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Reduction of collision-induced time jitters in dispersion-managed soliton transmission systems

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January 1, 1996 / Vol. 21, No. 1 / OPTICS LETTERS
39
Reduction of collision-induced time jitters indispersion-managed soliton transmission systems
Akira Hasegawa and Shiva Kumar
Department of Communication Engineering, Faculty of Engineering, Osaka University, 2-1 Yamada-oka, Suita, Osaka 565, Japan
Yuji Kodama
Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Received September 18, 1995
Nonadiabatic effects caused by lumped amplifiers in fiber soliton transmission systems are reduced by theuse of dispersion-decreasing fibers between amplifiers. As a practical application, we consider a stepwisedispersion-decreasing fiber with
M
steps and show that the increase in
M
is almost equivalent to the reductionof amplifier spacing,
Z
a
, to
Z
a
M
in reducing collision-induced jitters in soliton-based wavelength-divisionmultiplexing systems.
1996 Optical Society of America
In soliton-based wavelength-division multiplexing sys-tems with lumped amplifiers, soliton collisions inamplifiers resultin apermanent frequencyshiftofsoli-tons, which causes unacceptably large jitters in pulsearrival times. If, however, the amplifier spacing ismade much shorter than the collision distance, theeffect of lumped amplifiers on the collision is aver-aged out, and the frequency shift becomes negligiblysmall.
1
From a realistic design point of view it isdesirable to provide amplifications at the loss dis-tance or longer that can be comparable with or longerthan the collision distance. In this Letter we presenta dispersion-management scheme to reduce the fre-quency shift, even when the amplifier spacing is com-parable with or longer than the collision distance.The renormalized amplitude
u
of the light-wave en- velope
q
a
1
Z
u
in a fiber with loss
G
compensatedby lumped amplifiers placed at
Z
nZ
a
,
n
0, 1, 2 ...
satisfies
2
i
≠
u
≠
Z
0
1
12
≠
2
u
≠
T
2
1
a
12
Z
d
Z
j
u
j
2
u
0.
(1)Here
a
1
Z
is given by
a
1
Z
a
1
0
exp
2G
Z
2
nZ
a
,
nZ
a
,
Z
,
n
1
1
Z
a
,
(2)with
a
1
0
defined as
a
1
0
∑
2
G
Z
a
1
2
exp
2
2
G
Z
a
∏
1/2
.(3)
d
Z
is the group-velocity dispersion taken as a func-tion of
Z
, and
Z
0
is the inhomogeneous distance along the fiber, defined as
Z
0
Z
Z
0
d
Z
d
Z
.
(4)It is clear from Eq. (1) that
u
satisfies the idealnonlinear Sch¨rodinger equation if
d
Z
is chosen tobe proportional to
a
12
Z
; thus all the ideal solitonproperties are recovered, even in a system with lumpedamplifications; that is, there will be no radiation- orcollision-induced frequency shifts at amplifiers. Although some attempt to produce fibers with con-tinuously decreasing dispersion has been made,
3
fabri-cation of a fiber having a precise dispersion variationgiven by
a
12
Z
for an extended distance is impracti-cal, especially when we allow a
Z
a
sufficiently largethat
G
Z
a
may become much larger than unity. Con-sequently we consider a fiber with stepwise decreasing dispersion having
M
steps between amplifiers. Thenwe write Eq. (1) in the form
i
≠
u
≠
Z
0
1
12
≠
2
u
≠
T
2
1
a
M
2
Z
j
u
j
2
u
0,
(5)where
a
M
2
Z
a
12
Z
d
Z
is a stepwise functiondefined below. We note, however, that if
Z
Z
0
and the stepwise dispersion is chosen to representthe average value of
a
12
Z
between
mZ
a
M
,
Z
,
m
1
1
Z
a
M
,
m
0, 1, 2 ... ,
Eq. (5) matches exactlyasystem withamplifier spacingdecreased to
Z
a
M
butwith a constant dispersion between amplifiers.
4–6
We now consider the frequency shift
D
K
Z
of asoliton in one channel induced by a collision withanother channel with a frequency separation given by
D
B
. The total frequency shiftat
Z
`
thatis due to acollision may be obtained by the adiabatic perturbationtechnique
7
:
D
K
`
2
1
D
B
Z
`2`
d
Z
0
d
a
M
2
Z
dZ
0
3
Z
`2`
sech
µ
T
1 D
BZ
0
2
∂
sech
µ
T
2 D
BZ
0
2
∂
d
T
Im
(
32
p
2
`
X
n
1
Z
a
B
n
2
p
n
p
2
n
Z
a
D
B
4
sinh
p
2
n
Z
a
D
B
2
)
,(6)
0146-9592/96/010039-03$6.00/0
1996 Optical Society of America
40
OPTICS LETTERS / Vol. 21, No. 1 / January 1, 1996
where
B
n
1
Z
a
Z
Z
a
0
d
Za
12
Z
exp
µ
2
2
p
niZ
0
Z
a
∂
.(7)Note here that, if we choose
d
Z
a
12
Z
,
B
n
van-ishes, at except
B
0
1
, and therefore
D
K
`
0
(ideal dispersion management). Dividing the fiber be-tween amplifiers into
M
fiber sections of equal length
Z
a
M
with constant dispersion produces subharmonicswith period
Z
a
(i.e., spatial frequency components aremultiples of
2
p
Z
a
) in addition to the dominant peri-odicity of
Z
a
M
. If we choose fiber sections such thatthey are of equal length,
Z
a
M
in
Z
0
coordinates, re-duction of the subharmonic components with period
Z
a
can be achieved. We thus propose to choose the length
Z
l
and dispersion
D
l
of the
l
th section of the fiber suchthat
Z
0
Z
l
1
1
2
Z
0
Z
l
Z
a
M
,
l
0, 1, ...
M
2
1.
(8)Using Eq. (8), and taking
D
l
to provide dispersioncorresponding to the average of the ideal dispersion
D
l
1
Z
l
1
1
2
Z
l
Z
Z
l
1
1
Z
l
a
12
0
exp
2
2
G
Z
d
Z
,
(9)we find
Z
l
and
D
l
to be
Z
l
2
12
G
ln
Ω
1
2
l
1
2
exp
2
2
G
Z
a
M
æ
,
(10)
D
l
Z
a
M
Z
l
1
1
2
Z
l
.(11)Figure 1 shows the profile of
d
Z
and the fiber lengthof each section for
M
4
. Using Eqs. (9) and (10), wefind the Fourier coefficient
B
n
to be
B
n
a
12
0
2
M
exp
2
i
p
n
M
M
2
1
X
m
0
exp
2
i
2
p
nm
M
3
2
i
M
2
m
a
sin
n
p
M
1a
exp
2
i
p
n
M
G
Z
a
1
i
p
nD
m
,(12)where
a
1
2
exp
2
2
G
Z
a
. From Eq. (12) we notethat
B
n
has a dominant peak at
n
M
, indicating thatsubharmonic components with period
Z
a
are reduced.We have numerically evaluated the total frequencyshift
D
K
`
from Eq. (6), for various values of thenumber of steps
M
between amplifiers for the casein which the fiber is divided into equal lengths in
Z
(the real distance) and that in
Z
0
[where
Z
l
is givenby Eq. (10)] for the choice of
Z
a
2.45
,
G
0.185
, and
D
B
5
. Figure 2 shows the result. We see that if the fiber sections are of equal length in
Z
0
, for
M
largerthan 6,
D
K
`
is practically reduced to zero.We carried out the numerical simulations withthe following parameters of the fiber: wavelength
1.56
m
m
; pulse width 5 ps; loss rate
2
g
0.0461 km
2
1
(0.2 dB
km); dispersion
k
00
1 ps
2
km
, which corres-ponds to the dispersion distance
z
0
8.16
km;
A
eff
25
m
m
2
; nonlinear coefficient
N
2
3.18
3
10
2
16
cm
2
W
; and amplifier spacing
z
a
20
km. Thechannel separation is 0.281 THz. We have neglectedthe amplifier noise and higher-order terms in thenonlinear Schr¨odinger equation by assuming thattheir effects can be reduced to a low level by filters.Figure 3 shows the soliton collision without dispersionmanagement. Here, as well as in Fig. 4, the slowersoliton is artificially removed at
Z
9.8
to show thebehavior of the faster soliton more clearly. Because of periodic amplifications, dispersive waves are emittedby solitons. In addition, a collision-induced shift of
Fig. 1. Stepwise dispersion profile with fiber sectionsequally spaced in
Z
0
(solid curves) and ideal exponentialdispersion profile (dashed curves).Fig. 2. Frequency shift
D
K
`
versus steps
M
for the caseof fiber sections equally spaced in
Z
0
(solid curve) and in
Z
(dashed curve).
Z
a
2.45
,
G
0.185
, and
D
B
5.0
.Fig. 3. Absolute amplitude
j
q
j
as a function of distance
Z
and time
T
with no dispersion management. The slowersoliton is artificially removed at
Z
9.8
to clarify the meanposition of the faster soliton.
January 1, 1996 / Vol. 21, No. 1 / OPTICS LETTERS
41
Fig. 4. Same as Fig. 3 but with dispersion management.Fiber sections are equally spaced in
Z
0
;
M
4
.Fig. 5. Mean position of the faster soliton versus distancefor different values of
M
. Fiber sections are equallyspaced in
Z
0
.
the soliton’s mean position can be seen. Figure 4shows the soliton collision with four fiber sectionsbetween amplifiers with fiber lengths and dispersionas given by Eqs. (10) and (11). As can be seen, bothradiation- and collision-induced temporal shifts of thesoliton are reduced. The mean position of the fastersoliton is evaluated and plotted in Fig. 5. We see that
M
$
4
is sufficient to reduce the effect of collisionswithin a tolerable level.We note that the present dispersion-managementscheme involves only passive elements and thereforeis easier to implement than a fiber with distributedgain. If the dispersion-management scheme and theamplifier spacing are designed for a particular pulsewidth and channel separation, the same transmissionlink can be used over a wide range of pulse widths andchannel separation.For a choice of
Z
a
.
1
(amplifier spacing larger thanthe dispersion distance), the amplifiers produce notonly a large collision-induced shift in soliton positionbut also radiation. In such a case, the perturbationexpressions of
D
K
becomes less accurate. However,inasmuch as the dispersion-managed fiber reduces theradiation as well,
6
the present result is accurate forlarge
M
, even for
Z
a
.
1
. It should also be noted thatthe radiation at lumped amplifiers is also reduced bythe use of fiber sections equally spaced in
Z
0
ratherthan in
Z
. Y.Kodama acknowledges thesupport oftheMinistryof Education, Science and Culture of Japan under theInternational Scientific Research Program; S. Kumaracknowledges the award of a fellowship from Ministryof Education, Science and Culture of Japan.
References
1. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, J.Lightwave Technol.
9,
362 (1991).2. A. Hasegawa and Y. Kodama, Opt. Lett.
15,
1443 (1990);Phys. Rev. Lett.
66,
161 (1991).3. A. J. Stentz, R. W. Boyd, and A. F. Evans, Opt. Lett.
20,
1770 (1995).4. W. Forysiak, F. M. Knox, and N. J. Doran, Opt. Lett.
19,
174 (1994).5. W. Forysiak, F. M. Knox, and N. J. Doran, J. LightwaveTechnol.
12,
1330 (1994).6. A. Hasegawa, Y. Kodama, and S. Kumar, ‘‘Adiabaticsoliton transmission in fibers with lumped amplifiers,’’submitted to Opt. Lett.7. A. Hasegawa and Y. Kodama,
Solitons in OpticalCommunications
(Oxford U. Press, Oxford, 1995),p. 181.

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