Electromagnetic waves in plasma physics
Investigating EM wave propagation is one of the most important diagnostics in
laboratory as well as space plasmas.
radio wave propagation in
and through the Earth’s
plasma environment
radioastronomy (Jansky in 1930s):
bremsstrahlung and
cyclotron/synchrotron emissions of
charged particles
Dish and dome of the
Metsähovi Radio
Observatory
in Kirkkonummi
As we shall see there is a huge collection of different wave modes in a
magnetized plasma
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
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, Maxwell’s equations reduce to
where we simply denote B =
0H
Giving the wave Eqs. For the EM field
Called homogeneous wave equations
Solutions of these equations are waves propagating at speed
Solutions of the vacuum wave equations:
1. Plane wave:
Ex ( z, t )
propagation assuemd to ±z – direction
Ex 0 ( x ct )
Where Ex0(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
Ex (z,t)
E 0 co s( kz
Or in vector form:
where following notations have been used:
E0: amplitude of the wave
=2 f: angular frequency
k=2 / : wave number
Now c= /k is the phase speed of the wave
t)
• phase velocity: the velocity at which
the planes of constant phase move
vp
k
red dot located at a fixed wave phase
eˆ k
• group velocity: the velocity at which the envelope of the wave packet moves
vg
k
(k )
Group velocity is also a velocity with which energy propagates
• In a dispersive medium, the phase velocity varies with frequency and is not
necessarily the same as the group velocity of the wave
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/r3; the exact
solution can be expressed in terms of Bessel functions)
Dispersion relation
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
a dispersion relation.
and wave number k is called
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 vp= /k
- group velocity vg= / 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
Let’s 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:
In this case the phsyical (measurable) field is the real part of its
complex presentation
If E0 and B0 are constants, the fluctuating fields are called harmonic and the
differential operations reduce to multiplications:
and Maxwell’s equations
become algebraic equations
ik
ik
In general:
t
i
these will be used often!!
Assume next, that the wave propagates in a linear medium that is homogenous and
isotropic, so that and are constant scalars. We also assume that there are no
free currents or charges (J=0 and =0) and that the conductivity =0.
kE 0
kB 0
Let’s assume
k E
B (*)
k B
E
0
(**)
k E =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 k E, also E and B are also perpendicular to each other
(**)
k (*)
k ( k E)
k B
2
E
=0 for a transverse wave
using vector identity:
2
E
2
k E
k (k E) (k E)k k 2 E
k
i.e. we have found the
dispersion relation!
k
From
vp
vg
we can now calculate the group and phase speeds:
1
k
1
k
If we have =
0
and =
0 (vacuum)
Index of refraction n is defined as: n
(for =
0
and =
0
vp=c and vg =c.
c
vp
0
0
n=1)
If phase speed and group speed are same (like above), there is no dispersion in
medium
Waves in a linear medium with finite conductivity
Let’s now consider the medium where ,
and
are non-zero constants, but =0.
Now Maxwell’s Eqs with Ohm’s law give (Exercise):
This is called telegraph equation
(a standard class-room example of using Fourier
transformations to solve partial differential equations)
Instead of making the Fourier transformations, let’s start from Maxwell’s
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 || ez, E || ex, and H || ey
Giving a dispersion relation:
Thus, for a real
we have a complex k, that can be given as
k | k | ei
And solved for
The solution is:
Ex
E0 ei (|k | z cos
t)
e
|k | z sin
Where sin has to be positive. In this case the wave is damped when it propagates
into the medium that is physically sensible solution (if sin is negative wave grows
exponentially, note the factor e |k |sin z ).
From the dispersion relation (
phase speed v p
Re(k )
) we now get the
| k | cos
The distance after which the wave has decayed to 1/e of its original amplitude is
called skin depth:
1
Im(k )
1
| k | sin
Impedance of the medium is defined as:
Examples:
a good conductor
2
45 ;
For copper
An insulator:
>>
, whence
; vp
tan
f
50 Hz
1 cm v p
3 m/s
f
50 MHz
1 m vp
3 km/s
=0, > 0 and = 0,
=0 i.e. wave is not damped when it propagates in to the insulator;
Z
Where
0
Z0
0
Z0
0
0
376.73
is the impedance of free space
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.
Electric field is now: Eˆ
Eˆ 0 e i ( kz
Eˆ x =E x ei x
i
Eˆ y E y e y
where
Let’s choose:
ˆ
E
Ex e
y
i ( t kz )
x
ex
t)
( Eˆ x e x
Eˆ y e y ) e i ( kz
are phases in
the x and y directions
respectively
x,
are complex vector
;
Ey e
and define a complex number
x
t)
y
0
i ( t kz
)
ey
Eˆ y / Eˆ x
( Exe x
|
| ei
E y e i e y )e
i ( t kz )
1) If is real, Ey and Ex 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 Eˆ x , Eˆ y have different phases and
amplitudes and the wave is elliptically polarized
3) A special case of the elliptical polarization is obtained when
In this case there is a phase shift
i.e. Ey = Ex = E0
E
E0 (e x ie y )e
=
i
/2 and the amplitudes are equal
i ( t kz )
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
Comments on unit systems
In plasma physics one should learn how to smoothly deal with electromagnetic
quantities and natural constants. It it often useful and instructive to:
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.
measuring
cgs
SI equivalent
magnetic flux
density
Gauss (G)
10-4 Tesla (T)
energy
erg
10-7 Joule (J)
heat flux
calorie
4.1868 J
pressure
barye
0.1 Pascal (Pa)
magnetic flux
Maxwell (Mx)
10-8 Weber (Wb)
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
Maxwell’s 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