total permittivity $\epsilon({\mathbf{r}})=\epsilon_{p}({\mathbf{r}})+\delta\epsilon({\mathbf{r}}),$ into a periodic term $\epsilon_{p}({\mathbf{r}})$ which corresponds to the unperturbed system, and a defect term $\delta\epsilon({\mathbf{r}})=(\epsilon_{def}-\epsilon_{p})\Theta(r-|{\mathbf{R}}_{def}-{\mathbf{r}}|)$. An expansion of the $z$-component of the electric field in terms of maximally localized Wannier functions leads to a generalized eigenvalue problem with sparse matrices [@Busch2003]. The defect mode frequencies are obtained from the solution of this sparse system (Fig. \[fig:pointDefect\]). The first eight Wannier functions have been taken into account. There are monopole like defect modes for a defect rod permittivity of $\epsilon_{def}<12$ and doubly degenerated, dipole like modes for higher permittivities in the defect rod $\epsilon_{def}>12$. The results are in complete agreement with plane wave calculations [@mpb-art]. To analyze contribution of the individual Wannier functions to the defect mode, the band index contribution $$C_{n}=\frac{1}{M}\sum_{{\mathbf{R}}}|{E_{n \mathbf{R}}}|^{2}$$ where the normalization factor is given by $M=\sum_{n{\mathbf{R}}}|{E_{n \mathbf{R}}}|^{2}$ is shown in figure \[fig:bandContri\] for the defect modes $\epsilon_{def}=1$ and $\epsilon_{def}=30$. One can see, that $C_{n}$ rapidly decrease for higher band indices. Since the defect mode frequencies are in between the 1st and the 2nd band, only the lower band Wannier functions contribute to the defect modes. Therefore it is justified to cut of the band index which reduces the numerical load.

Conclusion\[sec:Conclusion\]
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