Good morning and welcome to new session on
introduction to Photonics, so next couple
of lectures we going to sort of switch tracks
to a new module and before I do that, I thought
we should go back and review what we have
seen so far, so we started looking at science
of light as far as this course this is concern
from the prospective of ray optics, where
we look at propagation of light in terms of
straight lines that we call as rays.
And then we said okay, that is going to be
limited in terms of with its utility because
light may be treated as waves in which case
you have a face and you know wavelength that
can be used to describe light and so we would
be looking at wave optics and we are at a
point where we are at sort of at a crossroads
between wave optics and actually treating
it as you know, particles in terms of photons.
I say that because as you know in the last
few lectures we have been looking at the concept
of coherence of light and we defined what
temporal coherence and spatial coherence means
and that is to say in practical light sources,
there is a certain level of randomness associated
with the propagation of light, the certain
level of randomness associated with the generation
of light itself, we will come back and look
at that in the next few weeks, but as of now,
we are just characterizing light propagation
and we are saying that, you know, quantifying
that randomness is what we are trying to do
in terms of temporal coherence where we say
okay, light may actually be polychromatic,
may have multiple colours and because of that,
you may not be able to get perfect correlation
when you take one way front and interfere
with another, a delayed way front right, that
is what we have been seeing with Michelson
interferometer and of course we saw that we
could define a coherence time and through
that a coherence length and of course some
of you there are taking this course here have
been able to do this experiment in the lab
where your actually experiencing or figuring
out how to quantify temporal coherence.
Okay, and you also looked at spatial coherence
which essentially says that okay, your light
waves not going to be a plain electromagnetic
wave all times right, so there is no such
thing as a perfect plain wave in practical
sources so you typically, deal with a sub
tension of wave vectors or sub tension of
angles as far as the source is concern and
we are trying to quantify that through this
spatial coherence length which we have mentioned
as rosy over here and that spatial coherence
length determines is basically inversely proportional
to the angles sub tender by the source, so
in other words, we say if you have a large
number of angles sub tenders by the source
as in the case of a LEDs a very good example.
Then you may have very little spatial coherence
that means it is far away from being a plain
wave or on the other side, if you have very
good collimated source which can be direction
of, which can be, you know determine by particular
angle, just one particular angle.
Then, we say it is a highly spatially coherent
light wave, so this is actually just talking
about the randomness with which they are encountering
this light and these are essentially measures
of that randomness.
So what we will, what is a natural step from
here is to essentially treat this light waves
as consisting of photons.
Okay, and each of those photons may have a
sub wavelet and a particular direction and
so on, so you may actually be able to characterise
these sources a little more easily when you
go to the photon picture, but of course that
is founded by the fact that, when you are
talking about light emission or absorption
of light that tends to be quantized in nature.
Okay, that is what Max Planck initially discovered,
and then followed by Einstein, who said okay,
the propagation of light waves can be explained
in terms of photons themselves right.
So we going to come to that, but before we
do that if we look at what we wanted to do
initially is to from the wave of this picture,
we wanted to go to the electromagnetics picture
right, so let us spend a little bit of time
understanding that electromagnetic picture,
and through that we will actually, you know,
do one demonstration where you are quantifying
the modes of an optical fibre and through
that quantification you are also going to
understand how an optical fibre can essentially
act like a special filter, okay, so that is
what we are going to do you know in this lecture
is just capture a little bit of electromagnetic
optics of course, that is actually a fairly
large field once again, there are a lot of
finer aspects to it, but we will just pick
up some essential concepts from there before
we move on to treating light as quantum, you
know, consisting of quantum object called
photons.
Okay, so let me start here the learning outcome
for the next few lectures is actually identify
the fundamental principles of photon optics
and quantify photon properties, but before
we do that let just look at electromagnetic
properties of light.
Okay, so now we are treating light as electromagnetic
waves right, so, and of course if we are treating
a light as electromagnetic waves, what is
this I have to satisfy?
It has to satisfy the Maxwells equation, so
you first of all have to satisfy the Maxwells
equation which you know is probably remember
from one of your electromagnetic courses consist
of Coulombs law diversions of the electric
displacement current density is given by Robey,
your magnetic flux density is conservative,
magnetic fields are conservative in nature,
so the diversion of magnetic flux density
is zero.
Then you have your famous Faradays law which
says that you have a circulation of an electric
field whenever there is a time varying magnetic
field and then you have Amperes law which
was modified by Maxwell to include displacement
current density, because originally, Amperes
was looking at only from a perspective of
explaining the circulation of magnetic fields
through the current carrying conductor, but
of course Maxwell completed this picture,
saying that it could actually be through a
displacement current as well, right, so all
of this constitutes the Maxwells equation
with of course the constuents relationships
which says that your electric displacement
current density is related to the electric
field intensity in terms of through this constant,
sorry through this parameter called Epsilon
which is permittivity and similarly, we said
the magnetic flux density is related to the
magnetic field intensity through mu, which
is the permeability right.
So in other words, permittivity and the permeability
constitute the response of a given medium
to an applied electric or magnetic fields
right, so that is what is, it has to satisfy
and of course this is represented by the wave
equation right in the wave equations are essentially
for the electric and magnetic fields can be
written in terms of is called the Helmholtz
equation del Square E Plus, K square E equals
to zero and del Square H plus K square H equals
to zero, where K equals to omega root mu abs
lone, we that before represents the wave number
for a given medium right, so what we understand
from this, now of course , one of the things
that we are starting to incorporate here is
the fact that we are considering this electric
and magnetic fields okay which was previously
represented by certain wave, you know we said
wave having a certain amplitude and phase
but now we are talking about electric and
magnetic fields and there is a specific notation
that I have used vector notation right which
essentially says okay there could be a specific
orientation for this electric and magnetic
fields and if we look at that more carefully
let us say for the case of for a plain M wave,
let say is propagating in a positive direction,
which is essentially a solution of C wave
equations, you will find that the electric
field vector can be represented in terms of,
if it is a plain wave, propagating along a
positive these direction where are the field
components X and Y right.
So it can be X plus in terms of AX, EX plus
AY, EY and you could have a certain face difference
between the two components also in a generic
case, so you say, if J5 and this is propagating
in the positive C direction so it is going
to accumulate face as it propagating and once
you express the electric field like this,
than you have specific conditions where if
5 equal to 0 that is EX and EY are in face,
what do you get, you have a certain polarisation,
certain orientation with which the electromagnetic
wave is propagating, so that corresponds to
linear polarisation or if 5 equals to + -5
x 2 and EX equal to EY, then that electric
field vector is going to propagate like this,
is going to go around and you know the terminus
of the electric field is going to draw a circle.
So that is what we called as circular polarisation
and of course we can go into very specifics
and say if it is +5 x 2 that corresponds to
a left circular polarisation -5 x 2 that corresponds
to a right circular polarisation and any other
case, you probably have an elliptical polarisation,
so the whole concept of polarisation of light
comes about, you know, when you go into, when
you start treating this light waves as electromagnetic
waves.
Okay and there are certain manipulations you
can do, you know depending upon the response
of your medium, if it is a isotropic response
than the polarisation remains the same but
if you are going into a material which responds
differently for different directions or response
differently for different electric field vectors,
then you may start having, you know certain
polarisation changes and some manipulation
that we could possibly do, all of that will
come back to it later on, you just enough
to understand at this point that light actually
could have polarisation, could carry a certain
polarisation, right.
This is all stuff that you would have seen
in one of your courses in electromagnetic
fields, right.
Basic courses in electromagnetic fields, now
what we understand by solving this wave equation
is that, this key point that for given structure,
you have certain field configurations that
satisfy Maxwells equation, so everything as
far as electromagnetic waves are concerned
as to satisfy Maxwells equations and so for
a given structure it has to satisfy Maxwells
equation as well and that is what is being
represented by this wave equation, so what
will we find is for a given structure only
specific field configurations are allowed,
okay, are allowed by the Maxwells equations,
so what do we called this specific field configurations,
so these are called the Eigen modes, okay,
or simply modes of the structure.
Okay.
So you have this concept of Eigen modes or
modes of a structure that come into the picture
because these electromagnetic fields will
have to satisfy Maxwells equation within that
structure, okay.
So let us take you an example, a very very
simple example of how this happens, what is
the implication of this?
Okay, for a specific structure, so let us
take the example of a total internal reflection,
a waveguide, so we already saw in ray optics,
we took an example of an optical fibre, right,
to say that the optical fibre can be used
as a guide, where did we use that ride?
In that case, we used that for endoscopy,
right, so we defined basically a largest guide
which can be designed to support endoscopy
right, so let us take a similar example, let
us go back and revisit that example and now
we will revisit it from the perspective of
modes of the waveguide okay.
So let us say you have, once again like we
done before, you have a medium with reflective
index N1, another medium with reflective index
N2, such that N1 is greater than N2, you going
from a denser medium into a rarer medium and
we saw that if we come in with a light ray
which is a greater than a specific angle,
right, theta greater than theta C, theta C
defined as sign in words of N2 over N1, right,
so if we come in with that, what happens?
You have total internal reflection and then
we said okay, if I put another parallel interface
over hear and outside of that interface, once
again you have this medium N2, then you could
support these bounds right and you can make
a waveguide out of it right.
Now, of course that is because, that is theta,
this is also theta, right and the reflected
angles also theta, so that same condition
is maintained as long as those two interfaces
are parallel to each other.
Okay, so we say that could act like a waveguide,
now the question is, the key question is,
this seems to suggest, does any angle theta
greater than theta C right, according to this
picture, any angle greater than theta C should
be guided right, any ray with an angle such
that theta is greater than theta C should
be guided, so it is that true, does any angle
theta greater than theta C, survive in this
waveguide.
Okay, so let us examine that a little bit
and to examine that let just go back and look
at this propagation little more closely.
Okay, so clearly this wave vector over here
can be, you know decomposed into two constituent
wave vectors right, one pointing up and one
pointing to the right side, right and similarly
when we look at this wave vector this will
have a component, that is pointing down another
component pointing to the right side and similarly
over here also you have the same case.
So after every bounce, we see that there is
one component pointing consistently along
the direction of the waveguide, okay and that
component is called that travelling wave component,
okay because it is pointing to a wave that
is travelling in that direction, but what
about this other component, what about this
component?
Now, when we look at that component, we see
that before the bounce it is pointing up and
after the bounce it is pointing down, right.
So essentially the fields, there are these
two different fields that are interfering
with each other in this region, so we are
defining a particular ray right, so what is
that mean, that essentially means that we
are having these wave fronts, if I have to
draw this wave fronts, so we have these wave
fronts like these and these wave fronts after
the bounce go like this, right.
So essentially we are saying this incident,
the fields corresponding to do incident wave
and the reflected wave are essentially interacting
with each other.
Okay, so when the interact with each other
what happens, depending upon their relative
phase, they will have to have constructive
or destructive interference and key point
here is for an electromagnetic waves to be
supported in the structure, it has to satisfy
Maxwells equations and Maxwells equations
says that, I have to satisfy certain conditions
at this boundary, but that same condition
would have to be satisfied at this boundary
as well because the angle is the same, angle
of incidence is the same and then this material
is the interfaces are the same right.
So whatever interference condition or whatever
boundary condition that I am satisfying at
this boundary is the same boundary condition
that I will have to satisfy here, the same
boundary condition that I will have to satisfy
here every time it goes through the bounce.
Okay, in other words the electromagnetic field
configurations that I am satisfying in this
structure is symmetric about the centre and
it is got to be consistent on either of these
boundaries, one way of saying consistency
is that, if I am having constructive interference
over here, I should have constructive interference
over here as well, okay, so over the round
trip I should have constructive interference
criteria, if I am looking at the incident
wave and the reflected wave, the phase that
I have accumulated between the two has to
satisfy the constructive interference criteria.
Okay, so that can be simply written as, so
what is the wave component corresponding to
this, let us say, we said this is actually,
let us say this is the Z direction, the travelling
direction and X and Y component are perpendicular
to that, so let us define this as X and Y
is actually coming out of the screen, right.
So that would be Y, now what we are saying
is that the phase accumulated for this vertical
component, over a round trip has to be integral
multiples of two pie, so that it satisfies
constructive interference criteria, so I can
write this as, if I say KX corresponds to
the wave component in the vertical direction
and let us say this waveguide is got a dimension
D, so KX multiplied by 2D has to be integral
multiples of two pie, so it has to go through
constructive interference criteria, so the
same pattern, essentially the same field configuration
is consistent as this wave is going across.
Now, if I were to write KX in terms of K,
let say K represents this direction of the
wave vector, right, that is going, so if this
is K, what is KX?
If this is theta, this angle, what is this
angle?
That is also theta, so what is KX in terms
of K?
That will be K cost theta right and so I can
just write it as K cost theta multiplied by
2D, is what you have on the left hand side.
Okay, K sign theta would correspond to the
other vector, the traveling wave vector, but
this one that is pointing upwards will be
this K cost theta right.
So this can be written as two pie over lambda
and the it is not the free space lambda, it
is not the free space wavelength, what is
the wavelength of light in this medium?
Lambda over N1, so you have N1 here 2D cost
theta, so those is equal to two pie M right
and I can cancel this, so I get this 
expression which says cost theta would have
to be M times lambda divided by 2 times, N1
times D right, so that has to be satisfied,
so there is a yes, so there is a phase change
at the boundary, that is a good point, somebody
is just pointing out, but we are neglecting
that for, you know, for a fact, but yes that
is actually there is a component that basically
says, there is a phase change at the boundaries
also that has comes into the picture, right.
So we will have to take that into account
also, but I am neglecting that just to drive
home a particular point, so I am not being
completely regress about this, okay, but the
key point is only certain angles theta M right,
so only certain angles theta M which satisfy
this condition, will actually survive this
waveguide, so it will survive in the waveguide,
everything else will have to be just radiated
out, okay or it will not even enter the waveguide,
okay, so that is what we are talking about,
so we are saying only specific angles are
supported within this waveguide and this is
different from the picture that we had just
looking at ray optics, if we say, you know
it just goes through total internal reflection,
it can bounce and there we say any angle theta
greater than theta C should go through these
bounces and it should survive, but what you
will find his only certain angles are supported
and this is something that could be demonstrated,
so what we will do, is will send red light
through these waveguide and will show that,
the field configurations that are corresponding
to only specific angles are existing within
that waveguide.
Okay.
So that is actually a beautiful demonstration
we will do today, okay, but come back and
let us look at this a little more closely,
so what we are saying is, theta M is different
for different values of M. Okay, when M equal
to 1, right, that would have to correspond
to the smallest value of cost theta M, right,
because as M goes greater and greater, then
you will have, you know much more solutions
that are possible and cost theta M correspondingly
will be lesser, right, because cost theta
M goes to 1 when theta equals to 0 and cost
theta M when it is 5 by 2 that goes to 0,
right.
So what we are saying here is theta M can
take values, what are the boundaries for theta
M?
It has to be less than pie by 2 for sure,
right, but what is the lowest value it can
take, theta C, right, because anything less
than theta C is not even totally internal
reflected, so it will escape out, so theta
M is bounded within these values and if we
looks specifically for M equal to 1, cost
theta M has to be the lowest value or in other
words theta M is highest, right and for it
to satisfy this equation for larger values
of theta M, for larger values of M, theta
M will have to be smaller and smaller.
Okay, until it gets to theta C.
So and take, highest is 5 by 2, so what is
that condition correspond to 5 by 2, so if
this is my waveguide, 5 by 2 corresponds to
a condition where I am almost going straight
down that waveguide, right, so your fundamental,
so your, for M equals to 1, when we say, you
are going straight down the waveguide, that
will correspond to, what is call the fundamental
mode of the waveguide.
Okay, and for, so that corresponds to M equal
to 1 and for as you go to larger values of
M, the theta M value gets smaller and smaller,
so that will correspond to bouncers like this,
so this will happen for larger values of M
and the largest M that would be supported
in this waveguide will just satisfy this condition
of total internal reflection.
Okay, that is theta C, okay, so do you understand
that is fairly simple way of looking at modes
in a waveguide and specifically we are talking
about modes in an optical fibre.
In fact the effect of this 5 at the boundaries,
which we are neglecting we can actually bring
that back into this expression, we will actually
pose that out as a tutorial problem for you
guys to look at little more detail, so the
number of modes supported, what do you think
it depends on?
It depends on this lambda over D value, right,
so you will find that if D approaches lambda,
okay, only one value of theta M can be satisfied
and you will have only the fundamental mode
propagating, so you will find that as D approaches
lambda you will have only the fundamental
mode, but if D is far far greater than lambda,
then that allows multiple values of M, right
and that would also allow multiple values
of theta M to satisfy this expression.
Okay.
So when D is far far greater than lambda,
you have large number of modes another word
it is called the multi-mode waveguide, right,
so it is basically a multi-mode waveguide,
as D approaches lambda, the size of the waveguide
approaches a wavelength, you could do have
only one mode propagating in this waveguide
but as you open up the waveguide, you know
it is much larger compare to the wavelength,
then you have a large number of modes, that
are propagating.
In this picture by the way is not very different
from what you might have encountered in electromagnetic,
in terms of rectangular hollow metallic waveguides,
rectangular waveguides, cylindrical waveguides,
they all have the same picture basically,
but what we are considering here is an optical
fibre, which is a dielectric cylindrical waveguide.
Okay, but the principles are the same, since
you have boundaries in the structure, you
have this concept of reflection that comes
into the picture and whenever you have reflection,
you have interference and that interference
actually gives the concept of modes of the
waveguide.
Okay, right.
Higher the order most cheaper the bounce,
okay, so for example and this is something
we can probably demonstrate also, you send
red light through a multimode waveguide and
then you bend the waveguide, okay if you bend
the waveguide what happens, you are essentially
changing this total internal reflection condition
at this point, so when you bend the waveguide
some of those modes steeped, the steepest
modes that were guided in this waveguide,
may not satisfy the total internal reflection
condition, so they will escape out, so you
will see red light, normally when you have
a straight waveguide, you will not see any
red light coming off because everything is
actually guided within the structure, but
the moment you bended, I am talking about
a multimode waveguide, you will have some
of these modes escape the waveguide and you
can see that corresponding radiation okay,
so we will try to demonstrate that later on
today.
Okay, so let us move on, so let us try to,
get to, a little more specifics and so when
we are considering an optical fibre, we are
considering a core with refractive index N1
surrounded by a cladding with reflective index
N2, okay and if you analyse this optical fibre,
what we want to understand is, what are the
modes that can propagate as a function of
different frequencies, so in one side I will
write propagation constant okay, that takes
a value of 0 to 1 okay, so 1 means it has
got a very highly provolty that will be supported
by the waveguide and zero means is very low
provolty that will be going to be supported
by the waveguide, we will that as a function
of frequency, okay, if you look at this picture
and this frequency, we will in case of an
optical fibre, we will look at is as normalize
frequency, what we call as a V number, which
is defined as 2 pie A over lambda, it has
got a 1 over lambda dependence end that is
why it is called frequency term.
Root of N1 square minus N2 square, okay, have
we seen this before root of N1 square minus
N2 square, that is the numerical approach,
a very good, right, so you are familiar with
that and 2A, A corresponds to the radius,
so 2A corresponds to the diameter, which we
have mentioned as D in the previous picture,
okay and of course this function of lambda,
now the propagation constant which we will
mention as B is define as ineffective minus
N2 divided by N1-N2, okay, so it is defined
like this so that the analysis becomes little
easier, the solving the wave equation becomes
a little easier, okay, so that is why we are
defining it in these two parameters but B
going 0 to 1 can be represented in terms of
ineffective, ineffective B is zero, ineffective
would have to be N2, right and if B is 1 ineffective
would have to be N1, okay.
So what is this represent, this represent
a condition where you have a waveguide like
this, with a core like this, but your wave
is like this, so it is primarily propagating
with the refractive index N2, whereas this
condition corresponds to, if you have a waveguide
like this, this condition corresponds to the
fact that all your electromagnetic energy
is well confined within this waveguide, so
you are going from very poor confinement in
which case all the energy is supported in
the cladding to a condition, where everything
is confined to the core.
If you do, if you solve the Maxwells equations,
a wave equations and will look at specifically
what are the different modes that are supported,
what you will find is, something like this,
you a fundamental mode, which is also called
the LP 01 okay and then at a specific value
of 2.405, where does that come from?
It comes from the fact that when you solve
the wave equation and look at field configurations,
the field configurations will be expressed
in terms of Bessel functions and 2.405 corresponds
to the first 0 of the zero thought a Bessel
function, so you will find that, the next
mode comes about like this and that is called
the LP 11 mode and then add another value
of 3.83, you will see a couple of other modes
come into the picture, one is called LP 21
mode and other is called LP 02 mode.
So essentially what we are saying is, if I
manage to stay within this region, V number
of less than 2.405, then I have only one mode
gets propagated, okay or alternatively if
I have a condition where D which is 2A in
this case, 2A is much much greater than lambda,
then the value of V is very high, right V
could be 10 or maybe 100 also and if you have
that region, so you were looking at somewhere
over here and within that you going to have
all these modes, they are going to be, you
know 100 of modes here, so very high value
of V would corresponds to a multimode fibre
and then you could upvas have a fibre where,
for a given wavelength, the V is define somewhere
over here and you may be able to see in this
case only these specific modes.
The interesting part is, these modes which
are generally called as LP MN modes, M corresponds
to number of Maxima in 5 directions, what
is 5 here?
5 would corresponds to the azimuthal direction,
okay and N corresponds to number of Maxima
in R direction, R would corresponds to that
radial vector, okay, that is pointing radially
outside, so LP 01 for example, will have a
field configurations like this, it will be
fairly dense at the centre and then it will
fall off as you go outside in the radial direction,
so this will be LP 01, whereas LP 11 will
look fairly dense in the centre bird it will
fall off and then it will have two lobes.
Essentially those, so why is it called LP
11 because when you are looking at the azimuth
direction, you look at the half azimuth plane
okay, so within that half azimuth, because
one is a reflection of the other, within the
half azimuth plane, you will have one Maxima,
right and radially outwards you have only
one Maxima okay, so that is why it is called
the LP 11 mode and so you can imagine what
a LP 21 mode looks like, it is basically,
it is got these four lobes like this, once
again when we look at the half azimuth, you
are going through two Maxima, so that is why
you are calling LP 2 and then going radially
out words you have only one Maxima, so this
is an LP 21, so what is the LP 02 mode?
How is it going to look like, zero means there
is no variation in the azimuth direction,
so it is all uniform but it has two Maxima
in the radial direction, so how is that going
to look like, you going to have one central
peak here surrounded by another ring over
here okay, so that will correspond to LP 02
and what will find is, this is not just some
Wegh theory, you will find that you can see
all these modes, if you choose a fibre with
whose V number is somewhere, you know greater
than four right, so and what we see here is
the V number is not like a define quantity
for one specific waveguide, any specific waveguide
you have 2A and this NA defined, those are
all material parameters and structure parameters,
but wavelength depending upon the wavelength
that which we use it, the V number can change.
In other words let us say you have 10 micron
waveguide, you can come in with a wavelength
of 10 micron and you will find that it actually
like a single mode waveguide, you can come
in with a wavelength of 1 micron and you will
find that it has a different V value, it has
got a higher V value, so it will actually
support more, so you cannot generally call
a fibre single mode or multimode okay, so
it is single mode at a particular wavelength,
multimode at a particular wavelength okay,
so these are the concepts that will going
to see in the demonstration today, so you
overshot my time but let us stop here.
Thank you.
