The discrete dipole approximation: an overview and recent developments
📝 Abstract
We present a review of the discrete dipole approximation (DDA), which is a general method to simulate light scattering by arbitrarily shaped particles. We put the method in historical context and discuss recent developments, taking the viewpoint of a general framework based on the integral equations for the electric field. We review both the theory of the DDA and its numerical aspects, the latter being of critical importance for any practical application of the method. Finally, the position of the DDA among other methods of light scattering simulation is shown and possible future developments are discussed.
💡 Analysis
We present a review of the discrete dipole approximation (DDA), which is a general method to simulate light scattering by arbitrarily shaped particles. We put the method in historical context and discuss recent developments, taking the viewpoint of a general framework based on the integral equations for the electric field. We review both the theory of the DDA and its numerical aspects, the latter being of critical importance for any practical application of the method. Finally, the position of the DDA among other methods of light scattering simulation is shown and possible future developments are discussed.
📄 Content
The Discrete Dipole Approximation (DDA) is a general method to compute scattering and absorption of electromagnetic waves by particles of arbitrary geometry and composition.
Initially the DDA was proposed by Purcell and Pennypacker (PP) [1], who replaced the scatterer by a set of point dipoles. These dipoles interact with each other and the incident field, giving rise to a system of linear equations, which is solved to obtain dipole polarizations. All the measured scattering quantities can be obtained from these polarizations.
The DDA was further developed by Draine and coworkers [2][3][4][5], who popularized the method by developing a publicly available computer code DDSCAT [6]. Later it was shown that the DDA also can be derived from the integral equation for the electric field, which is discretized by dividing the scatterer into small cubical subvolumes. This derivation was apparently first performed by Goedecke and O’Brien [7] and further developed by others (see, for instance, [8][9][10][11]). It is important to note that the final equations, produced by both lines of derivation of the DDA are essentially the same. The only difference is that derivations based on the integral equations give more mathematical insight into the approximation, thus pointing at ways to improve the method, while the model based on point dipoles is physically clearer.
The DDA is called the coupled dipole method or approximation by some researchers [12,13]. There are also other methods, such as the volume integral equation formulation [14] and the Digitized Green’s Function (DGF) [7], which were developed completely independently from PP. However, later they were shown to be equivalent to DDA [8,15]. In this review we will use the term DDA to refer to all such methods, since we describe them in terms of one general framework. However, it is difficult to separate unambiguously the DDA from other similar methods, based on the volume integral equations for the electromagnetic fields, such as a broad range of Method of Moments (MoM) with different bases and testing functions [16][17][18][19]. In our opinion, one fundamental aspect of the DDA is that the solution for the “physically meaningful” internal fields or their direct derivatives, e.g. polarization, plays an integral role in the process. In other words, any DDA formulation can be interpreted as replacing a scatterer by a set of interacting dipoles; this is further discussed in Section 2. An example of method that is not considered DDA is the MoM with higher-order hierarchical Legendre basis functions [17].
The DDA is a popular method in the light-scattering community and it has been reviewed by several authors. An extensive review by Draine and Flatau [4] covers almost all DDA developments up to 1994. A more recent review by Draine [5] mainly concerns applications and numerical considerations. DDA theory was discussed together with other methods for light scattering simulations in reviews by Wriedt [20], Chiappetta and Torresani [21], and Kahnert [15] and in books by Mishchenko et al. [22] and Tsang et al. [23]. Jones [24] placed the DDA in context of different methods with respect to particle characterization. However, many important DDA developments since 1994 are not mentioned in any of these manuscripts. Those that are mentioned are usually considered as side-steps, and are not placed into a general framework. Moreover, to the best of our knowledge numerical aspects of the DDA have never been reviewed extensivelyeach paper discusses only a few particular aspects. In this review we try to fill this gap.
A general framework is developed in Section 2 to ease the further discussion of different DDA models. This framework is based on the integral equation because it allows a uniform description of all the DDA development. However, connection to a physically clearer model of point dipoles is discussed throughout the section. The sources of errors in the DDA formulation are also discussed there.
In Section 3 the physical principles of the DDA are reviewed and results of different models are compared. In Subsection 3.1 different improvements of polarizabilities and interaction terms are reviewed from a theoretical point of view. Different expressions for Cabs also are discussed. Comparison of simulation results using different formulations is given in Subsection 3.2. Subsection 3.3 covers the special case of a cluster of spheres that allows particular improvements and simplifications. In section 3.4 different significant modifications are reviewed, which do not fall completely into the general framework described in Section 2.
Enhancements of the DDA for some special purposes also are discussed.
Different numerical aspects of the DDA are reviewed in Section 4. These are concerned primarily with solving very large systems of linear equations (Subsection 4.1). Subsection 4.2 describes the simplest iterative procedure to solve DDA linear system, which has a clear physical
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