& 0
\end{array} \right)\frac{e^{ik|\bm{r}-\bm{r'}|}}{|{\bm{r}-\bm{r'}}|}.
\nonumber\\ \label{Eq:freeGEH}\end{aligned}$$ In this work we focus solely on scatterers made from reciprocal constituents, as is commonly true for the metallic scatterers that constitute metamaterials. Since the materials that compose our scatterers (typically gold and silver) satisfy reciprocity microscopically, the polarizability tensor must also lead to a scattering theory that satisfies reciprocity. To derive reciprocity constraints on $\boldsymbol{\alpha}$, it is sufficient to examine the Green function in the presence of just one point scatterer at the origin. This Green function that quantifies the field at $\bm{r}_2$ due to a source at $\bm{r}_1$ in presence of a single scatterer at $\bm{r}_s$ can be written as [@Lagendijk96; @Pedro98; @tfootnote] $$\boldsymbol{G}(\bm{r}_1,\bm{r}_2)=\boldsymbol{G}^0(\bm{r}_1,\bm{r}_2)+\boldsymbol{G}^0(\bm{r}_2,\bm{r}_s)\boldsymbol{\alpha}\boldsymbol{G}^0(\bm{r}_s,\bm{r}_1),
\label{Eq:reciprocity}$$ Reciprocity requires for any Green function $\boldsymbol{G}$ (similarly split in four blocks) that $$\begin{aligned}
\left(
\begin{array} {c c}
\boldsymbol{G}_{EE}(\bm{r}_2,\bm{r}_1) & \boldsymbol{G}_{EH}(\bm{r}_2,\bm{r}_1)\\
\boldsymbol{G}_{HE}(\bm{r}_2,\bm{r}_1) & \boldsymbol{G}_{HH}(\bm{r}_2,\bm{r}_1)
\end{array}\right) = \qquad\qquad \nonumber\\
 \qquad\qquad \left(
\begin{array} {c c}
\boldsymbol{G}_{EE}(\bm{r}_1,\bm{r}_2) & -\boldsymbol{G}_{EH}(\bm{r}_1,\bm{r}_2)\\
-\boldsymbol{G}_{HE}(\bm{r}_1,\bm{r}_2) & \boldsymbol{G}_{HH}(\bm{r}_1,\bm{r}_2)
\end{array}\right)^T \nonumber  \\
\label{Eq:reciprocTRANSPOSE}\end{aligned}$$ which is equivalent to noting that swapping source and detector leaves the detected field unchanged, up to a change in sign. Specifically, Lorentz reciprocity requires a transpose for the diagonal 3$\times$3 blocks, meaning that swapping a detector and source of like character leaves the 