Page 539 - Fiber Optic Communications Fund

520 Fiber Optic Communications

and from Eq. (11.138), we have
Δz
2
2
(t, Δz)= (t, 0)− |A(t, z)| dz = (t, 0)− Δz |A(t, 0)| , (11.140)
∫
eff
0
where
exp (Δz)− 1
Δz eff = . (11.141)

Substituting Eqs. (11.140) and (11.139) in Eq. (11.136), we find
2
q (t, Δz)= q (t, 0) exp (−iΔz |q (t, 0)| + Δz), (11.142)
b b eff b
where
q (t, 0)= A(t, 0) exp [i(t, 0)]. (11.143)
b
l
With q (t, 0)= q (t, Δz), Eq. (11.142) becomes
b
b
l l 2
q (t, Δz)= q (t, Δz) exp (−iΔz |q (t, Δz)| + Δz). (11.144)
b b eff b
Fig. 11.24 illustrates the unsymmetric SSFS. This technique can be summarized as follows:
̂
(i) Initial field q (t, 0) is known. First, the nonlinear and loss effects (N) are ignored and the output of a
b
l
lossless, linear fiber q (t, Δz) is calculated using the Fourier transformation technique.
b
̂
(ii) Next, fiber dispersion (D) is ignored. The NLSE is analytically solved with the initial condition q (t, 0)=
b
l
q (t, Δz) and the field envelope at Δz, q (t, Δz) is calculated using Eq. (11.144).
b b
(iii) q (t, 2Δz) is calculated with q (t, Δz) as the initial condition by repeating (i) and (ii). This process is
b
b
repeated until z = L. The step size Δz should be chosen sufficiently small that the absolute value of
the nonlinear phase shift Δ accumulated over a distance Δz should be much smaller than .From
Eq. (11.140), it follows that
2
|Δ| = |(t, Δz)− (t, 0)| = Δz |A(t, 0)| ≪. (11.145)
eff
A disadvantage of the unsymmetric SSFS is that the step size has to be really small since the error scales
2
since as Δz [24]. The step size can be made significantly larger using the symmetric SSFS, which is described
as follows. From Eq. (11.125), we have
Δz
[ ]
q (t, Δz)= exp [N (t, z)+ D (t)]dz q (t, 0). (11.146)
b ∫ b b b
0
q b (t, 0)
l
l
l
= q (t, 0) Dispersion q (t, ∆z) Nonlinear q b (t, ∆z) Dispersion q (t, 2∆z) Nonlinear operation q b (t, 2∆z)
b b b
only operation only exp(∫ ∆z N
exp(D ∆z) exp(∫ ∆z b (t, z)dz) exp(D b ∆z) 0 b (t, z)dz)
N
b
0
l
q b (t, L) Nonlinear operation q (t, L) Dispersion
b
exp(∫ ∆z b (t, z)dz) only
N
0 exp(D b ∆z) q (t, L – ∆z)
b
Figure 11.24 Unsymmetric split-step Fourier scheme for backward propagation.