Biomedical Engineering Reference
In-Depth Information
The cutoff of each mode is obtained by solving the
characteristic equation for k
C
for that mode. The cutoff
frequency of a mode can be zero if k
C
¼
0. One and only
one mode can exist in an optical fiber with u
c
¼
0. This
mode is the hybrid HE
11
mode and it exists for all fre-
quencies. If the fiber is designed to operate with only the
HE
11
mode present, it will operate as a single-mode
optical fiber. The single-mode fiber has a very small core
diameter and small refractive index difference between
its core and cladding. These parameters are chosen to
ensure that all other guided modes are below their cutoff
frequency. To better understand the relationship be-
tween mode cutoff and the physical parameters of a fiber,
a cutoff parameter k
c
a, which is usually called the ''
V
''
number of the fiber, is defined as
Table 6.2-4 Dependence of the number of propagating modes
in a fiber on the physical properties of the fiber
Increasing physical
parameters
Number of
propagating modes
Core radius, a
Increases
Core refractive index, n
1
Increases
Cladding refractive index, n
2
Decreases
Source wavelength, l
0
Decreases
0.2%. These types of fibers are used for long-distance
communications where their very large information-
carrying capacity is needed. Single-mode fibers are used
in a large majority of the fiber-optic telecommunication
systems being installed today. A typical commercial step-
index multimode fiber, on the other hand, will propagate
hundreds of mode groups and have a relatively large
core diameter and refractive index difference between
the core and cladding. Step-index multimode fibers
are used in short-distance data links where their lower
information-carrying capacity is not an issue.
a
n
1
n
2
q
p
m
0
3
0
k
c
a ¼
u
c
(6.2.34)
Noting that
p
m
0
3
0
¼
2p
l
0
u
c
(6.2.35)
we finally obtain
q
n
1
n
2
V
hk
c
a ¼
2p
a
l
0
(6.2.36)
6.2.4 The graded-index fiber
The number of propagating modes in a step-index fiber is
proportional to the
V
number as shown in
Figure 6.2-16.
Table 6.2-4
illustrates how increasing
a, n
1
,n
2
,
or l
0
in-
fluences the number of propagating modes in the fiber.
Notice that for
V <
2.405 it is possible to design a single-
mode fiber that supports only the HE
11
mode. Typical
commercial single-mode fibers have a core diameter of
approximately 8-10
m
m and a refractive index difference
between the core and the cladding of approximately
The graded-index fiber, because of its relatively large
bandwidth and core diameter, is used in many local-area
networks where moderate information-carrying capacity
is needed. The multimode step-index fiber bandwidth is
severely limited (less than 100 MHz-km) due to modal
delay distortion. The grading of the refractive index
profile of a fiber core has the effect of increasing the
bandwidth of a fiber to up to 2 GHz-km by equalizing the
group delays of the various propagating mode groups.
Let's consider a multimode fiber with an in-
homogeneous core as shown in
Figure 6.2-17
. The wave
equation (
6.2.10
) is rewritten below showing the varia-
tion of the refractive index with
r:
30
TE
03
TM
03
HE
23
25
EH
22
HE
42
FðrÞ¼
0
HE
13
EH
12
HE
32
d
2
FðrÞ
dr
2
þ k
2
ðrÞ
b
2
y
2
r
2
þ
1
r
dFðrÞ
dr
20
EH
41
HE
61
TE
02
TM
02
HE
22
15
(6.2.37)
EH
31
HE
51
HE
12
EH
11
HE
31
where
10
TE
01
TM
01
HE
21
EH
21
HE
41
kðrÞ¼
2p
l
0
nðrÞ¼k
0
nðrÞ
(6.2.38)
5
HE
11
and
2
4
6
8
10
k
2
¼ k
2
ðrÞ
b
2
(6.2.39)
Figure 6.2-16 Types of modes in step-index fibers.
