Dynamics of Gaussian Pulse inside Optical fiber with particular consideration of Pulse Broadening

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


7

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,


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


9

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


10

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


11

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

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

z/LDT/T0

|U(z

,T)|2

Documents

Dynamics of Gaussian Pulse inside Optical fiber with particular consideration of Pulse Broadening