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Methods for calculating light scattering
by single particles
N.G. Khlebtsov
Part I. Exact methods
The general formulation of the problem of light scattering (diffraction) is rather simple. The field
~
E 0 is incident on a scatterer of the volume V , and creates the field ~
E i inside the scatterer and an
additional (diffraction) field ~
E s outside it. Thus, from Maxwell's equations one should find the total
field ~
E equal to ~
E i inside V and to ~
E 0 + ~
E s outside V and satisfying the boundary conditions on
V . Despite the simplicity of the scheme, a concrete solution of the problem essentially depends on
the geometry of the scatterer and its structure. For instance, even for spherical scatterer with an
anisotropic tensor of the refractive index, a general solution cannot be obtained in a closed form [7].
Therefore, in the theory of light scattering by small particles one has developed various methods whose
applicability regions and efficiency depend on the concrete conditions. In this part we discuss the exact
methods which include both the analytical and numerical approaches, since from a contemporary point
of view an efficient numerical algorithm realized at a computer is equivalent to an analytical solution
which as a rule also needs non­trivial calculations.
1 Separation of variables method
1.1 Spherical particles
The classic solution for homogeneous sphere was suggested in 1899 by A.E.N. Love [330], in 1908 by
G. Mie [342] and in 1909 by P. Debye [199]. In 1910 Lord Rayleigh [422] obviously knowing nothing
about the works of Mie and Debye removed a mistake in the paper of Love. Traditionally the theory of
light scattering by a sphere is called the Mie theory. The solution for a focused Gaussian beam (called
the generalized Mie solution) was first derived in the work [230] (see [236]) and developed further in
[236, 333]. The same problem was recently considered in [61] and its detailed analysis see in [329].
With the appearance of powerful PCs the computation using the Mie theory practically became a
routine. Nevertheless, even in 1994 we find a paper [19] published on the Mie theory where a closed
integral representation of the extinction cross­section was obtained (in other words the Mie series was
transformed into an integral). To some extent unusual results for magnetic spheres were presented in
[279].
The generalization of the Mie theory for core­mantle spheres was made in 1951--2 by A.L. Aden
and M. Kerker [133] and K.S. Shifrin [125] and for the case of an arbitrary number of layers in
[343, 490]. Detailed results for layered spheres can be found in [274, 276, 277]. A core­mantle sphere
as the model of biological particles was first considered in [168, 169, 198], a detailed study of this and
three­layered model of cells was done in [46, 49, 50]. For an inhomogeneous sphere with an arbitrary
radial distribution of the refractive index, the general formal solution was obtained in [504, 505] (see
also [325, 435]), and in [501] all refractive index profiles for which the radial solutions are expressed
via the known functions were given. A thorough investigation of the problem of light scattering by
inhomogeneous spheres was done by the group of A.P. Prishivalko [71, 72] (nearly all literature on this
question is cited in the monograph [72]) and in the recent paper of A.Ya. Perelman [67].
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The exact solutions of the Mie problem for an optically active sphere (i.e. a sphere of optically
active material [7]) and for a sphere with an optically active mantle were first obtained in 1974--5 by
S. Bohren [158, 159]. As it was mentioned, for a sphere of anisotropic material, the solution of the
general type like the Mie theory does not exist. Nevertheless for spheres with the axial anisothropy
of material a successful attempt to separate variables was undertaken in 1989 by J.S. Monzon [371]
who not only got the expansions of the fields, but also obtained numerical results. The approximate
methods of treatment of anisotrophic particles are considered below.
V.I. Rosenberg [76] generalized the Mie solution for the case of an arbitrary number of spherical
particles by using the theorems of summation of the spherical harmonics [196]. Last years the problem
of light scattering by clusters of spherical particles is very extensively studied (see the papers [21, 99,
107, 110, 111, 114, 195, 219, 220, 221, 222, 280, 286, 317, 354, 355, 362] and references therein), but
the work [76] remains apparently unknown to western scientists.
1.2 Cylinders
For an infinitely long circular cylinder, the solution of the light scattering problem for the perpendicular
incidence was first derived by Rayleigh [418] who returned to this problem once again in his last work
[417] written a short time before his death. The problem for oblique incidence was solved in 1955
by J.R. Wait [491] (Kerker [274] at p.256 mentions also the paper of Blank, 1955); a bit later the
same result was obtained in [172]). A detailed investigation of light scattering by a cylinder was
made in works [41, 42, 131, 153, 154, 189, 190, 233, 234, 326, 327, 430, 438] (homogeneous particles),
[134, 215, 270, 274, 278] (multi­layered cylinders). Though in principle the separation of variables is
possible also for non­circular cylinders (see, e.g., the paper of Rayleigh [419]), numerical methods in
this case are more efficient (see below and the papers [380, 424]). For an anisotropic cylinder with the
coaxial symmetry of the refractive index tensor, no principal differences appear [7, 31, 34, 72]. The
exact solution for a thin bianisotropic cylinder was given in [311], for an optically active cylinder in
[160], and for magnetized cylindrical plasma in [400].
1.3 Spheroids
In the scalar Helmholtz equation the variables are separated in 11 physically interesting coordinate
systems [55], however for the vector field containing 3 scalar functions, the complete separation of
variables is possible only in 6 systems: Cartesian, 3 cylindrical, conical and spherical ones ([55],
Ch.13). Therefore, the formal solution suggested in 1927 by F. Moeglich [368] for the spheroidal
coordinates was practically usefulness until the work of Sh. Asano and G. Yamamoto in 1975 [141]
where the method of separation of variables was adapted for numerical calculations by cutting the
infinite linked (only partly separated) systems of equations (in earlier papers [432, 433] one considers
either scalar scattering, or the axial incidence of the electromagnetic wave). In 1980--1983 V.G.
Farafonov [87, 88, 93] improved the method of Asano--Yamamoto by use of an original scheme of
separation of the fields in two types with the invariant angular parts [27]. At the moment the use of
both approaches gave large volume of data on the light scattering by spheroidal particles [13, 14, 15,
16, 17, 89, 90, 91, 92, 138, 139, 140, 141, 297, 323, 413, 430].
2 Point matching method
In this method the fields outside and inside a particle are represented by expansions in term of proper
basis functions (practically always the vector spherical harmonics). Since the particle surface does not
coincide with the coordinate surface, it is impossible to satisfy the boundary conditions exactly, and
2

the expansions are limited by a finite number of terms, and the boundary conditions are considered
in a finite set of points (matching points). From the derived system of equations one finds the
expansion coefficients. The first results for scalar scattering by a spheroid were obtained in [234],
but a generalization for the electromagnetic case appeared only in 1973--1974 [373, 374, 382, 385]. A
review of evolution and applications of this method can be found in [100, 148]. A similar version of
the projected(?) Galerkin method was developed in [8] (the corresponding papers are also mentioned
in [100]). Now the point matching method is nearly totally displaced by more efficient algorithms.
3 Integral equation method
This method occupies a particular place among the methods of solution of diffraction problems. It is a
general approach where using the Green functions the boundary­value problem is transformed into the
integral form including the boundary conditions and the radiation conditions at infinity [83, 401, 506].
Therefore, this method is not only a numerical approach, but in fact a basis for formulation of other
methods (e.g., the method of moments [243, 495], the T­matrix method [147, 463], Rayleigh­Debye­
Gans theory [25, 441]).
Pioneer rigid calculations using this method were performed in 1964--1965 [136, 137, 424]. A
detailed discussion of the questions connected with the application of the integral equations to light
scattering problems can be found in [24, 401, 464]. For solution of the integral equations one uses
different methods: method of moments, Galerkin method, etc. [228, 243, 265, 380, 434, 467, 495, 500].
From the integral equations one can calculate scattering and absorption by particles of different shape
and structure: cubes [272], disks [301, 500], cylinders of any shape (cross­section) [380, 424, 434], finite
thin rods [476], any structure aggregates of spherical particles [175, 265] and so on.
In 1972--1973 in a series of two papers A.R. Holt and V. Santoso [258] formulated the quantum
mechanics problem of scattering by a central potential (Schroedinger equation) in the form of the
Fredholm equation of the 2nd kind. It was shown that the algorithm is stable, since it automatically
satisfies the variational principle. In the subsequent papers applications to the problems of light
scattering by cylinders [257, 476, 478] and spheroids [260, 477, 478] were developed. In 1990 this
method was formulated in general case for anisotropic spheroids [387], but numerical results were
presented only for anisotropic single axis sphere.
4 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)
3

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 self­interaction 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 T­matrix 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.
4

5 T­matrix method (EBCM)
The T­matrix or transition matrix is well known in the quantum theory of scattering [56] where one
also uses the Heisenberg S­matrix connected with the T­matrix by the relation “
S = 1+2 “
T . In contrast
to the coordinate representation in the integral equation method, where the scattered and incident
fields are connected by the Green function, the T­matrix connects the coefficients of the expansions
of the scattered and incident fields in terms of a complete system of the vector basis functions. In the
theory of light scattering the T­matrix first appeared in the pioneer works of P.C. Waterman in 1965--
1979 [496, 497, 498, 499]. The initial formulation used the Huygens principle for perfectly conducting
[496] and dielectric [498] particles was generalized in [497] where also different ways of formulation of
the method equations without the Huygens principle and fictive sources were considered. Although the
papers of Waterman [496, 497, 498, 499] contained exhaustive information on principal and technical
features of the method, it attracted the attention of scientists in the optics only after the publication
of the paper of P.W. Barber and C. Yeh in the Applied Optics in 1975 [147] where the method was
formulated using the Shelkunov equivalence theorems and called the Extended Boundary Condition
Method -- EBCM. Now both names are nearly equally used. The further works of Barber's group
[142, 143, 144, 145, 146, 227, 324, 328, 493, 494, 508] also had a noticeable influence on the growth of
the method popularity. The generalization of the T­matrix method for multi­particle configurations
were developed in [300, 395, 464], for multi­layered scatterers in [300, 396, 493], and chiral particles of
non­spherical shape in [314, 319]. For strongly aspherical weakly absorbing particles, the convergence
of the T­matrix method is rather weak, therefore an iterative modification of the algorithm was
suggested in [267] with further development and applications in [266, 268, 269, 316, 318]. From the
computational point of view, the method convergence problems for strongly aspherical particles of
large size are connected with the errors of calculations of the integrals defining the T­matrix [49, 60],
therefore a simple and effective way of solution of this problem is the transition to representation of
numbers in the computer with extended accuracy [359, 361].
The analysis of the scattering characteristics of polydisperse ensembles of particles with random
orientation needs two additional integrations -- over size and orientation of scatterers and hence larger
computational resources. Therefore, very important is the derivation of equations for the analytic
orientational averaging of all physically interesting optical characteristics: the integral cross­sections
and Mueller matrix [54, 98, 281, 345, 350, 348], the extinction matrix of the transfer equation [344, 348]
and small­angle scattering flux [57, 388]. Using the apparatus of the quantum theory of angular
momentum we demonstrated that the equations for the orientational averaging of the T­matrix and
its bilinear forms (and hence the physically observer quantities) do not depend on the particle nature
and scattering waves [98, 281]. In the works of M.I. Mishchenko [344] and L.E. Paramonov [59, 60] (see
also the references in [60]) this approach was generalized for disperse systems with ordered orientation
of particles. The numerical algorithms built on the equations of the analytical orientational averaging
were found to be the most efficient ones for the analysis of light scattering characteristics of ensembles
of non­spherical particles with random orientation (see the works of Mishchenko's group at the NASA
Goddard Institute for Space Studies [345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357,
358, 359, 360, 361, 362] and also the papers [47, 48, 49, 60, 63, 64, 78, 102, 103] in Russian).
The important advantage of the T­matrix method is its natural adaptation to the problems of the
multi­scattering wave theory [331, 468, 469, 511, 471, 480, 481, 482, 485, 486, 487], radiative transfer
equations [60, 472, 473], scattering by cluster of particles [195, 353, 354, 355, 362], particles of irregular
shape [193] and Chebyshev T n ­particles [503], Raman scattering [188], extinction by polydisperse and
polymorphic metal zoles [102, 103, 282, 283] and so on. Thus, the T­matrix approach is practically
universal for all application fields of the light scattering theory. Concluding the discussion of this
method, we insistently refer a reader to the reviews [362, 483] where the exhausted literature on the
5

theory and applications of the T­matrix method is collected.
Part II. Approximate methods
6 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 Ü 1 and kajmj Ü 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 expan­
sions of the fields in terms of ka. For non­absorbing 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 ef­
fects can be found in [86, 94, 105, 235, 250, 251, 252, 376, 377, 409, 410, 459, 461] (see also the review
in [100]).
7 Rayleigh--Debye--Gans approximation (RDG)
In a large number of important cases the relative refractive index of particles is close to 1. Such
particles are called ``optically soft'', and the corresponding approximation could be named that of
optically soft particles. This approximation is in particular well applicable to scattering of X­rays and
neutrons [56, 211, 238]. The condition jm \Gamma 1j Ü 1 is not sufficient for development of the theory
since an important role is also played by the ratio of the size to the wavelength and the phase shift
ae = 2ka(m \Gamma 1). The RDG approximation is applicable when two conditions are satisfied
jm \Gamma 1j Ü 1; jaej Ü 1: (1)
6

For the theory based on the assumption (1), there are a set of the names: Rayleigh--Gans [11],
Rayleigh--Debye [274, 397], Rayleigh--Gans--Born [474], Rayleigh--Gans--Debye [78, 144, 409, 410],
Rayleigh--Gans--Rokard [132], Rayleigh--Debye--Gans (RDG) [100, 462], Born (or the first Born) [56,
135] approximation. We believe that the term RDG is the most suitable for the following reasons.
The basic ideas of the method including the derivation of successive approximations from the integral
equation for the scattered field were formulated by Rayleigh in 1881 in the paper [418], and the ex­
pression for the scattering cross­section for a sphere was obtained by him in 1914 [416]. Independently
of Rayleigh, in 1915 Debye [201] derived the general expression for the intensity of X­rays scattered by
an ensemble of randomly oriented particles. Later the approach was applied by Debye to scattering
by polymer solutions [200, 202] which was very important for development of this field [26, 130, 274].
The contribution of Gans is of course essentially smaller than those of Rayleigh and Debye as in his
paper [223] he actually repeated the Rayleigh result for a sphere [416] (in the third time the same
results was obtained by Rokard, see [124], p.249). An analog of the Rayleigh iterations in the quantum
mechanics was developed in 1926 by M. Born [163]. This approach became very popular one in this
field, and the name of Born is rightfully connected with this approximation. The name of Gans is
reasonable to leave in the term the RDG theory just for the reason that due to van de Hulst [11] the
term Rayleigh--Gans approximation became widespread, although he did not consider this term to be
a good one.
The RDG approximation can be obtained in several ways [25, 124, 274, 441]. For instance, one can
use the general integral relation expressing the field ~
E s scattered in the direction ~s = ~ k s =k through the
integral containing the tensor Green function for the vector Helmholtz equation \Gamma(~r; ~r 0 ) [56] and the
field inside the particle ~
E i , and make the latter equal to the field of the incident wave ~
E 0 exp(i ~ k 0 ~r).
Another way is based on the physical interpretation of scattering as a result of interference of the
fields of independent dipoles excited by the incident wave in the particle volume V . In both cases the
scattered field can be represented as
~
E s = ~
A exp(ikr)=r (2)
with the vector amplitude ~
A = k 2 (1 \Gamma ~s \Delta ~s) ~
E 0 U(~q), where ~q = ~ k s \Gamma ~ k 0 in the scattering vector with the
module q = 2k sin(`=2), U(~q) is the Fourier transform of the dielectric permittivity tensor (normalized
by the volume of polarizability of an optically soft particle, j(~r) = (''(~r) \Gamma 1)=4ú). For homogeneous
particles U(~q) = jV G(~q), where the interference function is
G(~q) = 1
V
Z
exp[i( ~ k 0 \Gamma ~ k s )~r]d 3 ~r: (3)
Thus, the polarization characteristics of this approximation do not differ from those of the Rayleigh
approximation for optically soft particles, and the angular ones differ only by the interference function.
The interference functions for particles of several shapes can be obtained in the form of simple
analytical expressions. Rayleigh got the formula for a sphere in the form of a series in [418] and using
the cylindrical functions of the order 3/2 in [422]. He also derived the formula for an infinitely long
cylinder in the case of normal incidence [418]. Finite cylinders were considered in the paper [217]
and in the book of van de Hulst [7], prisms in [82, 364], ellipsoids in the papers of A.L. Patterson in
1939 [389] (Kerker [274] at p.484 wrongly cites the paper [366] as the first one). For spheroids, the
formula for the interference function was obtained in [237, 238] and later in [426, 428, 447]. The RDG
approximation can be easily generalized for inhomogeneous particles, in particular with the spherical
symmetry of inhomogeneity [11, 135, 216], and for particles of arbitrary shape with the rotational
symmetry [369]. The tables of the interference functions see in [226, 274]. The papers [274, 444]
suggest a small modification of the RDG where the wavenumber inside the particle takes into account
the refractive index.
7

As one usually measures the intensity of scattered light from an ensemble of particles with a
random orientation, the averaged square of the interference function hG 2 (q)i = P (q) called the form­
factor or Debye scattering form­factor (sometimes it is written simply as G(q)) has the practical
interest. A detailed literature analysis of calculations of the form­factors for different models including
polydisperse ensembles and polymer solutions is given in [100], and here we just list some investigations
in this direction [82, 149, 150, 151, 152, 156, 197, 206, 225, 238, 245, 271, 274, 294, 296, 297, 298, 299,
335, 363, 364, 365, 366, 367, 378, 379, 381, 397, 398, 402, 426, 428, 445, 446, 465, 479]. The accuracy
of the RDG approximation was estimated for spheres [197, 236, 274, 275, 475], cylinders [171, 438]
and spheroids [144, 145, 297, 412, 475].
For anisotropic particles, the RDG theory was built in the assumption that the induced elementary
dipoles are the product of the optical polarizability tensor and the incident field. In Eq. (2) the
vector amplitude is not parallel to the incident field, and hence to describe the angular dependence
of scattering one needs four Debye form­factors corresponding to four combinations of the polarizator
and analyzator states relative to the planes of incidence and scattering (P vv ; P vh ; P hv ; P hh , where
the subscripts v and h have the common sense [384] and for a system with the random orientation
P vh = P hv ). Scattering by anisotropic spheres, cylinders, discs and spheroids was considered in
[94, 96, 100, 130, 246, 255, 256, 261, 262, 375, 399, 406, 407, 408, 409, 410, 411, 415, 423, 479]. After
the pioneer work [457] a large attention in the literature on polymers is paid to the scattering by
specific anisotropic structures -- so called spherulites [4, 5, 403, 429, 456].
The extinction cross­section in the RDG approximation is proportional to the volume and in
general case does not differ in form from the extinction cross­section in the Rayleigh approximation.
The scattering cross­section should be found from integration of the intensity over all solid angles,
since the optical theorem in the RDG approximation gives either the extinction cross­section or 0 (for
non­absorbing particles). The first calculations of the RDG scattering cross­section for the spheroidal
model of bacteria cells were performed by in 1961 by A.L. Koch [295]. A systematic study of the integral
functions in the method of the turbidity spectrum of spheroidal particles (the scattering cross­section
and wave exponent [334] was made in [100, 104, 116, 118, 119, 120]. These works contain the general
integral representations, results of numerical computations and asymptotic analytic expressions found
for the region of small sizes and the intermediate region between the RDG and anomalous diffraction
theories as well as in the cases of weak and strong eccentricity. Later the analysis of the integral
characteristics of RDG--spheroids was also made in [44, 49, 53, 75].
8 Generalizations of the RDG approximation
For non­spherical 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 approxi­
mation for isotropic non­spherical particles if to use a plane wave with a quasi­static (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, respec­
tively. 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].
8

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 spheroids 1 .
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 non­spherical 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 non­spherical 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]).
9 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 a AE – and the refractive index m ¸ 1. 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 cross­section from the optical
theorem [11, 56]
C ext = 2Re
Z
[1 \Gamma exp(\Gammaiae(z))]dS; (4)
where ae(z) is the phase shift of a ray propagating along the z­axis, and integration is over all the rays
intersecting the particle. The absorption cross­section is
C abs =
Z
[1 \Gamma exp(\Gamma2Imae(z))]dS: (5)
The range of applicability of the anomalous diffraction theory for the scattering amplitude is limited by
small angles, however the formules for the cross­sections became good approximations for a wide class
of particles. The AD approximation can be easily applied to non­spherical particles, which to some
extent explains its popularity. The formules for scattering and extinction cross­sections 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 cross­section in the general case of ellipsoids
1 In this paper at p.867 the discussion of our paper [96] and that of Seker (see reference [10] in [89]) makes the
impression that we and Seker utilized the erroneous decomposition of the quasistatic field using the unit vectors of the
spherical system instead of the spheroid axes, which is not correct.
9

is given in [84] (the same result was obtained later again in [49, 51]). The integral cross­section 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: core­mantle [22, 30, 100] and multi­layered [178] spheres (including non­concentric particles
[49]), multi­layered spheroids [29] as well as the particles with ledges [320] and cavities [321]. In
[186, 187] the theorem of summation of the cross­sections within the AD was derived and the formula
for prisma­like particles was obtained.
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 coauthors 2 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
cross­section 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 (small­parameter
estimates of the integral cross­sections) [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].
10 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 ap­
proximation 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 2
m+1 A(0) WKB = A(0) AD [56, 292].
2 See the references in [123] and the list of the further works of this group in the thesis of S.Yu. Shcheglov [127].
10

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 \Gamma 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 large­scale 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 large­scale anisotropic inhomogeneities.
11 Geometrical optics approximation (GO)
For particles of very large sizes, scattering can be considered as a result of interference of the rays
having multiple reflections and refractions according to the laws of geometrical optics. Diffraction
at the particle edges which gives a sharp maximum in the region of small angles can be taken into
account separately. The formulation of general principles, computational schemes and discussion of
obtained results within the GO can be found in books [7, 11, 124]. The papers [413, 414] presented
a method where the scattering field was expressed via the field at the particle surface using the
Huygens--Kirchhoff principle, and reflection and refraction laws were employed to find the unknown
field (the approach was called physical optics approximation). Utilizing the GO method the author
of [184] solved the general question why the mean extinction cross­section for an ensemble of large
non­spherical particles is always larger than this cross­section for equivolume spheres. Illustrations of
various applications of the method are given in [28, 30, 155, 192, 263, 412, 425, 427]. Note also a recent
paper [242] where the GO method is applied to analysis of light scattering by spheres with the size
parameter ka = 500 and the discovered theoretical and experimental relationships in the oscillation
of s and p components of intensity were used for successful solution of the inverse problem of sizing.
The accuracy of the GO method for non­spherical particles was studied in a recent paper [332] by
comparison with the exact T­matrix method results.
12 Perturbation method
The perturbation method is based on the small­parameter expansion of an unknown solution of the
light scattering problem in a vicinity of the exact solution. In application to non­spherical particles it
means that one searches for a solution in the form of small deviations from the Mie solution caused by
small deviations of the particles shape from a sphere. The first solution of this kind was obtained in
1960 by T. Oguchi [383] (the often cited paper [507] appeared 4 years later). The general approach to
solution of the problem by the perturbation method was developed in 1969 by V.A. Erma [213] (see
also the papers [212, 289, 372, 373] on concrete applications of the method and recent paper [12]).
11

With the development of more efficient algorithms for non­spherical particles the perturbation
methods lost its meaning. The only probably exclusion is the case of anisotropic particles for which
the rigid solution is either impossible or too complicated. For such problems the perturbation method
can give information of the first approximation, which is important for understanding of the basic
principles of influences of the material anisotropy. A good illustration here is a large number of
investigations made by the Minsk group of A.P. Prishivalko for weakly anisotropic spheres [32, 33, 35,
36, 37, 38, 39, 40, 72]. As the basic solution of this problem, one takes the Mie solution for an average
refractive index, and the anisotropy is represented by small deviations from it. More details can be
found in the book [72].
13 Hybrid and simple analytical approximations
Latimer [322, 323] suggested the approximations based on a combination of the RDG and AD theo­
ries for spheroids and the Mie theory to describe scattering and extinction of light by non­spherical
particles. The idea of these approximations is as follows: on the base of the RDG and AD theo­
ries to find an analogy between the formulas for the scattering characteristics for a sphere and an
oriented spheroid. From this analogy one postulates a certain relationship between the radius and
refractive index of an ``equivalent'' sphere from one side, and the spheroid parameters (including the
orientation) from another side. Then the scattering by a spheroid or an ensemble of spheroids is
replaced by scattering by the ``equivalent'' sphere calculated from the Mie theory. Such approxima­
tions are useful as simplified computational algorithms, though do not have rigorous enough physical
justifications. In certain sense such a method is analogous to a more pragmatic approach when the
exact formulas or results of calculations are approximated by simple approximate analytic expres­
sions. For example, such approach was used both for approximation of formulas of the Mie theory
[20, 73, 128, 214, 229, 247, 248, 249, 492, 512] and for description of scattering by spheroids [218] and
cylinders [437, 438, 439].
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