Nested antiresonant nodeless hollow core fiber Francesco Poletti* Optoelectronics Research Centre, University of Southampton, Southampton, SO17 1BJ, UK * [email protected]
Abstract: We propose a novel hollow core fiber design based on nested and non-touching antiresonant tube elements arranged around a central core. We demonstrate through numerical simulations that such a design can achieve considerably lower loss than other state-of-the-art hollow fibers. By adding additional pairs of coherently reflecting surfaces without introducing nodes, the Hollow Core Nested Antiresonant Nodeless Fiber (HC-NANF) can achieve values of confinement loss similar or lower than that of its already low surface scattering loss, while maintaining multiple and octavewide antiresonant windows of operation. As a result, the HC-NANF can in principle reach a total value of loss – including leakage, surface scattering and bend contributions – that is lower than that of conventional solid fibers. Besides, through resonant out-coupling of high order modes they can be made to behave as effectively single mode fibers. ©2014 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.2280) Fiber design and fabrication; (060.2400) Fiber properties; (060.4005) Microstructured fibers.
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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 laserinduced 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). 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23809
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.
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 antiresonant 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).
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 nearlyperiodical 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 v 2 , where v 2 rr n 2 1 / is the normalized frequency of the glass rods and rr 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23810
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:
m
2t n 2 1 , m 1, 2,3, m
(1)
where the air mode is phase matched to glass modes in the struts. At the low-loss operating wavelengths the glass membranes are in antiresonance and the electromagnetic field on at least one of the two air-glass interfaces is minimized. Therefore, SSL is typically extremely low in these fibers. However, two main issues affect the overall optical performances of ARFs: the presence of undesired and thicker nodes at the intersection between struts and the fact that it is difficult to arrange the antiresonant membranes in such a way to achieve a coherent light reflection in the radial direction. The first problem leads to the presence of spurious loss peaks and dips within the antiresonant windows [7, 8, 18, 20] which are detrimental to both the loss and the dispersive properties of the fibers. This can be partially attenuated by fiber designs that position the nodes as far away from the center as is possible, as in fibers with a negative curvature core surround [16, 19]. The second problem is seemingly far harder to tackle. While circularly symmetric Bragg fibers have been demonstrated as a solution to achieve tight modal confinement in the core with arbitrarily low confinement loss in the case of all solid fibers [29] or of hollow core fibers with an all solid Bragg stack [4], they cannot work in the case of structures made of a single glass and air. Here the glass rings need to be interconnected and the radial interconnecting struts unavoidably create nodes that significantly affect the loss [30]. Besides, in air-glass antiresonant fibers based on non-circularly symmetric claddings with many rings of holes such as those with a Kagome lattice, it has been demonstrated that most of the light confinement occurs due to antiresonance in the core surrounding ring with some contribution due to the second ring: the remaining part of the cladding is not effective at creating coherent reflections and it has almost no light-guiding role [8, 13]. This has generated recent interest in antiresonant fibers with a simplified cladding made of just one ring of capillaries such as those in Figs. 1(e), 1(f), 1(h) [16, 20–22, 31]. Since the coherent superposition of antiresonances in the radial direction cannot be achieved, the loss in all state-of-the-art antiresonant fibers is currently dominated by CL. 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 noncorrectly 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23811
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). 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: 3
[ m] . 0
sc [dB / km] F
(2)
While such a calibration in principle needs to be performed at any wavelength of interest, knowledge of the wavelength dependence of F allows us to calibrate the formula at one wavelength and adapt it to other wavelengths by introducing the term between brackets. In this work the calibration is done at λ0 = 1.55 µm, where by using η = 300 the good agreement between dashed and solid black line in Fig. 2 is achieved. Since it can be shown that F scales with λ2 [37] and R3 (R being the core radius, see Fig. 5(b)), the terms between brackets produces an overall SSL scaling like λ1 within the same fiber (similar to the experimentally measured λ-1.24 dependence [37] and in approximate agreement with more rigorous surface scattering calculations we performed, not shown), and like λ 4 when fibers are rigidly scaled to operate at different wavelengths like in Fig. 8. While this does not exactly match the wellknown λ3 law experimentally and numerically confirmed for PBGFs [9, 28], it is sufficiently close to allow us to draw qualitative conclusions from the simulation results reported here. With all these approximations, the SSL curves in this work should be regarded as a guideline only, differently from CL curves which are believed to be rather accurate, as shown below. 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.
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23812
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.
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 nontouching 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. 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23813
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 106 lower than in the center of the fiber. By contrast, this is reduced to below 108 for the NANF, which leads to a significant reduction in CL, as shown below.
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.
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.
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23814
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.
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
Abstract: We propose a novel hollow core fiber design based on nested and non-touching antiresonant tube elements arranged around a central core. We demonstrate through numerical simulations that such a design can achieve considerably lower loss than other state-of-the-art hollow fibers. By adding additional pairs of coherently reflecting surfaces without introducing nodes, the Hollow Core Nested Antiresonant Nodeless Fiber (HC-NANF) can achieve values of confinement loss similar or lower than that of its already low surface scattering loss, while maintaining multiple and octavewide antiresonant windows of operation. As a result, the HC-NANF can in principle reach a total value of loss – including leakage, surface scattering and bend contributions – that is lower than that of conventional solid fibers. Besides, through resonant out-coupling of high order modes they can be made to behave as effectively single mode fibers. ©2014 Optical Society of America OCIS codes: (060.2310) Fiber optics; (060.2280) Fiber design and fabrication; (060.2400) Fiber properties; (060.4005) Microstructured fibers.
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#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23807
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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 laserinduced 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). 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23809
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.
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 antiresonant 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).
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 nearlyperiodical 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 v 2 , where v 2 rr n 2 1 / is the normalized frequency of the glass rods and rr 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23810
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:
m
2t n 2 1 , m 1, 2,3, m
(1)
where the air mode is phase matched to glass modes in the struts. At the low-loss operating wavelengths the glass membranes are in antiresonance and the electromagnetic field on at least one of the two air-glass interfaces is minimized. Therefore, SSL is typically extremely low in these fibers. However, two main issues affect the overall optical performances of ARFs: the presence of undesired and thicker nodes at the intersection between struts and the fact that it is difficult to arrange the antiresonant membranes in such a way to achieve a coherent light reflection in the radial direction. The first problem leads to the presence of spurious loss peaks and dips within the antiresonant windows [7, 8, 18, 20] which are detrimental to both the loss and the dispersive properties of the fibers. This can be partially attenuated by fiber designs that position the nodes as far away from the center as is possible, as in fibers with a negative curvature core surround [16, 19]. The second problem is seemingly far harder to tackle. While circularly symmetric Bragg fibers have been demonstrated as a solution to achieve tight modal confinement in the core with arbitrarily low confinement loss in the case of all solid fibers [29] or of hollow core fibers with an all solid Bragg stack [4], they cannot work in the case of structures made of a single glass and air. Here the glass rings need to be interconnected and the radial interconnecting struts unavoidably create nodes that significantly affect the loss [30]. Besides, in air-glass antiresonant fibers based on non-circularly symmetric claddings with many rings of holes such as those with a Kagome lattice, it has been demonstrated that most of the light confinement occurs due to antiresonance in the core surrounding ring with some contribution due to the second ring: the remaining part of the cladding is not effective at creating coherent reflections and it has almost no light-guiding role [8, 13]. This has generated recent interest in antiresonant fibers with a simplified cladding made of just one ring of capillaries such as those in Figs. 1(e), 1(f), 1(h) [16, 20–22, 31]. Since the coherent superposition of antiresonances in the radial direction cannot be achieved, the loss in all state-of-the-art antiresonant fibers is currently dominated by CL. 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 noncorrectly 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23811
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). 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: 3
[ m] . 0
sc [dB / km] F
(2)
While such a calibration in principle needs to be performed at any wavelength of interest, knowledge of the wavelength dependence of F allows us to calibrate the formula at one wavelength and adapt it to other wavelengths by introducing the term between brackets. In this work the calibration is done at λ0 = 1.55 µm, where by using η = 300 the good agreement between dashed and solid black line in Fig. 2 is achieved. Since it can be shown that F scales with λ2 [37] and R3 (R being the core radius, see Fig. 5(b)), the terms between brackets produces an overall SSL scaling like λ1 within the same fiber (similar to the experimentally measured λ-1.24 dependence [37] and in approximate agreement with more rigorous surface scattering calculations we performed, not shown), and like λ 4 when fibers are rigidly scaled to operate at different wavelengths like in Fig. 8. While this does not exactly match the wellknown λ3 law experimentally and numerically confirmed for PBGFs [9, 28], it is sufficiently close to allow us to draw qualitative conclusions from the simulation results reported here. With all these approximations, the SSL curves in this work should be regarded as a guideline only, differently from CL curves which are believed to be rather accurate, as shown below. 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.
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23812
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.
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 nontouching 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. 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
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23813
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 106 lower than in the center of the fiber. By contrast, this is reduced to below 108 for the NANF, which leads to a significant reduction in CL, as shown below.
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.
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.
#220828 - $15.00 USD Received 11 Aug 2014; revised 5 Sep 2014; accepted 11 Sep 2014; published 22 Sep 2014 (C) 2014 OSA 6 October 2014 | Vol. 22, No. 20 | DOI:10.1364/OE.22.023807 | OPTICS EXPRESS 23814
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.
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
