High-Nonlinearity in Glass Fibers and Cross-Phase Modulation - Optical Signal Processing by Silicon Photonics - page 57

instability induced by XPM has fundamental importance as it suggests the possi-
bility of the soliton formation in the normal regime and it also has practical impli-
cations for the propagation of visible radiation in optical fibbers.

5.5.1 Theory

In order to present the major results in a possible simplest way, let us consider the
case of two optical fields propagating in a single mode fiber. The field amplitudes
are A 1 and A 2, respectively that satisfy the nonlinear Schrodinger equation [ 11 , 12 ]
modified for XPM by the addition of cross coupling term:

2 β j ∂ 2 Aj

∂ Aj
∂ z + V − 1

∂ Aj
∂ t

1
2 α j A j

1

2 2

(5.1)

i + = − γ j A j + 2 A 3 − j A j

gj

∂ t 2

where j = 1 or 2, V g shows the group velocity, α j is the absorption coefficient
( β j = dvg − 1 /d ω and β j < 0 for anomalous dispersion) and

ω j
CA eff

γ j = n 2 (5.2)

related to fiber nonlinearity that is responsible for the both SPM and XPM. In Eq.
( 5.2 ), A eff denotes the effective core area for silica fibers. And the last term in Eq. ( 5.1 )
is because of XPM that couples the two waves. It is the XPM-induced coupling due to
that modulation instability in the normal dispersion regime occurs provided that β j > 0
for both waves. For the sake of simplification, we neglect the fiber loss by setting
α j = 0 because inclusion of the fiber losses do not change the basic concept of the sub-
ject. Steady-state solution of Eq. ( 5.1 ) is given in the following equation showing that

P j e ( i φ j ) ,

(5.3)

A j = j = 1, 2

P j relates to optical power and phase

(5.4)

φ j = γ j P j + 2 P 3 − j Z

We can assume the stability of the steady state by letting

e ( i φ j )

(5.5)

A j = P j + a j

Showing α j as weak perturbation, linearizing equation ( 5.1 ) in a 1 and a 2 gives

∂ a j
∂ z

∂ a j
∂ t

+ V − 1
gj

i

2 β j ∂ 2 a j

1

− 2 γ j ( P 1 P 2 ) 2 a j + a j a 3 − j + a ∗ 3 − j =

∂ t 2 − γ j P j

(5.6)