important to mention, that the completeness and orthogonality of the Bloch modes $E_{n{\mathbf{k}}}({\mathbf{r}})$ and $H_{n{\mathbf{k}}}({\mathbf{r}})$ translates into the completeness and orthogonality of Wannier functions ${W_{n \mathbf{R}}}({\mathbf{r}})$ with respect to the corresponding inner product: $${\langle {W_{n \mathbf{R}}} | {W_{n' \mathbf{R'}}} \rangle}_{E/H}=\delta_{nn'}\delta_{{\mathbf{RR'}}}.\label{eq:orth_WannierTM}$$ Moreover, the translation property of the Wannier functions, $${W_{n \mathbf{R}}}({\mathbf{r}})={W_{n \mathbf{0}}}({\mathbf{r-R}})\label{eq:wan_translationBehavior},$$ follows from the periodicity of the envelope functions ${u_{n \mathbf{k}}}({\mathbf{r}})$.

In this work four different two-dimensional photonic crystals are analyzed for both fundamental polarizations. Square (Sq) and triangular (Tr) lattices of dielectric rods in air (D) and air rods in dielectric (A) are considered. In what follows we will refer to these systems as Sq-D, Tr-D, Sq-A and Tr-A, respectively. The radius of rods and the dielectric constant of dielectric material and air are chosen to be $r_0/a=0.2$, $\epsilon=12$ and $\epsilon=1$, respectively. We adopt the following definition of the second moment of the Wannier function ${W_{n \mathbf{R}}}({\mathbf{r}})$: $$\mathcal{S}_{n}(\{\phi_{n{\mathbf{k}}}\})={\langle {W_{n \mathbf{R}}} |}({\mathbf{r}}-{\mathbf{r}}_{0})^{2}{|{W_{n \mathbf{R}}} \rangle}_{E/H},\label{eq:def_SM}$$ with ${\mathbf{r}}_{0}$ being the Wannier center. Two positions of the Wannier center will be analyzed further, (i) in the center of the scatterer (“on-site”) and (ii) in the geometrical center between four (three) scatterers in the case of square (triangular) lattices (“between”). For example, in a 