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On effective index approximations of photonic crystal slabs

Summary: On effective index approximations
of photonic crystal slabs
Manfred Hammer and O. V. (Alyona) Ivanova
MESA+ Institute for Nanotechnology,
University of Twente, Enschede, The Netherlands
As a means to assess the quality of effective index approximations in simulations of pho-
tonic crystal slabs, we consider a reduction of 2-D Helmholtz problems for waveguide
Bragg gratings to 1-D wave propagation, and compare with rigorous 2-D reference solu-
tions. Variational procedures permit to establish a reasonable effective index profile even
in cases where locally no guided modes exist.
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 propaga-
tion 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 approxima-
tion 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


Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente
Hammer, Manfred - Department of Applied Mathematics, Universiteit Twente


Collections: Engineering; Mathematics