60 K.Busch et al. 4.3.1 Maximally Localized Photonic Wannier Functions A more natural description of localized defect modes in PCs consists in an expansion of the electromagnetic field into a set of localized basis functions which have encoded into them all the information of the underlying PC. Therefore,the most natural basis functions for the description of defect struc- tures in PCs are the so-called photonic Wannier functions,WnR(r),which are formally defined through a lattice Fourier transform WaR(r)=(2)Joz Vwsc Pke-ikR Enk(r) (4.4) of the extended Bloch functions,Enk(r).The above definition associates the photonic Wannier function WaR(r)with the frequency range covered by band n,and centers it around the corresponding lattice site R.In addition, the completeness and orthogonality of the Bloch functions translate directly into corresponding properties of the photonic Wannier functions.Computing the Wannier functions directly from the output of photonic bandstructure programs via(4.4)leads to functions with poor localization properties and erratic behavior (see,for instance,Fig.2 in [4.31]).These problems origi- nate from an indeterminacy of the global phases of the Bloch functions.It is straightforward to show that for a group of Nw bands there exists,for every wave vector k,a free unitary transformation between the bands which leaves the orthogonality relation of Wannier functions unchanged.A solution to this unfortunate situation is provided by recent advances in electronic bandstruc- ture theory.Marzari and Vanderbilt [4.32]have outlined an efficient scheme for the computation of maximally localized Wannier functions by determin- ing numerically a unitary transformation between the bands that minimizes an appropriate spread functional F Nw F=∑[aolr2no)-((n0))月=Mim. (4.5) Here we have introduced a shorthand notation for matrix elements according to (nRlf(r)ln'R)=/dPrWiR(r)f(r)ep(r)WwR(r), (4.6) for any function f(r).For instance,the orthonormality of the Wannier func- tions in this notation read as (nRIIn'R')=d'rWaR(r)Ep(r)Ww'R(r)=6nmRR, (4.7） /R2 The field distributions of the optimized Wannier functions belonging to the six most relevant bands of our model system are depicted in Fig.4.2 (see
60 K. Busch et al. 4.3.1 Maximally Localized Photonic Wannier Functions A more natural description of localized defect modes in PCs consists in an expansion of the electromagnetic field into a set of localized basis functions which have encoded into them all the information of the underlying PC. Therefore, the most natural basis functions for the description of defect structures in PCs are the so-called photonic Wannier functions, WnR(r), which are formally defined through a lattice Fourier transform WnR(r) = VWSC (2π)2 BZ d2k e−ikR Enk(r) (4.4) of the extended Bloch functions, Enk(r). The above definition associates the photonic Wannier function WnR(r) with the frequency range covered by band n, and centers it around the corresponding lattice site R. In addition, the completeness and orthogonality of the Bloch functions translate directly into corresponding properties of the photonic Wannier functions. Computing the Wannier functions directly from the output of photonic bandstructure programs via (4.4) leads to functions with poor localization properties and erratic behavior (see, for instance, Fig. 2 in [4.31]). These problems originate from an indeterminacy of the global phases of the Bloch functions. It is straightforward to show that for a group of NW bands there exists, for every wave vector k, a free unitary transformation between the bands which leaves the orthogonality relation of Wannier functions unchanged. A solution to this unfortunate situation is provided by recent advances in electronic bandstructure theory. Marzari and Vanderbilt [4.32] have outlined an efficient scheme for the computation of maximally localized Wannier functions by determining numerically a unitary transformation between the bands that minimizes an appropriate spread functional F F = NW n=1 n0| r2 |n0 − (n0| r |n0) 2 = Min . (4.5) Here we have introduced a shorthand notation for matrix elements according to nR| f(r)|n R = R2 d2r W∗ nR(r) f(r) εp(r) WnR (r) , (4.6) for any function f(r). For instance, the orthonormality of the Wannier functions in this notation read as nR| |n R = R2 d2r W∗ nR(r) εp(r) WnR (r) = δnmδRR , (4.7) The field distributions of the optimized Wannier functions belonging to the six most relevant bands of our model system are depicted in Fig. 4.2 (see
4 A Solid-State Theoretical Approach 61 19 Fig.4.2.Photonic Wannier functions,Wno(r),for the six bands that are most relevant for the description of the localized defect mode shown in Fig.4.3(a).These optimally localized Wannier functions have been obtained by minimizing the cor- responding spread functional,(4.5).Note,that in contrast to the other bands,the Wannier center of the eleventh band is located at the center of the air pore.The parameters of the underlying PC are the same as those in Fig.4.1. also the discussion in Sect.4.3.3).Their localization properties as well as the symmetries of the underlying PC structure are clearly visible.It should be noted that the Wannier centers of all calculated bands (except of the eleventh band)are located halfway between the air pores,i.e.inside the dielectric (see [4.32]for more details on the Wannier centers).In addition,we would like to point out that instead of working with the electric field [4.33,4.31, (4.1),one may equally well construct photonic Wannier functions for the magnetic field,as recently demonstrated by Whittaker and Croucher [4.34. 4.3.2 Defect Structures via Wannier Functions The description of defect structures embedded in PCs starts with the corre- sponding wave equation in the frequency domain 2 V2E(r) ( (Ep(r)+6s(r))E(r)=0. (4.8) Here,we have decomposed the dielectric function into the periodic part, Ep(r),and the contribution,s(r),that describes the defect structures. Within the Wannier function approach,we expand the electromagnetic field according to E(r)=∑EnRWnR(r), (4.9) n.R
4 A Solid-State Theoretical Approach 61 n=1 n=2 n=3 n=5 n=11 n=19 Fig. 4.2. Photonic Wannier functions, Wn0(r), for the six bands that are most relevant for the description of the localized defect mode shown in Fig. 4.3(a). These optimally localized Wannier functions have been obtained by minimizing the corresponding spread functional, (4.5). Note, that in contrast to the other bands, the Wannier center of the eleventh band is located at the center of the air pore. The parameters of the underlying PC are the same as those in Fig. 4.1. also the discussion in Sect. 4.3.3). Their localization properties as well as the symmetries of the underlying PC structure are clearly visible. It should be noted that the Wannier centers of all calculated bands (except of the eleventh band) are located halfway between the air pores, i.e. inside the dielectric (see [4.32] for more details on the Wannier centers). In addition, we would like to point out that instead of working with the electric field [4.33, 4.31], (4.1), one may equally well construct photonic Wannier functions for the magnetic field, as recently demonstrated by Whittaker and Croucher [4.34]. 4.3.2 Defect Structures via Wannier Functions The description of defect structures embedded in PCs starts with the corresponding wave equation in the frequency domain ∇2E(r) + ω c 2 (εp(r) + δε(r)) E(r)=0 . (4.8) Here, we have decomposed the dielectric function into the periodic part, εp(r), and the contribution, δε(r), that describes the defect structures. Within the Wannier function approach, we expand the electromagnetic field according to E(r) = n,R EnR WnR(r) , (4.9)