321Optical Review (2019) 26:319–331
1 3
ADA with the original ADA for randomly oriented particles
with different shapes and found some significant differences
in extinction and absorption efficiencies [18]. Xu and Lax
(2003) showed that ADA of extinction of light by soft par-
ticles was determined by a statistical distribution of the geo-
metrical paths of individual rays inside the particles; they
derived analytical formulas for this extinction using a Gauss-
ian distribution of the geometrical paths of the rays [19]. Tang
and coworkers investigated the feasibility and limitations of
using the ADA to calculate the extinction efficiency of non-
spherical particles; they used ADA instead of Mie extinction
efficiency to retrieve the particle size distributions based on a
genetic algorithm [20, 21]. Paramonov (2012) proved there is
an optical equivalence of randomly oriented ellipsoidal parti-
cles and polydisperse spheroidal and spherical particles using
the analytical potential of Rayleigh–Gans–Debye theory and
ADA; these results were used to develop an optical classifica-
tion of isotropic ensembles of ellipsoidal particles [22].
In the context of scattering theory, the extinction ker-
nel function of the spherical particles from Mie scattering
theory can be substituted by the new kernel function for
non-spherical particles derived from the ADA method.
Moreover, the functional relationship between the extinc-
tion efficiency factor and particle size distribution remains
the Fredholm integral equation of the first kind, for which
the Phillips–Twomey (P–T) method is the most appropriate
solution method [22, 23]. Undoubtedly, using both ADA
and P–T methods, it is feasible to retrieve non-spherical dust
particle size distributions using AOT data from a sun pho-
tometer or multi-wavelength lidar.
2 2 Light scattering theory
2.1 Mie scattering theory
At the beginning of the twentieth century, to solve the scat-
tering problem of uniform spherical particles, the German
scientist Mie established a scattering theory based on elec-
tromagnetic theory, which is also known as coarse-particle
scattering theory. Mie scattering theory provides an exact
solution to the scattering of planar waves by uniform spheri-
cal particles in an electromagnetic field, derived from Max-
well’s equations. In addition, Mie scattering theory defines
scattering laws for uniform spherical particles of arbitrary
diameter and arbitrary composition.
According to classical Mie scattering theory [16], we
can determine the scattering particle’s scale parameter, the
radius of particle; the complex refractive index of the scat-
tering particle, and both real and the imaginary parts of com-
plex refractive index.
In this paper, the complex refractive index m is assumed
to be 1.55–0.01i [24, 25], and the scale parameter range
is assumed to be 0–25. Figure1 shows the relationship
between the extinction and scattering efficiency factor and
the scale parameter x = 2πr/λ, where r is the particle radius
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5
0
0.5
1
1.5
2
Radius
r
/μm
Extinction efficiency factor
ADA/Mie extinction efficiency factor,
m
=1.01-0.1
i
(a)870nm
Mie
ADA
Error
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-1
0
1
2
3
4
5
Radius
r
/μm
Extinction efficiency factor
ADA/Mie extinction efficiency factor, m=1.55-0.01i
(b)340nm
ADA
Mie
Error
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
r
/μm
Extinction efficiency factor
ADA/Mie extinction efficiency factor, m=1.55-0.01i
(c)870nm
ADA
Mie
Error
Fig. 3 Comparison of the extinction efficiency factors and relative
errors for spherical particles using the Mie scattering theory and
the ADA method for (a) an incident light wavelength of 870nm and
complex refractive index of 1.01–0.1i, (b) 340 nm and 1.55–0.01i,
and (c) 870nm and 1.55–0.01i