the same sign (i) for all points ${\mathbf{r}}$ in the unit cell and (ii) for all wave vectors ${\mathbf{k}}$ in the first Brillouin zone (for each component of the vector Bloch function). Then maximizing the IM of the Wannier functions, $$\mathcal{I}^W_{n}(\{\phi_{n{\mathbf{k}}}\})=\int_{\text{UC}}d^{2}r
\left|{\mathbf{{\mathbf{W}}}_{n \mathbf{R}}}({\mathbf{r}})\right|^2$$ with respect to the Bloch phases $\phi_{n{\mathbf{k}}}$ is equivalent to maximizing the IM of the real part of the Bloch functions in the first unit cell around the Wannier center, $$\mathcal{I}^B_n(\{\phi_{n{\mathbf{k}}}\})=\int_{\text{UC}}d^{2}r\left(\text{Re}({\mathrm{e}^{\mathrm{i} \phi_{{\mathbf{k}}}}}{\mathbf{\tilde{B}}}_{n\mathbf{k}})\right)^{2}
\label{eq:blochCrit}$$ for each wave vector in the first Brillouin zone separately. This theorem holds in any spatial dimension, even though in this paper we consider two-dimensional systems only. Note that in the above expressions we have set the weight factor $X({\mathbf{r}})$ appearing in Eq. (\[eq:def\_IM\]) equal to unity for simplicity.

The proof is straightforward. Due to the translation property of the Wannier function, , it is sufficient to prove the equivalence for ${\mathbf{R}}={\mathbf{0}}$ only. In this case $${{\mathbf{W}}_{n \mathbf{0}}}({\mathbf{r}})=\frac{1}{\sqrt{N}}\sum_{{\mathbf{k}}\in\text{BZ}}
{{\mathbf{B}}_{n \mathbf{k}}}.$$

![Modulus square of Bloch-criterion (Eq. \[eq:blochCrit\]) optimized Wannier functions for a square lattice of dielectric rods in air (TM polarization). For n=1,2,4 the Bloch criterion is equivalent to the IM-criterion for real Wannier functions. For n=3,5,6 the Wannier functions show at least a tendency 