Nonlinear Fiber Optics - 4 ed. Agrawal

182 Chapter 6. Polarization Effects
For a linearly birefringent fiber (θ = 0), B = 2 3 ,C = 1 3
,D = 0, and Eqs. (6.1.19) and
(6.1.20) reduce to Eqs. (6.1.11) and (6.1.12), respectively.
Equations (6.1.19) and (6.1.20) can be simplified considerably for optical fibers
with large birefringence. For such fibers, the beat length L B is much smaller than typical
propagation distances. As a result, the exponential factors in the last three terms
of Eqs. (6.1.19) and (6.1.20) oscillate rapidly, contributing little to the pulse evolution
process on average. If these terms are neglected, propagation of optical pulses in an
elliptically birefringent fiber is governed by the following set of coupled-mode equations:
∂A x
∂z + β ∂A x
1x + iβ 2 ∂ 2 A x
∂t 2 ∂t 2 + α 2 A x = iγ(|A x | 2 + B|A y | 2 )A x , (6.1.22)
∂A y
∂z + β ∂A y
1y + iβ 2 ∂ 2 A y
∂t 2 ∂t 2 + α 2 A y = iγ(|A y | 2 + B|A x | 2 )A y . (6.1.23)
These equations represent an extension of the scalar NLS equation, derived in Section
2.3 without the polarization effects [see Eq. (2.3.27)], to the vector case and are referred
to as the coupled NLS equations. The coupling parameter B depends on the ellipticity
angle θ [see Eq. (6.1.21)] and can vary from 2 3
to 2 for values of θ in the range 0 to π/2.
For a linearly birefringent fiber, θ = 0, and B = 2 3
. In contrast, B = 2 for a circularly
birefringent fiber (θ = π/2). Note also that B = 1 when θ ≈ 35 ◦ . As discussed later,
this case is of particular interest because Eqs. (6.1.22) and (6.1.23) can be solved with
the inverse scattering method only when B = 1 and α = 0.
6.2 Nonlinear Phase Shift
As seen in Section 6.1, a nonlinear coupling between the two orthogonally polarized
components of an optical wave changes the refractive index by different amounts for
the two components. As a result, the nonlinear effects in birefringent fibers are polarization
dependent. In this section we use the coupled NLS equations obtained in the
case of high-birefringence fibers to study the XPM-induced nonlinear phase shift and
its device applications.
6.2.1 Nondispersive XPM
Equations (6.1.22) and (6.1.23) need to be solved numerically when ultrashort optical
pulses propagate inside birefringent fibers. However, they can be solved analytically
in the case of CW radiation. The CW solution is also applicable for pulses whenever
the fiber length L is much shorter than both the dispersion length L D = T0 2/|β
2| and the
walk-off length L W = T 0 /|Δβ|, where T 0 is a measure of the pulse width. As this case
can be applicable to pulses as short as 100 ps and sheds considerable physical insight,
we discuss it first.
Neglecting the terms with time derivatives in Eqs. (6.1.22) and (6.1.23), we obtain
the following two simpler equations:
dA x
dz + α 2 A x = iγ(|A x | 2 + B|A y | 2 )A x , (6.2.1)