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Scaling laws and vector effects in bandgap-guiding fibres

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Abstract

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

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Figures (3)

Fig. 1.
Fig. 1. Schematic diagram illustrating our new way of mapping the states of a photonic crystal, on normalised axes v 2 (or v) against w 2. Propagation is forbidden in the regions coloured red.
Fig. 2.
Fig. 2. Plots of DOS on normalised axes for d/Λ=(a) 0.96 and (b) 0.80, and for n=(i) 1.02 and (ii) 1.45 (the “scalar” and “vector” cases respectively). Regions with no propagating states (DOS≡0) are coloured red for emphasis. These correspond to cutoff (bottom right, β>kneff ), evanescent states (bottom left, β2<0) and bandgaps (elsewhere). The arrows on the top edge locate the resonances of the circular low-index regions, whose loci would be vertical lines on these plots.
Fig. 3.
Fig. 3. Plots of DOS on the low-index line as index contrast n is varied, for d/Λ=(a) 0.96 and (b) 0.80. The colour map for DOS is the same as that in Fig. 2. The shrinking bandgap around v=12 in (a) is the fundamental bandgap used in current air-guiding silica PCFs. The growing bandgap around v=12 in (b) is the robust type II bandgap of [13].

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

k t 1 k t 2 = k 2 n 1 2 β 2 k 2 n 2 2 β 2
( t 2 + k 2 n 0 2 β 2 ) h = ( t × h ) × ( t ln n 0 2 ) ,
f ( X , Y ) = { 0 low index regions 1 high index regions ,
T 2 Ψ + ( v 2 f w 2 ) Ψ = 0 ,
v 2 = Λ 2 k 2 ( n 1 2 n 2 2 ) .
w 2 = Λ 2 ( β 2 k 2 n 2 2 )
w 2 = ( 2 j l m d Λ ) 2 ,
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