Nonlinear Fiber Optics - 4 ed. Agrawal

6.6. Random Birefringence 215
PMD, with values of D p as low as 0.05 ps/ √ km [126]. Because of the √ L dependence,
PMD-induced pulse broadening is relatively small compared with the GVD effects. If
we use D p = 0.1 ps/ √ km as a typical value, σ T ∼ 1 ps for fiber lengths ∼100 km and
can be ignored for pulse widths >10 ps. However, PMD becomes a limiting factor for
lightwave systems designed to operate over long distances at high bit rates.
Several other factors need to be considered in practice. The derivation of Eq. (6.6.1)
assumes that the fiber link has no elements exhibiting polarization-dependent loss or
gain. The presence of polarization-dependent losses along a fiber link can modify the
PMD effects considerably [139]–[146]. Similarly, the effects of second-order PMD
should be considered for fibers with relatively low values of D p . Such effects have
been studied and lead to additional distortion of optical pulses [129].
6.6.2 Vector Form of the NLS Equation
As mentioned earlier, the polarization state of a pulse in general becomes nonuniform
across the pulse because of random changes in fiber birefringence. At the same time,
PMD leads to pulse broadening. Such effects can be studied by generalizing the coupled
NLS equations derived in Section 6.1, namely Eqs. (6.1.11) and (6.1.12), to the
case in which a fiber exhibits random birefringence changes along its length. It is
more convenient to write these equations in terms of the normalized amplitudes u and
v defined as
√
u = A x γLD e iΔβz/2 √
, v = A y γLD e −iΔβz/2 . (6.6.3)
If we also use soliton units and introduce normalized distance and time as
ξ = z/L D , τ =(t − ¯β 1 z)/T 0 , (6.6.4)
where ¯β 1 = 1 2 (β 1x + β 1y ), Eqs. (6.1.11) and (6.1.12) take the form
( ∂u
i
∂ξ + δ ∂u )
+ bu + 1 ∂ 2 (
u
∂τ 2 ∂τ 2 + |u| 2 + 2 )u
3 |v|2 + 1 3 v2 u ∗ = 0, (6.6.5)
( ∂v
i
∂ξ − δ ∂v )
− bv + 1 ∂ 2 (
v
∂τ 2 ∂τ 2 + |v| 2 + 2 )v
3 |u|2 + 1 3 u2 v ∗ = 0, (6.6.6)
where
b = T 0 2(Δβ)
, δ = T 0 d(Δβ)
(6.6.7)
2|β 2 |
|β 2 | dω
Both δ and b vary randomly along the fiber because of random fluctuations in the
birefringence Δβ ≡ β 0x − β 0y .
Equations (6.6.5) and (6.6.6) can be written in a compact form using the Jonesmatrix
formalism. We introduce the Jones vector |U〉 and the Pauli matrices as [130]
( ) ( ) ( ) ( )
u 1 0
0 1
0 −i
|U〉 = , σ
v 1 = , σ
0 −1 2 = , σ
1 0 3 = . (6.6.8)
i 0
In terms of the Jones vector |U〉, the coupled NLS equations become [148]
i ∂|U〉 (
∂ξ + σ 1 b|U〉 + iδ ∂|U〉 )
+ 1 ∂ 2 |U〉
∂τ 2 ∂τ 2 + s 0 |U〉− 1 3 s 3σ 3 |U〉, (6.6.9)