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Summary: On effective index approximations of photonic crystal slabs
Manfred Hammer, Alyona Ivanova
MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands
m.hammer@math.utwente.nl
To assess the quality of effective index approximations for photonic crystal slabs, we consider a reduction of
2-D problems for waveguide Bragg gratings to 1-D, and compare with rigorous 2-D solutions. A variational
procedure permits to establish reasonable effective indices even if locally no guided modes exist.
Summary
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. Ref. [1]). 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 usually described 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 (QUEP [2], 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, 3], based
on a variational view on the EIM, to uniquely define an effective permittivity even then.
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