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Dissertation at Uppsala University to be publicly examined in Polhemsalen, 10134, Ångström
Laboratory, Friday, September 30, 2005 at 14:00 for the Degree of Doctor of Philosophy. The
examination will be conducted in English
Abstract
Karlsson, R. 2005. Theory and Applications of Tri-axial Electromagnetic Field Measurements.
. Acta Universitatis Upsaliensis. Uppsala Dissertations from the Faculty of Science and Technology
. vi, 63 pp. Uppsala. ISBN 91-554-5436-4
Polarisation, which was first studied in optics, is a fundamental property of all electromagnetic
fields. A convenient way to describe the polarisation of 2D electromagnetic fields is given by
the Stokes parameters.
This thesis deals with different aspects of wave polarisation and electromagnetic field measurements.
A generalisation of the Stokes parameters to three dimensions is presented. The
theory has been used to develop methods and systems for obtaining the polarisation parameters
of electromagnetic waves. The methods can be applied for a wide range of electromagnetic
fields, measured both on ground and onboard satellites. The applications include, e.g. directionfinding,
polarisation analysis, radio telescopes, radar, and several examples in the field of wireless
communication.
Further applications are given in the analysis of satellite data, where a whistler wave is considered.
Whistlers are circularly polarised electromagnetic waves propagating in the magnetosphere
along the geomagnetic field. Dispersion in the magnetospheric plasma make the whistler
frequency components travel at different speeds and the signal takes the form of a chirp. From
instantaneous polarisation analysis of the whistler’s magnetic wave field, the normal to the polarisation
plane is obtained and found to precess around the geomagnetic field.
A statistical analysis of ionospheric stimulated electromagnetic emissions (SEE) is also presented.
SEE is generated by injecting a powerful high frequency radio wave into the ionosphere.
It is shown that the SEE features have a statistical behaviour indistinguishable from the amplitude
and phase distributions of narrow-band Gaussian noise. The findings suggests that SEE
cannot be explained by simple coherent processes.
An expression for the complex Poynting theorem is derived for the general case of anharmonic
fields. It is found that the complex Poynting theorem, in contrast to the Poynting theorem
for real fields and sources, is not a conservation law of the imaginary part of electromagnetic
energy.
Keywords: polarisation, polarization, radio waves, Stokes parameters, antennas, SEE, whistler
waves, direction-finding, LOIS, statistics, Poynting theorem
Roger Karlsson, Department of Astronomy and Space Physics. Uppsala University. Box 515,
SE-751 20 Uppsala, Sweden
c○ Roger Karlsson 2005
ISSN 1651-6214
ISBN 91-554-5436-4
urn:nbn:se:uu:diva-3344 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3344)

List of Papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I
II
III
IV
V
Parameters characterizing electromagnetic wave
polarization,
T. D. Carozzi, R. Karlsson, and J. Bergman,
Phys. Rev. E 61, 2024-2028, 2000.
Method and system for obtaining direction of an
electromagnetic wave,
J. Bergman, T. D. Carozzi, and R. Karlsson,
Metod och system för att erhålla riktning för en elliptiskt
polariserad elektromagnetisk vågutbredning, Swedish patent No
512 219, 2000; US Patent 6,407,702, 2002.
Multipoint antenna device,
J. Bergman, T. D. Carozzi, and R. Karlsson,
Antennanordning för användning av tredimensionell elektromagnetisk
fältinformation inherent i en radiovåg, Swedish patent No
517 524, 2002; International Patent Publication WO03/007422,
2003.
Three-channel digital radio vector field sensor: Description
and demonstration,
R. Karlsson, W. Puccio, J. Bergman, T. D. Carozzi, and B. Thidé,
Proceedings of the HF–04, Nordic Shortwave Conference, Fårö,
Sweden, 2004.
Present and future applications of the Information Dense
Antenna,
J. Bergman, T. D. Carozzi, and R. Karlsson,
Proceedings of the HF–04, Nordic Shortwave Conference, Fårö,
Sweden, 2004.
v

VI
VII
VIII
Precession of the whistler polarisation plane normal observed
on Freja,
R. L. Karlsson, T. D. Carozzi, J. E. S. Bergman, and A. I. Eriksson,
Submitted to Geophysical Research Letter, 2005.
Statistical properties of ionospheric stimulated
electromagnetic emission,
R. L. Karlsson, T. D. Carozzi, J. E. S. Bergman, L. Norin, B.
Thidé,
Submitted to Annales Geophysicae, 2005.
Complex Poynting theorem as a conservation law,
T. D. Carozzi, J. E. S. Bergman, and R. L. Karlsson,
Submitted to Journal of Mathematical Physics, 2005.
Reprints were made with permission from the publishers.
Papers not included in the thesis
Method for Three-Dimensional Evaluation,
J. Bergman, T. D. Carozzi, and R. Karlsson,
System för tredimensionell utvärdering av ett elektroniskt vektorfält, Swedish
patent No 523 086, 2004; International Patent Publication WO03/067710,
2003.
Three-channel digital radio receiver for simultaneous reception in three
orthogonal dimensions,
R. Karlsson and W. Puccio,
Proceedings of the HF–01, Nordic Shortwave Conference, Fårö, Sweden,
2001.
vi

Contents
1 Introduction 1
2 Plasma 5
2.1 The magnetosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Electromagnetic fields and wave polarisation 11
3.1 Electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Polarisation and Stokes parameters . . . . . . . . . . . . . . . . . . . . . 13
3.3 The coherency tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4 The spectral tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.5 Wave polarisation in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4 Electromagnetic field measurements 21
4.1 Method for estimating polarisation parameters . . . . . . . . . . . . . 21
4.2 System for estimating polarisation parameters . . . . . . . . . . . . . 23
4.3 The high frequency band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 LOIS—the LOFAR Outrigger In Scandinavia . . . . . . . . . . . . . 25
4.5 Wireless telecommunication . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Space applications 31
5.1 Antennas in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Satellite projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6 Whistler analysis using polarisation 37
7 Statistical approach to stimulated electromagnetic emissions 41
7.1 Characteristics and models for SEE . . . . . . . . . . . . . . . . . . . . . 41
7.2 Statistical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8 The complex Poynting theorem as a conservation law 45
8.1 The instantaneous Poynting theorem . . . . . . . . . . . . . . . . . . . . 45
8.2 The complex Poynting theorem . . . . . . . . . . . . . . . . . . . . . . . . 46
8.3 The time-harmonic complex Poynting Theorem . . . . . . . . . . . . 47
vii

9 Summary in Swedish 49
10 Conclusion and Outlook 53
References 57
viii

1. Introduction
The existence of radio waves was first predicted by James Clerk Maxwell
in the 1860’s. In 1886, Heinrich Hertz managed to generate and detect these
waves for the first time and they were first named Hertzian waves. Hertz did
not follow up his work by using the waves for sending and receiving information.
Instead, that important step was taken by a number of radio pioneers in
the 1890’s.
In 1893, Nicola Tesla demonstrated the first wireless communication and
described the equipment in articles. Tesla filed his first radio patent applications
in 1897 and three years later they were granted 1 . Tesla continued his
work on radio and in 1898 he demonstrated the first radio-controlled vessel 2 .
In 1895, Guglielmo Marconi presented a radio and made his first demonstrations
of radio communication. Marconi’s first American patent application,
filed in 1897, was turned down because of prior art of Tesla and other
inventors, and so were the succeeding applications. But in 1904, the US patent
office changed its decision and granted Marconi US patent 3 for the invention
of the radio. Marconi was also awarded the 1909 Nobel prize in Physics.
Among the radio pioneers were also Alexandr Stephanovitch Popov and
Edouard Branly. Already in 1885 Popov suggested that electromagnetic waves
could be used to transmit messages and in 1896 he demonstrated it.
For long time, Marconi was considered as the inventor of radio, but in 1943,
the US supreme court decided to uphold Tesla’s radio patent. Unfortunately,
Tesla never experienced the success. He died nine months earlier.
Even though Marconi cannot be considered as the inventor of radio, he
made important contributions to the development. In 1899, Marconi transmitted
radio signals across the English channel and in 1901, he was the first
person who succeded in transmitting radio wave across the Atlantic.
During the first time of radio transmission, spark machines were used to
generate Morse code. The Alexanderson alternator, named after its inventor
Ernst Alexanderson, made it possible to generate high-frequency oscillations
and allowed transmission of continuous radio waves. Reginald Fessenden invented
the heterodyne principle and the amplitude modulation (AM) of radio
1 US Patents 645,576 and 649,621
2 US Patent 613,809
3 US Patent 763,772
1

waves. In 1900, Fessenden managed to transmit voice the first time, but it
was not until 1906 that Fessenden made the first broadcast of voice by using
the Alexanderson alternator. Among the many inventions made by Ernst
Alexanderson, a tuning device for radios must be mentioned 4 . This device is
fundamental for radios also today.
On both sides of the Atlantic, many radio telegraph stations were built for
the transatlantic communication. These facilities were equipped with Alexanderson
alternators and huge antennas. All but one of these radio facilities are
gone today. The only remaining station is located at Grimeton, close to Varberg,
Sweden, see Figure 1.1.
Figure 1.1: Radio station Grimeton (SAQ) close to Varberg, Sweden.
A very important step in the development of radio was when Lee De Forest
in 1906 invented the triode, a three-electrode vacuum tube. The triode made
amplification of radio signals possible and soon became a fundamental part of
all radio equipment.
Radio transmission requires that the transmitted signal can be detected by
the receiving antenna. In other words, the transmitting antenna should be in
line of sight with the receiving antenna. But how could it work for transatlantic
radio transmissions? The explanation was given in 1902 when Heaviside and
Kennelly predicted that the radio waves could be reflected off by an ionised
layer in the upper atmosphere, making them propagate to other continents.
4 US Patent 1,008,577
2

Although the most fundamental radio inventions already were made, the
following decades offered a lot of new inventions and applications. In the
late 1920’s TV broadcast started and the 1930’s the radar (radio detection
and ranging) was invented. Sir Robert Alexander Watson-Watt is considered
to be the inventor of radar in the mid 1930’s. Watson-Watt also coined the
phrase ionosphere for the ionised upper part of the Earth’s atmosphere. Radar
can operate at any radio frequency, but the applications vary with frequency.
Common for all radar applications is that a radio wave is transmitted and the
echo is received, either by the same antenna or a separate antenna. The time
difference between the transmission and the echo specifies the distance to the
target and the signature of the echo can provide valuable information about
the target.
In the beginning, low frequencies were used for radio but with time, higher
and higher frequencies have come into use. Today, the radio frequency spectrum
is densely populated from the Long Wave (LW) radio band at 150 kHz
up to about 50 GHz. Frequencies below 150 kHz is used in, e.g., submarin
communication. Above 50 GHz, damping caused by rain or fog limits the applications.
The radio spectrum is used for radio and television broadcasting,
amateur radio, airplanes, ships, taxi, police, military communication, cellular
telephones, Bluetooth, Wireless LAN, satellite communication, radar, and
much more.
During the last 25 years, cellular telephone has developed strongly. The
first commercial systems were analogue (NMT 1981 in the Nordic countries;
ETACS and NTACS 1985 in the rest of Europe; AMPS 1983 in the US). In
1991 the first digital system was introduced (GSM 1991 in Europe and elsewhere;
PDC 1994 in Japan; IS, PCS (1996) in US), and recently the second
generation digital systems (3G) have started to operate.
Other mobile services have also been (or will soon be) introduced, for example
Bluetooth, WLAN (wireless local area network, the term Wi-Fi, wireless
fidelity is also used), Ultra wideband (UWB), ZigBee, and WiMax.
In many cases of radio broadcast and communication, only the amplitude
of the signal is measured. Since both the phase and the amplitude are equally
important for the characterisation of an electromagnetic wave field, half of
the information in the radio signal is then never used. Furthermore, in most
radio applications, only one component of the electromagnetic wave field is
measured. For most cases, that is enough, e.g., in receiving a signal with a
transistor radio. There are applications where two or even three components
of the wave field are measured, but it is implemented in far from all systems
which would benefit from it. By measuring two, or preferrably three, components
of the electromagnetic field it is possible to make use of its polarisation.
This thesis deals mainly with polarisation and merges science and technology
into applications beneficial for both of them. In Chapter 2, a short intro-
3

duction to plasma and the plasma environment surrounding the Earth is given.
Chapter 3 deals with electromagnetic fields and wave polarisation. Electromagnetic
field measurements on Earth and in space are considered in Chapters
4 and 5, respectively. In Chapters 6, polarisation is shown to be useful in
analysing satellite data and in Chapter 7, the statistical nature of radio wave
induced ionospheric radio emissions are investigated. Chapter 8 is devoted to
the complex Poynting theorem and finally, in Chapter 9 a summary in Swedish
is presented.
4

2. Plasma
When a gas is heated to a sufficiently high temperature, the molecules or atoms
in the gas start to dissociate and positively charged ions and free electrons are
formed. If the ionisation level is sufficiently high, this gas of charged particles
is a plasma.
Plasma is usually considered the fourth state of matter after solid, liquid
and gas. Much energy is needed to ionise a gas and this rarely happen naturally
on Earth. The only exception is lightning where the gas particles in the
conducting lightning channel become ionised. Plasma also exist in, e.g. manmade
devices for industrial processes, light tubes, and plasma displays. Even
though plasma is rare on Earth, it is abundant in space and 99% of the known
matter in universe is in plasma state.
Plasma behaves much more complicated than neutral gas because the free
charges interact electromagnetically over large distances. This leads to a lot
of phenomena which do not take place in a neutral gas. The physics needed to
describe many of the processes is also complicated. In general, the equations
describing a plasma cannot be solved exactly and approximations have to be
used.
Any ionised gas does not fulfill the requirements of being a plasma. The required
amount of ionisation can be estimated if we define a plasma in the
following way [10]: “A plasma is a quasineutral gas of charged and neutral
particles which exhibits collective behavior.” With collective behaviour is
meant that the motion of the plasma is affected not only by the local plasma,
but due to the long ranged electromagnetic force, also on remote properties.
The quasineutral requirement means that, locally the plasma is not neutral,
but globally, we can have approximately as many electrons as ions, n e ≃ n i .
We must allow for small variations in the plasma density. Otherwise, all the
electromagnetic forces would cancel.
In presence of only a homogeneous magnetic field B, a charged particle
with mass m and charge q will be affected by the Lorentz force qv × B and
the particle will move in a circular orbit. The frequency of the motion is the
gyrofrequency
ω c = qB m
(2.1)
Because of differencies in masses, the electron gyrofrequency is much higher
5

than the ion gyrofrequencies.
Plasma particles are influenced not only by magnetic fields. Presence of, e.g.
electric fields, gravitaional fields, and magnetic field gradients cause plasma
particles to drift.
A plasma has a natural oscillation frequency which depends on the plasma
density n = n e = n i . This frequency is the plasma frequency
ω 2 p = nq2
ε 0 m
(2.2)
which is very important in plasma physics.
2.1 The magnetosphere
From the upper part of the Sun’s atmosphere, a flow of plasma is emerging at
a supersonic speed of around 400 km/s. This is the solar wind and it carries a
magnetic field. When the solar wind comes close Earth, it feels the presence
of Earth’s magnetic field. A shock front is formed and behind the shock front
the solar wind flow is subsonic. A few Earth radii behind the shock front, the
magnetopause is located. The magnetopause is the boundary where the solar
wind pressure is balanced by the magnetic pressure of Earth’s magnetic
field, p m = B0 2/(2µ 0), where B0 2 is the geomagnetic field. Inside of the magnetopause,
the magnetosphere is located. A picture of the Magnetosphere is
shown in Figure 2.1, where different regions have been labelled. Charged particles
move under the influence of a magnetic field and form currents. The
flowing currents, on the other hand, generate magnetic fields. The magnetospheric
currents and magnetic fields are also shown in Figure 2.1. The subsolar
point on the Magnetopause is located about ten Earth radii from Earth, but because
of the dynamic nature of the solar wind, the distance varies with time.
The magnetosphere functions as a shield and protects Earth from the solar
wind. Mars, on the other hand, has a very weak magnetic field and the solar
wind impinges directly on Mars’ atmosphere. Probably Mars once had a much
denser atmosphere and the interaction between the solar wind the atmosphere
is believed to be a contributing factor to the loss of Mars’ atmosphere.
2.2 The ionosphere
The magnetosphere has a rather low plasma density, but the density increases
as we move closer to Earth. The plasma region of high density between about
60 km and 1000 km altitude is called the ionosphere. Figure 2.2 shows the
density profile of the ionosphere. The shape of the profile is caused by an in-
6

Figure 2.1: The Magnetosphere. From Kivelson and Russel [15].
creasing particle density as we approach the Earth from above in combination
with an ionisation—mostly by ultraviolett (UV) radiation—from the sun, but
also from cosmic rays and meteorites. The maximum density at about 300
km altitude can be explained as follows. At very high altitude the atmosphere
is very thin and even though the UV radiation is strong, there are few particles
which can be ionised. The plasma density is therefore low at these high
altitudes. For low altitudes, the particle density is high, but most of the UV
radiation have already been absorbed at higher altitudes, giving a low plasma
density. At mid-altitude regions, we have quite high particle density and still a
strong UV radiation, conditions which are favourable for a high plasma density.
The different layers of the ionosphere have been named in alphabetic order
starting with the E-layer for the layer first discovered. The names of the layers
are shown in Figure 2.2.
The ionised ionosphere can be used as a mirror for transmitting radio signals
around the Earth. For vertically injected signals, the frequency must be
lower than a critical frequency, which is equal to the plasma frequency of the
densest part of the ionosphere. Frequencies higher than the critical frequency
will penetrate the ionosphere and be lost. By increasing the ionospheric angle
of incidence, high frequencies, which otherwise would have passed through
7

Figure 2.2: The electron density of the ionosphere. From Kivelson and Russel [15].
the ionosphere can be made to reflect off, as shown in Figure 2.3. Long distance
radio transmissions are therefore possible in a wider frequency range.
Detailed descriptions of radio wave propagation in plasma are given by Budden
and Ginzburg [8, 12].
The plasma density of the ionosphere can be measured by utilising the ionosphere’s
ability to reflect radio waves of lower frequency than the critical frequency.
This is routinely done by ionosonds. A radio beam is then transmitted
into the ionosphere with a frequency which is swept from below 1 MHz up to
about 20 MHz. For a certain frequency, the time for the echo to return gives
the altitude for the reflection, while the frequency, according to Equation (2.2)
is a measure of the plasma density. In this way, only the plasma density in
the lower part of the ionosphere can be measured. The upper profile must be
measured with satellites equipped with top-side ionosonds.
8

Figure 2.3: Radio waves of different incident angles reflecting off the ionosphere.
From The ARRL handbook [13]
9

3. Electromagnetic fields and wave
polarisation
Charges placed close to one another interact and the situation is similar for
currents, which are formed by moving charges. By introducing electromagnetic
fields, the description of the interaction between charges and currents
can be simplified. In this chapter, we consider the electromagnetic fields and
focus on the polarisation, which is an important characteristic of electromagnetic
fields.
3.1 Electromagnetic fields
The electromagnetic fields consist of the electric field E(t, x) and the magnetic
field B(t, x), which both in general depend on time and space. These
fields are coupled and related to the electric charge density ρ(t, x) and the
electric current density J(t, x) by Maxwell’s equations
∇ · E = ρ/ε (3.1)
∇ · B = 0 (3.2)
∇ × E = − ∂B
∂t
(3.3)
∇ × B = µJ + µε ∂E
∂t
(3.4)
In a non-conducting and source free medium, Maxwell’s equations simplifies
to
∇ · E = 0 (3.5)
∇ · B = 0 (3.6)
∇ × E = − ∂B
∂t
(3.7)
∇ × B = µε ∂E
∂t
(3.8)
A wave equation can be derived from Maxwell’s equations by first taking
the curl of Equation (3.7) and using the vector identity ∇ × (∇ × a) =
11

∇·(∇·a)−∇ 2 a, where a is an arbitrary vector. Equation (3.8) then becomes
∇×(∇×E) = ∇·(∇·E)−∇ 2 E = −∇ 2 E = − ∂B
∂t = −µε∂2 E
∂t 2 (3.9)
or simplified
∇ 2 E − µε ∂2 E
∂t 2 = 0 (3.10)
This is the wave equation for the electric field propagating in a medium with
permittivity ε, and permeability µ. The wave equation for B is found in a
similar way by taking the curl of Equation (3.8). This equation is of exactly
the same form as the wave equation for the electric field. All components ψ
of E and B then satisfy the scalar wave equation
where
∇ 2 ψ − 1 v 2 ∂ 2 ψ
∂t 2 = 0 (3.11)
v = 1 √ µε
(3.12)
is the speed by which the pertubation propagates. A solution of
Equation (3.11) is
ψ = ψ(t, ˆn · x) (3.13)
where ˆn is a unit vector of unspecified direction. Defining ζ as the projection
of x along ˆn, ζ = ˆn · x, we have ∇ = ˆn∂/∂ζ, so that
∇ 2 ψ = ∂2 ψ
∂ζ 2 (3.14)
and the wave equation becomes
∂ 2 ψ
∂ζ 2 − − 1 ∂ 2 ψ
v 2 ∂t 2 = 0 (3.15)
This equation has a general solution of the form
ψ(t, ζ) = ψ 1 (ζ − vt) + ψ 2 (ζ + vt) (3.16)
where ψ 1 represents a pertubation propagating along increasing ζ and ψ 2 a
pertubation propagating along decreasing ζ. Assuming harmonic time varia-
12

tion, the solution becomes
ψ(t, ζ) = ψ 0 e i(±kζ−ωt) (3.17)
where k = ω/c is the wave number. By introducing the wave vector
k = kˆn = kˆk = ω c ˆk = √ µε ωˆk (3.18)
Equation (3.17) becomes
ψ(t, ζ) = ψ 0 e i(k·x−ωt) (3.19)
Since ψ denotes any components of E or B, the solution for the electromagnetic
fields can be written
E(t, x) = E 0 e i(k·x−ωt) (3.20)
B(t, x) = B 0 e i(k·x−ωt) (3.21)
These fields represent a plane wave propagating along ˆk = ˆn with E and
B perpendicular to each other and to the direction of wave propagation. The
fields E(t, x) nd B(t, x) are complex, while the observable physical fields
are real. The physical fields are found by taking the real parts of the complex
fields:
E physical (t, x) = Re{E 0 e i(k·x−ωt) } (3.22)
B physical (t, x) = Re{B 0 e i(k·x−ωt) } (3.23)
In the following, we will work with complex fields and in order to obtain the
physical fields, we have to remember to take the real part of the complex field.
3.2 Polarisation and Stokes parameters
Polarisation is an important property of an electromagnetic wave field. It describes
how the electric and magnetic field vectors change with time. As a
plane wave propagates, the tips of the field vectors describe figures in the
polarisation plane, which is the plane perpendicular to the direction of wave
propagation. In general, the figures are ellipses. There are two special cases,
where the major axes of the ellipse is equal and a circle is traced, and where
one of the major ellipse axes is zero and a straight line is traced. These three
cases correspond to elliptical, circular, and linear polarisation, respectively. In
all three cases, we will use the general term polarisation ellipse for the traced
figure.
The sense in which the polarisation ellipse is traced determines the sense
13

of polarisation. If the field vector sweeps clock-wise, as seen along the direction
of propagation, the sense is right-handed and if the field vector sweeps
counterclock-wise, the sense is left-handed.
A convenient way to describe the state of polarisation is in terms of the
Stokes parameters, which were introduced in 1852 by G. G. Stokes [28]. The
Stokes parameters are defined for quasi-monochromatic wave fields and we
will suppose that the wave field only contains frequencies within a narrow
(angular) frequency band ∆ω, centered around a mean frequency ¯ω. For a
quasi-monochromatic field, the bandwidth is small compared to the mean frequency:
∆ω
¯ω
= ∆ν
¯ν ≪ 1 (3.24)
The Stokes parameters are defined for a two-dimensional wave field and we
choose the direction of wave propagation to be along the z axis. An electromagnetic
field F(t, x) is then be represented by
( ) (
)
F x (t, x) F x,0 e i(kz−ωt+δx)
F(t, x) =
=
F y (t, x) F y,0 e i(kz−ωt+δy) (3.25)
where F x,0 and F y,0 are real valued and δ x and δ y are real valued phase factors
of the field components. The Stokes parameters are [7]
I = 〈F x F ∗ x + F y F ∗ y 〉 (3.26)
Q = 〈F x F ∗ x − F yF ∗ y 〉 (3.27)
U = 2 Re{〈F x F ∗ y 〉} (3.28)
V = 2 Im{〈F x F ∗ y 〉} (3.29)
where the brackets symbolise time average and ∗ complex conjugation. The
parameters satisfy the relation
Q 2 + U 2 + V 2 ≤ I 2 (3.30)
Here, I symbolises the intensity of the wave field. The identity in Equation
(3.30) only holds for a monochromatic wave field.
The degree of polarisation [7] is defined by
√
Q 2 + U 2 + V 2
p =
I
(3.31)
The degree of polarisation varies between zero and unity and a value p = 0
represents a totally unpolarised wave field and p = 1 a completely polarised.
The parameters Q and U represent linear polarisation and V circular polarisa-
14

tion.
A geometrical representation of the Stokes parameters, the Poincaré
sphere, was introduced in 1892 by H. Poincaré. In this representation, the
Stokes parameters describe a point with coordinates (Q, U, V ) on, or inside,
the Poincaré sphere. The point lies on the sphere for a completely polarised
wave field, inside the sphere for a partially polarised, and at the centre of the
sphere for a totally unpolarised wave field. The Poincaré sphere is shown in
Figure 3.1. The radius of the sphere is I and the degree of polarisation, p, is,
in general, less than I.
V
p
I
2χ
U
2ψ
Q
Figure 3.1: The Poincaré sphere visualises the state of polarisation with the Stokes
parameters Q, U, and V representing the coordinates of p, the degree of polarisation.
Three of the Stokes parameters can also be expressed in terms of the angles
ψ and χ, shown in the Poincaré sphere:
Q = √ Q 2 + U 2 + V 2 cos 2χ cos 2ψ (3.32)
U = √ Q 2 + U 2 + V 2 cos 2χ sin 2ψ (3.33)
V = √ Q 2 + U 2 + V 2 sin 2χ (3.34)
The angle χ is a measure of the degree of circular polarisation, or the ellipticity
of the polarisation ellipse, while ψ specifies the orientation of the polarisation
15

ellipse and can be found from
tan 2ψ = U Q
(3.35)
Figure 3.2 shows the polarisation ellipse, where ψ specifies the orientation of
the ellipse, and a and b are the two major axes of the ellipse.
y
b
a
2Fy,0
ψ
x
2F x,0
Figure 3.2: The polarisation ellipse.
If we consider a completely polarised state, linear polarisation is represented
by points on the equator of the Poincaré sphere, circular polarisation
by the poles of the sphere, where the north pole represents left-hand circular
polarisation and the south pole right-hand circular polariation. All other points
on the Poincaré sphere represent elliptical polarisation with left-hand polarisation
on the northern hemisphere and right-hand on the southern. Figure 3.3
shows a 2D projection of the polarisation on the Poincaré sphere.
3.3 The coherency tensor
The Stokes parameters can also be obtained from the coherency tensor, constructed
from the wave field F(t, x) in Equation (3.25). It is defines as
(
)
J = 〈FF † 〈Fx,0 2 〉 =
〉 〈F x,0F y,0 e −iδ 〉
〈F x,0 F y,0 e iδ 〉 〈Fy,0 2 〉 (3.36)
where † symbolises Hermitean conjugate and δ = δ y − δ x . Using the definitions
of Stokes parameters in Equations (3.26)–(3.29), the coherency tensor
16

2χ
2ψ
0 ◦
−90 ◦ 0 ◦
90 ◦ 180 ◦
−180 ◦ 90 ◦
−90 ◦
Figure 3.3: A 2D projection of the polarisation on the Poincaré sphere. Points on the
equator represent linear polarisation, while the two poles represent circular polarisation.
Points between the poles and the equator have elliptical polarisation. Points in
the upper hemisphere are left-hand polarised and in the lower right-hand polarised.
The orientation of the polarisation ellipse is specified by the angle 2ψ. After one turn
around the equator of the Poincaré sphere, the polarisation ellipse has only turned
180 ◦ , but the state of polarisation is back where it started.
can be expressed in terms of the Stokes parameters as
(
)
J = 1 I + Q U − iV
2 U + iV I − Q
(3.37)
The total intensity I is equal to the trace of the coherency tensor, I = Tr(J).
The coherency tensor has some interesting properties. In 1930, N. Wiener
[33] showed that the unit tensor and the three Pauli spin matrices form a base
in which the coherency tensor can be expanded. U. Fano [11] later showed that
the coefficients in this expansion are the Stokes parameters. The expansion is
J = 1 2 (I1 2 + Uσ 1 + V σ 2 + Qσ 3 ) (3.38)
[ ( ) ( ) ( ) ( )]
= 1 1 0 1 0
0 1 0 −i
I + Q
+ U + V
2 0 1 0 −1 1 0 i 0
17

3.4 The spectral tensor
The Stokes parameters can also be defined in the frequency domain. Let
˜F(ω, x) denote the Fourier transform of F(t, x). Then the spectral density
tensor is
S d = ˜F˜F † =
( )
˜Fx ˜F
∗
x
˜Fx ˜F
∗
y
˜F y ˜F
∗
x
˜Fy ˜F
∗
y
(3.39)
When the spectral density tensor is used the fields do not have to be quasimonochromatic.
We can define the spectral Stokes parameters I, Q, U, and
V, which are analogous to the ordinary Stokes parameters.
If the wave field has a small bandwidth ∆ω, we can integrate over that
bandwidth by applying the bandwidth operator (Paper I):
¯B =
∫ ¯ω+∆ω/2
¯ω−∆ω/2
dω (3.40)
This operator integrates over the small bandwidth ∆ω, centered around the
mean frequency ¯ω. Applying the bandwidth operator to the spectral density
tensor gives the spectral tensor:
S = ¯BS d =
∫ ¯ω+∆ω/2
¯ω−∆ω/2
S d dω (3.41)
3.5 Wave polarisation in 3D
The Stokes parameters were introduced to describe the polarisation of light. In
optics, the location of the source is known because it can easily be observed
and therefore only two field components in the polarisation plane are necessary
to describe the transverse electromagnetic wave field. If, on the other
hand, the location of the source is unknown, as for most radio waves, we must
consider all three field components.
Using the Fourier transform of a 3D wave field, ˜F(ω, x), the spectral density
tensor becomes
⎛
S d = ˜F˜F † =
⎜
⎝
⎞
˜F x ˜F
∗
x
˜Fx ˜F
∗
y
˜Fx ˜F
∗
z
˜F y ˜F
∗
x
˜Fy ˜F
∗
y
˜Fy ˜F
∗ ⎟
z ⎠ (3.42)
˜F z ˜F
∗
x
˜Fz ˜F
∗
y
˜Fz ˜F
∗
z
In Paper I, we have noted that the Pauli spin matrices, which were used
in the expansion of the coherency tensor in Equation (3.38) are generators of
the special unitary group SU(2). For 3D, the natural choice is to expand the
18

spectral density tensor in the generators of the symmetry group SU(3). This
has been done in Paper I and the unit tensor and the eight generators described
by Gell-Mann and Ne’eman [20] were used. The coefficients in the expansion
are nine generalised polarisation parameters. If the generalised polarisation
parameters are denoted by Λ i , the unit tensor in 3D by 1 3 , and the generators
of the SU(3) group by λ i , the spectral density tensor becomes
S d = 1 3 1 3 + 1 8∑
Λ i λ i (3.43)
2
i=1
where the normalisation 1/3 has been chosen to give the spectral intensity I
the same meaning as in the 2D case.
The explicit expression of the spectral density tensor is presented in Paper
I, where some interesting properties of the spectral density tensor also is
described. One such property is the pseudo vector V, formed from the antisymmetric
part of the spectral density tensor:
⎛
⎞
−Im{ ˜F y ˜F
∗
z }
⎜
V = 2 ⎝ Im{ ˜F x ˜F
∗
z }
−Im{ ˜F x ˜F
∗
y }
⎟
⎠ (3.44)
This pseudo vector is normal to the plane of the polarisation ellipse. The
pseudo vector V is also parallel to the vector described by Means [22]. The
magnitude of V is coupled to the spectral Stokes parameter V by the relation
|V| = |V| (3.45)
A complex electromagnetic wave field can be fully described in terms of
three amplitudes and three phases, so only at most six of the generalised polarisation
parameters can be independent. We also note that the spectral density
tensor does not change if an arbitrary phase is added to all three field components.
Therefore, only five of the nine generalised polarisation parameters can
be independent.
19

4. Electromagnetic field measurements
In this chapter, a method and a system for vector measurements of electromagnetic
field are described. Applications are considered in Section 4.3 for the HF
band (3-30 MHz), in Section 4.4 for a radio telescope, and in Section 4.5 for
wireless telecommunication in the microwave regime. Space applications are
presented in the next chapter.
4.1 Method for estimating polarisation
parameters
A method for estimation of polarisation parameters is described in Paper I,
Paper II, and Paper V. The method requires knowledge of an electromagnetic
vector field, such as E or B, and is based on the spectral density tensor. Expressed
in terms of the Fourier transform Ẽ of the electric field, the spectral
density tensor is
⎛
⎞
S d = ẼẼ† =
⎜
⎝
Ẽ x Ẽ ∗ x
Ẽ y Ẽ ∗ x
Ẽ x Ẽ ∗ y
Ẽ y Ẽ ∗ y
Ẽ x Ẽ ∗ z
Ẽ y Ẽ ∗ z
⎟
⎠ (4.1)
Ẽ z Ẽ ∗ x
Ẽ z Ẽ ∗ y
Ẽ z Ẽ ∗ z
The spectral density tensor contains five independent polarisation parameters,
which can be used to extract information from the wave field. The first
of these is the spectral intensity I, which is equal to the trace of the spectral
density tensor:
I = Tr(S d ) = |Ẽx| 2 + |Ẽy| 2 + |Ẽz| 2 (4.2)
The next thing to consider is the pseudo vector V, which is obtained from
the antisymmetric (imaginary) part of the spectral density tensor:
{
}
V = (V x , V y , V z ) = −2 Im ˆxẼyẼz − ŷẼxẼz + ẑẼxẼy (4.3)
This pseudo vector contains three independent parameters. Two of them specifies
the direction of the surface normal of the polarisation ellipse. Using simple
vector analysis, they can be expressed as the polar and azimuthal angle in
21

a sperical coordinate system:
cos θ = V y
V x
(4.4)
tan φ = V z
|V|
(4.5)
The third independent parameter contained in V is the degree of circular polarisation,
defined by
r c = V I
(4.6)
where the magnitude of V is given by Equation (3.45). Unless some additional
information about the wave field is available, the sign of V cannot be determined.
This means that there is an 180 ◦ ambiguity in the determination of the
surface normal of the polarisation ellipse. The pseudo vector V points in the
direction given by a right-hand polarised wave field, but the measured wave
could equally well be a left-hand polarised wave propagating in the opposite
direction.
There are a number of ways to solve the ambiguity. For example, if the signal
appear to come from a direction below ground, it is likely that it comes
from the opposite direction above ground. Apart from measuring, in our case,
three electric field components, knowledge of one of the magnetic field components
will in most cases give the answer. A better method is, of course, to
measure both the electric and magnetic fields, E and B, and then calculate
the Poynting vector N = 1
µ 0
E × B, which points in the direction of wave
propagation. Another method is to use several vector field antennas to form
an array.
So far, we have found four independent parameters. The fifth is the tilt angle
of the polarisation ellipse, given by Equation (3.35). From that equation, it is
clear that the tilt angle is related to linear polarisation by the spectral Stokes
parameters Q and U. Instead of expressing the fifth independent parameter as
the tilt angle, the total degree of polarisation, defined in Equation (3.31) can
be used.
Since the method for direction-finding is built on the normal vector of the
polarisation ellipse, the accuracy of the method scales with the degree of circular
polarisation. For linear polarisation, the ellipse has collapsed to a line
and the normal to that line can lie anywhere in a plane perpendicular to the
line. Reflections also affect the accuracy of the method, specially for the polar
angle, or the elevation angle. The azimuthal angle is not so sensitive to
reflections. A discussion about reflections is given in Section 2.2 of Paper V.
22

4.2 System for estimating polarisation
parameters
A system for estimation of polarisation parameters is described in Paper II.
The system is general in the sense that it can operate at any radio frequency
and can use any means for measuring an electromagnetic vector field. The system
outputs polarisation parameters such as the direction of arrival or the direction
of wave propagation, the intensity of the wave field at any frequency in
the band of operation, the degree of circular polarisation, and so on. The means
used for receiving the vector electormagnetic field is not specified. Any kind
of antennas can be used for extracting the electric or magnetic vector fields, or
both of them. Since an orthogonalisation can be applied to the received vector
fields, the antennas do not have to be orthogonal either.
In order to measure the electromagnetic fields for a particular frequency
band, suitable antennas must be used. In Figure 4.1, two antenna configurations
for measuring the electric or magnetic vector field is presented. In a)
three orthogonal electric antennas are shown and in b) three orthogonal magnetic
antennas. In c) the antennas in b) has been assembled to form a tri-axial
antenna, or a magnetic vector field sensor. Figure 4.2 shows two orientations
of the same configuration of electric vector field antennas. In the first case,
the coordinate system becomes simple, but the reflections from ground will
be different for vertical and horisontal antennas. In the second case, all antennas
are oriented with the same angle to ground and the electric fields will
experience the same reflection coefficients for all three antennas.
4.3 The high frequency band
Electromagnetic wave propagation in the high frequency (HF) band (3-30
MHz) and below is complex because of reflections from the ionosphere. The
ionosphere sometimes behaves like a perfect mirror and sometimes it is completely
irregular. In the first case, an electromagnetic wave is reflected from
the ionosphere mainly from one point, making it possible to obtain a good estimate
of the angle of arrival and the polarisation of the wave. When the ionosphere
is irregular, the received waves has been reflected from many points of
the ionosphere, making the direction of arrival fluctuate. In this second case,
averaging over time is required for an accurate estimate.
The accuracy of direction-finding systems in the HF-band are dependent on
the state of the ionosphere, the amount of reflections from ground and other
objects near the receiving antenna, and the equipment used to measure the
wave field.
Paper IV presents a vector field receiver developed at the Swedish Institute
23

z
x
y
x
z
y
y
x
z
(a)
(b)
Figure 4.1: Antenna configurations, which can be used to measure the vector fields.
In a) three orthogonal electric dipole antennas with pre-amplifiers are shown. These
antennas will be mounted with a common centre. In b) three orthogonal loops with
pre-amplifiers are displayed and in c) the three loops has been put together with a
common origin.
(c)
Figure 4.2: Electric vector field sensors with two different orientations of the antennas
with respect to ground. The first one has been used for field test and the second is used
for the LOIS test station, see Section 4.4.
24

of Space Physics in Uppsala, Sweden, see Figure 4.3. It is a fully digital receiver
for the frequency band 0–20 MHz and it samples the antenna signals directly
after the anti-aliasing filers and amplifiers. Digital mixers down-convert
the signals to complex base band in-phase (I) and quadrature (Q). The vector
receiver outputs a digital stream with 16+16 bits resolution and a bandwidth
between 1.5 kHz and 82 kHz. The vector receiver can be controlled remotely
and data can be streamed over the internet to the user.
DIGITAL MIXER CHIP
I x
Input
x channel
cos
High decimation
(LP) filter
FIR (LP) filter
Preamp
Anti−aliasing
(LP/BP) filter
Diff amplifier
10 dB gain
010
ADC
Phase/frequency
generator
High decimation
sin (LP) filter
Scale
Factor
(digital
gain)
FIR (LP) filter
Qx
DIGITAL MIXER CHIP
I y
Control
Input
y channel
cos
Unit
010
Scale
Factor
DSP
LAN
sin
Qy
FPGA
PC
DIGITAL MIXER CHIP
Iz
Input
z channel
cos
Scale
010
Factor
sin
Qz
Figure 4.3: Block diagram of a three-channel digital vector field receiver. The antennas
are external and can be exchanged. The three inputs of the instrument are marked with
arrows.
4.4 LOIS—the LOFAR Outrigger In
Scandinavia
The Low frequency array (LOFAR) will be built in the Netherlands, see Figure
4.4. It is the first of a new generation radio telescopes, which instead of
being composed of a small number of huge radio telescopes, will consist of
25

small and simple antennas located over an area covering the Netherlands and a
part of northern Germany. In total, 15,000 crossed 2 m long V-shaped electric
dipole antennas will be used. The sensitivity will be 100 times better than the
present radio telecopes and the amount of collected data will require computer
power of tens of Tera FLOPS.
LOFAR will be sensitive to see objects very far away and thereby it will
also see very far back in time, back to the early universe. LOFAR will also be
used to observe the sun, the solar wind, and the highest energy cosmic rays. Its
data transport network and computing facilities will make it possible to also
host sensors for geoscience, bioscience and weather research.
Figure 4.4: The locations of the radio telescopes LOFAR and LOIS.
LOIS is an extension to the Low frequency array. The LOIS will be located
in the southern Sweden with Växjö as a hub, Figure 4.4. It will contribute to
LOIS with up to a few thousands simple electric dipole antennas. These antennas
will be grouped three and three to form 3D vector sensors, as shown in the
right-hand part of Figure 4.2. LOIS will also contribute with a radar for active
experiments. The first step towards building LOIS has been to set up a test station
in Risinge outside of Växjö, see Figure 4.5. The test station is currently
26

housing five of the vector receivers described in Section 4.3. These receivers
are running continously and on request streaming data over the internet to the
user. Software for accessing these vector receivers can be downloaded from
(http://www.lois-space.net/software.html).
Figure 4.5: LOIS test station in Risinge outside of Växjö, Sweden. The figure only
show four tri-axial antennas, but a fifth has been installed since the photo was taken.
Photo by Jan Bergman.
4.5 Wireless telecommunication
For long time, polarisation has been underestimated in telecommunication.
Only two kinds of polarisations have been considered for base station antennas:
vertical and horisontal. Actually, a better formulation would be vertical or
horisontal. The reason is that when base station antennas utilise polarisation
diversity, only the strongest of the two components is selected.
The purpose of Paper III and Bergman et al. [6] was to introduce a new
dimension in telecommunication with focus on a full use of the polarisation.
This was achieved by, adding the third polarisation and also making proper
use of all three measured signals.
By measuring all three polarisations, the direction to the mobile terminals
(telephones) can be estimated. Even in a reflective environment, the estimation
of the direction to a mobile terminal could be of importance. Such examples
are, e.g., capability of directing the transmitted power mainly in the direction
of the mobile terminal, putting a null of the receiving antenna field pattern in
the direction of a mobile terminal in order to prevent it from disturbing the
27

communication to another mobile terminal at the same frequency, and higher
accuracy in positioning mobile terminals when information about the direction
to a terminal can be used together with time differences measured by several
base stations.
To measure more than one polarisation in mobile applications has also implications
on the received power. If two polarisations were measured instead
of only one, the received signal would, on the average, be twice as strong or
3 dB stronger. By also measuring the third polarisation, the received signal is
three times stronger, or 4.8 dB stronger. This means that weaker signals can
be detected and the signal to noise ratio increases.
The importance of polarisation diversity was demonstrated by a research
team from Bell Labs, which transmitted a Miro painting in colour over three
channels of orthogonal polarisation [3]. They demonstrated that the capacity
of a wireless transmission can be trippeled by using three orhogonal polarisations
of the electric field, or even incresed six times by using the three
polarisations of the magnetic field as well.
In an experiment performed by Redsnake Radio Technology 1 , Stockholm,
three parallel vertically oriented dipoles were compared with three orthogonal
dipoles, see Figure 4.6. The measurements were performed at 2.45 GHz in a
reflective indoor environment. The result was that the number of antennas determines
the capacity independently of how they are oriented. Three mutually
perpendicular antennas give the same capacity as three parallel antennas [1].
1 Redsnake Radio Technology AB was founded by Jan Bergman, Tobia Carozzi, Anders Engström,
Anders Eriksson, Roger Karlsson, and Johan Warnström in 2000. The business idea
was to improve the use of bandwidth in cellular systems by making use of the last information
carrying dimension, the polarisation. In 2002, the company was closed down.
28

Figure 4.6: The antennas used in the capacity measurements performed by Redsnake
Radio Technology AB [1]. The antenna, placed in the stand on the left-hand side, was
used to transmit three orthogonal polarisations. The three antennas on the right-hand
side were used as receiving antennas. Either the three vertical elements were used
for the antenna array with equal polarisations, or the antenna with three orthogonal
polarisations were used alone.
29

5. Space applications
In this chapter, the conditions for antennas in space will be considered, followed
by a presentation of satellite projects for which the vector field sensor
has been selected as payload.
5.1 Antennas in space
The conditions for measuring electromagnetic fields with satellites differ in
many ways from ground-based measurements. In space there is no ground or
other bodies in the vicinity of the spacecraft and the field pattern is relatively
undisturbed and free from reflections. The only disturbance is the structure
of the spacecraft itself. The importance of this effect depends on the size of
the spacecraft in comparison to the antennas. If the body of a spacecraft is
large, i.e., larger than λ/2, the spacecraft has the same function as the ground
for antennas placed above the Earth. For a spacecraft much smaller than the
wavelength, i.e.,

electric antennas are not a source of instability. Long antennas can still be a
problem because rapid change of the attitude or the orbit can cause them to
start oscillating, which in turn might disturb the stability of the spacecraft.
In general, antennas for measuring the magnetic field can be made much
smaller than electric antennas and they do not either require a cumbersome
deploy mechanism as is the case for most electric antennas. Because of the
small size of the magnetic field antennas, all three field components can relatively
easy be measured. Consequently, measurements of the magnetic field in
space do not suffer from the same restrictions as electric field measurements.
5.2 Satellite projects
Instruments built at the Swedish Institute of Space Physics in Uppsala (IRF-
U) for measuring the vector fields in the HF band have already been selected
as payloads. It is also under consideration for a number of other satellites. All
projects are listed below.
Obstanovka-1
The vector field sensor has been selected for the Radio Frequency Analyzer
(RFA), which is a part of the Obstanovka-1 instrument package [16] onboard
the International Space Station (ISS). The Space Research Institute (IKI) in
Moscow, Russia, is responsible for Obstanovka-1, which consists of seven
different instruments and is planned to be launched 2006/2007. Obstanovka is
Russian for environment and the instruments are going to monitor the plasma
environment of ISS.
The Obstanovka-1 instruments will be mounted on the Russian part of ISS.
Figure 5.1 shows ISS and the Russian modules are closest to the observer. On
the nearest of the thicker modules, the instruments will be mounted on the
side facing the Earth, as illustrated in Figure 5.2. An explanation of what the
instrument acronymes in Figure 5.2 represent is given in Table 5.1.
The RFA will measure the electromagnetic fields up to 18 MHz and is being
built by the Space Research Centre (CBK) of the Polish academy of sciences,
Warsaw, Poland, and IRF-U. The main unit consists of two parts, a
wave recorder (WRC) from Poland and a vector receiver (VRX) built by IRF-
U. The WRC has two receiver channels and samples the signals directly to
the memory. The three-channel VRX can perform frequency sweeps and data
reduction by digital down-conversion to base band.
An illustration of the RFA antennas is shown in Figure 5.3. These antennas
is under construction by CBK and will measure the vector electric and
32

Figure 5.1: The international space station (ISS). The Russian modules (RS ISS) are
closest to the observer. Photo from NASA.
Figure 5.2: A sketch of how the Obstanovka-1 instrument packages CWD-1 and
CWD-2 are going to be mounted on the second closest ISS module in Figure5.1.
The RFA antennas are shown in the lower right corner of the figure and the three
crossed orthogonal electric dipole antennas are labeled AD and the three orthogonal
loop antennas AM. Figure from IKI.
33

Table 5.1: Obstanovka-1 scientific instrument packages CWD1-1 and CWD1-2
Unit Name Country
LP1,-2 Langmuir probe Bulgaria
CWS-1,-2 Combined wave sensor Ukraine
DFM-1 Flux gate magnetometer Russia
DFM-2 Flux gate magnetometer Ukraine
RFA Radio frequency analyser Poland, Sweden
CORES Correlating electron spectrograph UK
DP-1,-2 Spacecraft potential monitor Bulgaria
SAS3 Signal analyser and sampler Hungary
magnetic fields. A switch inside of the main unit allows selection of any combination
of the six field components to the five receivers.
Figure 5.3: The RFA antennas with the three crossed dipoles to the left and the three
orthogonal loops to the right. The length of the electric dipoles is 1 m. Figure from
CBK.
COMPASS-2
A one-channel version of the Obstanovka-1 Radio frequency analyser has
been selected as payload onboard the Russian microsatellite COMPASS-2 [2].
COMPASS-2 is a disaster warning satellite designed to monitor and warn for
man-made and natural disasters, such as earthquakes, and to study the electrodynamic
coupling between the atmosphere, ionosphere and magnetosphere.
COMPASS-2 has been built by the Institute of Terrestial Magnetism,
Ionosphere and Radio Wave Propagation of the Russian Academy of Science
(IZMIRAN), Russia. The mass of COMPASS-2 is 85 kg and it will be
34

launched on 15 September from a Russian submarine into a circular orbit
of 400 km altitude and 79 ◦ inclination. Figure 5.4 shows the COMPASS-2
microsatellite.
Figure 5.4: COMPASS-2 microsatellite. Figure from IZMIRAN.
Microlink-1
Microlink-1, Figure 5.5, is a Swedish nano satellite which will be built
by the Ångström Aerospace Corporation (ÅAC) in Uppsala. The purpose
of Microlink-1 is to demonstrate advanced multifunctional micro- and
nanotechnology for space applications. With advanced technology is
meant that traditional aluminium structure will be replaced by multichip
modules built in silicon and containing most of the electronics and new
micropropulsion systems. The payload will be minaturised and sensors will
be developed with new technology and lightweight materials.
Microlink-1 has a mass of 10 kg and in its deployed configuration, excluding
the antennas, it has the size of approximately 100×100×10 cm 3 .
Microlink-1 is planned to carry in total four scientific instruments, listed in
Table 5.2.
Table 5.2: Microlink-1 scientific instruments. The EFVS antennas has been extracted
25 cm of maximum 100 cm.
Instrument Measures Range
Electric field vector sensor 3D electric field Few kHz–20 MHz
Langmuir probe Density and temp. DC–10 kHz
Anisotropic magnetoresistive mag. 3D magn. field DC–10 kHz
Flux gate magnetometer 3D magn. field DC–100 Hz
The Electric field vector sensor (EFVS), the Langmuir probe, and the
Anisotropic magnetoresistive magnetometer will all be developed by IRF-U,
while the Flux gate magnetometer will be built by Alfvén lab at the Royal
35

Institute of Technology in Stockholm. The electric field antennas of the EFVS
and the Langmuir probe booms are placed symmetrically at the corners of the
solar panels and span all three spatial dimensions. This antenna configuration
keeps the moment of inertia concentrated to the spin plane.
The time schedule depends on the financing and according to the present
situation, launch is planned 2007/2008.
Figure 5.5: Microlink-1. Figure from Ångström Space Technology Center (ÅSTC).
Future satellite projects
Apart from the projects mentioned above, which all are at advanced stages,
there are a number of projects possible for the future. The first of these is
Canopus-Volcano, which is a follow-up to COMPASS-2 with launch 2006.
Another project is Chandrayaan-1 which is an Indian moon mission. IRF-U
has together with an Indian research institute and an Indian university, both in
Pune, submitted a proposal for the Electromagnetic vector information sensor
(ELVIS). ELVIS measures the vector electric field from a few kHz up to
18 MHz and one component of the magnetic field from a few Hz up to 100
kHz. A transient detection mode is also implemented for electric fields up to
200 MHz. The Indian space board, ISRO, has not yet come to decision about
ELVIS.
Discussions are also taking place with the Taiwanese space board about an
instrument for a future Taiwanese satellite planned to be launched in 2008.
Additional contacts have also been made for satellite projects in Turkey, Mexico,
and Russia. IRF-U are also participating in a working group for development
of infrastructure and instruments for a future radio observatory on the
Moon (2020–2030).
36

6. Whistler analysis using polarisation
Whistler waves are circularly polarised electromagnetic waves propagating
along the geomagnetic field in the magnetosphere. They are generated by
lightning which forms a broadband electromagnetic pulse. The pulse starts to
propagate along the geomagnetic field. When the wave reaches the conjugate
ionosphere, it is reflected and propagates back again, see Figure 6.1. The magnetospheric
plasma is dispersive causing high frequencies to propagate faster
than low frequencies. The whistlers therefore aquire a typical time-frequency
signature—they sound like a whistle.
Figure 6.1: A whistler wave is generated by lightning (A). The whistler propagates
upward along the geomagnetic field, is reflected at the conjugate ionosphere (B) and
propagates back to A. Whistlers can be reflected many times between the hemispheres.
From Krall and Trivelpiece [17].
In Paper VI, a study of a whistler is presented. The whistler was measured
by the Swedish-German Freja satellite [19] with three AC magnetic field components
b(t), sampled at 4096 samples/s. Three components of the DC magnetic
field B 0 were measured simultaneously. The data was taken on June 23,
1993, at 21:49:29.2 UT, at an altitude of 1402 km and magnetic local time
15:18 and invariant latitude 49 ◦ .
A method for estimating the instantaneous polarisation properties of the
whistler has been used. From b(t), the windowed Fourier transform ˜b was
calculated using a Gaussian window of length 256 samples and a with full-
37

width half-maximum of 29.95 samples, and an overlap of 255 samples. A
spectrogram of the whistler is shown in Figure 6.2. The figure displays the
total spectral intensity, which is the sum of the intensity of the three field
components.
2
1.8
1.6
Frequency, [kHz]
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.05 0.1 0.15 0.2 0.25 0.3
Time τ, [s]
−6 −5 −4 −3 −2 −1
Spectral intensity, log (|b| 2 ) [nT 2 /Hz]
10
Figure 6.2: Spectrogram showing the total intensity of the whistler. Note how the
wave frequency varies with time due to dispersion in the magnetosphere. Figure from
Paper VI.
All combinations of auto- and cross-spectra are obtained by forming the
spectral density tensor in Equation (3.42) and Equation (2) in Paper VI. From
the spectral density tensor we obtain the normal to the polarisation plane
}
V = 2 Im
{−ˆx˜b y˜b∗ z + ŷ˜b x˜b∗ z − ẑ˜b x˜b∗ y (6.1)
This (pseudo) vector is parallel (anti-parallel) to the wave vector k for a righthand
(left-hand) polarised wave field. The magnitude of V is, according to
Equation (3.45), equal to the magnitude of the spectral Stokes parameter V.
The whistler is known to be circularly polarised and can be separated from
the background by its high degree of circular polarisation. We denote the normal
to the whistler’s polarisation plane V w .
A comparison between the geomagnetic field B 0 and V w shows that they
are approximately anti-parallel. The sign of the scalar product V w · B 0 gives
the sign of the spectral Stokes parameter V. The sign is found to be negative
and the whistler is therefore right-hand polarised and must be propagating
38

antiparallel to B 0 .
The normal to the polarisation plane deviates, on average, from the direction
antiparallel to B 0 with 6 ◦ and precesses in total four turns around the direction
antiparallel to B 0 . The precession frequency is estimated to about 20 Hz,
which is 10–100 times lower than the wave frequency. Figure 6.3 illustrates
the precession.
Three more whistlers have been studies and they also show precession of
the normal to the polarisation plane, although not as clear as in this example.
Theory can not give an unambiguous explanation to the observed precession.
Further studies are needed to see if the precession is a general feature of
whistlers. Three-dimensional ray tracing studies may also provide an answer
to how common the precession is.
Precession at 20 Hz
V w
b(t)
Wave
frequency
B 0
Polarisation plane
Figure 6.3: The whistler is propagating antiparallel to B 0 . The field vector b(t)
sweeps out the polarisation ellipse, which defines the plane of polarisation. The normal
of the polarisation plane, V w precesses around the direction antiparallel to B 0 at
a frequency of 20 Hz.
39

7. Statistical approach to stimulated
electromagnetic emissions
Transmitting a powerful radio wave vertically into the ionosphere gives rise
to secondary HF radiation from the ionosphere. The radiation has repeatable
characteristics and the power spectral density (PSD) is weak, asymmetric, and
has well structured sidebands around the injected frequency [30]. The secondary
radiation has been given the name stimulated electromagnetic emissions
(SEE). A comprehensive description of SEE is given in [18].
7.1 Characteristics and models for SEE
A power spectrum from the Sura ionospheric HF pumping facility close to
Vasil’sursk, Russia is shown in Figure 7.1. The spectrum displays SEE features
in a frequency band of ±120 kHz from the pump frequency of 6760
kHz. From right, the features are broad up-shifted maximum (BUM), upshifted
maximum (UM), narrow continuum (NC), down-shifted maximum
(DM), and second down-shifted maximum (2DM). The figure also shows a
structure which is displayed at the same frequency as where the broad downshifted
maximum (BDM) usually appear. This structure we denote tentative
BDM. The signals identified as interferences are marked by shading.
Most models of SEE are based on parametic processes, which are phase
coherent [31]. These processes also conserve energy and momentum. For a
three-wave process, the conservation of energy and momentum implies that
ω mother = ω daughter1 + ω daughter2 (7.1)
k mother = k daughter1 + k daughter2 (7.2)
Letting φ denote the phase of a wave, we have [32]
ω = − ∂φ
∂t
(7.3)
k = ∇φ (7.4)
which means that the total phase also is conserved for the interaction:
φ mother = φ daughter1 + φ daughter2 (7.5)
41

−20
Interference
−40
PUMP
Power [dBm]
−60
−80
Tentative BDM
2DM
DM
NC
UM
BUM
−100
−120
−100 −50 0 50 100
Frequency shift relative to the pump [kHz]
Figure 7.1: The SEE spectrum with the visible SEE components indicated. Interferences
from other radio sources are marked by shading. The data was taken at the Sura
ionospheric HF pumping facility, Russia at 15:46LT on September 22, 1998. From
Paper VII.
Since the pump wave is monochromatic and very phase coherent, the decay
products should also are phase coherent. From the PSD, it is impossible to say
if the SEE features are caused by coherent signals.
7.2 Statistical tests
A statistical investigation of the spectral SEE amplitudes and phases should
reveil the nature of the SEE characteristics. Such a study has been made in
Paper VII.
The opposite of a coherent signal is coloured narrow band Gaussian noise
which has Rayleigh distributed [5] amplitudes and uniformly distributed
phases. If the noise contains a sinusoidal, the amplitudes are instead Rice
distributed [27].
The stochastic/deterministic nature of the SEE features are investigated
with statistical tests of the null hypothesis:
Null Hypothesis (H 0 ) SEE is coloured complex Gaussian noise.
If SEE is generated by coherent processes, the null hypothesis will be
rejected. The following tests have been made:
42

Test 1 (T 1 ) χ 2 test to determine if the spectral component amplitudes are
Rayleigh distributed.
Test 2 (T 2 ) χ 2 test to determine if the spectral component phases are uniformly
distributed.
If no deviation from the Rayleigh distribution and the uniform distribution
can be found, we cannot reject H 0 and the SEE spectral components cannot
be distinguished from noise. If, on the other hand, any of the two tests show
significant deviation from noise, we perform a third test for sinusoidal components,
which is a sign of coherency.
Test 3 (T 3 ) Test for sinusoidal components.
In order to obtain estimates of the spectral amplitudes and phases, the data
was first divided into segments of 1024 samples. A windowed Fourier transform
was applied to each of these segment, giving in total 11,718 estimates of
amplitude and phase per frequency bin, or spectral component. The spectral
amplitudes and phases were binned into histograms with linear spacing and
the number of estimates per bin was compared to the predictions according to
the Rayleigh and uniform distributions.
The results of T 1 and T 2 are presented in Figure 7.2. Large values of the
test parameters indicate deviation from coloured Gaussian noise. If the test
parameter is larger than the p value of, say 5%, that spectral component is with
95% certainty not noise. Panel a) shows the result of T 1 and none of the SEE
features, except for the tentative BDM, show any significant deviation from
the amplitude distribution of narrow band Gaussian noise, while the tentative
BDM, the pump, and the interferences deviates significantly from noise. In
panel b), which displays the result of the phase test T 2 , only the pump and
one of the interferences deviate from the phase distribution of narrow band
Gaussian noise. Panel c) is shown to facilitate the identification of the SEE
features.
From the reults of T 1 and T 2 , we conclude that H 0 cannot be rejected for
the SEE features BUM, UM, DM, and 2DM. In other words, they behave
statistically as narrow band Gaussian noise.
For the tentative BDM, the pump and the interferences, it is interesting
to perform also the last test T 3 . The result of this test is shown in Figure 7
in Paper VII, from which we conclude that only the pump and one of the
interferences contain sinusoidals. The tentative BDM can therefore not be a
coherent feature. In fact, the tentative BDM appears to be a pump induced
disturbance and not a real SEE feature.
Since no coherent signatures were found for any of the SEE features, they
cannot be the results of simple parametric processes. The SEE features may
43

χ 2 amplitude test parameter
10 3
10 2
a)
p=10 −4 %
p=0.1%
p=5%
χ 2 phase test parameter
10 3
10 2
b)
p=10 −4 %
p=0.1%
p=5%
−60
Spectrum, power [dBm]
−70
−80
−90
−100
−110
c)
−120 −100 −80 −60 −40 −20 0 20 40 60 80 100 120
Frequency shift [kHz]
Figure 7.2: The result of the amplitude and phase tests. Panel a) shows the T 1 spectral
ampliude test parameter, panel b) the T 2 spectral phase test parameter, and panel c)
the average SEE power spectrum with vertical dotted lines marking the SEE features.
The power spectrum is identical to Figure 7.1. The p values of 5%, 0.1%, and 10 −4 %
are marked with the horizontal dashed lines in panels a) and b). From Paper VII.
still be generated by processes which are coherent on a small scale, but when
integrated over a large region they may give a random result.
44

8. The complex Poynting theorem as a
conservation law
The Poynting theorem is the energy conservation theorem in electrodynamics.
It is of fundamental importance and is described carefully in all books on the
subject, e.g. Jackson [14]. In Section 8.1 will consider the original form of
the Poynting theorem, the instantaneous Poynting theorem and show that it
describes the energy balance in an electrodynamic system. In Section 8.2, the
complex Poynting theorem (CPT) is derived and it is shown that the real part
of the CPT is a balance equation for real energy. The imaginary part of the
CPT is not a balance equation for reactive energy and we explain why.
8.1 The instantaneous Poynting theorem
The instantaneous Poynting theorem will be derived for real fields E(t, x)
and B(t, x), and a real current density J(t, x). We start with the two vector
Maxwell’s equations
∂E
∇ × B =µ 0 J + µ 0 ɛ 0
∂t
∇ × E = − ∂B
∂t
(8.1)
(8.2)
and take the scalar product of E and (8.1) and subtract the scalar product of
B and (8.2). We obtain
E · (∇ × B) − B · (∇ × E) = µ 0 E · J + µ 0 ε 0 E · ∂E
∂t + B · ∂B
∂t
(8.3)
Dividing with µ 0 and rewriting the left-hand side as −∇ · (E × B), gives
(
ε 0 E · ∂E
∂t + 1 B · ∂B )
+ 1 ∇ · (E × B) = −E · J (8.4)
µ 0 ∂t µ 0
Integration over the entire volume containing the sources and using the divergence
theorem results in the instantaneous Poynting theorem
∫ (
1 ∂
ε 0 E · E + 1 )
B · B dV + 1 ∮
∫
(E × B)·ˆnda = − E·JdV
2 ∂t
µ 0 µ 0
V
S
V
45

(8.5)
which states that the energy is conserved within the volome V . The first term
in Equation 8.5 represents the rate of change of the electromagnetic field energy,
the second term the energy flowing out of the volume V , and the right
hand-side the total work done by the fields on the sources.
8.2 The complex Poynting theorem
In this section, the analytic signal will be used and we define the fields and
sources E(t, x), B(t, x), and J(t, x) to be complex. Now we take the scalar
product of E and the complex conjugate of (8.1) and subtract the scalar product
of B ∗ and (8.2) we obtain a complex form of the Poynting theorem:
1
2
(ε 0 E · ∂E∗
∂t
+ 1 µ 0
B ∗ · ∂B
∂t
)
+ 1
2µ 0
∇ · (E × B ∗ ) = − 1 2 E · J∗ (8.6)
The real part of Equation 8.6 is
1 ∂
(ɛ 0 E · E ∗ + 1 )
B · B ∗ + 1 ∇ · (E × B ∗ + E ∗ × B)
4 ∂t
µ 0 4µ 0
= − 1 4 (E · J∗ + J · E ∗ ) (8.7)
The left-hand term in Equation (8.7) has the form of a time derivative of a
conserved quantity, the energy density
u c (t, x) = 1 (ɛ 0 |E| 2 + 1 )
|B| 2 (8.8)
4
µ 0
Since the second term is the Poynting vector in complex form, Equation (8.7)
is a balance equation for real energy density u c .
The imaginary part of Equation (8.6) is
1
4i
[ɛ 0
(E · ∂E∗
∂t
− E ∗ · ∂E
∂t
)
− 1 µ 0
(B · ∂B∗
∂t
− B ∗ · ∂B )]
∂t
+ 1
4iµ 0
∇ · (E × B ∗ − E ∗ × B) = − 1 4i (E · J∗ − J · E ∗ ) (8.9)
The temporal derivative in the term in the big paranthesis cannot be moved out
of the paranthesis and Equation (8.9) can therefore not be written as an energy
balance equation. In Paper VIII, this conclusion is motivated in a stricter form.
By defining the big paranthesis in Equation (8.9) as a temporal derivative of
a quantity U, it is still possible to write Equation (8.6) in the form of a balance
46

equation, which is one form of the complex Poynting theorem:
∂(u c + iU)
∂t
+ ∇ · N c = − 1 2 E · J∗ (8.10)
The conserved quantities are the real energy density u c and U, for the real
and imaginary parts, respectively. It should be noted that U is not the reactive
energy density.
8.3 The time-harmonic complex Poynting
Theorem
If we assume harmonic time variation for the fields and the source, i.e.,
E = E ω e −iωt (8.11)
B = B ω e −iωt (8.12)
J = J ω e −iωt (8.13)
we can show that the time-harmonic CPT can be derived from Equation (8.10).
With the electric and magnetic energy densities defined by
u e = ɛ 0
4 E ω · E ∗ ω (8.14)
u m = 1 B ω · B ∗ ω
4µ 0
(8.15)
the time-harmonic CPT can be written
∫
∫
1
E ω · J ∗ ω
2
d3 x + 2iω (u e − u m )d 3 x + 1 (E ω × B
V
2µ 0
∮S
∗ ω ) · ˆnda = 0
V
This is the form of the CPT found in the standard literature [14, 4].
The conclusion we make is the following. Although the instantaneous
Poynting theorem and the CPT are balance equations for (real) energy, the
imaginary part is not a balance equation for reactive energy. So, the CPT is
not a balance equation for electromagnetic energy density.
47

9. Summary in Swedish
I denna avhandling belyses hur polarisationen hos elektromagnetiska vågor
kan användas inom både vetenskap och radioteknologiutveckling. En elektromagnetisk
våg antar olika skepnad beroende på dess våglängd eller frekvens.
Lägst frekvens har radiovågor, sedan kommer med stigande frekvens mikrovågor,
infrarött, synligt ljus, ultraviolett, röntgen och gamma. Med elektromagnetiska
vågor kan man förmedla information över stora avstånd och detta används
framför allt inom radioområdet.
Synligt ljus är den form av elektromagnetisk vågrörelse som är lättast att
föreställa sig. Vi ser ljuset och kan avgöra dess frekvens genom dess färg. Vi
har också förmåga att skilja på olika ljusstyrkor, eller intensitet. Polarisationen
är en viktig egenskap hos en elektromagnetisk våg, men våra ögon kan inte
särskilja ljus med olika polarisation. Vi kan dock förvissa oss om att ljuset
innehar denna egenskap genom att sätta på oss ett par polaroidglasögon och
titta på solljus reflekterat fran en vattenyta eller en blöt väg. En ännu bättre
illustration av ljusets polarisation får man om man tittar genom två polaroidglas
och observerar att ljuset släcks ut helt då glasens polarisationsriktningar
är vinkelräta.
En elektromagnetisk våg är en vektorvåg, som har både storlek och riktning.
Den bär med sig ett elektriskt och ett magnetiskt fält, som båda är vinkelräta
mot utbredningsriktningen och definierar det s k polarisationsplanet. Det är
tidsvariationen av dessa fält i polarisationsplanet som utgör vågens polarisation.
För ljus är det uppenbart varifrån det kommer och detta ger då vågens utbredningsriktning.
Det räcker då med två fältkomponenter för att beskriva vågen
i dess polarisationsplan. Polarisationen beskrivs därför i två dimensioner
och ett praktiskt sätt att göra detta är att använda Stokes parametrar.
Polarisationen har också stor betydelse för radiovågor, men för dessa går
det inte att se varifrån de kommer. För att då ge en generell beskrivning av
polarisationen krävs därför tre fältkomponenter. En generalisering av Stokes
parametrar till tre dimensioner är därför önskvärd och det är detta denna
avhandling börjar med.
Nedan beskrivs översiktligt de arbeten som denna avhandling bygger på.
49

Arbete I: Parametrar för att karakterisera
elektromagnetisk vågpolarisation
I detta arbete beskrivs hur de vanliga tvådimensionella Stokes parametrarna
kan generaliseras till tre dimensioner och hur elektromagnetiska fält kan
karakteriseras på alternativa sätt. Ett av dessa karakteristika är polarisationsplanets
normalvektor, som oftast är parallell eller antiparallell med vågens
utbredningsriktning. Ytterligare två är de spektrala Stokes parametrarna som
förknippas med intensiteten och graden av cirkulärpolarisation. Arbetet
baseras på spektraltäthetstensorn, som utvecklas i en bas med nya generella
polarisationsparametrar som koefficienter. I två dimensioner är denna bas
Pauli-spinnmatriserna, som är generatorer till SU(2) gruppen, och koefficienterna
är Stokes parametrar. I tre dimensioner används på liknande sätt en bas
bestående av generatorerna till symmertigruppen SU(3). Koefficienterna i
denna utveckling är nya generella spektrala polarisationsparametrar. De nya
polarisationsparametrarna är härledda i frekvensdomänen, vilket möjliggör
tillämping på bredbandiga elektromagnetiska vågor.
Arbete II: Metod och system för att
erhålla riktning för en elliptiskt polariserad
elektromagnetisk vågutbredning
I detta patent presenteras en metod och ett system för att bestämma riktningen
till en radiokälla och dess polarisationsparametrar, såsom den spektrala intensiteten
och den spektrala graden av cirkulärpolarisering. Minst tre antenner
används för att ta emot det elektriska eller magnetiska fältet i minst en punkt
i rummet. Metoden separerar den mottagna signalens frekvenser och ger en
riktning per frekvenskomponent.
Arbete III: Antennanordning för användning
av tredimensionell elektromagnetisk
fältinformation inherent i en radiovåg
Här presenteras ett flertal antennanordningar för att mäta det tredimensionella
elektromagnetiska fältet. Det uppmätta fältet kan sedan bearbetas för att minimera
fädning, minimera interferens mellan mottagarkanalerna, etc. Antennanordningen
möjliggör också styrning av antennens strålningskänslighet i
olika riktningar.
50

Arbete 4: Trekanalig digital vektorfältsensor:
Beskrivning och demonstration
Detta arbete presenterar ett fungerande system för att i HF-området (3-30
MHz) mäta det elektriska eller magnetiska fältet och utifrån detta spektralt
bestämma det mottagna fältets polarisationsparametrar, såsom har beskrivits i
Arbete II. Systemet bygger på helt digital nedblandning av radiofrekvenserna
till basband. Den nedblandade signalen skickas sedan till operatörens dator där
processandet av data sker. Det grafiska användargränssnittet presenteras och i
detta visas alla de beräknade polarisationsparametrarna. Systemet kan också
demodulera den mottagna radiosignalen och gör det därför också möjligt att
lyssna på långvågs-, mellanvågs- och kortvågsradio.
Arbete 5: Nuvarande och framtida
tillämpningar av den informationstäta antennen
I detta papper ges en sammanfattning över teorin som presenterats i Arbete
I och II. Vidare beskrivs en rad tillämpningar av teorin och mätmetoden.
Tillämpningarna innefattar exempelvis riktningsbestämning för både bredbandiga
och icke-stationära signaler, modulationstekniker, signalkombinering
i MIMO (multiple input multiple output) system, riktningsbestämning genom
att använda brusmätningar, bestämning av absolut fas, radar, polarisation som
hjälpmedel för att detektera signaler gömda i brus och användning av ickeplana
vågor för att undgå detektion. Avslutningsvis ges en presentation av
forskningsprojekt som baseras på teorin: HF spektrometer på den internationella
rymdstationen, LOIS radioteleskop i södra Sverige och nanosatelliten
Nanospace-1, som numera heter Microlink-1.
Arbete 6: Precession av normalen till en
visslarvågs polarisationsplan observerat i
Frejadata
Precession hos normalen till en visslarvågs polarisationsplan har observerats
i satellitdata från Freja. Visslarvågor är cirkulärpolariserade och propagerar
i magnetosfären utmed det geomagnetiska fältet. Magnetometerdata samplat
med 4096 sampel per sekund har studerats och en instantan analysmetod har
applicerats på en uppmätt visslare. Metoden gör det möjligt att studera visslarens
instantana polarisationsegenskaper. Således har normalen till visslarens
polarisationsplan studerats och denna är i detta fall approximativt antiparallel
med det geomagnetiska fältet med en avvikelse på i medeltal 6 ◦ . Nor-
51

malen uppvisar en precession kring riktningen motsatt det geomagnetiska fältet.
Normalen precesserar totalt fyra varv och dess precessionsfrekvens är 20
Hz, vilket är väsentligt lägre än visslarens vågfrekvens.
Arbete 7: Statistiska egenskaper hos
jonosfärisk stimulerad elektromagnetisk
emission
En statistisk studie av de spektrala amplituderna och faserna hos stimularad
elektromagnetisk emission (SEE) har genomförts. Studien visar att
SEE-komponenterna BUM, UM, DM och 2DM inte signifikant skiljer sig
från amplitud- och fasfördelningarna för färgat smalbandigt brus. Det data
studien baseras på uppmättes vid Sura jonosfärsmodifieringsanläggning
utanför Vasil’sursk i Ryssland, efter att jonosfären under en tid bestrålats
med en kraftig radiovåg på 6760 kHz. Statistiska test har sedan applicerats
på de sekundära emissionerna från denna bestrålade jonosfär. Resultatet är
inte vad som förväntas enligt teoretiska modeller för SEE, där faskoherenta
parametriska processer ligger som grund. Eftersom den kraftiga utsända
radiovågen är helt koherent borde något av koherensen kvarstå även hos
sönderfallsprodukterna av de parametriska processerna. Den förmodade
förklaringen till att ingen koherens kunde påvisas är att processerna kan
vara koherenta på en liten skala. När man tar hänsyn till hela den bestrålade
volymen summeras bidragen från alla delar och koherensen går förlorad.
Arbete 8: Komplexa Poyntingteoremet som en
konserveringslag
Poyntingteoremet beskriver energibalansen i ett elektrodynamiskt system. Då
reella storheter används för att representera de elektromagnetiska fälten och
dess källor, beskriver Poyntingteoremet bevarandet av reell energi. Om istället
komplexa storheter används för att representera fälten och källorna, erhålls
det komplexa Poyntingteoremet. Dess realdel beskriver, precis som Poyntingteoremet
för reella storheer, bevarande av energi. I standardlitteraturen på
området behandlas inte det generella uttrycket för imaginärdelen av det komplexa
Poyntingteorement. Istället ansätter man ett harmoniskt tidsberoende
hos källor och fält och visar att imaginärdelen är proportionell mot skillnaden
mellan de elektriska och magnetiska energitätheterna. I Arbete VIII ges en
beskrivning av varför imaginärdelen av det komplexa Poyntingteoremet inte
utgör någon bevarelselag.
52

10. Conclusion and Outlook
With the increasing use of the radio spectrum, bandwidth is becoming more
and more important and also more expensive. It is therefore desirable to use
the bandwidth as effectively as possible. In the latest digital cellular telephone
systems, utilising Code division multiple access (CDMA), the usage of bandwidth
is enhanced by making the signal spectrum resemble a Gaussian process
and thereby approaching the theoretical maximum capacity over the radio
channel. In those systems, signals are coded and made to share frequencies
with other signals sent over the same channel. At the receiver, the signals are
identified by their code.
The capacity can be increased further by using Multiple input multiple output
(MIMO) systems, where the signal is transmitted with more than one antenna
and also received with more than one antenna. MIMO systems are in use
today in, e.g., the latest systems for cellular telephone, where spatial diversity
or polarisation diversity is implemented. By using spatial and polarisation diversity
together in MIMO systems, the capacity may increase further. In many
other areas of radio communication, the shortage in bandwidth should eventually
make it necessary to use the polarisation.
A number of new applications should be made possible in science, radar,
communication, and signal intelligence by measuring the full vector electromagnetic
field. In science the use of polarisation as a diagnostic can provide
useful information about the underlying physical processes. This was, for example,
demonstrated in the first polarimetric study of stimulated electromagnetic
emission [9]. In the areas of communication and signal intelligencewe
may see applications such as to use the polarisation to detect signals hidden by
noise (spread spectrum signals); use of non-planar waves to propagate signals
which make direction-finding techniques provide a false direction.
53

Acknowledgement
First of all I would like to thank my supervisor Bo Thidé. I also thank my tutor
Tobia Carozzi for good cooperation and valuable assistance, and Jan Bergman
who has been an invaluable part of our research team.
I am grateful to ALL the people at the Institute for Space Physics and the
Department of Astronomy and Space Physics. Apart from being colleagues,
they are also very good friends. A special thank to those who have supported
me much in my work for this thesis: Walter Puccio, Anders I. Eriksson, Sven-
Erik Jansson, Lars Norin, David Sundkvist, and Lennart Åhlén. I thank also
Bengt Holback for helping me with my personal mechanicar problems.
A thank to Christer Wahlberg, Vladimir Pavlenko, and Bo Thidé for giving
me the possibility to lecture advanced undergraduate courses. That experience
has given me a lot of confidence for the future. I thank Vladimir Pavlenko
also for being a source of inspiration with his kindness and deep knowledge
in physics.
I thank Erkki Brändas, Sven Kullander, Inger Eriksson, and Susanne
Söderberg at the Advanced Instrumentation and Measurement (AIM)
graduate school. I gratefully acknowledge AIM for financial support and
for finding me friends at the other departments of Uppsala university. I am
also greateful to AIM for pushing me to find my external supervisor Erland
Cassel, with whom I have spent many fruitful days together with, talking and
building antennas.
The time working with and for Redsnake Radio Technology AB was very
rewarding and I thank my colleagues Jan Bergman, Tobia Carozzi, Anders
Engström, Anders B. Eriksson, and Johan Warnström.
I am grateful to Mikael Lindholm and colleagues at Scandinova Systems
AB and I thank them for giving me the possibility to finish this thesis.
I have also enjoyed the time outside of work. All my friends are important
to me and I would like to thank specially Mattias Georgson and Urban Simu,
and also the former and present players of IBS Nabla.
A special thank to Paul, Isabel, and Titi. Paul, something fun always happens
when you are around and I will never forget how fun it was to have you
as colleague.
Finally I big thank to my family for all the support I have got through the
years.
55

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