Electromagnetic Waves in Plasma Physics

Electromagnetic waves in plasma physics
Investigating EM wave propagation is one of the most important diagnostics in

laboratory as well as space plasmas.
Dish and dome of the
Metshovi Radio
Observatory
in Kirkkonummi
radioastronomy (Jansky in 1930s):
bremsstrahlung and
cyclotron/synchrotron emissions of
charged particles
radio wave propagation in
and through the Earths
plasma environment
As a dielectric medium, the plasma can interact with EM waves by
-Reflecting
-Refracting, or
-Absorbing the waves
Energy and momentum are transferred between fields and particles leading to:
- Acceleration
- heating
- resonances
As we shall see there is a huge collection of different wave modes in a
magnetized plasma
In the following we will refresh some basic concepts of wave propagation. More
detailed description of different plasma wave modes found in plasmas follows later
during the course (refer also to some basic book of electromagnetic theory!).
EM waves in vacuum
In absence of charges and currents, Maxwells equations reduce to
where we simply denote B =
0
H
Called homogeneous wave equations

Giving the wave Eqs. For the EM field
Solutions of these equations are waves propagating at speed
propagation assuemd to z direction
Solutions of the vacuum wave equations:
1. Plane wave:
0
( , ) ( )
x x
E z t E x ct =
Where E
x0
(z) gives the initial wave form
Plane wave is defined as a wave whose phase is the same, at a given instant,
at all points on each plane perpendicular to some specified direction (e.g. now
E must have same the same phase at all points that have same z-value)
Example: a sinusoidal wave 0
( , ) cos( )
x
E z t E kz t e =
E
0
: amplitude of the wave
e=2tf: angular frequency
k=2t/: wave number
Now c=e/k is the phase speed of the wave
Or in vector form:
where following notations have been used:
phase velocity: the velocity at which
the planes of constant phase move
p k
k
e
= v e
In a dispersive medium, the phase velocity varies with frequency and is not
necessarily the same as the group velocity of the wave
red dot located at a fixed wave phase
group velocity: the velocity at which the envelope of the wave packet moves
( )
g
e = V
k
v k
Group velocity is also a velocity with which energy propagates
2. Spherical wave:
A wave can be considered as a plane wave only far from the source.
Sometimes it is necessary to consider spherical waves, i.e., waves,
for which there exists a spherical surface where E is constant.
An example is the field of a radiating dipole:
(this is an approximation up to terms of the order of 1/r
3
; the exact
solution can be expressed in terms of Bessel functions)
In plasma physics we often find ourselves investigating the ways the wave
propagation varies with the frequency of a wave.
The relationship between the angular frequency e and wave number k is called
a dispersion relation.
Dispersion relations are a central part of plasma physics since they contain the
information about the propagation of a given plasma wave mode
- phase speed v
p
=e/k
- group velocity v
g
=e/k
- frequency range where the wave is able to propagate
- reflection points
- resonance points (frequency at which energy can be transferred to plasma
particles)
- wave growth or damping
Dispersion relation
Lets now find dispersion relation for a plane wave and calculate the phase and
group speed
It is convenient to write down the plane
wave field as complex exponentials:
If E
0
and B
0
are constants, the fluctuating fields are called harmonic and the
differential operations reduce to multiplications:
these will be used often!!
i
i
i
t
e
V
V
c

c
k
k
In general:
and Maxwells equations

become algebraic equations
In this case the phsyical (measurable) field is the real part of its
complex presentation
Assume next, that the wave propagates in a linear medium that is homogenous and
isotropic, so that c and are constant scalars. We also assume that there are no
free currents or charges (J=0 and =0) and that the conductivity o=0.
0
0
c
e
ec
=
=
=
=
k E
k B
k E B
k B E
Lets assume c0 kE =0
Thus, both E and B must be perpendicular to k. such a wave is called transverse.
(A plasma also supports longitudinal (k||E) wave modes)
Since B is proportional to kE, also E and B are also perpendicular to each other
(*)
k(*)
2
( ) e ce = = k k E k B E
(**)
(**)
using vector identity:
2
( ) ( ) k = k k E k E k E
=0 for a transverse wave
2 2
k k ce ce = = E E
i.e. we have found the
dispersion relation!
k ce =
From we can now calculate the group and phase speeds:
1
1
p
g
v
k
v
k
e
c
e
c
= =
c
= =
c
If we have c =c
0
and =
0
(vacuum) v
p
=c and v
g
=c.
If phase speed and group speed are same (like above), there is no dispersion in
medium
Index of refraction n is defined as:
0 0 p
c
n
v
c
c
= =
(for c =c
0
and =
0
n=1)
Waves in a linear medium with finite conductivity
Lets now consider the medium where c, and o are non-zero constants, but =0.
Now Maxwells Eqs with Ohms law give (Exercise):
(a standard class-room example of using Fourier

transformations to solve partial differential equations)
This is called telegraph equation
Instead of making the Fourier transformations, lets start from Maxwells
equations and make the plane wave assumption
k, E and H are all again perpendicular
to each other: transversal wave
Choose the coordinate system so that: k || e
z
, E || e
x
, and H || e
y
The solution is:
Thus, for a real e we have a complex k, that can be given as

Giving a dispersion relation:
| |
i
k k e
o
=
And solved for
(| | cos ) | | sin
0
i k z t k z
x
E E e e
o e o
=
Where sino has to be positive. In this case the wave is damped when it propagates
into the medium that is physically sensible solution (if sino is negative wave grows
exponentially, note the factor ).
| |sin k z
e
o
From the dispersion relation ( ) we now get the
phase speed
Re( ) | | cos
p
v
k k
e e
o
= =
The distance after which the wave has decayed to 1/e of its original amplitude is
called skin depth:
1 1
Im( ) | | sin k k
o
o
= =
Impedance of the medium is defined as:
Examples:
a good conductor o >> ce, whence
2
45 ; ; tan
p
v o o eo o eo
oe
= = = =
For copper
50 Hz 1 cm 3 m/s
50 MHz 1 m 3 km/s
p
p
f v
f v
o
o
= ~ =
= ~ =
An insulator: o =0, c > 0 and =
0
,
o=0 i.e. wave is not damped when it propagates in to the insulator;
0 0
0
Z Z
c
c c
= =
Where is the impedance of free space
0
0
0
376.73 Z

c
= = O
Polarization of waves
An important parameter of electromagnetic waves is also its polarization.
Assume a transverse plane wave propagating along the positive z-direction and
consider the EM fields in plane z=0.

/ | |
i
y x
E E e
o
=
where
x
y
i
x x
i
y y
E E e
E E e
o
o
=
are complex vector
( ) ( ) ( )
( )
i t kz i t kz i i t kz
x x y y x x y y
E e E e E E e e
e e o o e
= + = + E e e e e
Lets choose: ; 0
y x x
o o o o = =
and define a complex number
( ) ( )
0

( )
i kz t i kz t
x x y y
e E E e
e e
= = + E E e e
Electric field is now:
o
x
, o
y
are phases in
the x and y directions
respectively
1) If is real, E
y
and E
x
are in the same phase (but have different amplitudes)
direction of E is (1,,0) (if = , E is in the y-direction).
This is a linearly polarized wave
2) If is a general complex number have different phases and
amplitudes and the wave is elliptically polarized
3) A special case of the elliptical polarization is obtained when i =
In this case there is a phase shift o = t/2 and the amplitudes are equal
i.e. E
y
= E
x
= E
0
( )
0
( )
i t kz
x y
E i e
e
= E e e
3a) i = + is called right-hand circularly polarized wave (positive helicity)
3b)
i = is called left-hand circularly polarized wave (negative helicity)
tip of the electric field
vector moves in a circle

,
x y
E E
Comments on unit systems
1. Check the physical dimension of relations
2. Approximate the quantity of results
At these lectures we use the SI-system (The International System of Units), but
note that many plasma physics books and almost all astrophysics books use the
cgs (centimeter-gram-second) Gaussian unit system.
10
-8
Weber (Wb) Maxwell (Mx) magnetic flux
0.1 Pascal (Pa) barye pressure
4.1868 J calorie heat flux
10
-7
Joule (J) erg energy
10
-4
Tesla (T) Gauss (G) magnetic flux
density
SI equivalent cgs measuring
Usefulness depends on what is measured:
e.g. magnetic field:
at the surface of the Earth:0.3-0.6 G
of the solar wind ~10
-9 T
In plasma physics one should learn how to smoothly deal with electromagnetic
quantities and natural constants. It it often useful and instructive to:
Maxwells equations transform from SI to cgs in the following way:
In plasma physics we
often give temperature
in eV: 1 eV 11600 K
eV: the amount of energy gained by an
electron when it accelerates through an
electrostatic potential difference of one volt

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Electromagnetic Waves in Plasma Physics

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Electromagnetic waves in plasma physics

Investigating EM wave propagation is one of the most important diagnostics in

Called homogeneous wave equations

and Maxwells equations

(a standard class-room example of using Fourier

The solution is:

Thus, for a real e we have a complex k, that can be given as

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