Page 462 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 443

Z
2 −Z
XPM (T, Z)= 2 |q (T − d Z, 0)| e dZ, (10.165)
p
p
∫
0
2
P (T, Z)= |q (T, Z)| = P (T, 0)e −Z. (10.166)
s s s
As in the case of SPM, the pulse width of the signal remains unchanged during propagation since we have
ignored dispersion. However, as can be seen from Eq. (10.165), the phase of the signal is modulated by the
pump. Hence, this is known as cross-phase modulation. The instantaneous frequency shift of the signal due
to XPM is
XPM Z |q (T − d Z, 0)| 2 −Z
p
p
=− =−2 e dZ. (10.167)
XPM ∫
T T
0
When the pump is sinusoidally modulated, its field envelope at the fiber input may be written as
√
q (T, 0)= P cos (ΩT), (10.168)
p p 0
2
2
|q (T − d Z, 0)| = P cos [Ω(T − d Z)]
p
p
p
p0
P p0
= {1 + cos [2Ω(T − d Z)]}. (10.169)
p
2
Substituting Eq. (10.169) in Eq. (10.165), we find
L
−Z
(T, L)= P {1 + cos [2Ω(T − d Z)]}e dZ
XPM p0 ∫ p
0
{ }
L ( )
−Z+i2Ω T−d p Z
= P L + P Re ∫ e dZ , (10.170)
p0 eff
p0
0
where
1 − exp (−L)
L = . (10.171)
eff

The first term on the right-hand side of Eq. (10.170) is the constant phase shift due to XPM, which is of no
importance. The second term denotes the time-dependent phase shift, which could potentially degrade the
performance. Ignoring the first term, Eq. (10.170) can be simplified as follows [14]:
{ [ [ ( ) ]]}
1 − exp − + i2Ωd p L
(T, L)= P Re e i2ΩT
XPM p0
+ i2Ωd
p
√
= P L XPM cos (2ΩT + ), (10.172)
p0 eff
where is the XPM efficiency given by
XPM
[ 2 ( ) −L ]
2 4sin Ωd L e
p
XPM (Ω) = 1 + −L 2 , (10.173)
2 2
2
+ 4Ω d p (1 − e )
{ −L ( ) } { }
e sin 2Ωd L 2Ωd p
p
= tan −1 − tan −1 . (10.174)
1 − e −L cos (2Ωd L)
p
When the walk-off parameter d = 0 or the modulation frequency Ω= 0, the XPM efficiency is maximum.
p
From Eq. (10.173), we find that XPM = 1 for this case. As the walk-off increases, the interaction between the