This review was originally published as B. T. Draine and P. J. Flatau. *Discrete dipole approximation for scattering calculations*, J. Opt. Soc. Am. A , 11: 1491-1499, 1994. It is extended and modified version.

## Abstract

The discrete-dipole approximation (DDA) for scattering calculations, including the relationship between the DDA and other methods, is reviewed. Computational considerations, i.e., the use of complex-conjugate gradient algorithms, fast-Fourier-transform methods, order of scattering approximaton are discussed. The method was reviewed by Draine and Flatau (1994) [2] and more recently by Yurkin and Hoekstra (2007) [12]. This review is intended to provide up to date information about extensions and applications of the DDA as well as basic information about the method.

## Introduction

The discrete-dipole approximation (DDA) is a flexible and powerful technique for computing scattering and absorption by targets of arbitrary geometry. Extension of the method allow calculations of light scattering on periodic targets or targets placed close to surfaces. The development of efficient algorithms and the availability of inexpensive computing power together have made the DDA the method of choice for many scattering problems incuding applications to photonics, radar backscatter, aerosol optics, cosmic dust. In this paper we review the DDA, with particular attention to recent developments incuding numerics and applications. This paper is modified and extended version of the review by Draine and Flatau (1994) [2]. Discrete dipole approximation found many applications. There are several theoretical advances of the method - notable are works on more general geometries and larger range of refractive index (see section on method extensions) and applications of T-matrix within the DDA to orientational averaging problem.

In Section on what is discrete dipole approximation we briefly summarize the conceptual basis for the DDA. The relation of the DDA to other methods is discussed. We also discuss extensions of the method to more complex geometries, high refractive index, and varying refractive index background.

DDA calculations require choices for the locations and the polarizabilities of the point dipoles that are to represent the target. In Section on dipole specification we discuss these choices, with attention to the important question of dipole polarizabilities. Criteria for the validity of the DDA are also considered.

Recent advances in numerical methods now permit the solution of problems involving large values of N, the number of point dipoles. These developments, particularly

the use of complex-conjugate gradient (CCG) methods and fast-Fourier-transform (FFT) techniques, are reviewed in section section_computational on computationa aspects.

We illustrate the accuracy of the method in section on method accuracy by using the DDA to compute scattering by a single sphere and by targets consisting of two spheres in contact and by comparing our results with the exact results for these geometries. Although these are only examples, they make clear that the DDA can be used to compute highly accurate results.

Applications of the DDA to phtonics, atmospheric optics, astrophysics, and marine optics are discussed in section on applications.

Information about exisiting DDA implementations and codes relevant to DDA such as conjugate gradient is given in Appendix A.

## What is the discrete dipole approximation

### Approximation

Given a target of arbitrary geometry, we seek 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. The DDA is one such method.

The basic idea of the DDA was introduced in 1964 by DeVoe [3] [4] 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 [8] 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, [1] [8] so the DDA is also sometimes referred to as the coupled dipole approximation. [9] [10] The theoretical basis for the DDA, including radiative reaction corrections, is summarized by Draine [1].

Nature provides the physical inspiration for the DDA: in 1909 Lorentz showed [7] 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 $r_j$ of the dipoles $j = 1, ... , N$ ) and the dipole polarizabilities $a_j$. For monochromatic incident waves the self-consistent solution for the oscillating dipole moments $P_j$ may be found, as is discussed in section on computational aspects; from these P 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 aj may be tensors, the DDA can readily be applied to anisotropic materials. [1] [11] The extension of the DDA to treat materials with nonzero magnetic susceptibility is also straightforward, [5], [6] although for most applications magnetic effects are negligible.

### Relation to other methods (to be added)

## Specification of the dipole array

## Computational considerations

## Accuracy and validations

## Applications

## Implementations

## Summary

*The discrete-dipole approximation and its application to interstellar graphite grains,*Astrophys. J. 333, 848-872 (1988).

*Discrete dipole approximation for scattering calculations*, J. Opt. Soc. Am. A , 11: 1491-1499, 1994.

*Optical properties of molecular aggregates. I. Classical model of electronic absorption and refraction,*J. Chem. Phys. 41, 393-400 (1964).

*Optical properties of molecular aggregates. II. Classical theory of the refraction, absorption, and optical activity of solutions and crystals,*J. Chem. Phys. 43, 3199-3208 (1965).

*General theory of the Purcell-Pennypacker scattering approach and its extension to bianisotropic scatterers,*Astrophys. J. 394, 494-499 (1992).

*Strong and weak forms of the method of moments and the coupled dipole method for scattering of time-harmonic electromagnetic fields,*Int. J. Mod. Phys. C3, 583-603 (1992).

*Scattering and absorption of light by nonspherical dielectric grains,*Astrophys. J. 186, 705-714 (1973).

*Evaluation of the scattering matrix of an arbitrary particle using the coupled dipole approximation,*J. Chem. Phys. 84, 2658-2667(1986).

*Light scattering by an arbitrary particle: a physical reformulation of the coupled dipoles method,*Opt. Lett. 12, 10-12 (1987).

*The ultraviolet extinction from interstellar graphitic onions,*Nature (London) 366, 227-228 (1988).

*The discrete dipole approximation: an overview and recent developments*, 2007.