A Macrocell Approach for the Analysis of 2D Photonic Crystals Devices

Development of fast, flexible and accurate methods for the analysis (and design) of photonic crystals devices is a relevant topic in order to optimize existing devices and/or developing new design solutions. Within this framework, in this communication we present an improved version of the Scattering Matrix Method (SMM) for the evaluation of the electromagnetic behaviour of two-dimensional finite-extent photonic crystals made of a finite set of parallel dielectric rods. In the SMM, each rod is characterized by a scattering matrix, and a rigorous modal expansion of the field scattered by each inclusion is exploited to formulate the overall scattering problem as the solution of a linear system. Such a method appears as a convenient choice to analyze devices based on truncation and insertion of "defects" in a periodic lattice. As a matter of fact, it gives the maximum flexibility as far as location and constitution of the single scatterer is concerned. Moreover, as it is based on modal expansion of the fields, it allows to dealing with a low number of unknowns. On the other side, as this latter number grows with the volume of the device, effectiveness of the method deteriorates in case of very large structures. In order to overcome this latter restriction, it proves convenient to note that the fine details of the electromagnetic field in those parts of the lattice which are located away from the "defective" region are not relevant to describe the behaviour of the device, also in view of the fact that the field is therein negligible. Accordingly, an alternative strategy can be devised based on the possibility of aggregating several neighbouring cylinders into larger circular inhomogeneous scatterers. By avoiding to compute the field inside these "macrocells", significant computational savings can be achieved (without any loss of accuracy) by applying the SMM to the modified lattice, which is made of single cylinders and macrocells, whose scattering matrix can be computed once for all 'off-line'. A rigorous analysis based on the properties of the fields scattered by the macrocell has been applied to determine the optimal dimensions of the aggregations and demonstrate the convenience of the proposed method. Moreover, an extended numerical analysis has been performed to prove the accuracy of the macrocell-SMM approach and the remarkable computational benefit arising from the reduction of the unknowns in the solution of the linear system.