Nonlinear Fiber Optics - 4 ed. Agrawal

236 Chapter 7. Cross-Phase Modulation
introduce normalized variables using
ξ = z/L D , τ =(t − z/v g1 )/T 0 , A j = γ 1 L D u j , (7.3.10)
Eqs. (7.1.15) and (7.1.16) can be written in the form
i ∂u 1
∂ξ − d 1 ∂ 2 u 1
2 ∂τ 2 +(|u 1| 2 + σ|u 2 | 2 )u 1 = 0, (7.3.11)
i ∂u 2
∂ξ + d 2 ∂ 2 u 2
2 ∂τ 2 +(|u 2| 2 + σ|u 1 | 2 )u 2 = 0, (7.3.12)
where d j = |β 2 j /β 20 |, β 20 is a reference value used to define the dispersion length, and
u 1 is assumed to propagate in the normal-dispersion region. We have also assumed
γ 2 ≈ γ 1 . The parameter σ has a value of 2 when all waves are linearly polarized but
becomes 1, the solution has the following form:
√
σd2 + d 1
u 1 (ξ ,τ) =r
σ 2 − 1 dn(rτ, p)[qdn−2 (rτ, p) ∓ 1]exp(iQ ± 1
ξ ), (7.3.13)
√
d2 + σd 1
u 2 (ξ ,τ) =r
σ 2 − 1
where the propagation constants Q 1 and Q 2 are given by
Q ± 1 =
Q ± 2 =
p 2 sn(rτ, p)cn(rτ, p)
exp(iQ ± 2
ξ ), (7.3.14)
dn(rτ, p)
r2
σ 2 − 1 [σ(d 2 + σd 1 )(1 + q 2 ) ∓ 2q(σd 2 + d 1 )] − r2 d 1
2 (1 ∓ q)2 , (7.3.15)
r2
σ 2 − 1 [(d 2 + σd 1 )(1 + q 2 ) ∓ 2qσ(σd 2 + d 1 )]. (7.3.16)
In these equations, sn, cn, and dn are the standard Jacobi elliptic functions [50] with
the modulus p (0 < p < 1) and the period 2K(p)/r, where K(p) is the complete elliptic
integral of the first kind, q =(1 − p 2 ) 1/2 , and r is an arbitrary scaling constant. Two
families of periodic solutions correspond to the choice of upper and lower signs. For
each family, p can take any value in the range of 0 to 1.
The preceding solution exists only for σ > 1. When σ < 1, the following single
family of periodic solutions is found [49]:
√
σd2 + d 1
u 1 (ξ ,τ) =r
1 − σ 2 psn(rτ, p)exp(iQ 1ξ ), (7.3.17)
√
d2 + σd 1
u 2 (ξ ,τ) =r
1 − σ 2 pdn(rτ, p)exp(iQ 2ξ ), (7.3.18)
where the propagation constants Q 1 and Q 2 are given by
Q 1 = − 1 2 r2 d 1 q 2 + r 2 (d 1 + σd 2 )/(1 − σ 2 ), (7.3.19)
Q 2 = 1 2 r2 d 2 (1 + q 2 )+r 2 σ(d 1 + σd 2 )/(1 − σ 2 ). (7.3.20)