450 CHAPTER 9. SOLITON SYSTEMS
An extension of the polarization-multiplexing technique, called polarization-multi-
level coding, has also been suggested [175]. In this technique, the information coded in
each bit is contained in the angle that the soliton state of polarization makes with one
of the principal birefringence axes. This technique is also limited by random variations
of fiber birefringence and by randomization of the polarization angle by the amplifier
noise and has not yet been implemented because of its complexity.
9.6.3 Impact of Higher-Order Effects
Higher-order effects such as TOD and intrapulse Raman scattering become quite im-
portant at high bit rates and must be accounted for a correct description of DM solitons
[182]–[185]. These effects can be included by adding two new terms to the standard
NLS equation as was done in Eq. (9.3.16) in the context of distributed amplification of
standard solitons. In the case of DM solitons, the Raman and TOD terms are added to
Eq. (9.4.5) to obtain the following generalized NLS equation:
i∂B
∂z−
β 2 (z)
2∂^2 B
∂t^2+γp(z)|B|^2 B=
iβ 3
6∂^3 B
∂t^3+TRγp(z)B∂|B|^2
∂t, (9.6.1)
whereTRis the Raman parameter with a typical value of 3 fs [10]. The effective non-
linear parameterγ ̄≡γpiszdependent because of variations in the soliton energy along
the fiber link. The Raman term leads to the Raman-induced frequency shift known as
the SSFS. This shift is negligible for 10-Gb/s systems but becomes increasingly im-
portant as the bit rate increases to 40 Gb/s and beyond. The TOD term also becomes
important at high bit rates, especially when the average GVD of the fiber link is close
to zero.
To understand the impact of TOD and SSFS on solitons, we use the moment method
of Section 9.5 and assume that the last two terms in Eq. (9.6.1) are small enough that
the pulse shape remains approximately Gaussian. Using Eq. (9.6.1) in Eqs. (9.5.1) and
(9.5.2), the frequency shiftΩ, and soliton positionqare found to evolve withzas
dΩ
dz=−
γpTR
E∫∞−∞(
∂|B|^2
∂t) 2
dt+∑
nδΩnδ(z−zn), (9.6.2)dq
dz=β 2 Ω+β 3
6 E∫∞−∞(
∂|B|
∂t) 2
dt+∑
nδqnδ(z−zn), (9.6.3)where the last term accounts for fluctuations induced by the amplifier noise. These
equations show that the Raman term in Eq. (9.6.1) leads to a frequency shift while the
TOD term produces a shift in the soliton position.
Consider the case of standard loss-managed solitons. If we ignore the noise terms
and integrate Eqs. (9.6.2) and (9.6.3) after usingB(z,t)from Eq. (9.5.16), the Raman-
induced frequency shiftΩand the soliton positionqare found to evolve along the fiber
length as
Ω(z)=−4 γTREs
15 T 03∫z0p(z)dz≡−8 fLM|β 2 |
15 T 04∫z0p(z)dz, (9.6.4)q(z)=β 2∫z0Ω(z)dz+
β 3 z
18 T 02+β 3∫z0Ω^2 dz, (9.6.5)