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00:00:18,590 --> 00:00:27,660
In the last lecture, we had approximated the
modal field of a single mode fiber by a Gaussian.
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We had studied the Gaussian mode and defined
the Gaussian spot size.
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00:00:33,600 --> 00:00:43,050
We had also studied how the power is distributed
between the core and the cladding of a fiber.
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00:00:43,050 --> 00:00:54,070
In this lecture we will define a spot sizes
in few more ways and see how the bending of
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00:00:54,070 --> 00:00:56,149
the fiber affects the performance.
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00:00:56,149 --> 00:01:03,640
So, there is another way of defining the spot
size which is Peterman 2 spot size, which
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00:01:03,640 --> 00:01:10,640
is related to transverse offset loss at a
joint between the 2 fibers, we will study
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00:01:10,640 --> 00:01:14,050
about these losses at the joints in the next
lecture.
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00:01:14,050 --> 00:01:21,050
So, Peterman 2 spot size is defined in terms
of transverse field pattern of the fundamental
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00:01:21,050 --> 00:01:30,840
mode psi r, and is given as WP2 is equal to
square root of 2 integration 0 to infinity
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00:01:30,840 --> 00:01:37,780
psi square rdr divided by integration 0 to
infinity psi prime square rdr.
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00:01:37,780 --> 00:01:49,170
Now, for a step index fiber I can write psi
r in the core as A J1 Ur over a, and in the
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cladding B K1 Wr over a, where a and b can
be related by the boundary conditions that
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is psi r is continuous at r is equal to a,
and if I use that then I can get the expression
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00:02:08,990 --> 00:02:17,920
for WP2 as square root of 2 J1 U over W J0
U times a.
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Here I have plotted the modal normalized modal
field of the fiber defined by n1 is equal
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00:02:26,500 --> 00:02:35,879
to 1.45, n2 is equal to 1.444, core radius
a is equal to 4 micrometer at wavelength lambda
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00:02:35,879 --> 00:02:38,379
naught is equal to 1.55 micrometer.
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00:02:38,379 --> 00:02:48,319
So, this blue line shows the exact modal field
of the fundamental mode of the fiber, and
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this marks this green line marks the pediment
to spot size of the mode calculated by this
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00:02:58,209 --> 00:02:59,469
modal field psi r.
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00:02:59,469 --> 00:03:08,790
Since WP2 for these fiber parameters and the
wavelength is 4.7 micrometer, and if I calculate
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the Gaussian spot size for these parameters
it comes out to be 4.75 micrometer.
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As we had done in the case of Gaussian, we
can also define an empirical relation between
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the Peterman 2 spot size WP2 and the normalized
frequency V.
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And it is given as WP2 over a is approximately
equal to 0.634 plus 1.619 over V to the power
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00:03:45,909 --> 00:03:56,510
3 by 2, plus 2.879 over V to the power 6,
minus 1.567 over V to the power 7, and this
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is quite good in quite accurate in the range
of V from 1.5 to 2.5.
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If I compare it with the Gaussian spot size
then in the case of Gaussian this term was
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0.65 and this term was not there.
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So, I am reducing this term from 0.65 to 0.634
and further I am reducing the spot size by
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00:04:29,060 --> 00:04:31,389
this much amount.
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00:04:31,389 --> 00:04:39,970
So, if I compare it with the Gaussian then
the Peterman 2 spot size would always be smaller
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00:04:39,970 --> 00:04:43,680
than the Gaussian.
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For large values of V this term will tend
to 0, the contribution of this term would
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be very small and the difference between the
2 would shrink; however, this much difference
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00:04:56,389 --> 00:04:58,120
would always be there.
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00:04:58,120 --> 00:05:04,870
So, now I compare the Gaussian spot size and
Peterman 2 spot size, and I can immediately
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00:05:04,870 --> 00:05:13,389
see that the difference is large at smaller
values of V, and it shrinks when I go towards
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higher values of V.
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00:05:16,729 --> 00:05:25,930
This Peterman 2 spot size is accepted as the
standard by a committee which sets the standards
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00:05:25,930 --> 00:05:34,500
for telecom fiber, and this committee is known
as CCITT according to its recommendations
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00:05:34,500 --> 00:05:39,930
for a single mode fiber to be used in long
haul telecommunication system, the Peterman
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2 spot size should lie between 4.5 and 5 micrometer
Yet another way of defining a spot size is
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00:05:53,729 --> 00:05:58,600
Peterman one spot size and this is related
to angular offset loss at the joint between
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00:05:58,600 --> 00:06:00,310
the 2 fibers.
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00:06:00,310 --> 00:06:07,300
And is defined as WP1 is equal to a square
root of 2 times integration 0 to infinity
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00:06:07,300 --> 00:06:14,470
size square r cube dr divided by integration
0 to infinity, psi square rdr.
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00:06:14,470 --> 00:06:25,909
So, here in this figure I have again plotted
the normalized modal field for the same fiber
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00:06:25,909 --> 00:06:31,000
and at the same wavelength and the Peterman
one spot size.
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The Peterman one spot size comes out to be
four point eight five micrometer and if I
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00:06:37,440 --> 00:06:46,530
compare it with the Gaussian was 4.75 micrometer
and Peterman 2 was 4.7 micrometer, for the
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00:06:46,530 --> 00:06:49,030
same fiber and the wavelength.
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00:06:49,030 --> 00:06:55,360
I also have Peterman 3 spot size which is
related to bend loss of the fiber and it is
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00:06:55,360 --> 00:07:03,680
defined as WP3 is equal to square root of
lambda naught divided by pi n1 beta minus
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00:07:03,680 --> 00:07:14,620
k naught n2, and in this figure I again see
the normalized modal field, and the Peterman
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00:07:14,620 --> 00:07:24,500
3 spot size which is shown to be 5.5 micrometer
calculated by this formula.
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00:07:24,500 --> 00:07:31,020
If I compare it with the other 3 spot sizes,
then I find that it is much larger.
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00:07:31,020 --> 00:07:40,120
Petermann 1is 4.85 micrometer, Petermann 2
is 4.7 and Gaussian is 4.75, but this is 5.5
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00:07:40,120 --> 00:07:46,669
micrometer which is quite large as compared
to the other three.
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00:07:46,669 --> 00:07:56,729
Here in this table I have listed some important
parameters of a single mode fiber in the range
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00:07:56,729 --> 00:08:02,289
of V from 1.5 to 2.4.
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These parameters are normalized propagation
constant b, you remember that U is defined
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as U is equal to a times square root of k
naught square n1 square minus beta square
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00:08:23,349 --> 00:08:33,950
or you can also write it as V times square
root of 1 minus b, and w is defined as a times
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00:08:33,950 --> 00:08:44,980
square root of beta square minus k naught
square and 2 square and it is defined as V
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times square root of b.
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00:08:46,740 --> 00:09:02,250
So, here I have listed the values of U, W
then Gaussian spot size and the Peterman 2
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spot size for different values of V. Please
note that these spot sizes I have calculated
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by using the empirical relations.
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00:09:17,990 --> 00:09:24,290
These values of different parameters become
very handy when we analyze a single mode fiber
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00:09:24,290 --> 00:09:34,960
or we design a fiber for a given application,
let us work out a few examples.
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00:09:34,960 --> 00:09:41,020
Here let us consider a step index optical
fiber with core refractive index and n1 is
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equal to 1.45, cladding refractive index n2
is equal to 1.444 and the core radius a is
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00:09:49,540 --> 00:09:53,080
equal to 4.2 micrometer.
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00:09:53,080 --> 00:09:58,330
Let us calculate the cut off wavelength of
the fiber.
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00:09:58,330 --> 00:10:04,680
So, I know that the cut off wavelength of
the fiber is given by lambda c is equal to
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2 pi over 2.4048 times a times square root
of n1 square minus n2 square.
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00:10:12,150 --> 00:10:19,000
So, if I plug in all these numbers in to this
formula, then I immediately get the cut off
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wave length as 1.446 micrometer.
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00:10:25,640 --> 00:10:31,390
The second thing is the wave length range
in which the fiber supports 2 modes.
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So, what is the range of wavelengths in which
the fiber supports 2 modes?
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So, first I need to find out what is the range
of V in which the fiber supports 2 modes.
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So, if I look at it b-V curves for a fiber.
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So, this is b is equal to 0 this is b is equal
to 1, this is LP01 mode this is LP01, this
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is LP11 and this is LP02 and LP21 and, but
I see this is 0, this is 2.404 and this is
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3.8317.
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So, if the fiber supports 2 modes then the
value of V should lie somewhere here.
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So, that these 2 modes are supported and the
other boards are cutoff.
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00:11:45,680 --> 00:11:55,570
So, the value of V should lie between 2.4048
and 3.8317.
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00:11:55,570 --> 00:12:03,770
So, what I need to just now find out the values
of lambda corresponding to these values of
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V, and V is given as 2 pi over lambda naught
times a times square root of n1 square minus
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n2 square.
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00:12:12,290 --> 00:12:19,670
So, V is equal to 2.4048 will correspond to
lambda naught is equal to 1.446 micrometer,
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which we had done in the in the previous part
itself.
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This is nothing, but the cut off wavelength
and corresponding to 3.8317 lambda naught
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comes out to be 0.907 micrometer.
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00:12:37,680 --> 00:12:45,630
So, lambda naught would should lie between
0.907 and 1.446 micrometer.
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So, that there are only 2 modes guided LP01
and LP11.
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00:12:52,140 --> 00:13:03,470
Third part is what is the mode effective index
at lambda naught is equal to 1.55 micrometer.
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At lambda naught is equal to 1.55 micrometer,
V is equal to 2.2435.
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So, there is only one mode because it is less
than 2.4048.
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So, there is only one mode this is LP01 mode.
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And I know from that table that from that
table also I can do or from the empirical
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formula also I can find it out, if I do it
from using empirical formula then b is given
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by a minus b over V whole square where a is
equal to 1.1428, b is equal to 0.996 and if
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I calculate the value of b from here it comes
out to be 0.4884.
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From here I can calculate the n effective
as square root of n2 square plus b times n1
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square minus n2 square, and it comes out to
be 1.4469.
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00:14:02,740 --> 00:14:09,670
Fourth part is what is the Gaussian spot size,
at lambda naught is equal to 1.55 micrometer.
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00:14:09,670 --> 00:14:20,420
So, I had already calculated the value of
V at 1.55 micrometer which is 2.2435, now
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I know the empirical relation for Gaussian
spot size which is given by this.
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So, if I just put the value of V here I can
immediately get the value of w over a as 1.15
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and since a is equal to 4.2.
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So, I can get the value of Gaussian spot size
as 4.85micrometer.
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Fifth part is what is the Peterman 2 sport
size at lambda naught is equal to 1.55 micrometer.
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So, I already have the value of V at this
wavelength which is 2.2435.
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Now, I just put it in the empirical relation
of Peterman 2 spot size, then the Peterman
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2 spot size comes out to be 4.75 micrometer.
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00:15:19,140 --> 00:15:26,700
Now let us find out what happens if I bend
a fiber.
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If I deploy the fiber in the field it is inevitable
to bend the fiber because I will have to take
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00:15:33,210 --> 00:15:40,250
fiber across the corners there are instances
when I will have to coil the fiber and keep
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it somewhere if there is some extra length
of the fiber.
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00:15:44,860 --> 00:15:56,050
So, it is important to find out what happens
when you bend the fiber, and we will see that
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bending of the fiber causes loss.
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How much loss does it cause, on what parameters
of the fiber it depends on let us study that.
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So, if I have a straight fiber this is the
core and this is the cladding, then I know
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that if I launch a wave like this, then if
this angle is greater than critical angle
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then there would be total internal reflection
and most of the power will remain in the core
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large amount of power will remain in the core,
and various little power will extend in to
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the cladding.
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00:16:40,970 --> 00:16:52,630
Now if I bend the fiber which is shown by
these blue lines, then just look at this wave
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itself, now this angle of this wave changes
with normal because normal does not remain
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this now, the new normal is this.
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00:17:01,680 --> 00:17:08,520
So, what I find that this phi is now smaller
than the critical angle.
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00:17:08,520 --> 00:17:15,680
If phi is smaller than phi c the critical
angle then this wave would be refracted in
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00:17:15,680 --> 00:17:18,390
to the cladding and there is no total internal
reflection.
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So, the power would be lost.
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00:17:20,650 --> 00:17:30,480
So, when you bend the fiber it causes loss
in the fiber how sensitive the fiber would
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00:17:30,480 --> 00:17:40,350
be for this bending it will depend upon how
close this phi is to the critical angle.
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00:17:40,350 --> 00:17:49,559
If this angle phi is very close to the critical
angle then even a small bending will cause
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00:17:49,559 --> 00:17:51,180
the bend loss.
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00:17:51,180 --> 00:17:59,830
If this angle is close to critical angle it
means that the mode is close to cutoff and
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00:17:59,830 --> 00:18:02,750
the field spreads more and more in the cladding.
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00:18:02,750 --> 00:18:09,889
So, I can also understand that if in this
way that if the mode is close to cut off,
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then also fiber is more sensitive to bending.
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00:18:16,190 --> 00:18:25,460
So, the bend loss of the fiber depends upon
various parameters like core radius, a relative
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00:18:25,460 --> 00:18:36,240
index difference between the core and the
cladding, delta and wavelength lambda naught.
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00:18:36,240 --> 00:18:43,669
For a single mode step index fiber the bend
loss can be given by this formula, which involves
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00:18:43,669 --> 00:18:55,649
Rc which is the radius of curvature, core
radius a parameter U, which is given by this
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00:18:55,649 --> 00:19:09,530
which contains a wave length n1 and the mode
effective index W. So, if I know all these
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00:19:09,530 --> 00:19:16,570
parameters then for a step index fiber a step
index single mode fiber I can find out the
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00:19:16,570 --> 00:19:24,919
bend loss in dB per meter using this formula.
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00:19:24,919 --> 00:19:32,419
Let us now look at the bend loss of a fiber
as calculated by the formula shown in the
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00:19:32,419 --> 00:19:35,130
previous slide.
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00:19:35,130 --> 00:19:43,820
Here I have plotted the loss of 100 turns
of a fiber, quite with the radius of 3.75
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00:19:43,820 --> 00:19:50,960
centimeter and this loss I have plotted as
a function of core radius, for different values
158
00:19:50,960 --> 00:19:58,120
of core cladding index difference relative
index difference, and in this figure I have
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00:19:58,120 --> 00:20:03,110
plotted it at wavelength 1300 nanometer.
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00:20:03,110 --> 00:20:10,210
Let us examine this and see what information
do we get from these plots.
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00:20:10,210 --> 00:20:18,820
If I fix the core radius a and change the
relative index difference.
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00:20:18,820 --> 00:20:29,029
So, if I go from here to here I increase the
index contrast, I see that the bend loss reduces
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00:20:29,029 --> 00:20:33,230
quite a lot if I increase the index contrast.
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00:20:33,230 --> 00:20:41,950
So, at about 3.5 micrometer core radius, delta
is equal to 0.2 percent gives you a loss which
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00:20:41,950 --> 00:20:52,759
is more than 1000 dB while a fiber with delta
is equal to my delta is equal to 0.3 percent
166
00:20:52,759 --> 00:20:57,760
gives you a loss which is close to 0.001 dB.
167
00:20:57,760 --> 00:21:00,330
So, there is a huge difference.
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00:21:00,330 --> 00:21:11,179
So, by changing the delta a little, this delta
causes a huge impact on the bend loss of the
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00:21:11,179 --> 00:21:21,429
fiber and it is understandable that if this
delta basically changes the changes the core
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00:21:21,429 --> 00:21:26,789
cladding index difference and it changes the
critical angle, the value of critical angle
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00:21:26,789 --> 00:21:30,130
is directly affected by this delta.
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00:21:30,130 --> 00:21:39,889
So, if you increase delta you increase delta,
your tolerance towards bending increases a
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00:21:39,889 --> 00:21:41,940
lot.
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00:21:41,940 --> 00:21:48,960
Now if I take one particular value of delta
and then change the core radius, then what
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00:21:48,960 --> 00:21:55,049
happens that if I increase the core radius
then bend loss decreases and it is understandable,
176
00:21:55,049 --> 00:22:02,940
it when I increase the core radius the power
is confined more and more in the core and
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00:22:02,940 --> 00:22:10,309
less power extends in the cladding, that is
my mode is going away from the cut off.
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00:22:10,309 --> 00:22:16,220
If the mode is moving away from the cut off
then it is less sensitive to bends.
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00:22:16,220 --> 00:22:23,059
So, the bend loss decreases.
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00:22:23,059 --> 00:22:27,340
Next is how wavelength affects the bend loss.
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00:22:27,340 --> 00:22:37,259
So, for that I have plotted these curves these
variations at wavelength lambda naught is
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00:22:37,259 --> 00:22:45,320
equal to 1550 nanometer also, and what do
I see that if I pick up one particular value
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00:22:45,320 --> 00:22:52,679
of delta and one particular value of a, then
I find that when I increase the wavelength
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00:22:52,679 --> 00:22:56,440
then the bend loss increases tremendously.
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00:22:56,440 --> 00:23:05,870
For example, here if I take the value of a
as 3 micrometer then the bend loss is somewhere
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00:23:05,870 --> 00:23:14,999
between 1 and 10 dB for delta is equal to
0.003, but here if I take value of a as three
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00:23:14,999 --> 00:23:20,889
micrometer then corresponding to delta is
equal to 0.003 the bend loss would be much
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00:23:20,889 --> 00:23:25,980
larger than 1000 dB.
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00:23:25,980 --> 00:23:34,659
So, if I increase the wavelength what happens
the mode modal field spreads in the cladding
190
00:23:34,659 --> 00:23:43,200
and that is why it is more sensitive to bending
the bend loss increases.
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00:23:43,200 --> 00:23:50,149
According to CCITT recommendations, the bend
loss of a fiber quilled with the radius of
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00:23:50,149 --> 00:24:01,539
3.75 centimeter at 1550 nanometer wavelength
with 100 turns should be less than 1 dB.
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00:24:01,539 --> 00:24:10,840
So, so CCITT recommendations say that at 1550
nanometer wave length I should operate below
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00:24:10,840 --> 00:24:19,340
this line of 1 dB, this loss should be less
than 1 dB for 100 turns.
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00:24:19,340 --> 00:24:27,169
So, I should choose the combination of delta
and a in such a way that I always remain below
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00:24:27,169 --> 00:24:28,220
this line.
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00:24:28,220 --> 00:24:39,999
Let us work out a numerical example, for this
if I consider a step index optical fiber with
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00:24:39,999 --> 00:24:51,350
parameters n1 is equal to 1.447, n2 is equal
to 1.444 and a is equal to 4.5 micrometer
199
00:24:51,350 --> 00:25:01,110
and calculate the bend loss at 1.3 micrometer
for 100 turns of the fiber wound with a radius
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00:25:01,110 --> 00:25:04,369
of 3.7 centimeter.
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00:25:04,369 --> 00:25:14,529
So, I know that V can be calculated for these
parameters by this formula, and if I calculate
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V it comes out to be 2.
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00:25:18,559 --> 00:25:25,419
Next what I do for this value of V. I calculate
the value of normalized propagation constant
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b, I can find it from the table or I can calculate
it from the empirical relation.
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00:25:33,649 --> 00:25:45,799
So, b for this value of V is 0.4162, correspondingly
U is equal to 1.5282 and w is equal to 1.2902.
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00:25:45,799 --> 00:25:50,820
Now Rc here is 3.75.
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00:25:50,820 --> 00:26:00,799
So, if I plug in these values in to this formula,
I get alpha as 0.205 dB per meter.
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00:26:00,799 --> 00:26:08,210
Now, what is the total length which is in
the coil when I coil the fiber and put and
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00:26:08,210 --> 00:26:14,690
take 100 turns with this radius, then the
total length of the fiber in the coil would
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be 2 pi Rc times 100, which is 23.56 meters,
and this is the loss in dB per meter.
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So, the total bend loss in the coil would
be 4.83 dB.
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00:26:29,179 --> 00:26:38,370
So, in this lecture we had seen a few more
spot sizes Peterman 1, Peterman 2 and Peterman
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00:26:38,370 --> 00:26:44,010
three and also seen the bend loss of the fiber.
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00:26:44,010 --> 00:26:51,059
In the next lecture we will study what happens
if I join 2 fibers together, what kind of
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00:26:51,059 --> 00:26:55,460
losses it causes and what are the sources
of these losses.
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00:26:55,460 --> 00:26:56,440
Thank you.