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When we use optical fiber in a system whether
it is telecommunication system or a sensor
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system, we come across the instances where
we need to join two fibers together. And when
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we make a joint between the two fibers then
there are always some losses, these losses
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are known as a splice losses. In this lecture
we will study about the splice losses.
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So, the fibers which we join can be identical
or non-identical for example, if I am joining
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two telecom fibers together then these fibers
are identical, but when I am putting some
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telecom component which is also made of fiber
itself then this fiber can be different from
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the telecom fiber its specs would be different
from the telecom fiber, and the two fibers
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I am joining now would be non-identical.
When I put two fibers together so how so ever
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precise alignment I do they are always some
degree of misalignment. And these misalignments
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even if very small they can cause significant
losses, and in case of two identical fibers
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even if there is perfect alignment, because
of the difference in the fiber parameters
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there are always some losses. Basically the
mode of one fiber would be different from
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the mode of the other fiber and because of
this mismatch between the modes of two fibers
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there would be a loss and that is known as
mismatch loss. So, what are the various types
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of misalignments let us first look into that,
and then calculate the expressions for loss
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corresponding to these misalignments.
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So, one type of misalignment is in the transverse
direction, that one fiber is shifted with
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respect to another in transverse direction.
Another is angular when you join two fibers
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there is some angle between them this is highly
exaggerated, this misalignment is of the order
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of a degree or so.
This can be caused due to your finite precision
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of the stages at which you are putting the
fiber or due to a not very sharp and in the
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transverse direction if this cleaving of the
fiber is not perpendicular to the axis of
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the fiber. So, ideally what you should have,
when you have two fibers and you want to join
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them then you first clean the fibers. So,
this surface should be perfectly perpendicular
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to the fiber axis, but if it is something
like this, then it will cause angular misalignment
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loss.
Third is that if there is a small gap between
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the two fibers, then there would be a loss.
So, I have transverse angular and longitudinal
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misalignment losses. Let us work out the expression
for loss in case of transverse misalignment
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and in this entire lecture I will use Gaussian
spots and Gaussian spot sizes of the modes
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of the fiber and I will restrict myself to
only single mode fibers.
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So, I have fiber 1 which has a spot size w1
is Gaussian spot size, and I have fiber two
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which has a spot size w2 this direction is
z and this is x.
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So, in whatever direction there is transverse
offset I can choose my x axis along that,
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and accordingly I will use my y axis perpendicular
to x and z. So, now, what I can do? I can
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write the Gaussian mode of this fiber and
let us say the mode of this fiber is centered
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at 00 and has size w1. So, the modal field
would be given by psi 1 x, y is equal to square
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root 2 over pi times 1 over w1 exponential
minus x square plus y square over w1 square.
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Now, this fiber is shifted with respect to
this by an amount u in x direction. So, the
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mode of this would be centered at u, 0 and
it has a size w2. So, the Gaussian mode corresponding
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to this would be given by this should be psi
2. psi 2 x, y is equal to square root 2 over
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pi 1 over w2 exponential minus x minus u square
plus y square over w2 square please make a
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correction this is psi 2. And in order to
get these I have used the normalization condition
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that mode psi squared dxdy integrated over
the entire range of x, y is equal to 1.
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Since I am using these two to calculate the
losses, these are basically area normalized.
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Now how do I calculate loss well calculation
of loss is very simple, I have a Gaussian
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of one fiber and Gaussian of another fiber
then the overlap between these two fields
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will give me the loss or the fraction of power
that is coupled from one fiber to another
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fiber.
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So, the fraction of power that is coupled
from one fiber to another fiber would be given
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by T is equal to integral minus infinity to
plus infinity over x and y, psi 1 psi 2 star
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dxdy mod square. So, this is the overlap integral
and if I take it mod square then it will give
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me the fraction of power coupled from one
fiber to another fiber, and this expression
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is true when these model fields are normalized
according to the condition which I have stated
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in the previous slide.
So, now if I use the expressions for psi 1
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and psi 2 which at I had written earlier and
do some mathematics then I will get the expression
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for T as, T is equal to 2 w1w2 over w1 square
plus w2 square whole square times exponential
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minus two u square over w1 square plus w2
square.
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From here I can calculate the loss in dB I
know this T is nothing, but Pout over Pin.
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So, loss in dB would be tan log one over T.
I can see from here that if the two fibers
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are identical and there is no offset loss
then this T should be equal to 1, and I can
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check that out if w1 is equal to w2 then this
term becomes one and when u is equal to 0
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this term becomes 1. So, from this expression
of fractional power coupled from one fiber
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to another fiber, I can estimate the loss
which is purely due to mode mismatch, and
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in order to get this loss which is known as
mode mismatch loss which is purely due to
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mismatch in the modes of two fibers I should
put u is equal to 0.
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So, in order to remove any contribution from
transverse misalignment. So, I put u is equal
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to 0, if I put u is equal to 0 here then alpha
in dB would simply be minus 20 log 2 w1w2
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over w1 square plus w2 square. I can divide
this numerator and denominator by w2 square
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then I will have w1 over w2 here and w1 over
w2 square here if I define this w1 over w2
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by f then it can be written as minus 20 log
2 f over 1 plus f square.
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Now let me plot this mode mismatch loss as
a function of w1 over w2, then I can see that
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if w1 is equal to w2 then there is no loss,
loss is 0 dB, but if w1 over w2 is smaller
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than one that is w2 is larger or if it is
greater than one that is w1 is larger, in
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both the cases I will have mode mismatch loss.
Let me zoom it to a level of 0.1 dB and when
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I do this then I find that that to keep the
loss below 0.1 dB, to keep this mode mismatch
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loss below 0.1 dB the mismatch between the
spots sizes should not be more than about
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15 percent. So, if you have a mode mismatch
up to 15 percent then your loss this mode
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mismatch loss would be less than 0.1dB and
this kind of analysis is very handy and very
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useful when you estimate the loss or when
you design your fiber components to be used
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in telecom systems. So, you always design
the components in such a way that the fiber
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you are using for them should not have mode
mismatch with the telecom fiber more than
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15 percent.
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Now, let us calculate the transverse offset
loss for two identical fibers. So, this is
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the fractional power and for two identical
fibers w1 is equal to w2 and let us say it
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is w then this T would become simply e to
the power minus u square over w square because
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this will become 1, and from here I can immediately
get the transverse offset loss in dB as alpha
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T is equal to 4.34 u square over w square.
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So, I can see from here that if w is small
then your transverse offset loss is large,
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if w is large then the tolerance towards transverse
offset loss is higher because the loss is
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smaller. So, to see this graphically I have
plotted this transverse misalignment loss
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or transverse offset loss as a function of
u that is transverse misalignment for different
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values of w. You see that this x axis has
the same range in all the figures, while if
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you look at the y axis that is the loss axis
you look at the numbers and these numbers
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decreased as I go for higher values of w.
So, for higher values of w the transverse
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offset loss is smaller.
Now, let me look at 0.1 dB level for a telecom
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fiber, which has a spot size typically 5 micrometer.
So, for w is equal to 5 micrometer, in this
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figure if I look at this line which corresponds
to loss is equal to 0.1 dB, then I find that
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that to keep the loss less than 0.1 dB u or
transverse misalignment should be less than
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0.76 micro meter. So, this is the degree of
precision with which I should align my fibers
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while making a splice while making a joint.
Another misalignment is in angular direction
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and it causes angular offset loss.
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So, I have fiber one with spot size w1, fiber
2 with spot size w2, and this is tilted with
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respect to fiber 1 by an angle theta. So,
again what I will do I will use Gaussians
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here, Gaussian mode of this fiber and Gaussian
mode of this fiber and I will have to take
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the overlap of them now how I will take the
overlap? Well I will represent the Gaussian
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of this in the rotated coordinate system.
So, this coordinate system is now rotated
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by an angle theta. So, I have this x prime
and y as and z prime x prime and z prime which
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is rotated by an angle theta.
So, I should do coordinate transformations.
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So, the coordinate transformations are like
this which relate x y z to x prime y prime
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z prime. Now I can write down the Gaussian
mode of fiber one like this, and Gaussian
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mode of fiber two like this z prime is equal
to 0 and this is a sum z. Where this n here
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and n here is the refractive index of the
material which goes in the gap between the
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two fibers, and usually this material is nothing,
but cladding material because when you join
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two fibers you align them and then fuse them
together. When you fuse them together the
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cladding material gets melted and it goes
in the gap. So, this n is nothing, but most
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of the time it is cladding material.
So, again I can find out the fractional power
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coupled from fiber 1 to fiber 2 by taking
the overlap of these Gaussians and then it
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comes out to be T theta is equal to 2 w1w2
over w1 square plus w2 square whole square,
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exponential minus k naught square n square
theta square, w1 square, w2 square divided
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by 2 w1 square plus w2 square. In order to
get this expression I have assumed that this
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the value of theta is a small so that sin
theta is equal to theta and cos theta is equal
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to 1. If you want to look into more details
about this then you can refer to the text
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book introduction to fiber optics by Ghatak
and Thyagrajan.
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So, this is the fractional power coupled from
one fiber to another fiber and from this now
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I can find out the angular offset loss for
two identical fibers. So, where I have now
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w1 is equal to w2 is equal to w, and this
gives me T theta is equal to exponential minus
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k naught n theta w by 2 whole square. So,
the loss angular offset loss in dB would simply
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be now given by 4.34, times k naught and w
theta by 1 whole square. So, this is the angular
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offset loss.
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Here I can see that if w is large then angular
offset loss is large. So, I have plotted it,
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I have plotted this for different values of
w and I can see that when I increase the value
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of w the angular offset loss increases. And
again I look at telecom fiber which has approximate
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value of w as 5 micrometer, and 0.1 dB loss
level. So, I can see that if I want to keep
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the angular misalignment loss below 0.1 dB,
then theta should be less than 0.6 degrees.
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So, with this precision I should align my
fibers or with this precision I should cleave
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my fiber. So, that it is perfectly perpendicular
it is perpendicular to the axis of the fiber.
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The last one is longitudinal offset loss,
which occurs when you join two fibers together
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and there is always some gap left between
them and when you fuse them then cladding
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material goes in the gap. This also causes
loss because the Gaussian mode when comes
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out of this fiber it diffracts.
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And when it reaches to this end of the fiber,
then this diffracted Gaussian would be different
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from the Gaussian mode of the fiber tube.
So, in order to estimate this loss what should
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I do? I should take the overlap between the
Gaussian of this fiber and diffracted Gaussian
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of this fiber. So, when I do this then I can
find out the longitudinal misalignment loss
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and it is given by alpha L in dB is equal
to 10 log 1 plus D by k naught n w square
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whole square.
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So, here I have plotted this for various values
of w, and I can see that that this longitudinal
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misalignment loss decreases as I increase
the value of w and it is understood that if
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I have small value of w then the diffraction
would be large, and if diffraction would be
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large then the Gaussian diffracted Gaussian
would be very much different from the Gaussian
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of fiber two; and it will cause more loss
a. So, for larger value of w the longitudinal
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offset losses is smaller.
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Again let me look into the values of loss
which are close to 0.1 dB, and for that I
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had I have increased the range of longitudinal
misalignment while plotting this loss curve
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for w is equal to 5 micrometer, and what I
see that even for longitudinal misalignment
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as large as 20 micrometer, the longitudinal
offset losses smaller than 0.1 dB. So, longitudinal
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offset loss is not a major concern for us
it is not a major concern for us; however,
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the transverse and angular offset losses are
quite large if the misalignment is large.
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Let us work out an example a numerical example.
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So, I consider two identical step index optical
fibers with n1 is equal to 1.45, n2 is equal
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to 1.444 and core radius a is equal to 4.2
micrometer and I want to calculate a lambda
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naught is equal to 1.55 micrometer, all the
three losses for various values of misalignments.
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So, let us see how do they come out. So, the
transverse offset loss for u is equal to 1
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micrometer well. So, at lambda naught is equal
to 1.55 micro meter, if I calculate the Gaussian
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spot size it comes out to be 4.81 micrometer.
The easiest way to find this out is you first
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find out the value of V and then using the
empirical relation you can find out the value
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of w for Gaussian spot size. So, it comes
out to be 4.81 micro meter, now you can calculate
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transverse offset loss by the formula alpha
is equal to 4.34 u square over w square, and
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it comes out to be 0.19 dB.
The next part is angular offset loss for theta
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is equal to 1 degree.
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So, again you find out the Gaussian spot size
w is equal to 4.81 micro meter comes out like
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this, and then you find out angular offset
loss by the formula 4.34 k naught and 2 w
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theta by 2 you see that instead of n I have
put n2 here because n2 is the cladding material
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and this is the cladding material which goes
inside the gap.
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So, angular offset loss comes out to be 0.26
dB; third is longitudinal offset loss for
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D is equal to longitudinal misalignment d
is equal to 10 micrometer. So, again the Gaussian
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spot size is 4.81 micrometer and if I calculate
the longitudinal offset loss using this formula
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then it comes out to be 0.023 dB. So, again
look at these figures that longitudinal offset
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loss is 0.023 dB the angular offset loss is
0.26 dB while the transverse offset losses
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0.19 dB. So, I can see that that I need to
when I join two fibers, I need to precisely
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align the two fibers in transverse and angular
direction in order to have as low loss as
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possible.
So, this is all in this section in this lecture
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and what is left is another very important
characteristic of a single mode optical fiber
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and that is waveguide dispersion. So, in the
next lecture we would look into the waveguide
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dispersion and subsequently the total dispersion
and this will give me what is the data carrying
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capacity of a single mode fiber.
Thank you.