Four Wave Mixing in Silicon Photonics - Optical Signal Processing by Silicon Photonics - page 105

By taking the coupled-amplitude wave equations that govern the nonlinear dynam-
ics of two optical waves interacting inside a SOI waveguide.
Generally, these equations consider not only linear loss, dispersion, and the
free-carrier effects, but also take into account for the intensity-dependent nonlinear
losses due to two photon and free-carrier absorption. Using approximations based
on physical insights, we simplify the equations as per situations of practical inter-
est and outline techniques that can be used to examine the impact of nonlinear
behavior as light propagates through a 1 cm silicon waveguide. In particular, prop-
agation of single pump and a signal through a waveguide of constant cross section
is described. A short description about the noise and losses are also discussed in
this section and NF is estimated for silicon waveguide under subject.

8.6 Mathematical Model of Wavelength Conversion
in SOI Waveguides

8.6.1 Propagation of Light in Silicon Waveguides

FWM is a process in which three or four waves co-propagate simultaneously
inside a silicon waveguide. Four waves are involved in the nondegenerate FWM
[ 18 , 19 ], while only three waves are present in the case of degenerate FWM [ 20 ].
These waves are pump, signal, and idler waves as shown in Figure, and there pho-
ton energies satisfy the energy conservation relation (Fig. 8.6 )

2 h ω p = h ω s + h ω i (8.1)

In this topic, the focus is on the degenerate FWM coniguration, where only a sin-
gle pump wave is involved along with a signal and idler waves. These are assumed
to be identically polarized in either the fundamental TE or TM mode. The TE and
TM modes have a polarization component along the 'z' direction so only a small
fraction of incident power is present and only the transverse component contains
the nonlinear effects. During their propagation, the two optical fields induce mate-
rial polarization P ( r , t ) . This polarization drives the evolution of total electric field
E ( r , t ) according to the Maxwell wave equation [ 4 ].

1
c 2

∂ 2 E
∂ t 2 =

1
ε 0 c 2

∂ 2 P
∂ t 2

∇ 2 E −

(8.2)

where ε 0 is the free-space permittivity and c is the speed of light in vacuum. The
linear and nonlinear polarizations constitute the total polarization in silicon as

P ( R , T ) = P L ( R , T ) + P NL ( R , T )

The main contribution sources to nonlinear polarization in silicon are the refrac-
tive index changes induced by the photo-generated free carriers, stimulated Raman
scattering (SRS), and polarization of bound electrons [ 21 ].