Hello 
and welcome to the optics and optical spectroscopy
and microscopy course, and in this lecture
what we are going to see is that how we are
going to develop the essential framework for
understanding a generic optical instrument
that we most probably would be using to investigate
a live system or in many biological system
or in any other soft condensed matter system
for that matter.
So, this would be an idealized equipment you
can think of, a schematic of equipment, then
we are going to parts out this equipment into
3 different parts and see what is there in
insides of these 3 different parts.
The idea here is that in the entire course
what we will be doing is that we will develop
a conceptual model of what an optical equipment
would be, that would be most probably used
to probe a living system or a soft condensed
matter system or for that matter any physical
or chemical study that somebody would like
to use 
and then what we will do is that this optical
equipment we will dissect out into different
parts as a layout now and then for each of
these parts we will go inside and investigate
what are the fundamental principles from ground
up we will go and then see what are the fundamental
principles on which this equipment is operating
on and what we can understand such that when
we develop a new technology based on optical
methodology that that we are studying in the
course, how we can leverage the various technical
advantages and overcome the limitations, okay,
so that kind of the spirit of the course and
which we will be moving forward.
So let us start by asking if I take an optical
instrument and then try to open up and see
what would be there in an instrument, what
do I expect to see?
Typically, so this is common for a spectrometer
or a microscope, mostly these are the two
things that we will be talking about, but
this is so general that, I am willing to bet
that any of the equipment that you see in
the lab lying around or in your office lying
around, you would be able to categorize this
equipment and other parts in one of these
main divisions.
So, if you open up an equipment, my hypothesis
as a claim here is that you would see a light
source 
and then 
opto-mechanical hardware, for routing the
light and then you will see 
detection systems okay.
So 
in our course, what we will do is that we
will follow this structure of first understanding
what is there in this light source?
How do we generate lights of various kinds,
mostly we will 
be here talking mostly about lasers of different
kinds 
and then once we generate the light, how do
we route them through various paths and optical
elements to impart some character to the light.
Then third is once this thus generated and
routed light interacts with the matter or
gets into the matter that we want to probe
or study, that the kind of detectors we use
or detector systems we use 
to 
probe and follow the signals that are generated
from the system of study, alright.
So, in order to do this in the course, it
turns out that we need to start from understanding
how the light itself is interacting with the
matter.
The premise of the courses is we are going
to probe the matter with using light and understand
how it interacts with the matter, in the process,
we try to unravel the principles of the dynamics
that might be happening in the inside of this
matter.
So, the first process in doing so is to understand
the light matter interaction itself.
The idea here is 
we will start developing or looking into the
formalisms that are existing to understand
the description of the light and matter of
interaction and how does the light interact
with the matter.
In doing so, what we will see is that we will
be able to appreciate and discover how the
operating principles of one of the prominently
used light sources are the lasers that will
be constantly using in all of our equipment,
at least that is part of this course.
So, as well, we would be able to understand
some of the signals that are generated because
of this light matter interaction, right.
So it is of very high importance for us to
pay attention to this, the formalism, and
this boils down to describing the matter per
se in terms of quantum mechanical states and
operators.
If you ask me, do I need to know quantum mechanics
to understand what is going to happen?
You do not need to, but the idea here is that
I do not assume that you would know, but if
you know it, it is better, but the point is
just because you do not know, should not preclude
you to follow this course.
The idea here is to provide you the basics
and give you the directions to follow to understand
and appreciate these principles fundamental
principles with the hope that tomorrow when
you are encountering with a problem that is
not necessarily taught in the course, but
something similar, you would be able to apply
that and then use your understanding and apply
your understanding there to appreciate what
could be happening there.
So, let us start by looking at the light matter
interaction itself.
For this, what we are going to do is as I
said, we are going to go into a little bit
of quantum mechanics.
In fact even before that, I am going to motivate
you to tell you a little bit about why we
even need to go into that, alright, and what
are the main differences and then again this
is not a full-fledged quantum mechanics course,
but I am just going to give you the glimpse
or the essence of what we need for this course
to understand how we can describe the interaction
of the light with the matter, good.
So, let us start by looking at this formalisms
itself.
Mainly as many of you might know, the quantum
mechanics per se was developed largely by
Schrodinger and Heisenberg in two independent
manner during the early days and then later
Dirac came up with a more generalized approach
and then showed how Schrodinger's approach
and Heisenberg's approach both of them are
really a special case or so to say a subset
of his more general treatment, and the notations
and the language that we will be using is
more close to Dirac's in this course.
Though I would have really, really stripped
down a lot of the details to keep it brief
and at the same time be informative, alright.
So, first let us look into how Dirac developed
this idea of quantum mechanics and what are
the basis?
I think if you were to in one line summarize
what triggered or what formed the basis of
his entire formalism, then I would say two
major principles.
One is called the absoluteness of size, it's
also called as absolute size, right, as against
relative size.
Second is the principle of superposition.
So, these are the founding stones so to speak
of Dirac's formalism.
So, let us look at these 2 to understand what
went into his theory.
So, the first thing that he realized is that
when you want to make a measurement like what
we are doing, I mean such as the one that
we are planning to do in terms of trying to
understand what happens to a system when we
are shining a light on to that.
What he realized is that the way we probe
the system, invariably interacts and causes
and disturbs the very system that we are trying
to probe.
Now, this is true uniformly across all scales
of the matter, but 
even though it is true all throughout, it
becomes particularly significant when the
sizes of the objects that we are dealing with
are comparable to that of the interaction
energies itself.
Okay let us simplify a little bit.
So, in classical mechanics, the mechanics
that we tend to take for granted in every
day, I mean in a day to day life that we deal
with, we implicitly assume for every object
that we try to study, we can always find a
smaller object with which we can probe the
system or the object that we are studying.
If you are studying a smaller object, then
you can find even more smaller object with
which you can interact.
That is the nature of implicit assumption
that we make, why do we make this and where
do we make this?
We make this because we neglect the interaction
or neglect the disturbance that this measurement
process or the interaction process itself
is bringing about to the system that we are
studying.
So, if I am trying to investigate how a cannon
ball is moving around in the outside world,
I kind of assume the way in probing that,
right, is in the light photon or using any
other means 
is not disturbing its trajectory.
That cannon ball is going on in its own trajectory
and just by me observing, I am not disturbing
that trajectory at all, right?
That is an assumption, and that assumption
is pretty good and valid when we are dealing
with sizes of that megascales.
However, that is not true and that is the
statement that Dirac made and that is an assumption
that he made, that is not true at all scales.
We cannot keep on extending it all throughout
to the micro cause.
So he said, at some level, this has to fail
and certainly it fails when it comes to the
level of atomic and subatomic particles and
objects that we are investigating in which
scale where the disturbance caused by the
interaction itself is non-negligible.
So, when that happens, we say that we have
reached the absolute size limit, we are not
able to find any more particles to investigate
the system without disturbing that.
Quantum mechanics deals invariably with systems
of that kind.
One of the major consequences of that is that
what we have taken in classical mechanics
has granted the causality, alright so, that
is not granted anymore at all because as you
can see the causal nature of this entire thing
assumes that you are not disturbing the system.
The moment the interaction disturbs the system,
there is no predictability You cannot predict
the trajectory while you are disturbing your
probe, trying to probe particularly that the
very act of you probing disrupt the system.
So, that is one and immediate consequence
to that is a statement of principle called,
I would like to call as uncertainty principle.
This tells you that because of this nature
of the quantum mechanics, the thing that you
have taken for granted, that is, the measurements
are known with infinite accuracy with very
high certainty, I call it as deterministic,
they are not deterministic anymore, they can
only assign a probability of you getting a
particular value for a measurement when you
are investigating a system.
So, that probabilistic nature brings in one
of very key principles that I am just going
to make use of to illustrate the point that
the quantum mechanics that we have stepped
down is not a very simple academic exercise,
I mean for that matter, it is not even an
esoteric exercise.
It in fact is very much integral to this entire
course, alright.
So one of the first statement of the principle
that I am going to make use of here is that
of telling you how accurate your measurements
of position and momentum could be, right?
When you are trying to make a prediction about
a trajectory, what you are trying to do is
that you are trying to make measurement of
the position of the particle as well as how
fast it is moving at that position, right.
So, if you know that with an infinite accuracy,
then you say that you know the trajectory
in a very deterministic manner, but the fact
when you go into this micro cause or microscopic
particles and objects and when we are trying
to probe this, this very deterministic nature
breaks down and it becomes very probabilistic.
As a result, several properties, new properties
that comes in, one of them being this uncertainty.
That is, if I were to measure my position
and then represent the uncertainty associated
with that position as delta x, then my uncertainty
associated with the momentum as delta p, then
what this principle states is that the product
of this 2 uncertainty has a limit.
The limit is given by Planck's constant by
4 pi.
So, what it tells you is that no matter whatever
the capacity of the equipment that you get,
it is not limited by your equipment or instrument,
but no matter howsoever an accurate or precise
system that you try to get, the accuracy with
which you know a position depends heavily
on how well if you are trying to make the
measurements simultaneously, the position
the momentum is tightly linked to how well
you know the momentum too.
The product 
of those 2 has a limit, I mean, it has a lower
limit of h by 4 pi, it cannot be any better
than this.
So, if you want to decrease this, right, in
which case you know this position really,
really accurately high position accuracy,
then what it tells you is that high position
accuracy 
implies low accuracy of momentum, right, because
only then this can be a constant here, I mean
the lower limit can be constant there.
So, now why am I saying this?
Remember, I told you that I am going to use
this to illustrate the point that it is not
an esoteric exercise, it is a very much needed,
very much related to what we are doing, alright.
So, part of this course is about microscopy,
right.
In microscopy, one of the important aspect
is our ability to tell apart how close or
how far two objects in space are, okay.
So, let us assume a point object here, it
has hardly any dimension, and then another
point object here, right.
So, often what do you like to ask is can I
tell this distance apart by looking into the
microscope and you say that you would be able
to tell these two things apart if your resolution
is at least this or lesser than that, meaning
lesser here means the objects that are even
smaller, I mean separated by smaller distance
can be still resolved.
So, now, this has been classically told us
these are the point sources 
and this has been classically equated to the
wavelength of light, I mean of the order of
the wavelength of light.
The idea that 
being there for a long period of time given,
so let me draw it out nicely for you here.
So we have a point source, another point source.
What we are asking is that since it is a bright
object, right, emitting light and if you had
to take capture an image of that and then
run a line profile that is to say, I m going
to draw a line across these 2 objects, and
then ask how does my intensity right, the
intensity varies as a function of space, my
x axis grows up like this, so what you will
see is that this is my intensity.
So, you would see the light going up, coming
back down and going up.
So, you could imagine that when you bring
this two point objects closer and closer,
there will be a point at which this would
be somewhere in between, there will be a point
at which they both would look or give rise
to one little peak her, not anymore 2 separate
peaks.
Now, clearly you can see this depends on how
wide these peaks are to start with, right,
how wide of a peak I get when I actually image
this point objected itself.
So, inherently, conventionally not inherently,
traditionally this has been called as a resolution
limit of a microscope, okay.
We will look at this in a much more detail
when we get into the microscopes and all that
stuff much later in the course, but it is
sufficient to say now here, when we have to
point sources and then we are looking at the
line profile, the profile being a line that
is drawn across an image of these 2 objects,
and then asking how the intensity changes
as a function of x axis and what I have done
is that I have brought these 2 two points,
right, closer and closer, right.
Next, the dotted lines are from points that
are of this origin and then the green lines
are of that origin, right.
So, let us say this is my green, blue dotted
lines would correspond to this distances,
right?
So now, I would say that it is customary to
call as a resolution limit and we say that
the particles are resolved whenever we are
able to find that the intensity has fallen
down to 1 over e between those 2 peaks, alright,
thanks to Abbe, this limit has been shown
as, I am going to just restrict to, of the
order lambda, where lambda is the wavelength
of light.
Now, what I am going to tell you is that this
very principle of there being a fundamental
limit with which we can resolve things, and
there is a fundamental limit to tell apart
2 things closely spaced in space using no
matter howsoever higher magnifying microscope
that we use, draws such basis right in from
the uncertainty principle that I described
before.
Let us look at that little bit more closely,
and where is that coming from?
So, ultimately, what we are actually asking
is that we are taking a lens and passing a
beam of light, and when you pass the beam
of light, and these are parallel rays that
I have drawn is called as a beam of light
that has these kind of rays, you call it as
collimated beam, clearly the reason being
that if you extend, if you let the beam pass
through a long distance, very very long distance
ideally, the collimated beams maintains its
width alright in space, no matter how much
ever distance it propagates.
Such kind of beams are called as collimated
beam, and then when you take the collimated
beam from simple high school physics, we know
that when you pass through a lens it focuses
at a distance, yeah the focal length of the
lens.
So what we want to ask is if a light consisting
of photons had to be focused, then what we
are trying to say is that the rays that the
lines that we draw here represents the boundaries
within which you have a higher probability
of locating that photon.
Now, that boundary is as we travel along in
this direction, we see that boundary is reducing,
we say that the light is getting focused.
When that happens, what we are actually doing
is that our ability to localize a photon to
our particular space increases progressively,
right?
Initially, your ability to localize the photon
if I were to slice up this region, right,
let us take a different color, so, let us
say the orange, and if I were to slice up
this regions in 10 sheets, then what I am
actually stating is that initially if it were
to be so dim a light, there is a one photon.
Then what we are trying to say here is that
that one photon could be present anywhere
within this shaded region, but as we progress
forward towards the focal length, the focal
point yes, then what we are seeing is that,
that distance has reduced from that to this,
right, some w1 to w2.
Like that, we can keep on going forward and
in geometry optics, we take it for granted,
it can be focused to an infinitesimally small
spot without any finite dimensions, but that
is not true.
In reality no matter however big or however
high focal length of lens that you have, however
good the lens that you have, when you focus
what you will see is that at the focus the
light occupies a definite width, definite
extent, it is not a no extent object.
It has a finite extent like the way we did
it previously to draw the profile, we could
draw the profile here and that could take
depending on the incoming a beam of light,
it could be a Gaussian or any profile, but
the key is it would have a finite width.
So, this is my distance from the center, this
is my distance 
from the optical axis.
So this is my optical axis.
What I am doing here is I am plotting the
intensity as a function of distance x, distance
from the optical axis okay.
Now, as you can see it is very high at the
center and then it drops down.
Now, what it tells us is that you have a very
high probability of locating the photon in
this place, but that it is not only present
here, there is also a final probability of
locating here, here, and so on, and I mean
if it had to be Gaussian, it goes to 0 only
at infinity.
That is that is to say your ability to localize
the photon are to form a tightest spot of
focus, right, where let us call this as our
omega f.
Now, there is a lower limit to omega f, right,
that is what it says, I mean no matter how
hard you try to focus with the best of the
lenses that you have, you still end up having
a finite width and that width is of the order
of lambda okay.
Now, why is that?
Now if you look at this diagram and all along
I have been talking about localizing the photon
in space, right.
So, I am actually trying to follow the position
of the photon, so that is x.
So, my uncertainty of localizing the photon
in space is given by delta x, which in our
case is basically omega f, right.
This is the width of the 
focal spot, then once we had transposed this
into the idea of uncertainty in position,
then we know the relationship, right.
The delta x of this photon, we are trying
to minimize this right.
We are trying to minimize it, so that it becomes
sharper and sharper, because the omega f becomes
smaller and smaller, this has a limit, what
is that limit?
We know that delta x photon times delta p
photon has a limit, right, given by uncertainty
principle has to be greater, the best can
be equal to that of h by 4 pi.
What does it mean?
It means the more and more you try to localize
the photon, the more and more you try to focus
it sharp, your uncertainty in momentum has
to increase okay.
So, now if that is the case, can I actually
write down an expression for p, so that I
can actually estimate how small a spot can
I found?
Like if I can actually get, I mean if there
is a limit to delta x, if I can write down
an expression for delta x in terms of properties
of the light photon, right, then I may be
able to see if there is a limit to it and
how does that compare to the conventional
or a traditional resolution limit that we
know of?
If I were able to show that these two are
same or equal or have the same order, then
my argument here is that you have a basis
for what has been taught or what has been
told to you as a resolution limit given by
Abbe, right, of microscope coming directly
from various different wave optics and other
equations.
Now here, we can trace down to the fundamental
principles of quantum mechanics predicting
what that limit would be, okay.
In the next class, we will write down the
p, the momentum of the photon and then calculate
the delta p, plug into this equation and then
write down an expression for x, delta x, which
is I mean clearly the lower, I mean the minimum
that you can actually go to, because it is
constrained by means it tells you a limit,
right?
So, it gives you a limit.
So you will get limit on delta x and see how
does that compare with the conventional resolution
limit, okay.
Thank you.
We will see you in the next class.
