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.

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.

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.

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.

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.

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.