Page 537 - Fiber Optic Communications Fund

518 Fiber Optic Communications

or
q b
=[N + D]q , (11.123)
b
(−z)
with the initial condition q (t, 0)= q(t, L). From Eq. (11.118), it follows that
b
−1
M q (t, 0) ≡ q (t, L)= q(t, 0). (11.124)
b
b
Thus, by solving Eq. (11.123), q (t, L) can be found, which should be equal to the fiber input q(t, 0). In other
b
words, if the fiber link inverse operator M −1 can be realized in the digital domain, by operating it on the fiber
link output, we can retrieve the fiber input q(t, 0). Since Eq. (11.123) is nothing but Eq. (11.111) with z → −z,
this technique is referred to as back propagation. Eq. (11.122) may be rewritten as
q b
=[N + D ]q , (11.125)
b
b
b
z
with q (t, 0) ≡ q(t, L)
b
2
D =−D = i , (11.126)
b 2 2
t
2
N =−N =−i|q | + . (11.127)
b b
2
The NLSE with reversed signs of dispersion, loss, and nonlinear coefficients is solved in the digital domain
to undo the distortion caused by the transmission fiber. Figs. 11.22 and 11.23 illustrate the forward and
backward propagation. Eq. (11.125) can be solved numerically using the split-step Fourier scheme [24].In
Eq. (11.125), the operators N and D act simultaneously and N changes with z, which makes it harder to
b
b
b
realize the operator M −1 numerically. However, over a small propagation step, Δz, D , and N may be approx-
b
b
imated to act one after the other. Hence, this technique is known as the split-step technique. This is an approxi-
mation, and this technique becomes more accurate as Δz → 0. First let us consider the unsymmetric split-step
fiber
q(t, 0) q(t, L) Rx.
Tx. front end DSP
β 2 ,γ,α
Figure 11.22 Propagation in a single-span fiber (forward propagation).

DSP
DBP

q (t, 0) q (t, L)
IF and b b
Rx. phase noise = q(t, L) = q(t, 0)
front end
removal
Virtual fiber
, –γ, –α
–β 2
Figure 11.23 Backward propagation in the virtual fiber.