], or alternatively enabling in-fiber guidance of the fundamental linearly polarized beam. Fabrication of iMM-HCFs with feature sizes and material combinations discussed here is feasible on the basis of either direct drawing a single preform or using pressure-assisted melt filling [6,9,10,12,20].