Effects of raman scattering and third order dispersion on soliton propagation in optical fiber

Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số 49, 06 - 2017 167
EFFECTS OF RAMAN SCATTERING AND THIRD ORDER 
DISPERSION ON SOLITON PROPAGATION IN OPTICAL FIBER 
Do Thanh Thuy, Nguyen Thanh Vinh, Bui Dinh Thuan* 
Abstract: In this study, the generalized nonlinear Schrödinger equation 
describing nonlinear wave propagation in optical fiber with all higher order 
effects such as higher order dispersion, self-phase modulation and Raman 
scattering is considered. Using the numerical method, the effects of the terms third 
order dispersion, Raman scattering on evolution of ultrashort pulses are 
investigated in detail. 
Keywords: Raman scattering; Third order dispersion; Soliton. 
1. INTRODUCTION 
The study of nonlinear effects on soliton propagation in Kerr media has become a topic 
of intense research activities because of its important application in the optical 
telecommunication [1, 2, 3]. Soliton in optical fibers is possible because of the exact 
balancing between the group velocity dispersion (GVD) and its counterpart self-phase 
modulation (SPM). Optical solitons are not only predicted theoretically [1, 4] but they 
have also been observed experimentally [5, 6]. Some possible applications of solitons such 
as optical pulse compression, all optical switching, logic devices, etc. have been proposed. 
For pulse widths T0 > 10 ps, the soliton dynamics is described by the nonlinear 
Schrödinger (NLS) equation for a scalar field. However, modeling the propagation of 
ultrashort pulses (femtosecond pulses having widths T0 <1 ps), the higher order effects 
in nonlinear media become important, and therefore the governing equation should still 
include third-order dispersion (TOD) and the self-frequency shift [7,8]. The effect of 
TOD is significant for fs pulses when the GVD is close to zero [1,6]. The Raman 
scattering can lead to self-frequency-shift to the long wavelength on the ultrashort 
optical pulse spectrum, which is termed red-shift. Fundamentally, when a femtosecond 
pulse propagates in nonlinear dispersion medium, the gain is different at different 
wavelength and the gain of red part, i. e. the long wavelength, is larger than that of the 
blue part. So the energy of the blue part is converted into the red part, which leads to the 
red-shift of pulse in the spectrum. 
The effects of Raman scattering on soliton propagation have been presented in [1], 
however, the combined effect of the dispersion and Raman scattering on soliton 
propagation in the optical fiber has not been adequately studied. In this work, the 
generalized nonlinear Schrödinger equation to study propagation of ultrashort optical 
pulses in the presence of Raman scattering and third - order dispersion effects is used. 
2. PROPAGATION EQUATION FOR ULTRASHORT PULSES 
To investigate the effect of dispersion and Raman scattering on the propagation 
dynamics of pulse, we use a generalized scalar nonlinear Schrodinger equation (GNLSE) 
to model the pulse propagation inside the fiber [1]. 
2 3
32
1 2 3
2
2
, , , ,
2 6
,
, , , 0.R
A z t A z t A z t A z ti
i i
z t t t
A z t
A z t A z t T A z t
t



   
   
 
  
 (1) 
Vật lý 
D. T. Thuy, N. V. Thanh, , “Effects of Raman scattering propagation in optical fiber.” 168 
where A=A(z,t) is a complex envelop function of optical field. This function varies slowly 
with time and position z along optical fiber; β1, β2 and β3 are first, second and third-order 
dispersion factors, respectively; γ is nonlinear factor of optical fiber; γTR term 
describes Raman scattering effect. 
Using the new parameters and variables 
2 2
20 0 0
2 0 20
31
3
0 2 0
1 1
, , , , , ,
, , ,
6
D N
D
P
U A z t L L N
PP
t z z
L
  
 
  

  
  
 (2) 
where LD is the dispersive length, LN is the nonlinear length, 0 is the pulse width and P0 is 
the peak power of the input pulse. We can rewrite the equation (1) in the normalized form: 
22 3
2 2
2 32 32
R
UU i U U
i N U U U  
   
   
     
. (3) 
In the general case it is very difficult to find analytic solutions of Eq.(3) and no such 
solution was known until now. For the numerical solution of Eq. (3) we consider the 
following expression: 
   , , , .U L N U U     



 (4) 
Here 
2 3
32 32
i
L 
 
 
 
 is an linear operator containing time derivative, 
 
2
22
R
U
N U i N U 

 
  
is a non-linear operator and is a function of ,U   . 
Equation (4) was solved by using the fourth-order Runge-Kutta method in the interaction 
picture using the following algorithm [9]: 
 1 ˆ, exp ,
2
U IFT L FT U

    
 
 
1
2 1 1 1 1
3 1 2 1 2
4 1 3
1 3
ˆ ˆexp , ;
2
ˆ , / 2 , / 2 ;
ˆ , / 2 , / 2 ;
ˆ ˆexp ,
2
ˆexp , .
2
K IFT L FT NU
K N A K A K
K N A K A K
K N IFT L FT A K
IFT L FT A K

   
    
    

   

  
  
  
  
  
  
 
We obtain the value of the envelope function in the location   
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số 49, 06 - 2017 169
 31 2 41ˆ, exp , .
2 6 3 3 6
KK K K
U IFT L FT U

     
 (5) 
In relations (5), FT and IFT denote the Fourier and inverse Fourier’s transforms, 
respectively. Errors in applying (5) are of orders 
5
 . 
3. NUMERICAL RESULTS AND DISCUSSION 
First, we consider the simplest case, which is to ignore the Raman scattering effect and 
only pay attention to the third-order dispersion. At this time equation (3) reduces to: 
2 3
22
32 32
U i U U
i N U U
  
  
  
. (6) 
Our purpose is to examine the influence of third order dispersion on soliton propagation 
in optical fiber, so first of all we consider the case when the pulse wavelength lies in the 
vicinity of the zero-dispersion wavelength, the β3 term provides the dominant contribution to 
the dispersive effects. In this case, we can not use abovementioned normalization. To 
normalize the equation (1), we must choose scaling parameters as follows 
3 3
' 0 0 0
'
3 3
, , .D
D
Pz
L N
L
  

 
and equation (6) is rewritten: 
3
2
3 3
1
( ) .
6
U U
sign iN U U
 
 
 
 (7) 
We will consider the propagation of the hyperbolic secant pulse with β3 = 0.1 ps
3/km. 
The pulse amplitude of vibration tail which is formed at the back edge of pulse decreases 
along optical fiber. 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. 
Figure 1. Propagation of the hyperbolic secant pulse with β3 = 0.1 ps
3/km 
over the distance  = 12. 
Vật lý 
D. T. Thuy, N. V. Thanh, , “Effects of Raman scattering propagation in optical fiber.” 170 
As we know, for the ultrashort pulses with the width 0 50 fs and the carrier 
wavelength 0 1.55 ,m  the higher-order parameters in (3) during their propagation in 
the medium SiO2 have the values 3 0.03, 0.1.R  In this case, the self-shift frequency 
effect dominates over the TOD and the self- steepening for the pulses with the width of 
hundred and ten femtoseconds. The self-frequency shift effect is the main cause of the 
pulse frequency spectrum in the propagation shifted to the low frequency domain. In other 
words, the medium was "amplified" the longer wavelengths of the pulse [7,8]. 
 For the ultrashort pulses with the width 0 50 fs . Equation (3) is rewritten: 
22
22
22
R
UU i U
i N U U U
 
  
   
. (8) 
Figure 2. Effects of Raman scattering on fundamental soliton. 
The nonlinear phenomena due to Raman scattering has made change both the pulse 
intensity and spectrum. The frequency of the pulse is shifted to low frequency domain 
(Figure 2d) and the intensity of the low frequency was being increased. This phenomenon 
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số 49, 06 - 2017 171
is called the phenomenon of soliton self - frequency shift. Thus for low - intensity pulses 
(N = 1, fundamental soliton) the effect of scattering Raman on pulse width is negligible 
(Figure 2b) and the change of the pulse peak intensity is not significant (Figure 2c). So, in 
this case, the propagating pulse is still considered as soliton. 
Figure 3. Effects of Raman scattering on third order soliton. 
Effects of Raman scattering on soliton propagation in optical fiber will be totally 
different when we consider for pulses of high intensity. Figure 3 shows propagation of the 
ultrashort pulse with the initial hyperbolic secant shape with the power parameter N = 3 
over the distance  = 0.8 . From this figure we see that, when the intensity of the pulse 
increase, in addition to the shift of the frequency of the pulses to the lower region also 
pulse splitting process has appeared (Figure 3c). During the propagation over a short 
distance, the pulse already splits to two parts: the main strong peak moves rapidly to the 
later time and the much lower peak. During the further propagation the main peak is 
strongly compressed and its width is much smaller than the input pulse. 
Vật lý 
D. T. Thuy, N. V. Thanh, , “Effects of Raman scattering propagation in optical fiber.” 172 
4. CONCLUSION 
In this paper, the influence of Raman scattering and higher-order dispersion effects on 
the propagation of optical pulses in a highly nonlinear fiber is investigated. It is shown that 
third order dispersion can lead to a pulse-breakup above a certain pulse power. The 
splitting is followed by an expansion of the spectrum towards longer wavelengths. The 
influences of higher - order effects such as Raman scattering on soliton are also 
investigated, and it is found that Raman scattering can significantly enhance pulse 
compression under certain conditions. 
Acknowledgements: This research was funded by Vietnam National Foundation for 
Science and Technology Development (NAFOSTED) under grant number 103.03-2014.62. 
REFERENCES 
[1]. G. P. Agrawal. “Nonlinear Fiber Optics”, Academic, San Diego, 2003. 
[2]. P. N. Butcher and D. Cotter, “The Elements of Nonlinear Optics”, Cambridge 
University Press, 1991. 
[3]. Y. S. Kivshar, G. P. Agrawal. “Optical Solitons”, 2003. 
[4]. J. H. B. Nijhof, H. A. Ferwerda, and B. J. Hoenders, “Derivation of the equation for 
an ultrashort pulse in a fibre”, Pure Appl. Opt., 4, (1994) 199-218. 
[5]. A. Hasegawa and Y. Kodama, “Solitons in optical communication”, Oxford 
University Press, New York, 1995. 
[6]. G.P.Agrawal and M.J.Potasek, “Nonlinear pulse distortion in single mode optical 
fibers at the zero-dispersion wavelength”, Phys Rev. vol33,no3. (1986) pp.1765 1776. 
[7]. M. Facão, M. I. Carvalho, “Soliton self-frequency shift: Self-similar solutions and 
their stability”, Physical Review E 81, (2010) 046604. 
[8]. H. P. Tian, Z. H. Li, Z. Y. Xu, J. P. Tian, and G. S. Zhou, “Stable 
soliton in the fiber-optic system with self-frequency shift”, J. Opt. 
Soc. Am. B 20, (2003), pp. 59–64. 
[9]. Zhongxi Zhang, Liang Chen, and Xiaoyi Bao, “A fourth-order Runge-Kutta in the 
interaction picture method for numerically solving the coupled nonlinear Schrödinger 
equation”, Optics Express, 8. (2010), pp. 8261-8276. 
TÓM TẮT 
ẢNH HƯỞNG CỦA TÁN XẠ RAMAN VÀ TÁN SẮC BẬC BA 
LÊN SỰ LAN TRUYỀN SOLITON TRONG SỢI QUANG 
Trong nghiên cứu này chúng tôi xem xét phương trình Schrödinger phi tuyến 
tổng quát mô tả quá trình lan truyền sóng phi tuyến trong sợi quang với các hiệu 
ứng bậc cao như là tán sắc bậc ba, tự biến điệu pha và tán xạ Raman. Sử dụng 
phương pháp số, ảnh hưởng của các số hạng tán sắc bậc ba và tán xạ Raman lên sự 
lan truyền của xung cực ngắn đã được nghiên cứu chi tiết. 
Từ khóa: Tán xạ Raman; Tán sắc bậc ba; Soliton. 
Nhận bài ngày 17 tháng 5 năm 2017 
Hoàn thiện ngày 16 tháng 6 năm 2017 
Chấp nhận đăng ngày 20 tháng 6 năm 2017 
Địa chỉ: Khoa Vật lý & Công nghệ, Trường Đại học Vinh. 
 *Email: thuanbd@vinhuni.edu.vn