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Generalizations of the RDG approximation

For non-spherical isotropic particles, the classic variant of the RDG approximation does not include the anisotropy of the shape and hence does not give the correct Rayleigh limit, say, for small ellipsoids. Nevertheless, we have discussed above the works on anisotropic particles where the optical anisotropy was involved in the RDG theory, but implicitly assumed that the anisotropy was connected only with the anisotropy of the material. There are obviously possibilities a bit to improve the RDG approximation for isotropic non-spherical particles if to use a plane wave with a quasi-static (electrostatic) amplitude as the internal field. We call this variant a modified RDG approximation (MRDG). The idea of such a modification comes back to the work of K.S. Shifrin [124], who developed the iteration method of solution of the integral equation for an effective field and got a generalized version of the RDG with an exact Rayleigh limit for spherical particles. In 1956 R. Burberg [173] and A.V. Shatilov [121,122] apparently were first who applied the MRDG to scattering by thin cylinders and spheroids, respectively. This variant of the theory was used in the works of J.-C. Ravey [406,407,408,409,410,411] on light scattering by ordered disperse systems and in our papers [96,100] to calculate the Mueller matrix (see also the recent paper [92] on the same topics) and in the theory of dichroism [97,101,108,113].

The quasistatic extension of the RDG was criticized by Stoylov and Stoimenova [462], but in our opinion that was groundless. First, the MRDG leads to the correct Rayleigh limit. Second, Ravey [409] demonstrated the coincidence of the results of the MRDG and the exact RGS approximation (in the small size domain) for extremely eccentric particles. At last the MRDG approximation was recently derived by V.G. Farafonov [89] as the limit of the rigid solution for strongly eccentrical spheroids¹. Thus, the paper [89] in fact closes the discussion on justification of the MRDG.

The second approach to generalization of the RDG is connected with the use of the Born iterations for the scattering integral equation [95,124,132,264]. We say about the second Born approximation which is also called the second approximation in Shifrin's method [132,292]. The first attempt of Y. Ikeda in 1963 [264] to apply this approach to non-spherical particles led to tedious results which did not find further application. In 1976 Ch. Acquista [132] made the Fourier transformation of the second approximation and obtained more compact formules as well as numerical results for spheres. In [95] we got a rigid integral equation for the scattering amplitude already in the complex Fourier space and then made its iterations. This approach was generalized for anisotropic particles in our works [97,101,285]. A method to increase the convergence of the Born iterations was suggested in [291]. Applications of the second approximation to different tasks see in [97, 101, 106, 109, 174, 191, 203, 204, 239, 240, 241, 253, 284, 285, 287, 292, 386, 461, 489]. For ensembles of non-spherical irregular particles, a new statistical approach within the RDG theory was suggested by K.S. Shifrin and I.A. Mikulinski [126,443]. In a recent review [162] the importance of this approach was especially pointed out (though without references to the papers [126,443]). In [185] the problem of light scattering by irregular particles was considered using a modified Mie theory (see also modelling of the Mueller matrix for irregular particles using the Mie theory in [254,394]).

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