Anomalous diffraction approximation (AD)
Using the Huygens and Babinet principles van de Hulst [11] considered the problem of light scattering and extinction by a particle having the size
and the refractive index 
.
The expression for the scattering amplitude in the small angle region
obtained by van de Hulst is a generalization of the Fraunhofer
diffraction formula and allows one to find the extinction
cross-section from the optical theorem [11,56]
where
![\begin{displaymath}
C_{\rm ext} = 2 {\rm Re} \int [1 - \exp(-i\rho(z))] dS,
\end{displaymath}](img35.gif){kind=small}
is the phase shift of a ray propagating along the 
-axis,
and integration is over all the rays intersecting the particle.
The absorption cross-section is
The range of applicability of the anomalous diffraction theory for the scattering amplitude is limited by small angles, however the formules for the cross-sections became good approximations for a wide class of particles. The AD approximation can be easily applied to non-spherical particles, which to some extent explains its popularity. The formules for scattering and extinction cross-sections for a spheroid were first derived by J.M. Greenberg in 1960 [232]. A simple method based only on the geometric consideration and using no complicated integrations in derivation of the extinction and absorption expressions was suggested in [79,80] (the same problem was later considered in [259]). Another approach was earlier developed in [170]. The extinction cross-section in the general case of ellipsoids is given in [84] (the same result was obtained later again in [49,51]). The integral cross-section for a cylinder was derived first in [194] (the same problem was considered in [68,290,292,293,438]). In frame of the AD approximation one also studied the integral characteristics of inhomogeneous particles: core-mantle [22,30,100] and multi-layered [178] spheres (including non-concentric particles [49]), multi-layered spheroids [29] as well as the particles with ledges [320] and cavities [321]. In [186,187] the theorem of summation of the cross-sections within the AD was derived and the formula for prisma-like particles was obtained.
![\begin{displaymath}
C_{\rm abs} = \int [1 - \exp(-2 {\rm Im} \rho(z))] dS.
\end{displaymath}](img38.gif){kind=small}
Detailed studies of the scattering coefficient and wave exponent of disordered spheroidal particles including numerical calculations and analytic asymptotics for large and small phase shifts and the limit cases of the eccentricity values were performed in [80,81,100,104,117,119,120,129] and later in [43,44,49,51,53,65,66,85].
Studies of the optical characteristics of polydisperse systems of spherical particles using the AD approximation was published in early works of K.S. Shifrin and his coauthors2 and in the papers of V.N. Lopatin and coauthors [45,48,49].
In 1969 F.D. Bryant and P. Latimer [170] suggested to use the ratio of the volume to averaged cross-section as a size parameter for disordered spheroids and the corresponding phase shift parameter. Similarly, for polydisperse systems A.P. Prishivalko (see, e.g., [69,70] and references in [127]) used a generalized size parameter equal to the ratio of the mean volume to the mean surface area. Both these ideas were utilized in the papers of V.N. Lopatin and his coauthors (``optical equivalence principle") [45,48,49,52] and further developed as a general approach by L.E. Paramonov (small-parameter estimates of the integral cross-sections) [58,60,62].
The accuracy of the AD approximation in description of extinction and absorption of light by different scatterers was investigated in a number of works (see, e.g., the books [11,23,49,123] and references therein). The general conclusion is that for the systems of optically soft particles the AD approximation qualitatively correctly shows all main laws of extinction, except for several minor features related to orientation, etc. [49]. Attempts to improve the AD approximation in the case of optically not very soft particles were undertaken in [244,293,370], but without success. This problem for spherical particles was solved by A.Ya. Perelman [390,391] after a sophisticated mathematical analysis of the Mie series, for cylinders a similar problem was solved in [439].
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