By W. Opechowski
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45a. In both case, the dielectric constant of the high-index material is 12. In many ways, photonic-crystal slabs are analogous to two-dimensional photonic crystals, such as those depicted in Fig. 1, and this analogy aids greatly in the visualization and analysis of slab systems. Two-dimensional calculations, however, cannot be applied directly to three-dimensional slab structures. In particular, the band structure computed for a two-dimensional structure, as shown in Fig. 2, applies in three dimensions only to a structure that is infinitely “extruded” in the third dimension.
Let us make two additional comments before we continue. First, in the special case where the crystal has translational symmetry ( a → 0 ), the irreducible Brillouin zone extends to k = ∞ , and is called a dispersion relation; this is commonly used to characterize waveguides. Second, when one is interested in just the extrema of the frequency bands, it is usually sufficient to consider only wavevectors along the edges of the irreducible Brillouin zone (or often merely the vertices)—this allows the band structure to be plotted in an ordinary two-dimensional diagram, many examples of which are found in this thesis.
4), but this merely means that we operate in a subspace of the set of all possible eigensolutions; it has no effect on most of the following discussion. Next, like in quantum mechanics, we look for time-harmonic states whose time dependence is e –iωt for some (angular) frequency ω —all possible solutions can be expressed via this form, since the equation is linear—and thus Eq. 5) becomes: 1 ω 2 ∇ × --- ∇ × H = ---- H . 6) This is a Hermitian eigenproblem over an infinite domain, and generally produces a continuous spectrum of eigenfrequencies ω .