Page 449 - Fiber Optic Communications Fund

430 Fiber Optic Communications

10.5 Self-Phase Modulation
To find the impact of fiber nonlinearity acting alone, let us ignore . To solve Eq. (10.81) under this condition,
2
we separate the amplitude and phase,
i
q = Ae . (10.84)
Substituting Eq. (10.84) into Eq. (10.81), we find
[ A ] [ ]
2
i + i A =− A + i A. (10.85)
Z Z 2
Separating real and imaginary parts, we obtain
A
=− A (10.86)
Z 2
or

A(T, Z)= A(T, 0) exp (−Z∕2), (10.87)
2 2 −Z
= A (T, Z)= A (T, 0)e . (10.88)
Z
Let the fiber length be L. Integrating Eq. (10.88) from 0 to L, we obtain
L
2
(T, L)= (T, 0)+ A (T, 0) e −Z dZ
∫
0
2
= (T, 0)+ A (T, 0)L , (10.89)
eff
where
1 − exp (−L)
L = . (10.90)
eff

Substituting Eqs. (10.87) and (10.89) into Eq. (10.84), we find
q(T, L)= A(T, L)e i(T,L)
2
= A(T, 0)e −L∕2 i[(T,0)+A (T,0)L eff ]
e
2
= q(T, 0)e −L∕2+i|q(T,0)| L eff , (10.91)
where
q(T, 0)= A(T, 0)e i(T,0) . (10.92)
2
Here, |q(T, 0)| represents the instantaneous power at the input. Since the phase of the optical signal is mod-
ulated by its own power distribution, this effect is known as self-phase modulation (SPM). From Eq. (10.91),
we find that
|q(T, L)| = |q(T, 0)|e −L∕2 . (10.93)
So, the amplitude of the signal decreases exponentially with distance, but the pulse width at the fiber output
remains the same as that at the fiber input. However, the spectral width at the output is larger than that at the
input. This is because the nonlinear mixing of the input frequency components due to SPM generates new
frequency components. Using Eq. (10.89), the instantaneous frequency at L is
d(T, L) d(T, 0) d|q(T, 0)| 2
(T, L)=− =− − L eff .
dT dT dT