# Discrete dipole approximation

File:Shape DDA1.svg
In the discrete dipole approximation a larger object is approximated in terms of discrete dipoles

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 cylinders, so approximate methods are in general required. However, the DDA employs no physical approximations and can produce accurate enough results, given sufficient computer power.

## Basic concepts

The basic idea of the DDA was introduced in 1964 by DeVoe [1] 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 [2] 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. [3] [4]

Nature provides the physical inspiration for the DDA: in 1909 Lorentz [5] 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 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.

Alternatively, the DDA can be derived from volume integral equation for the electric field.[6] 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.

## Extensions

The method was improved by Draine, 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. [7] [8] There are now several DDA implementations.[6] There are extensions to periodic targets [9] and light scattering problems on particles placed on surfaces. [10] A convergence theory of the DDA has been developed[11] and comparisons with exact technique were published.[12] The validity criteria of the discrete dipole approximation have been recently revised.[13] That work significantly extends the range of applicability of the DDA for the case of irregularly shaped particles.

## References

1. H. DeVoe, Optical properties of molecular aggregates. I. Classical model of electronic absorption and refraction, J. Chem. Phys. 41, 393-400 (1964).
2. E. M. Purcell and C. R. Pennypacker (1973). "Scattering and absorption of light by nonspherical dielectric grains". Astrophysical Journal. 186: 705. Bibcode:1973ApJ...186..705P. doi:10.1086/152538.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
3. S. B. Singham and G. C. Salzman, Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation, J. Chem. Phys. 84, 2658-2667(1986).
4. S. B. Singham and C. F. Bohren, Light scattering by an arbitrary particle: a physical reformulation of the coupled dipoles method, Opt. Lett. 12, 10-12 (1987).
5. H. A. Lorentz, Theory of Electrons (Teubner, Leipzig, 1909)
6. M. A. Yurkin and A. G. Hoekstra (2007). "The discrete dipole approximation: an overview and recent developments". Journal of Quantitative Spectroscopy and Radiative Transfer. 106: 558–589. arXiv:0704.0038. Bibcode:2007JQSRT.106..558Y. doi:10.1016/j.jqsrt.2007.01.034.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
7. Draine, B.T., and P.J. Flatau (1994). "Discrete dipole approximation for scattering calculations". J. Opt. Soc. Am. A. 11 (4): 1491–1499. Bibcode:1994JOSAA..11.1491D. doi:10.1364/JOSAA.11.001491.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
8. B. T. Draine and P. J. Flatau (2008). "The discrete dipole approximation for periodic targets: theory and tests". J. Opt. Soc. Am. A. 25. arXiv:0809.0338. Bibcode:2008JOSAA..25.2693D. doi:10.1364/JOSAA.25.002693.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
9. P. C. Chaumet, A. Rahmani, and G. W. Bryant, Generalization of the coupled dipole method to periodic structures, Phys. Rev. B 67, 165404 (2003).
10. R. Schmehl, B. M. Nebeker, and E. D. Hirleman, Discrete dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique, J. Opt. Soc. Am. A 14, 3026–3036 (1997).
11. M. A. Yurkin, V. P. Maltsev, and A. G. Hoekstra (2006). "Convergence of the discrete dipole approximation. I. Theoretical analysis" (PDF). Journal of the Optical Society of America A. 23 (10): 2578–2591. arXiv:0704.0033. Bibcode:2006JOSAA..23.2578Y. doi:10.1364/JOSAA.23.002578.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
12. A. Penttila, E. Zubko, K. Lumme, K. Muinonen, M. A. Yurkin, B. T. Draine, J. Rahola, A. G. Hoekstra, and Y. Shkuratov, Comparison between discrete dipole implementations and exact techniques, J. Quant. Spectrosc. Radiat. Transfer 106, 417-436 (2007).
13. E. Zubko, D. Petrov, Ye. Grynko, Yu. Shkuratov, H. Okamoto, K. Muinonen, T. Nousiainen, H. Kimura, T.Yamamoto, and G. Videen. Validity criteria of the discrete dipole approximation, Applied Optics 49, 1267-1279 (2010).