Coupled dipoles method (Purcell-Pennypacker, CDM)In the theory of light scattering by small particles this method became extensively applicable after the paper of E.M. Purcell and C.R. Pennypacker in 1973 [404], though without doubt the physical basis of the method was known and applied in other fields earlier (see, for instance, the review of A. Lakhtakia in 1992 [307] who cites the work of Gray published in 1916 and others). In the western literature this method is now called ``coupled dipole method" (CDM). The physical formulation of the method is rather simple. A particle is divided into a set of small domains so that each of them can be considered as a dipole scatterer. Each of the domains is exited by the field of the incident wave and by the the fields scattered by all other domains. Thus, a system of equations relative to the fields scattered by the elementary domains can be obtained, and summation of these fields gives the total scattering field. The accuracy of the method depends on the number, size and shape of the elementary domains as well as on some features of selected expression for their polarizability [209,307,313,317].Initially the CDM was applied to spherical elementary domains with the usual polarizability of a dipole sphere, but later two modifications were suggested: 1) radiation damping was included into the expression for the polarizability of a small sphere (in order to fulfill the optical theorem) [209]; and 2) the polarizability of the sphere was computed on the basis of the first term of the Mie expansion [210]. Since both modifications were made from physical reasons, the problem of confrontation of different modifications of the CDM and of the method of moments arisen. The problem was successfully solved by Lakhtakia [307,313,317]. First, he compared the CDM and solution of the volume integral equation by the method of moments, where the scatterer is also divided into domains. A subtle difference between both methods is in the fact that in the first method one works with the fields which excite a dipole (without the field of the dipole), whereas in the second method one uses the real fields in a given domain. In the rigid formulation both methods became equivalent (i.e. the systems of the algebraic equations could be obtained one from another). Further, Lakhtakia considered the problem of obtaining the dyadic polarizability for a small anisotropic region of an arbitrary shape and showed its two forms: ``strong'' one including contribution of self-interaction or radiation damping (they are analogous but not equivalent [209]), and ``weak'' one coinciding with the Purcell-Pennypacker formulation. If one uses the equivalent forms (strong or weak) of the CDM and the moment of moments, there is no principal difference between them. The important result of this analysis was the development of a general approach to calculation of the polarizability from rigid integral equations which was later applied to bianisotropic [308,313] and chiral [309] small particles as well as formulation of the generalized CDM for such scatterers [307,310,315,317]. Note that independently of this work a generalization of the CDM for optically active and chiral particles was made in [449,450,454], and use of the anisotropic elementary dipoles was discussed in [455]. Lakhtakia used the derived general equations for dyadic polarizability to generalize the classic theory of Maxwell Garnett for the effective optical constants of composite media [305,306,308,309,310,312,315,436]. Note that in works of A.G. Ramm [74,405] a general formalism of calculations of the polarizability was also developed for the body of an arbitrary shape, but the formalism did not include the radiation damping and was applied only to generalize the Rayleigh scattering theory. Solution of the equation systems in the CDM demands essential computer resources (memory and speed of CPU). In 1978 Y.L. Yung [510] attempted to use the variational principle for energy to increase the efficiency of the CDM and to bring the calculations to iterations, but his results did not find a further use. In 1987-8 Sh. Singham and C.F. Bohren [452,453] reformulated the equations of the CDM as a sequence of multiple interdipole scatterings, which significantly increased the efficiency of the algorithm. It should be noted that the priority in formulation of this idea belongs to R. Chiappetta's work of 1980 [181]. The approach was used in [182,183] to analyse scattering by spheroids and spiral structures. The paper [451] suggested a hybrid scheme where the particle was divided into domains larger than usually in the CDM and within which the fields were calculated exactly (or approximately), and then the iteration scheme of the CDM was applied or a lower order system of linear equations was solved. Generally, the same idea in a slightly modified form was realized in [165]. One of the advantages of the CDM is that the equations have a certain symmetry and allow one to use the powerful apparatus of the quantum theory of angular momentum for orientation averaging for ensembles of chaotical oriented [336,440,448] and aligned [466] particles. Though these works were done for macromolecular solutions (see also the application of an analogous approach in [157]), we believe that their ideas have a much wider application including the light scattering theory. For instance, in the following section one will see that the apparatus of the quantum mechanics theory of angular momentum is a powerful tool in the T-matrix method. The analysis of the cited literature (see also [21,171,174,302,304]) allows one to conclude that in coming years the CDM will be one of the most popular methods in the theory of light scattering by particles of complex shape and structure. As an illustration of its large possibilities we refer to the paper [484] where the rigid quantitative results were obtained for scattering by an anisotropic sphere. |
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