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=== Dyadic Green’s function ===
:<math>
\mathbf{G}(\mathbf{r}) = \left[ \nabla \nabla + k^2 \mathbf{I} \right] \frac{e^{ikr}}{r},
</math>
where <math>k</math> is the wavenumber, <math>\mathbf{I}</math> is the identity matrix, and <math>\mathbf{r}</math> is the vector from the source dipole to the observation point. Evaluating the derivatives leads to the explicit form:
:<math>
\mathbf{G}(\mathbf{r}) = \frac{e^{ikr}}{r^3} \left[
k^2 r^2 \left( \mathbf{I} - \hat{\mathbf{r}} \hat{\mathbf{r}} \right)
+ (1 - ikr) \left( 3 \hat{\mathbf{r}} \hat{\mathbf{r}} - \mathbf{I} \right)
\right],
</math>
where <math>\hat{\mathbf{r}} = \mathbf{r} / |\mathbf{r}|</math> is the unit vector pointing from the source to the observation point.
This Green’s tensor describes the electric field generated by a dipole in a homogeneous medium. It is used to compute the off-diagonal blocks of the interaction matrix in DDA, that is, the interaction between distinct dipoles <math>j \ne k</math>. The singular self-term <math>\mathbf{G}(\mathbf{r} = 0)</math> is excluded and replaced by a prescribed local term involving the inverse polarizability tensor <math>\boldsymbol{\alpha}_j^{-1}</math>.
Thus, the electric field at dipole <math>j</math> due to dipole <math>k</math> is given by
:<math>
\mathbf{G}_{jk} = \frac{e^{ikr_{jk}}}{r_{jk}^3} \left[ k^2 r_{jk}^2 \left(\mathbf{I} - \hat{\mathbf{r}}_{jk} \hat{\mathbf{r}}_{jk}\right) + \left(1 - ikr_{jk}\right)\left(3\hat{\mathbf{r}}_{jk}\hat{\mathbf{r}}_{jk} - \mathbf{I}\right)\right],
</math>
where <math>\mathbf{r}_{jk} = \mathbf{r}_j - \mathbf{r}_k</math>, <math>r_{jk} = |\mathbf{r}_{jk}|</math>, and <math>\hat{\mathbf{r}}_{jk} = \mathbf{r}_{jk}/r_{jk}</math>. Here <math>\mathbf{I}</math> is the identity matrix and <math>k = 2\pi/\lambda</math> is the vacuum wavenumber.
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