Nonlinear Fiber Optics - 4 ed. Agrawal

180 Chapter 6. Polarization Effects
6.1.2 Coupled-Mode Equations
The propagation equations governing evolution of the two polarization components
along a fiber can be obtained following the method of Section 2.3. Assuming that the
nonlinear effects do not affect the fiber mode significantly, the transverse dependence
of E x and E y can be factored out using
E j (r,t)=F(x,y)A j (z,t)exp(iβ 0 j z), (6.1.10)
where F(x,y) is the spatial distribution of the single mode supported by the fiber,
A j (z,t) is the slowly varying amplitude, and β 0 j is the corresponding propagation
constant ( j = x,y). The dispersive effects are included by expanding the frequencydependent
propagation constant in a manner similar to Eq. (2.3.23). The slowly varying
amplitudes, A x and A y , are found to satisfy the following set of two 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 + 2 )
3 |A y| 2 A x + iγ 3 A∗ xA 2 y exp(−2iΔβz), (6.1.11)
∂A y
∂z + β ∂A y
1y + iβ 2 ∂ 2 A y
∂t 2 ∂t 2 + α 2 A y
= iγ
(|A y | 2 + 2 )
3 |A x| 2 A y + iγ 3 A∗ yA 2 x exp(2iΔβz), (6.1.12)
where
Δβ = β 0x − β 0y =(2π/λ)B m = 2π/L B (6.1.13)
is related to the linear birefringence of the fiber. The linear or modal birefringence leads
to different group velocities for the two polarization components because β 1x ≠ β 1y
in general. In contrast, the parameters β 2 and γ are the same for both polarization
components because they have the same wavelength λ.
The last term in Eqs. (6.1.11) and (6.1.12) is due to coherent coupling between the
two polarization components and leads to degenerate four-wave mixing. Its importance
to the process of polarization evolution depends on the extent to which the phasematching
condition is satisfied (see Chapter 10). If the fiber length L ≫ L B , the last
term in Eqs. (6.1.11) and (6.1.12) changes sign often and its contribution averages out
to zero. In highly birefringent fibers (L B ∼ 1 cm typically), the four-wave-mixing term
can often be neglected for this reason. In contrast, this term should be retained for
weakly birefringent fibers, especially those with short lengths. In that case, it is often
convenient to rewrite Eqs. (6.1.11) and (6.1.12) using circularly polarized components
defined as
A + =(Ā x + iĀ y )/ √ 2, A − =(Ā x − iĀ y )/ √ 2, (6.1.14)
where Ā x = A x exp(iΔβz/2) and Ā y = A y exp(−iΔβz/2). The variables A + and A − represent
right- and left-handed circularly polarized (often denoted as σ + and σ − ) states,