1. Introduction

Optical fibres can guide light in a hollow core by the photonic bandgap mechanism [1] when the cladding does not support modes for certain ranges of wavelength and propagation constant. Light in the core within those ranges (the bandgaps) is unable to couple to the cladding and remains guided in the core with low loss. The best examples are silica hollow-core photonic crystal fibres (HC-PCF), which have a hexagonal arrangement of holes in the cladding. The core is formed by a larger hole which replaces 7 or 19 unit cells of the cladding and has a boundary that is antiresonant with the core mode to increase its confinement [2–4]. Other examples are “Omniguide” fibres where the cladding is a Bragg reflector made of soft glass and polymer [5] and polymer ring-structured Bragg fibres where the cladding has concentric rings of holes [6].

Another alternative hollow-core fibre design replaces the hexagonal lattice cladding with a kagome lattice [7,8], where the repeating unit is a Star of David. The kagome lattice does not support photonic bandgaps [1] and it has been proposed that these fibres guide through a series of high-order bandgaps [8] or by suppressed coupling to the cladding through a low density of states in the cladding [9]. Other characteristics of hexagonal-lattice HC-PCF such as coupling to surface modes localised in the core boundary were also not observed in the kagome-lattice fibres [4,8].

Here we present kagome-lattice fibres produced in polymer. Numerical simulations reveal that a key aspect of the guidance mechanism is the low overlap of the core modes and solid material in the microstructure which, in addition to the low density of states in the cladding [9], suppresses coupling of light to the cladding and allows it to be guided in the core.

We also consider the potential of these hollow-core microstructured polymer optical fibres (HC-mPOF) as a means to obtaining low loss transmission in a polymer fibre for wavelengths in the infrared. The most common polymer fibres are large-core multimode fibres made of polymethylmethacrylate (PMMA) and are operated at 650 nm with a loss of 0.15 dB/m [10]. Their use beyond the visible is limited by high material absorption, which reduces their compatibility with silica fibre technology and related components. Polymers more transparent in the infrared have been developed, such as CYTOP [10], but these come at a 100-fold increase in cost and are not widely available. Previous reports of HC-mPOF [6] demonstrated transmission with a loss 50x below the material absorption in the infrared, thus providing a promising route to useful PMMA fibres at these wavelengths. The key feature of HC-mPOF that allows transmission in the infrared is simply that the light propagates in air with very little overlap with the solid material, suggesting that kagome-lattice fibres may be the most appropriate approach.

2. Fabrication and characterisation

The preforms were prepared by stacking 4 mm PMMA tubes into a hexagonal array, and inserting into a larger tube. The central 4 mm tube was removed and replaced with short spacers at either end of the preform, leaving the central position empty to form the core. The preform dimensions ranged from 40–70 mm in diameter and 20–40 cm in length. They were stretched to 6 mm cane, sleeved to 12 mm and drawn to fibre [11] with different diameters in the range of 200–500 μm. Examples are shown in Fig. 1.

The kagome lattice is formed under draw conditions that allow the interstitial holes in the cladding to remain open; a slight collapse here produced dodecagonal rather than hexagonal cladding holes [Fig. 1(b)]. The air fraction of the cladding was 93% and the pitch Λ was 10–20 μm, depending on the fibre diameter. The strut thickness was as low as 300 nm, but in the core perimeter this was reduced by about a third to 200 nm, as the core expanded to a diameter of 2Λ and deformed the first ring of holes.

 

Fig. 1. SEM images of (a) a three ring fibre and (b) the unit cell in the cladding. (c) Optical micrograph of a seven-ring fibre illuminated from below with white light, showing guidance in the core.

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Fig. 2. Transmission spectra as a function of (a) wavelength and (b) normalised frequency of the three-ring fibres with Λ = 20 μm, length 1.5 m (black); Λ = 17.5 μm, length 0.5 m (blue); Λ = 14 μm, length 0.5 m (red); Λ = 12 μm, length 0.3 m (light blue). The transmission axis has an arbitrary offset. The inset shows the near field of the output.

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Fibres with three rings of cladding holes [Fig. 1(a)] were characterised using a supercontinuum source [12] and were found to support two wide transmission windows, as shown in Fig. 2. The first of these occurs at a normalised frequency of approximately kΛ = 60–140, for k the free-space wavenumber, and the second begins at kΛ = 200. The transmission did not appear to scale linearly with the pitch [Fig. 2(b)], with a smaller pitch resulting in transmission at shorter wavelengths than expected. This is most notable in the blue edge of the lower frequency transmission window which monotonically increases from kΛ = 120 to 160 as the pitch is reduced from 20 to 12 μm. The cladding modes’ properties will determine the transmission bands of the fibre and are derived from the modes supported by individual features of the microstructure in the cladding, such as the struts, in isolation [3]. The discrepancy can be accounted for by an additional 20% reduction in strut thickness as the pitch was decreased from 20 to 12 μm. Deformations during the fibre draw can lead to such non-linear scaling but this could not be verified directly owing to the small size of the struts. The scaling of transmission with respect to changes in refractive index was also investigated and a 60% reduction in wavelength was observed upon filling the fibre with water, as expected [13, 14].

Despite the large pitch and core size, the fibres were single-mode in the infrared for lengths greater than 1 m. High order modes could not be excited by adjusting the launch conditions and were only observed if the fibre was bent sharply near the output, indicating their high loss. The loss of the fibres was measured using the cutback method on 2 m lengths and reached approximately 10 dB/m in both transmission windows for the larger pitch fibres. Although still high, this is a factor of 3 improvement over previous HC-mPOF loss results [6].

3. Guidance mechanism

The numerical simulations were carried out using the adjustable boundary condition (ABC) method [15] on a structure with Λ = 10 μm, as shown in Fig. 3(a). The unit cell was created from the image in Fig. 1(b), thus the modelled structure included some draw-induced deformations but was simplified by neglecting irregularities in the fibre cross section. The fundamental mode was found to be extremely well confined to the core, with the intensity at the core boundary dropping by 40 dB [Fig. 3(b)]. The confinement loss of the fundamental mode was found to be in the range 1-10 dB/m, below but in approximate agreement with the experimental measurements; the modelled structure is expected to have a lower loss, arising from the omission of irregularities in the fibre cross section.

 

Fig. 3. (a). Structure used for the simulations (Λ = 10 μm) showing the near field of the fundamental mode on a linear (left) and logarithmic scale (right, 50 dB range shown, cf inset of Fig. 2). (b) The cross-sections of the near field taken from corner to corner and across the flat sides of the hexagonal core.

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To analyse the guidance mechanism, many cladding modes were solved and are shown in Fig. 4(a). The (ideal) kagome lattice does not support bandgaps [9], and this remains true after the partial collapse of the interstitial holes experienced and modelled here. The cladding modes can be separated into two groups of “strut” and “airy” modes [3, 16]. The strut modes have power predominantly in the solid parts of the microstructure and can be identified by the steeper dispersion curves that begin at mode effective indices n _eff > 1. The airy modes are concentrated in the holes, have flatter dispersion curves and asymptote to n _eff = 1 for short wavelengths. The probability that light from the core mode will be coupled into a cladding mode will depend on the mismatch of the two modes’ propagation constants Δβ = kΔn _eff, the spatial overlap of the two modes and their overlap with the perturbation that causes the coupling. In the case of photonic bandgaps, Δβ is sufficiently large within the bandgaps to prevent any coupling between core and cladding modes. In the case of a kagome lattice and the absence of photonic bandgaps, the other two factors must also contribute.

The Δβ between the core and cladding modes can be inferred qualitatively from Fig. 4(a). The airy modes form the densest bands but are below and separated from the core mode in n _eff, as would be expected given the smaller cladding hole size compared to the core. The strut modes intersect frequently with the core mode, however, they do not form dense overlapping bands, placing the core mode in a region of relatively low cladding mode density for some wavelength ranges. Three high-density strut mode regions are identified at approximately kΛ = 27–31, 45–60 and >120 corresponding to high loss for the fibre, whilst low-loss bands can be expected for the range kΛ = 31–45 and 60–120. The latter is in agreement with the experimental results of Fig. 2. Higher values of kΛ were not explored numerically using this approach as the calculations became prohibitively time consuming.

 

Fig. 4. (a). The core (blue) and cladding modes (black) of the modelled structure with example strut and airy modes indicated (red). The high-density mode regions in the vicinity of the core mode are highlighted in orange. Only modes of the same symmetry as the fundamental mode are shown, modes of other symmetries had identical distributions. Inset shows the calculated near field of the fundamental mode at 560 nm. (b) Examples of calculated cladding modes, also at 560 nm, close in n _eff to the core mode. The core mode falls between the 3^rd and 4^th example, indicated by the “x”.

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Quantitatively, the Δβ between the core and nearest cladding mode at each wavelength was on average 2,700 m^-1 in the low mode density regions and 1,800 m^-1 in the high mode density regions identified above, giving a difference of 50%. To estimate the mismatch required to effectively suppress coupling, the example of standard telecommunications fibre SMF-28 is used. Modelling using the ABC method [15] indicated that its fundamental mode is separated from the first high-order (leaky) mode by approximately Δβ = 10,000 m^-1. Considerations of coupling in silica hexagonal-lattice HC-PCF between the core mode and surface modes supported by the core boundary [17] indicated a similar value. Thus, the values here are not large enough to prohibit coupling altogether, but will clearly affect its extent. It must be noted that we have considered an average effect, modes with Δβ below the average will induce higher losses.

The next consideration is the overlap of the core and cladding modes. The core mode is physically separated from the cladding as a result of its high confinement [Fig. 3(b)] and the first ring of holes, which has been deformed during the draw. The hexagonal polymer core boundary is thinner than the struts in the remaining cladding and the innermost holes smaller, meaning the modes concentrated closest to the core mode will not necessarily be resonant with the remaining cladding and will have lower n _eff. The modes most likely to couple to the core mode (smallest Δβ) are the strut modes which occupy the solid parts of the microstructure – the airy modes are typically further separated from the core mode in propagation constant. The overlap between the core and strut modes can be approximated as the overlap between the core mode and the solid parts of the microstructure [18], and was found to have a minimum value of 0.01% for this fibre; values of 0.025% were typical across the wavelength range indicated in Fig. 4(a). This is an order of magnitude lower than the specific examples of hexagonal-lattice HC-PCF that appear in the literature, for which minimum values of 0.12–0.8% have been reported [4,19]. (No corresponding values for kagome-lattice fibres were reported in the literature.)

Finally the susceptibility to perturbations such as surface roughness [2] is considered. This can also be approximated by the overlap of the core mode and solid material [18] or perimeter of the holes, indicating that this fibre has a reduced susceptibility to perturbations as a result of the low overlap.

Thus, the guidance of light in the core is possible in the absence of photonic bandgaps through a reduced probability of coupling to the cladding modes. This is not only a consequence of the low cladding mode density [9]. The low overlap between the core and cladding modes and the reduced susceptibility to perturbations, arising ultimately from the low overlap of the mode field and polymer, make a key contribution to inhibiting the coupling.

4. Implications for polymer optical fibres

The low overlap of the core mode and solid material that enables the guidance of light in these fibres is especially important for polymer optical fibres made of PMMA which have a high material absorption in the infrared, as shown in Fig. 5, reaching 3000 dB/m in the wavelength range indicated. The loss values obtained here of 10 dB/m are high in themselves, but demonstrate a maximum 100x loss reduction over the material absorption. Transmission with a loss below the material loss was achieved here for wavelengths between 1120 – 1680 nm (limited by the range of the detector in the infrared), and no features in the loss spectrum corresponding to material absorption peaks were observed. The dominant loss mechanism for the fabricated fibres is believed to be irregularities in the fibre cross-section (i.e. from unit cell to unit cell in the cladding), which support modes different to the remainder of the cladding and increase the loss through resonant coupling to the core. Such issues are expected to be resolved through improved fabrication. In this section the extent to which the loss of HC-mPOF can be improved will be explored for both a kagome- and hexagonal-lattice claddings.

 

Fig. 5 The potential loss of HC-mPOF is estimated by applying the analysis of [2] and assuming 9-cell and 17-cell core sizes (thick curves). Estimates for a kagome-lattice fibre with 0.025% overlap are also shown (dotted curve) but it is unclear whether other loss mechanisms will prevail. The material absorption of PMMA is also shown (thin curve). The horizontal dashed line indicates the current operating loss of 0.15 dB/m (at 650 nm).

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Hexagonal-lattice HC-mPOF will be considered first, following the analysis applied to the silica hexagonal-lattice HC-PCF that have reached a fundamental loss limit imposed by scattering from frozen-in surface capillary waves [2]. The scattering is proportional to the glass transition temperature T _g and inversely proportional to the surface tension γ of the material. The processing temperature of polymer is much lower than silica (400 K compared to 1900 K) but the viscosity is also lower (0.032 N/m compared to 0.3 N/m) [20], meaning the overall contribution to the loss from surface capillary waves is increased when compared to silica. The result is shown in Fig. 5 under the assumption of an ideal hexagonal-lattice cladding for which all other loss mechanisms except material absorption are negligible. The core is formed by the omission of 7 or 19 cells of the periodic cladding, resulting in an approximately 1% and 0.1% overlap of the mode field with the solid material [19], and hence a two and three order of magnitude reduction in the effective material absorption.

This analysis reveals that the material absorption still remains dominant beyond the visible, whilst the scattering prevails at shorter wavelengths. An advantage over the current operating conditions for PMMA fibres (at 650 nm with a loss of 0.15 dB/m) is only gained with the larger-core example, and gives rise to three potential transmission bands at 700–1130, 1230–1330 and 1470–1600 nm, conveniently overlapping with the telecommunications wavelengths. The loss within these bands reaches as low as 0.05 dB/m.

As the material loss remains the dominant mechanism, the kagome-lattice fibres may offer an advantage over the hexagonal-lattice designs. A potential further reduction in the material loss can be achieved both through the intrinsic lower overlap of the material and light, or by increasing the core size without encountering surface modes [2,4,8]. The loss estimate for such a fibre with a 0.025% overlap of the core mode and material is shown in Fig. 5 and could potentially open up a much larger transparency window for PMMA fibres.

This result should be treated with caution as it assumes that the coupling-induced loss can be sufficiently reduced, which remains to be investigated. Coupling-induced loss differs from confinement loss experienced in other microstructured or bandgap fibres in that the light couples to the microstructured cladding, so this loss mechanism cannot be decreased arbitrarily by increasing the cladding thickness.

5. Conclusion

Hollow-core mPOF with a kagome lattice cladding were fabricated and characterised. Numerical simulations indicated that the fibres are able to guide through suppressed coupling between the core and cladding modes, a result of a low cladding mode density, a low overlap of the core and cladding modes and a reduced susceptibility of the core mode to perturbations. The latter two are a consequence of the low overlap of the core mode and the solid parts of the microstructure, in which the relevant cladding modes are concentrated and the perturbations such as surface roughness are most likely to occur. Although the fibres here and the numerical simulations had dodecagonal shaped cladding holes, this analysis should apply equally well to an ideal kagome lattice cladding.

The low overlap of the core mode and polymer holds great promise for reducing the loss of PMMA fibres. Although the hexagonal-lattice HC-PCF designs have been able to reach fundamental loss limits imposed by surface capillary waves [2], it is not as yet clear if this is possible with kagome-lattice fibres. Confinement loss can be eliminated by increasing the size of the cladding but it may not be possible to eliminate the coupling-induced loss. Ways through which the coupling may be reduced through changes in the cladding or by modifying the first ring of holes so as to form an antiresonant structure immediately surrounding the core [4] are currently being investigated. Such designs may lead to low-loss polymer fibres that can operate at any wavelength, whilst retaining the intrinsic advantages of polymers such as flexibility and low cost material and fabrication methods.