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Научни трудове на Съюза на учените в България-Пловдив. Серия В. Техника и технологии, естествен ии хуманитарни науки, том XVI., Съюз на учените сесия "Международна конференция на младите учени" 13-15 юни 2013. Scientific research of the Union of Scientists in Bulgaria-Plovdiv, series C. Natural Sciences and Humanities, Vol. XVI, ISSN 1311-9192, Union of Scientists, International Conference of Young Scientists, 13 - 15 June 2013, Plovdiv.
FINDING A SOLITON SOLUTION OF THE NONLINEAR AMPLITUDE EQUATION DESCRIBING THE EVOLUTION OF OPTICAL PULSES IN A SINGLE-MODE FIBER
Aneliya MM. Dakova(1), Diana Y. Dakova(2)
(1,2)University of Plovdiv "Paissi Hilendarski", Tzar Assen Str.24, BG-
4000 Plovdiv
e-mail(1): anelia.dakova@abv.bg e-mail(2): ddakova2002@yahoo.com
Abstract
In the present work is being reviewed the propagation of optical pulses in a nonlinear dispersive single-mode fiber. The linear dispersion is of second order and the nonlinearity is of Kerr type. An amplitude equation describing the evolution of the pulse envelope is used. It is found an exact analytical soliton solution of this equation.
Introduction
The development of communication systems and the need for fast and efficient transmission of information over long distances require the study of the propagation of light pulses in nonlinear dispersive media. It is particularly relevant the exploration of the evolution of optical pulses in a single-mode waveguides. In recent years, particular attention attracts the soliton regime of propagation. It is a result of the dynamic balance between the effects caused by the dispersion and the nonlinearity of the medium. The behavior of soliton-like impulses in Kerr-type media is well studied. It is described by the nonlinear Schrödinger equation [1,2]. The influence of various factors on the phase and the amplitude of the soliton has been studied.
In the present work we are interested in the longitudinal propagation of light pulses in a single-mode optical fiber with a negative dispersion of the group velocity. Their behavior is described by the nonlinear amplitude equation, in which are included the effects of diffraction. This equation is different from the nonlinear Schrödinger equation. In Its use is required by the need for more complete and accurate description of the evolution of the pulse. We consider that the axis of symmetry of the fiber coincides with the axis Oz of the introduced Cartesian coordinate system.
Basic equation
The general nonlinear amplitude equation describing the evolution of spatial pulses in isotropic dispersion media is discussed in [3,4]. In the present work we investigate the behavior of pulses in a single-mode fiber. Due to the waveguide nature of propagation of the optical pulses, the transverse structure of the laser beam is determined by the fiber modes characteristics. Therefore, we are interested only in the longitudinal dimensions of optical pulses. In this case, the
one-dimensional scalar nonlinear amplitude equation which describes the propagation of optical pulse in isotropic fiber recorded in a Cartesian coordinate system (O,x,y,z) is:
2 ik0
d A
+
1 d A
v,
gr
d t
d A (1 + ß ) d2 A
d z
gr
d f
+ k 02 n.
|A|2 A
(1)
2
where
k 0 = k (®0), ß = k 0 vgrk
v = — gr k '
n.
(2)
gr
k ' =
d k d z
k " =
d2 k
d z2
ngr = n + G)o
d n
da
, n = n (a0)
In the equation above A = A (t, z) is the scalar amplitude function of the envelope of the pulse; t is time; a>0, k, vgr, n, n2 are respectively the carrier frequency, the wave number, the group velocity, the linear and nonlinear refractive index of the medium. Parameter ¡3 characterizes the second order of linear dispersion of the medium. Since we are interested in a media with anomalous dispersion (k" <0), it is assumed that:
ß = -| ß\=- k o V2gr\k
To simplify equation (1) we change the variables [5]:
z = z0 z
z 0 = vgrt 0-.
t = 10 t '
A = A 0 A '
(3)
(4)
With t0, z0 and A0 are presented the initial duration, the longitudinal length and the amplitude of the optical pulses. For simplicity, in further consideration, we will not write prim ('). By the substitution (4) the equation (1) can be presented in dimensionless form:
f
- 2 ia
d A d A
\
■ +
d z d t
d2A „ I oh d2A « .n . - (1 -|ß|)—^ + y\A | A
d z
d t'
(5)
where
a = k0z0, y = a2n2|A0
(6)
Constant a characterizes the number of oscillations at level 1/e of the maximum of amplitude of the pulse and it is usually a>>1. Parameter y is the nonlinearity of the medium. It depends on n2.
Finding a solution of the scalar amplitude equation
We consider the amplitude equation (5). It is a nonlinear partial differential equation of a second order. We are searching for solution of the form [2,5]:
A (z, t) = O (£) exp( iat + ibz ) {1)
where E,=z-t, a and b are constants.
By replacing (1) in (5) and after couple of transformations we obtain the following ordinary differential equation for the unknown function 0=0(^J:
|ßO ''+ 2 i O '[ b + a (1 -|ß|)] -O [ b2 - a2 (1 - |ß|) + 2a( a + b )] + y®3 = 0
(8)
1
c
w = a
0
a = a
2
where with 0 and 0' are respectively presented the first and second derivative of 0 with a respect to the variable
Equation (8) is an ordinary complex differential equation of second order and the third degree. By the substitution (7) we assume that the unknown function 0(^) is real. Therefore, in order to find a solution of this equation, we equalize the real and imaginary parts on both sides of the equality. Thus, we obtain two ordinary differential equations.
The real part: № & 2 - a ^ - ||) + 2a (a + b )] + ^ 3 = 0 (9)
The imaginary part: °'[b + a (1 - I11)] = 0 (10)
First we consider the equation (10). For a random value of ^different from zero, 0' is also nonzero. In order to satisfy the equality, the expression in the brackets must be zero. From this condition we find a relation between the constants a and b:
b = - a (1 |) (11)
Now we turn to the equation (9). We divide both sides of the equation to || |and we assume that:
1 2 2 II 1 a2n2|A0|2 |^[b - a (1 -||)] + 2a(a + b) = n= const, -| =r = const (12)
The equation (9) is of the type: °'' = + 3 = 0 (13)
This is an ordinary differential equation of the second order and third degree for the unknown real function 0. The solution of this equation is the function [1,2,6]:
° = se° h (n)
r sec h ¡¡L, ) (14)
O „ =
This expression describes a soliton with amplitude: O o = p (15)
From the assumption (12) it is clearly seen that the constant r) can be expressed by a and b. By substituting (11) into (12) we obtain the following algebraic equation for the constant a:
a 2(1 -|ß) - 2aa + n = 0 ^
The solution of this equation is:
a
(1 -ß|)
a12 = T-ß\ (1 ±i1 ) (17)
We assume that the expression under the square root is zero. Thus, we find not only a single solution for the constant a, but also an expression for n We determine that:
2 /1 I - I n I ■
(18)
a = a /(1 - ß|), b = -a, rj = a2 /(1 - |ß)
We replace the results (14) and (18) in the substitution (7) and we find the solution for the amplitude equation (5):
aE t
A (z,t) = O o sec h ( . )exp[ - ia( z --—r-r)], £ = z - t (19)
Having in mind the change of variables (4), the expression above, in a Cartesian coordinate system can be written in the form of:
k(z - vgrt) vgrt A (z, t) = ® 0 sec h( . ^ ) exp[ -ik0(z--^p-r)], ® 0 =
0 1 -\\ i
21 \\
2
0 I
(20)
n 2(1 -|\)| A
The condition for forming a fundamental soliton with an amplitude ®0=1 is:
A0I2 = 2\\/[ n 2(1 -\\)] (21)
The obtained expression is different from the classical one, i.e. for a soliton whose behavior is described by the nonlinear Schrödinger equation [2,6]. For the observation of a single soliton with a carrying wavelength Ä=1400nm in a silica single-mode fiber with k0=4,488.106m~1, k"=-1,2.10'26s2/m, and a nonlinear refractive index n2=4,5.10'20m2/W, the initial intensity of the Schrödinger soliton is \A0\2shr=0,34.1014W/m2 and for pulses obtained in our work: \A0\2=1.101/W/m2. The critical intensity for ionization of the medium is \A0\2=2,2.1019W/m2. Therefore, to observe the soliton that we obtained, it is needed an initial intensity lower than the critical and significantly higher than that of the Schrödinger's soliton.
Summary
In the present work we obtained a soliton solution of the one-dimensional nonlinear scalar amplitude equation which describes the propagation of optical pulse in single-mode fiber with anomalous dispersion and Kerr-type nonlinearity. This solution is different from the soliton solution of the nonlinear Schrödinger equation. The critical intensity, for observation of the solitons we obtained, is higher by several orders of that of Schrödinger.
Acknowledgements
Special thanks are due to Dr. Lubomir Kovachev from the Institute of Electronics of Bulgarian Academy of Sciences-Sofia for the valuable advices and recommendations in the discussions on this work.
References
1. Agrawal, G. P., Nonlinear fiber optics, Academic Press, INC, New York (2007).
2. Abdullaev, F., Darmanian, S. A., Khabibullaev, P., Optical solitons, Berlin; London : SpringerVerlag (1993).
3. Boyd, R.W., Nonlinear optics, Academic Press, (2003).
4. Lubomir M. Kovachev and Kamen Kovachev, "Linear and Nonlinear Femtosecond Optics in Isotropic Media. Ionization-free Filamentation", Laser Pulses / Book 1, chapter, ISBN 978-953307-429-0, InTech, (2011).
5. Dakova, A., Dakova, D., "Nonlinear regime of propagation of femtosecond optical pulses in single-mode fiber", Proc. SPIE 8770, 17th International School on Quantum Electronics: Laser Physics and Applications (2013).
6. Hasegawa, A., Tappert, F., "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion", Appl. Phys. Lett., 23, 142 (1973).
