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 (see60 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 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 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 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) Wn
R
(r) , (4.6) for any function f(r). For instance, the orthonormality of the Wannier func￾tions in this notation read as nR| |n R
=  R2 d2r W∗ nR(r) εp(r) Wn
R
(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