We investigate the guidance properties of low-contrast photonic band gap fibres. As predicted by the antiresonant reflecting optical waveguide (ARROW) picture, band gaps were observed between wavelengths where modes of the high-index rods in the cladding are cutoff. At these wavelengths, leakage from the core by coupling to higher-order modes of the rods was observed directly. The low index contrast allowed for bend loss to be investigated; unlike in index-guiding fibres, anomalous “centripetal” light leakage through the inside of the bend can occur.
©2005 Optical Society of America
A photonic bandgap fibre is an optical fibre whose core is surrounded by an arrangement of holes or solid inclusions that has a photonic bandgap [1–3]. Light with wavelengths corresponding to the bandgaps cannot escape the core and is therefore guided along the fibre with low loss. If the low index material in the cladding is air, the high index material must form a connected structure of high-index rods and bridges joining them . However, if the low index material is a solid, the photonic crystal in the cladding can comprise an array of isolated high-index rods in a low-index background material [4–9]. The fibre’s low-index core can then be a defect in the photonic crystal formed by omitting one or more of the rods. Such fibres have been made by filling the holes in an index-guiding PCF with high-index fluid [4,5], or by drawing solid structures made from two dissimilar glasses . Although photonic bandgaps are often associated with large refractive index contrasts, we have recently demonstrated bandgap fibres with an index contrast (index difference/lower index) of only 1% , the rods being Ge-doped silica in an undoped silica background.
Such fibres with isolated high-index inclusions have been referred to as antiresonant reflecting optical waveguides (ARROWs) [5,8,9]. According to the ARROW picture of guidance, the high-index rods in the cladding allow light to leak out from the core if they are on resonance but reflect it back into the core if they are antiresonant. This is a useful simplification of the full bandgap description. The photonic states of an isolated rod, seen as a conventional step-index core, are its waveguide modes or resonances (the analytic continuation of the modes beyond cutoff). The states of the rods couple together to form bands of supermodes , each band’s mode effective index centred on that of the rod mode from which it is formed. The bandgaps are the ranges of effective index in between.
Resonance in the ARROW picture corresponds to a match in effective index between the bandgap-guiding core and one or more supermodes, causing leakage of light from the core and hence loss. Since the bandgap-guiding core is made from the same material that surrounds the rods, resonance occurs around the cutoffs of the higher-order modes of an isolated rod. Low-loss guidance is therefore expected in the bandgaps between the rod mode cutoffs, where no supermode matches the bandgap-guiding core in effective index .
The low-index regions in the cladding (in between the rods) also support supermodes. These regions are significantly smaller than the core and the mode effective indices of their supermodes are always much lower than that of the core mode. The mismatch in effective index means that they are never in resonance with the core mode and so do not affect the spectral positions of the high-loss bands. For this reason, the exact position or periodicity of the rods is of secondary importance, providing that the regions between them remain small compared to the core.
In this work, low-contrast bandgap fibres were used to study the consequences of the ARROW picture. Several fibres were made, and bandgap guidance over wide wavelength ranges was observed. The edges of the bandgaps corresponded to the cutoffs of higher-order modes of the rods as expected, and leakage via individual rod modes was directly observed. Ellipticity of the rods was investigated experimentally and through numerical modelling, giving features in the transmission spectra consistent with the ARROW description.
The low index contrast in these fibres allowed bend loss to be observed in bandgap fibres for the first time. Unlike conventional fibres and index-guiding PCFs, in which the light always escapes “centrifugally” from the outside of the bend [10,12,13–15], anomalous “centripetal” light leakage from the inside of the bend can occur in bandgap fibres.
The bandgap fibres have a triangular arrangement of rods of index nr and diameter d embedded in a material of slightly lower index n m [Fig. 1(a)]. The rods’ centre-to-centre spacing is Λ. Their elliptical shape is described by the aspect ratio b/a, which is 1 for ideal circular rods. The overall fibre structure is characterised by the size of the bandgap-guiding core in unit cells (the number of rods omitted) and the number of rings of rods in the cladding.
The cutoffs of the modes of a circular rod are conveniently expressed in terms of its normalised frequency or V-value 
where r=d/2 is the rod radius, λ the free space wavelength and NA is the numerical aperture
The confinement loss for a low-contrast bandgap fibre like that in Fig. 1(a) was calculated using the Adjustable Boundary Condition (ABC) method  as the number of rings of rods surrounding the core was increased. The loss is plotted in Fig. 1(b) against the V-value of the rods. The loss decreases exponentially with the number of rings, with >3 rings required for a <0.1 dB/m loss in the first bandgap. The cutoffs of the rods are also indicated, and do indeed mark the edges of the bandgaps. For example, the blue edge of the first bandgap lies at the cutoff of the LP11 mode, at V=2.405.
A benefit of low contrast and circular rods is the degeneracy of certain higher-order modes. For example, the vector TE01, TM01 and HE21 modes become degenerate and form the scalar LP11 mode, and the LP21 and LP02 modes are degenerate at cutoff . This gives a small number of cutoffs with wide bandgaps in between. Deviation from circularity lifts some of these degeneracies, widening or splitting the bands and fragmenting the bandgaps. For example, the two bands arising from the LP11 and LP01/21 modes become five bands as the mode degeneracies lift, Fig. 1(a), compromising the widths of the bandgaps.
The confinement loss for the same fibre as Fig. 1(b), with 5 rings but of elliptical rods with different aspect ratios, is shown in Fig. 1(c) and (d). The area of the rods is kept constant, and V is defined as in eq. (1) for a mean radius r=(ab) 1/2. The cutoffs of the modes of the rods (as given in  and confirmed by us using the ABC method) split and become more evenly distributed with decreasing b/a, greatly narrowing the transmission windows of the fibres. Note that the first bandgap is least affected by the deformations, so it is desirable to operate a fibre in this bandgap. Although higher-order bandgaps have lower confinement loss, that loss can be matched in the first bandgap simply by increasing the number of rings. This overcomes potential problems caused by fibre deformations to which the higher-order bandgaps are more sensitive.
The bandgap fibres were made as described in  by filling PCF preforms with commercially-available multimode and single-mode fibres. When the preforms were drawn to fibre, the cores of the multimode fibres became the high-index rods and the single-mode fibres (with negligibly small cores in the final fibre) formed the effectively-undoped core.
Three bandgap fibres were made from different commercial fibres , Fig. 2. Fibres B and C had a single-cell core surrounded by 4 and 7 rings of step-index rods (NA=0.18) respectively, with d/Λ=0.34. Fibre D had a 7-cell core surrounded by 6 rings of graded-index rods (NA=0.275), with d/Λ=0.36. (Note that graded-index cores have distinct cutoff properties, e.g. V=3.5 for the LP11 mode ). The fibres were drawn to various diameters, with the high-index rods becoming 2–3 µm in diameter. Preliminary results from fibre B and a similar fibre (labelled “A”) were reported in . A common imperfection evident in Fig. 2 was the ellipticity of the fibres, and hence of their rods.
4. Bandgap measurements
Broadband light from a supercontinuum source  was coupled into the fibres using a 60× microscope objective. The transmission spectrum for fibre B (b/a=0.8) is plotted in Fig. 3(a) and shows the first bandgap, and part of the second at shorter wavelengths. (The measured range was limited by the source spectrum). Near-field images of the output endface for wide illumination (of the core and the rods) at the input are inset, showing the bandgap-guiding core to be single-mode in the first bandgap, multimode in the second, and not guiding at all in between. High loss corresponded to V ≈ 2.4, i.e. the cutoff of the odd and even LP11 modes, as expected.
The transmission spectrum for a sample of fibre C (b/a=0.6) is plotted in Fig. 3(b) and shows the first four bandgaps. Thinner samples of this fibre had the first bandgap shifted to cover most of the visible spectrum (<480–820 nm) and were single-moded for this wavelength range.
The bandgap-guided core mode has a presence in the rods, the field taking on the shape of the rod mode to which it is closest in effective index - i.e. the one which is closest to being resonant. This is essentially the result of inefficient directional coupling between mismatched waveguides. If input light is tightly focused into the core to excite only the bandgap-guided core mode, for wavelengths in a bandgap only the rods closest to the core are illuminated. In contrast, for wavelengths outside the bandgaps no light remains in the core and all the rods are illuminated. This is the mechanism by which light leaks out of the core when the rods are resonant (in the ARROW picture): focusing into the core excites a supermode of the rods. The rod modes making up the supermodes were observed directly in a sample of fibre C and are shown in the insets of Fig. 3(b) [cf Figs. 1(a) and (d)].
The transmission of fibre D with Λ≈6 µm is shown in Fig. 3(c), as are near-field images at the output. The larger core enabled this fibre to support two modes in the first bandgap. The high ellipticity (b/a<0.6) of the rods, and their non-uniformity in shape across the fibre and in size along the fibre, are consistent with the apparent absence of higher-order bandgaps. These bandgaps are sensitive to deformations and their spectral positions will vary along the fibre length, blocking any transmission. Nevertheless the first bandgap is still present, illustrating the insensitivity of this bandgap to the poor quality of the fibre.
A cut-back loss measurement was performed on a different sample of fibre C and is shown in Fig. 4(a). The minimum loss of 1 dB/m at 550 nm (in the first bandgap) undoubtedly includes a contribution due to the imperfect structure of the fibre, Fig. 2, yet is still comparable to that of commercially available air-guiding bandgap fibres at that wavelength. A near-field image of the guided mode is shown. The six satellite lobes are typical for guidance in the first bandgap, their shapes corresponding to LP01 rod modes as expected.
Sections of fibre C that were tapered during the draw process were examined. The change in diameter along the fibre produced a corresponding change in the wavelengths of the bandgaps. Only wavelengths remaining within a bandgap all along the fibre could be guided through it, so the tapers behave as optical filters. Such filters with transmission windows as narrow as 25 nm FWHM were observed, and an example spectrum is presented in Fig. 4(b).
4. Bend Loss
In conventional fibres, pure bend loss arises from the resonant coupling of light from the core to radiation modes in the cladding on the outside of the bend. In a straight fibre this coupling does not occur because the radiation modes have a lower effective index n eff than the core mode. The effect of bending can be modelled by treating the bent fibre as a straight fibre with a modified refractive index profile:
where x is the linear coordinate outward from the local centre of curvature with an origin at the centre of the fibre, n 0(x) is the unmodified index profile, R is the radius of curvature, and χ a correction due to the elasto-optic effect (χ=-0.22 for silica) [13–15]. Hence the modified index increases linearly with distance on the outside of the bend, at some point (the radiation caustic) rising above the n eff of the core mode. At this point, coupling can occur to radiation modes that now have a matching n eff, Fig. 5(a). This effect is well understood in conventional fibres (and indeed index-guiding PCFs ), and “centrifugal” radiation at the outside of a bend can be observed experimentally. However, it is notable that bend loss in such fibres can never occur at the inside of the bend.
Since bend loss is only prominent for low index contrasts, our fibres allow it to be studied experimentally in bandgap fibres for the first time. We expect there to be anomalous features that cannot be observed in conventional fibres. In particular, the existence of cladding supermodes with higher n eff than the core mode should allow “centripetal” radiation at the inside of the bend in certain circumstances.
To a first approximation (for large bend radii) the effect of bending will be to change all refractive indices (including the n eff of the bands) across the fibre, as indicated in Fig. 5(c). Thus the n eff of the core mode intersects that of a band on both the inside and the outside of the bend. Which side the light leaks out to depends on which band is closer in effective index. Fig. 5(b) is a calculated plot of n eff for the bands of supermodes in a representative bandgap fibre cladding with n r=1.465, n m=1.450 (1% contrast), Λ=6.0 µm and d=2.0 µm; the ABC method was used. It shows that the lower-n eff band is closer to the core-mode n eff at the blue edge of the bandgap, and the upper band is closer at the red edge.
Near the blue edge of a bandgap, bend loss is qualitatively the same as in conventional fibres. The n eff of the core mode lies very near to the band beneath it. A bend raises the n eff of this band on the outside of the bend so the light escapes centrifugally from the outside. In contrast, bend loss is anomalous near the red edge of the bandgap. The n eff of the core mode lies very near to the band above it. A bend lowers the n eff of this band on the inside of the bend so the light escapes centripetally from the inside of the bend, Fig. 5(c), unlike in any other type of waveguide.
At wavelengths in the middle of the bandgap the core mode is far removed from the bands above and below and is unaffected by large-radius bends. Hence the net effect of a slight bend is to narrow the bandgap, with loss greatly increasing at the edges. The steeper slope of the red edge of the bandgap in Fig. 5(b) means that the n eff mismatch between core and band increases more rapidly at that edge, so the bandgap should narrow more slowly at the red edge than the blue edge as the fibre is bent.
As well as reducing the width of the bandgap, increasingly tighter bends will at some stage create enough difference between the refractive indices of individual rods to weaken the coupling between them. Rods at the same x remain strongly coupled, but coupling between rods with differing x will weaken. Discrete supermodes should then become distinguishable within the bands. These will appear as resonances in the middle of the bandgap and manifest themselves as spectral features with reduced transmission, beyond a general decrease in transmission across what remains of the bandgap. The bend loss was investigated experimentally on a sample of fibre C with Λ=7 µm, in the first bandgap. The fibre ends were fixed onto two stages and the fibre was bent by moving the stages towards each other. The fibre was kept in one plane and prevented from twisting.
Approximate bend radii (down to 11 mm) were determined and transmission spectra recorded for various stage positions, Fig. 6(a). The results were consistent with the discussion above. For large bend radii the low-loss range narrowed due to high bend loss at the edges, the blue edge being affected more than the red edge and the middle of the bandgap remaining unaffected, Fig. 6(b). Loss at the red edge of the bandgap will be to the inside of the bend. For decreasing bend radius, the expected discrete resonances appeared in the middle of the bandgap, with transmission across the entire bandgap becoming increasingly compromised.
Our description of bend loss should also apply to bandgap fibres with higher index contrast. They will however be less susceptible to bend loss, for the same reasons as conventional step-index fibres. A fibre with raised contrast will have a smaller transverse scale to maintain corresponding V-values, so a tighter bend is needed to reproduce a given scaled effective-index variation as depicted in Fig. 5(a). Equivalently, for the high-contrast fibre, the band plot of Fig. 5(b) will be stretched vertically even though horizontal positions on the plot will be little changed, making bend-induced mode coupling to radiation modes fall off more rapidly as we move in from the band edges. We believe this to be why bend loss in bandgap fibres has not been reported before now.
The guidance properties of low-contrast bandgap fibres were investigated experimentally and through numerical modelling, confirming the ARROW picture of bandgap guidance and giving insight into their sensitivity to deformations, leakage mechanisms and bend loss. Elliptical rods were considered for the first time, and it was found that the first bandgap was the most robust to deformations. The fibres transmitted light over a wide wavelength range, up to the fourth bandgap, with measured transmission losses comparable to commercial bandgap fibres at the same wavelengths. The wavelengths of the edges of the bandgaps corresponded to the cutoffs of higher-order modes of the rods. Coupling to such modes from the core was observed.
The bend loss of bandgap fibres could be investigated for the first time owing to the low index contrast in our fibres. Bend loss near the red edge of a bandgap is due to anomalous centripetal radiation towards the inside of the bend. The middle of the bandgap is insensitive to gradual bending, though tighter bends compromise transmission across the entire bandgap as the supermode structure of the cladding bands fragments. The ease of fabrication, low-loss and spectral transmission may see these fibres employed as filters or in other capacities in the near future.
We thank A.K. George, W.A. Lambson, C.K.W. Cheung, N.Y. Joly, H.R. Perrott, G.J. Pearce, K. Lyytikainen and A. Roberts for their contributions. N. Issa is acknowledged for providing the ABC code used in the numerical modelling. We acknowledge support from M.C.J. Large and M.A. van Eijkelenborg for A.A.’s visit to Bath. A.A. acknowledges financial support from the Research and Scholarships Office and the School of Physics, University of Sydney.
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