Welcome to the course on Modern Optics.
The course modern optics has the prime focus
for light matter interaction.
Light as you know is an electromagnetic wave
. When an electromagnetic wave travels through
a medium, several exciting phenomena happen;
this is basically due to the interaction of
the material medium with the electromagnetic
wave.
So, in order to understand this phenomena,
we will briefly touch upon and visit the basics
of the propagation characteristics of electromagnetic
waves ah in free space material media, the
medium could be lossy, it could be ah isotropic,
anisotropic and so on and so forth.
We will discuss all those things.
So, the content of ah the course is the Maxwell's
equation and electromagnetic waves this under
this we will discuss Maxwell's equation, the
origin, how Maxwell's equations ah came up,
ah the these are basically ah experimental
laws.
And then we will talk about the different
ah medium media, where the electromagnetic
waves propagate; followed by that the wave
equation in a dielectric medium.
So, so as you know that light is an electromagnetic
wave.
And this electromagnetic waves are best described
by Maxwell's equation.
And these equations are based on the experimental
laws .
You know these four Maxwell's equation del
dot D equal to rho; del dot B equal to 0;
del cross E equal to minus del B del t; and
the last curl equation del cross H equal to
J plus del D del t.
So, this first equation so called electric
field from a charge distribution has actually
come from the Gauss' law of electrostatics.
And the second law which comes from the from
that no magnet monopole exist this fact, which
is also a consequence of gau Gauss' law in
magneto statics.
The third law that is del cross E is an outcome
of the ah Faraday's law of electromagnetic
induction which tells that the time varying
magnetic field induces an electric field . The
last one that is del cross H equal to J plus
del D del t is ah is the one which where Maxwell
actually ah contributed by putting incorporating
that this displacement current.
So, this is current and time-varying electric
field produce magnetic field.
This fact is in incorporated in this law .
Now, this Maxwell's four equations looking
at the symbols and the notations rho is the
charge density; J is the current density;
E the electric field; and H, the magnetic
field.
D represents the the electric displacement;
and B the magnetic induction vector.
There are constitutive relations which are
associated with these equations particularly
with D, B and J . This D is equal to this
epsilon into E. This epsilon in general is
a tensor 3 by 3 - 9 component tensor; but
in a simple medium, it reduces to a scalar
quantity . Likewise B is also connected to
the magnetic field through this tensor mu,
which is in general a 3 by 3 - 9 component
tensor.
And for a simple medium, and for most dialectics
this mu is a scalar and has a constant value
. J is the current density which is again
connected to the electric field by the equation
sigma . This is actually the outcome of the
ohm's law . Now, this e dielectric permittivity
as I have told is the 3 by 3 tensor.
Magnetic permeability is also a nine component
tensor; sigma the conductivity of the medium.
These are the basic definitions and the terms
which are associated with the Maxwell's equation.
Now, going back with the history of Maxwell's
equation except the term, which represents
the displacement current that is this del
D del del t all other laws were already given
in these equations are were known before Maxwell
. Now, Maxwell introduced this concept of
this displacement current.
And by introducing this displacement current,
he could derive the wave equation and you
could predict that electromagnetic waves exist.
This we will see in the ah later stage that
how the experimental laws put together you
see the Maxwell's equation; and from there
you arrive that the wave equation.
From there you calculate a very important
quantity that is the velocity of the wave
and how you verify the experimental law.
Well propagation of electromagnetic waves
can happen in different medium natural medium;
it could be isotropic or anisotropic.
So, let us first of all try to look at what
is the isotropic.
If the material, if the if the medium through
which the electromagnetic wave propagates
is independent of the direction of the of
the propagation of the wave that is it behaves
identically whichever direction the electromagnetic
wave propagates, then it is called isotropic.
If the behavior the behavior of the material
in terms of the permittivity or more par ah
particularly the refractive index if it depends
on the direction of propagation of the electromagnetic
waves, then the material is called the anisotropic
material .
So, these are there are two broad varieties
of ah materials, homogeneous and inhomogeneous.
If the material part composition everywhere
will within the medium is the same, then we
call it is the homogeneous medium.
And if the composition varies from different
from place to place, for example, I start
with a pure at one end and in the medium the
other end it becomes ordinary glass, then
the material composition has varied and it
has become an inhomogeneous material.
Linear and non-linear, this is another important
aspect of the the medium.
If the properties of the medium particularly
the permittivity or the refract or the refractive
index if it depends if it is a function of
the of the electric of the intensity of the
electric field, intensity of the incident
electromagnetic wave, then it becomes non-linear
. If it does not behave with the intensity,
does not behave differently with the intensity
of the ah incident reflect ah incident electromagnetic
wave, then it is a linear medium.
So, these are the three broad varieties of
ah material medium through, which electromagnetic
waves are propagate and we will study ah each
of them in this discussion .
The medium could be dielectric medium also;
it could be interfaces and layers.
It could be an absorbing medium lossy medium
as well.
For example, when you take the interfaces
the of two dielectric media and if an electromagnetic
wave has to propagate to this medium, then
how this structure interacts with the electromagnetic
wave we will study that.
And if the if the medium is lossy for example,
any conducting medium, then how it works that
also we will study in this discussion .
Maxwell's equations in linear isotropic and
homogeneous dielectric this particularly the
first thing that we would like to emphasize
is that this is the case of this is the case
of a free space or vacuum that we will study
and we will arrive that the different ah equations.
Since, this medium is linear and homogeneous,
then the permittivity, permeability and the
conductivity all these parameters assume a
constant value . They are no more tensors,
if the medium is very simple.
And because it is an isotropic medium dielectric
medium, so these quantities are also scalars.
Assuming the medium to be charge free that
is if you assume that there is no free charge
in the medium, which is the most common case
then we put rho equal to 0 .
Putting that rho equal to zero that is charge
free reason, this equation becomes this, because
D you have simplified by epsilon into E, where
E is a con epsilon is a constant.
And similarly, del dot B has also become simplified,
because mu is again constant.
Del cross E this equation will take the form
of this mu times del H del t.
I have use that constitutive relations.
And for this equation, del cross H also, the
J is now again replaced by the constitutive
relations sigma into E. So, these are the
equations set of four equations, which now
represent the isotropic homogeneous and linear
dielectric medium.
And let us see how we organize this equation
to arrive at the wave equation.
So, let us take the curl of this equation
.
Let us take the curl of this equation . So,
this equation will give you del cross H del
del t of that.
Now, because time and space this derivative
they are independent, so we can put in this
form . And because del cross h is also known
to us from here, so we just substitute the
value of del cross h.
So, we end up with an equation which is the
curl equation from 1 d.
For the LHS, we use a very well known identity
that del cross del cross E is equal to this
using this identity.
Now, we have this equation del dot E equal
to 0.
If I use this in this equation, then this
quantity becomes 0 . So, this identity becomes
del cross del cross E is equal to minus del
square E the Laplacian of the electric field
vector.
So, arranging both the sides, we can write
this equation del square E equal to this quantity,
and the second order time derivative of the
electric field.
If we proceed in the similar way starting
with the third the fourth equation, and then
if we substitute the third equation into that,
then we will end up with a similar equation
for the electric field, del square H equal
to del H del t times this mu into sigma, and
the second order derivative time derivative
of the magnetic field.
So, we have a pair of two equations representing
electric field and the magnetic field .
Now, this pair of wave equations for the electric
and magnetic field, we can see that these
equations are the most general form.
Because if I if I use the specific properties
of mu, and sigma, epsilon etcetera, then they
will reduce and take the appropriate form
of the med of the medium through which will
study the properties of the electromagnetic
waves.
Electromagnetic waves in a linear isotropic
and homogeneous dielectric medium, these are
the most general form .
Now, the pair of wave equation this E field
and H field for we will study this equation
for a perfect dielectric conductivity, where
sigma equal to 0.
A dielectric perfect dielectric is represented
by this property that sigma equal to 0, then
this equation will reduce to this equation
because you have put sigma equal to 0 here.
And similarly for this equation, we will end
up with this equation.
So, the reduced form of the pair of wave equations;
for this case of perfect dielectric are represented
by these two equations .
So, for a perfect dielectric sigma equal to
0, the wave equations for E and H fields are
this as you have seen, these are vector equations.
But if you see that if I take the component
of E, for example, Cartesian components we
will see later the E x E y and E z that each
of the components will also satisfy this equation.
Similarly, for the magnetic field, in general
this is the vector wave equation, but each
of the components will satisfy this wave equation,
and then that equation which will be satisfied
by the components will be called the scalar
wave equation .
So, the general form of the vector wave equations
for E and H fields or now this but the Cartesian
components of this equation E x, E y, E z,
they will be again satisfied as I have mentioned
before.
If you consider 3 components of E or H field
in the respective equations, then the same
equation will represent the scalar wave equation.
Now, because each of the components, all the
6 components of the electric and magnetic
fields can be represented by these two equations;
we can generalize and we can write one compact
single equation for that in terms of psi.
So, there are psi the equation for psi will
represent the scalar wave equation, where
the where the psi can assume can take up any
value of this E x, E y, E z or H x, H y, H
z.
The Cartesian components are given by this
E x, E y, E z and so on spherical coordinate
in spherical coordinates these components
will be E r, E theta, E phi; and cylindrical
coordinates it will be like this.
Now, the solution of this equation represents
waves.
Maxwell predicted the existence of electromagnetic
waves, how he predicted after organizing this
wave equation, he could arrive at the at the
quantity v which represents the velocity in
terms of this.
So, this is the standard wave equation.
And this is the wave equation for electromagnetic
waves, where psi represents the component
of the electric fields in any of the coordinate
systems.
So, if you compare these two values, you get
v equal to 1 upon under root of mu and epsilon
all right.
So, comparing these equations with the form
of wave equation, we get that v equal to 1
upon mu naught epsilon 1.
By by this comparison, Maxwell could predict
that ah the existence of the electromagnetic
waves; and from this equation, he also predicted
the speed of the electromagnetic waves in
free space by knowing the values available
to him in that time .
For electromagnetic waves in free space, the
best value which was known to Maxwell was
for epsilon not, which is a scalar; and for
free space the value is this.
And for most dielectrics for free space the
mu naught the permeability is given by this
equation; these are very usual.
And using these two numbers, the speed of
the electromagnetic waves could be predicted
by plugging into this equation.
If I use these numbers, then the value of
the velocity of the speed of the electromagnetic
waves will come out to be this .
Now, in those days in around 1849, the speed
of electromagnetic wave, speed of light waves
were measured by several people FIZEAU's experiment
is very well known.
ah And he measured the speed of light as ah
this value.
You can see that these two numbers are ah
these two numbers are very close at least
by the order, and by the magnitude also.
Maxwell commented that these two numbers are
not accidental.
And during those time several other measurements
by these people, Weber agreed also this number
so well.
So, there were several experimental measurements
for measurement of the speed of light.
And Maxwell simultaneously predicted the the
estimated the value of the speed of light
and these numbers where matching.
So, by doing this, he could established that
electromagnetic waves that is the the ah light
is is the ah is the form of electromagnetic
waves .
Thank you.
