Download cut off frequency for different modes

MODAL PROPAGATION INSIDE AN OPTICAL
FIBER (WAVE MODEL-II)
1. CUT OFF FREQUENCY
As seen earlier,  has to be real for a propagating mode. The frequency range over
which  remains real therefore is important information. It can be shown that for  to
be real the frequency of the wave has to be greater than certain value, called the cut-off
frequency.
1. Cutoff frequency is defined as the frequency at which the mode does not remain
purely guided. That is, when a guided mode is converted into a radiation mode.
2. The cut-off is defined by w  0 (and not   0 as is usually done for the metallic
waveguides)
where, w2   2   2  2 , and   2 is propagation constant in cladding  2  .
If w is real we get the guided mode
and if w is imaginary we get radiating mode
3.
At cut off frequency
since w  0 ,
  2
At cut off frequency,  approaches  2 which means that the propagation constant of
the wave approaches to the propagation constant of a uniform plane wave in the
medium having dielectric constant  2 .
2. V Number of Optical Fiber
1. The V-number is one of the important characteristic parameters of a step index optical
fiber. V-number of optical fiber is defined as
1
2
(1)
V
a n21  n 2 2 2



  
2
1

1
2 2
2

a
Now since
u 2   2  2   2
w2   2  2   2
 u 2  w2   2 1   2  2
 12   22
If we multiply both sides by a 2 we get
a 2  u 2  w2   a 2 ( 12   22 )  V 2
(2)
At the cutoff when w  0 , we have ua  V number.
The V-number provides information about the modes on a step index fiber. As can be
seen, the v-number is proportional to the difference between the square of refractive
index of optical fiber i.e. numerical aperture, and radius a of the core and is inversely
proportional to wavelength  .
3. CUT OFF CONDITIONS FOR VARIOUS MODES
MODES
TE0 m , TM 0 m
CUTOFF
t0m
HE1m
t1,m 1
EH m
t m
 n2  n2

HE m  1 2 2  1
 n1

t  2,m
where t pq is the q th root of the J p Bessel function.
Figures of various modes:
HE11 Mode (fig 3)
TE01 Mode (fig 4)
TM 01 Mode (fig 5)
HE21 Mode (fig 6)
At that point we have
ua  2.4 since it is the maximum value for guided wave propagation.
For single mode propagation,
V  2.4
1
1
2
2
2 2
2
2 2
let n1  n2  0.1 (assumed)

a n1  n2  2.4 ,





a
 2   0.1  2.4

a 2.4
 
 0.6
a
  4 m
(3)

So the effective radius is very small for single mode optical fiber which is why the
normal light source will not become the source. Only LASERS type source is essential to
launch light inside single mode optical fiber and the cross section becomes so small that
it accepts only one ray corresponding to HE11 mode.
It can be noted from the table that for HE11 mode the cut-off for  a    t11  0 . That is,
HE11 mode does not have any cut off.
Explanation of figures: HE11 mode is a special mode as it always propagates.
From table, we note that, HE21 mode shows cutoff when  a    t2 2, m  t0 m . The cut
off for TE0m and TM 0m modes is given by  a    t0m . Hence we note that the cutoff
frequencies of TE0 m , TM 0 m and HE21 is the same. Since t01  2.4 (first root of J 0 Bessel
function), below  2.4 only HE11 mode propagates.
Important: (i) The lowest order mode on the optical fiber is the HE11 mode.
(ii) The fiber remains single mode if its v-number is less than 2.4.
4.
Weakly Guided Approximation
For a practical fiber, the difference is the refractive indices of the core & the cladding is
very small. This justifies certain approximation in the modal analysis. This
approximation is called the ‘weakly guiding approximation’. The characteristic equations
then can be simplified in this situation. The approximate but simplified analysis can
provide better insight into the modal characteristics of an optical fiber. In weakly guiding
situation there is substantial spread of fields in the cladding. The optical energy is not
tightly confined to the core and is weakly guided.
1. Primarily we are interested in velocity of different modes since this information helps
in obtaining the amount of the pulse spread, i.e., dispersion.
2. The phase and group velocity of a mode is given as
Phase velocity 


d
d
3. Variation of  as a function of frequency is the primary outcome of the model analysis.
Group velocity 
From equation (1) we note that V number 

2


1
2 2
2
a n n
2
1

a  N .A
(4)
c
For a given fiber the radius  a  is fixed and the numerical aperture  NA is also fixed. We
therefore get
 V   (V number is proportional to frequency of operation).

Since V is proportional to  , instead of writing variation of  in terms of  , we can
obtain variation of  as a function of V-number of optical fiber. The V-number
hence is called the Normalized frequency.
4. For a guided mode we have
2    1
Let us define the Normalized propagation constant b as
 2   22
(5)
b 2
1   22
We can see from the above equation that if mode is very close to cutoff, then   2 ,
and b  0 .
On the other hand when a mode is very far from cutoff then   1 and b  1 .
If we plot the normalized propagation constant, b, instead of the absolute value of
propagation constant  , its value lies between 0 and I. Also instead of frequency we use
V number since V  number is proportional to frequency.
A plot of b v/s V is called the b  V diagram. The b  V diagram is the characteristic
diagram for propagation of mode in a step index optical fiber.
b  V Diagram (fig 7)
Explanation of b-V diagram:
(a) b  V plot for every mode is a monotonically increasing function of the V-number.
Every plot starts at b  0 and asymptotically saturates b  1 . That suggests that the modal
fields get more and more confined with increasing V  number .
(b) The HE11 mode propagates for V  0 .
(c) TE01 , TM 01 and HE21 modes propagates for v  24 .
(d) All these mode propagates which have cut-off V-number less than the V-number of
the fiber.
5. For weakly guiding fibers, all the modes which have same cutt-off V-number
degenerate. That is, their b  V curves almost merge. For example, for a weakly
guiding fiber, the three modes TE01 , TM 01 and HE21 would degenerate. These modes
then have same phase velocity. The three modal patterns corresponding to the three
modes than travel together with same phase change. Consequently the three modal
patterns are not seen distinctly but we get a super imposed field distribution.
6. The superimposed field distributions are linearly polarized in nature. That is, the field
orientation is same every where in the cross-sectional plane. Fig 4 to 6 demonstrates
that. Since the fields are not linearly polarized, the modes are designated as the
Linearly polarized (LP) modes.
(a) If TE01 mode superimposes on to HE21 mode, the field distribution is horizontally
polarized.
(b) If TM 01 mode superimpose on to HE21 mode the field distribution is in vertically
polarized.
Electrical field direction changes in crosses sectional plane i.e. polarization of mode
changes in cross sectional plane depending upon whether we have vertical or
horizontal field. Now we have weakly guided approximation because the polarization
is either horizontal or vertical. So in weakly guided approximation the general modes
get converted into linearly polarized modes (LP modes).
4. LINEARLY POLARIZED MODES
1. LP Modes are not the fundamental modes TE , TM and hybrid modes are still the
fundamental modes but since we are unable to distinguish weakly guided fiber we get
a field distribution which will look like linearly polarized modes. So we get
LP m mode which has two indices.
HE11  LP01
(3-7a)
TE0m , TM 0m , HE2m  LP1m
(3-7b)
HE 1,m  EH 1,m  LP m ,   2
(3-7c)
So what we get on actual fiber are nothing but Linearly Polarized Modes where the
electric and magnetic field components having either x comp. or y components. It is
not going to vary in  r ,  , z  .
2. Now for qth modes there are
mode
q
modes. Since there is a degeneracy of 4 with every
2
q
 4  2q Modes
2
So total no of modes:

Q
M    2q   Q 2
q 1

4V 2
2
V2
2
Therefore if we have V number, then we can quickly calculate how many modes will
actually propagate on an optical fiber.
M 