August 10, 2007 / Vol. 5, No. 8 / CHINESE OPTICS LETTERS 441
Influence of group-delay ripple on timing jitter induced by
SPM and IXPM in systems w ith dispersion
compensated by CFBG
Xi Qin (
)
1,2
, Jihong Cao (
ÍÍÍ
)
1,2
, Yong Che n (
ííí
)
1,2
,
Feng Z han g (
)
1,2
, and Shuisheng Jian (
ßßß
)
1,2
1
Key Laboratory of All Optical Netw ork & Advanced Telecommunication Network of EMC,
Beijing Jiaotong University, Beijing 100044
2
Institute of Lightwave Technology, Beijing Jiaotong University, Beijing 100044
Received January 11, 2007
An analytical expression was proposed to analyze the influence of group-delay ripple (GDR) on timing jitter
induced by self-phase modulation (SPM) and intra-channel cross-phase modulation (IXPM) in pseudo-
linear transmission systems when dispersion was compensated by chirped fiber Bragg grating (CFBG).
Effects of ripple amplitude, period, and phase on timing jitter were d iscussed by theoretical and numerical
analysis in detail. The results show that the influence of GDR on timing jitter changes linearly with
the amplitude of GDR and whether it decreases or increases the timing jitter relies on the ripple p eriod
and ripple phase. Timing jitter induced by SPM and IXPM could be suppressed totally by adjusting the
relative phase between the center frequency of the pulse and the ripples.
OCIS codes: 060.2330, 190.3270, 230.0230, 230.1480.
Pseudo-linear transmission is a regime for transmission
of high-speed time division multiplexing (TDM) signals
where fast variations of each channel waveform with
cumulative dispersion allow important averaging of the
intra-channel effects of fiber nonlinearity
[1]
.Themain
nonlinearities in this kind of systems are self-phase mod-
ulation (SPM), intra-channel cross-phase modulation
(IXPM) and intra-channel four wave mixing (IFWM).
IXPM together with SPM cause timing jitter between
pulses, which is considered as one of the major limit-
ing factors for systems with transmission speed reaching
40 Gb/s or beyond. This makes timing jitter become a
hot area of research
[2,3]
. Theoretical analyses have shown
that timing jitter induced by SPM and IXPM in this kind
of transmission system results from a two-step process:
Firstly, SPM and IXPM induce frequency shift between
adjacent pulses. Then, the frequency shift is transformed
to timing shift between pulses through dispersion and
this finally results in timing jitter
[4,5]
.
In this kind of transmission system, chirped fiber
Bragg grating (CFBG) is a good alternative as disper-
sion compensator
[6−8]
. It has some advantages relative
to other dispersion compensation methods such as low
insertion loss, small package size, and reduced nonlinear
effects. It is likely that the group-delay ripple (GDR)
of CFBG could bring about some effects on timing jit-
ter since it changes the lumped dispersion of the grating.
However, to our knowledge, no qualitative or quantita-
tive analysis of this problem, has been published so far.
In this letter, an analytical method is proposed to an-
alyze the influence of GDR on timing jitter induced by
SPM and IXPM in pseudo-linear transmission systems
using CFBG as dispersion compensator.
The transmission function of the fiber grating is mod-
eled as
[9]
H (ω)=exp
j
g
2
ω
2
− j
Γ
T
2
0
cos(ωT
0
+ θ)+j
Γ
T
2
0
, (1)
where g is the average lumped dispersion of the grating.
Parameters Γ, 2π/T
0
,andθ are the amplitude, period,
and phase of the dispersion ripple, respectively. Using
the Jacobi-Anger expansion,
exp(jz cos θ)=
∞
n=−∞
j
n
J
n
(z)exp(jnθ), (2)
where J
n
(z) is the Bessel function of the first kind. Leav-
ing out the terms containing high-order Bessel function,
we get
exp
−j
Γ
T
2
0
cos(ωT
0
+ θ)
≈ J
0
(−
Γ
T
2
0
)+jJ
1
(−
Γ
T
2
0
)exp(jωT
0
+ jθ)
−jJ
−1
(−
Γ
T
2
0
)exp(−jωT
0
− jθ). (3)
Substituting Eq. (3) into Eq. (1), then
H (ω)=exp(j
g
2
ω
2
)exp(j
Γ
T
2
0
)
×
J
0
(−
Γ
T
2
0
)+jJ
1
(−
Γ
T
2
0
)exp(jωT
0
+ jθ)
−jJ
−1
(−
Γ
T
2
0
)exp(−jωT
0
− jθ)
. (4)
For the sake of analyzing convenience, we set exp(j
g
2
ω
2
)
as H
1
(ω), and J
0
(−
Γ
T
2
0
)+jJ
1
(−
Γ
T
2
0
)exp(jωT
0
+ jθ) −
jJ
−1
(−
Γ
T
2
0
)exp(−jωT
0
−jθ)asH
2
(ω). Then, H (ω)can
1671-7694/2007/080441-04
c
2007 Chinese Optics Letters