Page 447 - Fiber Optic Communications Fund

428 Fiber Optic Communications

Eq. (10.67) may be rewritten as
q(T, Z)= A(Z, T)e i(T,Z)
√ [ 2 ( 2 )]
P T T T + i Z
0 0
2
0
= exp − , (10.72)
2 2
i
4
|T |e 1 2(T + Z )
1 0 2
√ [ 2 2 ]
P T T T 0
0 0
A(T, Z)= exp − ( ) , (10.73)
4
2 2
|T | 2 T + Z
1
0 2
2
T Z
2
(T, Z)=− (Z)− 4 , (10.74)
1
2 2
2(T + Z )
0 2
The instantaneous power is
[ ]
2 2
P T 2 T T 0
0 0
2
P(T, Z)= A (T, Z)= exp − , (10.75)
2 4
|T | |T |
1 1
and the instantaneous frequency is (see Eq. (2.165))
T Z
2
(T)=− = . (10.76)
T (T + Z )
4
2 2
0 2
The negative sign is chosen because the carrier wave is exp (−i t). The actual instantaneous frequency is
0
+ .
0
The instantaneous power P(T, Z) and instantaneous frequency are plotted in Fig. 2.32 for an anomalous
dispersion fiber ( < 0). From Eq. (10.76) and Fig. 10.3, when < 0, we see that near the leading edge
2
2
(T < 0) is positive (blue shift) whereas it is negative (red shift) near the trailing edge (T > 0). These
changes in frequency occur continuously as the signal propagates down the fiber. Since the blue components
travel faster than the red components in the anomalous dispersion fiber, the frequency components near the
leading edge arrive early and the frequency components near the trailing edge arrive late. This explains why
the pulse is broadened at the fiber output. For a fiber with normal dispersion, the situation is exactly opposite.
10.4 Nonlinear Schrödinger Equation
Owing to the Kerr effect, an optical signal undergoes a phase shift that is proportional to the signal power as
given by Eq. (10.52). If this nonlinear effect is included, Eq. (10.58) is modified as (see Appendix B)
( ) 2
q q q 2 q
2
i + 1 − + |q| q =−i , (10.77)
z t 2 t 2 2
where is the nonlinear coefficient related to the Kerr coefficient n by (Appendix B)
2
n 0
2
= , (10.78)
cA eff
where A eff is the effective area of the fiber mode. Using the frame of reference that moves with the group
speed of the pulse,
T = t − z, (10.79)
1
Z = z, (10.80)