Discrete dipole approximation
Encyclopedia
The discrete dipole approximation (DDA) is a method for computing scattering
of radiation by particles of arbitrary shape and by periodic structures. Given a target of arbitrary geometry, one seeks to calculate its scattering and absorption properties. Exact solutions to Maxwell's equations
are known only for special geometries such as spheres, spheroids, or infinite cylinders, so approximate methods are in general required.
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 of course 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 (or Lorentz-Lorenz) relation, when the atoms are located on a cubic 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.
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.
, Flatau, and Goodman who applied Fast Fourier Transform
and conjugate gradient method
to calculate convolution
problem arising in the DDA methodology which allowed to calculate scattering by large targets. They distributed discrete dipole approximation open source code DDSCAT.
There are now several DDA implementations
. There are extensions to periodic targets and light scattering problems on particles placed on surfaces.
Comparisons with exact technique were published.
The validity criteria of the discrete dipole approximation have been recently revised. That work significantly extends the range of applicability of the DDA for the case of irregularly shaped particles.
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
of radiation by particles of arbitrary shape and by periodic structures. Given a target of arbitrary geometry, one seeks to calculate its scattering and absorption properties. Exact solutions to Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
are known only for special geometries such as spheres, spheroids, or infinite cylinders, so approximate methods are in general required.
Basic concepts
The basic idea of the DDA was introduced in 1964 by DeVoewho 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
Edward Mills Purcell
Edward Mills Purcell was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery of nuclear magnetic resonance in liquids and in solids. Nuclear magnetic resonance has become widely used to study the molecular structure of pure materials and the...
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 of course 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 (or Lorentz-Lorenz) relation, when the atoms are located on a cubic 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.
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.
Extensions
The method was improved by DraineBruce T. Draine
Bruce T. Draine is an American astrophysicist. He attended Swarthmore College from 1965 to 1969. He served in the U.S. Peace Corps in Ghana from 1969-71, where he taught secondary school physics and mathematics. He received his Ph.D. from Cornell University in 1978...
, Flatau, and Goodman who applied Fast Fourier Transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
and conjugate gradient method
Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is an iterative method, so it can be applied to sparse systems that are too...
to calculate convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...
problem arising in the DDA methodology which allowed to calculate scattering by large targets. They distributed discrete dipole approximation open source code DDSCAT.
There are now several DDA implementations
Discrete dipole approximation codes
This article contains list of discrete dipole approximation codes and their applications.The discrete dipole approximation is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. Given a target of arbitrary geometry, one seeks to calculate its scattering...
. There are extensions to periodic targets and light scattering problems on particles placed on surfaces.
Comparisons with exact technique were published.
The validity criteria of the discrete dipole approximation have been recently revised. That work significantly extends the range of applicability of the DDA for the case of irregularly shaped particles.
See also
- Computational electromagneticsComputational electromagneticsComputational electromagnetics, computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment....
- Mie theoryMie theoryThe Mie solution to Maxwell's equations describes the scattering of electromagnetic radiation by a sphere...
- Finite-difference time-domain methodFinite-difference time-domain methodFinite-difference time-domain is one of the primary available computational electrodynamics modeling techniques. Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.The FDTD method...