Welcome back, we are continuing module 2.
In this module, we have emphasized that electronics
are way too slow to measure both fast electromagnetic
oscillation as well as intensity profile.
If we try to measure a pulse with the help
of electronics, then what might happen a time
sensitive detector emits electrons in response
to photon.
Let us consider a photodiode whenever I shine
light on it, it will give me an electrical
response and that electrical response can
be monitored with the help of an oscilloscope.
Time sensitive detectors include photodiodes
including photodiodes and photomultiplier
tubes; they are way too slow that we have
already pointed out. They have very slow rise
and fall times and that gives you the response
time which is nanosecond.
So, the whole procedure how long does it take
to capture the light from here and plot it
in oscilloscope, this whole time scale is
going to be 1 nanosecond. Let us assume there
is 1 is nanosecond. Therefore, if I have a
pulse of 100 microsecond duration then with
the help of electronics, I will able to see
100 microsecond pulse I will be able to plot
it. Because for an 100 microsecond pulse electronics
is faster and I can plot it for 100 nanosecond
pulse I can plot it and I can see 100 nanosecond.
But if I shine femtosecond pulses to the photodiode
let us say 100 femtosecond pulse is going
to the photodiode then what will happen, photodiode
and oscilloscope both will show me a pulse
of 1 nanosecond that is all. And this is a
big surprise electronics cannot measure the
variation of 100 femtosecond variation of
the profile; it will just give me a slow response
of its own which is nanosecond response. So,
any femtosecond pulse we try to monitor with
the help of electronics that will become broad
electronic response in the oscilloscope.
We cannot just measure the intensity profile
with the help of photodiode coupled oscilloscope,
but if it is nanosecond pulse I can measure;
because the response time is equivalent to
the nanosecond. Slow detectors I have already
pointed out will measure time integrated intensity.
So, this signal if it is a slow detector,
then what I will get is a time integrated
over minus infinity to pulse infinity I(t)
dt this is what I will measure and this is
nothing, but energy content of each pulse.
What I measure is basically can be connected
to the energy pulses. Does detector output
voltage is proportional to the pulse energy
only. (Please look at the slides for mathematical
expression)
The slow response time of available time sensitive
detectors does not permit us to make time
domain intensity profile measurement of ultrafast
pulse. We cannot measure femtosecond or picoseconds
pulses with the help of the electronics.
We have to use the pulse intensity profile
to measure its own intensity profile and this
is this kind of measurement is called intensity
auto-correlation measurement. So, let us consider
100 femtosecond pulse, what I have to do?
I have to use this profile to measure another
100 femtosecond pulse. I am correlating this
pulse with this pulse that is why it is call
correlation measurement 
and this is the only way one can measure an
ultrafast pulse with the help of slow detector.
Then we do not have any issue we have a slow
detector, but we are correlating with two
short pulses and get this autocorrelation
measurement.
One of the simplest technique by doing autocorrelation
measurement is that we can use 50-50 beam
splitter this we will study very soon in the
pulse measurement content in the context of
pulse measurement. I can have 50 50 beam splitter
which will split the beam by 50 percent of
its parent intensity and then I can reflect
back and then I get this two pulses.
Now depending on the path length difference
between these two arm, this is more like a
Michelson interferometer and depending on
the path length between here to here. This
path length difference will decide what is
the delay between two pulses? If I use a bbo
crystal and produce 800, 400 nanometer from
800 nanometer pulse.
I can check the energy with photometer, I
can plot this energy as a function of tau
of 400 nanometer beam, and I can get points
at different delay and this is a profile which
is called autocorrelation profile that can
be directly connected to the intensity. So,
this is our way to measure the pulse direct
measurement. With the help of electronics
it is not possible.
We have seen that a Gaussian envelope can
be used to represent an optical pulse field
envelope that, we have seen. We will draw
one more important concept that we should
remember if I plot a pulse I will plot it
like the dotted line is the field envelope
and this green line is the carrier wave oscillation,
and this blue line is representing the intensity
profile of the pulse which is similar to the
field envelope profile, but they are slightly
different. And whenever we say pulse duration,
we have to remember it is the full width of
max of the intensity profile. This delta t
is a pulse duration contain the full width
of max of the intensity profile. So, now,
this is representing an isolated propagating
pulse and this Gaussian function representing
the envelope function.
Intensity can be given by this because I which
is EE*. We have seen already in this module
that this is the expression which is valid
for the intensity profile and then we can
always get Fourier transform of this time
domain pulse to get the frequency domain.
So, which means that I can get Fourier transform
of this time domain pulse and I get the frequency
domain. And once we get the frequency domain
field, then I can again take the power spectrum.
So, if you look at this two pulses, the intensity
profile is centered at equals 0. This is t
equal 0 profile looks like e to the power
minus 2 at square. (Please look at the slides
for mathematical expression)
On the other hand corresponding spectrum after
Fourier transform corresponding spectrum which
we measure is S omega[S(ω)] not E omega[E(ω)].
S omega when you measure is also Gaussian.
A Gaussian Fourier transform is another Gaussian,
but it is centered at omega naught now. We
have to remember that the expression for this
Gaussian is going to e to the power minus
omega minus omega naught whole square by 2
a. So, the expression we get for the spectrum
has different positions; one of the spectrum
is centered at omega naught and temporal intensity
profile is centered at t equal 0. (Please
look at the slides for mathematical expression)
Now these are the things which we are already
familiar with in this module and we have seen
that delta t delta omega are related. This
is called time bandwidth product will come
back to this time bandwidth product one more
time to understand more details of it.
In the previous slide; I have shown that,
the spectrum was represented in the omega
domain which is centered at omega naught.
But often when we talk about spectrum, we
represent spectrum in the lambda domain which
is the wavelength domain. And it is instructive
to understand how to convert this omega domain
spectrum to the lambda domain spectrum and
that can be done very easily omega which is
angular frequency which can be represented
by 2pi mu is optical frequency which is nothing,
but 2 pi c by lambda. (Please look at the
slides for mathematical expression)
Now if take the derivative d omega then we
get minus 2 pi c by lambda square d lambda
and this is an important realization whenever
we are converting frequency domain to the
lambda domain, we have to think about along
this line only. Now you know that area under
the curve would be the same area under the
curve representing the probability representing
how many such system are contributing so,
that will be constant always area under the
curve does not matter which domain represent
that should be the same. (Please look at the
slides for mathematical expression)
So, with this idea, we can make this area
under the curve to be the same which is integrated
area under the curve for both domains. And
once we represent it, then we get S lambda
which is spectrum represented in lambda domain
can be converted to the spectrum represented
in the omega domain or vice versa; both can
be converted with the help of this equation.
So, this is a simple mathematical trick we
need to employed to convert the omega domain
spectrum to the very frequently use lambda
domain spectrum.
While dealing with pulse intensity profile
and the spectrum, we have shown that the intensity
profile; this is delta t full width half max
that is called pulse duration. And in the
omega domain, this is omega naught corresponds
to t equal 0 and delta t corresponds to delta
omega and this delta omega is related to the
bandwidth and delta t is related to the pulse
duration which is intensity full width half
max or the temporal duration of the pulse.
So, this is pulse duration and this product
has a particular meaning for a particular
pulse delta t delta nu instead of omega we
can write down delta n u; we can convert omega
to delta nu very easily by 2 pi mu[μ]. So,
delta omega is going to be always 2 pi mu
that is all we can convert that very easily.
So, this 2 delta t pulse duration and delta
nu band width are related by time bandwidth
product. (Please look at the slides for mathematical
expression)
This concept is very important concept for
the ultrafast optics and spectroscopy. The
product of the temporal and the spectral width
is called time bandwidth product we have seen
that for a transform limited Gaussian pulse
the time bandwidth product is going to be
0.441, this is the time bandwidth product
for a Gaussian pulse.
This relationship which is similar to time
energy uncertainty principle in quantum mechanics;
time energy uncertainty principle in quantum
mechanics looks like delta E delta t is going
to be h cut by 2. So, it is quite similar
to this uncertainty principle, which suggest
that the shorter pulse requires a broader
spectrum. Time bandwidth product is minimum
for a transform limited pulse. (Please look
at the slides for mathematical expression)
What does it mean to have a transform limited
pulse? That for a given spectrum this is the
shortest duration pulse. I have often seen
student will confuse in this time bandwidth
product concept when they are thinking about
time bandwidth product; we have to remember
that delta nu is the cause, I need the bandwidth
to produce a pulse and this is called effect.
So, the right terminology is always to say
that in order to get a short pulse I need
large bandwidth sometimes by mistake we say
that.
If I can reduce the pulse duration the bandwidth
will change; I cannot change the bandwidth.
Bandwidth is something which is a characteristic
of the source. So, for a particular source
if I have a source for a particular source
delta nu is constant with this delta nu I
can have infinite possibilities infinite possibilities
of pulses I can have a pulse like this, I
can have a pulse like this, I can have pulse
like this many other possibilities, but I
will have one possibility one pulse having
the shortest duration.
And in that case only delta t delta nu is
going to be 0.441 otherwise delta t delta
nu is going to be greater than 0.441. So,
for a given source for a given bandwidth when
I have this delta t delta nu equals 0.441
it means that I have been able to obtain one
shortest pulse for a given source. If I do
not get the shortest pulse, then the pulse
duration always be longer and that is the
consequence of this time bandwidth product.
But question is this derivation we have seen
in this module already and the question may
come whether we always need to use a Gaussian
pulse to represent the intensity profile because
this equation is valid only for Gaussian pulse,
this equation is not valid for any other pulse.
And one more point I would like to make here
is that all though this equation is similar
to the Heisenberg time energy uncertainty
principle in quantum mechanics, but we cannot
get this number from this equation. We have
to use the derivation which we have shown
in this module for the Gaussian pulse.
And question is it always necessary to use
a Gaussian pulse to represent the experimental
ultrafast pulse. So, far we have seen the
Gaussian pulse envelop is called when we consider
Gaussian pulse envelop the time bandwidth
product becomes delta nu delta t equals greater
than equals 0.441 condition obtained for the
shortest duration pulse for a given spectrum.
Now, Gaussian envelops are most commonly used
in ultrafast optics because subsequent analytical
math becomes very simple and closely represent
an experimental ultrafast pulse; however,
it is not necessary that we have to consider
a Gaussian field envelop. One can assume many
other pulse envelop which may closely represent
an experimental pulse. Selection of the appropriate
field envelop depends on the experimental
observation. For example one can say that
a rectangular pulse can also be another way
to represent ultrafast pulse.
I have nothing 0 then suddenly I have this
V naught value then again coming down 0. So,
this is your 0 and this is your V naught.
A rectangular pulse is commonly used in electronics
and in signal transmission lines. We can also
consider this kind of rectangular pulse to
represent an ultrafast pulse theoretically;
we can take a look at the time bandwidth product
for this rectangular pulse. Procedure is known
to us now a rectangular pulse is depicted
here and V(t) is represented as the temporal
profile of the pulse. It is centered at t
equals 0 and this pulse can be represented
by 
this two equations. (Please look at the slides
for mathematical expression)
From theory of interference of plane waves
it is already known that a pulse is originated
due to superposition of many pure waves frequency
components. So, that is true for even rectangular
pulse which is shown here.
So, in order to produce this kind of rectangular
pulse in femtosecond domain let us say, it
is an hypothetical idea we are discussing
here. We are producing this kind of rectangular
pulse in femtosecond domain then we have to
consider where this pulse was produced with
the help of many frequency components because
only optical way one can produce pulse is
the interference. And we can immediately convert
this time domain pulse to the frequency domain
by Fourier transform which would look like
minus infinity to plus infinity, time domain
representation multiplied by e to the power
minus I omega t dt. So, I can get this frequency
domain representation. (Please look at the
slides for mathematical expression)
As a representative example, we have considered
here an ideal rectangular pulse which is centered
at t equals 0. So, Fourier transform of this
function will get V(ω) as shown in slide.
(Please look at the slides for mathematical
expressions)
And further we can write down this integration
we can simplify this. So, V(ω) can be solved
step by step as shown in the slides. (Please
look at the slides for mathematical expressions)
And if you look at the cardinal sin function
the power spectrum is going to be the power
spectrum can be represented by this P(ω)
which is nothing, but V(ω) square modulus
and of V(ω) which is nothing, but V naught
T whole square cardinal sin function square
omega T 
by 2. (Please look at the slides for mathematical
expressions)
The power spectrum plotted here and this is
the plot we have and it is evident that V(t)
is the rectangular pulse contains many frequency
components all these frequency components
extended up to plus infinity. So, theoretical
from 0 to plus infinity all this frequency
components we have, the amplitude of a cardinal
sin square function decreases by a factor
of half when this omega T by 2 becomes 1.39.
This is very well known number we will use
this number in non-linear optics as well any
cardinal sin function when you write down
this sin x by x which is nothing, but cardinal
sin function. (Please look at the slides for
mathematical expressions)
This cardinal sin function will drop to its
maximum intensity when x is 1.39. So, we can
use this only positive component of the frequency
see negative frequency is nothing, but mathematical
artifact is just as given to me because we
have taken both complex conjugate and the
complex number. So, we have to avoid considering
this negative frequency component which does
not mean anything. We have to consider only
positive frequency component and we would
like to find out what is the full width of
max for this and for that we are saying that
it is delta omega T by 2.
That is nothing, but 1.39 coming from this
characteristic of cardinal sin function and
from that we get the time bandwidth product
of an rectangular pulse which is nothing,
but 0.443 which is which is quite similar
to the ultrafast pulse for a Gaussian pulse.
We will stop here and we will continue this
module in our next lecture.
