Nonlinear Fiber Optics - 4 ed. Agrawal

266 Chapter 7. Cross-Phase Modulation
where Ω = ω 1 − ω 2 is the pump-probe frequency difference, P 1 = 〈A 1 |A 1 〉 is the pump
power and ˆp = 〈A 1 |σ|A 1 〉/P 1 is the Stokes vector representing the pump SOP on the
Poincaré sphere. The effective nonlinear parameter, γ e = 8γ/9, is reduced by a factor
of 8/9 because of the averaging over rapid variations of the pump SOP. Residual birefringence
enters through the vector b ′ that is related to b by a random rotation on the
Poincaré sphere. Since such rotations leave the statistics of b unchanged, we drop the
prime over b ′ in what follows.
The pump equation (7.7.9) can be easily solved. The solution shows that the pump
pulse shifts in time as its walks away from the probe pulse but its shape does not
change, i.e., P 1 (z,τ) =P 1 (0,τ − z/L W ). The total probe power also does not change
as it is easy to show from Eq. (7.7.10) that the probe power P 2 = 〈A 2 |A 2 〉 satisfies
∂P 2 /∂z = 0. However, the orthogonally polarized components of the probe pulse can
exhibit complicated dynamics as its state of polarization evolves because of XPM. It is
useful to introduce the normalized Stokes vectors for the pump and probe as
ˆp = 〈A 1 |σ|A 1 〉/P 1 , ŝ = 〈A 2 |σ|A 2 〉/P 2 , (7.7.11)
and write Eq. (7.7.10) in the Stokes space as [108]
∂ŝ
∂z =(Ωb − γ e P 1 ˆp) × ŝ. (7.7.12)
The two terms on the right side of this equation show that the SOP of the probe rotates
on the Poincaré sphere along an axis whose direction changes in a random fashion as
dictated by b. Moreover, the speed of rotation also changes with z randomly.
It is useful to note that Eq. (7.7.12) is isomorphic to the Bloch equation governing
the motion of spin density in a solid [108]. In that context, P 1 corresponds to a static
magnetic field and b has its origin in small magnetic fields associated with nuclei
[111]. In a fashion similar to the phenomenon of spin decoherence, the probe SOP
evolves along the fiber in a random fashion and leads to intrapulse depolarization, a
phenomenon produced by the combination of XPM and PMD. Physically speaking,
the XPM interaction between the pump and probe makes the probe SOP vary across
the temporal profile of the pump. Fluctuating residual birefringence of the fiber then
transfers this spatial randomness into temporal randomness of the signal polarization.
The distance over which the SOPs of the pump and probe become decorrelated because
of PMD is governed by the PMD diffusion length defined as L diff = 3/(D p Ω) 2 . This
distance is