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An approximate method for finding the band structure of simple photonic bandgap fibres is presented. Our simple model is an isolated high-index rod in a circular unit cell with two alternative boundary conditions. Band plots calculated this way are found to correspond closely to calculations using an accurate numerical method.
©2006 Optical Society of America
1. Introduction
A photonic bandgap fibre has a cladding that is two-dimensionally microstructured, typically with features repeating in a regular lattice. Light is confined to the core by a photonic bandgap of the cladding, instead of the more usual total internal reflection. Conceptually the simplest bandgap fibres are made of a low-index background material with a periodic array of isolated high-index rods in the cladding [1–11]. The core is the low-index site of a missing rod. In this paper the expression “bandgap fibre” will (unless otherwise qualified) refer only to fibres of this type. They are exemplified by recent all-solid-silica bandgap fibres [3–5] where both low- and high-index materials are fused silica, Fig. 1, the rods being doped with Ge to raise the index. However, the first examples were index-guiding silica-air photonic crystal fibres (PCFs) that were converted into bandgap fibres by filling the holes with a high-index fluid [1]. Proposed applications of solid bandgap fibres include spectral filtering, dispersion compensation and high-power systems [4,7,8].
Fig. 1. SEM image of an all-solid silica bandgap fibre [5]. The light grey regions are Ge-doped raised-index rods. The fibre’s bandgap-guiding core is the site of a missing rod in the centre.
These fibres have been modelled in two ways. Firstly, numerical techniques for solving Maxwell’s equations are applied to find the photonic states supported by the fibre. Numerical errors can be reduced to any desired level (at the cost of computing time) so the results of well-executed calculations can be regarded as exact. By applying such a method to an infinite defect-free cladding, combinations of frequency ω=ck and propagation constant β where photonic states exist (photonic bands) or do not exist (photonic bandgaps) can be mapped on a band plot. Guided modes in a core introduced as a defect can be represented as curves on the same plot. Such plots contain detailed information about the fibre.
However, simplified models that nevertheless retain key features of the physical system in question can be much more effective aids to understanding. The second approach to modelling considers the physical origins of the bandgaps from the resonant properties of an isolated high-index rod, treating the fibre as an anti-resonant reflecting optical waveguide (ARROW) [9–11]. The bands correspond to coupled supermodes of the array of rods, separated in ω-β space by bandgaps. The bandgaps can therefore be located by calculating the modes of an individual rod. This yields many important insights, but only limited (and approximate) information about the fibre.
Here we report an intermediate approach, where the states of a rod in its unit cell are calculated for two alternative boundary conditions. The method is approximate but (unlike the ARROW picture) does explicitly consider the sizes of the low-index regions separating the rods, and provides a close representation of the exact band plot. The method is numerical, but only to the extent of finding the roots of one-dimensional analytical expressions.
2. Method of calculation
An infinite periodic cladding structure with pitch Λ is defined by a hexagonal unit cell containing a central circular rod of radius a=d/2 and uniform refractive index n hi in a background of lower index nlo , Fig. 2(a). The difference nhi - nlo is small in all-solid-silica bandgap fibres [3], so we can apply the scalar weak-guidance approximation [12].
The states of the cladding can in principle be found by solving the scalar wave equation for the field distribution Ψ in a unit cell, with boundary conditions given by Bloch’s theorem. We are interested in how the modes of individual rods broaden into bands when the rods are brought together to form the cladding array. In particular, we want to identify the β and ω for which photonic states do not exist (ie the bandgaps). This can be done by mapping the top (maximum β) and bottom (minimum β) of the bands. Drawing on ideas from molecular and solid-state physics, we argue that the top and bottom of a band are defined by the Bloch states with the most bonding and anti-bonding character respectively [13,14].
In the case of a square lattice of rods, the states at the band edges are those determined by the boundary conditions dΨ/ds=0 and Ψ=0 at the edge of the unit cell centred on a rod, where s is normal to the cell boundary. The condition dΨ/ds=0 defines states with lines of mirror symmetry along the cell boundaries; we argue that these are the most bonded states of the band, with the least spatial field variation, and so have the highest β. The condition Ψ=0 defines states where the cell boundaries are anti-mirror lines; these are the most anti-bonded states, with the greatest field variation, and so have the lowest β.
The case of the hexagonal lattice is more problematic because the nature of bonding and anti-bonding states is not as clear. Mirror and anti-mirror lines on the cell boundary also pass through the centres of adjoining cells, thus imposing an unwanted symmetry restriction on the states found. If the boundary conditions are applied to a single unit cell, the symmetry restriction is lifted but the solutions may not be Bloch states. Nevertheless, the results presented below show that naive application of the boundary conditions identified for the square lattice to the hexagonal unit cell still provides a remarkably good approximation to the real band structure.
Fig. 2. (a) A rod of radius a in a hexagonal unit cell of width Λ, with local co-ordinate s normal to the cell boundary. (b) The rod in a circular unit cell of radius b, with radial co-ordinate r.
We further approximate the hexagonal unit cell to a circular one, Fig. 2(b), greatly simplifying the analysis by making the problem separable in polar co-ordinates r and θ. To preserve areas, so that the rod occupies the same fraction f=(π/2√3)(d/Λ)2=(a/b)2 of the unit cell, the radius of the circular unit cell is b=(√3/2π)1/2Λ. The approximation is justified not only by the rough similarity between a circle and a hexagon, but also by the key finding of the ARROW model that the bandgap frequencies are primarily determined by the properties of the individual rods rather than their arrangement in a lattice [9–11]. Indeed our results are probably valid for many lattice types or even for a disordered array, though we expect the best correspondence to be with the most circular unit cell, the hexagon. This step was applied in the first analysis of index-guiding PCFs [15] and should be reasonable if d/Λ is not too big.
The solutions derived in the circular cell have labels l and m defining the azimuthal and radial variations respectively. Although these labels cannot properly be used either in a square or hexagonal lattice, they are well defined for highly-confined states above cutoff because these are simply the weakly-coupled LPlm modes of the rods. The method therefore enables us to track these states as they move towards and below cut-off when frequency is reduced.
Thus we find our approximations for the states at the top and bottom of the band by solving the scalar wave equation for the structure in Fig. 2(b) with our two alternative boundary conditions. The resulting fairly cumbersome but analytical characteristic equations satisfied by β for given ω, l and m are presented in the Appendix. The region between the two β(ω) curves satisfying these equations is filled with states derived from the LPlm rod mode. By mapping such regions on a band plot for all relevant values of l and m, we can therefore identify the bandgaps (the spaces left over) as well as the states making up the bands.
Fig. 3. Plots of band structure for the example bandgap fibre described in the text. The bandgaps are shown in red. The rod modes from which the bands arise are labelled along the top. (a) DOS calculated using the plane-wave method, with light grey corresponding to high DOS. The yellow curve is the “fundamental” core-guided mode. (b) Band edges calculated using the method of Section 2 and filled in grey between top and bottom edges. The edges of the LP04 band (considered in Fig. 4) are drawn thicker.
3. Comparison with exact numerical calculation
The above procedure applied to an example fibre (d/Λ=0.41, nlo =1.458 and nhi =1.48716) is compared to an “exact” numerical calculation using a vector plane-wave method [16] in Fig. 3, on axes of (β-knlo )Λ representing β and kΛ representing ω. As in previous papers [5,6,17] we plot this information as the photonic density of states (DOS). As well as indicating the bandgaps (where the DOS is zero), such a plot also reveals structure within the bands. Features can be seen that correspond to particular classes of photonic states (eg resonances of high- or low-index parts of the cladding, with anti-crossing behaviour at their intersections), helping to give insight into the origins of the band structure [17].
There is an excellent qualitative and quantitative match between the two methods. The approximate band edges follow lines of high or low DOS in the exact plot, indicating that such features do indeed correspond to photonic states with distinct symmetries. The locations and depths of the bandgaps are well-represented, including the way odd-numbered bandgaps are deeper than their even-numbered neighbours [5]. The approximation also provides a convenient way to identify l and m values for the bands calculated by the exact method.
In [17] we suggested that the band plots for a hollow-core bandgap fibre were a hybrid of states concentrated in high- and low-index regions. On the axes of Fig. 3 these correspond to curves that are respectively steep (nearly vertical) or flatter (shallow hyperbolas, which would be horizontal lines on a graph of w 2 from [17]). Intensity distributions for the LP04 band are plotted in Fig. 4 for kΛ=115 and 170 (where both band edges follow the steep high-index trend) and kΛ=145 (in between, where they follow the flatter low-index trend). Whereas in the former cases the light is concentrated in the high-index rod, in the latter case it is concentrated in the low-index background, showing that the LP04 rod states have indeed become strongly hybridised with resonant states concentrated in the background regions between rods. This behaviour is typical of the bands we examined.
Fig. 4. Normalised radial intensity plots |Ψ|2 within a circular unit cell for the states at the top (red) and bottom (blue) of the LP04 band for frequencies kΛ of (left to right) 115, 145 and 170.
The effect on the band plot of changing the unit cell size b while keeping the rod size a fixed is shown in Fig. 5. As b (and hence the rod separation) increases there is: a scaling in frequency so that ka is invariant (as expected from the ARROW picture [9]); a compression of the flatter (low-index) features towards the cutoff line (because the low-index resonators in the structure are getting bigger [17]); and a narrowing of the frequency widths of the bands near the cutoff line β=knlo (because the LP lm modes of adjacent rods are becoming more weakly coupled).
Fig. 5. Approximate band structure for the fibre cladding of Fig. 3 but with d/Λ of (left to right) 0.6, 0.4 and 0.2. The horizontal axes have been scaled so that the same range of ka is shown. The rod modes above cutoff can be identified by comparing the middle graph with Fig. 3.
This last point can be understood from the analogous tight-binding model of solid state theory [13], or alternatively by considering the waveguide directional coupler, where supermode splitting is proportional to a coupling coefficient [12]. Hence the frequency width of a band along the cutoff line is related to how effectively the states in that band can transfer light away from the core, eg as bend loss. Analytical expressions for this width can be found from the characteristic equations in the Appendix for small d/Λ:
where the rod filling fraction f and the V-value V (representing frequency) are defined in Section 2 and the Appendix respectively. As expected [5,11], for small f the width decreases rapidly with increasing l, and more gradually with increasing frequency for a given l. Table 1 gives the widths of the first five m=2 bands for the fibre of Fig. 3 from both Eq. (1) and a plane-wave calculation (scalar, to avoid confusion due to polarisation splitting), showing a good match and also reflecting our previous findings on l and bend loss [5].
Table 1. Calculated frequency widths of some m=2 bands at cutoff, in kΛ units. The plane-wave value for the LP22 band* is imprecise because (see Fig. 3) this band crosses the LP03 band at cutoff.
4. Extensions and limitations
Our method can be extended to cases where the rod’s radial index distribution is more complicated than a single step. For example, for graded-index rods (such as those used in our experimental work [5]) the analysis remains one-dimensional in r and simple numerical techniques for solving ordinary differential equations can be applied.
This is not necessary for index distributions n(r) of several uniform regions bounded by steps, such as a high-index ring [6], which can be solved analytically in each region. The resulting characteristic equations will of course be more cumbersome because each extra layer introduces a new boundary. However, a cladding with an array of finite-thickness high-index rings can be modelled using a thin ring whose radial index profile is a Dirac δ function, weighted so that its profile volume matches that of the actual thick ring [18]. Unlike a thick ring, the thin δ-function ring does not introduce new boundaries but instead modifies the boundary we already have (as well as giving the media either side of r=a the same index).
Fig. 6. (a) Band plot for the thin-ring cladding described in the text. (b) Cutoff V-values of the modes of an isolated thick ring as a function of the inner to outer radius ratio c/a. m=1 modes with l>6 are omitted for clarity.
The band plot of Fig. 6(a) is the result for d/Λ=0.4, for a profile volume matching a ring of thickness 0.1Λ and index nhi =1.48716 in a background of index nlo =1.458. It illustrates the limit of the behaviour described in [6] (though with a different value for d/Λ), with all m>1 rod modes (and hence the bands derived from them) pushed below cutoff. Hence the low-loss frequency ranges of core-guided modes are delimited only by m=1 modes of the rings. For comparison Fig. 6(b) shows the LP lm mode cutoff V-values for an isolated thick ring versus the ratio of the inner and outer radii c and a, where V 2=k 2(a 2-c 2)(-). In the thin-ring limit (as c/a→1) the cutoffs of the m=1 modes approach V=2√l while those of the m>1 modes are banished to infinity.
A vector version of the method could be developed; as it stands, our method only represents low-contrast structures. The approximations it embodies also become less accurate for larger d/Λ: the circular unit cell becomes less appropriate as the rod approaches the unit cell boundary and the corners of the hexagonal unit cell become a more significant fraction of the background.
5. Conclusions
We have described a simple semi-analytical model for photonic bandgap fibres with a cladding of high-index rods in a low-index background. The hexagonal lattice and unit cell are replaced with a circular unit cell and 1-D boundary conditions. This makes the calculations analytical except for simple numerical root-finding in a function of one variable. The method reproduces the band structure of the cladding with remarkable accuracy despite the crude approximations, at least when d/Λ is not too large. We expect this method to be of value where speed of calculation is more important than accuracy, and as an aid to intuition.
Appendix
The general solution of the scalar wave equation [12] within the circular unit cell of Fig. 2(b), corresponding to the LP lm mode of the rod and assuming that β<knhi , is
multiplied by any linear combination of cos(lθ) and sin(lθ). Jl , Yl , Il and Kl are Bessel functions, A to H are unknown constants and U, W and V are the familiar parameters of waveguide theory [12] applied to the rod
with Q being useful where β<knlo .
Application of
at r=b and the continuity of Ψ and Ψ′ at r=a as boundary conditions gives analytical characteristic equations satisfied by β for given ω, l and m, which can be written as
For the state at the top of the band
and for the state at the bottom of the band
where represents the derivative of Jl etc, α=b/a=1/√f represents the relative size of the rods, and the coefficients A to H in Eqs. (2), (6) and (7) are
Eqs. (6) and (7) look awkward mainly because different expressions are needed for different ranges of W 2. The particular forms of the equations are chosen so that the g(V, W 2) are continuous and finite as W 2 varies through W 2=0, which simplifies the application of root-finding algorithms. The functions are well-behaved where U 2>0, which is sufficient for our purposes since there are no propagating states with U 2≤0(β≥knhi ).
Eq. (1) for the frequency widths of the bands is obtained from the characteristic equations with W 2=0 and α large.
Acknowledgements
We would like to thank JC Knight, F Luan, A Wang and JM Stone for useful discussions. The work was supported by the UK Engineering and Physical Sciences Research Council.
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