1. Introduction

Hollow-core optical fibers have been studied and developed for several decades. In principle, by guiding light in air rather than in a solid material, these fibers could enable one to exploit the ultra-low Rayleigh scattering and nonlinear coefficients of air – orders of magnitude lower than for any glass – and thus allow propagation at ultra-low loss and non-linearity. Besides, they provide significantly higher propagation speeds (i.e. reduced latency) and laser-induced damage thresholds than all-solid fibers, they can transmit at wavelengths where the solid state cladding is opaque and are in principle more robust to environmental perturbations such as mechanical vibrations, magnetic fields and ionizing radiations than solid counterparts. Finally, they are an ideal platform to enhance and study gas-light interactions.

Due to these attractive properties, over the years, several ways of guiding light in a hollow core fiber have been proposed, ranging from metallic mm waveguides [1], to hollow dielectric fibers [2], metal coated dielectric fibers [3] and multimaterial hollow core Bragg type fibers [4]. In addition to these types, microstructured hollow fibers made of a single glass and air have made a particularly significant progress in the last twenty years and will be the focus of this work. There are two main types of single-material hollow core fibers: one is based on photonic bandgap guidance (photonic bandgap fibers – HC-PBGFs [5, 6] or more simply PBGFs to simplify the notation in this paper, an example of which is shown in Fig. 1(a)), while the other, still at the center of intense study and technological development, relies for guidance on a combination of inhibited coupling to low density of states cladding modes and anti-resonance [7, 8]. For simplicity we will refer to these latter fibers in general terms as hollow core anti-resonant fibers – HC-ARFs (or more simply ARFs, again, to simplify notation). Within this broad category, many structurally very different fiber types have been proposed. Figure 1(b)-1(h) shows a non-exhaustive catalogue of some of the most relevant, presented in chronological order and ranging from those with a Kagome cladding and a straight (b) or hypocycloid (d) core surround, to simplified anti-resonant fibers like the hexagram (c) or the double antiresonant fiber (h), to fibers with a ‘negative curvature’ core surround (e)-(g).

 

Fig. 1 Scanning Electron Micrographs (SEMs) of some representative hollow core fibers: (a) PBGF [17]; (b-h) ARFs. In particular, (b [18],) and (d [19],) have a Kagome cladding and straight vs hypocycloid core surround, respectively; (c [20],) and (h [15]) are simplified anti-resonant fibers with a hexagram and a double antiresonant cladding, respectively; (e [21],), (f [16],) are simplified hollow core fibers with ‘negative curvature’ core surround, like also (g [22],), which however presents a cross-section without nodes and will be referred to, in the text, as antiresonant nodeless tube-lattice fiber (ANF).

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It is generally well accepted that PBGFs offer the lowest loss amongst hollow core fibers, with a minimum reported attenuation around 1-2 dB/km at telecoms wavelengths [9], but over a bandwidth of only around 10-30% of the central operating wavelength, while ARFs offer bandwidths as wide as an octave but with a higher straight loss and a more pronounced bend sensitivity. Recent works on PBGFs have thus targeted bandwidth enhancement [10] as well as a further loss reduction [11] and modal purity improvement [12] which are necessary to enable high capacity data transmission applications, while the latest work on ARFs has focused on reducing the straight and bend loss [13, 14] and on addressing novel transmission regions in the UV and IR spectral regions [15, 16].

In this work we present a novel fiber which combines the best characteristics of PBGFs (low propagation loss and bend robustness) and ARFs (wide bandwidth and low modal overlap with the cladding) and adds the possibility to achieve robust single mode guidance. Numerical simulations predict that the fiber proposed here has the potential to reach total levels of loss, including leakage, surface scattering and bending, that are lower than that of state-of-the-art conventional solid fibers.

The paper is organized as follows: Section 2 reviews the two main guidance mechanisms for single material hollow core fibers and validates the numerical tools against fabricated fibers of both types. Section 3 presents the novel fiber, compares it against other types of hollow core fibers and studies its performances, scaling laws and bend loss. Section 4 studies the effect that structural variations have on the fiber performance with the aim of identifying optimum parameter ranges. Section 5 briefly discusses two possible applications of the fiber for data transmission and power delivery, and Section 6 summarizes the work.

2. Photonic bandgap versus antiresonance guidance

The guidance mechanism in PBGF is by now fairly well understood. An out-of-plane photonic bandgap that extends below the air-line and can guide an air-mode in a suitably engineered defect is formed for some frequencies and angles of incidence through the nearly-periodical arrangement of glass rods in the cladding [5, 23]. The presence of thin glass struts to interconnect the nodes is necessary for structural stability but detrimental for the fiber’s guidance properties [24] as it eliminates high order bandgaps and only leaves one spectral region available for air guidance. The air bandgap is typically located at frequencies $1\le v\le 2$, where $v=2\pi r_r\sqrt{n^2-1}/\lambda$ is the normalized frequency of the glass rods and r_r their effective radius. At frequencies within the bandgap the nodes are in antiresonance and therefore they are effective at ‘repelling’ light or radially backscattering it to the core. As a result, the loss contribution due to light leakage from the core, the confinement loss (CL), can be reduced to arbitrarily low values by simply adding as many rings of resonators around the central defect as it is necessary [25]. The thin glass struts however are typically fabricated as thin as possible in order to widen the bandgap [26] and therefore they are not in antiresonance at the operating wavelength. The electromagnetic field intensity on their surfaces is thus higher than around the rods, especially for the struts surrounding the core where the modal intensity is higher. This causes significant scattering at the air-glass surfaces, which are intrinsically ‘rough’ due to frozen in thermodynamic fluctuations [27]. As a result, surface scattering loss (SSL) is the dominant source of loss in these fibers [9, 28].

A very different physical picture occurs in ARFs. Here all the thin membranes of equal thickness t and refractive index n surrounding the core are designed such that the operational wavelength λ sits in between the high loss resonant wavelengths of the fiber at:

2.1 Validation of numerical models

The fundamental difference in guidance mechanism discussed above leads to the already discussed and commonly accepted conclusion that CL-dominated ARFs are inherently lossier than SSL-dominated PBGFs. However, PBGFs have a bandwidth that is typically between 2 and 10 times narrower than ARFs. This stems from the bandgap-distorting effect of the struts [24] combined with the potential presence of surface modes, parasitic modes located on non-correctly terminated the core surrounds [24, 32] and which can further reduce the useable bandwidth through anti-crossing with the air guided modes [33].

Figure 2 compares the measured loss of a typical ARF (here based on a simplified hexagram cladding [20]) to that of a state-of-the-art PBGF [10]. The loss-bandwidth trade-off can clearly be seen. The current record low-loss all-solid standard step-index fiber (SSIF) is also shown for comparison [34]. These experimental losses have been used to validate and optimize the simulation tools used throughout this work for the calculation of the main fiber’s properties, in particular CL and SSL. All simulations reported in this work are based on a full vector finite-element based modal solver (Comsol Multiphysics).

 

Fig. 2 Loss comparison: A. ARF (hexagram fiber [20]); B. PBGF [10]; C. record low loss SSIF [34]. The solid lines are measured losses; dotted and dashed lines are simulated CL and SSL, respectively. Note how in a ARF the loss is dominated by CL while in a PBGF SSL is the dominant loss mechanism.

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The use of perfectly matched layers (PMLs, in this work used with their standard cylindrical definition) to surround the simulation area enables the direct calculation of the CL as the imaginary part of the eigenvalue returned by the modal solver [35]. For ARF we have found that great care must be employed to optimize both mesh and PML parameters in order to achieve accurate results. Typically, accurate simulations were found to require the use of quadratic finite elements, of a carefully optimized PML and of extremely fine meshes, with maximum element size of λ/4 and λ/6 in air and in the thin glass regions, respectively. The choice of such simulation parameters can produce excellent agreement with the measured loss, as shown by the dotted green line in Fig. 2, for example, although it typically produces sparse matrices of several million degrees of freedom, which are 1-2 orders of magnitude computationally more demanding than for conventional PCFs.

SSL is a more complicated quantity to estimate. In principle, a statistical treatment of the scattering process that includes the power spectral density (PSD) of the surface roughness should be employed [28]. Such PSD is a difficult quantity to measure in practice though, due to the small r.m.s. roughness of the surfaces, which are also hard to access. In this work we adopt a simplified method that has been found to provide fairly accurate results in a number of tested PBGF cases [36]. It relies on neglecting the dependence of the surface roughness on spatial frequencies and on the size of holes and membranes, and on postulating that the average roughness is process-independent, i.e. all fabricated fibers are assumed to have the same roughness. Under these assumptions we estimate the SSL by multiplying the normalized electric field intensity at the interfaces, the F-parameter of [9], by a normalization factor η:

The plots in Fig. 2 confirm that CL dominates the loss for ARFs where SSL is negligible, while the opposite is true for PBGFs, as previously contended based on physical arguments. The good agreement between simulations and measurements for both hollow core fibers and loss mechanisms supports the numerical results presented later on in this work.

3. Hollow core nested antiresonant nodeless fiber (HC-NANF)

Numerous works in the literature have investigated the use of core boundaries with a negative curvature to locate undesirable nodes in positions where the modal field intensity is lower and therefore to reduce their detrimental effects [13, 16, 19, 21, 34, 38]. A further improvement has been proposed by Kolyadin et al. and it consists of surrounding the core by a ring of non-touching tubes [22] (Fig. 1(g)). In this way the nodes are completely eliminated and the CL can be further decreased as compared to a similar fiber with touching nodes, as will be shown later on. In such a fiber, shown in the top half of Fig. 3(a), the tubes are azimuthally separated by a distance d and light localization in the core occurs due to the two Fresnel reflections from the inner and outer surface of the thin glass tubes that form the core surround. Despite this, its CL is still considerably higher than its SSL.

 

Fig. 3 Comparison between antiresonant nodeless fiber (ANF), (top) and the version with nested elements (NANF) proposed in this work (bottom): (a) structure; (b) 3-dB contour plots and (c) cross-sectional profile of the fundamental mode’s z-Poynting vector, showing how the addition of the nested ring decreases the field on the outer cladding from 6 to over 8 orders of magnitude below its maximum value.

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The improved antiresonant fiber design that we propose in this work maintains the nodeless tube lattice structure but it adds one or more nested tubes with the same thickness as the outer ones and attached to the cladding at the same azimuthal position, as shown in the bottom part of Fig. 3(a), The separation between inner and outer tubes along the radial direction is a further structural parameter, z. We will refer to this design as Hollow Core Nested Antiresonant Nodeless Fiber (HC-NANF or simply NANF in this paper). A similar design but with touching tubes has been recently proposed by Belardi et al. [14]; the differences with the fibers in this work are discussed in Section 4.1. The advantage brought by the nested elements as compared to the standard antiresonant nodeless tube-lattice fiber (HC-ANF or more simply ANF) of Kolyadin et al. (Fig. 1(g)) is evident by comparing the modes guided in the two structures. Figure 3(b) shows how the additional anti-resonant membrane is effective in improving the confinement of the mode to the core, both along radial directions where the resonant tubes are present (y axis) and in between them (x axis). Figure 3(c) compares the normalized fundamental mode (FM) intensities in the two fibers calculated along two orthogonal directions (red: x; blue: y). For the ANF it can be seen that the field decays more in a direction passing through the holes than through an antiresonant ring. For the particular structure chosen in the example it touches the outer glass boundary with an intensity around 10⁻⁶ lower than in the center of the fiber. By contrast, this is reduced to below 10⁻⁸ for the NANF, which leads to a significant reduction in CL, as shown below.

3.1 Loss comparison with alternative ARFs

To quantify the advantages of the nested structure over other types of ARF, we plot in Fig. 4 the results of a comparison, where we have simulated 6 different structure, all with the same core radius R = 15 µm (defined as the maximum radius of a circle that can be inscribed inside the core) and the same uniform core surround thickness t = 0.42 µm that generates the first high loss resonant peak at around λ = 0.85 µm. The reference structure is an ideal Bragg fiber, with an annular glass ring suspended in air and separated from the outer jacket by an optimum distance corresponding to the half wave stack condition [20]. The thin black line shows its loss (here CL, since SSL is negligible) calculated with a standard matrix method [39], while the red curve is the result of an FEM simulation, again, in perfect agreement. The thick black and orange lines show the loss of a hexagram and of a Kagome lattice fiber with negative curvature, respectively. As can be seen, both structures have CL that over narrow spectral ranges are lower than the reference Bragg fiber, but that overall oscillate around the same values (around a few dB/m for this core diameter). Most importantly, the presence of nodes generates large loss oscillations with spectral periods of a few nanometers. Note that ways to further reduce the loss in Kagome fibers by increasing the curvature of their core surround have been proposed [13], but such an analysis is outside the scope of this work.

 

Fig. 4 Confinement loss comparison between 6 different ARFs: hexagram (black); Kagome with negative curvature core (orange); idealized Bragg (red); tube lattice fiber (green), ANF (cyan); NANF (blue). All fibers have the same core diameter R = 15 µm and uniform strut thickness t = 0.42 µm. The dashed line indicates the SSL for the ANF, identical to that of the NANF.

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The green curve shows that a simple tube lattice design (here with 6 tubes) allows some loss improvement over the reference Bragg case. This is interesting and somehow unexpected, indicating that the curvature/length of the core surround can have an influence on the CL. Even more interestingly, we have found that by separating the tubes in order to eliminate the nodes (ANF, cyan curve) CL nearly one order of magnitude lower than when nodes are present can be achieved. The minimum loss becomes in this case ~0.1 dB/m, still considerably higher than the scattering loss for this fiber, shown as a dashed line and around the sub-dB/km range. Finally, the blue curve shows the loss of the NANF: its CL, with a minimum value of only ~0.2 dB/km, is 3 orders of magnitude lower than the ANF and more than 4 orders of magnitude lower than the reference Bragg fiber with the same core diameter. By significantly improving the modal confinement, the addition of the nested element helps achieving CL as low as the SSL, which for this fiber is nearly identical to that of the ANF shown in light blue.

This is the first demonstration of a hollow core fiber with simultaneously ultra-low values of CL and SSL. Since in NANFs it is reasonable to expect a lower SSL than in PBGFs, where not all of the glass membranes are in antiresonance, the result in Fig. 4 hints at the remarkable possibility of an antiresonant fiber with a lower loss than a PBGF.

We now study the loss dependence of NANF on wavelength λ and core radius R. It is well known that the loss in circularly symmetric Bragg type fibers operating at λ<< R follows a λ^a/R^{(a + 1)} scaling law, where a = 2 for a simple circular hole in glass [2], a = 3 for a tube glass in air [40], a = 4 for the Bragg fiber of Fig. 4 (tube in air plus additional glass jacket) and more in general with a incremented by 1 at each additional air-glass interface [20].

Figure 5 shows the results of numerical investigations aimed at studying the CL in NANFs. By running a broad wavelength scan from 0.4 to 2.5 µm such that three antiresonance regions of a NANF are simulated (Fig. 5(a) shows a fiber with t = 0.75 µm but the same conclusion has been found for other values of t), we have found that the minimum loss in each antiresonant transmission window is well fit by a λ⁷ curve. This indicates that by operating in higher order antiresonant regions the CL can be decreased significantly. However, since as any other scattering-mediated mechanism, SSL is known to increase at shorter wavelengths (see Eq. (2) and [37]), the total loss will follow a characteristic V-shape, dominated by SSL and CL at short and long wavelengths, respectively. The optimum operational window of the NANFs will be close to the wavelength where CL equals SSL (λ~0.6 µm in between second and third windows in the example). The loss dependence on the core radius R, shown in Fig. 5(b), is found to follow a R⁻⁸ curve, and interestingly it reduces faster than the SSL which goes as ~R⁻³ like in other hollow core fibers, e.g. hole in glass, PBGF,… [2, 41]. Therefore, for any given wavelength, enlarging the core while keeping the same membrane thickness is a simple way to reduce the fiber CL to values as low as its SSL.

 

Fig. 5 Simulated dependence of the loss contributions of a typical NANF on (a) wavelength at a fixed core radius, and (b) core radius at a fixed wavelength (λ = 1µm). For the CL an overall λ⁷/R⁸ dependence is observed while SSL goes approximately like 1/λR³.

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In order to put the loss of the NANF into perspective, we compare it in Fig. 6 to those of the ARF, PBGF and SSIF previously discussed. The 4 NANFs shown in the graph have a uniform boundary thickness t = 0.55 µm that generates a minimum loss at around 1.5-1.7 µm, and core diameters D = 2R of 26, 30, 40 and 50 µm. The role of the other structural parameters, here d = 5t and z/R = 0.9, will become clear later on. For comparison, the core diameter D of the ARF and of the PBGF are 50 µm and 26 µm, respectively. Note that the fabricated ARF has a thinner membrane thickness (~0.32 µm) that shifts the antiresonant window to shorter wavelengths. It can be seen that for the same core size (D = 26 µm) the NANF has similar loss as the PBGF but a bandwidth that is over two times as broad. Most importantly, while in PBGFs enlarging the core size for the same operational wavelength requires removing additional elements from the core and adds considerable complexity in the fabrication procedure [42], in the case of the NANF core scaling can happen seamlessly and with no additional complexity. Simulations indicate that a fiber with D = 30 µm would have a loss around 1 dB/km, for D = 40 µm losses would already be below those of SSIF (~0.1 dB/km), and further increasing D would allow further loss reduction, according to the D⁻⁸ scaling law previously discussed.

 

Fig. 6 Comparison between the measured loss of a typical ARF (hexagram fiber [20], D = 50 µm), a state-of-the-art wide bandwidth PBGF [10] with D = 26 µm, the record low-loss SSIF [34] and the simulated CL of NANFs with D = 2R = 26, 30, 40 and 50 µm, all having t = 0.55 µm, d/t = 5 and z/R = 0.9. The SSL of these fibers (not shown) is comparable to their CL.

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3.2 Bend Loss and high order mode suppression

Since the NANFs have core diameters significantly larger (i.e. 10-20 times) than the operational wavelength, it is worth studying their resilience to external perturbations. Here we focus on macro bend loss. To calculate the bend loss of the fibers we use the standard conformal transformation to modify the refractive index of an equivalent straight fiber [43]:

 

Fig. 7 Bend loss of NANF: (a) wavelength dependence for fundamental mode (solid lines) and lowest loss high order mode (dotted lines) of a D = 40 µm fiber; (b) dependence on the radius of curvature R_c for the 30, 40 and 50 µm diameter fibers of Fig. 6 at λ = 1.8 µm. The markers indicate the critical radius at which the loss doubles as compared to a straight fiber. All simulated fibers have t = 0.55 µm, d/t = 5 and z/R = 0.9.

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So, overall, despite the large core to wavelength ratio, the NANFs can be bent to reasonably small bend radii. Their bend performance sits somewhere in between those of PBGFs, extremely low even for D/λ ~15 [46] and those of ARFs, which are significantly larger [16, 31]. This seems to indicate that the additional nested element plays an active role also in improving the bend robustness.

Apart from limited exceptions with small cores [47] or with resonant elements in the cladding [12], all known hollow core fibers guide several air modes. This typically results from the fact that in order to reduce their loss (SSL and CL) the fibers are fabricated with core defects with as large D/λ as is possible. The number of guided modes strongly influences their ultimate application. In general, this depends on the ratio between the size of the hollow core – determining the effective index (n_eff) of the air modes, a very good approximation of which is given in [2] – and the average size of the holey features in the cladding – determining the lower n_eff edge of the photonic bandgap/antiresonant region and therefore the number of core-guided core modes the structure can effectively support. PBGFs are known to support ~12, ~40 and ~80 vector modes for a 7, 19 and a 37 missing cell core, respectively [41]. The loss of these modes increases with the mode order, typically by a few times for each subsequent mode group, so that LP₁₁-like and LP₂₁-like modes have a loss ~2.5 and ~5.5 times higher than the LP₀₁, respectively [36, 46]. While this might be exploited to increase the transmitted capacity through mode division multiplexing (MDM) [42], in general it means that over short lengths the fibers are multimoded, and therefore not ideal for applications where a high modal purity is desirable, for example in gyroscopes or in gas cells. NANFs on the other hand, offer the possibility of a much greater extinction of the higher order modes (HOMs) and therefore a better modal purity. This can be seen in the example in Fig. 7(a) where a nearly 100 time lower loss is obtained in the fundamental mode (solid line) as compared to the lowest loss HOM (dotted lines) when the fibers are kept straight. The ratio further increases as the fiber is bent and can be engineered to match the application requirements by structural adjustments, as discussed in more detail in Section 4.2.

3.3 Wavelength scaling

Having analyzed the loss contributions and the loss scaling of NANFs, we now investigate how the NANF loss scales with wavelength. Taking the D = 40 µm, t = 0.55 µm fiber of Fig. 6 and 7 as a reference we scale t from 0.2 to 2.25 µm and D by the same factor, such that the ratio D/t remains constant. This homothetic scaling corresponds to a set of fiber structures that can be fabricated from the same preform. The glass infrared absorption (α_glass) for pure silica glass has been extracted from [48] and has been directly included in the FEM calculations. Figure 8 plots the confinement loss and infrared absorption (solid colored lines) and the surface scattering loss (dashed black lines) for 7 different fibers. Note that for each fiber only the fundamental antiresonance region is plotted, but as shown in Fig. 5(a) guidance also occurs at shorter wavelength in higher order antiresonant spectral regions.

 

Fig. 8 Example of wavelength scaling. 7 NANFs with the same d/t = 5 and z/R = 0.9 are rigidly scaled from R = 7 um to R = 80 um such that R/t is conserved. Only the fundamental antiresonance window in shown. The solid lines represent CL, the dashed black lines SSL. The IR glass absorption loss for silica extracted from [47] is directly included in the simulations and is plotted in a dashed gray line with a pre-multiplication factor accounting for a modal overlap with glass, ζ.

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As can be seen, sub 1 dB/km loss is predicted (at these core diameters) from ~0.8 to ~2.7 µm, and ~0.1 dB/km seems feasible from 1.5 to 2.3 µm. In this wavelength range, with further fiber optimization it is possible to equalize CL and SSL. For wavelengths shorter than 1 µm the SSL begins to dominate and it limits to ~10 dB/km the achievable loss in the visible range of the spectrum. The cross-over point between where infrared absorption loss starts to dominate over CL and SSL occurs at around 2.4 µm, where with appropriate designs leading to a further reduction in CL (see Fig. 5 and Section 4.4) the minimum possible loss in NANFs is predicted to occur. Thanks to the fact that all the core boundaries operate in antiresonance and that no nodes are present, all the fibers have a very low fraction of power guided in the glass, which at the center of the antiresonant window is around ζ = 5·10⁻⁵. This compares to ~5·10⁻⁴ and 1·10⁻⁴ for optimized 19c and 37c PBGFs, respectively, and it enables light transmission at wavelengths where the glass is even more opaque. For example, losses below 1 dB/m can be expected up to a wavelength of ~4.7 µm, while in principle losses of only 4-5 dB/m can be expected up to ~7 µm, where silica absorption is around 60 dB/mm.

Another advantage of NANFs over PBGFs is that they would allow the use of mid-infrared transmitting glasses which, with the exception of some fluorides, all have refractive indices higher than silica. While a higher index difference with air is detrimental for PBGFs as it increases the negative effect of the interconnecting struts and it ultimately leads to a complete closure of the fundamental bandgap leaving only narrower high-order bandgaps available for light guidance [49], in NANF no such negative effect is present. The larger index difference only affects the resonant frequencies by shifting the low loss regions according to Eq. (1), but it has no predicted effect of the loss.

4. Performance optimization

After a general introduction on the main general properties of NANFs, in this section we turn our attention to the role that different structural parameters have on the fiber performance. In particular we study the optimum azimuthal spacing between the resonant nested elements d, how the radial separation between inner and outer ring z affects the HOM extinction, and the optimum number of azimuthal and radially nested elements.

4.1 Optimum azimuthal separation of the nested elements

Figure 9 shows the result of a parametric study where we have modified the separation between the antiresonant elements d in a 6 nested element NANF while keeping all the other fiber parameters constant: core radius R = 15 um, ring thickness t = 0.42 um, z/R = 0.9. The curve for d = 0 corresponds to the idealized fiber where the tubes have an infinitesimally small contact point, as already studied by Belardi et al. in a fiber with 8 rather than 6 resonant nested tubes [14]. This fiber is not achievable in practice, since surface tension would tend to smooth out sharp angles at the tube connections points and create localized nodes, but it is a useful reference structure that separates fibers with and without nodes. Curves with negative values of d correspond to fibers where the outer antiresonant rings overlap to form nodes of increasingly larger size as d decreases. Note that this simplified geometrical representation does not conserve the glass mass and it is not an accurate representation of a realistic fabricated fiber. In practice, fiber with negative d would have even larger nodes than those simulated here, with an even more significant expected performance degradation. Fibers with positive d on the other hand are expected to be realizable from a suitable preform, since differential pressure and surface tension would naturally assist these circular shapes to form.

 

Fig. 9 Effect of antiresonant element separation d: (a) shows the spectral dependence of the CL of the 40 um NANF of Fig. 6 as d/t is changed from negative (nodes) to positive (no nodes); (b) shows a more detailed scan over d/t for a number of wavelengths. Values of d/t between 1 and 6 are found to give the best compromise between bandwidth and minimum loss.

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The figure shows that the presence of nodes significantly degrades the fiber loss. While in principle fibers with d = 0 could still achieve ultra-low loss, albeit over a much reduced frequency range, for fibers with d/t = −0.3 or −0.5 the presence of small nodes increases the minimum loss by orders of magnitude. Separating the antiresonant elements has thus the double benefit of reducing the loss and widening the useable bandwidth. It can be seen that as d/t increases from 0 to 7 the bandwidth keeps increasing. At the same time though, the minimum loss starts to increase after reaching a minimum. This can be explained by the fact that since the core radius is fixed, an increase in d corresponds to a decrease in the diameter of the resonant rings. This brings the outer jacket closer to the core, therefore increasing the CL. Besides, the larger tube separation also increases the leakage rate through the gap. Finally, the total perimeter of the tubes is also reduced, which increases the spatial overlap between air modes and ring guided modes, increasing their coupling strength and thus the loss of the air modes [7, 23]. Figure 9(b) summarizes the loss dependence on the tube separation for a NANF with z/R = 0.9 and for different wavelengths. While it emerges that no value of d/t minimizes the loss at all wavelengths and in general the numbers will be slightly different for fibers with other values of z, the structures with d/t between 1 and 6 are found to provide the best compromise between loss and bandwidth.

More physical insights can be gained by analyzing the field intensity of the fundamental core mode at the tube touching or minimum distance point, as shown in Fig. 10. For touching tubes (negative d/t) a large fraction of the field (here normalized to the maximum value it assumes over the entire cross-section) concentrates in the glass, a clear indication of phase matching to glass guided modes which leads to high loss; for small, positive d/t, the tubes, in antiresonance, expel the field from their surrounding areas, which results in a very small fraction of power leaking through the tubes, hence low loss. Leakage and hence CL increase for tube separations larger than the optimum value.

 

Fig. 10 Effect of nested element separation on the modal intensity. (b) shows the E_x-field along the blue line interconnecting the center of the tubes shown in (a), as a function of d/t. The red dashed line is the field at the intersection between red and blue lines in (a), which is somehow proportional to the leakage loss through the tubes and reaches a minimum for d/t in the range 1-3. The fibers are the same as in Fig. 9 and λ/t = 3.3.

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4.2 Radial separation of nested elements and high order mode control

All the structures presented so far had an inner nested tube with roughly half the diameter of the external one, which for the parameters considered corresponded to z/R = 0.9. Here we study the effect of changing z/R, i.e. the size of the inner nested antiresonant element.

In analogy to the case of waveguides made of circular holes in dielectrics [2], all core defects with R>>λ in principle support a multitude of air guided modes. However, if the core defect is surrounded by smaller holes, only the core modes with an effective index higher than the fundamental air-mode of the side holes will in practice be guided with low loss. The remaining core modes can be resonantly coupled to lossier air-guided modes of the cladding and suffer as a result from a considerably higher loss. Therefore, the ratio between the diameter of the core and that of the cladding holes is key to determine the number of low loss core-guided modes. In PBGFs, for example, air-guided modes localized inside the cladding holes form the bottom of the photonic bandgap [23, 24, 32]. Only core modes with an effective index higher than the cladding air modes can be guided with low loss. Increasing the size of the defect compared to the cladding hole size (e.g. by moving from fibers with a 3 cell defect to those with 7, 19, 37, … cell defects) creates defect modes with an increasingly larger effective index as compared to the index of the air cladding modes, such that a larger number of core modes can be guided, see for example [47].

Figure 11 shows how this resonant coupling effect can be exploited to make NANFs effectively single moded. The figure shows effective index and loss of the first 5 core guided modes (that for simplicity in the following will be named using the LP notation: LP₀₁, LP₁₁, LP₂₁, LP₀₂, LP₃₁, although a vector nomenclature would be more appropriate) as z/R is scanned from 0.1 (inner circles almost touching the outer ones) to 1.3 (small inner circles). Contour plots of the first few core modes are shown on the left hand side. As can be seen and expected, the effective index of the core modes does not change with the size of the nested antiresonant elements, as it is only dependent on the core radius, here kept constant. The value of z however has a strong impact on the air-modes guided inside the antiresonant tubes, referred in the following as cladding modes (CMs). Some CMs are shown on the right hand side and their effective index is indicated by dashed black lines in the effective index plot. For z/R in the 0.1-0.7 range the curves, with a negative slope, correspond to CMs localized inside the small tube, while for z/R above 0.7 the curves, with a positive slope, correspond to modes guided in between the two tubes. The effective indices of the CMs always remains below the LP₀₁ effective index and importantly they do cross the LP₁₁ line for z/R = 0.3 and 1.05. As a consequence, fibers with nested resonators of approximately these dimensions will have a very efficient resonant out-coupling of the LP₁₁ mode which will considerably increase its loss. Since the loss of other HOMs is found to be already significantly higher than that of the fundamental one even without coupling to cladding modes (loss plot in Fig. 11), NANFs with z/R ~0.3 or ~1.05 behave as effectively single mode fibers. Simulation results shown in Fig. 11 indicate that the high order mode extinction ratio (defined as the ratio between the loss of the lowest loss HOM and of the FM) that can be achieved in these fibers ranges from ~10 for z/R around 0.7 to 400 and 600 for z/R = 0.25 and 1.1, respectively. Such high differential loss causes strong modal filtering and robust single mode behavior even in short fiber lengths. The minimum FM loss is obtained for 0.6 < z/R < 0.8, although it is possible to achieve HOM extinctions greater than 200 at d/t = 1 with a FM loss only a factor of 2 times higher than the minimum.

 

Fig. 11 Effect of changing the size of the inner nested tube in a NANF with a fixed R, t and d. When z is changed, the cladding modes (CM) shown on the right resonantly interact with different core modes (shown on the left). Operation around z/R ~0.25 or 1.1 enables maximum suppression of the lowest loss HOM (LP₁₁), which can result in HOM extinction ratios in excess of 500.

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The small ripples that can be seen in the loss curve of LP₀₁ and LP₁₁ modes originate from anticrossings with ring guided modes of high azimuthal number [50], whose n_eff is shown by the green dots in the n_eff plot. Given the high aspect ratio between length and thickness of the glass tubes, the overlap between air and glass modes are small but still sufficient to create the oscillations observed in the loss plot. It should be finally stressed that not only is the resonant out-coupling of HOMs a broadband effect, as shown in Fig. 7(a), but since the size of the nested elements is proportional to the size of the core, modal filtering and single mode operation can be engineered to happen for all fiber dimensions.

4.3 Number of nested elements

Theoretical work has shown that for the standard nodeless antiresonance tube fiber (ANF) the best HOM extinction is achieved with 7-8 antiresonant elements [50]. This stems from the same physical principle discussed above of effective index matching between HOM and CMs. Such a conclusion has quite likely influenced subsequent experimental works on antiresonant fibers with a negative curvature core surround, all of which have chosen structures formed of either 8 (or in some cases 10) tube elements [14, 16, 22]. In order to achieve the highest HOM extinctions in NANFs however, the size of the external tubes needs to be enlarged in order to phase-match a core HOM to a CM located in the area between the tubes or inside the smaller tube. As a result, the optimum number of elements in NANFs is lower. Fibers with 6 elements have been found to offer improved performances as compared to fibers with 8 or 10 elements. For example, while a 6-element NANF with z/R of 0.9 can have a HOM extinction of 50-100, a NANF with the same core size but with 8 or 10 elements will have a HOM extinction of only 2.5 or less than 2, respectively. Moreover, thanks to the larger separation between core surround and outer jacket and to a longer tube perimeter, the CL of the fundamental mode in a 6-element NANF is lower, as shown in Fig. 12.A further reduction in the number of resonant elements (i.e. less than 6) can in principle be obtained (see for example [15]), but one must be careful not to create air regions inside the resonant elements that can support modes with a similar or slightly larger effective index than the core, which could otherwise affect the fundamental mode loss. In practice, unless more than one tube is nested, see the next Section, a 4 element NANF would seem difficult to realize with low loss.

 

Fig. 12 Comparison between the CL of NANFs with 6, 8 and 10 nested elements but with the same R = 20 um, t = 0.55 um, d/t = 5. The SSL of all three fibers are identical and shown by the dashed black curve.

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4.4 Multiple nested elements

One of the advantages of NANFs as compared to other types of antiresonant fibers, for example those with a Kagome’ cladding, is that the effect of each antiresonant interface can add coherently, as can be seen by the considerable loss improvement of NANF over ANFs shown in Fig. 4. It is therefore to be expected that by nesting additional tubes even lower losses can be achieved. This is confirmed by the simulation results shown in Fig. 13.Here the fiber with 6 antiresonant elements and no nested tubes of Fig. 4 is compared to NANFs with one (1N) and two (2N) nested rings. As can be seen, for a large spectral region the CL of the 1N-NANF is already lower than its SSL, which dominates between λ~1 and ~1.4 µm. Therefore, when the fibers are kept straight the ~30 times CL improvement achieved through the introduction of an additional nested element is only marginally beneficial, as its main result is in widening the SSL dominated spectral region. However, under operation in a tight bend (for example, here a 3 cm bend radius is shown) the CL of the 1N-NANF would once again become dominant over SSL, and the 2N-NANF can provide significant advantages, enabling a SSL-dominated loss of ~0.1 dB/km even at such small bend radii.

 

Fig. 13 Loss comparison between NANFs with 1 and 2 nested elements in straight (solid lines) and bent (R_c = 3 cm, dot-dashed lines) configuration. All fibers have R = 15 µm, t = 0.42 µm and the same SSL, shown by the dashed black line. The additional nested element further lowers the CL and could be useful to improve bend performances of NANFs. A ANF with the same R and t is also shown for comparison.

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The conclusion of this simple study is therefore that whilst adding additional nested elements would require an increased fabrication complexity and it might not always provide significant advantages, it is a solution worth considering in applications where the bent performance of the NANF is critical.

5. Applications

The simulations reported so far have indicated that the NANF has the potential to guide light with propagation losses below the 0.15 dB/km, which represents a fundamental physical limit in current all-solid telecoms fibers, and over a broader useable bandwidth (Fig. 6). Differently from most forms of hollow-core fibers currently in existence, the NANF is also able to operate with effectively single-mode behavior, which is particularly desirable to suppress signal distortions due to intermodal cross-talk. Due to these characteristics, the fiber would seem an ideal transmission medium for a variety of applications, ranging from low-latency, high capacity data transmission, high power laser delivery, operation in environments with strong ionizing radiations, mid-IR gas spectroscopy, interferometric applications, and in general in applications where a low dependence on power, magnetic field and temperature fluctuations is essential. In the following some remarks will be made about two applications in particular: data transmission and high power delivery.

5.1 High capacity, low latency data transmission

Combining a nonlinearity that is typically three orders of magnitude lower than in a glass-guiding fiber with a potentially very low transmission loss, hollow core fibers can in principle offer a significant increase in transmission capacity over existing fibers, at transmission speeds which are ~30% higher [10, 41]. Before this work the accepted understanding was that PBGFs were the ideal hollow fibers for data transmission, and they represented the only known solution to reach loss levels compatible with conventional SSIFs. The results presented in Fig. 6 show however that NANFs have the potential to reach similar loss levels, or even lower. Here we compare the structure, fabrication requirements and optical properties of a PBGF with a large 37 cell core where all the structural features have been optimized to achieve the minimum possible loss, to those of a NANF providing similar simulated loss of ~0.2 dB/km. While to achieve this loss the PBGF needs to operate at around 2 μm wavelengths [9, 11, 51] and requires a core diameter as large as 50 μm, the NANF can operate over a broader wavelength range. Here we choose the fiber with D = 40 μm of Fig. 6 with a minimum loss of ~0.15 dB/km at 1.6 μm which would be compatible with current telecoms systems. Figure 14 compares their structures and shows their FMs when kept straight (a) and under a tight bend of Rc = 3 cm (b), while Table 1 summarizes the main differences.

 

Fig. 14 Structural comparison between a 37 cell core PBGF optimized for ultra-low loss operation at 2 µm wavelengths with R = 50 μm (top) and a NANF providing similar loss with R = 40 μm (bottom). (a) shows 3-dB contour lines of the fundamental mode in both fibers when straight, while (b) shows the mode at a 3 cm bend, where the HC-BPGF is more effective at containing the mode inside the core.

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Table 1. Comparison between Ideal PBGF and NANF with Similar Loss

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From a structural point of view, the PBGF requires more than stacked 250 capillaries, all of which need to assume a very high air-filling fraction in the fiber (diameter over pitch d/Λ~0.99), with a resulting minimum membrane thickness of only ~40 nm around the core. In contrast, the NANF only requires 12 capillaries which can have a more relaxed inner to outer diameter ratio of ~0.96 hence requiring less pressurization during fabrication, and a glass thickness of around 500 nm.

From a transmission point of view, while the predicted loss is similar in both cases (~0.2 dB/km), the theoretical 3-dB bandwidth is of ~400 nm for a surface-mode free ideal PBGF, likely less in practice if uncontrolled core distortions end up supporting surface modes in the fiber, which compares to ~600 nm for the NANF. Despite its larger core the PBGF is more robust to bending, with a critical bend radius of only 2 cm as compared to 8.5 cm for the NANF. This can be appreciated by the modal distortion induced by a 3 cm bend radius shown in Fig. 14(b) and stems from the larger number of radially positioned resonators for the PBGF. If operation under relaxed bends is allowed though, the NANF offers the additional advantage of an effectively single mode operation, achieved through a large HOM extinction and particularly attractive for single mode data transmission applications. The fraction power in the glass also favors NANFs which as a result can operate at slightly longer wavelengths in the infrared and/or suffer from lower nonlinear signal degradation due to the glass nonlinearity. In conclusion, numerical simulations indicates that for data transmission applications NANFs might provide a valid alternative to PBGFs.

5.2 Power delivery

The very small fraction of power guided in the glass (<10⁻⁴) which comes from the simultaneous antiresonant behavior of all glass membranes, combined with a low propagation loss and an effectively single mode behavior suggest that NANFs could find application as flexible power delivery waveguides. This could be especially important for pulsed operation where the peak powers involved induce detrimental non-linear spectral broadening and temporal pulse distortion, or could even exceed the damage threshold of the material.

Differently from PBGFs, the output mode field diameter (MFD) of the fibers can be easily tailored to match exactly that of an active/passive solid fiber that operates, for example, at a fiber laser wavelength of 1.06 µm, 1.55 µm or 2 µm. From the loss scaling with R of ARFs, the fibers fabricated to date need to have core diameters in the range 30-100 µm to keep propagation loss low. This can often be too large for a direct interconnection with a solid fiber. In Fig. 15 we show that thanks to the additional nested element, an adequately low loss for a few-meter propagation can be achieved in NANFs even at small core diameters.

 

Fig. 15 Example of three NANFs operating at 1.06 µm in the first antiresonant band and with different MFD. Solid lines indicate the CL of straight fibers while dotted lines are for fibers coiled at a 2.5 cm diameter. A continuous tuning of the MFD is possible to match the output MFD of a solid active/passive fiber.

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The figure shows the loss of three NANFs with a fundamental antiresonant region around 1.06 µm and small MFDs, in the region of 9, ~12 and ~16 µm. The loss of both straight and bent fibers is shown in solid and dotted lines, respectively. A coiling diameter of 2.5 cm was chosen to demonstrate that for such moderate MFDs the fundamental mode loss is almost unaffected even by tight bends. The bend robustness of these NANFs is significantly superior to that of any solid fiber with a comparable effective area, and losses of 10-100 dB/km can be achieved. Even lower losses, of the order of 1 dB/km have been observed in simulated fibers operating in the first antiresonant region (not shown).

6. Conclusions

We have proposed a novel type of hollow core fiber based on nested antiresonant tube elements and no nodes, and shown through simple physical understanding and numerical simulations how this NANF is predicted to overcome the known limitations of both state-of-the-art PBGFs and ARFs. The nested tubular elements are useful to decrease the CL of the fiber to levels comparable to its already low values of SSL. The absence of nodes in the structure provides a low-loss antiresonant region as wide as an octave and, to the best of our knowledge, with the lowest simulated loss for any realistically achievable hollow core fiber. The minimum loss is achieved at an optimum azimuthal separation between the antiresonant tubes, 1 < d/t < 6. Besides, by controlling the radial separation between nested elements, a very high HOMs suppression factor (in excess of 500) can be achieved, which indicates that the fibers should behave as effectively single moded over an adequate length. The simulated optical performances of these NANFs are remarkable. Thanks to a loss scaling following a λ⁷/R⁸ trend for a fiber with one nested tube, simulations indicate that by choosing a suitably large core radius they should be able to achieve lower losses than even standard solid fibers, at comparable wavelengths but over a broader wavelength range. Although their bend loss performance is somewhat inferior to that of PBGFs, they are still more bend robust than solid fibers with a comparable MFD and could therefore lend themselves to applications, for example, in high power delivery, where single modedness and high nonlinear and damage threshold are important. For applications where bend loss is critical on the other hand, we have shown how the introduction of additional nested elements can be beneficial.

By combining the best properties of PBGFs and ARFs in a radically novel structure, NANFs are predicted to attain remarkable optical performances which could further extend the applicability and reach of hollow core fibers to fields as diverse as data transmission, gas sensing, power delivery, nonlinear optics and interferometric devices.. From a rheological point of view, the fluid dynamics of the fiber draw indicates that the proposed NANFs should be amenable to fabrication, and perhaps even easier to fabricate than PBGFs. While comparison with experimental results is certainly needed in order to fully corroborate all these claims and predictions, the use of numerical tools validated over the years with many drawn hollow core fibers leaves us confident about the validity of the results presented here.