Friends, in this lecture I would like to continue
our discussion on electromagnetic wave propagation
in plasma. Here, we will discuss the electromagnetic
wave dispersion relation, phase velocity and
attenuation constant, deduce an expression
for a skin depth in over dense plasmas. We
also study the propagation of or the flow
of energy with an electromagnetic wave, in
observing media collision plasmas we will
calculate the power loss in decibels. And
if time permits, I will discuss the reflection
of electromagnetic waves at normal incidence
from plasmas.
The reference for today’s presentation is
again the same book electromagnetic theory
for telecommunications by professor CS Liu,
and myself published by Cambridge university
press.
We will discussing the plane wave solution,
like E is equal to A exponential minus i omega
t minus k dot r under what condition this
solution will satisfy Maxwell’s equations.
We consider third and forth Maxwell’s equations
and those equation on replacing del operator
by i k and time derivative as minus i omega
take the following form the third Maxwell’s
equation takes the form k cross E is equal
to omega mu 0 into H and the forth equation
is k cross H is equal to minus omega epsilon
0 epsilon effective into E they are coupled
in E and H.
So, what I did was? I took k cross of the
first equation and use the second one and
I got this relation k cross k cross E is equal
to minus omega square mu 0 epsilon 0 epsilon
effective into E. First of all, if I take
k dot of this equation then left hand side
will be 0, because this is a vector perpendicular
to k as well as k cross E. So, if I dot this
equation with k, so multiply this equation
by k dot 
left hand side is 0, so right hand side should
also be 0; means epsilon effective into k
dot E should be 0. Because, omega is a constant,
epsilon 0 mu 0 are constant, so they cannot
be 0 either this should be 0 or this should
be 0.
So one simple consequence of this equation
is that my solution given above will satisfy
these equations only when either k dot E is
0 or epsilon effective is 0.
Let us see, one possibility is that epsilon
effective you make 0, in view of this if I
down my k cross k cross E equation again I
can write down this k cross k cross E equation
as let me write this, I think I here k cross
E is equal to minus omega square mu 0 epsilon
0 into epsilon effective into E this was the
basic equation, whose k dot I took and I got
this condition that I can satisfy this equation
and this is 0. When I put this equal to 0,
what does this say this left hand side is
identically equal to k, this k into k dot
E minus E into k dot k which is k square and
because I am choosing epsilon effective equal
to 0 this should be 0.
This says that, this is a vector in the direction
of E this is a vector in the direction of
k these two can be equal, only when E is parallel
to k so important consequence is that E is
parallel to k and if you go to third Maxwell
equation which I had written as curl of rather
sorry not curls k cross E is equal to omega
mu 0 H. So if E is parallel to k this term
will be 0, so H will be 0 this is a very important
consequence.
So whenever epsilon effective is 0, those
waves will have electric field parallel to
the direction of propagation and will have
no magnetic field. So they cannot be called
as electromagnetic wave. Because, there is
no magnetic field they will be called as purely
electric waves in plasma jargon they are called
electrostatic waves. and if they are travelling
in the direction of k, this is my direction
of k vector then the electric field of these
waves will oscillate in the same direction
like this.
Now, what will happen in such a case they
are called longitudinal waves and in some
region the electric field will be in this
direction at the same instant in some direction
will be like this in some wave it will be
like this. So, these electrons will be compressed
somewhere they will be rarefied somewhere
so this gives rise to charge oscillations
or density oscillations. These waves are also
known as density oscillations or propagating
density perturbations they have given a name
electrostatic wave.
This is a very important class of waves in
plasma physics, lately these waves have been
found to be very useful in electron acceleration
and electron acceleration reaching the values
of about 1 g e v have been achieved by these
waves. So for particle acceleration these
waves are very useful, well we shall discuss
those utilities of these waves later on. second
possibility to satisfy the k dot of this equation
being 0 is that k dot E is 0. If, I choose
k dot E equal to 0 then in this equation in
this equation sorry if, I simplify this term
will be simply equal to minus k square E because
this is 0 this is the expansion of this term.
So if I choose k dot E equal to 0 this is
0 only this will survive, so put this in place
of here and equate through this. because in
this case epsilon effective is non zero. So,
right hand side is not 0, so what you get
here is minus k square E is equal to minus
omega square mu 0 epsilon 0 into epsilon effective
into E these waves have k given by this relation.
So, I will write down k for these waves 
is equal omega k square is equal to omega
square mu 0 epsilon 0 into epsilon effective.
Let us understand the character of these waves,
first of all mu 0 epsilon what? Epsilon 0
is the free space permittivity in m case units
its value is 10 to the power 9 upon 36 pi,
mu 0 is called free space magnetic permeability
whose value is 4 pi into 10 to the power minus
7 in m case units both in M K S units. If,
I put these values epsilon 0 mu 0 turns out
to be 1 upon c square, where c is the velocity
of in free space given by 3 into 10 to the
power 8 meter per second.
So, I will put this in here and I will get
k square is equal to omega square by c square
into 1 minus for epsilon effective is omega
p square upon omega square 1 plus i mu by
omega this is the dispersion relation but,
how about the character of these waves whether
they are transverse or non transverse.
So, let us go step by step from forth Maxwell
equation I had written, k cross H is equal
to minus omega epsilon 0 epsilon effective
into E this says that E is perpendicular to
k and H both. Whereas, from third Maxwell
equation k cross E was written is equal to
omega mu 0 H; this says that H is perpendicular
to k and E both.
So this equation tells E is perpendicular
to k so suppose my wave is going in Z direction
like this or some direction like this of k
in the and the electric field has to be perpendicular.
This will oscillate like this then magnetic
field has to be perpendicular to both of these
like this this is B field. So these are purely
transverse waves, electric field is perpendicular
to k vector, magnetic field also perpendicular
to k vector and E and B are both perpendicular
to each other.
Before, I proceed further I would like to
recall a quantity called pointing vector.
I will not drive this here but, I will borrow
this result from general theorem that energy
flow with a wave is denoted as turns out to
be equal to E cross H. Here, real part of
E has to be multiplied with the real part
of H and as we have been doing earlier this
implies that this is equal to half real part
of E cross H plus E star cross H, so let me
write down the values of E and H and see,
what is the pointing vector?
Pointing vector well, for this pointing vector
I can write down E explicitly as A exponential
minus i omega t minus k dot r and H, I just
wrote down from third Maxwell equation is
equal to k cross E upon omega mu 0. So what
you will see here, in the pointing vector
S is equal to half real part of first term
is E cross H, so which is equal to E cross
k cross E upon omega mu 0. the second term
would be E star cross k cross E upon omega
mu 0, you may note one thing in here E has
an exponential factor as exponential minus
i omega t minus k dot r this is also has.
So, the product if you multiply them will
be at 2 omega frequency time average of this
function will be 0, this is the only term
whose exponential time dependence will cancel
the two factors because this will have minus
i omega t and if you take complex conjugate
it will give you minus i omega t plus i omega
t so they will cancel.
So time average power vector will be always
the term that will contribute as time average
is equal to let me write down with a different
ink because this is a important expression.
This quantity is equal to 1 upon twice omega
mu 0 and real part of this quantity, I will
write down this as vector triple product as
k vector here into E dot E star 
minus this E vector and k dot E star but,
we have already seen k dot E is equal to 0.
So this term vanishes, because they are transverse
waves k is perpendicular to E, so this is
the only term and you know E star is the real
quantity.
So this equation simply gives you time average
pointing vector simply 
as average is equal to modulus of E square
upon twice omega mu 0 into real part of k,
if k is complex only real part of k has to
be written there this is a very important
result.
If, somehow we find that k is imaginary purely
imaginary then there is no power flow, because
time average pointing factor or energy flow
is 0, the magnitude of this quantity is also
called intensity of the wave or intensity
of radiation.
So let me write down this intensity of radiation
as average magnitude which is equal to modulus
of E square upon twice omega mu 0 into k real
part k r. Well, when we discuss different
cases of wave propagation and different kind
of plasmas we will make use of this expression.
So let me consider, the wave propagation as
collision less plasma where collisions are
small, what you get? k square is equal to
omega square by c square 1 minus omega p square
over omega square this can also be written
as omega square is equal to omega p square
plus k square c square. If, I want k to be
real then k square should be positive.
So, if I want my wave to carry energy then
omega must be bigger than omega p that is
very important condition plasma does not allow
the propagation of all sorts of waves it will
allow propagation of only those waves whose
frequency is bigger than omega p and. If omega
somehow less than omega p then k will be purely
imaginary and from that point on energy should
be reflected back. Sometimes your plasmas
suppose the plasma starts form here but, the
density is low here and density increases
in the plasma suppose density increases like
this.
If you are launching an electromagnetic wave
on a plasma, whose frequency is constant but,
plasma frequency increases then it will go
up to a region where omega p becomes equal
to omega and beyond this omega p is bigger
than omega the wave will not propagate it
will come back this is a very important phenomenon
called total reflection of electromagnetic
wave from critical density plasmas or from
critical layer which has been used in measuring
the density profile of the ionosphere. I shall
discuss this when I discuss the applications
of these waves in a either today or in my
next next talk. but let me mention something
about this, that I can plot a dispersion relation
omega versus k.
And this will be of this form omega normally
is plotted on the y axis and k on the x axis
it starts from omega p here, because waves
of frequency less than omega do not propagate.
So, k is not real for them and this will go
like this and asymptotically this dispersion
relation approaches a line called omega equal
to k c line called light line. And we can
define a quantity called reflective index,
a reflective index is defined as velocity
of light in the medium in free space to velocity
of light in the medium but, we know v p is
already omega by k. So this is c upon omega
upon k or k c by omega and if I put the value
of k it turns out to be equal to 1 minus omega
p square upon omega square under root. So
this quantity is less than 1, for omega bigger
than omega p and its imaginary which has no
meaning for omega less than omega p, so we
should not talk in terms of.
Because space velocity actually does not have
any meaning there. So, let me introduce a
term called skin depth characterize the penetration
of fields in a over dense plasma a plasma
where omega p is bigger than omega is called
over dense plasma and a plasma where omega
p is less than omega is called under dense
plasma.
So, in an over dense plasma over dense where
I am choosing omega p bigger than omega what
is my k is equal to omega square sorry omega
by c into 1 minus omega p square by omega
square to the power half so k is purely imaginary.
I can call this is equal to i times alpha,
where alpha I will call as omega by c into
omega p square by omega square minus 1 to
the power half. If, I put k is equal to i
alpha in my electric field expression, what
do I get but, k is a vector quantity I am
little worried so to appreciate the physical
consequence of k being imaginary. I will consider
the wave propagation one dimension for 1 D
wave propagation.
What I am going to do? I will will write down
say electric field is equal to A exponential
minus i omega t minus k z, let my wave is
be travelling in the z direction this is my
k vector or the wave is going in this direction
say z axis space variation is only with z.
but k is imaginary, if I put k is equal to
i alpha here.
This equation can be written as E is equal
to A e to the power minus alpha z exponential
minus i omega t. So, in an over dense plasma,
phase term does not have any z dependence?
No z dependence, in phase and this entire
thing is called amplitude then 
and amplitude will fall off with z exponentially.
Alpha has the dimension of one upon length
and I introduce a quantity called skin depth
which is equal to delta and this is 1 upon
alpha by definition. And if I put the value
here of alpha from the previous expression
it is c upon omega p square minus omega square
under root. Well this is in the limit when
there are no collisions, I consider collision
less skin depth then the collision frequency
was taken to be 0. However if nu is included
what happens let us see.
So when collisions are finite, what do you
expect? your k is equal omega by c, 1 minus
omega p square upon omega square 1 plus i
nu by omega to the power half for propagation
of waves of frequency omega p, in a plasma
nu is always less than omega p indeed much
smaller than omega p in all plasmas, nu is
always less than omega p and we have learned
that if we want the wave to travel in a plasma
with very little attenuation omega should
be bigger than omega p.
So in the limit where nu is much less than
omega you can simplify this expression. I
can write down k is equal to k real plus i
time k imaginary imaginary part comes because
of this nu term.
Simplify this you get k real nearly equal
to omega by c into 1 minus omega p square
by omega square to the power half and imaginary
part of k is k i of the order of if you simplify
this expression it becomes nu upon 2 c into
omega p square over omega square multiply
divided by k r, well it is not k r this is
1 minus omega p square by omega square to
the power half this sort of expression you
will get. You may check here, that nu if you
increase a plasma with high collisionality
will give you higher value of k imaginary
and imaginary part of k always gives rise
to damping of the wave it is also called damping
coefficient or attenuation constant and in
the vicinity of omega equal to omega p this
term is most dominant.
So, when wave goes from a low density plasma
into high density plasma two things happen
collision frequency increases. Because, this
depends on plasma density and because of this
omega being closer to omega p. Now, denominator
becomes smaller and there is enhancement because
of that also, this k i is very strongly increasing
function with plasma density. So, if you are
launching a wave into a inhomogeneous plasma
whose density suppose a plasma has a density
variations like this density 0 here. Suppose
a plasma density is increasing like this you
are launching a wave, what will happen? Initially
collision frequency will be very tiny I am
talking of strongly ionized plasma where collisions
frequency depends on electron density or ion
density.
So, as the density increases nu will increase
omega p will increase this vector will decrease
and there is a considerable enhancement in
k i. So, the wave gets very very strongly
damped as it travels deeper and deeper in
to a denser plasma so there is a message contained
in this expression.
Another thing that you may note here, that
if I am talking of very low frequency response
at omega much less then nu, what will happen?
These are the kind of waves that are of interest
in the E L F range, because the frequency
of E L F wave in less than collision frequency.
So what will happen, in this case k square
is equal to omega square by c square 1 minus
omega p square over omega square into 1 plus
i nu upon omega, what I have to do? I am choosing
nu much bigger than omega, I can ignore this
1 and this becomes of the order of omega square
by c square into omega p square upon omega
nu into i. I have ignored one as compared
to this term because this term becomes very
large when omega is very small.
So this is the typical thing and k i becomes
of the order of take the under root it becomes
omega p by c into omega upon nu to the power
half and this I will give you 1 plus i upon
root 2 this is the under root of i i.
So k has a real part and imaginary part of
equal magnitude. I can write this is equal
to k real plus i times k imaginary of the
same values. And skin depth in this case is
defined as 1 upon k i imaginary part of k
that will be equal to reverse of this, so
c upon omega p into twice nu upon omega to
the power half. The important thing is the
dependence on this skin depth on frequency
this skin depth decreases with frequency.
So this goes as omega to the minus half, please
note at omega much bigger than nu we found
that the skin depth was like c upon omega
p kind of thing. whereas this is like this
here.
So, if I plot a graph of skin depth versus
frequency. Delta versus omega at very low
omega this will be large and as omega approaches
omega p again this will be large. this is
actually a logarithmic plots kind of thing,
so this is skin depth would be quite large
at very low frequencies like if you are sending
a wave of radio frequency like 10 megahertz
in a solid or in a high density plasma, in
that case the skin depth would be quite large
in several centimeters. Whereas, the same
metal at microwave frequencies will have a
very tiny skin depth may be of the order of
a few microns. So, skin depth is a very strong
function of frequency. I think, I would like
to you know mention a quantity because when
we talk of absorption of waves via collisions
we always talk in terms of power loss in decibels.
So let me introduce this quantity here, whenever
k is complex, k magnitude wise is equal to
k real plus i times k i your electric field
can be written as E is equal to a amplitude
exponential minus k i z, for wave propagation
on z axis exponential minus i omega t minus
k r into z this sort of expression you get.
And as I just told you that the pointing vector
or intensity of radiation which is time average
pointing vector is equal to modulus of E square
divided by twice mu 0 omega into k real is
the expression I have given to you.
So, if I put the values please note here when
I take the modulus square only A square will
not come exponential factor was also real
and square of this will also come. So, what
you get? your intensity will depend on z.
Now, because of this vector this vector.
And the result would be that I can write down
my intensity as 
this will be A square exponential minus twice
k i into z upon twice mu 0 omega multiplied
by k r, which is omega by c into 1 minus omega
p square by omega square to the power half;
this is a typical expression. In a weakly
collision plasma, where I have assumed omega
bigger than nu.
Well, all these vectors are constant in a
plasma of uniform density only this is z dependence.
So, I can write this is equal to some constant
I 0 exponential minus twice k i z means the
amplitude intensity of radiation decreases
with z. Power loss in decibel is defined like
this, as alpha which is equal to 10 log to
the base 10 of i 0 upon i. And if you evaluate
this quantity like this it becomes 20 into
k i into z into log to the base 10 of log
of E. Log of E is to the base 10 is 1 upon
2.3, so if you divide this by 2.3 this will
be something like 8 point something 8.7 or
something, 8.7 i guess into k i into z.
So in many experiments you measure the power
loss in decibels over a certain distance z
and hence k i can be calculated.
I will give you an example I will give you
an example A wave suffers say 3 d b power
loss 
in a distance of say 2 meter, estimate k i.
what should I do? It is very trivial alpha
is 3 here, so expression was alpha is equal
to 8.7 multiplied by k i into z is 2, so k
i turns out to be equal to alpha is equal
to 3 upon 8.7 multiplied by z which is 2,
one can evaluate this quantity which is I
think is equal to point this is I think 0.17
or so just just a very rough I have written
and one can estimate so it is a simple calculation.
The advantage here is, that once you measure
the power loss in decibels you can calculate
k i and k i if you put the expression for
k i from there you can deduce the collision
frequency.
So absorption measurement gives you information
about the collision frequency, if you know
plasma density that you can measure by some
other method. And hence this is a important
diagnostic technique also. I must also mention
that when we were discussing the conductivity
of a collisional plasma, rf conductivity of
a collisional plasma we found that heating
of the medium takes place only when collisions
are there. and here we have noted that damping
of the wave takes place, when k i is finite
or collisions frequency is finite. So collisions
are responsible for both damping of the wave
and heating of the plasma, after all where
the energy goes when the electromagnetic wave
travels in a plasma and if it is attenuated
it is attenuated by the electrons, which oscillate
in the presence of the electric field of the
wave and which undergo oscillations.
But when they collide with scatterers like
ions or neutral atoms then the their velocity
has a component in phase with the electric
field and that gives rise to net power dissipation.
So, electrons are heated and obviously when
electrons collide with ions or neutral atoms
they impart part of their energy to those
and they can also be heated but, the temperature
of electrons in such plasmas is always bigger
than the ion temperature or neutral temperature.
So, I think these are some basic characteristics
of electromagnetic wave propagation in plasmas.
I would like to finally, discuss the implication
of complex k on wave reflection, because we
have just learnt that if wave is launched
on to a over dense plasma then propagation
constant is purely imaginary and hence time
average pointing vector is 0. The question
arises is there some electric field still
in the over dense plasma or not, so to address
this problem I would like to discuss the problem
of wave reflection in a from a plasma boundary.
Consider, a boundary between a plasma here
and free space there, plasma is characterized
by a single quantity called epsilon effective
and free space has this permittivity equal
to unity it is a free space. What we do? We
are launching an electromagnetic wave here,
partly the wave will be transmitted in the
plasma partly it will be reflected.
So there are three waves a incident wave a
reflected wave and a transmitted wave and
let me consider this direction as z direction
and the interface as z equal to 0. So, the
incident wave if I write E incident is equal
let me take the this wave is polarized perpendicular
to the direction of propagation, so either
E x is finite or E y is finite or both can
be finite. but without any loss of generality
I will choose this to be x polarized and that
the amplitude of this incident wave is A 0
and this is a wave travelling the positive
z direction.
So, I will write down this is equal to minus
i omega t minus free space, where this is
effective permittivity is one, I can write
down the propagation constant as omega by
c and similarly, reflected wave will also
have A 0, suppose the amplitude is reduced
by a factor R A subscript means amplitude
reflection coefficient.
The frequency would be same but, the direction
of propagation is reversed. So, I will write
down plus omega by c z this is a reflected
wave and the transmitter would be E T again
should have polarization to satisfy the boundary
conditions, let the amplitude will be A 0
into T transmission amplitude of T A exponential
minus i omega t, the wave will have a different
k vector this k z. Where, k is omega by c
epsilon effective to the power half.
So, now I know the fields I must apply the
continuity conditions well there are two unknowns
here, R A and T A. I require two boundary
conditions; one boundary condition is that
the tangential component of electric field
must vanish here must be equal on both sides
should be continuous and second tangential
component of magnetic field should also be
continuous.
So, let me apply this to boundary conditions.
the first boundary condition continuity of
E tangential E x at boundary which is z equal
to 0, in these expressions put z equal to
0 and so the on the left hand side, the field
is E I incident field plus reflected field
x component of this x component of this should
be equal to x component transmitted field
x z equal to 0 this gives me 1 plus R A is
equal to T A.
Second thing is magnetic field continuity
of magnetic field. Now, magnetic field we
already know for a wave v is equal to k cross
E upon omega, what you get from here? For
the incident wave B, well please understand
k is in the x z direction E is in the x direction
so k cross E will be in the y direction. So,
I will write down simply v y is equal to for
the incident wave this would be k is omega
by c so this will be 1 upon c, A 0 exponential
minus i omega t minus omega by c into z is
the incident.
Let me call this as I B y of the reflected
wave I can write down similarly, as minus
1 by c because k reverse sign into A 0 exponential
minus i omega t minus omega by c into z. And
B y of the transmitted wave would be this
R also there I forgot R A this will be k upon
omega into A 0 into T A exponential minus
i omega t minus omega by c z. Just I have
used this expression to write the fields due
to three waves in three different forms.
Apply form there you can write down the H
which is equal to v upon mu 0 and the continuity
condition is that H y due to the incident
wave plus H y due to the reflected wave should
be H y due to the transmitted wave apply this
at z equal to 0.
The condition turns out to be 1 minus R A
is equal C k upon omega into T A the other
equation was 1 plus R A is equal to T A these
two equations can be added to obtain amplitude
transmission coefficient, which is equal to
2 upon 1 plus eta. Where, eta is equal to
C k upon omega of the refractive index and
similarly, amplitude reflection coefficient
is 1 minus eta upon 1 plus eta, so you get
amplitude reflection coefficient to be positive
because eta is less than 1. So far, omega
bigger than omega p, if collisions are weak
then eta is nearly real and R A is less than
1 but, positive. there is no phase change
on reflection on the other hand, so no phase
change on the reflection on the reflection
on the other hand.
If, omega is less than omega p in that case
eta is imaginary, when eta is imaginary R
A this can be written as suppose i alpha prime
for instance, that becomes 1 minus i alpha
prime upon 1 plus i alpha prime. So modulus
of this quantity will be one take modulus
of quantity which is 1 plus alpha prime square
under root denominator will also have the
same magnitude so there is 100 percent reflection
but, there is a phase change on reflection.
Well, we shall elaborate on this issues later
another important issue is that in this case
t is non zero amplitude transmission coefficient
is 2 upon 1 plus eta, if eta I put i alpha
it becomes 2 upon 1 plus i alpha prime which
is non zero; means if, I have a plasma over
dense which is omega p bigger than omega a
wave is coming in here, wave will be reflected
back from there there is no power transmission,
because k is imaginary there but, field is
not zero, there is always a field in this
region and that is called evanescent field;
So evanescent field exists here and how much
is the evanescent field.
Let me write down this expression, in terms
of T i will write down my transmitted field
is E T, x direction that is the direction
of polarization, amplitude was A into T A.
T A, I have taken as 2 upon 1 plus alpha prime,
and in the second medium the this goes a exponential
minus omega by c into alpha prime, this is
the value of k imaginary into exponential
of minus i omega t.
So your wave into z also there, its having
a amplitude which is decaying with z. So,
if you have the boundary here. If, I plot
magnitude of this quantity which is called
amplitude of the wave, this is i alpha actually
i is there. So, if I plot here say modulus
of E T as a function of z, because of this
factor it will fall down. So this is a wave
whose real part of k is 0, so as average is
0 in the transmitter medium, I am talking
about the steady state, and eventually. So,
there is no power transfer over there, if
there is no collision. I think further details
of such situations, we will discuss when we
consider specific cases; that is for that
is all for today. Thank you. .
