Discrete dipole approximation: Difference between revisions

Revision as of 18:17, 21 July 2025

Revision as of 21:47, 23 July 2025

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:<math>

\alphaalpha_j = \frac{3V_\mathrm{dipole}}{4\pi}\frac{\varepsilon_j - 1}{\varepsilon_j + 2},

</math>

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To improve the accuracy of the method 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.

=== Size parameter ===

The size parameter is a dimensionless quantity used in scattering theory to characterize the size of a particle relative to the wavelength of the incident light. For a sphere, it is defined as:

:<math>x = \frac{2\pi r}{\lambda} = k r</math>

where: <math>x</math> is the size parameter (dimensionless), <math>a</math> is the radius of the sphere, <math>\lambda</math> is the wavelength of light in vacuum,

:<math>k = \frac{2\pi}{\lambda}</math> is the wavenumber.

In case of a sphere, the size parameter determines the scattering regime:

* If <math>x \ll 1</math>, Rayleigh scattering dominates.

* If <math>x \sim 1</math>, the scattering is described by Mie theory.

* If <math>x \gg 1</math>, the geometric optics approximation becomes valid.

For nonspherical targets with the same volume as a sphere, the effective radius <math>r_{\text{eff}}</math> is often used in place of <math>r</math>, with:

:<math>x = \frac{2\pi r_{\text{eff}}}{\lambda}</math>

=== Effective radius and dipole discretization ===

The '''effective radius''' <math>r_{\text{eff}}</math> is defined as the radius of a sphere whose volume equals the total volume of the dipole array:

:<math>r_{\text{eff}} = \left( \frac{3 V_{\text{tot}}}{4\pi} \right)^{1/3} = \left( \frac{3 N d^3}{4\pi} \right)^{1/3}</math>

where: <math>N</math> is the total number of dipoles, <math>d</math> is the dipole spacing, <math>V_{\text{tot}} = N d^3</math> is the total volume represented by the dipoles.

=== Dipole-scale size parameter ===

Each polarizable point (dipole) occupies a cubic volume with side length <math>d</math>. Analogous to the global size parameter <math>x = 2\pi r / \lambda</math> used for whole particles, one can define a local size parameter for each dipole:

:<math>x_d = |m|kd = \frac{2\pi |m| d}{\lambda}</math>

This local parameter quantifies the ratio of the dipole size to the wavelength of light inside the material. For the DDA to be accurate, the field should vary slowly over the size of each dipole. This condition is satisfied when:

:<math>x_d = |m|kd \lesssim 0.5</math>

This ensures that each dipole is optically small, fields vary slowly over the dipole and the polarizability formula used for each dipole is accurate. Notice that a similar parameter plays a crucial role in the anomalous diffraction theory of van de Hulst, where the total phase shift experienced by light rays traveling through or around the particle is given by:

:<math>\delta = 2\pi (m - 1) \frac{r}{\lambda}</math>

This describes the optical path difference introduced by the particle (or in the case of DDA by a dipole).

==Fast Fourier Transform for fast convolution calculations==