Hello and welcome to the course on optical
spectroscopy and microscopy, and so far, we
have been looking at how well we can localize
the photons, particularly we were talking
about the relationship between the quantum
mechanical principles, particularly the uncertainty
principle and how it manifests in a routine
experiment that is done in the lab or in a
routine instrument that is operated day in
and day out by everybody.
For example, in this specific case, we are
talking about a microscope and in a microscope
we are trying to see how well we can separate
two objects that are closely spaced. Let us
say we said that we went ahead and defined
what is the resolution limit in this particular
case.
The resolution limit is defined as the closest
the two objects can be and still we are able
to say these 2 objects are 2 distinct objects,
or in other word, let us say if you were to
plot out if you have to take 2 points in space,
alright, point A and point B and then if I
were to image this right, if I have capture
an image of this points in space, what I would
do is that I would be generating equivalent
points in my images and what I am going to
do is an experiment wherein I am going to
reduce this distance, let us call this distance
as delta x.
How small a distance, I mean what is the smallest
distance these 2 points can be and still in
my image I am seeing it as 2 distinct points.
Now that is called as a resolution limit and
I said we will talk about this in more detail
and in a much more elaborate manner later
in the course, but then I said this is given
by conventional means as resolution limit
or where we can actually plot out the intensity,
right. We can actually draw a small line profile
across this image and then say I am going
to plot the intensity as a function of space.
So if I do that then what I will see is this
x and then the intensity here you will see
there will be a spike, each spike corresponding
to the position of the dots here. Now, as
we bring this closer and closer, so let us
I think I was taking an orange line there,
what is going to happen is this is going to
come very close, at some point it is going
to look like that before actually merging
into one single line.
The resolution limit, thanks to Abbe, is defined
as 
so if you plot out, if you change the x and
then if you plot out the intensity, then resolution
limit is defined as distance by which the
intensity has fallen down by 1 upon e, okay.
So now this distance yeah we would call it
as a resolution limit and 
so we said that this resolution limit is of
the order of the wavelength right, that is
what we have been seeing there right. The
resolution limit is of the order of wavelength
of the light.
So now today, what we are going to do is we
are going to continue that formulation and
then see how we can actually obtain this a
relationship from the fundamental principles
of quantum mechanics. The principle that we
are talking about is the principle of uncertainty
that is connecting, that is linking the uncertainty
between the position and measurement of the
position and the momentum okay of any particle.
So in this particular case, what we are trying
to do is we are trying to localize a photon.
I said okay now this resolution limit is tightly
linked to the idea how well we can focus the
light in space or how well we can localize
the light photon in space. One of the ways
we can localize the light photon is by focusing
through a lens and we were actually constructing
lens and then drawing the ray diagrams to
say how to estimate the size of the beam,
our size of the focus spot, alright. So if
you remember, so will reconstruct the lens
here, when reconstructing the lens what we
see is high school physics tells us that the
rays that are traveling incoming rays converge
at a point which is a f from the lens.
They are converging to a point at a distance
f where f is termed as the focal length of
the lens. Now what I said is that while that
is true in paper, but in reality what you
always see is that no matter how so ever good
lens that you take, it never is a point, I
mean there is a finite extent, point would
mean that there is no width to that spot at
all, that is not true, there is a finite extent
to which this light rays converge before they
start to diverge after the focal point. So
the right picture would be to actually draw
it out something like this and what we are
trying to measure is how small a spot or the
width wf can be okay.
So if I get it smaller and smaller, I can
localize the photon to a smaller and smaller
spot. So this wf to me is equivalent to delta
x, good. So if I need to know how small a
delta x I can which means I want to minimize
my inaccuracies in the photon's localization.
I need to maximize my uncertainty on the momentum
side, right, because our delta x times delta
p is greater than or equivalent to h by 4
pi.
So now what we do is that we write down what
the momentum of the photon itself is. So the
P the momentum of the photon, photon's momentum
is given by h cross times k vector or kr or
in a simple term we can write it as h by lambda
times k dash okay, sorry no I am going to
use a different notation here so that we do
not get confused. So we can write it as kappa,
we can write it as kappa cap where kappa cap
is the unit vector along the direction of
propagation of the photon, right.
So normally you would be more familiar with
writing the photons momentum, right. We write
as h by lambda, there we are actually talking
about this, that is the modulus of P which
is having only the magnitude, right. We are
used to writing it as h by lambda, but for
this discussion, I need to pay attention to
the direction, so what I am going to do is
I am not going to write the modulus of P,
but actually I am writing the P, the vector,
the momentum vector itself.
For that I need to include the direction and
that direction is coming through the kappa
okay, kappa cap the unit vector pointing towards
the direction of the propagation of the photon
itself. Now given this, then how do you write
the uncertainty associated with the photons
momentum? So there are 2 things to think about,
one is h is the Planck's constant, so that
is a constant, so there is nothing uncertain
about it, but there is lambda. It is very
tempting to think that the uncertainty in
the momentum could come from the uncertainty
in the lambda itself, but let us take a step
back and then see what it means.
What it means is that the impinging light
into the lens has multiple color, each lambda
correspond to different colors. So you can
think of that is a spread in colors of different
light and then we are trying to focus, as
a result you are having this spread in the
localization of the photon per se. However,
that runs into a big problem there. The problem
being that I could in principle provide you
a monochromatic light, a light with one single
pure wavelength to very great accuracy okay.
Still we would see that there is a definite
amount of extent to which you can actually
focus and not any below than that, which means
the uncertainty has to come from somewhere
else, it is coming in reality, so that uncertainty
if you watch carefully, one of the other parameter
that could change is this k, the Kappa vector,
the unit vector that is directed along the
direction of propagation of the photon itself.
Where is the uncertainty in that?
The uncertainty if you look at the ray diagram
of the light that is emerging out from the
lens, it would become much more clear. So,
we have a lens and the light ray, I am going
to go into the geometric optics where it is
point it is easier to draw and propagate.
So there are two different directions the
light rays converge onto this point, one from
the top we will call that a top and the other
from the bottom. So a photon that you find
it here could in principal could have come
from a path that is indicated by the light
ray on the top.
So it could have come from a light ray that
is on the top indicated by this ray or could
have come from a light that originated from
the bottom alright, the yellow is not very
visible, so let us take the red okay. So we
would not be able to tell where it came from,
or in for that matter, it could have come
from any line, any light ray that is in between
these two okay. So what we are going to say
is that there uncertainty regarding the light
ray or the direction of propagation of the
light because of which the photon is localized
in this place.
So we need to consider the momentum, the direction
of the momentum vector, particularly the K
vector, Kappa vector that is along this to
this. So, now we are going to take two extreme
cases right. The maximum uncertainty is between
the top to the bottom ray, anything else is
lesser than that. So our idea here is to maximize
uncertainty in the momentum so that we can
minimize, we can get the minimum limit associated
with the delta x uncertainty with the position.
So to do that, what we are going to do is
we are going to resolve this momentum, right,
the momentum of the top vector we will call
that as Pt, so that is the top vector and
the bottom one we will call that as Pb. See
ideally I mean these two are in the magnitude
wise exactly same, the only difference is
in the directions, right, because they are
the same photons and the focus by the same
light, I mean same lens. So now what we need
to do is we need to figure out what is happening
at this place.
We have to take the difference between Pt
minus Pb and that would be our, this difference
would be our delta P. Now how do we do this?
We realize that it can be simplified nicely,
moment we recognize that this component, right,
the component on the top ray and on the bottom
ray can be resolved into its components, this
vector can be resolved into these components
okay. So I am going to redraw this diagram
for clarity.
So what we are looking at is this triangle,
that is my optical axis which is nothing but
a line that connects the centre of the lens
with the focal point about which all the rays
converge on to it, so and then we have an
angle theta and it is called as the angle
of focus alright. So now what I am trying
to say is that I am going to resolve this
momentum vectors in 2 components. So since
this is two, this is theta, this is going
to be theta and this component that is parallel
to the optical axis would be Pt cos theta
and this would be Pt sin theta.
Similarly we could resolve the light ray that
is traveling from bottom towards the focus
in 2 components, right. So there is a component
that is in the upward direction because it
is converging up so 
and the parallel component, this is our theta.
So, now this would be my Pb cos theta and
the other component would be Pb sin theta.
Using this we can actually substitute in here,
when you calculate the delta P, what you will
realize is that the component Pb sin theta
and Pt sin theta, cos terms will cancel out,
how does it work.
So now, let us see. If you write down this
expression delta P which is modulus of the
Pt which is Pt cos theta + Pt sin theta minus
Pb cos theta + Pb sin theta. Now if you look
at the diagram carefully, you will realize
that Pb sin theta and Pt sin theta are in
opposite direction. So if you take the negative
of that, so what would happen is that Pt cos
theta canceling out with Pb cos theta because
of the negative sign. However Pt sin theta
minus Pb sin theta, the term that is left
out, you will realize is they are pointing
in opposite direction.
So as a result what you will get is 2Pb sin
theta, they will not cancel, they would actually
add up because it is sin of minus theta, so
it is minus sin theta, so minus minus plus,
so that will you will get the component as
delta P equal to two 2Pb sin theta.
Now that is interesting because we can take
this and then substitute in our expression
connecting in the uncertainty principle. So
what we have is delta x times delta P equals
h by 4 pi, not equals but it is greater than
or equal to h by 4 pi, and we have an expression
for delta P, the expression is two modulus
2P sin theta right. So delta x will be greater
than h by 4 pi times 1 divided by 2P, momentum
of the photon Pb or Pt does not matter but
both of them are equal, sin theta and this
is nothing but greater than or equal to h
by 4 pi 1 divided by modulus P2 sin theta,
which can, this whole this term okay, can
be written as h by 8 pi.
Now we know that the momentum P for a photon
is h by lambda right. So if you substitute
that what we will have is a sin theta term
unhindered divided by the modulus P if you
write it, it will be h by lambda. So all of
this would imply or can be simplified as will
boil down to lambda, the one the denominator
goes up, divided by 8 pi sin theta. The first
thing we realize is this is of the order of
lambda, not just that, if you actually look
at the expression, expression for the resolution
limit.
I intentionally did not write down that expression
before, but now we can go back and look at
the textbooks or any optical literature sites
talks about diffraction limit, you would come
across the resolution limit to be resolution
limit would be given as lambda over 2 times
the numerical aperture okay. So what is numerical
aperture, numerical aperture in turn would
be given as lambda divided by 2 into mu sin
theta. Now compare okay, this expression,
you will see there is a striking similarity
right.
The uncertainty principle tells you that the
delta x 
can only be greater than this, right, and
then your resolution limit well what people
have gotten is this. The only difference that
if you take the ratio between these two, the
ratio would be actually 8 pi by 2 mu factor
okay. We predict the resolution limit to be
lower from quantum mechanics than what has
been told, but that is only a difference in
the factor, the functional form, the dependence
on the lambda, the dependence on the sin theta
that, the theta is the angle of focus is very
very striking.
This notwithstanding if you are still wondering
why and how this quantum mechanics are treatment
of the matter using quantum mechanics is relevant
to studying spectroscopy and microscopy, I
am going to give you one more, I mean one
more striking example and then show you that
it is really important to pay attention to
this fact that we need to take an approach
of understanding or using quantum mechanics
to understand how light interacts with the
matter. The second one again makes use of
uncertainty principle.
This starts with a slightly modified version
of an experiment, I am going to call that
as thought experiment, I am going to propose
it as a thought experiment and then I am going
to introduce a different form of an uncertainty
principle and then show you how that can be
useful in predicting the results there. So
in summary, what I have told you is you can
use uncertainty principle to arrive at an
expression for the resolution limit or the
localization limit of a photon which is very
strikingly similar to a conventional expression
that is thrown out of a kind of nowhere.
I mean you can derive it from the first principles
of diffraction, but it is a lot more complicated
and involved while this is extremely straightforward
and comes out naturally okay, and I am now
going to go into the second experiment and
we will describe the experiment before actually
going into the details of writing down this
expression and showing how it is really predicted
by quantum mechanics. The experiment is as
follows.
Let us take a laser light source okay. So,
this could be a very ubiquitous laser such
as helium-neon laser operating 
with a central wavelength of 633 nanometers
wavelength, right. It is a red light that
read HeNe that you use to do several things
in the lab. So now the light beam is coming
out of the laser. What I am going to do is
I am going to intervene this light beam with
shutter, a mechanical shutter or a light chopper
whichever you feel like right. So, I am going
to represent the shutter as something like
this.
So when it is closed, no light, and when it
is open, so okay let us represent the shutter
with a different color of the pen, so red.
So when the shutter is on, no light, but you
can have a different configuration where the
shutter is off okay, so you close down the
shutter, so in which case let us represent
right this is off state and this is on, off
the shutter. These are the states of the shutter
and what you will see is that I am going to
have a small detector here, a small device
that tells you whether the light is present
or not okay.
If 
I plot the detector output, whenever there
is off state, I will have the light coming
through and when the shutter is on, I will
have no light coming through okay, so intensity
of the light as a function of time. Now, I
can keep doing this off and on, off and on,
right. So, then what I am going to get? So,
if I repeatedly operate the shutter, what
is going to happen is that the shutter is
off, light is on, shutter is on, no light,
again as and when you keep opening and closing,
you will get the light off and on and closing
on.
So now, I start to increase the speed at which
I am actually closing and opening the shutter,
what you would expect is that it will become
shorter, the duration will become shorter
if assuming that I am uniformly closing it
off and on. Like that, one can conceptually
ask a question how fast can I actually keep
going, is there a limit to the speed at which
actually I can operate? Given that it is a
mechanical shutter, you can see that okay
there is a limitation in terms of the instrument
that you can construct and then there may
be a limit.
However, conceptually if we think of that,
there is no limit, you can actually operate
at any speed that you would like, right, because
it is like a mass less shutter and you would
be able to really really go very high speed,
but the question that you would like to ask
is how does the detector output look like
when we go actually at a very very very high
speed? Would I keep getting this light all
the time? It turns out that answer is a little
bit a little bit more complicated.
However, what is I mean grossly true is that
at some limit or some speed beyond which you
will not get any light output at all despite
the fact you are opening the shutter, well
that is very surprising, right? You keep,
you are opening and closing, only thing that
you are doing is you are increasing the speed,
at one speed, you start getting any light
out from the device. The fast operating shutter
starts to act as if it is a stop after laser
light.
There are quite a few qualifiers to this statement,
but that is roughly the on an average behavior
that you would observe, why is that? What
is happening and how is it relevant to the
observations of the experiments that we are
going to do and study in the course? We will
do that in the next lecture and what I am
going to do is I am going to make use of a
different form of an uncertainty principle
and tell you this is exactly what you would
expect.
In fact, I would even say when you would expect
this. I would even be able to predict at what
speed you will be able to expect this and
what governs this, alright. With that, I will
hope to see you in the next lecture. Bye.
