JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
6
Dynamics of Gaussian Pulse inside Optical fiber with particular consideration
of Pulse Broadening
Chandra Kamal Borgohain* ,Satyajit Bora* and Chakresh Kumar**
Abstract:The main aim of this paper is to implement the effect of pulse broadening of Gaussian pulse propagation inside optical fiber.
Analysisof Gaussianpulses; chirped Gaussian pulses and super Gaussian pulses are done by modifying the Nonlinear Schrodingerequation
(NLSE).Further we have analyzed the pulse broadening at different propagation distances.The highest amplitude of the pulse observed at zero
kilometer, and the amplitude decreases when the pulse travels for longer distances.And the boarding factor of chirped Gaussian and super
Gaussian pulseswith the propagation distance is also observed.
Keywords:Nonlinear Schrödingerequation (NLSE), Gaussian pulse,Pulse broadening,Dispersion, Broadening factor
1. Introduction
With the advancement of fast data communication
process and information technology, optical fibers
are finding immense utilization in many different
fields, which can fulfill the demand of high bit rate
communication. The propagation of light inside
optical fiber can be described by the nonlinear
Schrodinger equation(NLSE). Pulses propagating
in optical fiber tend to broaden as they travel. This
is due to the nonzero line width of the source and
dispersion of the fiber material. The other causes of
pulse broadening are associated with the fact that
time of flight of a pulse along a ray depends on the
ray trajectory. Pulsestravel along optical rays
usually goes faster than pulses travel along rays of
large amplitude.As fiber dispersion and
nonlinearity leads to pulse degradation it is
necessary to analyze the behavior of various
optical pulses which may lead to lesser
degradation.The polarization mode dispersion
infibers is strong enough to cause the pulse
broadening and distortion [3].Simulation results
show that super Gaussian pulse is broader and
deformedthan Gaussian pulse.Also pulse
broadening strongly depends on frequency chirp.
It is seen that there is a large pulse deformity for
negative chirped pulse than for positive chirped
pulse.
2. Propagation Regime
The Nonlinear Schrodinger equation (NLSE)governs the propagation of optical pulse inside single mode fiber [1]
i∂A/∂z=-iαA/2+(β2/2)(∂2A/∂T
2)-γ|A|
2A (1)
Where
Ais the amplitude of the pulse envelope that
varies slowly.Tis measuredin a frame of
reference moving with the pulse (at the group
velocityvg). The effects of fiber losses,
dispersion, and nonlinearity on pulses are
governed by the three terms in the right hand
side of the Equation (1) respectively.
————————————————
*Pursuing B.Tech Electronics and Communication Engineering at Tezpur (central) University,India. *Pursuing M.Tech Electronics and Communication Engineering at Tezpur( cantral)University. **Assistant Professor, University School of Information & Communication Technology, Guru Gobind Singh Indraprastha University, Dwarka, New Delhi-110078, India
JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
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2.1 Gaussian pulses
For a Gaussian pulsethe incident field is considered as
[1],
U (0, T)=exp(-T2/2T0
2) (2)
Here,T0 is the half-width. In practice, it is usual to
use the full width at half maximum (FWHM) in
place of T0. In case of a Gaussian pulse, the two are
related as
TFWHM=2(ln2)1/2T0≈1.665T0(3)
The amplitude at any point zalong the fiber is
given by
U(z,T)=(T0/(T02-iβ2z)1/2)exp(-T2/2(T02-iβ2z))(4)
A Gaussian pulse maintains its shape on
propagation. Its width T1 increases with z as
T1(z)=T0[1+(z/LD)2]1/2 (5)
Where LD is the dispersion length, which
determines the extent of broadening. For a given
fiber length, short pulses broaden more of a
smaller dispersion length. At z = LD, a Gaussian
pulse broadens by a factorof (2)1/2. The expression
of U(z,T) can be written in the form
U(z,T)=|U(z,T)|exp*iΦ(z,T)+ (6)
Where
Φ(z,T)=(-sgn(β2) z/LD)T2)/(1+(z/LD)2T02)
+(1/2)tan-1(z/LD) (7)
Φ(z,T) istime dependent which implies that the
instantaneous frequency differs across the
pulsefrom the central frequency ɷ0. Whereδɷ is
the time derivative ∂Φ/∂T and can be written as
δɷ(T)= -∂Φ/∂T=(sgn(β2)(2z/LD)T)/
(1+(z/LD)2T02) (8)
We have considered the dispersion length LD=T02/ abs(β2), normalized time T=(t- β2z)/T0.
2.2 Chirped Gaussian Pulses
In the case of linearly chirped Gaussian pulses,
the incident field can be written as
U(0,T)=exp(-(1+ic)T2/2T02) (9)
WhereCis the chirpparameter. It is common to
refer to the chirp as positive or negative
depending on whether C is positive or negative.
The numerical value of C can be calculated from
the spectral width of the Gaussian pulse.
U(0, ɷ) = (2πT02/(1+ic))1/2 exp (-ɷ2T02/2(1+ic)
)(10)
The spectral half-width is given by
∆ɷ=(1+c2)
1/2/T0 (11)
In the presence of linear chirp the spectral width
is enhanced by a factor of (1+c2)1/2. Equation
(11)can be used to estimate |c| from
measurements of∆ɷand T0. To obtain the
transmitted field,
U(z,T)=T0/[T02-iβ2z(1+ic)]1/2exp(-(1+ic)T2)/
2[T02-iβ2z(1+ic)])(12)
Thereby, a chirped Gaussian pulse holds its
Gaussian shape on propagation.The width
T1(after propagating a distance z)is related to T0
(the initial width)by the relation,
(T1/T0)=*(1+cβ2z/T02)2+(β2z/T02)2]1/2 (13)
This above expression implies that broadening
depends on the relative signs of the Group
Velocity Dispersion parameter β2 and the chirp
parameter C. whereas for β2C>0 a Gaussian
pulse broadens monotonically with z, and for β2
C<0 it goes through an initial narrowing stage.
In the case β2 C<0, the pulse width becomes
minimum at a distance,
JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
8
Zmin=|c|LD/(1+c2)(14)
The minimum value of the pulse width at z=zmin
can be expressed as,
T1min=T0/(1+c2)1/2 (15)
2.3 Super-Gaussian Pulses
A super-Gaussian shape can be usedto model the
effects steep leading and trailing edges on dispersion-
induced pulse broadening. Fora super-Gaussian
pulse, generalized equation is of the form
U(0,T)=exp[(-(1+ic)/2)(T/T0)2m
] (16)
Wheremcontrols the degree of edge sharpness. For
m=1 the case of chirped Gaussian pulses can be
recovered.
Ϭ=[<T2>-<T>
2]
1/2 (17)
The expression for broadening factor can be
analytically evaluate to
Ϭ/Ϭ0=[1+(Ґ(1/2m)cβ2z)/(Ґ(3/2m)T02)
+(m2(1+c
2)Ґ(2-1/2m))/Ґ(3/2m))(β2z/T0
2)
2]
1/2
(18)
Here,Ϭ0 is the initial RMS width of the pulse at
z=0 andҐis the gamma function.
2.4 Higher-Order Dispersion
2.4.1 Third-Order Dispersion
The dispersion induced pulse broadening is due to the
lowest order group velocity term proportional to
β2.The dispersive effects can be considered by
including both β2 and β3terms by neglecting the
nonlinear effects.
U(z,T) satisfy the equation,
i∂U/∂z =(β2/2)∂
2U/∂T
2 +(iβ3/6)(∂
3U/∂T
3) (19)
Table 1:Different values of input parameters
Operating
wavelength
1550 nm
Pulse width T0
1 Ps
β2
20 ps2/km
Chirp factor
c
-3,-1,0,1,3
m
1,3,5,7
Distances z
0,5,20,25,60,100,
250,450,550,750
km
γ
2.4×10-3
(Wm)-1
Vg
2×108 m/s
3. Simulation and Results
Figure1:z=0km
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
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Figure2: z=5km
Figure3: z=20km
Figure4:z=25km
Figure 5:z=60km
Figure 6: z=100km
Figure7:z= 250km
Figure8: z=450km
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
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Figure9: z= 550km
Figure10:z=750km
For Gaussian pulse the effect of pulse broadening for
different propagation distances are shown above.
Figure.1 shows the pulse broadening at z=0km.It is
seen that pulse broadening is very less as a result
highest amplitude is observed. Pulse broadening for
different propagation distances observed at
5km,20km,25km,60km,100km,250km,450km,550km
,750km.From the simulation it is analyzed that for
larger distances the pulse broadening become
increases.
Figure.11: Variation of Broadening factor with
distance for a chirp Gaussian pulse.
It can be observed from the Figure11 that for chirp
parameter c>0 thebroadening factor increases linearly
from the leading to the trailing edge. While for c<0,
the broadening factor first decreases and then
increases.Increasing rate of broadening factor is high
for higher values of |c|.
Figure.12:Variation of Broadening factor with
distance for a super Gaussian pulse.
.
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
-10 -8 -6 -4 -2 0 2 4 6 8 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
u(z
,T)2
T1/T0
0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
3
3.5
4
Distance (z/LD)B
roadenin
g F
acto
r (T
1/T
0)
c = -3
c = 3
c = 0
c = -1
c = 1
0 0.5 1 1.5 21
2
3
4
5
6
7
Distance (z/LD)
Bro
adenin
g F
acto
r
m = 7
m = 5
m = 3
m = 1
JOURNAL OF TELECOMMUNICATIONS, VOLUME 23, ISSUE 2, JANUARY 2014
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Figure.12 shows the broadening factor of super
Gaussian pulse as a function of propagation
distance for different values of m. Here, m=1
corresponds to Gaussian pulse. The pulse edges
become increasingly steeper for larger values of
m.The magnitude of the pulse broadening depends
on the sign of the product β2c.In particular even
super Gaussian pulses exhibit initial narrowing
when β2c<0.
The 3D representation of a Gaussian pulse is
shown below,
Figure.13: 3D representation of third-order Gaussian
pulse.
4. Conclusion
In this paper by using the appropriate parameter
values, the effect of pulse broadening on Gaussian
pulse propagation has been investigated. The Pulse
broadening and pulse deformity have been
observed during the propagation process. It is also
observed that the pulse broadening becomes
higher for longer distances are case of Gaussian
pulse, i.e. the intensity of the pulse decreases. For
chirped Gaussian pulses the analysis is performed
for chirp parameter c=-3,-1,0,1,3.Among them the
satisfactory results are obtained for positive chirp
(c=0,c=1,c=3) and for negative chirp (c=-1,c=-
3).Again in case of super Gaussian pulses the
pulse broadening is observed more for higher
values of m.
References
[1+ G.P Agarwal, “Nonlinear fiberoptics, optics and photonics “Third edition Academic press, ISBN-10:81-312-0119-8, 2006.
[2+ Govind p. Agrawal, “Fiber optic communication systems”, Second edition., John Wiley & Sons, Inc,ISBN 0-471-21571-6, 2002. [3]S.Vinayagapriya,A.Sivasubramanian,”PMD induced broadening on propagation of chirped super Gaussian pulse in single mode optical fiber, International Conference on Signal processing,Image processing and pattern Recognition[ICSIPR],2013
-10-5
05
10 12
34
56
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z/LDT/T0
|U(z
,T)|2