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Rayleigh and Rayleigh-Gans-Stevenson approximations

The main ideas of all the approximate methods are related to certain regions of the values of the basic diffraction parameters - the size parameter $ka$ and the relative refractive index $m = n/n_0$. For instance, if $ka \ll 1$ and $ka \vert m\vert \ll 1$, one has the Rayleigh scattering [418,419,421] when the particle can be interpreted as an elementary dipole whose polarizability can be evaluate from the electrostatic equations. Apparently, Love [330] was first who paid attention to the fact that the solution for a small sphere is identical to the Hertz oscillator. The possibility to use the electrostatic approximation to calculate the polarizability allows one to consider practically any shape scatterers - from the regular ones (ellipsoid, cylinder, disc) [7,274] to those with an arbitrary shape, structure and anisotropy [308,309,313,405] (in the last case the polarizability tensor should be found numerically) - using the theory of the Rayleigh scattering.

The accuracy of the Rayleigh approximation for spheres was most carefully investigated in [303]. Because of its simplicity this approximation was utilized in a huge number of works to estimate the influence of the optical, geometrical and morphological parameters of particles on their scattering and extinction characteristics [7,11,25,124]. Typical examples are the papers on estimates of the extinction spectra of disperse systems of small particles [161,205,207,208,231,273,427,431]. The Rayleigh approximation is one of the main approaches in the theory of light scattering by gases, liquids and solutions in the regions distant from the critical points [18,26,130]. It should be noted that in the literature the terms including the name of Rayleigh (Rayleigh scattering, Rayleigh line, etc.) are often applied to quite different physical phenomena. We refer a reader to the paper of Yung [509] where this question is discussed in detail.

In 1953 A.F. Stevenson in two papers [458,460] generalized the Rayleigh theory, using the expansions of the fields in terms of $ka$. For non-absorbing particles, the Rayleigh term is proportional $(ka)^2$, and the next one found in the general case by Stevenson is proportional to $(ka)^4$. For some simple shapes (ellipsoid, thin disc, etc.) one gets explicit, but rather complicated formules (the generalization of the Stevenson theory for the case of inhomogeneous particles and some simplifications see in [488]). This theory is called in the literature the Rayleigh-Gans-Stevenson approximation (RGS) keeping in mind the paper of R. Gans on ellipsoids published in 1912 [224]. Note, however, that the principal questions of light scattering by small spheroids were in our opinion considered in the paper of Rayleigh [419]. Applications of the RGS theory to different light scattering problems including orientation effects can be found in [86, 94, 105, 235, 250, 251, 252, 376, 377, 409, 410, 459, 461] (see also the review in [100]).

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