On photonic band gaps in two-dimensional photonic crystal fibres. Analysis in the vicinity of the low-dielectric light line

We consider `off-axis' electromagnetic wave propagation down the homogeneous direction of a low-loss two-dimensional periodic dielectric (or photonic crystal fibre) near the light line of the low-dielectric material constituent. Numerous physical and numerical experiments demonstrate the presence of photonic band gaps in the vicinity of this `critical' light line. We mathematically analyse the existence of photonic band gaps near the line and characterise them in terms of frequency gaps in the spectrum of the Maxwell equations restricted to the line. We apply the results to both one-dimensional photonic crystal fibres, and a genuinely two-dimensional photonic crystal fibres with `thin' inclusions, namely ``ARROW'' fibres. In the case of ARROW fibres, by an asymptotic analysis, in terms of the small inclusion parameter, we demonstrate the existence of low frequency photonic band gaps.