We propose a guidance mechanism in hollow-core optical fibres dominated by antiresonant reflection from struts of solid material in the cladding. Resonances with these struts determine the high loss bands of the fibres, and vector effects become important in determining the width of these bands through the non-degeneracy of the TE and TM polarised strut modes near cut-off. Away from resonances the light is confined through the inhibited coupling mechanism. This is demonstrated in a square lattice hollow-core microstructured polymer optical fibre.
© 2008 Optical Society of America
Hollow-core fibres transmit light in an air core [1, 2], and two guidance mechanisms have been identified to explain their optical properties. Hollow-core fibres with a photonic crystal in the cladding, consisting of e.g. a hexagonal lattice of holes, operate using a photonic bandgap [3, 4]. The bandgaps prevent light of certain frequency ranges from coupling from the core to the cladding as the cladding supports no guided modes in these bandgaps.
The second mechanism has thus far been observed only in kagome lattice fibres, where the repeating unit in the cladding has a Star of David shape [5, 6]. In this case, the cladding does not support bandgaps so the light in the core could propagate through the lattice and not be confined. However, it is largely prevented from coupling to the cladding in certain wavelength regions due to a low density of cladding modes  and the core mode’s extremely low overlap with the solid material in the fibre . The latter minimises the overlap between the core mode and the cladding modes and any perturbations that would facilitate their coupling, both of which are concentrated in the solid material . The resulting inhibited coupling leads to the confinement of the core mode. Moreover, the overlap integral between the core mode and cladding modes is further reduced due to mismatches in phase , and regions of high-mode density and high loss were found to correspond to resonances of the struts in the cladding .
Here we present a fibre with a square lattice of cladding holes. This fibre falls into the same class as the kagome lattice fibres, as its cladding does not support photonic bandgaps. A discussion of its guidance mechanism in terms of antiresonances and the inhibited coupling mechanism will be presented, and its transmission properties shown to depend on the thickness of the struts in the cladding. The role of vector effects, arising from the index contrast in the fibre will be identified and discussed.
The fibres were made from polymethylmethacrylate (PMMA) and the preform was assembled from tubes 4 mm in diameter arranged in a square lattice and inserted into a larger tube. A single tube was removed from the centre to form the core. The preform was drawn to cane of 6 mm diameter, sleeved to 12 mm diameter, and drawn to fibre of 200 to 640 µm diameters (Fig. 1).
During the draw process the surface tension deforms each tube, straightening the walls between the four points of contact with the neighbouring tubes (3 points for alternate tubes surrounding the core). This leads to the square lattice composed of rectangular polymer struts. The core has an octagonal shape, with alternate struts stretched to approximately half the nominal width due to expansion of the core, as shown in Fig. 1.
Detailed measurements of the fibre dimensions were taken using a scanning electron microscope to reveal that the strut thickness δ ranged from 250–820 nm from the smallest to largest diameter fibre; a variation in δ of 8% was measured in the fibres’ cross sections. The core size and pitch Λ ranged from approximately 23–60 µm and 6–18 µm respectively. The four thinner struts surrounding the core were as thin as 120 nm.
An alternative approach to fabricating a square lattice is to pressurise the tubes in the preform – this will form a lattice oriented at 45° to the current result. However, this would produce a doubled strut thickness as each strut will be formed by the walls of two adjacent tubes, and the accumulation of material at the strut intersections will lead to the formation of solid islands. For these reasons, this approach was not used for the fibres presented here.
3. Analysis of guidance mechanism
The idealised fibre structure shown in Fig. 1(a) was modelled using the Adjustable Boundary Condition method  and the fundamental mode was found to have many avoided crossings with cladding modes. This is typical of the inhibited coupling mechanism expected, and confirmed the absence of photonic bandgaps.
Given the large aspect ratio of the struts (Λ/δ=25), they can be approximated as planar waveguides of infinite width . Considering them as isolated waveguides and neglecting any coupling between them, they can be described using the normalised frequency V,
for n co and n cl the core and cladding indices of the struts when viewed as independent waveguides, and n the refractive index of PMMA. A mode supported by such a waveguide can be described using the U parameter 
for n eff the mode effective index.
These struts will become transparent to light in the core (i.e. reflection and guidance will be minimised) when the strut width corresponds to a phase change of an integer multiple of π , as given by
for κ the transverse wavenumber and m a positive integer, which can be expressed as
when n is approximated as 1.49 for PMMA. The expression V/π=m/2 in Eq. (3) corresponds to the cut-off frequencies of the modes supported by such a planar waveguide (i.e. the struts), occurring when U=V and n eff=n cl. These values of V will correspond to high loss features in the transmission spectrum, as the cladding struts will be transparent to the light in the core and hence be unable to confine it .
This argument presented thus far fails to predict behaviour at long wavelengths as the fundamental strut mode corresponding to m=0 has no cut-off and its U asymptotically approaches V as wavelength increases. To expand the argument further vector effects must be considered. Although the scalar approximation is not made in this argument, and the TEm and TMm polarised strut modes are always degenerate far from and at cut-off (m denoting the number of nodes in the mode field across the strut), they are not degenerate near cut-off and the difference between them increases with increasing index contrast . The TMm modes remain very close to cut-off for a significant range of V/π above m/2, thus extending the range of wavelengths over which the struts would be effectively transparent to the core mode (Fig. 2). This results in a widening of the high loss regions, now bounded by the TE cut-offs on the low frequency side, and the “effective” TM cut-offs on the high frequency side. The TM0 mode is sufficiently close to cut-off for V>0, which results in high loss at finite wavelengths.
To consider the “effective” cut-off and the non-degeneracy near cut-off in more detail, two approaches may be taken. Firstly, near cut-off, the deviation from cut-off differs between TE and TM modes as described by 
where N differs for TE and TM modes, as defined by
Numerically, it was found that this can be approximated as n eff ∝-N 2 λ near cut-off.
This indicates that as the refractive index contrast in the structure is increased, the TM modes’ deviation from cut-off is delayed by a factor dependent on the index contrast, through the term N. Therefore the “effective” cut-off for TM modes will always occur at higher frequencies for higher index contrast structures, manifesting in the width of the high loss regions. Examples of this deviation can be seen in Fig. 2, where the TE and TM modes are compared for different index contrasts.
Secondly, the difference between the polarisations can be visualised by considering the fraction of power of the strut mode that is concentrated in the strut η . In the inhibited coupling argument, the cladding modes most likely to couple to the core mode were concentrated in the struts, and hence their overlap with the core mode was approximated as that of the core mode and the solid material in the fibre . This becomes invalid when the strut modes extend increasingly away from the struts as they approach cut-off. The TM modes become less confined at higher frequencies than the TE modes, as shown in Fig. 2(c).
The arguments above do not take into account the coupling between struts and the finite length of the struts. The boundary conditions imposed by the finite length result in modes with a different number of nodes along the struts’ length being non-degenerate; the number of such modes will increase with frequency. However, given the aspect ratio of the struts, this will only be a small effect. Although both the coupling and finite length will result in a wider distribution of modes (and closure of photonic bandgaps that would have arisen if, e.g. a 1D grating structure was considered), guidance in the core is still possible through the inhibited coupling mechanism.
4. Fibre characterisation
The transmission of the fibres described in Section 2 was characterised using a supercontinuum light source (480–1700 nm) and is presented in Fig. 3. The range of sources, detectors and fibre sizes allowed the frequency range of V/π=0.16–1.81 to be investigated, in which multimode transmission was observed to occur in three transmission bands.
The loss was measured using the cutback method on 4 m lengths of fibre using the supercontinuum source, a HeNe laser (633 nm) and a frequency doubled Nd:YAG laser (532 nm). Loss minima occurred in approximately the middle of each transmission band and were found to be in the range of 4–7 dB/m; a factor of 2.5 improvement over previous loss figures for hollow-core microstructured polymer optical fibres (HC-mPOF) . It should be noted that no loss features associated with material absorption were observed, despite the material absorption reaching 3000 dB/m in the wavelength range used .
The values of frequency V/π=0.5, 1, 1.5 predicted by Eq. (3), applying to TE mode cut-offs, are indicated in Fig. 3 and accurately predict the low-frequency edge of the high loss regions in the spectrum. To determine the high-frequency edges of the high loss regions, the “effective” cut-off of the TM modes must be formalised. In previous work , a difference in propagation constant between two modes Δβ of order 104 m-1 was used as a sufficiently small difference to allow coupling between the modes. This translates to an effective cut-off at (V-U)/π of order 10-3 [rather than the actual cut-off at (V-U)/π=0]. Such a value accurately predicts the high-frequency edges of the high-loss region and the low-frequency edge of the first transmission band. At this proximity to cut-off, the fraction of power of the strut modes concentrated in the struts themselves is 20%, and the modes are delocalised. This places the effective TM cut-offs at V/π≈0.16, 0.59, 1.07, 1.56.
The error bars associated with the V/π values account for the 8% deviation in the strut width δ in the fabricated fibres, which results in a narrowing of the transmission windows, and the variation of the PMMA refractive index from 1.50 to 1.48 across the wavelength range used. The corresponding width of the error bars in frequency space at high frequencies explains why the 4th transmission near V/π=1.75 was not observed. In addition, the increased number of modes confined along the length of the struts at high frequencies will contribute to increasing the loss through variations in the strut lengths within each fibre. Still higher frequencies were outside the experimental range.
To verify the vector effects directly, the transmission of a 0.3 m length of fibre was recorded before and after filling the fibre with water (Fig. 4). The high loss region near V/π=0.5 was observed to narrow as its high frequency edge shifted to lower frequency by V/π=0.03. (The entire spectrum was also shifted in wavelength as a result of the change in index contrast [13, 14].) The size of the shift was confirmed by calculations similar to those of Fig. 2, and corresponds to the change of the effective cut-off of the TM1 mode when N was changed from 0.67=1.0/1.49 for polymer struts with air cladding, to 0.89=1.33/1.49 for polymer struts with a water cladding. The lower frequency edge of the high loss region corresponding to the TE1 mode did not shift.
Some aspects of the guidance mechanism discussed here have been identified in kagome lattice fibres . However, a kagome lattice will have shorter struts (hence a lower strut aspect ratio) and a larger number of strut intersections than a square lattice of identical strut width and pitch. The assumptions of independent planar waveguides of infinite width do not hold as strongly for kagome lattices as they do for square lattices. The coupling between the struts and boundary conditions imposed by the struts’ length will thus play a more significant role in kagome lattices. Square lattices made by inflating the tubes as discussed at the end of Section 2 will encounter the same limitations. The thickness of the struts would double whilst the pitch remained constant, thus halving the aspect ratio. Apart from the shift in the fibre transmission spectra arising from the thicker struts, the introduction of additional resonances from the solid islands may contribute to the guidance through additional resonant confinement or loss.
We have fabricated and characterised a large pitch square lattice hollow core optical fibre for the first time as well as explained the guidance, with a simple but efficient model, in terms of antiresonances with the struts in the cladding and the inhibited coupling mechanism. Vector effects play a role in the width of the high loss regions, as the high-frequency edges are determined by the TM modes. The TM mode cut-offs deviate from the TE modes’ by an amount depending on the index contrast, as demonstrated. We have also confirmed that the broadband guidance enabled by the inhibited coupling mechanism achieved in kagome lattice fibres is not limited to kagome lattices but can be extended to other fibre geometries.
SEM images were taken at the Electron Microscope Unit of the University of Sydney. The authors thank Felicity Cox for assistance with characterisation and Maryanne Large and Martijn van Eijkelenborg for useful discussions.
References and links
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