In this section we investigate photonic bandgaps in two-dimensional photonic crystal lattices. We start by plotting a band diagram for a periodic lattice with negligible refractive-index-contrast. We then introduce a plane-wave expansion method for calculating the eigenmodes of a general 2D photonic crystal, and then develop a perturbation approach to describe bandgap formation in the case of photonic crystal lattices with small refractive index contrast. Next, we introduce a modified plane-wave expansion method to treat line and point defects in photonic crystal lattices. [1,2] Finally, we introduce perturbation formulation to describe bifurcation of the defect states from the bandgap edges in lattices with weak defects.

The two-dimensional dielectric profiles considered in this section exhibit discrete translational symmetry in the plane of a photonic crystal, and continuous translational symmetry perpendicular to the photonic crystal plane direction (Fig. 6.1). The mirror symmetry described in Section 2.4.7 suggests that the eigenmodes propagating strictly in the plane of a crystal can be classified as either TE or TM, depending on whether the vector of a modal magnetic or electric field is directed along the ẑ axis.

Two-dimensional photonic crystals with diminishingly small index contrast

In the case of a 2D discrete translational symmetry, the dielectric profile transforms into itself ε(r + δr) = ε(r) for any translation along the lattice vector δr defined as δr = ā1N1 + ā2N2,(N1, N2) ⊂ integer.