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Wikipedia / Discrete dipole approximation

The discrete dipole approximation (DDA), also known as the coupled dipole approximation, is a numerical method for computing the scattering and absorption of electromagnetic radiation by particles of arbitrary shape and composition. The method represents a continuum target as a finite array of small, polarizable dipoles, and solves for their interactions with the incident field and with each other. DDA can handle targets with inhomogeneous composition and anisotropic material properties, as well as periodic structures. It is widely applied in fields such as nanophotonics, radar scattering, aerosol physics, biomedical optics, and astrophysics.

Basic concepts

The basic idea of the DDA was introduced in 1964 by DeVoe who applied it to study the optical properties of molecular aggregates; retardation effects were not included, so DeVoe's treatment was limited to aggregates that were small compared with the wavelength. The DDA, including retardation effects, was proposed in 1973 by Purcell and Pennypacker who used it to study interstellar dust grains. Simply stated, the DDA is an approximation of the continuum target by a finite array of polarizable points. The points acquire dipole moments in response to the local electric field. The dipoles interact with one another via their electric fields, so the DDA is also sometimes referred to as the coupled dipole approximation.

Nature provides the physical inspiration for the DDA - in 1909 Lorentz showed that the dielectric properties of a substance could be directly related to the polarizabilities of the individual atoms of which it was composed, with a particularly simple and exact relationship, the Clausius-Mossotti relation (or Lorentz-Lorenz), when the atoms are located on a cubical lattice. We may expect that, just as a continuum representation of a solid is appropriate on length scales that are large compared with the interatomic spacing, an array of polarizable points can accurately approximate the response of a continuum target on length scales that are large compared with the interdipole separation.

For a finite array of point dipoles the scattering problem may be solved exactly, so the only approximation that is present in the DDA is the replacement of the continuum target by an array of N-point dipoles. The replacement requires specification of both the geometry (location of the dipoles) and the dipole polarizabilities. For monochromatic incident waves the self-consistent solution for the oscillating dipole moments may be found; from these the absorption and scattering cross sections are computed. If DDA solutions are obtained for two independent polarizations of the incident wave, then the complete amplitude scattering matrix can be determined. Alternatively, the DDA can be derived from volume integral equation for the electric field. This highlights that the approximation of point dipoles is equivalent to that of discretizing the integral equation, and thus decreases with decreasing dipole size.

With the recognition that the polarizabilities may be tensors, the DDA can readily be applied to anisotropic materials. The extension of the DDA to treat materials with nonzero magnetic susceptibility is also straightforward, although for most applications magnetic effects are negligible.

There are several reviews of DDA method.

The method was improved by Draine, Flatau, and Goodman, who applied the fast Fourier transform to solve fast convolution problems arising in the discrete dipole approximation (DDA). This allowed for the calculation of scattering by large targets. They distributed an open-source code DDSCAT. There are now several DDA implementations, extensions to periodic targets, and particles placed on or near a plane substrate. Comparisons with exact techniques have also been published. Other aspects, such as the validity criteria of the discrete dipole approximation, were published. The DDA was also extended to employ rectangular or cuboid dipoles, 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 at that dipole by its polarizability:

In anisotropic case (diagonal polarizability)

is diagonal

This leads to componentwise relations:

, so

acting on the j and the fields radiated by all other dipoles:

due to a unit dipole at the origin.

Dyadic Green's function

The free-space dyadic Green's function used in the discrete dipole approximation (DDA) can be expressed as the action of a differential operator on the scalar Green's function:

is the wavenumber is the identity matrix is the vector from the source dipole to the observation point. Evaluating the derivatives leads to the explicit form:

is the unit vector pointing from the source to the observation point.

is given by

is the vacuum wavenumber.

Define

is:

are defined as:

Polarizability

of light in vacuum. The dielectric function is related to the complex refractive index 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, which connects the polarizability to the dielectric function:

is the lattice spacing. The Clausius–Mossotti polarizability for each dipole is

is the relative permittivity is constant across all dipoles.

This formula assumes that each dipole occupies a volume embedded in an otherwise uniform dielectric medium. In most implementations of DDA the formulation is expressed in Gaussian units has dimensions of volume (cm3 is the total number of dipoles. The total target volume is thus

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:

is the wavelength of light in vacuum,

is the wavenumber.

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

, Rayleigh scattering dominates.
, the scattering is in the regime of Mie scattering.
, the geometric optics approximation becomes valid.

Effective radius and dipole discretization

, with:

is the total volume represented by the dipoles. This gives effective size parameter

, in such a case effective radius is the same as effective size parameter.

Dipole-scale size parameter

used for whole particles, one can define a local size parameter for each dipole:

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:

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:

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

Explicit Matrix Form of the DDA System

The Discrete Dipole Approximation (DDA) linear system is expressed as:

is the incident electric field vector. We have

Let N:

complex matrix, defined by:

, respectively.

Typically dipoles are arranged on a regular grid. This implies translational invariance:

is symmetric:

by grouping all x-components together, then y-components, then z-components:

Similarly, the incident field can be grouped as:

Because the system is linear, we can equivalently rewrite it in block matrix-component of the resulting field:

The expanded form of the equations is:

can be computed as a convolution when the dipoles are arranged on a regular grid, allowing the use of Fast Fourier Transforms (FFTs) to accelerate the solution.

matrix. This gives:

, all off-diagonal elements vanish, and the expressions reduce to a simple element-wise division:

Note on practical implementation. In Fortran. In contrast, Python (NumPy.

Conjugate gradient iteration schemes and preconditioning

. Among the earliest implementations were those based on direct matrix inversion, as well as the use of the conjugate gradient (CG) algorithm of Petravic and Kuo-Petravic. Subsequently, various conjugate gradient methods have been explored and improved for DDA applications. explicitly.

. These operations can be computed efficiently using convolution and FFT-based techniques when the dipole geometry is grid-based.

Fast Fourier Transform for fast convolution calculations

The use of the fast Fourier transform (FFT) to accelerate convolution operations in the discrete dipole approximation (DDA) was introduced by Goodman, Draine, and Flatau in 1991. Their method employed the three-dimensional FFT algorithm (GPFA) developed by Clive Temperton. This extension was achieved by reversing and mirroring the Green's function tensor blocks so that all positive and negative spatial offsets (lags) were represented in a single array, with an inserted zero plane between the positive and negative sides along each axis. This arrangement ensured that the discrete convolution of the Green's function with the polarization vector could be performed as a cyclic convolution using FFTs, avoiding aliasing from wraparound effects. The sign-flipping of the Green's function in the frequency domain and the block extension procedure became standard steps in efficient DDA implementations. Several alternative formulations have since been proposed.

A similar variant to that of Goodman, Draine, and Flatau was adopted in the 2021 MATLAB implementation by Shabaninezhad and Ramakrishna sizes used in DDSCAT differs from DDSCAT axes, which is mathematically equivalent to performing a full 3-D FFT on the padded domain.

Sequence of 1D FFTs was used by MacDonald in his Ph.D. thesis.

and block size, and was used originally in the discrete dipole approximation.

Thermal discrete dipole approximation

Thermal discrete dipole approximation is an extension of the original DDA to simulations of near-field heat transfer between 3D arbitrarily-shaped objects.

Discrete dipole approximation codes

Most of the codes apply to arbitrary-shaped inhomogeneous nonmagnetic particles and particle systems in free space or homogeneous dielectric host medium. The calculated quantities typically include the Mueller matrices, integral cross-sections (extinction, absorption, and scattering), internal fields and angle-resolved scattered fields (phase function). There are some published comparisons of existing DDA codes.