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=== Polarizability ===
{{Main|Polarizability}}
In the discrete dipole approximation, the electromagnetic response of a target is modeled by replacing the continuous material with a finite array of point dipoles. Each dipole represents a small volume of the material and acts as a polarizable unit that interacts with both the incident field and the fields radiated by all other dipoles. The key parameter that describes how each dipole responds to the local electric field is its polarizability <math>\alpha_j</math>. For a homogeneous material, the polarizability of a dipole is determined by the material’s complex dielectric function <math>\varepsilon(\lambda)</math>, which depends on the wavelength <math>\lambda</math> of light in vacuum. The dielectric function is related to the complex refractive index <math>n = n' + i n''</math> through <math>\varepsilon = n^2</math>. The goal in DDA is to assign to each dipole a polarizability <math>\alpha_j</math> such that the array of dipoles reproduces, as accurately as possible, the scattering and absorption behavior of the original continuous medium. For isotropic materials, a common starting point is the [[Clausius–Mossotti relation|Clausius–Mossotti relation]], which connects the polarizability to the dielectric function:
:<math>
\alpha_j = 3V_j \frac{\varepsilon - 1}{\varepsilon + 2},
</math>
where <math>V_j</math> is the volume associated with dipole ''j''. This formula assumes that each dipole occupies a spherical volume embedded in an otherwise uniform dielectric medium. In most implementations of DDA the formulation is expressed in Gaussian units (CGS). In these units, the polarizability <math>\alpha_j</math> has dimensions of volume (cm³). In SI units, additional factors involving <math>4\pi\epsilon_0</math> appear in the expressions for <math>\alpha_j</math> and the field interactions.
To improve the accuracy of the method, particularly for targets with high refractive indices or for fine discretizations, various corrections to <math>\alpha_j</math> are applied. These include: the lattice dispersion relation (LDR) polarizability (Draine & Goodman, 1993), which adjusts <math>\alpha_j</math> to ensure that the dispersion relation of an infinite lattice of dipoles matches that of the continuous material; the radiative reaction (RR) correction, which compensates for the fact that each dipole radiates energy and is influenced by its own radiation field.
==Fast Fourier Transform for fast convolution calculations==
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