We experimentally demonstrate solid-core photonic crystal fibers that guide via the inhibited coupling mechanism. We measure an overall transmission window of more than an octave, as well as an uninterrupted width of almost one octave. The fiber is fabricated in polymer, with high-index ring-shaped inclusions. This type of fiber was conceived based on a simple model which shows that the cutoffs of the modes of a thin ring cluster around the cutoffs of planar waveguide modes. The model shows that such ring based fibers are closely related to kagome and square lattice hollow core fibers, and have transmission bandwidths that could in principle reach 1.6 octaves. Measured transmission properties are in good agreement with rigorous modelling.
© 2010 Optical Society of America
Since the original demonstration of photonic crystal fibers (PCFs) many variations on the essential idea have been investigated . The original PCFs featured a hexagonal array of cylindrical air holes running down the fiber length. In these fibers the core has a higher refractive index than the cladding, and light is guided by modified total internal reflection (MTIR). In other fibers the core has a lower refractive index than the cladding, ruling out light guidance by MTIR, whether it be a solid core or an air core. These fibers have a cladding consisting of high-index regions (which may be connected) delimited by low-index regions. The high-index inclusions support their own guided modes and the guidance properties of the fibers are shaped by the modes of the inclusions, particularly their cutoffs. Resonant effects result in these fibers having high transmission spectral regions separated by regions of high loss.
Many applications of solid-core fibers with low-index cores derive from the ability to tailor dispersion, particularly in nonlinear optics where the dispersion determines the properties of solitons, which require anomalous dispersion, or four-wave mixing, which has a phase matching condition. One of the limitations is that the transmission bands in these fibers are narrow, making them unsuitable for broadband nonlinear applications such as supercontinuum generation. While nonlinear phenomena can extend over several transmission windows [2, 3], more flexibility can be achieved if the transmission bands are broadened.
Two guidance mechanisms have been identified in low-index core PCFs. The first is photonic bandgap guidance and occurs when the high transmission regions correspond to photonic bandgaps – an absence of cladding modes. The Anti-Resonant Reflecting Optical Waveguide (ARROW) model provides a good description of this behaviour [?, 4]. Bandgaps correspond to anti-resonances between the core and the high index inclusions in the cladding. These inclusions scatter strongly leading to confinement to the core. At the resonances associated with modal cutoffs, the inclusions are transparent (similar to Fabry-Perot cavities on resonance) so the light can leave the core by coupling to the inclusions, leading to poor transmission. The second mechanism is inhibited coupling guidance where the high transmission regions do not correspond to bandgaps. Cladding modes exist in those regions, but light from the core mode is unable to couple to most of these owing to a lower density of cladding modes and a low overlap between these and the core mode [6–8]. Coupling to some modes of the inclusions still occurs, and hence the ARROW model can be applied to predict the high loss regions [8–10].
Bandgap guidance has been observed in both solid and hollow-core fibers. In hollow-core fibers only one bandgap of 20 % width compared to the central wavelength (0.3 octaves) is typically observed , although larger bandgaps have recently been predicted . In solid-core examples many bandgaps are observed with the primary (longest wavelength) bandgap being the widest at approximately 60 % of the centre wavelength (0.85 octaves) . Cladding structures demonstrated have included high index circular or elliptical rods [13, 14], annuli  and rectangles , with only a slight index difference between these inclusions and the background material. Inhibited coupling has only been observed in hollow core kagome- and square-lattice fibers [10, 17], the cladding of which is essentially composed of thin struts resembling planar waveguides, and typically provides wider transmission windows of 70 % (one octave) , but with a higher loss compared to the hollow-core bandgap fibers. Ultimately the different shape and arrangement of the cladding features and the cutoffs of the modes they support leads to this difference in behaviour. In practice, the width of the transmission windows (be they via bandgaps or inhibited coupling) depends on how it is measured, e.g. on the loss and length of the fiber. The widths quoted here were estimated from transmission or loss spectra presented in the respective references.
In this paper we investigate cladding inclusions that consist of high index rings. Two dimensional photonic crystals composed of lattices of air rings , and metal-coated silicon pillars  were demonstrated to possess broader bandgaps relative to solid cylindrical holes or cylinders. In a fiber geometry, dielectric rings of only slightly higher refractive index than the background have been shown to reduce bend loss which is associated with bandgaps being “deeper”  rather than wider (in bandwidth). Here we show that if the cladding inclusions consist of a ring of high refractive index in a low-index background with a lower index inside the ring, then the low order modal cutoffs cluster. This leads to wide spectral regions with high transmission in a solid-core fiber via the inhibited coupling mechanism. We confirm this result experimentally in microstructured polymer optical fibers (mPOF) [?], which exhibit transmission windows spanning more than an octave.
2.1. Slab waveguide model
We consider a fiber with a cladding consisting of a hexagonal lattice of rings with lattice constant Λ, in a uniform background of index n3. The rings have an inner radius a and outer radius b and an index n2 higher than the background. They are filled with a third material with index n1, so that n1 < n3 < n2 (see Fig. 1). For the ARROW mechanism to operate, the thickness of the ring needs to be sufficient for it to support at least one guided mode. We consider fairly sparse inclusions, with Λ/b = 6.75, so the density of inclusions is low (lower, for example, than in ) and modes of individual high index rings are only weakly coupled. The fiber transmission is then largely determined by the modes of the individual rings [?] rather than their supermodes. Indeed, from the tight binding model, bands of the cladding can be seen as resulting from coupling between the modes of individual rings, with the width of the bands being proportional to the overlap between modes of two neighbouring rings [22, 23]. With the inclusions being sparse, coupling is low and bands are close to the dispersion curves of the individual inclusions.
While below we analyze these fibers using a rigorous multipole treatment, we first discuss a simplified treatment which gives physical insight and explains why we can expect such fibers to have unusually wide transmission bands.
Since the properties of the fiber derive from the mode structure of the individual inclusions, we consider the modes of the ring in Fig. 1(b) in the limit of a thin ring, i.e. b − a ≪ b. In this limit the ring looks locally like an asymmetric planar waveguide with thickness b – a. The modes of such planar waveguides are well understood: they support transverse electric (TE) and transverse magnetic (TM) modes. The TEp and TMp modes have p nodes in their electric and magnetic field, respectively. In contrast, the modes of the rings have two indices, associated with the number of nodes in the radial (p) and azimuthal (m) direction. Since for a vanishingly thin ring the curvature has vanishing importance (for b/λ ≳ 1), we conclude that for such rings the azimuthal index m is relatively unimportant, and that the modes’ dispersion relation essentially only depends on the radial index p.
The cutoffs of the modes of a single ring, obtained from full analytic expressions  are shown as a function of normalised ring thickness a/b and normalised frequency b/λ in Fig. 2(a). The parameters used were n1 = 1.00, n2 = 1.5802, and n3 = 1.4898, and were chosen to match the polymers used to fabricate the fibers (see Section 3) at a wavelength of 633 nm, without material dispersion. All results below are obtained using these values unless otherwise specified. In Fig. 2(a) modes with azimuthal numbers m = 0 are indicated with black curves while the curves for m = 1 to m = 6 are indicated in increasingly lighter shades of grey. The cutoffs of the TE modes of planar waveguide with width b – a are shown in red. The asymmetric planar waveguide has well known closed-form solutions for its cutoffs. In terms of its dimensionless frequency V, the pth cutoff Vp for TE modes is given byEq. (1) by a factor n2/n1 in the argument of the arctangent, but on the scale of Fig. 2 the resulting cutoff frequencies are very close to those of the TE modes and are omitted in the figure for clarity. The right-hand side of Eq. (2) is a constant which depends on the refractive indices and the mode number, but not on wavelength or the ring diameter. Thus we find that in Fig. 2(a) the cutoffs of the planar waveguide modes are given by hyperbolas. In particular, for a given mode as a/b → 1, the cutoff frequency b/λ → ∞. At fixed frequency, as a/b → 1, b – a eventually becomes much smaller than the wavelength and the waveguide supports fewer and fewer modes.
The white region near the top of Fig. 2 (a → b), indicates that the ring needs to have a sufficient thickness for it to support guided modes. In this region the ARROW mechanism does not operate, but light can be guided by MTIR. Indeed, the low frequency homogenized refractive index of the inclusions is given byFig. 2 this condition is satisfied for a/b ≳ 0.43. The fact that fibers can be index-guiding at long wavelengths does not imply that index guidance is the dominant guidance mechanism at all wavelengths as has been discussed in Ref. .
For a = 0 the cutoffs are those of a standard step-index fiber, which are not strongly related to the cutoffs of a planar waveguide. However, as a/b increases and the ring gets thinner the cutoffs cluster, approaching those of the asymmetric planar waveguide, especially at high frequencies as seen in Fig. 2(a).
As a result of this clustering, we see the opening of some substantial frequency regions without modal cutoffs, most prominently for a/b ≈ 0.9 and b/λ ≈ 5. As the transmission bands are delimited by the cutoffs of the ringed inclusion, transmission bands are expected over these frequencies, which appear to cover more than an octave. However, this is not quite true: the dots in Fig. 2(a) indicate the positions of the cutoffs for m = 7, ···,20, for a/b = 0.9. While the modes with higher azimuthal dependence would clearly fill this gap, they will be strongly confined to the rings and will have a high spatial frequency. Following the inhibited coupling mechanism arguments, both the strong confinement  and high spatial frequency  result in a low overlap and very little coupling with the guided modes of the core, and thus the fiber’s guidance properties are not affected by these ring modes. Therefore, based on this simple model of a solid-core fiber with ring-shaped inclusions we conclude that these fibers should exhibit wide transmission bands via the inhibited coupling mechanism which can exceed an octave in width.
Figure 2(b), like Fig. 2(a), shows the a/b ratios at which modes of a single ring-shaped inclusion with same parameters as Fig. 2(a) are cut off, but as a function of normalised wavelength λ/b and for m = 0 to m = 3. The cutoffs ratios a/b of the planar waveguide model form straight lines (omitted in Fig. 2(b) for clarity) and are the asymptotes the ring modes tend towards. Using wavelengths rather than frequencies makes it easier to look at details of the first transmission window, in particular for comparison with further simulation and experimental data in Sections 2.1 and 3. In Fig. 2(b) we also identified a number of modes by their scalar LP notations for further discussion and consistency with prior publications (e.g. ). The step-index fiber scalar modes LPxy cluster around the p = y – 1 planar waveguide mode. The azimuthal index m corresponds to the vector step-index fiber modes, with m = 0 corresponding to TE0y/TM0y modes, and m > 0 to HEmy and EHmy.
2.2. Full PCF simulations
As a first step to test this prediction we performed rigorous numerical calculations of a full PCF rather than a single ring, based on the multipole method [?, 25–27], and from now on including material dispersion. Assuming only the lowest order ring modes m = 0 and m = 1 affect guidance in the core, we expect to see a high transmission region extending over an octave for all values of a/b ≳ 0.6, as seen in Fig. 2. For a/b = 0.6 the relevant cutoffs for m = 0 and m = 1 are λ/b ≈ 0.3 and λ/b ≈ 0.7 respectively. Accordingly, we simulate fibers with large values of a/b, with b chosen so that the transmission window is near the visible, to be compatible with fabrication in polymer, and with relatively large values of Λ/b to minimize broadening of bands due to coupling between inclusion modes. We note that all three fibers having a/b > 0.43 are expected to be index guiding in the long wavelength limit as discussed in Section 2.1.
Figure 3 shows the simulated transmission loss for the lowest loss modes for three selected geometries labelled I-III (corresponding to fibers fabricated as described in the next section). The figure shows losses of all calculated modes of same symmetry as the fundamental mode (in McIsaac’s notation classes 3 and 4 for C6v symmetry, ), and the lowest loss mode corresponds to the fundamental core mode. Fibers I (a/b = 0.71) and II (a/b = 0.68) have similar spectral features, with an overall low-loss transmission window covering 450 nm-950 nm and 630 nm-1330 nm respectively, interrupted by loss peaks. Superimposed onto Fig. 2(b) are the segments in parameter space corresponding to the a/b values of fibers I-III simulated in Fig. 3 and their low loss wavelength region. The modal fields around the loss peaks and an inspection of Fig. 2(b) show these interruptions arise from coupling to m = 2 and m = 3 modes of the high index rings. The m = 2 loss peak is prominent, with increased loss on the short wavelength side and decreased loss on the long wavelength side due to Fano-type resonances . As expected, the coupling of the core mode to m = 3 resonances is much weaker, and only occurs for an extremely narrow wavelength range. Coupling of the core mode to higher m resonances is too weak to be seen in the calculations. Fiber III has larger a/b = 0.92, so that cutoffs of different m have already clustered towards the slab modes. The fiber thus features a wide wavelength range with low losses between 370 nm and 690 nm, uninterrupted by higher order resonances. Above 980 nm the rings do not support modes but the homogenized index of the overall inclusions is lower than that of the background and the fiber becomes index-guiding.
3. Fabrication and characterisation
The fibers modelled above were fabricated in polymer using methods developed for microstructured polymer optical fibers (mPOF) [?]. Polycarbonate (PC) tubes with a/b ≈ 0.7 were sleeved using polymethyl methacrylate (PMMA) to form tubes representing the unit cell of the cladding. The primary preform was formed by stacking 58 of these unit cell tubes, with a solid PMMA rod in the centre to form the core. The stack was inserted inside a larger PMMA consolidating tube and stretched to cane with vacuum applied to close the interstitial air gaps between unit cells. The cane was sleeved using PC to form the secondary preform, and this was drawn to fiber with diameters in the range of 100 – 250 μm. Pressure was used to control the size of the holes and hence the a/b value of the fibers. The dimensions of the structure of the fibers were measured using a field emission gun scanning electron microscope (FEG-SEM). An image of a fiber end-face is shown in Fig. 4(d).
The fibers were characterised using a supercontinuum source and an optical spectrum analyser. Short fiber lengths (4 cm) were used so as to clearly resolve spectral features associated with the modes of the rings, minimising features of the absorption of the polymer. The transmission spectra of several fibers are shown in Fig. 4. These fibers correspond to Fibres I–III as simulated in Fig. 3 and the parameter space they cover is indicated in Fig. 2(b). The edges of the transmission windows and the positions of the anticipated loss features associated with m = 2 – 5 resonances expected from simulations are indicated in the figure, and show good agreement with the experimentally obtained spectra. In particular the width and depth of the loss features clearly decrease with increasing m, and become lost in the noise for m = 5. Loss features for m > 3 were too narrow to be seen in our full multipole simulations, and the corresponding values used in Fig. 4 are taken from exact cutoff calculations of modes of a single ring inclusion of each fiber, including material dispersion. It is noted that the exact position of the resonances and of the transmission window depends on both the a/b and b – a values of each fiber, which were difficult to measure with very high accuracy, in particular as b – a is of the order of 100’s of nm. The m = 2 and m = 3 resonances are clearly seen in Figs. 4(a) and (b), which correspond to a/b values of 0.71 (Fibre I) and 0.68 (Fibre II) respectively. An overall transmission band ≈ 1.33 octaves wide is observed in Fig. 4(b), with the largest uninterrupted section 0.97 of an octave wide. In Fig. 4(c) (Fibre III), the a/b value is larger at 0.92 such that the higher m resonances have merged with the lower order ring modes into the p = 0 band, and are no longer distinguishable. This results in a wide loss region corresponding to the p = 0 band, with guiding by MTIR occurring beyond its red edge, and guiding by inhibited coupling below its blue edge, as expected from the calculations. Although this shorter wavelength transmission band is almost an octave in width (and with no interruptions), this was not observed experimentally, being beyond the spectral range of the source used.
The experimentally observed loss features caused by the m = 2 and m = 3 (and to a lesser extent, m = 4) resonances seen in Fig. 4 are wider than expected from calculations. The observed transmission losses are also substantially larger than those predicted in Fig. 3 by the simulations for perfect structures. This is attributed to a variation between individual rings in the fabricated fibers. Further simulations showed that adding slight random variations of only 5% in a and b can increase losses by a factor of 10 (in dB/m) in the middle of the transmission windows, and by several orders of magnitude near the edges.
As observed for values of a/b ≈ 0.7, the overall transmission windows are very wide but contain interruptions, whereas no interruptions were observed for the larger value of a/b ≈ 0.9, as the higher m resonances have merged into the loss band. Using ARROW arguments, the maximum possible width of the transmission windows (and position of any interruptions) are determined by the mode cutoffs. This maximum width ignores any coupling between the rings, resulting band formation and broadening of the high loss regions. Figure 5(a) shows the maximum possible width of the longest wavelength transmission window as a function of a/b derived from LP02 and LP21 cutoffs from Fig. 2, using the same indices and no material dispersion. The transmission window is seen to increase in width as the modal cutoffs of rings approach those of planar waveguides (a/b → 1), reaching an uninterrupted width of 1.55 octaves at a/b = 0.964. These large values of a/b correspond to small values of λ/b < 0.1. Figure 5(b) shows the same for a fixed value of b = 1.8 μm, with the material dispersion of PMMA and PC taken into account. In this case the width reaches a maximum near a/b ≈ 0.7 and then decreases. The decrease occurs as the material dispersion shifts the blue edge of the transmission windows to longer wavelengths. Given the widths are of order one octave, it is reasonable to expect material dispersion to play a role.
In the limit of a/b → 1, the width of the transmission bands can be determined from the cutoffs of the planar waveguide modes given in (1) and (2). The width of a transmission window (in octaves) lying between the pth and (p + 1)th cutoffs isFig. 5(a). The curves in Fig. 5(a) seem to approach a lower value, and this is due to modes in the p = 0 group not being exactly degenerate for a/b ≠ 1.
More generally, Eqs. (1)–(4) are not specific to the parameters used in this paper, and present the cutoffs and upper limits to the transmission window width for any refractive index combination. Allowing this additional degree of freedom and by referring to Eq. (1)p may be minimised by increasing the value of n2 – n3, i.e. increasing the index of the ring compared to the background, or reducing the value n3 – n1, i.e. having symmetric planar waveguides with the index inside the ring the same as the background. The symmetric planar waveguide gives V0 = 0 as the fundamental mode has no cutoff. However, at sufficiently long wavelengths the fundamental mode would extend sufficiently far from the rings enabling the core mode to couple strongly to it, and also the inclusions to couple strongly to each other, depending on their separation. This coupling forms a band of modes and the higher order modes within that band do have a cutoff. These effects will result in an effective cutoff for the fundamental planar waveguide mode as, for example, investigated in  which obtained a value of V0 = 0.32π for the parameters used in that work. The exact value of V0 depends on the index contrast in the structure. Using this value, the long wavelength transmission window is limited to log2(π/0.32π) ≈ 1.6 octaves for similar symmetric slab index profiles.
The symmetric case was considered in  for a denser structure than considered here and the same conclusions were drawn: the longest-wavelength transmission window is the widest, with its long wavelength edge determined by the band arising from the fundamental ring mode. The arguments used above to infer a maximum bandwidth – symmetric waveguide and/or high index contrast between rings and background; large a/b; small λ/b to operate in the longest wavelength transmission window – strongly resembles the hollow-core kagome and square lattice structures already demonstrated [7,8,10,17,18], suggesting these are in some sense close to optimal. The only significant difference between those and this work is that the kagome and square lattices are connected structures.
We have shown that a PCF with high-index ring-shaped inclusions can guide via the inhibited coupling mechanism with transmission windows larger than an octave. The geometry of the rings is important in positioning the cutoffs of the modes of the rings. Sufficiently thin rings resemble planar waveguides and the cutoffs of their modes cluster around those of planar waveguide modes. This clustering is less pronounced in slightly thicker rings, and whilst they still produce wide transmission windows they contain interruptions caused by higher order ring modes. A consideration of the structures more generally identified how to maximise the width of the transmission windows and position the resonances. Since the transmission windows and cladding resonances have a strong influence on waveguide dispersion, ring-based PCFs might be an interesting and versatile platform to extend recent experimental work on tunable nonlinear pulse propagation in PCFs experiments and applications .
This research was supported by the Australian Research Council (ARC). CUDOS is an ARC Centre of Excellence. B.T.K. is supported by an ARC Future Fellowship and A.A. by an ARC Australian Research Fellowship. The SEM facilities at the Australian Centre for Microscopy and Microanalysis at the University of Sydney were used in this work and the authors thank Ian Kaplin and Richard Lwin for assistance. This work was performed in part at the OptoFab node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nanofabrication and microfabrication facilities for Australian researchers. The work of S.C. is supported by a New Economy Research Fund (NERF) grant from The Foundation for Research, Science and Technology of the New Zealand government.
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