We describe a new method which allows us to investigate photonic crystals modelled as stacks of gratings consisting of metallic, circular cylinders. Our formulation enable us to compute photonic band diagrams of two- and three-dimensional arrays of inclusions and to estimate accurately the edges of the band gaps in the case of lossless materials. In turn, this provides useful estimates of the gap edges for lossy materials.
The technique, which is applicable to structures whose layers do not interpenetrate (thereby permitting the use of plane wave expansions for fields between layers) has, at its heart, an eigenvalue problem which is derived by applying a quasi-periodicity condition. The resulting dispersion relation (which involves the scattering matrices of the single representative layer) can then be reduced to an algebraic eigenvalue problem. The solutions of the eigenvalue equation also provide a generalised technique to determine the field modes of the structure. We note that our technique is intrinsically more stable than similar methods as it avoids the inversion of particular scattering matrices, the condition or stability of which cannot be guaranteed. Also, the accuracy of the method is not affected by the high contrast between the cylinders and the surrounding medium. We use the reflection and transmission scattering matrices of a single grating, to generate the corresponding scattering matrices for a finite stack of gratings, and demonstrate closed form expressions for solutions of the recurrence relations. By considering the limit as the stack length increases without bound, we calculate the reflection scattering matrix R¥, which is the fixed point of the recurrence relation for the reflection scattering matrices. By means of R¥ it should be possible to define an effective permittivity at general points in the Brillouin zone. In the quasistatic limit, we obtain an effective permittivity identical to that obtained using the electrostatic theory.
The method can be applied to the study of both ordered and disordered photonic crystals, and can yield analytic insights into problems such as localization and homogenisation. At long wavelengths, the individual grating layers homogenise and the structure behaves as a one-dimensional stack in which Anderson localization of waves is evident, until the wavelength is sufficiently large so as to render the crystal homogeneous. This method can also be generalized to coated cylinders and to crossed stacks with rotation of layers (one layer to the next). A particular choice of crossing angles of 0° and 90° produces approximate polarization insensitivity.