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preamble, added information about anisotropy and nonhomogeneous targets |
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There are now several DDA implementations,<ref name=Yurkin2007a/> extensions to periodic targets,<ref name=chaumet2003/> and particles placed on or near a plane substrate.<ref name=schmehl1997/><ref name=yurkin2015/> Comparisons with exact techniques have also been published.<ref name=penttila2007/>
Other aspects, such as the validity criteria of the discrete dipole approximation, were published.<ref name=zubko2010/> The DDA was also extended to employ rectangular or cuboid dipoles,<ref name=smunev2015/> which are more efficient for highly oblate or prolate particles.
== Theory ==
In the discrete dipole approximation, a target object is represented as a finite array of ''N'' point dipoles located at positions <math>\mathbf{r}_j</math> (<math>j = 1, 2, \dots, N</math>). The polarization vector <math>\mathbf{P}_j</math> of each dipole is related to the local electric field <math>\mathbf{E}_j</math> at that dipole by its polarizability tensor <math>\boldsymbol{\alpha}_j</math>:
In anisotropic case (diagonal polarizability)
:<math>\mathbf{P}_j = \boldsymbol{\alpha}_j \cdot \mathbf{E}_j</math>
where <math>\boldsymbol{\alpha}_j</math> is diagonal
:<math>
\boldsymbol{\alpha}_j =
\begin{pmatrix}
\alpha_{x,j} & 0 & 0 \\
0 & \alpha_{y,j} & 0 \\
0 & 0 & \alpha_{z,j}
\end{pmatrix}.
</math>
This leads to componentwise relations:
:<math>
\begin{aligned}
P_{x,j} &= \alpha_{x,j} E_{x,j},\\
P_{y,j} &= \alpha_{y,j} E_{y,j},\\
P_{z,j} &= \alpha_{z,j} E_{z,j}.
\end{aligned}
</math>
For isotropic materials, <math>\alpha_{x,j} = \alpha_{y,j} = \alpha_{z,j} = \alpha_j</math>, so
:<math>\mathbf{P}_j = \alpha_j \mathbf{E}_j</math>.
The local electric field <math>\mathbf{E}_j</math> acting on the ''j''‑th dipole is given by the sum of the incident field <math>\mathbf{E}_{\mathrm{inc}}(\mathbf{r}_j)</math> and the fields radiated by all other dipoles:
:<math>\mathbf{E}_j = \mathbf{E}_{\mathrm{inc}}(\mathbf{r}_j) + \sum_{k \ne j} \mathbf{G}(\mathbf{r}_j - \mathbf{r}_k) \cdot \mathbf{P}_k</math>
Here, <math>\mathbf{G}(\mathbf{r})</math> is the dyadic Green's function describing the field at position <math>\mathbf{r}</math> due to a unit dipole at the origin.
==Fast Fourier Transform for fast convolution calculations==
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