Four Wave Mixing in Silicon Photonics - Optical Signal Processing by Silicon Photonics - page 108

where k j is the propagation constant and β 2 is the group velocity parameter. As
the nonlinear part of the phase match is positive, phase matching can be realized
by locating the pump wave in the anomalous dispersion regime (β 2 < 0 ) so that
the linear phase mismatch can compensate the nonlinear one. At the telecommu-
nication wavelength of 1,550 nm, the material dispersion of crystalline silicon is
normal ( D =− 2 π C β 2 /λ 2 =− 880 PS /( NM × KM )) Because of the strong modal
confinement in silicon waveguides, the waveguide dispersion can counteract
the effect of the normal material dispersion [ 27 ]. It is verified simulationsly that
through proper design of waveguide shape and size, anomalous GVD in the range
of 200 − 1,200 ps / nm × km can be obtained at 1,550 nm [ 28 ].
In addition to waveguide geometry and material dispersion, the TPA-induced free
carrier dispersion (FCD) may also influence the phase matching condition by chang-
ing the refractive index locally. Due to the free-carrier plasma effect, the refractive
index of silicon decreases linearly with increasing carrier density. However, the
quadratic wavelength dependence of the free-carrier plasma effect

e 2 λ 2
8 π 2 c 2 ε 0 n

N e
m ce

N h
m ch

≈− 8.2 × 10 − 22 λ 2 N

(8.9)

n FC =− +

will alter the local dispersion and phase mismatching conditions as described in
the following FCD equation:

1
c

d ∆ n FC
d λ

=− 5.46 × 10 − 30 λ N

(8.10)

∆ D FC =

8.7 Noise Figure

In silicon waveguides, the gain is generated by FWM process and the attenuation
is caused by nonlinear losses of TPA, FCA, and linear loss. The noise is associated
with photon fluctuation created by the gain and the loss in the optical amplifica-
tion process. Noise is a term generally used to refer to any undesired disturbances
that mask the received signal in a communication system. The NF is thus the
ratio of actual output noise to that which would remain if the device itself did not
introduce any type of noise. It is a number by which the performance of a radio
receiver can be specified.
There are three main types of noise due to spontaneous fluctuations in optical
fiber communication system;

• Thermal noise
• Dark current noise
• Quantum noise.

The shot noise and thermal noise are the two fundamental noise mechanisms
responsible for current fluctuations in all optical receivers even when the incident
optical power P in is constant.