Scaling laws for photonic bandgaps in photonic crystal fibres are described. Although only strictly valid for small refractive index contrast, they successfully identify corresponding features in structures with large index contrast. Furthermore, deviations from the scaling laws distinguish features that are vector phenomena unique to electromagnetic waves from those that would be expected for generic scalar waves.
©2004 Optical Society of America
Hollow-core bandgap-guiding optical fibres were first demonstrated in 1999 . Light is confined to the core of the fibre by a photonic bandgap in the cladding, which prevents light propagating in the cladding within a range of propagation constant β (longitudinal component of wavevector) that is allowed in the core . Bragg fibres have a cladding structure arranged in concentric layers around the core [3,4], whereas bandgap-guiding photonic crystal fibres (PCFs) have cladding features that essentially repeat transversely [1,5–7] although strict periodicity is not necessary . In both cases light can be guided in a core with a refractive index that is lower than the average index in the surrounding cladding structure, which is clearly not possible in conventional fibres that guide by total internal reflection.
Fibres incorporating air within the cladding structure can be designed to have bandgaps for light in free space , ie for β<k where k is the free-space wave constant 2π/λ. These were the first optical fibres to truly guide light in air with low loss and they have widespread applications [5,6]. They are typically made from fused silica with an array of air holes, giving a respectable index contrast (ratio) of 1.45 between high- and low-index regions. However, this contrast is much less than is needed to give bandgaps in 2-D structures for propagation in the plane of periodicity . Indeed, bandgap guidance can occur in fibres with very small index contrasts within the cladding (between undoped and doped silica, for example [4,10]).
Low-contrast structures can have out-of-plane bandgaps because a large contrast in transverse rather than absolute wavevector is needed. Because the light propagates along the fibre, much of its wavevector is taken up by the longitudinal component β. By choosing β to be close to kn 2 (where n 1 and n 2 are the indices of the high- and low-index materials respectively), the contrast in transverse wavevector kt can be arbitrarily large even if the index contrast n=n 1/n 2 itself is small. However, in this case β cannot fall far below kn 2, which makes all-solid claddings with low index contrast unsuitable for guiding light in a hollow core [4,10].
The modal magnetic field distributions h(r) in a z-independent dielectric structure of arbitrary index contrast satisfy a vector wave equation 
where n 0(r) is the index distribution and ∇t is the transverse Laplacian operator. The equation obeys the well-known length scaling law : a solution for a transverse scale represented by Λ is replicated in an identical structure with a different Λ if the wavelength is scaled proportionately, to keep kΛ constant. Given the present widespread interest in bandgap fibres with different index contrasts (eg, replacing silica with high-index glasses, or filling the holes with other materials [7,10,13]), a similar scaling law for index contrast would be very useful.
Equation (2) does not yield such a scaling law (only a less interesting one for absolute index at fixed contrast ). However, waveguides with low index contrast can be analysed by scalar methods, and support modes Ψ that are essentially linearly-polarised (LP modes). These satisfy a scalar wave equation obtained by neglecting the so-called vector term on the RHS of Eq. (2). This does give a simple scaling law for index contrast, which has long been applied to conventional “weakly guiding” fibres via the V parameter ; it also applies to bandgap fibres with low index contrast.
Although the vector term cannot be neglected for bandgap fibres with a large index contrast (such as silica/air), we find that the analogous scaling law still matches corresponding features in the band structures of fibres with different index contrasts.
2. Scaling laws for the scalar wave equation
The scalar wave equation can be re-cast in terms of normalised transverse co-ordinates X=x/Λ and Y=y/Λ. We consider binary step-index structures, though generalisation to graded index structures is straightforward . Introducing an index distribution function
which defines the microstructure but not its scale or index contrast, the resulting normalised scalar wave equation is
where ∇ T =∂ 2/∂X 2+∂ 2/∂Y 2 and
The frequency parameter v 2 and eigenvalue w 2 are directly analogous to the parameters V 2 and W 2 of conventional waveguide theory , except that Λ defines any suitable transverse scale in a structure that may not have a waveguide core at all. The analysis is quite general and applies equally to the guided modes of a bandgap fibre and the Bloch states of a defect-free 2-D photonic crystal. We retain w 2 as a squared quantity, so we can study states either side of the “low-index line” β=kn 2 (the “air line” for a silica-air PCF) in a continuous manner. In particular, w 2 is negative for modes guided in a hollow-core bandgap fibre.
From the form of Eq. (4), the eigenvalues w 2 of its modes Ψ are determined by v 2 and f alone. This leads to the general scaling laws:
if k, Λ, n 1 or n 2 vary, the photonic states scale so that v 2 and w 2 are invariant.
For example, Fig. 1 is a schematic plot of v 2 against w 2 for a 2-D photonic crystal, showing a photonic bandgap “finger” and the effective-index line that governs guidance by total internal reflection in a PCF with a high-index core . For scalar waves, such a map of photonic states applies universally for all index contrasts n; a single plot summarises the behaviour of a given microstructure f, regardless of scale, wavelength or refractive indices. (In principle the plot could extend to negative v 2 and include results for an inverse structure with n 2>n 1. However, it is not clear that there is much advantage to be gained by doing so.)
If a bandgap lies on the low-index line (w 2=0) for a given n, then (in the scalar case) the bandgap remains on the low-index line for any other n. The bandgap is located in frequency by requiring that v 2 in Eq. (5) be held constant; the gap does not disappear as n→1, it’s just that ever bigger kΛ is needed to attain it. In the scalar case, the only feature on the plot that varies with n is the line defining in-plane propagation, β=0. This “in-plane line” becomes steeper as n increases, so if n is large enough for the in-plane line to intersect a bandgap finger, we have an in-plane bandgap . If n is further increased so that a horizontal line between the in-plane line and the low-index line lies entirely within a bandgap, we have quasi-metallic behaviour , sometimes referred to as an omnidirectional bandgap .
Although the scaling laws are general, we focus on holey glass photonic crystals like the cladding of a hollow-core PCF. Different expressions for invariant quantities are convenient in different circumstances. For example, if the glass index n 1 is varied and n 2=1 is fixed the photonic states scale so that v=kΛ(n 2-1)1/2 and w 2=Λ2(β2-k 2) are invariant. Alternatively, if n 1 is fixed and the material n 2 in the holes is varied the invariant quantities can be expressed as v=kΛn 1(1-1/n 2)1/2 and u 2≡v 2- w 2=Λ2(k 2 -β2), u being a combination of v and w that does not explicitly depend on n in this case.
We emphasise that the scalar wave equation (and therefore the scaling laws derived from it) is accurate for the smallest index contrasts only. However, for step-index structures the vector term in Eq. (2) only exists at boundaries, so the scalar wave equation accurately represents wave propagation elsewhere. Since bandgaps arise from interference and resonance effects among such generic waves, the scaling laws of Eq. (5) should be at least approximately valid.
3. Density-of-states maps
The scaling laws were examined by plotting the photonic density of states (DOS) for different index contrasts. We used our fixed-frequency interior-eigenvalue plane-wave method  to calculate the DOS, although here we express DOS per interval of (βΛ)2. Under this definition the vacuum DOS is independent of βΛ, which is useful if states around β=0 are considered. As in  we normalise the DOS to the vacuum value to gauge whether it is enhanced or suppressed.
We modelled a triangular array of circular holes in glass, representing realistic air-guiding fibre claddings. Two relative hole sizes (different functions f) were considered, d/Λ=0.96 and 0.80, where d is the hole diameter and Λ is their pitch (centre-to-centre spacing). The first value approaches the maximum air filling fraction with the glass regions connected, the preferred regime for air-guiding silica PCFs [5,6]. The second value is one for which we identified robust “type II” bandgaps in PCFs made from high-index glasses .
The DOS is plotted on axes of v against w 2 in Fig. 2. We chose v instead of v 2 because a linear frequency scale is more familiar, and also avoids compressing low-frequency features of the plots. However, it does convert straight lines in Fig. 1 (like the in-plane line) into parabolas. Our plots are oriented to correspond as closely as possible to Fig. 1 of ; recall that for empty holes v=kΛ(n 2-1)1/2 and w 2=Λ2(β2-k 2). Each pixel is a single calculated value of DOS, with no interpolation. For each d/Λ, results for n=1.02 (indistinguishable from the scalar case) are compared to those for n=1.45 (the index contrast between silica and air).
In Fig. 3, the DOS on the low-index line (w 2=0, the delimiting case for air-guiding) is plotted on axes of v against index contrast n, so the left hand edge corresponds to the scalar case. This shows how the scaled bandgaps evolve with increasing n.
The results confirm that bandgaps can exist for arbitrarily small n≥1 . The clear correspondence in Fig. 2 between features in each pair of plots justifies the validity of the scaling laws to identify corresponding features even where a scalar analysis is not exact. If the vertical axes had been kΛ instead of v, the features would be mismatched vertically by a factor of 5.2 and the correspondence would have been far from evident. The validity of the scaling laws is also expressed by the horizontal orientation of the features in Fig. 3.
Many of the bandgaps (like the “fundamental” gap for d/Λ=0.96, used in air-guiding PCFs [5,6]) are wider in the scalar (n→1) case than the vector case, and close up completely as n increases. This may come as a surprise , since in more “traditional” photonic crystals (3-D structures, or in-plane propagation in 2-D structures) large index contrasts are needed to achieve bandgaps at all . However, it can easily be explained. In the scalar case the two available states of polarisation are degenerate. The vector term on the RHS of Eq. (2), treated as a perturbation, breaks this degeneracy and causes a polarisation splitting that increases with index contrast. (The splitting has an analogy in the splitting of the TE01, HE21 and TM01 second-order modes of a step-index fibre .) One can then imagine two copies of each bandgap finger, one for each polarisation, shifted with respect to each other. The absolute bandgap, being the overlap between the fingers, must be reduced by the shift.
Other gaps appear and open up for larger n, including the type II gaps  for d/Λ=0.8 that come to dominate photonic band plots of structures with lower air filling fractions and high n. (They presumably arise because the vector term is no longer just a perturbation.)
Comparison of the scalar and vector plots demonstrates the basic validity of the scaling laws; the contrast between them highlights features in the band structure that are specifically due to the vector nature of the electromagnetic field and the particular form of Maxwell’s equations, as opposed to generic behaviour that might be expected for any kind of wave.
A final property is highlighted by the monochrome colour map for non-zero DOS in Fig. 2, which reveals beautiful “flowing” patterns. Features of high DOS appear to follow two families of curves, one nearly vertical and the other parabolic, with anti-crossing behaviour at their intersections. One interpretation is that the first set are resonances in the circular low-index (eg air) regions, whereas the second set are resonances in the background high-index (eg silica) regions. (This goes beyond , which doesn’t apply to air-guiding fibres, considers resonances of circular regions alone, and then only when they are high-index.) It is clearest in the first plot of Fig. 2, presumably because the scalar case avoids confusion due to polarisation splitting, and large holes are better at isolating high-index resonators.
Our interpretation is supported by a calculation of the resonances of the circular low-index regions: if the resonance field distributions are Bessel functions falling to zero at the boundary, their propagation constants give constant values of w 2:
where jlm are the zeros of the Bessel functions. The resonances map onto vertical straight lines in Fig. 2 at the values of w 2 indicated by arrows, which correspond closely to the vertical family of curves. Similarly locating the parabolic family is not so straightforward because the high-index resonators are not circular, and modelling them is beyond the scope of this paper. However, maybe a useful qualitative (or even quantitative) model of bandgap formation in air-guiding bandgap fibres could result from considering the interactions of such resonators?
Scaling laws are derived for the photonic states in 2-D structures, such as bandgaps in airguiding PCF claddings. Although only strictly valid for small index contrasts, they correctly locate corresponding features in the band structure of photonic crystals with significant index contrast. The deviations of real band structure from the scalar case indicate features that arise specifically from the vector nature of the electromagnetic field. In particular, raising the index contrast initially causes bandgaps to become narrower rather than wider.
This work was supported by the UK Engineering and Physical Sciences Research Council. We would like to thank M. Banham for confirmatory calculations.
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