Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings

A two-scale asymptotic theory is developed to generate continuum equations that model the macroscopic behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The resulting equations include tensors that represent effective refractive indices near band edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale modulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes. The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are verified against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided modes. The second example is the propagation of electromagnetic waves localised within a planar array of dielectric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dynamic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour