High-Nonlinearity in Glass Fibers and Cross-Phase Modulation - Optical Signal Processing by Silicon Photonics - page 59

Fig. 5.2 Gain spectra g( Ω )
for different power levels P 1
and P 2 of the two beams [ 9 ]

where

4 γ j
β j

= (5.13)

b j P j

So for frequencies such that |Ω| <Ω c then a weak modulation of the steady state
will be affected by the gain given below

(5.14)

g (Ω) = 2Im ( K )

where Im is used for the imaginary part.
Figure 5.2 shows the gain spectra against different values of the ratio of power
i.e., P 2 / P 1 by taking P 1 = 100 W. As the gain spectrum is symmetric on both sides
g ( − ) = g ( ) so only one part is shown.
The parameters of fiber related to a realistic situation of propagation of silica
fibers in visible region near about 0.53 μ m. The wavelengths of two propagating
fields differ very slightly so that β j , γ j , and V gj are nearly the same for j = 1 and 2.
In this respect, g can be evaluated as a closed form and can be expressed as

1
2

(5.15)

Ω c − Ω 2

g (Ω) = β 1 |Ω|

Where c is given in Eq. ( 5.12 ). The maximum gain occurs at = C / √ 2 and the
maximum gain increases as we increase P 1 and P 2 . Since the gain is because of
XPM and it vanishes when either the powers P 1 or P 2 become zero means both the
optical fields should be present.
If K has an imaginary part against different values of the , then the steady
state becomes unstable as a 1 and a 2 will exponentially grow with the fiber length.
This phenomenon shows the modulation instability [ 2 - 10 ] because it leads to