becomes: $$\begin{aligned}
\rho_{\mathbf{\bar{G}}}^{-1}=&\left(\frac{\rho_h}{\rho_c}-1\right)\frac{2\pi}{9a^2}\left[r_d\frac{J_1(|\mathbf{\bar{G}}|r_d)}{|\mathbf{\bar{G}}|}+\right. \notag
\\
&\left.2r_0\frac{J_1(|\mathbf{\bar{G}}|r_0)}{|\mathbf{\bar{G}}|}\sum_{j=1}^{4}{\cos(j\bar{G}_ya)}\right],\notag
\\
& \,\rm{\ for\ } \mathbf{\bar{G}}\neq0.
\label{sc_coeff2_a}\end{aligned}$$ On the other hand, for the case of two defects separated by a single row in a supercell containing nine cylinders ([Fig. \[sc\_supercells\]]{}(b)), [Eq. [(\[sc\_coeff2\])]{}]{} reads: $$\begin{aligned}
\rho_{\mathbf{\bar{G}}}^{-1}=&\left(\frac{\rho_h}{\rho_c}-1\right)\frac{2\pi}{9a^2}\notag
\\
&\left[r_{dU}\frac{J_1(|\mathbf{\bar{G}}|r_{dU})}{|\mathbf{\bar{G}}|}e^{i\bar{G}_ya}+r_{dL}\frac{J_1(|\mathbf{\bar{G}}|r_{dL})}{|\mathbf{\bar{G}}|}e^{-i\bar{G}_ya}+\right.\notag
\\
&\left. 2r_0\frac{J_1(|\mathbf{\bar{G}}|r_0)}{|\mathbf{\bar{G}}|}\left(1/2+\sum_{j=2}^{4}{\cos(j\bar{G}_ya)}\right)\right],\notag
\\
 &\,\rm{\ for\ } \mathbf{\bar{G}}\neq0.
\label{sc_coeff2_b}\end{aligned}$$ The corresponding expressions for $b_{\mathbf{\bar{G}}}^{-1}$ are analogous to [Eq. [(\[sc\_coeff2\_a\])]{}]{} and [Eq. [(\[sc\_coeff2\_b\])]{}]{}.

[Fig. \[sc\_fig2\]]{} shows the projected band diagrams obtained using the PWE method for (a) a SC with two equal defects corresponding to the absence of cylinders ($r_{dU}=r_{dL}=0$), (b) a SC with two defects of very different radius, $r_{dU}=0$ (one empty row) and $r_{dL}=1.2\,\rm{mm}$, (c) a SC with a single empty defect (of radius $r_d=0$) and (d) a SC with a single defect of $r_{d}=1.2\,\rm{mm}$. In [Fig. \[sc\_fig2\]]{} we also represent the supermodes for the most relevant allowed bands, which have been calculated using [Eq. [(\[sc\_bloch\])]{}]{}. In this work we will focus in two of the obtained bands for the case of two linear defects (Figs. \[sc\_fig2\](a) and (b)), one represented as a dark gray solid line (green in the online version) which we will call S (as it corresponds to a symmetric supermode when both 