Parameters characterizing electromagnetic wave polarization
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Carozzi, T, Karlsson, R and Bergman, J (2000) Parameters characterizing electromagnetic wave
polarization. Physical Review E, 61 (2). pp. 2024-2028.
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Parameters characterizing electromagnetic wave polarization
T. Carozzi,
*
R. Karlsson,
†
and J. Bergman
‡
Swedish Institute of Space Physics, Uppsala Division, SE-755 91 Uppsala, Sweden
共Received 23 December 1998; revised manuscript received 16 September 1999兲
In this paper, generalizations of the Stokes parameters and alternative characterizations of three-dimensional
共3D兲 time-varying electromagnetic fields is introduced. One of these characteristics is the normal of the
polarization plane, which, in many cases of interest, is parallel 共or antiparallel兲 to the direction of propagation.
Others are the two spectral density Stokes parameters which describe spectral intensity and circular polariza-
tion. The analysis is based on the spectral density tensor. This tensor is expanded in a base composed of the
generators of the SU共3兲 symmetry group, as given by Gell-Mann and Y. Ne’eman 关The Eight-fold Way
共Benjamin, New York, 1964兲兴 and the coefficients of this expansion are identified as generalized spectral
density polarization parameters. The generators have the advantage that they obey the same algebra as the Pauli
spin matrices, which is the base for expanding the 2D spectral density tensor with the Stokes parameters as
coefficients. The polarization parameters introduced are formulated in the frequency domain, thereby further
generalizing the theory to allow for wide-band electromagnetic waves in contrast to the traditional quasi-
monochromatic formulation.
PACS number共s兲: 42.25.Ja, 02.20.Qs
I. INTRODUCTION
The standard description of wave polarization in the trans-
verse plane of propagation of an electromagnetic wave is
given by the Stokes parameters 关1兴. As is well known, these
parameters can be found from the two-dimensional coher-
ency tensor, constructed from the transverse components of
the wave field. This two-dimensional coherency tensor has
some interesting properties: in 1930, Wiener used the unit
matrix and the three Pauli spin matrices as base to expand
the coherency tensor 关2兴. Fano later showed that the coeffi-
cients in this expansion are the Stokes parameters 关3兴. The
description in terms of the Stokes parameters is straightfor-
ward when the direction of arrival is known and the proce-
dure of obtaining them is described by Born and Wolf 关4兴.
If the direction of arrival from the source is unknown a
priori, we must consider all three field components. In this
case, the two-dimensional coherency tensor cannot be used
and the Stokes parameters cannot be found directly. Instead,
one can use the three-dimensional coherency tensor, or the
corresponding tensor in the frequency domain, the spectral
tensor, to obtain a complete wave characterization.
There exist several different techniques for finding the
degree of polarization, and the axes and the surface normal
of the polarization ellipse for n-dimensional fields with n
⬎ 2; see Refs. 关5–14兴. Means 关5兴 obtains the surface normal
of the polarization ellipse directly from the antisymmetric
part of the spectral tensor, and does not introduce the Stokes
parameters. Roman 关6兴 and Samson 关7兴 utilizes generaliza-
tions of the Stokes parameters to higher dimensions than 2.
Roman generalizes to three dimensions 共3D兲, by expanding
the coherency tensor in terms of nine Hermitean matrices
that constitute a Kemmer algebra. Samson 关7兴 generalizes the
Stokes parameters to an arbitrary dimension n, where the
spectral tensor is expanded in a base of n
2
trace orthogonal,
Hermitean tensors. In three dimensions (n⫽ 3), the tensors
in the expansion represent one set of generators of the special
unitary symmetry group SU共3兲.
The way of characterizing wave polarization presented in
the present paper essentially combines the methods 关5–7兴,
but instead of concentrating on the spectral tensor and use
trace-orthogonal Hermitean tensors or Hermitean tensors
obeying Kemmer algebra, we use the concept spectral den-
sity tensor and the Hermitean SU共3兲 generators given by
Gell-Mann 关15兴. Because the SU共3兲 generators obey the
same algebra as the Pauli spin matrices, the expansion of the
2D spectral density matrix in terms of the Pauli spin matrices
can be extracted from the 3D spectral density tensor as a
limiting case. From the 3D spectral density matrix we can
obtain the normal of the plane of polarization and the two
spectral density Stokes parameters which describe spectral
intensity and circular polarization.
The parameters introduced in this paper provide a power-
ful description of arbitrary fields for theoretical work but
they are also useful in instrumentation where coherent detec-
tion of all three components of the electric or magnetic field
is possible and where the measured field is completely arbi-
trary. Useful areas of application could be, e.g., space based
radio frequency instruments or characterizing uncollimated
light beams.
II. DESCRIPTION OF 3D WAVE POLARIZATION
We are interested in characterizing the sense of polariza-
tion of a wave field, and also to obtain the normal of the
polarization plane. In order to do so we introduce a set of
new 3D polarization parameters and show how they are con-
nected to the usual 2D Stokes parameters and also comprise
the normal of the polarization plane.
*
Electronic address: tc@irfu.se
†
Electronic address: rk@irfu.se
‡
Electronic address: jb@irfu.se
PHYSICAL REVIEW E FEBRUARY 2000VOLUME 61, NUMBER 2
PRE 61
1063-651X/2000/61共2兲/2024共5兲/$15.00 2024 ©2000 The American Physical Society
A. Spectral density matrix of a vector field
Consider an arbitray electric or magnetic field f(r,t). We
wish to investigate the wave polarization properties of this
field at a spatial point. By Fourier transforming the field in
the time domain, we decompose the field into its spectral
components, which depend on the wave polarization. We
define the Fourier transform of the field as
F
共
r,
兲
⫽
冕
⫺ ⬁
⬁
f
共
r,t
兲
e
i
t
dt 共1兲
and represent this field in terms of three real amplitudes and
three real phases, according to
F
共
r,
兲
⫽
冉
F
x
共
r,
兲
F
y
共
r,
兲
F
z
共
r,
兲
冊
⫽
冉
F
1
共
r,
兲
e
i
␦
1
(r,
)
F
2
共
r,
兲
e
i
␦
2
(r,
)
F
3
共
r,
兲
e
i
␦
3
(r,
)
冊
. 共2兲
If, from the outset, the field components are not orthogonal,
orthogonalization is performed so that Eq. 共2兲 describe three
orthogonal (x,y,z) components of the vector field.
The polarization properties of the wave field can be stud-
ied from a second rank tensor formed from the wave field.
We will use the spectral density tensor, defined as
S
d
共
r,
兲
⫽ FF
†
⫽
冉
F
x
F
x
*
F
x
F
y
*
F
x
F
z
*
F
y
F
x
*
F
y
F
y
*
F
y
F
z
*
F
z
F
x
*
F
z
F
y
*
F
z
F
z
*
冊
, 共3兲
where † symbolizes Hermitean conjugate.
Usually the coherency tensor is used to analyze the polar-
ization properties of fields in the time domain, while in the
frequency domain, the corresponding tensor used is the spec-
tral tensor. Usage of these tensors requires that the field in
question is quasimonochromatic, i.e., are almost monochro-
matic with a limited bandwidth. On the other hand, when the
spectral density tensor is used, there are no limitations on the
field. The spectral tensor is formed from the spectral density
tensor by applying the operator
B
ˆ
⫽
冕
¯
⫺ ⌬
/2
¯
⫹ ⌬
/2
d
共4兲
that integrates over a small bandwidth ⌬
, centered around
the angular frequency
¯
. We find the spectral tensor,
S(r,
¯
,⌬
), to be given by
S
共
r,
¯
,⌬
兲
⫽ B
ˆ
S
d
共
r,
兲
⫽
冕
¯
⫺ ⌬
/2
¯
⫹ ⌬
/2
S
d
共
r,
兲
d
. 共5兲
The components of the spectral tensor have the physical di-
mension power, while the spectral density tensor has units
power over frequency. One sees that the spectral density ten-
sor is more fundamental than the spectral tensor since no
frequency band needs to be specified.
B. Spectral Stokes parameters
By choosing a coordinate system with the z axis along the
direction of wave propagation, a transverse wave field can be
represented as
F
⬘
⫽
冉
F
x
⬘
F
y
⬘
0
冊
⫽
冉
F
1
⬘
e
i
␦
1
⬘
F
2
⬘
e
i
␦
2
⬘
0
冊
. 共6兲
The spectral density tensor obtained from the field in Eq. 共6兲,
by omitting the zeros in the third row and the third column
and introducing the phase difference
␦
⬘
⫽
␦
2
⬘
⫺
␦
1
⬘
, takes the
form
S
d
⬘
共
r,
兲
⫽
冉
共
F
1
⬘
兲
2
F
1
⬘
F
2
⬘
e
⫺ i
␦
⬘
F
1
⬘
F
2
⬘
e
i
␦
⬘
共
F
2
⬘
兲
2
冊
. 共7兲
From Eq. 共7兲, we can introduce the spectral density Stokes
parameters
I
共
r,
兲
⫽
共
F
1
⬘
兲
2
⫹
共
F
2
⬘
兲
2
, 共8a兲
Q
共
r,
兲
⫽
共
F
1
⬘
兲
2
⫺
共
F
2
⬘
兲
2
, 共8b兲
U
共
r,
兲
⫽ 2F
1
⬘
F
2
⬘
cos
␦
⬘
, 共8c兲
V
共
r,
兲
⫽ 2F
1
⬘
F
2
⬘
sin
␦
⬘
. 共8d兲
The usual Stokes parameters I, Q, U, and V, see Born and
Wolf 关4兴 for the definition, can be found from the spectral
density Stokes parameters in Eq. 共8兲 by applying the operator
defined in Eq. 共4兲. For example, I is equal to B
ˆ
I.
We immediately see that the spectral density tensor can
be expressed in terms of the spectral density Stokes param-
eters as
S
d
⬘
⫽
1
2
冉
I⫹ QU⫺ iV
U⫹ iVI⫺ Q
冊
. 共9兲
Note that the total spectral intensity I is equal to the trace of
the spectral density tensor, I⫽Tr(S
d
⬘
). We also know that
the tensor in Eq. 共9兲 can be written as a linear combination of
the three generators of the special unitary symmetry group
SU共2兲, i.e., the three Pauli spin matrices
i
, and the unit
matrices 1
2
关2兴, with the spectral density Stokes parameters
as scalar coefficients 关3兴:
S
d
⬘
⫽
1
2
共
I1
2
⫹ U
1
⫹ V
2
⫹ Q
3
兲
. 共10兲
The determinant of the spectral density tensor in Eq. 共7兲 is
equal to zero, and so also the determinant of Eq. 共9兲. The
following relation is thus obtained:
Q
2
⫹ U
2
⫹ V
2
⫽ I
2
. 共11兲
For the usual Stokes parameters, we instead of Eq. 共11兲 have
the relation Q
2
⫹ U
2
⫹ V
2
⭐I
2
关4兴, where the equality only
hold for a monochromatic field.
PRE 61 2025PARAMETERS CHARACTERIZING ELECTROMAGNETIC . . .
C. Generalized polarization parameters
To generalize Eq. 共9兲 to three dimensions, we use the
generators of the SU共3兲 symmetry group to form a new rep-
resentation of the spectral density tensor. The unit matrix in
three dimensions, 1
3
and the generators,
ˆ
i
, i⫽ 1,...,8,
given by Gell-Mann 关15兴, will be used. The spectral density
tensor is formed as a linear combination of the unit matrix
and the generators, and for the scalar coefficients we use the
symbols ⌳
i
, i⫽ 0,...,8:
S
d
⫽
1
3
⌳
0
1
3
⫹
1
2
兺
i⫽1
8
⌳
i
ˆ
i
⫽
冉
1
3
⌳
0
⫹
1
2
⌳
3
⫹
1
2
冑
3
⌳
8
1
2
⌳
1
⫹ i
1
2
⌳
2
1
2
⌳
4
⫹ i
1
2
⌳
5
1
2
⌳
1
⫺ i
1
2
⌳
2
1
3
⌳
0
⫺
1
2
⌳
3
⫹
1
2
冑
3
⌳
8
1
2
⌳
6
⫹ i
1
2
⌳
7
1
2
⌳
4
⫺ i
1
2
⌳
5
1
2
⌳
6
⫺ i
1
2
⌳
7
1
3
⌳
0
⫺
1
冑
3
⌳
8
冊
. 共12兲
We call the coefficients ⌳
i
generalized spectral density po-
larization parameters. It can be seen that the trace of Eq.
共12兲 is equal to ⌳
0
and we identify the spectral intensity I
⫽ ⌳
0
. The normalization in Eq. 共12兲 is such that it reduces to
the spectral Stokes parameters for the case of a plane wave
chosen to propagate along the z axis. In this case ⌳
4
⫽ ⌳
5
⫽ ⌳
6
⫽ ⌳
7
⫽ 0 and ⌳
8
⫽ (1/
冑
3)⌳
0
. Inserting these expres-
sions into Eq. 共12兲, we obtain
S
d
⫽
1
2
冉
⌳
0
⫹ ⌳
3
⌳
1
⫹ i⌳
2
0
⌳
1
⫺ i⌳
2
⌳
0
⫺ ⌳
3
0
0
0
0
冊
, 共13兲
which is the same as Eq. 共9兲, except for the name of the
parameters.
III. PHYSICAL INTERPRETATION
OF THE PARAMETERS
The spectral density tensor, which was introduced previ-
ously, can be seen to consist of several parts which can be
ascribed specific physical meaning that we now will discuss.
A. The normal of the polarization plane
To the antisymmetric part of a tensor, a dual pseudovector
is associated, see Arfken and Weber 关16兴. The pseudovector
of the spectral density tensor in Eq. 共12兲 is given by
i(⫺ ⌳
7
,⌳
5
,⫺ ⌳
2
). We introduce a similar vector V
V⬅
共
⌳
7
,⫺ ⌳
5
,⌳
2
兲
, 共14兲
which by definition is real. Comparing the expressions for
the spectral density tensors in Eqs. 共3兲 and 共12兲, we obtain
⌳
7
⫽⫺2Im
兵
F
y
F
z
*
其
, 共15a兲
⌳
5
⫽⫺2Im
兵
F
x
F
z
*
其
, 共15b兲
⌳
2
⫽⫺2Im
兵
F
x
F
y
*
其
. 共15c兲
The time-dependent field vector f(r,t) traces the polariza-
tion ellipse. The complex vector F(r,
) and its complex
conjugate F
*
(r,
) form a polarization plane in space. This
plane is the same plane as the polarization ellipse defines.
Because
V• F⫽ V• F
*
⫽ 0, 共16兲
V is perpendicular to F, and thereby also perpendicular to the
polarization plane. It can be shown 共see Lindell 关17兴兲 that a
vector normal to the plane of polarization is parallel to
iF⫻ F
*
, and the magnitude of this vector is 2/
times the
area of the polarization ellipse. Using Eqs. 共2兲, 共3兲, and 共12兲,
we obtain
iF⫻ F
*
⫽
共
⌳
7
,⫺ ⌳
5
,⌳
2
兲
⫽ V, 共17兲
i.e.,
兩
V
兩
is equal to 2/
times the area of the polarization
ellipse.
The normal of the polarization plane gives, in the case of
transverse waves, the direction of wave propagation. Defin-
ing right- and left-hand polarization as the polarization seen
by the wave itself and not by an observer looking at the
approaching wave, the normal of the polarization plane V is
parallel to the direction of propagation for a right-hand po-
larized wave and antiparallel for a left-hand polarized. The
orientation of the plane of polarization can be specified with
two angles. Referring to Fig. 1, the first angle is the angle
␣
between the plane of polarization and the xy plane. The sec-
ond is the angle

between the intersection of the planes and
the x axis.
B. Number of independent parameters
In general, six parameters are needed in order to charac-
terize a wave field, for example three complex numbers or
three real amplitudes and three real phases. The four 2D
spectral Stokes parameters I, Q, U, and V, characterize the
polarization in the transverse plane of propagation. The
propagation direction adds another two parameters to the
spectral Stokes parameters, giving a total number of six pa-
rameters.
2026 PRE 61T. CAROZZI, R. KARLSSON, AND J. BERGMAN
When the spectral density tensor is formed, the overall
phase is lost and the field can be multiplied with an arbitrary
phase without changing the spectral density tensor. The num-
ber of independent parameters can therefore only be five and
for the spectral density Stokes parameters the loss of an in-
dependent parameter is described by Eq. 共11兲. This also
means that only five of the nine generalized spectral density
polarization parameters can be independent.
By considering the scalar invariants of the spectral density
tensor, we can verify that there is five independent param-
eters. The scalar coefficients in the secular equation, det(S
d
⫺ 1
3
)⫽
3
⫺ I
1
2
⫹ I
2
⫺ I
3
⫽ 0, of a tensor S
d
are indepen-
dent of the base vectors, and are called scalar invariants. The
scalar invariants of a tensor are the trace (I
1
), the sum of the
three cofactors (I
2
), and the determinant (I
3
). The 2D and
3D spectral density tensors use different base vectors for the
wave field, but are equivalent and must have the same invari-
ants. Further, the symmetric and the antisymmetric parts of
the spectral density tensor are never mixed and in turn they
must have their own invariants.
The first invariant, the trace, gives just the spectral inten-
sity. Calculating the second invariant gives two equations,
one for the symmetric part and one for the antisymmetric.
The third invariant adds another two equations, resulting in a
number of four equations that reduce the number of indepen-
dent generalized spectral density polarization parameters
from 9 to 5. One of these equations is specially interesting,
since it gives the magnitude of the spectral density Stokes
parameter V:
兩
V
兩
⫽
冑
⌳
7
2
⫹ ⌳
5
2
⫹ ⌳
2
2
⫽
兩
V
兩
. 共18兲
This tells us that ⌳
7
, ⌳
5
, ⌳
2
, and
兩
V
兩
all describe circular
polarization, and that
兩
V
兩
is an invariant.
IV. EXAMPLE
We now present a simple example on how the polariza-
tion parameters discussed in this article can be used to char-
acterize an electric field: consider two circularly polarized
monochromatic beams of unit amplitude which intersect one
another at right angles, see Fig. 2. As a simplification, we
model the beams as plane waves and allow only one degree
of freedom, namely, the relative phase
␦
, between the two
plane waves. Let one of the wave vectors lay along the x axis
and the other along the y axis, intersecting each other at the
origin. Assume right-hand circularly polarized beams, with
fields represented by
E
1
T
⫽
共
0 ⫺ 1 ⫺ i
兲
and E
2
T
⫽ e
i
␦
共
10⫺ i
兲
.
共19兲
At the intersection point the total electric field can be written
as
E⫽ E
1
⫹ E
2
共20兲
and the resulting spectral density tensor is
冉
1 ⫺ cos
␦
⫺ i sin
␦
⫺ sin
␦
⫹ i
共
1⫹ cos
␦
兲
⫺ cos
␦
⫹ i sin
␦
1 ⫺ sin
␦
⫺ i
共
1⫹ cos
␦
兲
⫺ sin
␦
⫺ i
共
1⫹ cos
␦
兲
⫺ sin
␦
⫹ i
共
1⫹ cos
␦
兲
2
共
1⫹ cos
␦
兲
冊
. 共21兲
FIG. 1. Orientation of the polarization plane. The line ab is the
intersection between the plane of polarization and the xy plane, and
the angle between these planes is
␣
. The angle

specifies where in
the xy plane the two planes intersect.
FIG. 2. Two right-hand circularly polarized beams intersecting
each other at right angles. In this example, beam 2 has a phase shift
␦
with respect to beam 1. As a function of the phase shift,
␦
, the V
vector traces out an ellipse depicted at the origin. The figure shows
the case when
␦
⫽
/4 and V⫽ (1⫹1/
冑
2,1⫹ 1/
冑
2,1/
冑
2).
PRE 61
2027PARAMETERS CHARACTERIZING ELECTROMAGNETIC . . .
