Today, I begin a new topic that is electromagnetic
wave propagation in plasma, in fact plasma
is known for very to be very rich for wave
phenomena, and hence I would like to discuss
this in some detail.
And, I will begin my presentation with elaborating
on the frequencies of interest, then I will
talk about plane wave representation of an
electromagnetic wave. Then I will talk about
effective plasma permittivity that brings
equivalence of plasmas to dielectrics, then
we will discuss transverse, and longitudinal
waves as two possible solutions of Maxwell's
equations. And then, we will discuss propagation
of electromagnetic waves, and we will derive
an expression called dispersion relation,
and then we will discuss phase velocity and
attenuation constant. And finally, we will
discuss the propagation of waves in over dense
plasmas, and hence we introduce a term called
is skin depth.
Well, the reference for today’s presentation
is this book electromagnetic theory for telecommunications
that professor C S Liu and I wrote this was
published by foundation books.
Let me begin with the frequencies of interest;
well these frequencies of interest 
range from very low frequencies called ultra
low frequencies U L F, which has a range from
1 to about 30 hertz, hertz means cycle 1 cycle
per second this is the frequency.
Now, these waves are used for underground
exploration suppose this is our earth this
is earth and this is air and there may be
some object buried inside the earth it may
be a oil reservoir or it may be a mineral
and we would like to probe at what depth is
this mineral and what is the identity of the
mineral. So, what you have to do? Because,
earth is a conducting medium in m t s units
earth conductivity is about one and in wet
soils it can as much as four in m t s units.
So what you have to do you have launch those
waves that can penetrate deeper and go to
the buried material.
So you have to launch waves from here and
then the waves are reflected back at the surface
you measure a quantity called ratio of the
electric field of the wave to magnetic field
of the wave this E by H ratio can provide
information about the material its the depth
and its conductivity or effective plasma conductivity.
So these waves are very useful how the plasma
arrives in this domain is in the generation
of these waves. These ultra low frequency
waves obviously one can produce by an antenna
on the ground but, the efficiency is very
low. So what we do? There are some natural
sources in the earth’s ionosphere which
is above the surface of the earth.
So in the ionosphere there are already some
natural sources that produce these radiations.
So, we employ those U L F waves to examine
that E by H ratio on the surface of the earth,
second frequency band of interest is called
E L F waves this E L F wave range is from
30 hertz to about 300 hertz. Let me just check
30 hertz to about 1 kilo hertz rather this
band of frequency is useful in communicating
with submarines.
So, this is suppose the surface of the ocean
and there is a submarine sitting underneath
or any objects submerged in water. And if
we want to communicate with this then what
people do there is a ionosphere also up in
the atmosphere of the earth so this is ocean
and this is ionosphere. Ionosphere has some
currents you can modulate those currents and
generate these waves those waves will penetrate
into the water. And you can send signals command
signals to the person sitting in these objects
like submarines. So, E L F waves are used
for submarine communication.
Then there is another frequency of interest
to plasma and that is called V L F waves,
the frequency range is between one to 30 kilo
hertz, you know our earth is like a sphere.
If, north pole is somewhere here and south
pole somewhere here, so there is a line magnetic
lines of force like these.
These are the lines of force magnetic lines
of force, what happens that? In this region
of space which is about 90 kilometers and
above the earth surface of the earth, the
air is ionized and we call this region as
ionosphere and when you move to distances
of the order of earth radius and larger than
the region of space is called magnetosphere.
So, in order to probe those regions of space
ionosphere and magnetosphere waves of these
frequencies between one to 30 kilo hertz have
been found to be very useful they are given
a name called whistlers.
So whistler waves have been used for exploration
of ionosphere and magnetosphere 
then comes the famous frequency meant for
radio communication called medium frequency
waves; medium frequency waves M W, I will
call and these waves have a frequency range
between 300 kilo hertz to 3 mega hertz.
Now, we have earth which has a curvature.
If, you have a antenna placed somewhere here
this will emit waves these waves travel along
the surface of the earth when earth bends
these waves also bend and they can carry undistorted
or noise free signal to long distances up
to a few 1000 kilometers in this frequency
range and they have been used for radio communication.
Then comes the short wave communication, the
frequency used are called short waves, the
frequency bend is 3 mega hertz to 30 mega
hertz in this bend also ionosphere plays a
very important role. Suppose, this is my earth
surface, this is the ionosphere, and I want
to communicate from here, and and send my
signal to far away distances like somewhere
here, what should I do? I will launch a wave
in the ionosphere, and we learn that ionosphere
is a optically rarer medium.
So when the wave enters the ionosphere it
gets turned like this and it can reach a far
distant point like this so ionosphere plays
a very important role in short wave communication.
Then we have a very important industrial application
of waves in this frequency or radio frequency
fields. In this band and that is called material
processing. So what you do in material processing
you have a substrate here like gold whose
film you have to deposit on a substrate.
So, this is a target and this is substrate,
you bring in a plasma you create a plasma
here by means of a radio frequency field in
this frequency range typical frequency that
is used in most of these systems is about
13.6 megahertz. So, it is a plasma here, what
plasma does? The plasma ions which come in
contact with the target they cause very rapid
ablation of material top clear of the material
here and then the ions which are released
they pass through here and the film is deposited
over the substrate.
So, plasma plays a very important role in
increasing the sputtering or ablation yield
of target material and deposition of the film
on the substrate. Then comes the V H F band
very high frequency band and ultra high frequency
band, this is between 300 between 30 megahertz
to 300 megahertz and this goes from about
300 megahertz to about 1 gigahertz; gigahertz
means 10 to the power 9 cycles per second.
These waves have been used for television
communication, telephones and also used for
plasma heating in a big device called tokomak
tokomak heating. Let me also mention that
the waves of frequencies below 30 megahertz
called short waves are also useful are also
used in plasma heating. So, plasma heating
or rather heating the fusion devices a very
major application of radio frequency fields.
Then comes the famous microwave range 
usually we talk this about 1 to 30 gigahertz.
Where, the at 30 gigahertz the wave length
of these waves in free space is 1 centimeter
and at 1 gigahertz it will be 30 centimeter.
These waves have lot of applications in domestic
for domestic use like heating, microwave heating,
micro oven is very famous but, then they are
also very useful in material processing in
plasma heating and in communication.
And then we go to higher frequencies or shorter
wavelengths called millimeter waves then sub
millimeter waves 
whose wavelength is in the millimeter range
and these wave have wavelength in the sub
millimeter range less than 1 millimeter and
then we go over to what we call as terahertz
waves, whose frequencies vary from about 10
to the power 12 hertz to about may be 5 times
10 to the power 12 hertz.
Well, in both these bands this goes from 30
gigahertz to about 300 gigahertz and this
goes from 300 gigahertz to about 1 terahertz
these frequencies are not produced by conventional
sources to produce these waves one employs
very special kind of plasma sources called
gyrotrons and free electron laser.
Gyrotron and free electron laser 
much of plasma physics research has gone into
these two devices in a gyrotron the basic
principle is that if you can launch an electron
beam in a magnetic field like this, then the
electrons gyrate in the field lines like this
and millions and trillions of such electrons
gyrates in free lines. And if, they can radiate
coherently then we get microwaves at the cyclotron
frequency or two time cyclotron frequency.
So this is a very important device and in
free electron laser you launch an electron
beam electron beam and you pass them through
a very special kind of magnetic field structure
where lines of force are like this alternately
up and down. This region of space we call
having a wiggler magnetic field, when electron
beam travels through them they produce radiation
and this radiation is tunable. The frequency
of this radiation can be controlled by controlling
the energy of these electrons and this could
be starting from sub millimeter waves to optical
frequencies it is a very versatile and very
efficient device.
So, collective behavior coherent oscillation
coherent radiation by electron beam induced
by a wiggler magnetic field is a very important
plasma physics subject and this device is
very useful and it has got a promise for plasma
heating in tokomak as well as for communication.
So, with these applications in mind. Now,
I would like to go over to discuss the propagation
of electromagnetic waves in plasmas, first
of all I like to mention a few things regarding
what is a wave.
So, let me start with a very simple basic
definition of a wave. Normally, wave is called
a propagating disturbance a propagating disturbance
what is this mean? In the simplest possible
terms suppose this is the direction of propagation
of my disturbance this is my position Z is
equal to 0. If, I have some signal here may
be disturbance in the case of sound wave it
could be a pressure disturbance for the case
of electromagnetic waves it could be an oscillatory
electrical magnetic field.
So, suppose this is my signal here, I have
no electric field, magnetic field here or
there only in this region this is there after
a while the same thing appears here little
later this will appear somewhere here, then
I call this as a propagating disturbance.
A field that is confined in certain region
of space here after a while it appears here,
then after a while it appears there then there
then we say its called a propagating disturbance
for a wave.
The velocity of propagation of a signal suppose
is v p. Now, the disturbance in the case of
electromagnetic wave is in electric disturbance
or a magnetic disturbance or a combination
of both.
So, let me consider this to be the electric
field at Z equal to 0. so let me the first
space here is allocate for the Z coordinate
and then the time coordinate. So electric
field in general in the simplest possible
situation depends on position and time and
at this suppose this is simplest representation.
Suppose, I say is A exponential minus i omega
t. Suppose, the electric field that is produce
by any antenna or any source in this region
Z equal 0 is having this dependence well,
why I am choosing this dependence? Because,
any general time dependence of electric field
can be represented as a sum of many such monochromatic
fields by using Fourier series or Fourier
integral representation.
So, if we can understand the propagation of
a one frequency component, we can study the
propagation of a group of such frequencies.
So, most of the electromagnetic theory is
really concentrated about the propagation
of a monochromatic or a single frequency wave.
So I will consider this kind of dependence
there is another reason for this as we shall
learn plasma is a dispersion medium.
If you send a time bound signal like this,
when it travels in a plasma it gets distorted
and time bound signal if you carry out the
Fourier analysis is equivalent to having many
frequencies and different frequencies signals
travel with different velocities and consequently
the signal gets distorted. So, in order to
avoid such subtle effects to begin with I
consider single frequency so that we can assign
a single velocity of propagation to this field.
So, now consider an a signal consider a signal
at Z equal to 0 having this kind of field
and my issue is that if the signal is travelling
along Z axis this by Z direction, so what
is the value of the signal at position Z at
time t.
Well, I do not want to solve any equation
for this I just want to use this concept of
a propagating disturbance introducing this
value. I know that the signal had arrived
from Z equal to 0, the time it has taken is
equal to Z by v p, because the distance it
has travelled is Z and time is v p.
So whatever was the value of the signal is
here at time t must had been here at earlier
time, so this should be the same as E at 0
at time t minus this, because this is the
additional time it has taken to travel.
So whatever field you observe at Z at time
t should be same as at Z equal to 0 at time
t minus this quantity. But, here I have already
stated that if you know that at time t the
field is so much then in place of t, replace
this t by t minus Z by v p, and you will get
the field there so I will write this as such.
You will get E at Z t is equal to E at 0 at
time t minus Z upon v p. And if I put the
expression for E in terms of a then this becomes
E to the power minus i omega in place of t,
I will write t minus Z upon v p. I can multiply
omega in the interior and I can write this
as A exponential minus i omega t minus k Z,
where I have defined k is equal to omega upon
v p this by definition.
So on physical grounds I have written down
a dependence of electric field on position
and time, if my wave is travelling in the
positive Z direction with velocity v p and
the wave frequency is omega, this entire exponent
from here to here is called phase of the wave.
And the quantity which is here is called amplitude,
one may note here in this particular case
phase does not depend on x and y.
So, if I consider a plane perpendicular to
Z axis means over which Z is constant then
this is the wave front equation of the wave
front. So, a plane perpendicular to Z axis
on which Z is constant my wave was propagating
like this. So any plane this is my Z axis,
any plane perpendicular to this will have
same value of electric field here, here here
here here because on this planet Z does not
change only x and y change but, x and y do
not appear in the phase and hence phase is
constant.
So this is called the equation of the wave
front Z is equal to constant. Because, if
Z is constant then phase will not change at
a given time. So, the wave one may note here
my wave was travelling in Z direction, a plane
perpendicular to the direction of propagation
is called wave front this is wave front. Well,
this is what we call as the one dimensional
representation of a plane wave, the wave front
is a plane surface and hence this wave is
called plane wave but, here I assumed that
my wave was travelling around Z direction.
Now, I would like to write this in a little
more general case suppose my wave is travelling
at an angle to Z axis, suppose this is my
x axis, this is y axis and Z axis is perpendicular
of the board of the paper so this is my x
axis y axis and Z axis I may denote like this.
Suppose my wave is traveling at an angle like
this, I call this as unit vector n. So, I
want to find out if field at position O is
given to me or the origin is given to me and
it is also given to me that my signal is travelling
in the direction of this arrow, what is the
field at a point here at position r from the
origin.
So my point is that E at the origin at time
t is suppose A exponential minus i omega t
then question is what is E? At position r
at time t, what you can do first of all consider
a draw a plane passing through point r. So,
if I draw a plane passing through r but, perpendicular
to this direction of wave propagation. So
then I will draw a plane like this and extend
this unit vector n, this will hit this unit
vector n at a point M.
So, what I except is that the electric field
at this point is the same at this point because
this plane is perpendicular to direction of
propagation and hence this is the wave front
and on the wave front, the value of the electric
field at every point is the same so E at r
at time t will be same as E at M point at
time t. Now, the field at M is reaching from
the origin along this path, so this should
be same as E at the origin at time t minus
that distance O M by the velocity of propagation
of the signal v p.
Now, O M I can easily calculate because the
distance from here to here, this distance
is r if this angle between the two is theta,
I can write O M is equal to r Cos theta but,
r Cos theta can be vectorially written as
n unit vector along O M dot r. So, O M I will
write down a simply n dot r and then I use
this stop expression I will get this is equal
to A exponential minus i omega t minus k dot
r. where, k I have written as magnitude wise
is equal to omega upon v p as before but,
I have put a unit vector n.
So, k is known as the propagation vector whose
magnitude is omega upon v p, the velocity
of propagation of the signal and direction
is the direction of wave propagation n. So
this is called a 3 dimensional representation
of a plane electromagnetic wave but, we do
not know what is the connection between the
direction of k and direction of a and what
is the value of v p, the entire information
about the medium is contained in this quantity
v p this is the medium which gives rise to
specific values to phase velocity to v p.
Now, please understand here we are considering
the signals whose amplitude is constant only
phase changes. So this is the velocity of
phase propagation amplitude is not changing.
So, v p is really not called the velocity
of signal propagation this is called the velocity
of phase propagation. Because, only phase
moves only phase changes with time so v p
is the velocity of phase propagation or phase
velocity.
So, I think we have obtained on physical grounds
an expression for electric field E as a function
of position and time, I will call this is
the simplest possible expression for field
that represents a propagating disturbance
of constant amplitude and constant frequency.
Now, let us see under what conditions this
solution satisfies the Maxwell’s equations.
So, before I undertake the general conditions
under which this solutions satisfies the Maxwell’s
equations.
I would like to introduce a quantity called
effective plasma permittivity. let me examine
the Maxwell’s equations from this perspective
well, in general my electric field is a function
of position and time, well I will like to
keep my r dependence little more general then
the plane wave representation suppose like
this. Well, I can write down this as omega
t minus k dot r then a can taken as a constant
but, there is a very special space dependence
it could be more general dependence. So suppose
I am specifying only by time dependence and
to keep my space dependence of electric field
is more general.
Let us see, in terms of this representation
how do the Maxwell’s equation look, let
me begin with the last Maxwell equation.
The generalized ampere’s law which says
that curl of H is equal to J plus delta D
by delta t, when I write down the time dependence
of electric field like this I also imply the
time dependence of J, H and D also of the
same form. Because, these equation should
be true in the steady state and every term
should have same time dependence. So, whenever
the D or J or H they vary with time like this
it means delta delta t of this function if
you take you replace delta delta t by minus
i omega and J, I write J for plasma is equal
to sigma E conductivity times, the electric
field and D for plasma is written as epsilon
0 into E.
So, just substitute them here and you will
get sigma E minus i omega for this operator
delta delta t and D is epsilon 0 into E, epsilon
0 is called free space permittivity. I can
take minus i omega epsilon 0 common, in the
interior you will get one plus i sigma upon
omega epsilon 0 into E. 
So this is a single term, so these two terms
have been combined into a single term which
is a coefficient of E, the term in this bracket
is given a symbol epsilon effective.
And I can write down this as 
epsilon effective is equal to 1 plus i sigma
upon omega epsilon 0, in terms of this my
Maxwell equation becomes curl of H is equal
to minus i omega epsilon 0 epsilon effective
into E.
If you look at the generalized amperes law
in a dielectric, what would you have? In a
dielectric, what do you get curl of H is equal
to J but, there is no current density, so
J is 0 is equal to delta D by delta t but,
in a dielectric D can be written as free space
permittivity into relative permittivity epsilon
r into E. And delta delta t for monochromatic
fields can be written as minus i omega, so
it becomes minus i omega epsilon 0 epsilon
r into E.
Now, compare this equation with this equation
the two equations are exactly same with the
only difference that epsilon r, in the case
of a dielectric is replaced by epsilon effective
in the case of a plasma. So plasma behaves
mathematically as a dielectric with only difference
that wherever epsilon r occurs, relative permittivity
occurs, you just put epsilon effective so
epsilon effective is called also effective
relative permittivity of the plasma. So this
quantity epsilon r is a very important quantity
here and equivalent of this is epsilon effective.
In case of metals or semiconductors, what
you have? In metals 
you have D is equal to epsilon 0 into epsilon
L into E, where epsilon L is the relative
permittivity due to lattice in case of solids
and J of course, we write as sigma E. So when
you look at the fourth Maxwell equation, curl
of H becomes J plus delta D by delta t, if
you put J equal to sigma E delta delta t is
minus i omega. I can write down this is minus
i omega epsilon 0 into epsilon L plus i sigma
upon omega epsilon 0 into E.
So, this quantity from here to here is called
effective permittivity of metal relative permittivity
of metal. So epsilon effective is equal to
lattice permittivity plus conductivity into
I upon omega epsilon 0.
So, the only difference in conductors and
plasma is in this term epsilon L for plasmas
this term is unity for lattice, for solids
its epsilon L for intense for gold at optical
frequency, infra red frequency. Lower frequencies
epsilon L is like 9, for silver its 4, for
microwave at microwave frequencies for germanium
like 14, for indium antimonite at microwave
frequencies the value of this expression is
like 70.
So, this varies from material to material
but, for plasmas this quantity is unity. So,
we have found some equivalence of plasma to
a dielectric or a conductor to a dielectric.
But this is only one equation. I would like
to see whether this epsilon effective brings
the equivalence in other equations also.
Now, the Maxwell’s equations other Maxwell
equations are divergence of D is equal to
rho then divergence of B is equal to 0 and
curl of E is equal to minus delta B by delta
t, for non magnetic materials B is written
as mu 0 into H, so no dielectric properties
involved in these two equations.
So, this is the only equation which may be
different from in different media, from conductors
to semiconductors plasmas they may be different
then from dielectrics but, we will see that
if you use the concept of effective plasma
permittivity this equation can also be written
in the same form as in a dielectric, let me
write down this. In dielectric, D is equal
to epsilon 0 epsilon r into E, where as rho
equal to 0, so this equation simply becomes
divergence of epsilon 0 into epsilon r into
E equal to 0. how about in the plasma? In
a plasma rho comes, so let me write down in
a plasma.
In a plasma or a conductor, what we do? For
plasmas we write D is equal to epsilon 0 E,
how about rho we know that plasma in equilibrium
is quasi neutral, so charges appear only when
there is flow of current so and rho is governed
by the equation of continuity which is delta
rho by delta t plus divergence of J is equal
to 0.
J, I am writing sigma E and E varies in time
as exponential minus i omega t, so rho must
also depend on time in the same exponential
fashion, so replace this delta delta t after
minus i omega plus divergence of sigma E 
equal to 0. So we obtain rho from here, which
is rho is equal to 1 upon i omega divergence
of sigma E and this I am putting in this equation
divergence of D which is equal to epsilon
0 E.
So, I can combine these two divergence terms
take this from right to the left and what
you get is divergence of epsilon 0 if you
take common in the interior you will get 1
plus i sigma upon omega epsilon 0 into E equal
to 0. The equation is the same as in a dielectric,
because this term I call as epsilon effective,
so epsilon effective comes in automatically
there.
So, we have shown that epsilon effective which
is written as 1 plus i sigma upon omega epsilon
0, makes a plasma equivalent to a dielectric.
Now, conductivity expression if I substitute
rf conductivity we derived an expression in
a un magnetized plasma it was n e square upon
m negative sign here, I there omega plus i
mu this is the conductivity substitute this
back in there then it becomes epsilon effective
is equal to 1 minus n e square upon m epsilon
0 omega square into 1 plus i nu upon omega.
Here, n is the electron density, E is the
magnitude of electron charge, m is the electron
mass, epsilon 0 is free space permittivity
and the value of this expression we already
seen earlier, n e square upon m epsilon 0
under the root this quantity has the dimension
of frequency and it is defined as omega p,
if you put the value of E m and epsilon 0
then this equal to about 50 times under root
of electron density, when n in per meter cube
n in per meter cube.
So, this is a simple expression I get and
let me rewrite this effective plasma permittivity.
As epsilon effective is equal to 1 minus omega
p square upon omega square into 1 plus i mu
by omega.
In most dielectrics, imaginary part of relative
permittivity is small, so in plasmas also
if you are talking of high frequency waves
through a plasma mu is quite small so character
is very similar however there is a big difference.
Usually, the dielectric constant of a dielectric
or relative permittivity of a dielectric is
bigger than 1, in case of plasma it is less
than one so that is a big difference and it
has lot of implications on wave propagation.
With this term let me examine the validity
of our plane wave solution under what conditions
this will satisfy Maxwell’s equations.
So, I am examining the plane wave solution,
two Maxwell’s equations my electric field
I am taking as A exponential minus i omega
t minus k dot r, well in my definition of
epsilon effective I kept the r dependence
of electric field very general. But, when
I am trying to study the plane wave propagation
I am writing my r dependence like a plane
wave so writing like this. So, using A as
a constant amplitude. Now, this is a dot product
of two vectors,so if I elaborate this it will
comes out to be a exponential minus i omega
t minus k x into x minus k y into y minus
k z into Z this is the meaning of this dot
product.
Explicitly, I have written this for a purpose.
Because, if I ever encounter the derivative
of E with respect to x for instance then I
will simply get i k x into E. Because, all
field quantities like E, H, B, D, J etcetera
all of them for a monochromatic wave response
of propagation they depend in the same way.
And hence whether the derivative occurs over
E or H or v we must replace this del operator
delta delta x operator by i k x or simply
I am saying that is del operator is written
as which is written as x component of this
delta delta x plus y component is written
as delta delta y plus z component is written
as delta delta z, it turns out to be simply
i into k vector.
So, in subsequent discussion when we examine
the validity of this solution from the perspective
of Maxwell’s equations, wherever del operator
comes in the Maxwell’s equations. I will
replace this by i k vector and similarly,
wherever time operator comes I will replace
by minus i omega and let us see, how the Maxwell’s
equations look.
I will write down third and fourth Maxwell’s
equations, curl of E is equal to minus delta
B by delta t; this is known as the third Maxwell’s
equation or faradays law of electromagnetic
induction. Just replace this by i k cross
E delta delta t by minus i omega, so it becomes
plus i omega B is mu 0 into H or a plasma
I can cancel this I from here, there the fourth
Maxwell’s equation is curl of H is equal
to J plus delta D by delta t but, we just
note down this equation as minus i omega epsilon
0 epsilon effective into E and curl of H,
I can write down as i k cross H.
I can cancel this i from here with this i
there, what I can do? I can take k cross of
this equation. So, I will get k cross of k
cross E 
is equal to omega mu 0, k cross H then use
k cross H from the lower equation it becomes
is equal to minus omega square mu 0, E 0,
epsilon 0, epsilon effective into E. And k
cross k cross E if you use vector repul product
can be written as k vector into k dot E minus
k square E vector, this is the equivalence
of wave equation it is an algebraic equation,
because I have already replaced del operators
by i k, otherwise this like a wave equation.
The character of this equation will reveal
that this equation supports two kinds of waves
electrostatic waves, and electromagnetic waves,
and also for electromagnetic waves this will
give you a dispersion relation, and entire
information about the waves is contained in
this equation. I think, we are close to closing
this lecture for now. In our next lecture,
we shall discuss the implications of this
equation, and elaborate on the character of
electromagnetic waves. Thank you.
