Page 499 - Fiber Optic Communications Fund

480 Fiber Optic Communications

the Stokes intensity so that the depletion of the pump due to Stokes wave amplification can be ignored. Eq.
(10.407) may be approximated as
d p
≅−  , (10.408)
p p
dZ
− p Z
 (Z)=  (0)e . (10.409)
p
p
From Eq. (10.406), we have
d
s − p Z
s
R p
 s = g  (0)e − . (10.410)
Integrating Eq. (10.410) from 0 to L, we obtain
( )
 (L)
s
In = g  (0)L eff,p − L, (10.411)
s
R p
 (0)
s
where the effective length of the pump, L , is given by
eff,p
1 − exp (− L)
p
L eff,p = . (10.412)
p
Rearranging Eq. (10.411), we find
 (L)=  (0)e − s L+g R  p (0)L eff,p . (10.413)
s
s
The Stokes wave is amplified if g  (0)L > L. If a signal (Stokes wave) is down-shifted in frequency
R p eff,p s
by about 14 THz, it would have the highest amplification since g (Ω) is maximum when the frequency shift
R
is about 14 THz (see Fig. 6.21).
Example 10.8
Stokes and pump beams co-propagate in a fiber of length 2 km. The Raman coefficient of the fiber g =
R
−1
1 × 10 −13 m/W, input Stokes’s signal power =−10 dBm, input pump power = 20 dBm, = 0.046 km ,
s
−1
2
= 0.08 km , and effective area of the fiber = 40 μm . Calculate the Stokes signal power at the fiber output.
p
Solution:
The Stokes and pump powers in a fiber may be approximated as
P p,s
 ≅ . (10.414)
p,s
A
eff
So, Eq. (10.413) can be rewritten as
P (L)= P (0)e − s L+g R P p (0)L eff,p ∕A eff , (10.415)
s
s
P (0) (dBm) = 20 dBm. (10.416)
p
P p (0) (dBm)∕10
P (0)= 10 mW
p
= 100 mW, (10.417)
P (0) (dBm) =−10 dBm, (10.418)
s