ANALYSIS OF FOUR-WAVE MIXING IN
OPTICAL FIBER LINKS WITH NON-
UNIFORM CHROMATIC DISPERSION
J. R. Souza
1
and P. B. Harboe
2
1
Center for Telecommunication Studies
Pontifical Catholic University of Rio de Janeiro
Rua Marque ˆ s de Sa ˜ o Vicente
225, Rio de Janeiro–RJ, CEP 22451-041, Brazil
2
Telecommunication Engineering Department
Federal Fluminense University
Rua Passo da Pa ´ tria
156, Nitero ´ i–RJ, CEP 24210-240, Brazil
Received 24 March 2003
ABSTRACT: The four-wave mixing (FWM) process in single-mode optical
fibers is investigated by considering that a zero-dispersion wavelength var-
ies along the length of the fiber link, which is represented as a series of
segments with different lengths, propagation constants, and zero-dispersion
wavelengths. The results indicate that new optical frequencies are efficiently
generated for particular combinations of the input-signal wavelengths, even
when the phase-matching condition is not fully met. © 2003 Wiley Periodi-
cals, Inc. Microwave Opt Technol Lett 39: 102–105, 2003; Published on-
line in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/
mop.11139
Key words: nonlinear fiber optics; four-wave mixing; chromatic disper-
sion; WDM systems
1. INTRODUCTION
Very high-capacity, long-haul optical communication systems are
made possible by the extremely wide bandwidth of optical fibers,
which is best exploited by wavelength-division multiplexing
(WDM) [1]. The performance of long-distance optical communi-
cation systems is limited, however, by chromatic dispersion and
nonlinear effects of fiber, which interact and accumulate along the
length of the optical link. Chromatic dispersion, which broadens
the pulses, can be reduced by using dispersion-shifted fibers at the
1550-nm wavelength range. But low chromatic dispersion en-
hances some nonlinear effects of fiber, especially four-wave mix-
ing (FWM), as the phase mismatch due to dispersion is also
reduced [2].
Four-wave mixing is a nonlinear process associated with the
third-order electric susceptibility
(3)
of the optical fiber, where
any three frequencies f
i
, f
j
, and f
k
interact through
(3)
to generate
a new optical wave of frequency f
ijk
= f
i
+ f
j
- f
k
[2, 3]. In
WDM systems, every three channels will originate, nine new
optical frequencies, due to FWM. Whenever these new frequencies
are generated within the system bandwidth and coincide with one
or more channel frequencies, the system performance is degraded
by the resulting cross talk. Also, the power of the signal carriers is
reduced in the FWM process, contributing to further performance
degradation. FWM is then one of the major limiting factors in
long-haul WDM systems. The effects of FWM on WDM systems
can be reduced either with the introduction of dispersion into the
system, or by maximizing the channel separation.
This paper addresses the effect of the FWM phenomenon on
two-channel systems operating close to the zero-dispersion wave-
length of the optical fiber. We investigate the influence of both the
variation of the dispersion along the fiber link, which can be seen
as a form of dispersion management, and the channel separation.
The analysis is based on the undepleted pump model, according to
which the signal power is not diminished by the generation of
FWM frequencies [2]. The paper is organized as follows: the next
section introduces the basic formulation for the analysis of the
four-wave mixing process in segmented optical-fiber links. In
section 3, results are presented for the efficiency of FWM gener-
ation, and also for the ratio between the FWM and signal powers
at the end of the fiber link. Conclusions are drawn in section 4.
2. FORMULATION
In a real optical-fiber link, the zero-dispersion wavelength may
vary along the link length, due to the perturbations introduced
during the fabrication process. To circumvent the difficulty of
treating this situation in an exact form, the fiber link is represented
by a series of segments of length L
i
, zero-dispersion wavelength
0i
, and propagation constant
i
( i = 1, 2, . . . , N), as illustrated
in Figure 1. This same figure can also illustrate a dispersion-
managed link, where spans of fiber with different chromatic dis-
persion are used to introduce a low net dispersion in the system,
and thus reduce the efficiency of four-wave mixing.
The FWM analysis is carried out for each of the segments in
Figure 1, subjected to the appropriate boundary conditions [4]. The
amplitudes of the optical carriers are usually much higher than
those of the new optical waves generated by FWM. So, in a
first-order approximation, it is assumed that the former are not
affected by the FWM process.
The fundamental equation describing the FWM process in each
of the N fiber segments in Figure 1 is written as [4 – 6]:
d
dz
E
F
i
z +
2
E
F
i
z = j
n
2
F
cA
eff
D
3
E
p
i
z
i-1
E
q
i
z
i-1
E
r
i
* z
i-1
exp
-
3
2
+ j
i
z - z
i-1
(1)
where i = 1, 2, . . . , N; z is the distance along the fiber link,
E
F
( z ) is the FWM field component generated at the frequency
f
F
= f
p
+ f
q
- f
r
=
F
/2, E
s
(0) ( s = p, q, r ) is the signal
amplitude at frequency f
s
at z = 0, is the loss coefficient of the
fiber, D is the FWM degeneracy factor ( D = 3 for p = q r , and
D = 6 for p q r ), n
2
is the fiber nonlinear refractive index,
c is the speed of light in vacuum, A
eff
is the modal effective area
of the fiber,
i
=
i
( f
p
) +
i
( f
q
) -
i
( f
r
) -
i
( f
F
) is the
phase mismatch. In addition, z
i
= ¥
k=1
i
L
k
, and L
k
is the length
of the k
th
fiber segment, z
0
= 0.
The boundary conditions at the interface between two fiber
segments, at z = z
i
, are described as [4]:
E
F
i
z
i
exp j
i
F
L
i
= E
F
i+1
z
i
,
E
s
i
z
i-1
exp
-
2
+ j
i
s
L
i
= E
s
i+1
z
i
, s = p, q, r, (2)
with i = 1, 2, . . . , N - 1, and E
F
1
(0) = 0.
For each segment, the solution of Eq. (1) at z = z
i
is calculated
as
Figure 1 Optical-fiber link composed by segments of different zero-
dispersion wavelengths
102 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 39, No. 2, October 20 2003