106
3 Optical Signal Attenuation and Dispersion
for a mode of order (ν, m) for a step-index waveguide is
α νm = α 1
P core
P
+ α 2
P clad
P
(3.14)
where P is total power in a given mode and the fractional powers P core /P and P clad /P
are shown in Fig. 3.5 as a function of the V number for several low-order modes.
From the relation P = P core + P clad , it follows that Eq. (3.14) can be written as
α νm = α 1 + (α 2 − α 1 )
P clad
P
(3.15)
The total loss of the waveguide can be found by summing over all modes weighted
by the fractional power in that mode.
For the case of a graded-index fiber the situation is much more complicated. In
this case, both the attenuation coefficients and the modal power tend to be functions
of the radial coordinate. At a distance r from the core axis the loss is [14]
α(r ) = α 1 + (α 2 − α 1 )
n
2
(0) − n
2
(r )
n 2 (0) − n
2
2
(3.16)
P
core /P
1.0
0.8
0.6
0.4
1.0
0.2
0
1.0
0.8
0.6
0.4
0.2
0
2
4
6
P
clad /P
V
LP 01
LP 11
LP 21
LP 02
Fig. 3.5 The fractional powers P core /P and P clad /P for several low-order modes