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In the last lecture we had seen that in a
single mode fiber in addition to material
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dispersion, we have what is known as wave
guide dispersion.
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And these material dispersion and wave guide
dispersion they occur due to finite line width
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of the source, because of different wavelength
components that are present in the light coming
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out of the source.
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So, these 2 dispersions are also known as
chromatic dispersions.
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Wave guide dispersion occurs because the mode
propagation constant of LP01 mode of a single
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mode fiber depends upon the wavelength.
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So, in the last lecture we had worked out
the expression for the wave guide dispersion
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by first calculating the group velocity of
the mode.
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In the process of working out the wave guide
dispersion, we had assumed that n1 and n2
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that is the refractive indices of the core
and the cladding do not depend upon the wavelength.
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So, we had not considered the wavelength dependence
of the refractive index of the material of
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the fiber, in order to explicitly bring out
the effect of wave guide dispersion.
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So, in this lecture we will see on what parameters
of the fiber the wave guide dispersion depends
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on and how can we tailor the wave guide dispersion.
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So, the expression that we worked out for
wave guide dispersion coefficient is given
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by Dw is equal to minus delta n2 over c lambda
naught times Vd2(bV) over dV square.
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We know how b depends on V.
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So, if we work out this V times d2(bV) over
dV square then we can find out wave guide
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dispersion.
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So, apart from this which depends on the value
of V only, in this expression we also have
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the cladding refractive index and the value
of delta that is the relative index difference
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between the core and the cladding.
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When we plot this dispersion coefficient as
a function of wavelength, what we find that
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wave guide dispersion is always negative for
silica glass fiber, for these particular values
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of different parameters corresponding to step
index fiber and if I look at material dispersion
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for few silica glass it goes like this.
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So, I see that it is negative for wavelength
shorter than 1.27 micrometer, and it is positive
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for wavelengths longer than 1.27 micrometer.
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So, the total dispersion which is the summation
of the material dispersion and wave guide
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dispersion is 0 around the wavelength 1.31
micro meter, and below this wavelength the
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total dispersion is negative and above this
wavelength the total dispersion is positive.
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What is the meaning of negative and positive
dispersion let us understand that.
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Does it mean that negative dispersion compresses
the pulses and positive dispersion, expands
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the pulses broaden the pulses no, move the
dispersions will broaden the pulses?
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If the input pulses unchirped if the input
pulses unchirped then both the dispersions
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negative dispersion and positive dispersion
will broaden the pulses.
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Let us understand the meaning of negative
and positive dispersion.
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So, to understand this first of all I should
understand why the broadening happens.
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The broadening happens because the spectral
components have different velocities; the
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mode which is excited at different spectral
components has different velocities.
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Now when these components have different velocities
then I can have 2 possibilities, one possibility
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is the shorter wavelength components travel
faster then longer wavelength component, and
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another possibility is longer wavelength components
travel faster than shorter wavelength components
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ok.
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So, what happens is that if I plot to understand
this if I plot group velocity as a function
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of wavelength for a typical silica glass fiber;
then it goes like this the step index fiber.
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Now let me pick up a point somewhere here
which is less than 1.3 micrometer wavelength
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then I know at this wavelength I have just
seen that at this wavelength the dispersion
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is negative.
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If I take my source at this wavelength, and
then this is the center wavelength and then
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I see the components in the vicinity wavelength
components in the vicinity of the center wavelength
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then I find that the shorter wavelength components
have smaller velocity, while longer wavelength
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components has larger velocity.
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So, shorter wavelength components travel slower
here and longer wavelength components (Refer
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Time: 06:29) travel faster.
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So, if I launch a pulse at this center wavelength
and this is unchirped pulse, then this the
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fiber at this wavelength gives you negative
dispersion and pulse broadens because the
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shorter wavelength components travel slower
than longer wavelength components.
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So, the pulses broaden, but what I also see
in this broadened pulse that the frequency
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content in the leading and trailing edge is
different; what you see that in the leading
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edge the frequency is lower while in the trailing
edge the frequency is high.
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So, the pulse not only gets broadened, but
also gets chirped and chirping is like this.
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While if I take my laser source somewhere
here, then I know at this wavelength the dispersion
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is positive and if I look at the group velocity
of the shorter and longer wavelength components
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in the vicinity of this around the center
wavelength, then here the shorter wavelength
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components travel faster and the longer wavelength
components travel slower.
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So, now, if I launch a pulse around this center
wavelength again the unchirped pulse, then
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it will experience positive dispersion and
it will give you a broadened pulse like this
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which is chirped in the opposite direction
ok.
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So, here what I get the leading edge has higher
frequency component, while the trailing edge
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has lower frequency components.
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So, because of this I have negative or positive
dispersion, but both the dispersions will
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broaden the pulse if the input pulse is unchirped.
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So, this is the intensity of the pulse.
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So, if I is I launch a pulse at 0 is equal
to 0 at the input end of the fiber unchirped
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pulse, and I see how it evolves along the
length of the fiber then I see that it will
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get broadened.
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If I capture this pulse at 0 is equal to 0
at the input end, and z is equal to l that
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is the output end then it would look like
this.
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So, the pulse clearly gets broadened, but
the total energy in the pulse will always
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remain the constant if I assume there is no
attenuation.
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In the absence of attenuation the total energy
in the pulse remains the constant.
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So, when the pulse broadened its amplitude
goes down or its peak intensity goes down
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so, as to have the total energy conserved.
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Now, this wave guide dispersion depends upon
various wave guide parameters, because it
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is purely due to the wavelength dependence
of propagation constant of the mode.
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Now if my fiber is such that beta depends
upon lambda, and if I change the fiber parameter
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this dependence of beta on lambda can change
or dependence of n effective on lambda gets
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changed then the dispersion will be different.
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In order to show this, I have taken several
examples.
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So, one of the examples is that I take a fiber
with delta is equal to 0.5 percent, and a
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is equal to 3 micro meter.
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And now if I calculate the effective index
of the mode at different wavelengths and plotted,
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that effective index varies something like
this.
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If I change the parameters of the fiber make
delta from 0.5 percent to 0.75 percent, and
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bring down a from 3 to 2.5 in order to keep
V value almost at the same level to keep the
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fiber always single moded.
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If I am increasing delta will have to bring
down the value of a.
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So, now, again if I plot n effective as a
function of lambda naught then t goes like
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this.
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I see that the slope of this is different
from the slope of this and hence d2(bV) over
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dV square for both these curves would be different
and this will lead to change in wave guide
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dispersion.
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So, it will change the wave guide dispersion
if you change the fiber parameters.
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In order to demonstrate this, I take numerous
examples now this is from the software.
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So, where I can change the relative index
difference delta I can change the core diameter,
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and then see what happens to dispersion.
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So, this is when delta is equal to 0.0015
that is 0.15 percent, and core diameter is
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10 micro meter then I see that at around 15-16
nanometer wavelength my wave guide dispersion
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is minus 4.28 picoseconds per kilometer nanometer,
and total dispersion is about 18.22 picoseconds
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per kilometer nanometer.
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Graphically if I look at it the wave guide
dispersion is shown by this blue line and
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total dispersion is shown by this red line.
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So, the zero dispersion wavelength for the
total dispersion is around 1.3045.
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If I change the fiber parameter if I increase
delta to 0.2 percent from 1.15 percent to
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0.2 percent, and core diameter I change from
10 to 9 micro meter, then what I see that
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the wave guide dispersion increases to minus
5.17 and the total dispersion becomes 17.33,
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and the total dispersion wavelength changes
to 1.31 micro meter.
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So, when I change delta and a change the wave
guide dispersion, I further increase delta.
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So, it is now 0.3 percent and core radius
is 8 then this is the wave guide dispersion
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this is the total dispersion, if I further
change it then I increase the wave guide dispersion
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to about point about minus 8 picoseconds per
kilometer nanometer if I further increase.
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So, I see that if I increase the value of
delta then my wave guide dispersion is changing
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quite a lot, if I am increasing delta then
of wave guide dispersion is increasing, and
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the total dispersion is decreasing.
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Here at delta is equal to 0.7 percent and
core diameter 5 micro meter 0 dispersion wavelength
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could be moved to 1.426 micro meter which
was earlier 1.3 micro meter or 1.3 micro meter.
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So, I can drastically change the wave guide
dispersion if I change the fiber parameters.
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What I also notice in all these curves are
that for a given fiber, wave guide dispersion
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is larger the magnitude of wave guide dispersion
is larger at longer wavelength as compared
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to at shorter wavelengths why it is so?
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It is understandable because at longer wavelength
the field spreads more into the cladding at
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shorter wavelengths field is more and more
confined into the core.
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So, at shorter wavelengths the mode does not
see much effect of the cladding.
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So, the dispersion is small because it does
not see the composite structure.
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So, a strongly, but when you increase the
wavelength then the field spreads into cladding
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also.
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So, the fraction of power that goes into cladding
increases.
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So, the fiber modes is more and more the composite
structure, and the wave guidance is effected
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more and more so with wavelength.
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So, that is why at longer wavelengths I see
much stronger effect as compared to at shorter
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wavelengths.
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Now, can I tune my fiber parameters in order
to get a desired value of wave guide dispersion
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is it possible.
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So, for that if I look at this expression
of wave guide dispersion.
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Then it goes as Dw is equal to minus delta
n 2 divided by c lambda naught times Vd2(bV)
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over dV square.
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Now for a given value of V this is fixed.
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If I want to obtain wave guide dispersion
certain value of wave guide dispersion at
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a given value of lambda naught then and I
always use silica glass fiber when I use silica
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glass fiber, then at a given wavelength this
n2 and lambda naught they are now fixed n2
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is fixed at a given lambda naught then what
I can do.
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Now, if I change delta for a given value of
V, then I can obtain our desired value of
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Dw.
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So, if I fix Dw then I change the value of
V and change the value of delta then an appropriate
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combination of V and delta can give me a desired
value of Dw.
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So, I can understand it with the help of an
example, if I want to target my Dw as 22 picoseconds
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per kilometer nanometer at lambda naught is
equal to 1560 nano meter wavelength, then
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n2 at this wavelength is around 1.4439.
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This is the refractive index of silica glass
at 1516 nanometer wavelength.
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So, this is now fixed n2 divided by t lambda
naught is fixed as soon as I fix this wavelength
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1516 nanometer.
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So, it is 1.4439 divided by 3 into 10 to the
power 8 meter per second times 1560 into10
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to the power minus 9 meters this is the wavelength.
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So, if I work this out then this comes out
to be 0.0031 second over meter square.
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So, this is fixed now I vary the value of
V let us say from 1.5 to 2.5 and I get this
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V times d2(bV) over dV square for example,
from marques formula which I have shown in
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the next in the previous lecture that it gives
you much accurate value of V times d2(bV)
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over dV square for any given value of V lying
between 1.5 and 2.5.
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So, I get this from marques formula, and then
vary delta and then calculate delta in order
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to obtain Dw is equal to 22 picoseconds per
kilo meter nanometer.
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So, I do that and I plot it as a function
of V.
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So, if I vary V.
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So, what values of delta will give me wave
guide dispersion of minus 22 picoseconds per
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kilo meter nanometer at lambda naught is equal
to 1560 nanometer.
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So, this is how it varies the delta varies
with V like this in order to obtain this value.
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Correspondingly because I am changing delta
for a given V then a has to change to keep
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the V constant, correspondingly the core radius
varies like this.
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So, you are increasing the value of delta
then correspondingly the core radius should
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decrease in order to keep this V constant.
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So, what I see that if I vary delta then to
keep the delta less than 1 percent, I would
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like to keep delta less than 1 percent I would
not like to have very large value of delta
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because if I increase delta then attenuation
in the fiber increases ok.
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So, if I typically keep the value of delta
is smaller than 1 percent, hen these are the
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combinations of delta and a that will give
me the wave guide d of minus 22 picoseconds
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per kilo meter nanometer.
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So, I see the value of a would be around 2
micro meter.
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So, in this way I can obtain any desired value
of wave guide dispersion as I want to certain
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extent not any desired value.
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Because if I want to have much larger value
of wave guide dispersion then perhaps I will
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have to have very large value of delta and
correspondingly very small value of a, which
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is not desirable because very high value of
delta will give you very large attenuation
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and very low value of delta will very low
value of a will make problems will create
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problems in coupling light from source laser
source into the fiber.
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So, we will have to be practical.
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Let me work out an example, I consider a step
index optical fiber with core refractive index
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1.45 cladding refractive index 1.444 and core
radius 4.2 micro meter and I want to calculate
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broadening due to wave guide dispersion at
lambda naught is equal to 1.55 micro meter
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and I using Marques formula.
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So, I want to calculate wave guide dispersion
coefficient.
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So, solution is well what I need to do first?
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I need to find out this V times d2(bV) over
dV square.
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So, I first need to find out what is the value
of V at lambda naught is equal to 1.55 micro
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meter.
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So, V is given by this.
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So, if I calculate V then it comes out to
be 2.2435 and marques formula for V times
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d2(bV) over dV square is this.
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So, if I put this value of V then the express
the value of this comes out to be 0.2714 delta
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for this is I need delta also.
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00:25:19,419 --> 00:25:26,139
So, delta if I work out for these values of
n1 and n2 then it comes out to be 0.0041,
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now I have everything in place I need to just
plug in these values into this formula and
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get the wave guide dispersion coefficient.
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00:25:34,649 --> 00:25:42,919
So, Dw it is given by minus delta n2 over
c lambda naught times Vd2(bV) over dV square
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00:25:42,919 --> 00:25:49,410
comes out to be minus 3.48 into 10 to the
power minus 6 seconds per meter square, and
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00:25:49,410 --> 00:25:53,929
I can convert it into picoseconds per kilo
meter nanometer and it comes out to be minus
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00:25:53,929 --> 00:26:00,190
3.48 picoseconds per kilometer nanometer.
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00:26:00,190 --> 00:26:11,549
So, till now we had considered a step index
optical fiber, and wave guide dispersion in
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step index optical fiber.
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We can also work out this wave guide dispersion
in graded index optical fiber.
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So, in graded index optical fiber the only
thing is that you need to calculate the effective
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index as a function of wavelength.
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00:26:33,239 --> 00:26:41,779
So, this we have done using the software lite
sim where there is an option of calculating
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00:26:41,779 --> 00:26:43,950
dispersion for graded index optical fiber.
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00:26:43,950 --> 00:26:52,240
So, I just want to tell you how the profile
refractive index profile of the fiber changes
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the wave guide dispersion.
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So, here I have power lap profile.
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00:26:57,910 --> 00:27:04,320
So, this is triangular profile which corresponds
to q is equal to 1, I have taken delta is
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equal to 1 percent and core diameter 4, and
this is how the wave guide dispersion total
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dispersion varied.
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00:27:14,460 --> 00:27:21,350
So, at 1550 nanometer wavelength if I pick
out the values of wave guide and total dispersion,
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and the wave guide dispersion in minus 23
picoseconds per kilo meter nanometer and total
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00:27:26,239 --> 00:27:36,850
dispersion is about minus 2 picoseconds per
kilo meter nanometer very small.
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00:27:36,850 --> 00:27:46,350
And the 0 dispersion wavelength shifts to
about 1565 nanometer.
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If instead of taking this triangular profile,
I take parabolic profile I am sorry it has
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to be parabolic profile it is not triangular
profile.
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00:27:57,179 --> 00:27:59,470
So, it has to be parabolic profile.
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00:27:59,470 --> 00:28:07,770
So, in a parabolic profile if I again take
the value of delta as 1 percent and the core
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diameter I have now made 3.
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00:28:11,940 --> 00:28:21,270
So, now, the total dispersion waveguide dispersion
they vary like this, and at 1550 nanometer
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wavelength wave guide dispersion is minus
24 picoseconds per kilo meter nanometer and
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00:28:26,070 --> 00:28:30,379
total dispersion is about minus 3 picoseconds
per kilo meter nanometer.
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00:28:30,379 --> 00:28:37,600
So, by changing the profile you can change
the wave guide and total dispersion.
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00:28:37,600 --> 00:28:46,989
This is cubic profile which corresponds to
q is equal to 3, delta here is 0.6 percent
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00:28:46,989 --> 00:28:53,940
while the core radius is 2 micro meter, and
lambda naught at lambda naught is equal to
247
00:28:53,940 --> 00:29:02,700
1550 nanometer the wave guide dispersion is
minus 21.68 picoseconds per kilometer picoseconds
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00:29:02,700 --> 00:29:08,759
per kilometer nanometer and total dispersion
is very close to 0, it is 0 picoseconds per
249
00:29:08,759 --> 00:29:09,909
kilo meter nanometer.
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00:29:09,909 --> 00:29:13,940
So, 0 dispersion wavelength here I have got
1550 nanometer.
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00:29:13,940 --> 00:29:20,700
So, so in this way I can even shift my 0 dispersion
wavelength, I can tune my 0 dispersion wavelength
252
00:29:20,700 --> 00:29:29,019
added at a desired value if I change the fiber
profile.
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00:29:29,019 --> 00:29:37,499
So, I have seen the step index fiber, I have
seen graded index fibers, I can also have
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00:29:37,499 --> 00:29:43,869
multi layer fibers where core is not made
up of one layer, but I have 2 layers in the
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00:29:43,869 --> 00:29:49,539
core one high index layer then surrounding
it is small index layer, and then you have
256
00:29:49,539 --> 00:29:50,539
a cladding.
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00:29:50,539 --> 00:29:55,629
So, I can have a multi layer and with the
help of multi layer I just want to show you
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00:29:55,629 --> 00:30:01,250
that you can drastically change the wave guide
dispersion, just look at the values of wave
259
00:30:01,250 --> 00:30:06,509
guide dispersion here they are less than minus
75 picoseconds per kilo meter nanometer.
260
00:30:06,509 --> 00:30:11,700
So, the magnitude is larger than 75 picoseconds
per kilometer nanometers.
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00:30:11,700 --> 00:30:17,419
So, you can enhance the wave guide dispersion
quite a lot.
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00:30:17,419 --> 00:30:28,730
So, with the help of multilayer structures
you can considerably tweak with the wave guide
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00:30:28,730 --> 00:30:30,530
dispersion.
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00:30:30,530 --> 00:30:41,659
So, in this lecture what we had seen that
wave guide dispersion depends upon fiber parameters
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00:30:41,659 --> 00:30:44,549
profile of the fiber.
266
00:30:44,549 --> 00:30:54,210
So, by changing the fiber parameters and refractive
index profile of the fiber, we can tune waveguide
267
00:30:54,210 --> 00:30:59,549
dispersion to whatever value we want to certain
extent.
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00:30:59,549 --> 00:31:00,939
Thank you.