Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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10.2 The Stokes parameters 385
of a circularly polarized electromagnetic wave with δ =±π/2 + 2mπ, m = 0,
±1, ±2,... and a
1
= a
2
= a the parameters S
1
and S
2
vanish so that, according
to (10.28), cos(2β) = 0. This in turn means that P is on the north pole or on the
south pole of Poincare’s sphere, denoting right-handed and left-handed circular
polarization, respectively.
Partially polarized light is a mixture of natural and polarized light. If E
p
and E
u
represent the constituent flux densities of the polarized and the unpolarized light,
then the degree of polarization D
p
is defined by
D
p
=
E
p
E
p
+ E
u
(10.29)
As can be shown, for partially polarized light the degree of polarization can also
be expressed in terms of the Stokes parameters by means of
D
p
=
S
2
1
+ S
2
2
+ S
2
3
1/2
S
0
(10.30)
For completely polarized light we will now describe the state of polarization in
terms of the Stokes vector consisting of the four components (S
0
, S
1
, S
2
, S
3
). For a
concise description we introduce the following notation. Linearly also called plane
polarized light is said to be in the P-state while right and left circularly polarized
light are in the R-state and L-state. Analogously, elliptically polarized light is in
the E-state.
Often the Stokes parameters are expressed in normalized form by dividing all
elements by S
0
. For completely polarized light a table of Stokes vectors may be
computed. We will briefly show how this can be done for linearly polarized light. If
a
1
and a
2
label the amplitudes of the horizontal and the vertical component of the
electromagnetic wave, we have a horizontally polarized wave if a
1
> 0 and a
2
= 0.
Utilizing (10.26) we see that the normalized Stokes vector is now given by
1
S
0
S
0
S
1
S
2
S
3
=
1
1
0
0
(10.31)
In this case the light is said to be in the horizontal P-state.
Analogously to (10.31) other polarization states can be identified. Table 10.1
summarizes some polarization states which may be easily obtained by applying
the appropriate values of the variables (a
1
, a
2
,δ)or(a, b,ψ) to (10.26) or (10.28).
More complete tables may be found in Shurcliffe (1966) or Deirmendjian (1969).
386 Effects of polarization in radiative transfer
Table 10.1 Normalized Stokes vectors for various types of polarized light.
Upper panel: linear polarization. Lower panel: circular and general
elliptical polarization
horizontal P-state vertical P-state 45
◦
P-state −45
◦
P-state
1
1
0
0
1
−1
0
0
1
0
1
0
1
0
−1
0
R-state L-state E-state
1
0
0
1
1
0
0
−1
1
cos(2β) cos(2ψ)
cos(2β) sin(2ψ)
sin(2β)
The previous section largely followed the standard textbook by Born and Wolf
(1965) where additional information may be found. We have also made use of the
discussions on polarization given in the textbooks by Shurcliffe (1966), Fowles
(1966) and Hecht (1987).
10.3 The scattering matrix
10.3.1 Representation of the electric vector in the scattering plane
Let us consider an electromagnetic wave of wavelength λ which is scattered by
a spherical particle. In order to describe the scattering process various coordinate
systems will now be introduced as shown in Figure 10.4. The basic Cartesian
(x, y, z)-coordinate system is fixed in space with origin in the center of the scattering
sphere. By drawing meridians from the z-axis to the x-axis and to the y-axis we
construct a sphere whose equatorial plane coincides with the (x, y)-plane. The
incident wave propagating in the direction k
i
is defined in terms of the Cartesian
coordinate system (x
i
, y
i
, z
i
) with unit vectors (i
i
, j
i
, k
i
). Similarly, the scattered
wave moving in direction k
s
is defined with respect to the Cartesian coordinate
system (x
s
, y
s
, z
s
) with unit vectors (i
s
, j
s
, k
s
). The scattering plane is formed by
the straight lines in directions k
i
and k
s
. The intersection of the scattering plane with
the sphere defines the scattering angle . The straight lines pointing in the directions
k
i
and k
s
together with the z-axis form two vertical planes whose intersections with
10.3 The scattering matrix 387
x
y
z
i
i
j
i
k
i
l
i
r
i
i
i
j
i
l
i
r
i
i
s
j
s
k
s
l
s
r
s
i
s
j
s
l
s
r
s
Incident wave
Scattered wave
γ
γ
β
β
Θ
Fig. 10.4 Definition of various coordinate systems needed to describe the scatter-
ing on a sphere, see also text.
the sphere define two meridians. The unit vectors l
i
and l
s
are tangential to the
meridians while the unit vectors r
i
and r
s
are perpendicular to the corresponding
vertical planes. The two systems with the unit vectors (i
i
, j
i
, k
i
) and (i
s
, j
s
, k
s
) are
chosen in such a way that i
i
and i
s
are parallel to the scattering plane while j
i
and
j
s
are perpendicular to this plane.
We are now ready to apply the results of the Mie theory. In the following the
subscripts i and j refer to the components of the incident electric vector along
the unit vectors i
i
and j
i
. According to (10.2) the incoming electric vector may be
expressed by
E
i
=
a
i
i
exp(−iδ
1
)i
i
+ a
i
j
exp(−iδ
2
)j
i
exp
[
i(k
0
z − ωt)
]
=
E
i
i
i
i
+ E
i
j
j
i
exp
[
i(k
0
z − ωt)
]
(10.32)
where a
i
i
and a
i
j
are real amplitudes. With the help of (9.102) we obtain
E
s
i
E
s
j
=
exp[ik
0
(z
s
− z
i
)]
ik
0
z
s
S
2
(cos )0
0 S
1
(cos )
E
i
i
E
i
j
(10.33)
388 Effects of polarization in radiative transfer
x
y
z
ϕ
i
ϕ
s
∆ϕ
Incident wave
Scattered wave
Θ
ϑ
s
ϑ
i
β
γ
Fig. 10.5 Definition of the variables (ϑ
i
,ϑ
s
,ϕ) for the formulation of the scat-
tering matrix with respect to the fixed (x, y, z)-coordinate system.
The amplitude functions S
1
(cos ) and S
2
(cos ) are defined in (9.101). Hence the
scattered electric vector depends only on the scattering angle . The advantage of
this representation is the rotational symmetry of the scattered light with respect to
the scattering plane. The disadvantage is that the scattering plane is not fixed in
space, but it is continually changing its position. This implies that the corresponding
coordinate systems describing the incident and the scattered light are not fixed in
space either. In order to avoid this disadvantage we will transform (10.33) in such
a way that only variables appear which can be expressed with respect to the fixed
(x, y, z)-coordinate system. The required variables are (ϑ
i
,ϑ
s
) and ϕ = ϕ
i
− ϕ
s
as shown in Figure 10.5.
With the help of Figure 10.4 we easily find the transformations
E
i
i
E
i
j
=
−cos γ sin γ
sin γ cos γ
E
i
l
E
i
r
E
s
l
E
s
r
=
cos β sin β
sin β −cos β
E
s
i
E
s
j
(10.34)
10.3 The scattering matrix 389
Substituting these relations into (10.33) yields
E
s
l
E
s
r
=
exp[ik
0
(z
s
− z
i
)]
ik
0
z
s
cos β sin β
sin β −cos β
S
2
(cos )0
0 S
1
(cos )
×
−cos γ sin γ
sin γ cos γ
E
i
l
E
i
r
=
exp[ik
0
(z
s
− z
i
)]
ik
0
z
s
S
11
S
12
S
21
S
22
E
i
l
E
i
r
(10.35)
where the following abbreviations have been introduced
S
11
=−cos β cos γ S
2
(cos ) + sin β sin γ S
1
(cos )
S
12
= cos β sin γ S
2
(cos ) + sin β cos γ S
1
(cos )
S
21
=−sin β cos γ S
2
(cos ) − cos β sin γ S
1
(cos )
S
22
= sin β sin γ S
2
(cos ) − cos β cos γ S
1
(cos )
(10.36)
The spherical triangle with the sides (, ϑ
i
,ϑ
s
) will now be used to determine
the relationship between the scattering angle and the angles (ϑ
i
,ϑ
s
,ϕ), see Figure
10.5. Spherical trigonometry provides the law of cosines for the sides
(a) cos = cos ϑ
i
cos ϑ
s
+ sin ϑ
i
sin ϑ
s
cos ϕ
(b) cos ϑ
s
= cos cos ϑ
i
+ sin ϑ
i
sin cos γ
(c) cos ϑ
i
= cos cos ϑ
s
+ sin ϑ
s
sin cos β
(10.37)
the law of sines
sin ϕ
sin
=
sin γ
sin ϑ
s
=
sin β
sin ϑ
i
(10.38)
and the law of cosines of the angles
cos ϕ =−cos β cos γ + sin β sin γ cos
cos β =−cos γ cos ϕ + sin γ sin ϕ cos ϑ
i
cos γ =−cos β cos ϕ + sin β sin ϕ cos ϑ
s
(10.39)
(10.37a) is a fundamental equation for cos and has already been derived earlier,
see (2.65). Substituting (10.38) into (10.37b,c) yields
cos ϑ
i
sin ϕ = cos ϑ
s
sin ϕ cos + cos β sin γ sin
2
cos ϑ
s
sin ϕ = cos ϑ
i
sin ϕ cos + sin β cos γ sin
2
(10.40)
390 Effects of polarization in radiative transfer
which may be reformulated as
cos ϑ
i
sin ϕ = cos β sin γ + sin β cos γ cos
cos ϑ
s
sin ϕ = cos β sin γ cos + sin β cos γ
(10.41)
Introducing the abbreviation
cos χ = cos ϑ
i
cos ϑ
s
cos ϕ + sin ϑ
i
sin ϑ
s
(10.42)
after a few easy manipulations we find the following relations
cos β cos γ =
cos cos χ − cos ϕ
sin
2
,
cos β sin γ =
(cos ϑ
i
− cos ϑ
s
cos ) sin ϕ
sin
2
sin β sin γ =
cos χ − cos cos ϕ
sin
2
,
sin β cos γ =
(cos ϑ
s
− cos ϑ
i
cos ) sin ϕ
sin
2
(10.43)
Thus, the matrix coefficients of (10.36) assume the form
S
11
= T
1
cos ϕ + T
2
cos χ , S
12
= (T
1
cos ϑ
i
+ T
2
cos ϑ
s
) sin ϕ
S
21
=−(T
1
cos ϑ
s
+ T
2
cos ϑ
i
) sin ϕ, S
22
= T
1
cos χ + T
2
cos ϕ
(10.44)
where we have used the abbreviations
T
1
(ϑ
i
,ϑ
s
,ϕ) =
S
2
(cos ) − cos S
1
(cos )
sin
2
T
2
(ϑ
i
,ϑ
s
,ϕ) =
S
1
(cos ) − cos S
2
(cos )
sin
2
(10.45)
Detailed calculation steps will be left to the exercises.
Equation (10.44), together with (10.37a), (10.42) and (10.45), shows that the
matrix elements S
ij
are functions of the variables (ϑ
i
,ϑ
s
,ϕ) only so that now the
scattering process is completely described in terms of quantities which are known
in the fixed (x, y, z)-coordinate system, see Figure 10.5.
From (10.45) we see that for = 0or = π the denominators of T
1
and T
2
vanish. Therefore, we must show that in these cases the values of both quantities
are non-singular and well-defined. From the definitions (9.94) and (9.101) it can
10.3 The scattering matrix 391
be seen that
τ
n
(cos )
=0
= π
n
(cos )
=0
, τ
n
(cos )
=π
=−π
n
(cos )
=π
=⇒
S
1
(cos )
=0
= S
2
(cos )
=0
, S
1
(cos )
=π
=−S
2
(cos )
=π
(10.46)
Thus in the limits → 0 and → π we easily find from (10.45)
T
1
=0
= T
2
=0
=
1
2
S
1
=0
, T
1
=π
=−T
2
=π
=−
1
2
S
1
=π
(10.47)
verifying that T
1
and T
2
are nonsingular.
10.3.2 Transformation of the Stokes vector
For the consideration of polarization effects in the RTE we need a transformation law
for the Stokes parameters in analogy to the transformation law (10.35) for the elec-
tric vector. Equation (10.26) introduced the Stokes parameters in terms of the ampli-
tudes a
1
, a
2
and the phase angle δ. Instead of using the symbols (S
0
, S
1
, S
2
, S
3
), in
honor of Stokes, it is customary to use the symbols (I, Q, U, V ), defined by
I = E
i
E
∗
i
+ E
j
E
∗
j
, Q = E
i
E
∗
i
− E
j
E
∗
j
U = E
i
E
∗
j
+ E
j
E
∗
i
, V =−i(E
i
E
∗
j
− E
j
E
∗
i
)
(10.48)
whereby the electric vector components are given by
E
i
= a
i
exp(−iδ
i
)exp
[
i(k
0
z − ωt)
]
, E
j
= a
j
exp(−iδ
j
)exp
[
i(k
0
z − ωt)
]
(10.49)
As before, the subscripts i and j denote the directions parallel and perpendicular
to the scattering plane, see Figure 10.4. Substituting (10.49) into (10.48) yields the
Stokes parameters as
I = a
2
i
+ a
2
j
, Q = a
2
i
− a
2
j
U = 2a
i
a
j
cos δ, V = 2a
i
a
j
sin δ with δ = δ
j
− δ
i
(10.50)
Of course, (10.26), (10.48) and (10.50) are equivalent. Since the intensity compo-
nents of a radiation source can be measured directly, it seems practical to introduce
the so-called modified Stokes parameters
I
i
= CE
i
E
∗
i
= Ca
2
i
, I
j
= CE
j
E
∗
j
= Ca
2
j
U = 2Ca
i
a
j
cos δ = 2C(E
i
E
∗
j
), V = 2Ca
i
a
j
sin δ = 2C(E
i
E
∗
j
)
(10.51)
Both sets (10.50) and (10.51) give a complete description of polarization of an
electromagnetic plane wave since I = I
i
+ I
j
and Q = I
i
− I
j
. For dimensional
392 Effects of polarization in radiative transfer
reasons, in (10.51) we have included a proportionality factor C to indicate that the
Stokes parameters carry the dimension of intensity. The factor C depends on the
system of electromagnetic units. As already mentioned, the expression ‘intensity’
is used quite loosely and, in reality, could be a flux density which is expressed in
units of energy per unit time and unit area.
Next we are going to carry out the linear transformation
I
s
l
I
s
r
U
s
V
s
=
A
(k
0
r)
2
I
i
l
I
i
r
U
i
V
i
(10.52)
where, as before, the superscripts i and s denote the incoming and the scattered
light and the subscripts l and r refer to parallel and perpendicular to the meridional
planes. Furthermore, in the denominator we have replaced z
s
by the radius r of the
scattering sphere. The constituents required to find the 4 × 4 matrix A are provided
by (10.35). Analogously to (9.161) we call the term
˜
P(cos ) =
A
k
2
0
V
(10.53)
the scattering matrix for the intensities whereby V is a small volume element
around the scattering sphere.
Before proceeding, we wish to recall a few helpful formulas from complex vari-
able theory. Let (z
1
, z
2
) represent complex variables. Then the following relations
hold
(z
1
z
2
) =(z
1
)(z
2
) −(z
1
)(z
2
), (z
1
z
2
) =(z
1
)(z
2
) +(z
1
)(z
2
)
(z
1
z
∗
2
) =(z
∗
1
z
2
), (z
1
z
∗
2
) =−(z
∗
1
z
2
)
(10.54)
Utilizing these relations, it is straightforward to show that A may be written as
A =
S
11
S
∗
11
S
12
S
∗
12
(S
11
S
∗
12
) −(S
11
S
∗
12
)
S
21
S
∗
21
S
22
S
∗
22
(S
21
S
∗
22
) −(S
21
S
∗
22
)
2(S
11
S
∗
21
)2(S
12
S
∗
22
) (S
11
S
∗
22
+ S
12
S
∗
21
) −(S
11
S
∗
22
− S
12
S
∗
21
)
2(S
11
S
∗
21
)2(S
12
S
∗
22
) (S
11
S
∗
22
+ S
12
S
∗
21
) (S
11
S
∗
22
− S
12
S
∗
21
)
(10.55)
This form of the transformation matrix for the Stokes vectors was first derived by
Sekera (1956). The detailed derivation of (10.55) will be left as a problem of the
exercises.
10.3 The scattering matrix 393
We will conclude this section with a few additional remarks about the Stokes
parameters. In the previous sections we have considered strictly monochromatic
waves. We have shown that every wave of the type (10.2) is elliptically polarized.
This implies that with increasing time the endpoint of the electric (and also of
the magnetic) vector at each point in space periodically describes the circumfer-
ence of an ellipse. In special cases the ellipse degenerates into a straight line or
into a circle. Moreover, the amplitudes a
1
and a
2
and the difference in the phase
angles δ are constants, i.e. they are independent of time. However, no perfectly
monochromatic radiation exists. Even in the best so-called monochromatic sources
there is always some finite frequency spread centered about a mean frequency. Let
us briefly consider a hypothetical quasi-monochromatic light source having the
following property: the oscillation and the subsequent field varies sinusoidally for
a certain time and then changes phase abruptly. This time is known as the coher-
ence time. The sequence keeps repeating indefinitely, and the phase change after
each coherence time occurs randomly. This type of a field may be regarded as an
approximation to that of a radiating atom. The abrupt changes of phase may result
from collisions.
To continue the discussion on Stokes parameters we consider the complex ampli-
tudes a
i
exp(−iδ
i
) and a
j
exp(−iδ
j
), which are no longer constants but functions
of time, i.e. a
i
= a
i
(t), a
j
= a
j
(t), δ
i
= δ
i
(t) and δ
j
= δ
j
(t). Over time intervals of
the order of the period of the oscillation 2π/ω they vary slowly. However, for a time
interval large in comparison with the period, the amplitudes fluctuate in some way,
perhaps independently or perhaps with some correlation. If the complex amplitudes
are completely uncorrelated, the light is natural or unpolarized. In this case, over
sufficiently long periods of time, vibration ellipses of all shapes, handedness and
orientation will have been traced out so that there exists no preferred polarization
ellipse. In contrast, if a
i
exp(−iδ
i
) and a
j
exp(−iδ
j
) are completely correlated, the
light is called polarized. This definition includes strictly monochromatic light,but
it is somewhat more general: (a
i
, a
j
,δ
i
,δ
j
) may separately fluctuate provided that
the ratio a
i
/a
j
of the real amplitudes and the phase difference δ = δ
j
− δ
i
are inde-
pendent of time. If a
i
exp(−iδ
i
) and a
j
exp(−iδ
j
) are partially correlated, the light
is said to be partially polarized. Ignoring some statistical fluctuations, such a par-
tially polarized beam is characterized by a preference in handedness, or ellipticity,
or azimuth, which is the angle between the major axis of the ellipse and an arbitrary
reference direction.
The Stokes parameters of a quasi-monochromatic beam (omitting again the
constant C) are defined by
I =
1
E
i
E
∗
i
2
+
1
E
j
E
∗
j
2
, Q =
1
E
i
E
∗
i
2
−
1
E
j
E
∗
j
2
U = 2
1
E
i
E
∗
j
2
, V = 2
1
E
i
E
∗
j
2
(10.56)
394 Effects of polarization in radiative transfer
where the symbols ... refer to a time average over an interval which is long in
comparison with the period of the vibration. It can be shown that in this case the
inequality
Q
2
+ U
2
+ V
2
≤ I
2
(10.57)
must hold. When this situation occurs we speak of partial polarization.Aswe
already know from (10.27), the equality sign refers to completely polarized light.
Let us briefly look at the Stokes parameters from the experimental point of view.
Since the fluctuations are extremely swift in comparison with the duration of any
measurement, it is possible to measure only mean amplitudes and mean phases so
that all measured Stokes parameters refer to mean quantities.
We will now consider the Stokes vector of natural or unpolarized light such as
the Planckian black body radiation and the parallel (unscattered) solar radiation.
Due to the randomness of the fluctuations, experimental techniques fail to measure
any differences between the phase angles and the intensity components. Thus the
intensity in any direction in the transverse plane is the same so that the Stokes
parameters Q = U = V = 0. This is a necessary and sufficient condition for light
to be natural. That Q is zero follows directly from (10.56). U and V vanish because
cos δ and sin δ average to zero independently of the amplitudes. Thus the Stokes
vector for natural light is (I, 0, 0, 0), that is the intensity I by itself is sufficient to
specify unpolarized radiation.
It can be shown, see for example Chandrasekhar (1960) and Born and Wolf
(1965), that in all cases partially polarized light of intensity I may be regarded to
consist of one part of completely (elliptically) polarized light I
pol
and another part
of completely unpolarized light I − I
pol
. These parts are independent of each other,
and this representation is unique. The Stokes vector for the completely polarized
part of the radiation is specified by (I
pol
, Q, U, V ) with I
pol
= (Q
2
+ U
2
+ V
2
)
1/2
while the unpolarized part of the radiation is described by the Stokes vector (I −
I
pol
, 0, 0, 0).
Let us consider two or more quasi-monochromatic beams, traveling in the same
direction, which are superposed incoherently. This means that there is no fixed
permanent relationship among the phases of the separate beams. The total irradiance
is the sum of the irradiances of the individual beam. Because of the definition of the
Stokes parameters they are additive if a collection of incoherent sources is involved.
We have previously shown that for monochromatic light the Stokes vectors of
the scattered and the incoming light are related by a transformation matrix of the
type (10.55). Each one of the 16 elements of this matrix is a real number. If the
light is partially polarized, which is the case for most physical atmospheric sit-
uations, the same transformation matrix is still valid. The reason for this is that