Page 476 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 457

Pulse Pulse Pulse
1 2 3
Ch. 1 Ch. 2 Ch. 3
Echo Echo
FWM FWM pulse pulse
sideband sideband
f + f – f 3 f 1 f 2 f 3 f + f – f 1 frequency t 1 + t 2 – t 3 t 1 t 2 t 3 t 3 + t 2 – t 1 time
3
2
1
2
Figure 10.24 The analogy between interchannel FWM and intrachannel FWM.

second-order echo pulses are very small, and they are not visible in the linear plot shown in Fig. 10.23. We
ignore second-order echo pulses in the analysis of IFWM.
Suppose the pulse centered at mT is q . The nonlinear interaction between q , q , and q due to IFWM is
m
b
m
n
l
∗
described by q q q and the resulting echo pulse is centered near (l + m − n)T . For example, in Figs. 10.22
l m n
b
and 10.23, the first-order echo pulses centered around −50 ps and 50 ps and generated due to the nonlinear
interaction of signal pulses centered at −25 ps, 0 ps, and 25 ps. The echo pulse at 50 ps is generated by a
∗
nonlinear interaction of the form q q q where l = 1 (25 ps), m = 0 (0 ps), n =−1(−25 ps), and l + m − n = 2
l m n
(50 ps). The nonlinear interaction of the first-order echo pulses and signal pulses leads to second-order echo
pulses at −75 ps and 75 ps (see Fig. 10.22).
When l = m, it is known as degenerate IFWM, similar to the degenerate interchannel FWM. Otherwise, it
is called non-degenerate IFWM. The echo pulse centered around 50 ps is generated not only by signal pulses
∗
centered at −25 ps, 0 ps, and 25 ps (q q q ) due to non-degenerate IFWM, but also by pulses centered at
1 0 −1
2 ∗
0 ps and 25 ps (q q ) due to degenerate IFWM. Note that the echo pulses are generated in the locations of
1 0
signal pulses as well. In Fig. 10.23, the nonlinear interaction of the signal pulses centered at −25 ps, 0 ps,
∗
and 25 ps (q q q ) leads to an echo pulse around 0 ps (l = 1, m =−1, n = 0, l + m − n = 0). The coherent
1 −1 0
superposition of the signal pulse and echo pulse around T = 0 ps leads to the distortion of the signal pulse at
T = 0 ps, as shown in Fig. 10.23. Section 10.9 provides the mathematical description of IFWM.
10.8.3 Intra- versus Interchannel Nonlinear Effects
Fig. 10.25 illustrates the difference between intrachannel and interchannel nonlinear impairments. The pulse
located at the center interacts nonlinearly with the pulses within the trapezoids. This interaction includes
both intrachannel and interchannel nonlinear effects (SPM, IXPM, IFWM, XPM, FWM). The area of the
trapezoids depends on the system parameters such as fiber dispersion, nonlinear coefficient, launch power,
and transmission distance. For example, if the launch power is higher and/or the transmission distance is
longer, the area of the trapezoids would be larger. The nonlinear interaction of the pulse located at the center
with the pulses within the ellipse corresponds to intrachannel impairments (SPM, IXPM, IFWM).

10.9 Theory of Intrachannel Nonlinear Effects
The optical field envelope in a fiber-optic transmission system can be described by the nonlinear Schrödinger
equation (NLS) (see Eq. (10.81))

2
q (Z) q 2 (Z)
2
i − + |q| q =−i q, (10.242)
Z 2 T 2 2