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WKB and eikonal approximations (high energy - HEA)

The approaches mentioned in the section title belong to one family which is called in Newton's book [56] the short waves or high energy approximation (HEA). It means the scattering by soft particles (or potentials) in the case of the wavelengths short in comparison with the particle size (or correspondingly the case of scattering of high energy particles by the potentials). In a certain sense these approaches are a generalization of the AD theory.

The Wentzel-Kramers-Brillouin (WKB) method was suggested first in optics by Rayleigh [420], but systematically used and got its name in the quantum mechanics [56,292,442]. The WKB approximation can be considered as a modification of the RDG theory, where in the interference integral (3) besides the geometrical phase difference one also includes the change of the phase of a ray on its path in the particle to the scattering element. This leads to an increase of the particle size region in calculations of the angular intensity, but only for small angles. On the other side, the application of the optical theorem to the WKB amplitude of the forward scattering gives the formula of the AD up to a factor $\frac{2}{m+1} A(0)_{\rm WKB} = A(0)_{\rm AD}$ [56,292].

In the eikonal approximation the field inside the particle is also approximated by the plane wave with the phase shift which is proportional to $(m^2-1)$ and hence differs from the WKB phase shift only by the factor $(m+1)/2$. The eikonal approximation formula for the scattering amplitude can be slightly modified and represented in the form suggested by Glauber [56], i.e. as an integral over the impact parameters. This approach was used in light scattering by small particles in the works of T.W. Chen [176,177,178,179] and of the French group [164,166,167,180,392,393] to increase the angle region where one can calculate the intensity of scattered radiation. The eikonal approximation was also useful in solution of the problem of multiple scattering in media with large-scale inhomogeneities [1,2,9,10,77].

The main disadvantage of all versions of the short wave (AD, WBK, eikonal, etc.) approximation consists of their scalar nature and ignoring the wave polarization (an attempt [393] to get the vector generalization of the eikonal approximation can be hardly called successful). The generalization of the AD theory for anisotropic particles suggested in 1979 by G. Meeten [338] and used in [337,339,340,341,502,513] and our works [101,112,115,284,288] is applicable only to the optically soft particles, but in the isotropic case this approximation is reduced again to a scalar variant. In this connection a particular interest is deserved by a recent pioneer work [10] where the eikonal approximation principles are applied to solution of the complicated problem of multiple scattering of electromagnetic waves in a medium with large-scale anisotropic inhomogeneities.

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