Today, I would like to discuss the propagation
of electromagnetic waves in inhomogeneous
plasmas. Well, I will discuss a few cases
where plasmas with gentle density gradient
are found, and then discuss the propagation
of waves on the basis of WKB solution. And
then I will consider the case of oblique propagation
of an electromagnetic wave to a density gradient
in a plasma, and finally I like to discuss
the application of this analysis to a technique
called ionosonde to determine the electron
density profile in the earth ionosphere.
The reference for today’s presentation is
the book electromagnetic theory for telecommunications,
that professor C S Liu, and myself wrote and
published by Cambridge university press.
Well, let me cite a few example where realistic
plasmas are of inhomogeneous density variation,
for instance ionosphere look at the earth
ionosphere earth ionosphere is the region
of earth atmosphere which is ionized. So,
if you take earth like this then the ionosphere
starts at a height typically about 90 kilometers
but, the density suddenly does not change
from zero in the atmosphere to finite value
in the ionosphere, it gradually builds up
rather the density increases as you go up
so electron density end increases with height
its nearly zero at the boundary and as you
moves up it increases.
So, if you want to launch an electromagnetic
wave from the ground based transmitter, the
wave will penetrate into the ionosphere but,
the ionosphere is not of constant density
but, of gradual density variation, so this
is a typical example of inhomogeneous plasma.
Currently there is lot of interest in laser
interaction with targets for instance consider
a thin metal foil, if you launch a laser beam
on the metal foil and the metal foil and the
laser intensity is large, the laser penetrates
just a little in the material but, it can
convert this into a plasma and plasma expands
in time. So, after a little while of onset
of the laser, if you examine the density variation
of or the character of plasma that is formed
outside this is called a plasma plume.
The plasma is created at this boundary here
and this is expanding outside, so if you look
at this plasma plume, the density here is
the minimum and as you move into the plume
the density increases, so this is another
example of gradual density variation. Third
example which is of current interest is, what
we call as laser, this is the example of a
foil; metal foil and then there is an example
of laser gas jet interaction. what you do
in laser gas jet interaction?
You have a container that has a gas at a high
pressure and there is a small nozzle in here,
through which the gas comes out, the gas forms
sort of a plume here like this and you will
shine a very high power laser into this gas
then it forms a plasma here, in the region
through which the laser propagates. If you
look at the density in the plasma as you move
from one end to the other end, the density
varies at the edge here density variation
is rather rapid but, then it becomes quite
uniform then falls off. but its not that rapid
it increases in several wave lengths. So,
I can still treat the plasma boundary to be
a diffuse boundary rather than a sharp boundary.
So this is another example, where density
variation is important in many cases the density
variation can be taken be linear, so we say
that the density variation profile is linear
n profile 
like density can vary like n is equal to n
0 z upon L n means If I plot a density on
the y axis electron density and z is that
distance from the plasma boundary this I have
written for z bigger than 0 and this is 0
for z less than 0.
So, if you have this kind of profile then
the density if you plot this as a function
of z will be a straight line like this in
some cases you may have exponential density
profile, where density varies as some constant
n 0 exponential of z upon L n some constant
L n minus 1 in that case the density will
vary little more rapidly like this. So this
is called exponential density profile, this
is called linear density profile, this is
exponential both kind of profiles are observed
in realistic laboratory plasmas and as well
as in space plasmas.
Another important density variation that occurs
especially in laser plasma interaction is
called a parabolic profile, what you have
for instance you have gas jet target and laser
here like this. This is the laser intensity
variation, so when the laser travels through
a gas jet it produces a plasma on the axis
of the laser, whose density increases in this
direction as well as in this direction this
is say z axis and so I am considering a coordinate
system where x axis is here and z axis is
there.
So, in this situation density can vary with
x and the profile usually that one considers
say of this form density is minimum on the
axis laser axis and it increases with x like
1 plus x square upon some constant L n square,
n 0 is the density on the laser axis really
what happens that when the laser propagates
through a plasma its intensity being largest
on the axis and less outside it exerts a radiation
pressure force on the electrons in these directions
and the electrons move out, sometimes they
can carry the ions also along with them.
So there is a reduction in plasma density
on the axis, so this is called a parabolic
density profile. And this has been found to
be very suitable for guiding the laser. So,
this is a important problem in plasma physics
to examine the propagation of laser in inhomogeneous
plasma.
Before, I go into discussing the wave theory
of wave propagation, let me tell you something
regarding the qualitative aspects or physical
aspects of what do you expect really in such
situations. What you can consider is to be
simple, consider a general density variation
with z for instance, what you can do? You
can think as if suppose this is my z equal
to 0 plane from where the plasma begins this
is free space here and the density is increasing
for instance with z.
What you can do? Mathematically you can divide
this plasma into many layers of increasing
density but, in each layer the density is
constant, you know the refractive index of
a plasma is eta which is 1 minus omega p square
upon omega square to the power half omega
p depends on electron density.
So when density changes from place to place
omega p also increases, as you z increases
as a result eta decreases. So if you are launching
a wave in free space from here, at some angle
for instance then this ray as it goes from
a rarer medium optically rarer medium to a
denser medium sorry optically denser medium
to optically rarer medium, because plasma
is a rarer medium having refractive index
less than 1.
So this will move away from the normal, I
may draw this like this it will move towards
this strikes the boundary between this layer
and this layer again this will move further
away from here so it may go like this. So
the ray bends away from the normal as it goes
from one medium to another medium this is
what you physically expect, well one thing
is very important in here and that is called
the Snell’s law.
Which tells that if a ray goes from one medium
at an angle theta 1 and then it gets out in
second medium with angle of transmission equal
to theta 2, if the refractive index of medium
number 1 is eta 1 that of second medium is
eta 2 then n 1 sin theta 1 is equal to eta
2 sin theta 2 this is Snell’s law, to which
if you multiply omega by c, where omega is
the frequency of the light of the e m wave
that you are launching.
So then omega by c you multiplied here and
multiply omega by c here you know the product
of omega by c into refractive index is called
k. So this equation is equivalent to saying
that k 1 sin theta 1 is equal to k 2 sin theta
2, what is k 1? I am considering a situation
where density is changing in z and plasma
is uniform in the x direction.
So what you are seeing here, that k 1 into
sin theta 1 is the component of k vector of
the wave along the interface or perpendicular
to z axis. So I can call this is as k 1 x
is equal to k 2 x. Similarly, if you apply
the same Snell’s law at the boundary between
second layer and third layer this will be
equal to k 3 x and so on means as an electromagnetic
wave moves from 1 medium to another medium
this k 1 x, k 2 x, k 3 x will remain same.
So, I can call them as a simply k x how about
k z in any in any layer, so in any layer suppose
the refractive index is eta z then total k
is omega by c into eta z, so total k is decreasing
as eta decreases with increasing density but,
k x remains constant in order to have k vary
k z must change.
So what you really have, because k x square
plus k z square is equal to k square which
is equal to omega square by c square into
refractive index square this equation tells
me that k z square in any region, will be
equal to omega square by c square eta square
minus k x square this is a constant omega
is a constant, eta is the decreasing function
of z if the plasma density is increasing function
of z so k z will decrease.
So, what you are having? If your medium begins
at this point and if you are launching a ray
here like this, the ray as it travels into
a plasma of increasing density ray will bend
like this and eventually will get out like
this this is the direction of ray that we
expect the issue is that at somewhere k z
will become 0, whenever k x becomes equal
to omega by c eta then k z becomes 0, so the
point is called turning point where k z becomes
0, the wave vector travels in the x direction
at this point. This remember this is my z
direction this is my x direction. So at the
turning point 
this is called turning point, I will call
this z is equal to z t turning point.
K z is 0 which means eta becomes c upon omega
into k x, if the wave is coming from free
space this is free space at an angle of incidence
is theta i then k x can be written as omega
by c sin theta i because the magnitude of
k vector in free space is omega by c, if the
angle of incidence is theta i then the component
of this k vector along x direction will be
omega by c into sin theta i. So eta becomes
is equal to sin theta i at the turning point.
What is the consequence of this, what is eta?
A refractive index of a plasma is eta is equal
to 1 minus omega p square by omega square
under root and I am saying put this is equal
to sin theta i. If, you solve this equation
it gives you at the turning point plasma frequency
is equal to omega Cos theta i.
If, theta i is large then this quantity will
be smaller, so at large angle of incidence
this is my boundary between free space and
plasma and the rate trajectory is like this.
And If I had a higher angle of incidence like
this then this ray will come back early, this
is the first ray, this is the second ray,
the first ray was coming at a small angle
of incidence this is angle of incidence, the
second ray was coming at a large angle of
incidence so it travels only a little in the
plasma it comes out.
So this is a important consequence of physical
consideration of the or application of Snell’s
law in the propagation of waves in a inhomogeneous
plasma. Well one thing I would like to mention
in here, that if you want to determine the
rate trajectory the equation of that trajectory
of the ray can also be deduced on physical
grounds what you except that a ray is travelling
in a medium actually travels in the direction
of propagation I will call the direction of
propagation is n vector, this is the direction
of propagation 
which is the same thing as k vector upon magnitude
of k this is called the unit vector in the
direction of wave propagation.
Here, so far we have discussed the propagation
of waves with velocity v p phase velocity
we have not talked about group velocity, probably
we will discuss it today or some other time
the concept of group velocity. But, what really
happens if you launch a wave into a plasma
and the wave amplitude is limited in time
it is not a continuous wave.
If you launch a pulse, then you can write
down the wave field at the entry point as
some function of time and exponential minus
i omega t then later when this wave propagates
through a plasma then the phase changes with
z it is amplitude also changes with z, the
velocity of amplitude propagation is called
group velocity. So we denote this quantity
called v g velocity of amplitude propagation,
where as the velocity of phase propagation
is called v p velocity of phase propagation.
There is a relationship between the 2 v p
is denoted as omega by k and v g will be shown
to be equal to delta omega by delta k it is
a vector sign put there and vector sign there.
v g has three components v g x, v g y, v g
z delta omega by delta k x is called v g x,
delta omega by delta k y is called v g y,
delta omega by delta k z is called v g z.
In a isotropic plasma omega does not depend
on direction of k depends only on magnitude
of k, so in that case this becomes is equal
to delta omega by delta k is scalar into delta
k vector delta k scale upon delta k vector.
And if you differentiate k to k vector, what
do you get? Because k is equal to under root
of k x square plus k y square in general plus
k z square, so if you obtain delta k upon
delta k x, this will be equal to simply k
x upon k. Similarly, delta k upon delta k
y turns out to be equal to k y upon k, and
similarly delta k by delta k z you can write.
So, what happens that the group velocity can
be written as delta omega by delta k magnitude
into k vector upon k, which is n unit vector
the direction of a propagation in a isotropic
plasma.
Well, what is the consequence of this? If
your ray is travelling like this this is the
direction of your ray in a plasma, inhomogeneous
plasma then suppose I am considering the wave
propagation in x z plane this is my z direction
and this is my x direction. So my wave is
going in the x z plane, I am excepting that
this velocity is in the x z plane also, v
g has x component and z component also, what
happens? I can write down the rate trajectory
because when ray goes it the point moves from
here to here.
So the distance travelled by the light by
the electromagnetic wave in the x direction.
I will call as d x by d t is equal to v g
x the distance travelled by the ray in time
d t will be d x this will be proportional
to velocity put this is equal to delta omega
by delta k into k x upon k take x component.
And similarly, d z by d t is equal to v g
z which is equal to delta omega by delta k
into k z by k. If you divide these two equations
you can write down d x by d z.
And the result would be 
d x upon d z is equal k x upon k z. please
remember I just mentioned if the plasma has
variation in density along z axis then k x
is a constant only k z depends on z. So you
can easily integrate this equation and you
can write down d z into k z as a function
of z that you know and integrate this is equal
to k x is a constant into sorry I made a mistake
this is inverse k z inverse into d z is equal
to k x inverse into d x plus a constant of
integration. k z let me write down explicitly
k z as a function of z is equal to omega square
by c square refractive index is square as
a function of z minus k x square under the
roo.
So this equation can be easily integrated
and this is called the ray, the equation of
the ray. So I think based on Snell’s law
and physical considerations we have learnt
that if we know the profile of refractive
index variation with position, we can deduce
the ray equation by integrating this equation.
And obviously we have to know the angle of
incidence because k x depend on depends on
the angle of incidence of the ray. well with
this in production let me go over to discuss
the phenomenon of wave propagation in a inhomogeneous
plasma.
I will consider the propagation of wave in
one dimension first. In one dimension, what
you have? That I having a plasma where omega
p square depends only on z and my wave is
also going in the z direction, this is the
direction where density is changing or omega
p is changing, so this is the gradient of
n parallel to z and my wave is also going
in the same direction.
My electromagnetic wave I can write down,
E is equal to some amplitude exponential minus
i omega t at z equal to 0, say for instance
I would like to find out how much the field
looks at higher values of z, to be specific
i because I have already learnt that when
a wave travels in a plasma then the electromagnetic
wave is transverse. So the A vector the amplitude
has to be either in the y direction or x direction,
so without any lose of generality I choose
my x axis along the amplitude of the wave
or A in the direction of x.
So, I will choose this E for z greater than
0 let me call the initial amplitude to be
A 0 and let the amplitude afterwards becomes
A which is a function of z, I do not know
what is the x z dependence and time dependence
I will take as i omega t. So first let me
deduce the equation governing A and then under
certainly approximations we will solve that
equation.
The relevant Maxwell’s equations are curl
of E is equal to minus delta B upon delta
t this is the third Maxwell equation, because
the time variation I have a specified as exponential
minus i omega t, delta delta t i replace by
minus i omega, so it becomes i omega mu 0
H. The fourth Maxwell equation is curl of
H is equal to J plus delta D by delta t again
replace delta delta t by minus i omega J by
i omega as sigma E. So combine these two terms
and you will get minus i omega epsilon 0 epsilon
effective into E, this is how? The Maxwell’s
equations resemble a dielectric so this is
the effective plasma permittivity.
Now, these two equations can be combined by
taking curl of the first equation, so this
becomes curl of curl of E right hand side
becomes i omega mu 0 curl of H for curl of
H, I use the second equation so the right
hand side becomes omega square by c square
epsilon effective into E and this I can break
using vector identity into gradient divergence
of E minus del square of E.
The issue is, what is the value of divergence
of E? In my particular case because I am considering
the wave propagation to be along z axis. So,
I will choose delta delta z to be non zero
but, I will choose delta delta x to be 0 and
delta delta y to be 0, regarding the electric
field by incident electric field is in the
z x direction, so I want to choose E parallel
to x axis.
So, If I take the x component of this equation
you know that del operator means delta delta
x is 0, so this term vanishes means in this
particular case of wave propagation along
the density gradient this term does not contribute
at all. So, thus and del square becomes how
much D to d z square because other derivatives
are 0, so then this equation in this particular
case becomes a differential equation in one
variable.
And it becomes d to E x d z square plus omega
square by c square epsilon effective which
is a function of z into E x is equal to 0.
There is no approximation so far but, if epsilon
effective is general function of z, a general
solution of this equation is very difficult,
if this quantity varies very gradually with
z then one can solve this equation in one
approximation that I write down E x into a
terms of two functions; one called amplitude
function. So, let me write down this quantity
say A1 some function of z into e to the power
some i another quantity phi some function
of z, where A1 is purely real and phi is purely
real. I can call actually there is no need
to put may be subscript one here but, does
not matter.
So if this is the kind of dependence no approximation,
I am just writing any complex quantity E x
can be written as some amplitude which depends
on z and some phase term but, from over a
plane wave solution and a homogeneous medium
we know amplitude is a constant this is a
rapidly varying function of z as exponential
of i k z.
So, here I am saying that when the plasma
is inhomogeneous a still main z dependence
comes through phi and v z dependence comes
through the amplitude. So I am going to assume
that delta A 1 by delta z is small is much
less than delta phi by delta z, actually I
should compare this with something so I will
call this 1 upon A 1 and this as 1 upon phi.
I will say that this is a much stronger z
dependence than this 1.
So when I substitute this expression in the
wave equation, I will say that first derivative
I will certainly retain of A1 but, I will
ignore the second order derivative of this
so I am going to neglect d 2 A1 by d z square,
this neglect of second order of derivative
of A1 with respect to first order derivative
of A1 or second order derivative of phi with
respect to z is called W K B approximation.
So first of all, we will employ this approximation
and obtain the values of A1 and phi and then
justify under what conditions this assumption
is justified.
So, let me substitute this if E x, I am choosing
is equal to a let me call A1 a function of
z exponential of i phi then differentiate
with respect to z obviously there is a time
dependence is also there minus i omega t is
already there. So delta E x by delta z will
be how much this will be equal to delta A1
by delta z into this entire function plus
i delta phi by delta z into exponential minus
i omega t into exponential of i phi this is
one derivative.
Second order derivative would be d 2 E x by
delta z square is equal to d 2 A1 by d z square,
which I am going to ignore in little while
plus if you differentiate this you will get
I, d 2 phi by d z square there is A1 here
also, I forgot this write A1 here please write
A1 there into A1 plus I will get i delta phi
by delta z into delta A1 by delta z.
Then you start differentiating the exponential
term, so you will get i delta phi by delta
z into this terms so it becomes two times
and then you multiply the differential coefficient
of this with this term, so you will get minus
d to phi d z sorry delta phi by delta z whole
square into A1 multiplied by this exponential
terms minus i omega t into exponential i phi.
And this has to be put equal to in the wave
equation minus omega square by c square epsilon
effective into E x which is equal to this
whole expression so A1 exponential of i minus
i omega t exponential of i phi.
Now, please remember there are this is a common
factor on the right hand side as well as on
the left hand side this factor is common.
So they will cancel out this from here through
here and from here through here, they just
cancel each other on both sides cancel them
and equate the real part on the left with
the real part on the right and imaginary part
of these terms on the left should be zero,
because there is no left imaginary part on
the right after these exponential have been
canceled.
So what do you get, on equating the real part
means this this I am ignoring this is called
W K B approximations we ignore. So real parts
is simply this term equate this term to this
term and imaginary part is this term plus
this term is zero.
Let me write this this gives me delta phi
by delta z 
is equal to omega by c epsilon effective to
the power half. So phi can be easily obtained
from this equation, so the phase of the wave
a special part of the phase is omega by c
under root of epsilon effective to the power
half d z, because we had been calling this
quantity as refractive index. I can also write
this as omega by c refractive index into d
z. If, the medium are homogeneous this is
a constant you can take it out and d z simply
becomes z integration, how about the this
is by equating the real part on the left to
the real part on the right.
When you equate the real imaginary parts,
you will get imaginary parts give you d to
phi by d z square of A plus twice delta phi
by delta this is A1 rather actually this A1
everywhere. I made a mistake this is A1 into
delta phi by delta z into delta A1 by delta
z is equal 0, because there is no imaginary
term on the right hand side these two equations
can be combined into one, If I multiply both
terms by a quantity called A1.
So, let me multiply this by A1, so If I multiply
this by A1 this becomes A1 square and A1 i
multiply here but, this 2 A1 into d A1 by
d z becomes d 2 A square by d z and this can
be written simply as delta delta z of A1 square
delta phi by delta z is equal to 0, you can
just check it, which means this is a constant
term?
A1 square delta phi by delta z is equal to
constant, let me put the value of delta phi
by delta z from this expression here its omega
by c into epsilon effective to the power half,
which simply says that because omega by c
is a constant.
So this equation tells you that as the waves
travels A1 square into epsilon effective to
the power half is equal to constant. Initially,
if the wave started from free space I will
call this quantity initial amplitude of the
was A 0 and epsilon effective was unity in
free space.
So this is square A 0 square for A1 and this
what you get? So what you are getting here
the A1 amplitude of the wave in the medium
is initial amplitude A 0 divided by epsilon
effective to the power one-fourth 
or A 0 upon refractive index to the power
1 by 2.
This expression can be physically understood,
if you recall that as the that in a plasma,
the time average pointing vector which is
called intensity of the wave was equal to
modulus of E square upon twice mu 0 c into
refractive index. So, what is happening here?
As E if you put the value of E it becomes
simply A1 square, so it becomes A 0 square
upon eta will cancel out because 1 upon eta
will be from here ,and eta will cancel upon
twice mu 0 c.
So, what is happening? If you are launching
a wave into a plasma whose density is zero
here, and density increasing in the interior
then what you are excepting that the wave
amplitude after all total power of the wave
if reflection is ignored passing through any
unit area should remain constant. after all
whatever electromagnetic energy is entering
here must pass through here must pass through
here must pass through here everywhere it
must pass.
So, if there is no absorption as we are ignoring
absorption here by taking eta to be real as
the wave travels its energy crossing should
remain same. So, if this has to remain constant
but, eta is decreasing as the plasma density
is increasing the refractive index decreases.
So, E must increase that is why this amplitude
is increasing so just by to converse the energy
a denominator a refractive index under root
comes in the denominator this ensure the energy
conservation in the medium. So it is a important
physical interpretation of this dependence.
Well, it has a very important implication
let me elaborate on this suppose I have a
plasma whose density I am plotting like this,
suppose the density is increasing like this.
So, If I plot omega p here and plot z here
omega omega p square rather. because this
is proportional to density so this is my density
profile written in terms of plasma frequency
square and suppose at some point omega p becomes
is equal to omega, so this point is called
omega square means at this point. So, if the
wave is coming from here it will penetrate
up to this distance at this point omega p
becomes is equal to omega, up to this at this
point your refractive index becomes zero beyond
this refractive index is imaginary the wave
does not travel.
So what is happening, If I plot the wave wave
amplitude will be like this we have just seen,
so the if the wave at a given instant of time
if you see the oscillator electric field of
the wave plot it will be like this.
Actually from W K B theory, when eta goes
to zero your amplitude goes to infinity because
eta to the power half comes in the denominator
and there actually this theory fails, because
the amplitude variation becomes quite rapid
so d to a by d z square that we had ignored
its neglect cannot be justified so theory
does not hold but, if you do a little more
careful calculation then the amplitude is
not infinite. But, it is contained but never
the less as the wave travels, its amplitude
becomes larger and larger this is called plasma
swelling of plasma induced swelling of the
wave amplitude.
Plasma induced wave swelling, I am plotting
field at a given instant of time as a function
of z wave swelling this is a important consequence
of W K B theory that wave amplitude becomes
stronger and stronger as the wave approaches
the critical layer where plasma frequency
equals the wave frequency.
Now, a few remarks regarding the oblique propagation
for that is called a 2 D propagation means,
the wave is not traveling in the direction
of density gradient but, at an angle to density
gradient.
So, I have a plasma where density gradient
is like this around z axis for instance but,
my wave is coming at an angle I am launching
a wave like this. So, I would like to find
out as this wave travels into the medium,
how does it its field vary with z. Well, let
me specify the x axis also, I will choose
this as the x axis and this is my z axis and
I will choose my fields to vary in x and z
but, being constant in y this is called two
dimensional propagation.
So, I am choosing delta delta y to be 0 but,
delta delta x is non zero and delta delta
z is also non zero but, my properties of a
medium the density of the plasma is a function
of z alone. So from Snell’s law we except
that the wave vector of the wave in the x
direction will not change with distance it
will remain the same as it was in the beginning
this one thing that we can recall from there
but, we should be able to deduced the same
thing from the Maxwell’s equations and we
shall do that.
Another thing that I would like to mention
here there are two possibilities that this
wave may be polarized this wave may be polarized
perpendicular to the x z plane means there
are two possibilities; that the electric field
of the wave is parallel to y axis and second
possibility is the electric field is in the
x z plane is in the x z plane there are two
possibilities let us consider either of the
two. For the sake of simplicity, I will consider
the wave propagation with y axis this polarization
is called S polarization 
and the other one is called P polarization
plane of incidence polarization.
So, the issue is if my wave is travelling
in with electric field in the y direction
like this and the medium properties do not
change with y, you will just see that the
wave will maintain its y polarization whereas,
if the electric field of the wave is in this
direction and the wave travels something very
different happens what happens lets physically
see.
When the wave travels in a inhomogeneous medium
and it curves like this and if the electric
field is polarized in the plane of incidence
like this then as the ray bends this electric
field also bends, because it has to remain
perpendicular to the direction of propagation
its bends like this and if this point this
becomes like this, so this is case of P polarized
light. Now, there is a gradual rotation of
the electric field of the wave as it travels
in the medium and as you know from physical
considerations that this is a layer called
turning point, where omega p is equal to omega
Cos theta I, where this wave was having an
angle of incidence equal to theta i.
This is not equal to omega P equal omega.
But, what happens at the turning point, the
electric field of the wave is parallel to
density gradient because this is the direction
of density gradient. So when the electrons
oscillate parallel to density gradient they
always give rise to density oscillations.
Normally, we know that if the electromagnetic
wave travels in a plasma it does not give
rise to density oscillations but, because
of the background gradient and density density
gradient in background plasma density and
when the electric field is parallel to the
density gradient then it gives rise to density
oscillations. We have learnt that a plasma
has a natural frequency of oscillation is
equal to omega P, so if this wave can tunnel
from here to a region somewhere here there
will be a region called omega P equal to omega
critical layer.
So, if the gap between these two regions,
these two layers is not large, then this wave
can penetrate from here to here, and can resonantly
excite a plasma wave there. So, there is a
very special thing that happens here; therefore,
P polarize light as the wave rotates or changes
its orientation, the electric field also rotates
and it may give rise to large density oscillations
or the conversion of the wave into plasma
wave can occur. Well certainly this is not
within the limits of W K B theory, but this
phenomena we shall discuss sometime during
this course in some detail. So, this is a
basic difference between the S polarized light,
and P polarized light; the polarized light
is polarized perpendicular to the ray like
this, and its orientation does not change,
it does not give rise to any density oscillations.
However, in the region away from the turning
point like in this region etcetera, the W
K B theory for P polarize light, and S polarize
light is the same. So, we shall essentially
consider or rather limit our discussion to
region away from the critical layer or away
from the turning point, away from the turning
point 
S polarization, and P polarization have similar
character. And the mathematical analysis will
be taken up later to discuss the propagation
of waves at oblique angle, in a inhomogeneous
plasma. I think today, I stop at this stage.
Thank you very much.
