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  .
2. 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.
The HE11 mode has 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 will be 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 HE11 mode.
(ii) The fiber remains single mode if its v-number is less than 2.4.
2.
Weakly Guided Approximation
For a weekly guided fiber we have n1  n2 or we have seen from the ray model that to
reduce the dispersion n1 should be as close to n2 as possible. In practice therefore we
prefer n1  n2 given 1   2 In this situation there is substantial spread of fields in the
cladding. The optical energy is not tightly confined to the core and is weakly guided. The
propagation problem generally is solved for the ‘weekly guided approximation’.
n2  n2
________ (3-4)
  1 2 2  1
2n1
3. Primarily we are interested in velocity of different modes because it decides the
dispersion of different modes and the amount of the pulse spread.


d
Group velocity 
d
Variation of  as a function of frequency is the primary outcome of the model analysis.
1
2
From equation (3-1) V number 
a  n12  n22  2


4. Phase velocity 
a  N .A
(3-5)
c
For a given fiber the radius of fiber is fixed  a  and the numerical aperture  NA is also
fixed. We therefore get
 V   (V number is proportional to frequency of operation).

So instead of writing variation of  in terms of  we obtain variation of  as a
function of v-number of optical fiber. The v-number is called the Normalized
frequency.
5. For a guided mode we have
2    1
Let us define the Normalized propagation constant b as
b
 2   22
12   22
(3-6)
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 instead of 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. We get a diagram which 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 (fig3.7)
Explanation of figure: As we seen in the figure HE11 mode starts from zero and another
set starts from V  2.4 i.e. TE01 , HE21 , TM 01 modes and then other clusters of modes and so
on . It is a typical b  V diagram that starts from b  0 and saturates at value b  1 .
As we increase the value of…….. we can draw that value of V . The vertical line finds
out how many modes lie below this V number. Count these modes. These are the modes
which are actually going to propagate on optical fiber. If we find out the total number of
modes which are propagating, the cutoff frequency is less than V number of optical fiber
and so we find it.
6. For conditions of Weakly Guided Approximation, if refractive index of core and
cladding approach each other then cutoff frequency also approaches the root of Bessel
functions, the three second set of curves as in (fig 3-7). If we go for weakly guided
approximation then these three modes TE01 , TM 01 , HE21 lose their identity. Actually
we always see the combination of these modes because if three curves are together,
variation of  with respect to  , then the phase velocity and group velocity is same
for the three modes.
That’s why we see the combination of these three modes and these are not seen
distinctly in an optical fiber.
7. As in figure (3.4 to 3.6):
(a) If TE01 mode superimposes on to HE21 mode  field distribution is in horizontal
direction.
(b) If TM 01 mode superimpose to HE21 mode  field distribution is in vertical
direction.
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.
(3-7a)
HE11  LP01
(3-7b)
TE0m , TM 0m , HE2m  LP1m
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
  4  2q Modes
2
q
modes. Since there is a degeneracy of 4 with every
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 