Page 467 - Fiber Optic Communications Fund

448 Fiber Optic Communications

× 10 –4
2.5

∆f = 100 GHz
2
XPM PSD (mW/GHz) 1.5 1 ∆f = 200 GHz

∆f = 300 GHz
0.5

0
–20 –10 0 10 20
Frequency (GHz)

−1
Figure 10.15 The power spectral density of the XPM distortion vs. frequency. Parameters: = 0.046 km , P =
p0
−3
2
4mW, P = 0.1mW, L = 80 km, = 1.1 × 10 , =−21 ps /km, and = 0. Bit rate = 10 Gb/s.
s0 2 3
2
For the OOK–NRZ signal, a = a . Therefore, the PSD of P (t) is given by Eq. (4.20), i.e.,
n
n
p
2 2 [ ]
⟨|P ()| ⟩ P p0 T b 2 2 ()
p
lim = sinc (T ∕2) 1 + . (10.207)
b
T→∞ T 4 T
b
Fig. 10.15 shows the signal distortion due to XPM using Eqs. (10.203) and (10.207), and ignoring the discrete
part of the spectrum (second term in Eq. (10.207)). As can be seen, the PSD of XPM distortion decreases as
the channel spacing Δf(= Ω ∕(2)) increases.
p
This analysis can be modified by taking into account the pump envelope change due to dispersion [18].The
amplitude and phase fluctuations of the modulated signal due to XPM can also be calculated using a first- or
second-order perturbation theory [20].
10.7.2 Four-Wave Mixing
Four-wave mixing refers to the generation of a fourth wave at the frequency Ω due to the nonlinear interaction
n
of three waves at frequencies Ω , Ω , and Ω . To study the impact of FWM alone, let us ignore the SPM and
k
j
l
XPM terms in Eq. (10.140). In addition, in order to simplify the analysis, let us assume that the signals in each
channel are CW (constant envelope), so that the second and third terms on the left-hand side of Eq. (10.140)
can also be ignored. With these simplifications, Eq. (10.140) becomes
N∕2−1 N∕2−1 N∕2−1
q ∑ ∑ ∑
n ∗ iΔ jkln Z
+ q = i q q q e . (10.208)
n
j k l
Z 2
j=−N∕2 k=−N∕2 l=−N∕2
j+k−l=n
No SPM, no XPM