Nonlinear Fiber Optics - 4 ed. Agrawal

6.1. Nonlinear Birefringence 181
respectively, and satisfy somewhat simpler equations:
∂A +
∂z + β ∂A +
1
∂t
∂A −
∂z + β ∂A −
1
∂t
+ iβ 2
2
+ iβ 2
2
∂ 2 A +
∂t 2
∂ 2 A −
∂t 2
+ α 2 A + = iΔβ
2 A − + 2iγ (
|A+ | 2 + 2|A − | 2) A + , (6.1.15)
3
(
|A− | 2 + 2|A + | 2) A − , (6.1.16)
+ α 2 A − = iΔβ
2 A + + 2iγ
3
where we assumed that β 1x ≈ β 1y = β 1 for fibers with relatively low birefringence.
Notice that the four-wave-mixing terms appearing in Eqs. (6.1.11) and (6.1.12) are
replaced with a linear coupling term containing Δβ. At the same time, the relative
strength of XPM changes from 2 3
to 2 when circularly polarized components are used
to describe wave propagation.
6.1.3 Elliptically Birefringent Fibers
The derivation of Eqs. (6.1.11) and (6.1.12) assumes that the fiber is linearly birefringent,
i.e., it has two principal axes along which linearly polarized light remains linearly
polarized in the absence of nonlinear effects. Although this is ideally the case for
polarization-maintaining fibers, elliptically birefringent fibers can be made by twisting
a fiber preform during the draw stage [11].
The coupled-mode equations are modified considerably for elliptically birefringent
fibers. This case can be treated by replacing Eq. (6.1.1) with
E(r,t)= 1 2 (ê xE x + ê y E y )exp(−iω 0 t)+c.c., (6.1.17)
where ê x and ê y are orthonormal polarization eigenvectors related to the unit vectors ˆx
and ŷ used before as [12]
ê x =
ˆx + irŷ
√ ,
1 + r 2
ê y =
r ˆx − iŷ
√
1 + r 2 . (6.1.18)
The parameter r represents the ellipticity introduced by twisting the preform. It is
common to introduce the ellipticity angle θ as r = tan(θ/2). The cases θ = 0 and π/2
correspond to linearly and circularly birefringent fibers, respectively.
Following a procedure similar to that outlined earlier for linearly birefringent fibers,
the slowly varying amplitudes A x and A y are found to satisfy the following set of
coupled-mode equations [12]:
∂A x
∂z + β ∂A x
1x
∂t
∂A y
∂z + β ∂A y
1y
∂t
+ iβ 2 ∂ 2 A x
2 ∂t 2 + α 2 A x = iγ[(|A x | 2 + B|A y | 2 )A x +CA ∗ xA 2 ye −2iΔβz ]
+ iγD[A ∗ yA 2 xe iΔβz +(|A y | 2 + 2|A x | 2 )A y e −iΔβz ], (6.1.19)
+ iβ 2 ∂ 2 A y
2 ∂t 2 + α 2 A y = iγ[(|A y | 2 + B|A x | 2 )A y +CA ∗ yA 2 xe 2iΔβz ]
+ iγD[A ∗ xA 2 ye −iΔβz +(|A x | 2 + 2|A y | 2 )A x e iΔβz ], (6.1.20)
where the parameters B, C, and D are related to the ellipticity angle θ as
B = 2 + 2sin2 θ
2 + cos 2 θ , C = cos2 θ sinθ cosθ
2 + cos 2 , D =
θ 2 + cos 2 θ . (6.1.21)