On effective index approximations of photonic crystal slabs

The propagation of light through slab-like photonic crystals (PCs) is frequently described in terms of effective indices (effective index method EIM, cf. e.g. Refs. [1, 2, 3]). One replaces the actual 3-D structure by an effective 2-D permittivity, given by the propagation constants of the slab modes of the local vertical refractive index profiles. Though the approach is being described usually for the approximate calculation of waveguide modes, it is just as well applicable to propagation problems. Our aim is to check the approximation by analogous steps that reduce finite 2-D waveguide Bragg-gratings, which in turn can be seen as sections through 3-D PC membranes, to 1-D problems, which are tractable by standard transfer matrix methods. A 2-D Helmholtz solver ([4], reference) allows to solve the 2-D problem rigorously, i.e. to assess the quality of the EIM approximation. The EIM-viewpoint becomes particularly questionable if locally the vertical refractive index profile cannot accommodate any guided mode, as e.g. in the holes of a PC membrane. We check numerically a recipe [1, 5] to uniquely define an effective permittivity even for these cases, based on a variational view on the EIM.