Digital Signal Processing Reference
In-Depth Information
Consider four continuous waves (CWs), oscillating at frequencies
ω
1
,
ω
2
,
ω
3
and
ω
4
and linearly polarized along the same axis
x
. The total electric field can be
written as:
4
1
2
x
′
E
=
E
j
exp
i
k
j
z
− ω
j
t
+
cc
j
=
1
where the propagation constant
k
j
=
n
j
ω
j
/
c
,
n
j
being the mode index.
Substituting the value of E from above equation in
P
NL,
4
1
2
x
′
P
NL
=
P
j
exp
i
k
j
z
− ω
j
t
+
cc
j
=
1
It is found that
P
j
= (
j
=
1
∼
4
)
consists of a large number of terms involving the
products of three electric fields. For example,
P
4
can be expressed as
3
ε
0
4
x
(
3
)
xxxx
2
E
4
+
2
2
2
2
P
4
=
E
4
+
2
E
1
E
2
E
3
exp
(
i
θ
+
) +
2
E
1
E
2
E
3
exp
(
i
θ
−
) +···
|
E
4
|
|
E
1
|
+ |
E
2
|
+ |
E
3
|
where
θ
+
and
θ
−
are defined as;
θ
+
= (
k
1
+
k
2
+
k
3
−
k
4
)
z
− (ω
1
+ ω
2
+ ω
3
− ω
4
)
t
,
θ
−
= (
k
1
+
k
2
−
k
3
−
k
4
)
z
− (ω
1
+ ω
2
− ω
3
− ω
4
)
t
.
8.3 Phase Matching Condition for FWM
The first four terms containing E
4
in P
4
are responsible for the self-phase modula-
tion (SPM) and cross phase modulation (XPM) effects but the remaining terms
result from the frequency combinations (sum or difference) of all four waves. How
many of these are effective during a FWM process depends on the phase mismatch
between
E
4
and
P
4
governed by
θ
+
,
θ
−
or a similar quantity.
If three optical fields with carrier frequencies
ω
1
,
ω
2
,
and
ω
3
co-propagate
inside the fiber simultaneously,
χ
(3)
generates a fourth field whose frequency
ω
4
is related to other frequencies by a relation
ω
4
= ω
1
± ω
2
± ω
3
.
Many frequencies
corresponding to different plus and minus sign combinations are possible in prin-
ciple. In practice, some of these combinations do not build up because of phase
matching requirement [
4
].
Significant FWM occurs only if the phase mismatch nearly vanishes. This
requires matching of the frequencies as well as of the wave vectors. The latter
requirement is often referred to as phase matching. In quantum-mechanical terms,
FWM occurs when photons from one or more waves are annihilated and new pho-
tons are created at different frequencies such that the net energy and momentum
Search WWH ::
Custom Search