1
00:00:18,779 --> 00:00:29,039
After having understood the modal field patterns
of a step index optical fiber now let us have
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00:00:29,039 --> 00:00:37,320
a look at what is the fractional power in
the core, how the power is distributed between
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00:00:37,320 --> 00:00:39,440
the core and the cladding.
4
00:00:39,440 --> 00:00:48,559
So, again we are analyzing this kind of step
index optical fiber, and we know that the
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00:00:48,559 --> 00:00:57,010
power is proportional to if psi is the modal
field which is basically the electric field,
6
00:00:57,010 --> 00:01:05,159
then the power is proportional to the electric
field square integrated over the entire transverse
7
00:01:05,159 --> 00:01:06,840
cross section.
8
00:01:06,840 --> 00:01:15,550
So, if psi is the modal field then the power
would be proportional to integral over r phi
9
00:01:15,550 --> 00:01:20,390
mod psi square rdrd phi.
10
00:01:20,390 --> 00:01:28,630
So, in the core it would be given by some
constant Q and this integration over r would
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00:01:28,630 --> 00:01:37,280
be from 0 to a, and if I want to calculate
the power in the cladding then this integration
12
00:01:37,280 --> 00:01:45,200
over r will go from a to infinity because
I am considering a case where the cladding
13
00:01:45,200 --> 00:01:46,570
is infinitely extended.
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00:01:46,570 --> 00:01:55,310
So, let us calculate the power for even modes,
for even modes the solutions psi are in the
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00:01:55,310 --> 00:02:04,539
region's core and the cladding are given by
this, where A and B are related to each other
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00:02:04,539 --> 00:02:11,390
by boundary conditions, here I have used the
boundary condition that psi r is continuous
17
00:02:11,390 --> 00:02:12,630
at r is equal to a.
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00:02:12,630 --> 00:02:21,780
So, A Ji U is equal to B Kl W and let me equate
two some other constant C, so I can get these
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00:02:21,780 --> 00:02:25,110
A and B in terms of a single constant C.
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00:02:25,110 --> 00:02:40,220
So, P core is this and if I now substitute
the modal field of the core, then it would
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00:02:40,220 --> 00:02:47,520
be of the form Jl square Ur over a rdr and
then phi.
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00:02:47,520 --> 00:02:55,849
So, phi part would be cosine square l phi
t phi, and this would be the constant which
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00:02:55,849 --> 00:02:58,760
comes out.
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00:02:58,760 --> 00:03:05,630
C square times Q and then this there would
be Jl square U for a given mode even Jl square
25
00:03:05,630 --> 00:03:11,270
U would be constant because beta is constant
for a given mode.
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00:03:11,270 --> 00:03:21,360
So, this I can simplify like this cosine phi
solution the phi integral is a straight forward,
27
00:03:21,360 --> 00:03:32,700
and then it can be written in this particular
form by doing some mathematical manipulations
28
00:03:32,700 --> 00:03:41,530
if you are interested in the details of this,
then you can refer to the book optical electronics
29
00:03:41,530 --> 00:03:48,440
by Ghatak and Thyagarajan or to introduction
to fiber optics by Ghatak and Thyagarajan
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00:03:48,440 --> 00:03:49,440
any of these books.
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00:03:49,440 --> 00:03:52,330
So, there are the details.
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00:03:52,330 --> 00:04:02,489
Similarly, in the cladding I can get the power
as this where G is a constant then I can find
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00:04:02,489 --> 00:04:10,670
out the fractional power as power in the core
divided by total power and I can express this
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00:04:10,670 --> 00:04:16,220
in this particular form and again the details
can be seen in this book.
35
00:04:16,220 --> 00:04:23,520
So, I have the fractional power in the core
given by this expression.
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00:04:23,520 --> 00:04:33,010
Now, I am interested in how does this look
like when I vary V, when I vary the parameters
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00:04:33,010 --> 00:04:42,009
of the fiber and in this way I vary the value
of V normalized frequency, then how the power
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00:04:42,009 --> 00:04:47,660
in the core and the cladding is distributed
how this distribution changes.
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00:04:47,660 --> 00:04:57,710
So, for that let me plot this eta frictional
power as a function of V and this has been
40
00:04:57,710 --> 00:05:00,140
taken from this paper.
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00:05:00,140 --> 00:05:02,180
I have adopted this.
42
00:05:02,180 --> 00:05:06,530
So, it is slightly modified.
43
00:05:06,530 --> 00:05:15,979
So, what I see that if I increase the value
of V then the fractional power in the core
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00:05:15,979 --> 00:05:18,000
increases.
45
00:05:18,000 --> 00:05:22,520
This is a very simple observation here.
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00:05:22,520 --> 00:05:38,110
Another thing that I see is that for LP01
mode and LP11 mode LP12 mode this starts from
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00:05:38,110 --> 00:05:47,900
0, while for these modes this starts from
0.5 and what I see that if I do a little mathematics
48
00:05:47,900 --> 00:05:49,390
I can show.
49
00:05:49,390 --> 00:05:56,930
I can show that for l is equal to 0 mode,
l is equal to 0 mode as the mode approaches
50
00:05:56,930 --> 00:06:07,850
to cut off then the fractional power can be
given by eta tending to 0 for l is equal to
51
00:06:07,850 --> 00:06:14,750
0, and eta tends to l minus 1 over l, for
l greater than or equal to 1.
52
00:06:14,750 --> 00:06:20,069
So, I can see for LP01 more the cut off is
0.
53
00:06:20,069 --> 00:06:25,819
LP01 mode the cut off is 0 and l is equal
to 0.
54
00:06:25,819 --> 00:06:32,590
So, as I approach to cut off the power is
0.
55
00:06:32,590 --> 00:06:38,750
And if I take LP11 mode then again eta is
0.
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00:06:38,750 --> 00:06:41,419
LP11 mode cut off is 2.4048.
57
00:06:41,419 --> 00:06:50,050
So, as I approach to cut off at 2.4048 it
will start from 0; however, for l is equal
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00:06:50,050 --> 00:06:54,830
to 2 mode this would be half.
59
00:06:54,830 --> 00:07:01,240
So, l is equal to 2 mode LP21 mode the cut
off is around 3.8.
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00:07:01,240 --> 00:07:07,780
So, around 3.8 it will start from 50 percent.
61
00:07:07,780 --> 00:07:18,639
So, this is how the power is power of different
parts is distributed among between the core
62
00:07:18,639 --> 00:07:23,819
and the cladding.
63
00:07:23,819 --> 00:07:31,800
As and I can see that as we increases there
is more and more power confined in the core
64
00:07:31,800 --> 00:07:37,260
and therefore, the fractional power in the
core increases as we increases this, this
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00:07:37,260 --> 00:07:41,120
I know from the theory of planar wave guide
also.
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00:07:41,120 --> 00:07:52,800
We have seen that now let us see some examples
for a fiber with n1 is equal to one point
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00:07:52,800 --> 00:08:00,830
four five n2 is equal to 1.44 and a is equal
to 4.5 micrometer, if I calculate the value
68
00:08:00,830 --> 00:08:09,430
of V then it comes out to be 4.135 at lambda
is equal to 1.55 micrometer and then if I
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00:08:09,430 --> 00:08:17,910
see the fractional power in the core for various
supported modes then it is given like this.
70
00:08:17,910 --> 00:08:21,600
For LP01 mode 95 percent power is in the core.
71
00:08:21,600 --> 00:08:28,889
So, if I look at the mode intensity plots
then 95 percent power is in the core, this
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00:08:28,889 --> 00:08:35,610
dashed this dashed line corresponds to the
core cladding interface.
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00:08:35,610 --> 00:08:44,450
So, most of the power is inside the core for
LP11 mode about 86 percent power is in the
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00:08:44,450 --> 00:08:47,610
core, and 14 percent goes out.
75
00:08:47,610 --> 00:08:59,630
For LP21 mode 68 percent power is in the core
and for LP02 mode only 38 percent power is
76
00:08:59,630 --> 00:09:01,010
in the core.
77
00:09:01,010 --> 00:09:13,050
So, I see that as I go towards higher order
modes the fractional power in the core decreases.
78
00:09:13,050 --> 00:09:25,400
Because higher order modes are closer to cut
off; if I look at how the power at different
79
00:09:25,400 --> 00:09:29,250
wave lengths is distributed between the core
and the cladding.
80
00:09:29,250 --> 00:09:40,060
So, for that I consider a fiber with n1 is
equal to 1.45, n2 is equal to 1.44 and a is
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00:09:40,060 --> 00:09:43,020
equal to 4 micrometer and then here.
82
00:09:43,020 --> 00:09:50,199
I have tabulated the fractional power in the
core for LP01 which is the fundamental mode
83
00:09:50,199 --> 00:09:53,490
and LP11 which is the first higher order mode.
84
00:09:53,490 --> 00:10:04,630
So, I see that at 0.65 micrometer wave length,
90 more than 98 percent power for LP01 mode
85
00:10:04,630 --> 00:10:13,900
is located in the core and 96 percent power
is located in the core for LP11 mode.
86
00:10:13,900 --> 00:10:24,150
So, as I increase the wavelength I see the
general trend that the power in the core for
87
00:10:24,150 --> 00:10:26,170
a given mode decreases.
88
00:10:26,170 --> 00:10:31,810
So, as I increase the wavelength the mode
spreads out, because I know if I increase
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00:10:31,810 --> 00:10:36,810
the wavelength which means I am decreasing
the value of V and the mode approaches to
90
00:10:36,810 --> 00:10:42,089
cut off and when a mode approaches to cut
off then the field spreads out.
91
00:10:42,089 --> 00:10:47,340
So, the fractional power in the core would
decrease.
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00:10:47,340 --> 00:10:58,140
If you look at the modal fields of LP11 mode
at 0.65 micrometer wavelength then it looks
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like this, we have the value of V at this
wavelength is 6.57.
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00:11:03,699 --> 00:11:06,140
So, high value of V.
95
00:11:06,140 --> 00:11:11,580
So, very, so most of the power is in the core.
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00:11:11,580 --> 00:11:18,010
But if you go to 1.7 micrometer wave length
the value of V is 2.5 which is very close
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to cut off the cut off is 2.4048, and then
I see that the field spreads out and there
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is only 37 percent power in the core.
99
00:11:35,420 --> 00:11:40,870
What happens if I change the numerical aperture
of the fiber?
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00:11:40,870 --> 00:11:50,400
So, for that I take a fiber with cladding
refractive index 1.444, core radius four micrometer
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and I analyze it at 1.55 micrometer wavelength.
102
00:11:55,890 --> 00:12:00,160
So, now, I change the value of numerical aperture.
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Then what I see as I increase the value of
numerical aperture, then the fractional power
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00:12:07,449 --> 00:12:17,199
in the core increases and this is intuitively
correct I expect this intuitively that.
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As the index difference between the core and
cladding increases, then more and more power
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00:12:24,520 --> 00:12:29,490
is pushed inside the core.
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00:12:29,490 --> 00:12:32,790
The index contrast between the core and the
cladding increases.
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00:12:32,790 --> 00:12:41,400
So, intuitively I think that the core cladding
interface this boundary is hard to penetrate
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for the field and so, less field extends in
the cladding region and there is more and
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00:12:49,170 --> 00:12:51,490
more field in the core region.
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00:12:51,490 --> 00:12:58,510
So, that is why the power in the core region
increases the fractional power in the core
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00:12:58,510 --> 00:13:05,199
is very high if the value of numerical aperture
is high.
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00:13:05,199 --> 00:13:17,010
When the numerical aperture is low very low
it is 0.05 just 0.05 it is very weekly guiding
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fiber.
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00:13:18,590 --> 00:13:27,120
So, it hardly guides the light only 14 percent
of light is in the core and 86 percent light
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00:13:27,120 --> 00:13:31,640
spreads out in the cladding.
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00:13:31,640 --> 00:13:42,650
So, it is really very poor guidance the value
of V is very small 0.87 quite close to 0,
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and when the numerical aperture is high then
of course, the entire field is pushed in the
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00:13:49,300 --> 00:13:50,929
core.
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00:13:50,929 --> 00:13:57,510
The value of V is very large about 5.7.
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00:13:57,510 --> 00:14:05,930
Now after this I would like to focus on single
mode fiber and because single mode fiber is
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the fiber which is used in long haul telecommunication
system, and I would like to first look at
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00:14:13,540 --> 00:14:21,820
some very important parameters of a single
mode fiber what are these parameters.
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00:14:21,820 --> 00:14:27,670
One is cut off wavelength what is the cut
off wavelength of the fiber, what is the spot
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00:14:27,670 --> 00:14:38,760
size or mode field diameter or effective area
of the fiber, what is bend loss and what are
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00:14:38,760 --> 00:14:42,199
the dispersion properties of this fiber.
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00:14:42,199 --> 00:14:49,339
So, we would look into these in subsequent
lectures as we go along.
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00:14:49,339 --> 00:14:57,730
So, let us first look at cut off wavelength,
what do I mean by cut off wavelength?
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00:14:57,730 --> 00:15:05,669
If you look at a fiber, then the single mode
condition is given by V is less than equal
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00:15:05,669 --> 00:15:23,049
to 2.4048 corresponds to the limiting case,
and V is given by 2 pi over lambda naught
131
00:15:23,049 --> 00:15:30,570
times a times square root of n1 square minus
n2 square.
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00:15:30,570 --> 00:15:42,410
So, if I find out the wavelength corresponding
to this value of V, 2.4048, then this wavelength
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will come out to be 2 pi divided by 2.4048
times a times square root of n1 square minus
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00:15:50,199 --> 00:15:58,170
n2 square, or I can also express it in terms
of numerical aperture because the square root
135
00:15:58,170 --> 00:16:03,699
of n1 square minus n2 square is nothing, but
numerical aperture or I can also represent
136
00:16:03,699 --> 00:16:09,980
it in terms of relative index difference delta,
because delta is n1 square minus n2 square
137
00:16:09,980 --> 00:16:13,060
over two n1 square.
138
00:16:13,060 --> 00:16:21,949
So, in this way I can calculate the value
of lambda c and what happens that for all
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the wave lengths greater than lambda c the
fiber is single mode, and for wavelengths
140
00:16:29,600 --> 00:16:41,579
is smaller than lambda c or shorter than lambda
c the fiber is multimode or few moded.
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00:16:41,579 --> 00:16:49,549
This wavelength lambda c give you a demarcation
between the single mode operation and multimode
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00:16:49,549 --> 00:17:01,010
operation although this wavelength corresponds
to the cutoff of LP11 mode, but in general
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00:17:01,010 --> 00:17:05,020
we call it the cut off wavelength of the fiber.
144
00:17:05,020 --> 00:17:12,949
So, when I say cut off wavelength of the fiber
then it means the cut off wavelength of LP11
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00:17:12,949 --> 00:17:18,530
mode of the fiber.
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00:17:18,530 --> 00:17:29,160
Next thing is propagation constant if I look
at the b-V curve of LP01 mode, because it
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00:17:29,160 --> 00:17:30,220
is a single mode fiber.
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00:17:30,220 --> 00:17:40,190
So, only LP01 mode is guided, if I look at
the b-V curve of LP01 mode it looks like this.
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00:17:40,190 --> 00:17:48,810
This I have obtained by solving the transcendental
equation corresponding to l is equal to 0
150
00:17:48,810 --> 00:17:56,320
mode and extracting the first root of that
which corresponds to LP01 mode.
151
00:17:56,320 --> 00:18:03,590
So, this exact means by numerically solving
the transcendental equation or Eigenvalue
152
00:18:03,590 --> 00:18:04,660
equation.
153
00:18:04,660 --> 00:18:07,690
So, it goes like this.
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00:18:07,690 --> 00:18:17,530
So, do I need to refer back always to this
curve if I am given a fiber and wave length
155
00:18:17,530 --> 00:18:24,640
then I know the value of V, and I want to
find out the value of b or propagation constant
156
00:18:24,640 --> 00:18:35,150
then I need to refer back to this curve always
can I do something that I can fit some equation
157
00:18:35,150 --> 00:18:36,150
to this.
158
00:18:36,150 --> 00:18:44,780
So, that I just use that equation and extract
the value of b if I know the value of V. And
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00:18:44,780 --> 00:18:58,140
it can be done what is done what is seen that
if you fit this equation to this curve, then
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00:18:58,140 --> 00:19:07,490
it fits very well for a is equal to this and
b is equal to this in the range 1.5 to 2.5
161
00:19:07,490 --> 00:19:11,960
in the range of V which goes from 1.5 to 2.5.
162
00:19:11,960 --> 00:19:20,730
So, this is an empirical relation between
b and V, which is given by a minus b over
163
00:19:20,730 --> 00:19:28,390
V whole square where a is equal to 1.1428
and b is close to 1.
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00:19:28,390 --> 00:19:39,770
So, I can always use this empirical relation
if the value of V lies in this range to obtain
165
00:19:39,770 --> 00:19:47,610
the propagation constant of LP01 mode of the
fiber, and it becomes very handy while doing
166
00:19:47,610 --> 00:19:51,410
calculations.
167
00:19:51,410 --> 00:20:03,520
How good this approximation is, how good this
empirical relation is for that I have superposed
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00:20:03,520 --> 00:20:13,680
the values of b obtained by this empirical
relation, on the values of V which I have
169
00:20:13,680 --> 00:20:21,620
obtained by solving the transcendental equation,
and I find that the agreement is quite good
170
00:20:21,620 --> 00:20:29,770
if I look at this graph I can also have the
quantitative estimate of this fitting how
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00:20:29,770 --> 00:20:31,950
good this fitting is.
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00:20:31,950 --> 00:20:44,260
For that I will do in the next slide, but
if I know now this empirical relation, then
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00:20:44,260 --> 00:20:51,880
my life becomes very easy I need not to go
to solve the transcendental equation again
174
00:20:51,880 --> 00:21:00,230
and again for example, if I have a fiber with
n2 is equal to 1.45, delta is equal to 0.64
175
00:21:00,230 --> 00:21:08,600
percent, a is equal to 3 micrometer and I
consider a wavelength of 1546 nanometer.
176
00:21:08,600 --> 00:21:18,530
So, these values correspond to a value of
V which is 2 and if I put it here then I find
177
00:21:18,530 --> 00:21:28,100
out the value of b immediately as 0.41616
and from there I can extract the value of
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00:21:28,100 --> 00:21:30,720
n effective.
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00:21:30,720 --> 00:21:42,020
Now, coming back to how accurate it is, I
have tabulated the values of b obtained by
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00:21:42,020 --> 00:21:49,860
solving the transcendental equation and obtained
by the empirical relation given in the previous
181
00:21:49,860 --> 00:21:58,940
slide, and I have looked at the percentage
difference between the two; and I see that
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00:21:58,940 --> 00:22:08,170
the maximum difference is 0.23 percent in
the range of V going from 1.5 to 2.4.
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00:22:08,170 --> 00:22:17,740
So, in this range it is a very good approximation
I can use this empirical relation without
184
00:22:17,740 --> 00:22:22,740
any problem.
185
00:22:22,740 --> 00:22:33,130
Next important parameter of a single mode
optical fiber is the spot size, when power
186
00:22:33,130 --> 00:22:43,900
exits from a single mode fiber, and if I capture
the field or the intensity.
187
00:22:43,900 --> 00:22:52,370
Which comes out of the fiber just at the output
end of the fiber as it exits from the fiber,
188
00:22:52,370 --> 00:22:53,640
then it looks like this.
189
00:22:53,640 --> 00:23:04,120
When I look at it then it very much resembles
to with the spot which comes out of a laser
190
00:23:04,120 --> 00:23:05,450
beam.
191
00:23:05,450 --> 00:23:18,830
So, it immediately tends me to think that
weather can I weather can I approximate it
192
00:23:18,830 --> 00:23:30,160
by a Gaussian because the output of a Gaussian
is output of a laser is Gaussian of a single
193
00:23:30,160 --> 00:23:32,490
mode laser.
194
00:23:32,490 --> 00:23:42,450
So, what I try to fit a Gaussian to this kind
of distribution and a Gaussian I represent
195
00:23:42,450 --> 00:23:49,080
as shi r is equal to A e to the power minus
r square over w square, where a represents
196
00:23:49,080 --> 00:23:58,360
what is the value at the center and w is the
width of the Gaussian.
197
00:23:58,360 --> 00:23:59,740
So, what I do?
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00:23:59,740 --> 00:24:11,920
I plot the exact field in blue line, which
I obtained in the form of Bessel functions
199
00:24:11,920 --> 00:24:19,640
and then I plot in green color the best fitted
Gaussian.
200
00:24:19,640 --> 00:24:27,690
How do I fit I will explain in the next slide
and what I see that this fitting is very good,
201
00:24:27,690 --> 00:24:38,700
this fitting comes out to be very good at
least for this case.
202
00:24:38,700 --> 00:24:50,090
So, I can very well use this Gaussian approximation
to represent this field, and why I want to
203
00:24:50,090 --> 00:24:52,440
do this?
204
00:24:52,440 --> 00:25:01,670
Because using Bessel functions all the time
is not a convenient thing and working with
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00:25:01,670 --> 00:25:04,170
Gaussian is very easy.
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00:25:04,170 --> 00:25:17,200
So, I express my modal field with a Gaussian
and then with the help of this I can estimate
207
00:25:17,200 --> 00:25:26,030
or define the spot size the size of this spot
as you know that the field goes down to 0
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00:25:26,030 --> 00:25:33,380
when r tends to infinity, but what is the
size, what is the region in which the maximum
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field is there.
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So, that we can define by r is equal to w
because it r is equal to w the modal field
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00:25:43,121 --> 00:25:49,120
drops down to 1 over e of its peak value,
and intensity drops down to 1 over e square
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of its peak value.
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00:25:50,370 --> 00:25:53,990
So, it can give me a good estimate of the
size of the spot.
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00:25:53,990 --> 00:26:00,670
So, this w is known as Gaussian spot size,
and two w twice of this value is basically
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the mode field diameter.
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00:26:04,220 --> 00:26:11,890
And I can find out the effective mode area
by pi w square.
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00:26:11,890 --> 00:26:16,140
How do I find out the best fitted Gaussian?
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The best fitted Gaussian has two parameters.
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00:26:19,330 --> 00:26:28,820
So, I need to optimize 2 parameters a and
w, a is easy to do because it can be found
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00:26:28,820 --> 00:26:35,750
out what is the maximum value of the field,
and to find out w what I do I take the overlap
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of this Gaussian with the exact field which
is defined in terms of Bessel function.
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00:26:42,050 --> 00:26:50,480
So, I take this overlap of the Gaussian with
this field and I vary the value of the w,
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00:26:50,480 --> 00:26:54,010
in order to obtain the maximum value of eta.
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00:26:54,010 --> 00:27:02,559
The value of w which gives me the maximum
value of overlap that gives me the best fitted
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00:27:02,559 --> 00:27:12,470
Gaussian.
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00:27:12,470 --> 00:27:24,110
Then I tend to think just in the same way
I have defined an empirical relation between
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the propagation constant and V, can I do the
same for the spot size also.
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00:27:28,770 --> 00:27:34,600
So, that I need not to all that time fit this
Gaussian.
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00:27:34,600 --> 00:27:44,230
So, there is an empirical relation which relates
the spot size with normalized frequency V,
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00:27:44,230 --> 00:27:47,870
and it has been given by D. Marcuse.
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00:27:47,870 --> 00:27:58,480
So, with this relation I can find out the
spot size of a given fiber if I know the value
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00:27:58,480 --> 00:28:08,720
of V. And this empirical relation is quite
accurate in the range of V from 0.8 to 2.5.
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00:28:08,720 --> 00:28:10,640
So, if I plot it.
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00:28:10,640 --> 00:28:23,080
So, it goes like this and this is an obvious
result it as I increase the value of V as
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00:28:23,080 --> 00:28:30,929
I know the confinement in the core increases
and therefore, the value of w over a would
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00:28:30,929 --> 00:28:32,800
decrease.
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00:28:32,800 --> 00:28:45,890
Now let me find out how accurate this approximation
is how accurate this empirical relation is.
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00:28:45,890 --> 00:28:52,070
So, for that I take two representative points.
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00:28:52,070 --> 00:29:02,620
One is for smaller values of V that is at
V is equal to 0.9, and if I see the modal
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00:29:02,620 --> 00:29:09,840
field here it looks like this, and I find
out the value of w by two methods one is by
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00:29:09,840 --> 00:29:16,179
Bessel functions the exact value, and another
is from empirical relation.
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00:29:16,179 --> 00:29:28,030
So, I find that at V is equal to 0.9 the exact
value of w is about 30.7 micrometer, while
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00:29:28,030 --> 00:29:33,250
empirical relation gives me a value of 31.8.
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00:29:33,250 --> 00:29:44,140
I take another representative point at V is
equal to 1.8 this is the model field and the
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00:29:44,140 --> 00:29:55,020
value of w obtained from exact and empirical
formula give me almost the same value they
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00:29:55,020 --> 00:29:56,680
are in good agreement.
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00:29:56,680 --> 00:30:06,701
So, I find that for lower value for smaller
values of V there is some discrepancy, but
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00:30:06,701 --> 00:30:17,540
this empirical formula works very well when
I go towards higher values of V.
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00:30:17,540 --> 00:30:29,750
Now, let me look at the fractional power in
the core and see how good the Gaussian approximation
250
00:30:29,750 --> 00:30:33,160
and empirical fitting is.
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00:30:33,160 --> 00:30:39,640
So, what I have done well I have calculated
now, the fractional power in the core which
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00:30:39,640 --> 00:30:46,320
is given as P core over P total, while considering
the Gaussian spot size.
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00:30:46,320 --> 00:30:50,480
So, as I have said that it is very easy to
work with Gaussians.
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00:30:50,480 --> 00:30:57,230
So, now, calculating these integrals would
not be very difficult if you use shi as a
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00:30:57,230 --> 00:30:58,230
Gaussian.
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00:30:58,230 --> 00:31:04,020
So, when I do this then it comes out to be
1 minus e to the power minus 2 a square over
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00:31:04,020 --> 00:31:11,190
w square, and when I plot it goes like this.
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00:31:11,190 --> 00:31:16,270
To find out how accurate it is I again take
two representative point that V is equal to
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00:31:16,270 --> 00:31:28,090
0.9 and at b is equal to 1.8, and I find that
at smaller values of V there is a huge discrepancy,
260
00:31:28,090 --> 00:31:38,710
it is because at lower value of V there is
a small discrepancy between the exact value
261
00:31:38,710 --> 00:31:47,860
and the and the Gaussian and empirically fitted
value, but this small discrepancy is amplified
262
00:31:47,860 --> 00:31:58,340
because this appears as the it appears in
the form of e to the power minus something.
263
00:31:58,340 --> 00:32:10,510
So, this gets amplified a lot; however, at
the higher values of V the discrepancies is
264
00:32:10,510 --> 00:32:17,730
small and therefore, the discrepancy in fractional
power is also very small.
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00:32:17,730 --> 00:32:19,950
So, the agreement is good.
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00:32:19,950 --> 00:32:30,200
So, in the next lecture I would look in to
some other ways of defining the spot size,
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00:32:30,200 --> 00:32:41,140
and I would also look into various examples
to understand the cut off and spots sizes.
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00:32:41,140 --> 00:32:41,920
Thank you.