§¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVI, sè 2A-2007 59
INFLUENCE OF THIRD-ORDER DISPERSION ON
SOLITON'S TRANSMISSION IN OPTICAL FIBERS

Dinh Xuan Khoa
(a)
, Bui Dinh Thuan
(a)Abstract. In this article, we use split-step Fourrier method to investigate the
influence of loss and third-order dispersion on a transmission soliton in optical fibers.

1. Introduction
Soliton in optical fibers is formed by the balance between phase self-
modulation and dispersion caused by group velocity. The low-loss transmission of
soliton in optical fibers is described by non-linear Schrodinger equation. The loss
and dispersion are main factors which diminish transmission distance and durable
of soliton. Normally, we consider only second-order dispersion factor. However,
when pulse's width is small, higher-order dispersion factors are not negligible.
Besides, in optical fibers, material dispersion equals zero at the wavelength of 1300
nm. In this case, the change of pulse's form and its durable depend on high-order
dispersion.
In order to investigate transmission of soliton in optical fibers, we can use
back-scattering and perturbation methods. But if the loss is considered, back-

−−=
∂
∂
t
A
ATAA
t
i
AAi
t
A
t
A
i
t
A
A
z
A
R
2
2
0
2
3
3
3
2
2
2
60
Let us consider an optical pulse with the width about some ps. In this case,
self-steeping and Raman scattering effects can be negligible. Applying the
transformation t'=t-z/v
g
, (v
g
is group velocity) equation (1) is re-written as follows:
AAi
t
A
t
Ai
A
z
A
2
3
3
3
2
2
2
'
6
1
'
2

does not change during transmission process. These pulses are solitons. So that we
can normalize the equation (2) to obtain

(
)
( )
(
)
(
)
( ) ( )
τξτξ
τ
τξ
τ
τξ
τξ
ξ
τξ
,,
,,
2
1
,
,
2
3
3
2
2

10,0 =U
,
(
)
0
0,0 AA =

2
0
2
2t
z
L
z
D
β
ξ
==
,

(
)
00
/
'
t
vzt
t
t
g

β
ξ
=

3
3
0
2
'
β
α
t
=Γ

and equation (2) is re-written

(
)
( )
(
)
( ) ( )
τξτξ
τ
τξ
τξ
ξ
τξ
,,
,

(
)
( ) ( ) ( )
tzAANtzAL
z
tzA
i ,.
ˆ
,
ˆ
,
+=
∂
∂
. (5)
L
ˆ
is an operator containing time derivative,
N
ˆ
is a non-linear operator and
is a function of A(z,t).
Solution of equation (5) has the form
( ) ( )
[
]
( )
tzAzANzLitzzA ,
ˆˆ
exp,


















∆
∆






∆
=∆+
ˆ
.
2

cancellation between the phase self-modulation and group velocity dispersion. This
case rarely happens.
Fig. 2 and Fig. 3 are 3D and 2D graph of the transmission of soliton in
optical fibers when the loss and third-order dispersion are considered. In this case, second-order dispersion factor is different from zero but is not so
big in comparison with third-order dispersion factor (β
3
= 0,1 ps
3
/km; α=0.2 dB/km).
Due to third-order dispersion, pulse's shape is deformed during its transmission.
Peak power is moved to the positive direction of the axis due to the change of v
g
;
pulse's width is widened; vibration tail is formed at sides after transmitting along
optical fibers. If the initial pulse has small width, the value of B factor is large, the
influence of third-order dispersion is considerable.
Fig2.
3D graph of the transmission of

soliton with
Γ
=1.1510
-6
, B= 0.83

Fig3.
2D graph of transmission of

2
=0 (at
dispersion wavelength equals zero) and β
3
= 0.1 ps
3
/km. The amplitude of pulse
decreases quickly, pulse amplitude of vibration tail which is formed at the back side
of pulse decreases along optical fiber. The maximum of amplitude moves to the
positive direction of the axis. Numerical calculation shows that the deformation of
pulse caused by third-order dispersion at dispersion wavelength equals zero can
limit the efficiency of optical fibers information system. 4. Conclusion
In this article, Split-Step Fourrier method is used to investigate the
influence of loss and third-order dispersion on pulse's transmission in optical fibers.
It shows that the deformation of pulse caused by third-order dispersion at dispersion
wavelength equals zero can limit the efficiency of optical fibers information system.

( )


Đại học Vinh Tạp chí khoa học, tập XXXVI, số 2A-2007 64
References

[1] Cao Long Van, Đinh Xuan Khoa, M. Trippenback, Introduction to Nonlinear
Optics, Vinh, 2003.
[2] Dinh Xuan Khoa and Bui Dinh Thuan Soliton study of schrodinger equation by
hirota method, Vol 15, number 2, June 2005.
[3] P. N. Butcher and D. Cotter, The elements of nonlinear optics, Cambridge
University Press, New York, 1990.
[4] A. Hasegawa and Y. Kodama, Solitons in optical communication, Oxford
University Press, New York, 1995.
[5] G. P. Agrawal and M. J. Potasek, Nonlinear pulse distortion in single mode
optical fibers at the zero-dispersion wavelength, Phys Rev. vol33, no3. pp.1765
1776, 1986. Tóm tắt

ảnh hởng của tán sắc bậc ba
lên soliton lan truyền trong sợi quang

Trong bài này sử dụng phơng pháp split-step Fourrier, chúng tôi đã khảo
sát ảnh hởng của mất mát và tán sắc bậc 3 lên soliton lan truyền trong sợi quang. (a)