Nonlinear Fiber Optics - 4 ed. Agrawal

6.2. Nonlinear Phase Shift 183
dA y
dz + α 2 A y = iγ(|A y | 2 + B|A x | 2 )A y . (6.2.2)
These equations describe nondispersive XPM in birefringent fibers and extend the
scalar theory of SPM in Section 4.1 to the vector case. They can be solved by using
A x = √ P x e −αz/2 e iφ x
, A y = √ P y e −αz/2 e iφ y
, (6.2.3)
where P x and P y are the powers and φ x and φ y are the phases associated with the two
polarization components. It is easy to deduce that P x and P y do not change with z.
However, the phases φ x and φ y do change and evolve as
dφ x
dz = γe−αz (P x + BP y ),
dφ y
dz = γe−αz (P y + BP x ). (6.2.4)
Since P x and P y are constants, the phase equations can be solved easily with the result
φ x = γ(P x + BP y )L eff , φ y = γ(P y + BP x )L eff , (6.2.5)
where the effective fiber length L eff =[1−exp(−αL)]/α is defined in the same way as
in the SPM case [see Eq. (4.1.6)].
It is clear from Eq. (6.2.5) that both polarization components develop a nonlinear
phase shift whose magnitude is the sum of the SPM and XPM contributions. In
practice, the quantity of practical interest is the relative phase difference given by
Δφ NL ≡ φ x − φ y = γL eff (1 − B)(P x − P y ). (6.2.6)
No relative phase shift occurs when B = 1. However, when B ≠ 1, a relative nonlinear
phase shift between the two polarization components occurs if input light is launched
such that P x ≠ P y . As an example, consider a linearly birefringent fiber for which B = 2 3 .
If CW light with power P 0 is launched such that it is linearly polarized at an angle θ
from the slow axis, P x = P 0 cos 2 θ, P y = P 0 sin 2 θ, and the relative phase shift becomes
Δφ NL =(γP 0 L eff /3)cos(2θ). (6.2.7)
This θ-dependent phase shift has several applications discussed next.
6.2.2 Optical Kerr Effect
In the optical Kerr effect, the nonlinear phase shift induced by an intense, high-power,
pump beam is used to change the transmission of a weak probe through a nonlinear
medium [4]. This effect can be used to make an optical shutter with picosecond response
times [6]. It was first observed in optical fibers in 1973 [13] and has attracted
considerable attention since then [14]–[25].
The operating principle of a Kerr shutter can be understood from Figure 6.1. The
pump and probe beams are linearly polarized at the fiber input with a 45 ◦ angle between
their directions of polarization. A crossed polarizer at the fiber output blocks probe
transmission in the absence of the pump beam. When the pump is turned on, the
refractive indices for the parallel and perpendicular components of the probe (with