Fig. 1. SEM image of an all-solid silica bandgap fibre. The light-coloured regions are the Ge-doped raised-index rods. The fibre’s bandgap-guiding core is the site of a missing rod in the centre.

Fig. 2. Transmission spectra (against frequency, with an auxiliary wavelength scale at the top) of 2 m of the fibre of Fig. 1 subject to different bend radii R. We observed transmission in bandgaps 3 to 7 (marked) in the straight fibre. At a bend radius of 15 cm there is dramatic loss in gaps 4 and 6 but little in gaps 3 and 5. When the bend radius is decreased to 7.5 cm some light is lost in all bandgaps, gap 3 being the least affected.

Fig. 3. Calculated DOS for a triangular lattice of graded-index rods in a low-index background as described in Section 2. The axes are effective index n _eff =β/k and normalised frequency kΛ, with the cutoff line n _eff =n _BG =1.458 marked. Red regions represent bandgaps (zero DOS) and are numbered in order of increasing kΛ. The grey-scale shading of non-zero DOS (low DOS in black, high DOS in white) highlights the continuity of features within and across bands. The yellow curve is the core line in bandgaps 3–6 for a core formed by the omission of one rod. The designations of the scalar rod modes from which the bands evolve are labelled in the form LP _lm . Also defined are the downward and upward effective index mismatches Δn- and Δn+ between the core line and the bands, to aid the discussion in Section 4. Across each bandgap, Δn- is smaller than Δn+ except near the long-wavelength edge.

Fig. 4. Schematic plots of effective index n _eff against displacement r from the fibre axis along the radius of curvature. The index n _fm of the fundamental core mode is marked in red. (top) A step-index fibre when (a) straight and (b) bent, where Δn is the mismatch between n _fm and the cladding index n _cl . A radiation caustic appears where n _eff =n _fm , at a distance from the axis proportional to Δn and the radius of curvature R. (bottom) The cladding bands of a bandgap fibre when (c) straight and (d) bent. Two radiation caustics appear, one on each side of the axis, again at distances proportional to R and the appropriate Δn. Bend loss is predominantly to the radiation caustic closer to the core, determined by whichever Δn is smaller.

Fig. 5. Critical bend radius R _c calculated from the data in Fig. 3 using Eq. (2) for gaps 3–6.

Fig. 6. Calculated unnormalised intensity distributions |Ψ|² at cutoff for the first four LP _l2 modes of a step-index rod with radius ρ, showing the different rates of decay |Ψ|²~1/r ^2l into the background. For l=0 and l=1 this integrates so that the fraction of the power in the core at cutoff is zero, but for higher values of l the field decays quickly enough for a non-zero fraction 1 - 1/l of the power to exist within the rod even at cutoff. The plot also illustrates how only l=0 modes have non-zero intensity in the centre of the rod. (The m=2 modes were chosen for this illustration because of their prominence in Fig. 3, and also because the LP₀₁ mode has no cutoff.)

Fig. 7. (upper) Images (on a logarithmic grey scale) of the light patterns in the cladding excited by bend loss, for the wavelengths near the bandgap edges indicated at the top. The plots show whether light is coupled to l=0 or l=1 rod modes by the bend, depending on whether the patterns have central peaks. (lower) Calculated patterns expected at the same wavelengths, together with the inferred LP designations of the rod modes from which the adjacent band of cladding states is formed.

Fig. 8. Light patterns across the fibre after bending, for wavelengths near the blue (left) and red (right) edges of gap 5. The top of each image is towards the outside of the bend. Near the red edge of the bandgap, bend loss is towards the inside of the bend.