Four Wave Mixing in Silicon Photonics - Optical Signal Processing by Silicon Photonics - page 101

are conserved during the parametric interaction. The main difference between a
FWM process and a stimulated scattering process is that the phase matching con-
dition is automatically satisfied in the case of Raman or Brillouin scattering as a
result of the active participation of the nonlinear medium. In contrast, the phase
matching condition requires a specific choice of input wavelengths and fiber
parameters before FWM can occur with high efficiency.
The phase matching condition can be approximately satisfied if the zero-dis-
persion wavelength of the fiber is chosen to coincide with the pump wavelength.
The fiber nonlinearity generates the phase-conjugated signal at the frequency
ω c = 2 ω p − ω s , provided that the phase matching condition k c = 2 k p − k s .

8.4 Degenerate and Nondegenerate FWM

There are two types of FWM terms in P 4, the term containing θ + corresponds
to the case in which three photons transfer their energy to a single photon at the
frequency ω 4 = ω 1 + ω 2 + ω 3 . This term is responsible for the phenomena such
as third-harmonic generation ( ω 1 = ω 2 = ω 3 ), or frequency conversion when
ω 1 = ω 2 = ω 3 . In general, it is difficult to satisfy the phase matching condition for
such processes to occur in optical fibers with high efficiencies. FWM is also pre-
sent if only three waves interact. In this case the term F 0 = f 1 + f 1 − f 2 couples
three components, thus generating the so-called Degenerate FWM, showing iden-
tical properties as in case of four interacting waves.
FWM can take place in any material; it refers to interactions of the waves via
a third-order nonlinear polarization. Degenerate four waves mixing (DFWM) can
yield phase conjugation and is useful, for example, for correcting aberrations by
using a phase conjugate mirror (PCM).
FWM is a fiber-optic characteristic that affects WDM systems, where multiple
optical wavelengths are spaced at equal intervals or channel spacing. The effects
of FWM are pronounced with decreased channel spacing of wavelengths and at
high-signal power levels. High chromatic dispersion decreases FWM effects, as
the signals lose coherence. The interference FWM causes in WDM systems is
known as interchannel crosstalk. FWM can be mitigated by using uneven channel
spacing or fiber that increases dispersion.
The term containing θ − corresponds to the case in which two photons at fre-
quencies ω 1 and ω 2 are annihilated, while two photons at frequencies ω 3 and ω 4 are
created simultaneously such that:

ω 3 + ω 4 = ω 1 + ω 2

The phase matching requirement for this process is Δ k = 0, where

∆ k = k 3 + k 4 − k 1 − k 2
= ( n 3 ω 3 + n 4 ω 4 − n 1 ω 1 − n 2 ω 2 )

c = 0

and n j is the effective mode index at the frequency ω j (Fig. 8.2 ).