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62 K.Busch et al. with unknown amplitudes EnR.Inserting this expansion into the wave equa- tion (4.8)and employing the orthonomality relations,(4.7),leads to the basic equation for lattice models of defect structures embedded in PCs ∑{nrr+D}EnR=()'∑AER (4.10) n n'R The matrix A depends only on the Wannier functions of the underlying PC and is defined through A2=- d2r WiR(r)V2Ww'R(r). (4.11) R2 The localization of the Wannier functions in space leads to a very rapid decay of the magnitude of matrix elements with increasing separationR-Rbe- tween lattice sites,effectively making the matrix A sparse.Furthermore, it may be shown that the matrix A is Hermitian and positive definite. Similarly,once the Wannier functions of the underlying PC are determined, the matrix DR depends solely on the overlap of these functions,mediated by the defect structure: d2r WaR(r)6E(r)Ww'R(r). (4.12) R3 As a consequence of the localization properties of both the Wannier functions and the defect dielectric function,the Hermitian matrix D,too,is sparse. In the case of PCs with inversion symmetry,Ep(r)=Ep(-r),the Wannier functions can be chosen to be real.Accordingly,both matrices,A and DR become real symmetric ones. Depending on the nature of the defect structure,we are interested in (i)frequencies of localized cavity modes,(ii)dispersion relations for straight waveguides,or (iii)transmission and reflection through waveguide bends and other,more complex defect structures.In the following,we consider each of these cases separately. 4.3.3 Localized Cavity Modes As a first illustration of the Wannier function approach,we consider the case of a simple cavity created by infiltrating a single pore at the defect site Rdef with a material with dielectric constant edef,as shown in the inset of Fig.4.3(a).In this case,we directly solve (4.10)as a generalized eigen- value problem for the cavity frequencies that lie within the PBG,and recon- struct the cavity modes from the corresponding eigenvectors.In Fig.4.3(a) we compare the frequencies of these cavity modes calculated from (4.10) with corresponding calculations using PWM-based super-cell calculations.62 K. Busch et al. with unknown amplitudes EnR. Inserting this expansion into the wave equation (4.8) and employing the orthonomality relations, (4.7), leads to the basic equation for lattice models of defect structures embedded in PCs n,R δnnδRR + Dnn RR EnR = c ω 2 n,R Ann RREnR . (4.10) The matrix Ann RR depends only on the Wannier functions of the underlying PC and is defined through Ann RR = − R2 d2r W∗ nR(r) ∇2 WnR (r) . (4.11) The localization of the Wannier functions in space leads to a very rapid decay of the magnitude of matrix elements with increasing separation |R − R | between lattice sites, effectively making the matrix Ann RR sparse. Furthermore, it may be shown that the matrix Ann RR is Hermitian and positive definite. Similarly, once the Wannier functions of the underlying PC are determined, the matrix Dnn RR depends solely on the overlap of these functions, mediated by the defect structure: Dnn RR = R2 d2r W∗ nR(r) δε(r) WnR (r) . (4.12) As a consequence of the localization properties of both the Wannier functions and the defect dielectric function, the Hermitian matrix Dnn RR , too, is sparse. In the case of PCs with inversion symmetry, εp(r) ≡ εp(−r), the Wannier functions can be chosen to be real. Accordingly, both matrices, Ann RR and Dnn RR become real symmetric ones. Depending on the nature of the defect structure, we are interested in (i) frequencies of localized cavity modes, (ii) dispersion relations for straight waveguides, or (iii) transmission and reflection through waveguide bends and other, more complex defect structures. In the following, we consider each of these cases separately. 4.3.3 Localized Cavity Modes As a first illustration of the Wannier function approach, we consider the case of a simple cavity created by infiltrating a single pore at the defect site Rdef with a material with dielectric constant εdef, as shown in the inset of Fig. 4.3(a). In this case, we directly solve (4.10) as a generalized eigenvalue problem for the cavity frequencies that lie within the PBG, and reconstruct the cavity modes from the corresponding eigenvectors. In Fig. 4.3(a) we compare the frequencies of these cavity modes calculated from (4.10) with corresponding calculations using PWM-based super-cell calculations.