FOP-2011, 3-5 DECEMBER, IIT, DELHI

PULSE COMPRESSION OF SIT SOLITONS THROUGH NONLINEAR TUNNELING EFFECTS

ABSTRACT

We have investigated the Nonlinear Schrödinger-Maxwell-Bloch (NLS-MB) equation with variable coefficients which represents the propagation of optical pulses in an inhomogeneous erbium doped fiber system. Firstly, the nonlinear Schrödinger equation with distributed coefficients is solved analytically by Darboux Transformation based on Lax pair. We claimed two soliton solution for further investigation of soliton propagation through nonlinear tunneling. To analyse this case, we consider a special form of dispersion and nonlinearity coefficients.

INTRODUCTION

Optical soliton propagation in erbium-doped system is governed by the Maxwell-Bloch (NLS-MB) equations. They showed that if the energy difference between the two levels of the media coincides with the optical wavelength, then coherent absorption takes place. The media becomes optically transparent to that particular wavelength, called the self-induced transparency (SIT) [1]. Nonlinear tunneling effect which is occur in nonlinear media has been discussed by several authors [2,3]. Very recently in [4], W.P. Zhong et. al. has been investigated in detail about the nonlinear tunneling effects of solitons with distributed coefficients in the presence of external harmonic potential. The study of SIT (NLS-MB) soliton propagation through the nonlinear tunneling effect have not well been explored, hence we are interested in such a study to achieve the pulse compression and necessary physical background.

M.S. Mani Rajan1*, A. Mahalingam2

1,2Department of Physics, Anna University, Chennai-600 025, India.E-mail: senthilmanirajanofc@gmail.com

THEORY

The generalized inhomogeneous NLS-MB (GINLS-MB) equation describing the SIT

soliton propagation through the nonlinear erbium doped fiber is of the following form

where q(z, t) is the complex envelope of the field, p(z, t) is the measure of the polarization of

the resonant medium and (z, t) denotes the extent of the population inversion, which are given by

and respectively, being the wave functions of the two energy levels of

the resonant atoms , the functions the functions D(z), R(z), and are real and arbitrary which account

for the varying dispersion for which the dispersion curve has a harmonic oscillator potential form,

varying nonlinearity, nonlinear focus length and arbitrary gain respectively. Second term of Eq. (1)

results from the group velocity and the Parameter V(z) denotes the velocity of propagation. The

angular bracket represents averaging over the entire frequency. Thus

where h(w) is the uncertainty in the energy levels.

02)(2

)(2

)())()()((

2 piqzi

qqzRqzD

qtzzDzViqi tttz

)1()),(),()(2))(2( tztzqzRtzDpipt

),(),(),( ** tzqtzpqtzpt

*21 2

2

2

1 21 and

)2(,;, dhtzptzp )3(1 dh

LAX PAIRThe Lax pair of a given system confirms its integrability and provides a means for obtaining soliton

solutions. We can construct the linear eigenvalue problem for Eq. (1) as follows:

where U and V are

with the transformation

Eq. (1) can be obtained from the compatibility condition and this condition is satisfied by

considering the flow to be non-isospectral

The Lax pair confirms the complete integrability of Eq. (1).

)4(, VU zt PJiU

*

2

)(2)()()()()(

p

piQ

iJzVziPzDzJzDiV

10

01J

)()()()*)(2*)(*)()()(2

1exp(

))(2)()(())(2

1exp()()()(

22

22

zVztqzRqzViqzDiqzDtziD

R

qzViqzDiqzDtziD

RzVztqzR

Q

t

t

0))(2

1exp(*

))(2

1exp(0

)(

)(

2

2

tziq

tziq

zD

zRP

and))(2

1exp(),(),( 2tzitzp

D

Rtzp )5(),(),( tz

D

Rtz

)6()()(2

1zzV

z

DARBOUX TRANSFORMATIONThe Darboux transformation has been proved to be an efficient way to find the soliton

solution for integrable equations

From above equation, we can obtain

where i, j = 1, 2, and

Two-soliton solution

where

)(]1[ SI

)7(),(, 211 diagHHS

**

11*1

)( jiijijS

2

2

2

1det H

)8()22()(

)(2),( 2111

1

F

GASeche

zR

zDqtzq Bi

(9) and)))2()((

)2(()))((

))2()(((

11212

22

112

21222

11)(22)22()(2

22

112

211212212

ATanhie

ASecheie

ASecheeeeG

ABi

BiABi

AABiBiABiAABi

)))(()2()2()))(2(

))(2())(2(2()2((22

212

2221121

21212

12

ACoshASechAACosh

AACoshCosACoshF

Nonlinear tunneling of a Schrödinger soliton through dispersion (or nonlinear) wells and

barriers was discussed in several papers. It was found that in certain circumstances, depending

on the ratio of soliton amplitude to barrier height, the soliton can tunnel through the barrier in

a lossless manner. The tunneling effects of optical solitons governed by the NLS-MB system

have been less investigated until now. The propagation of solitons with varying GVD and the

nonlinearity coefficients can simply be interpreted as a tunneling through the barriers. To

achieve the pulse compression, we consider a system with decaying nonlinearity and

dispersion barrier riding on the exponential background as below [4]

In the above expression, positive or the negative sign indicates the dispersion potential

barrier or the well respectively. Here, we have to consider the dispersion barrier i.e, corresponds

to the positive sign. Also z represents the barrier’s amplitude, d is the parameter relating to the

barrier’s width, z0 denotes the position of the barrier, and r is a decaying parameter. d0 and R0

are constant parameters, having values are d0>0, R0>0.

TUNNELING EFFECT

)10())(()exp()(D 00 zzSechzrdz

)exp()( 0 zrRzR

RESULTS AND DISCUSSIONS

1(a) 1(b)Fig. 1(a) Two soliton solution with ω1=4.7, ω2=1.5, a1= -0.05, a2 = 0.09, b1 =0.8 and b2= -1 with r = -

0.02. Fig. 1(b) Corresponding contour plot.

we chose the position of the barrier, z0=4. Therefore dispersion barrier is formed at z= z0.

When soliton passing through the dispersion barrier compression and amplification

is occurs as shown in Fig 1(b). There is no any compression till reach the position of the barrier. This means that, when the pulse

passes through the dispersion barrier with an exponential decay of coefficients, it can be compressed to a desired extent by the choice of the barrier parameters. Soliton propagation through tunneling has potential applications in the areas of optical fiber compressors, optical fiber amplifiers, nonlinear optical switching, optical communications, and long-haul telecommunication networks for achieving pulse compression.

CONCLUSION With the help of the Darboux transformation, we have successfully derived the one and

two soliton solution for the GINLS-MB equation. Via symbolic computation, we have carried out our study from an analytic viewpoint.

According to the given two-soliton solution we have discussed the nonlinear tunneling effect.

In practical applications, the model is of primary interest not only for the compression and amplification of optical solitons in inhomogeneous systems, but also for the stable transmission of soliton control. The properties are meaningful for the investigation on the stability of soliton propagation in the optical soliton communication. This is why we are interested in the study and the ultimate goal of our research.

ACKNOWLEDGEMENT We wish to thank for DST India for providing financial support for

this research work.REFERENCES[1] G. P. Agrawal, Nonlinear Fiber Optics (San Diego: Academic Press), 2005.

[2] V.N. Serkin and T.L. Belyaeva, JETP. 74, 573 (2001).

[3] W.J. Liu , B. Tian , P. Wang , Y. Jiang, K. Sun , M. Li and Q. X. Qu J. Mod. Opt. 57, 309 (2010).

[4] W.P. Zhong and M. R. Belić, Phys. Rev. E. 81, 056604 (2010).