Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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9.5 Mie’s scattering problem 355
The scattered electric vector can be written as
E
s
= E
s
ϑ
e
ϑ
+ E
s
ϕ
e
ϕ
= E
s
l
e
ϑ
+ E
s
r
e
ϕ
since E
s
ϑ
= E
s
l
, E
s
ϕ
= E
s
r
(9.100)
In order to have a shorthand notation we introduce the amplitude functions S
1
and
S
2
by
S
1
(cos ϑ) =
∞
n=1
2n + 1
n(n + 1)
a
s
n
π
n
(cos ϑ) + b
s
n
τ
n
(cos ϑ)
S
2
(cos ϑ) =
∞
n=1
2n + 1
n(n + 1)
a
s
n
τ
n
(cos ϑ) + b
s
n
π
n
(cos ϑ)
(9.101)
By using (9.19), and (9.100) and (9.101) the first two equations of (9.96) can be
written in matrix notation as
E
s
l
E
s
r
=
exp
[
ik
0
(r − z)
]
ik
0
r
S
2
(cos ϑ)0
0 S
1
(cos ϑ)
E
i
l
E
i
r
(9.102)
In Chapter 1 the symbol was introduced for the scattering angle, see
Figure 1.17. In the spherical coordinate system used to derive (9.102) the scattering
angle corresponds to the polar angle ϑ. In order to obtain consistency with the earlier
notation, from now on we will again use the symbol for the scattering angle.
9.5.3 Rayleigh scattering
Rayleigh (1871) has developed a scattering theory for particles which are small in
comparison with the wavelength. With the help of the formulas which he derived
he was able to explain the blue color of the sky. Since the Mie theory is valid
for spherical particles of any size, it is possible to derive the Rayleigh formulas
as special cases of the Mie equations rather than following Rayleigh’s original
work. Thus, in the following we will assume that the radius a of the scattering
particle (e.g. an air molecule) is small compared to the wavelength λ of the incident
electromagnetic wave.
Before we begin with the simplification of the Mie equations we need to state a
number of formulas. Employing the definition of the spherical Bessel functions as
stated in (9.36), with the help of the sum formula for the ordinary Bessel function
of the first kind, (see e.g. Abramowitz and Stegun, 1972) we have
j
n
(x) =
√
π
2
x
2
n
∞
k=0
−
1
4
x
2
k
k!(n + 3/2 + k)
(9.103)
356 Light scattering theory for spheres
From (9.56) we first find
[
xj
n
(x)
]
= xj
n−1
(x) −nj
n
(x) (9.104)
and then the sum representation
[
xj
n
(x)
]
=
√
π
x
2
n
∞
k=0
−
1
4
x
2
k
k!
1
(n + 1/2 + k)
−
n
2(n + 3/2 + k)
(9.105)
which will be used to treat some special cases as required for the Rayleigh theory.
We also need the definition of the spherical Hankel function of the first kind which
is given by
h
1
n
(x) = j
n
(x) −i (−1)
n
j
−(n+1)
(x) (9.106)
The sum representation follows directly
h
1
n
(x) =
√
π
2
x
2
n
∞
k=0
−
1
4
x
2
k
k!(n + 3/2 + k)
−
i
√
π
2
(−1)
n
x
2
−(n+1)
∞
k=0
−
1
4
x
2
k
k!(−n + 1/2 + k)
(9.107)
from which we obtain the derivative formula
xh
1
n
(x)
=
√
π
x
2
n
∞
k=0
−
1
4
x
2
k
k!
1
(n + 1/2 + k)
−
n
2(n + 3/2 + k)
− i(−1)
n
√
π
x
2
−(n+1)
∞
k=0
−
1
4
x
2
k
k!
×
1
(−n − 1/2 + k)
−
n + 1
2(−n + 1/2 + k)
(9.108)
Now we are ready to obtain the simplified formulas which will be needed soon.
The arguments of the Bessel functions are of the form x = ka where, as usual,
k is the wave number and a is the radius of the scattering particle. If a λ then
the argument approaches zero. Thus it is sufficient to discontinue the various sums
after the first term. Evaluation of (9.103) for small arguments yields
j
n
(x) ≈
√
π
2
x
2
n
1
(n + 3/2)
(9.109)
9.5 Mie’s scattering problem 357
Similarly, we obtain from the derivative formula (9.105)
[
xj
n
(x)
]
≈
√
π
x
2
n
1
(n + 1/2)
−
n
2(n + 3/2)
(9.110)
while (9.107) and (9.108) result in
h
1
n
(x) ≈
√
π
2
x
2
n
1
(n + 3/2)
−
i
√
π
2
(−1)
n
x
2
−(n+1)
1
(−n + 1/2)
(9.111)
and
xh
1
n
(x)
=
√
π
x
2
n
1
(n + 1/2)
−
n
2(n + 3/2)
− i(−1)
n
√
π
x
2
−(n+1)
1
(−n − 1/2)
−
n + 1
2(−n + 1/2)
(9.112)
We will now investigate the coefficients a
s
n
and b
s
n
as given by (9.92) and find
out in which way these simplify if we apply the above approximation formulas.
The numerators of both coefficients contain powers of (x/2)
n
. Moreover, the real
part of the denominator also contains powers of (x/2)
n
while the imaginary part is
a function of x
−1
. Thus for very small arguments of the functions and for n > 1we
may set approximately
a
s
n
≈ 0, b
s
n
≈ 0 for n > 1 (9.113)
implying that in (9.101) the S
1
and S
2
series may be discontinued after the first term.
Now let us find explicit expressions for a
s
1
and b
s
1
. Inspection of the definitions of
the spherical Bessel functions shows that these contain the gamma functions. By
using the formulas
(1/2) =
√
π, (x + 1) = x(x) (9.114)
we find from (9.109) and (9.110) for small values of the argument
j
1
(x) ≈
x
3
,
[
xj
1
(x)
]
≈
2x
3
(9.115)
Likewise, from (9.111) and (9.112) we obtain
h
1
(x) ≈
x
3
−
i
x
2
,
[
xh
1
(x)
]
≈
2x
3
+
i
x
2
(9.116)
358 Light scattering theory for spheres
Substitution of these expressions into (9.92) yields the approximate expressions
for a
s
1
a
s
1
≈
2ρ
0
3
N
3
− N
2
ρ
0
3
−
i
ρ
2
0
N −
2ρ
0
3
+
i
ρ
2
0
N
3
=−
ρ
3
0
N
2
− 1
ρ
3
0
N
2
− 1
−
3
2
i
N
2
+ 2
ρ
6
0
(N
2
− 1)
2
+
9
4
(N
2
+ 2)
2
(9.117)
Finally, by ignoring in (9.117) the sixth power of ρ
0
gives
a
s
1
≈
2
3
iρ
3
0
N
2
− 1
N
2
+ 2
(9.118)
By the same procedure we find that the coefficient b
s
1
vanishes, that is
b
s
1
≈ 0
(9.119)
Thus for Rayleigh scattering we obtain from (9.101) the amplitude functions
S
1
(cos ) ≈ iρ
3
0
N
2
− 1
N
2
+ 2
π
1
(cos ), S
2
(cos )≈ iρ
3
0
N
2
− 1
N
2
+ 2
τ
1
(cos )
(9.120)
From (9.94) we get for n = 1
π
1
(cos ) = 1, τ
1
(cos ) = cos (9.121)
so that the amplitude functions assume the simple forms
S
1
(cos ) ≈ iρ
3
0
N
2
− 1
N
2
+ 2
, S
2
(cos ) ≈ iρ
3
0
N
2
− 1
N
2
+ 2
cos (9.122)
For the = 0 we observe that S
1
= S
2
.
Utilizing the above approximations, for Rayleigh scattering the relationship
(9.102) between the components of the incoming and the scattered electric vec-
tor with reference to the scattering plane are given by
E
s
l
E
s
r
=
ρ
3
0
k
0
r
N
2
− 1
N
2
+ 2
exp
[
ik
0
(r − z)
]
cos 0
01
E
i
l
E
i
r
(9.123)
Since ρ
0
= k
0
a = 2πa/λ it is seen that for Rayleigh scattering the electric field
vector of the scattered wave is proportional to λ
−2
so that the radiance of the
scattered light is proportional to λ
−4
. Thus in the atmosphere scattering of the
visible sunlight on air molecules is most efficient for the shorter wavelengths, i.e.
9.6 Material characteristics and derived directional quantities 359
x
y
z
P
P
E
i
r
A
˜
Θ
Fig. 9.5 Hypothetical experiment to define the extinction coefficient.
for blue light. This explains why the sky is blue, or in other words, why the sun
appears red during dawn and dusk. In the next section we are going to discuss the
scattering properties in a little more detail.
9.6 Material characteristics and derived directional quantities
9.6.1 Extinction, scattering and absorption coefficients
Let us consider a single fixed particle of arbitrary shape and composition being
illuminated by an electromagnetic wave as shown in Figure 9.5. The origin of the
(x, y, z)-coordinate system is somewhere within the particle. One part of the energy
which is incident on the particle is scattered and, in general, another part is being
absorbed. At a point
˜
P which is located at a distance r =z from the origin, the
radiative flux φ incident on the surface A is given by
φ =
A
EdA (9.124)
Now we remove the particle so that none of the incident light is lost. Then at point
˜
P the observed flux is given by
φ
i
=
A
E
i
dA (9.125)
360 Light scattering theory for spheres
The extinction due to the presence of the particle causes a change φ
e
in the radiant
flux as given by
φ
e
= φ
i
− φ =
A
(E
i
− E)dA (9.126)
The change of the radiant flux can be described with the help of the extinction
cross-section C
ext
which is defined by means of
C
ext
=
1
E
i
A
(E
i
− E)dA
(9.127)
Thus C
ext
has the dimension (m
2
).
Similarly, the scattered flux φ
s
can be expressed by means of
φ
s
=
0
4πr
2
E
s
dA (9.128)
where the integration extends over the spherical surface defined by the radius r .
Analogously to (9.127) we define the scattering cross-section by means of
C
sca
=
1
E
i
0
4πr
2
E
s
dA
(9.129)
Finally, we need to define the absorption cross-section. Since the extinction may
be treated as a combination of scattering and absorption, we may introduce
C
abs
=
1
E
i
0
4πr
2
(E
i
− E
s
− E)dA
(9.130)
as the defining equation for C
abs
so that
C
abs
+ C
sca
= C
ext
(9.131)
Hence in the absence of absorption C
ext
= C
sca
.
If we divide the quantities C
ext
, C
sca
and C
abs
by the geometric cross-section
G of the particle, we obtain the dimensionless efficiency factors for absorption,
scattering and extinction
Q
abs
=
C
abs
G
, Q
sca
=
C
sca
G
, Q
ext
=
C
ext
G
(9.132)
In case of the Mie theory which was developed for spherical particles, we use
G =π a
2
since the cross-section of a sphere with radius a is a circle having the
same radius.
In the following analysis we will assume that the particles have spherical shape.
Reference to equation (9.102) shows that each component of the electric vector can
9.6 Material characteristics and derived directional quantities 361
1
)=σ(Θ
2
) σ(Θ
1
)=σ(Θ
2
)
Θ
2
Θ
1
P
P
E
i
E
i
σ(Θ
/
˜
Fig. 9.6 Influence of the origin on the phase.
be expressed in the form
u
s
= S()
exp
[
ik
0
(r − z)
]
ik
0
r
u
i
(9.133)
Here, u
s
and u
i
are the amplitudes of the scattered and the incident wave and S()
is the amplitude function. The amplitude function is in general complex and can
also be written in the form
S() = s()exp
[
iσ ()
]
(9.134)
where σ is the phase of the scattered wave. The quantity s is positive and σ is real.
In general s() is independent of the choice of the origin while σ () depends
on this choice. The only exception is the case with =0 since a displacement of
the origin does not change the scattering angle. Figure 9.6 illustrates the situation
with =0 (left panel) and =0 (right panel).
At every point outside the particle the electromagnetic wave can be separated into
two parts representing the scattered wave in direction P and the incoming wave.
In order to determine the extinction caused by the presence of the particle we need
to know the amplitude of the wave at point
˜
P which is located at a distance r =z
from the scattering center where the scattering angle is zero, see Figure 9.5. Since
in this case the directions of the incoming plane wave and the scattered spherical
wave are the same it is impossible to distinguish between the two waves. In order
to make a distinction possible one chooses a point which is very close to
˜
P so that
the scattering angle for all practical purposes is still zero. For the coordinates of
362 Light scattering theory for spheres
this point we have z x and z y so that we may use the approximation
r =
-
x
2
+ y
2
+ z
2
≈ z +
x
2
+ y
2
2z
≈ z +
x
2
+ y
2
2r
(9.135)
Adding the amplitude of the incoming wave and the scattered wave we may write
u = u
i
+ u
s
= u
i
1 + S(0)
exp
[
ik
0
(r − z)
]
ik
0
r
≈ u
i
1 + S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
(9.136)
The radiance of the radiation is equal to the square of the amplitude. Since
no integration over the scattering angle needs to be carried out to obtain the flux
density E we have
E = E
i
1 + S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
1 + S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
∗
= E
i
(1 + a + ib)(1 + a − ib) = E
i
(1 + a
2
+ 2a + b
2
) with
a =
S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
, b =
S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
(9.137)
Since a
2
and b
2
are proportional to r
−2
we ignore these two terms for large r . Hence
we obtain for the total flux density
E = E
i
1 + 2
S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
(9.138)
from which we find the radiative flux with respect to the surface A as
φ =
A
Ed A =
A
E
i
dA+
A
E
i
2
S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
dA (9.139)
By comparing (9.127) with (9.139) we obtain for the extinction coefficient
C
ext
=−
A
2
S(0)
exp
ik
0
(x
2
+ y
2
)/2r
ik
0
r
dA (9.140)
which is an useless expression unless we are able to carry out the integration. To
accomplish this, first of all we use the substitutions
α
2
=
k
0
x
2
2r
, β
2
=
k
0
y
2
2r
=⇒ dA = dx dy =
2r
k
0
dαdβ (9.141)
9.6 Material characteristics and derived directional quantities 363
After a little algebra we obtain
C
ext
=−
α
2
α
1
β
2
β
1
4
k
2
0
[S(0)(cos α
2
sin β
2
+ cos β
2
sin α
2
)]dβdα
−
α
2
α
1
β
2
β
1
4
k
2
0
[S(0)(cos α
2
cos β
2
− sin β
2
sin α
2
)]dβdα
(9.142)
The integrals in (9.142) can be evaluated if the limits of the double integral are
extended to infinity. This procedure is legal since only points x z and y z
effectively contribute to the result. Observing
∞
−∞
sin x
2
dx =
∞
−∞
cos x
2
dx =
%
π
2
(9.143)
we obtain the final result for the extinction cross-section
C
ext
=−
4π
k
2
0
[
S(0)
]
(9.144)
The radiance of the light wave components I
s
l
and I
s
r
are obtained by squaring
the corresponding components of the electric vector. Thus we find from (9.102)
I
s
l
=
1
k
2
0
r
2
S
2
()S
∗
2
()I
i
l
, I
s
r
=
1
k
2
0
r
2
S
1
()S
∗
1
()I
i
r
(9.145)
The total scattered light intensity is given by
I
s
= I
s
l
+ I
s
r
=
1
2k
2
0
r
2
[S
1
()S
∗
1
() + S
2
()S
∗
2
()]I
i
(9.146)
with I
i
l
=I
i
r
=I
i
/2 which is true for natural light. In the general case where the light
is polarized, the zeros in the matrix on the right-hand side of (9.102) are replaced by
other quantities. In which way polarized light must be treated in radiative transfer
will be discussed in the following chapter.
We define the Mie intensity functions by means of
i
1
(cos ) = S
1
(cos )S
∗
1
(cos ), i
2
(cos ) = S
2
(cos )S
∗
2
(cos )
(9.147)
so that I
s
can be written as
I
s
=
i
1
(cos ) +i
2
(cos )
2k
2
0
r
2
I
i
(9.148)
The functions i
1
and i
2
refer to light vibrating perpendicularly and parallel to the
scattering plane. For a given scattering angle , the flux densities can be stated as
E
s
=
i
1
(cos ) +i
2
(cos )
2k
2
0
r
2
E
i
(9.149)
364 Light scattering theory for spheres
Hence, with reference to equation (9.129) the scattering cross-section is given by
C
sca
=
2π
0
1
−1
i
1
(x) +i
2
(x)
2k
2
0
dxdϕ with x = cos (9.150)
In this equation we substitute for i
1
and i
2
the definitions S
1
and S
2
as given by
(9.101) and find
C
sca
=
π
k
2
0
1
−1
∞
n=1
∞
m=1
c
n
c
m
a
s
n
a
s∗
m
+ b
s
n
b
s∗
m
[
π
n
(x)π
m
(x) +τ
n
(x)τ
m
(x)
]
dx
+
π
k
2
0
1
−1
∞
n=1
∞
m=1
c
n
c
m
a
s
n
b
s∗
m
+ b
s
n
a
s∗
m
[
π
n
(x)τ
m
(x) +τ
n
(x)π
m
(x)
]
dx
(9.151)
with c
i
= (2i + 1)/[i(i + 1)]. According to (9.94) the functions π
n
(x) and τ
n
(x)
may be written as
π
n
(x) =
P
1
n
(x)
√
1 − x
2
, τ
n
(x) =−
-
1 − x
2
dP
1
n
dx
(9.152)
Recalling the orthogonality relation (9.50) we find the condition
1
−1
[
π
n
(x)π
m
(x) +τ
n
(x)τ
m
(x)
]
dx
=
1
−1
1
1 − x
2
P
1
n
(x)P
1
m
(x) +(1 − x
2
)
dP
1
n
dx
dP
1
m
dx
dx =
2n
2
(n + 1)
2
2n + 1
δ
n,m
(9.153)
Furthermore, we easily obtain the orthogonality condition
1
−1
[
π
n
(x)τ
m
(x) +τ
n
(x)π
m
(x)
]
dx =−
1
−1
d
dx
P
1
n
(x)P
1
m
(x)
dx = 0 (9.154)
so that the scattering cross-section C
sca
can finally be written as
C
sca
=
2π
k
2
0
∞
n=1
(2n + 1)
a
s
n
a
s∗
n
+ b
s
n
b
s∗
n
(9.155)
Thus C
sca
can be expressed entirely by the coefficients a
s
n
and b
s
n
.
Next we will show that C
ext
can also be formulated in terms of these coefficients.
From textbooks on spherical harmonics and from mathematical tables, such as