Nonlinear Fiber Optics - 4 ed. Agrawal

230 Chapter 7. Cross-Phase Modulation
is obtained by setting the time derivatives in Eqs. (7.1.15) and (7.1.16) to zero. If fiber
losses are neglected, the solution is of the form
Ā j = √ P j exp(iφ j ), (7.2.1)
where j =1or2,P j is the incident optical power, and φ j is the nonlinear phase shift
acquired by the jth field and given by
φ j (z)=γ j (P j + 2P 3− j )z. (7.2.2)
Following the procedure of Section 5.1, stability of the steady state is examined
assuming a time-dependent solution of the form
A j = (√ P j + a j
)
exp(iφ j ), (7.2.3)
where a j (z,t) is a small perturbation. By using Eq. (7.2.3) in Eqs. (7.1.15) and (7.1.16)
and linearizing in a 1 and a 2 , the perturbations a 1 and a 2 satisfy the following set of two
coupled linear equations:
∂a 1
∂z + 1 ∂a 1
+ iβ 21 ∂ 2 a 1
v g1 ∂t 2 ∂t 2 = iγ 1 P 1 (a 1 + a ∗ √
1)+2iγ 1 P1 P 2 (a 2 + a ∗ 2), (7.2.4)
∂a 2
∂z + 1 ∂a 2
+ iβ 22 ∂ 2 a 2
v g2 ∂t 2 ∂t 2 = iγ 2 P 2 (a 2 + a ∗ √
2)+2iγ 2 P1 P 2 (a 1 + a ∗ 1), (7.2.5)
where the last term is due to XPM.
The preceding set of linear equations has the following general solution:
a j = u j exp[i(Kz− Ωt)] + iv j exp[−i(Kz− Ωt)], (7.2.6)
where j = 1or2,Ω is the frequency of perturbation, and K is its wave number. Equations
(7.2.4) through (7.2.6) provide a set of four homogeneous equations for u 1 , u 2 , v 1 ,
and v 2 . This set has a nontrivial solution only when the perturbation satisfies the following
dispersion relation:
where
[(K − Ω/v g1 ) 2 − f 1 ][(K − Ω/v g2 ) 2 − f 2 )=C XPM , (7.2.7)
f j = 1 2 β 2 jΩ 2 ( 1 2 β 2 jΩ 2 + 2γ j P j ) (7.2.8)
for j = 1 or 2. The coupling parameter C XPM is defined as
C XPM = 4β 21 β 22 γ 1 γ 2 P 1 P 2 Ω 4 . (7.2.9)
The steady-state solution becomes unstable if the wave number K has an imaginary
part for some values of Ω. The perturbations a 1 and a 2 then experience an exponential
growth along the fiber length. In the absence of XPM coupling (C XPM = 0), Eq. (7.2.7)
shows that the analysis of Section 5.1 applies to each wave independently.
In the presence of XPM coupling, Eq. (7.2.7) provides a fourth-degree polynomial
in K whose roots determine the conditions under which K becomes complex. In general,
these roots are obtained numerically. If the wavelengths of the two optical beams