Soliton propagation in a medium with Kerr nonlinearity and resonant impurities: A variational approach

Soliton propagation in a medium with Kerr nonlinearity and resonant impurities:

A variational approach

D. P. Caetano, S. B. Cavalcanti, and J. M. Hickmann

Departamento de Fı´sica, Universidade Federal de Alagoas, Cidade Universita´ria, AL 57072-970 Maceio´, Brazil

A. M. Kamchatnov

Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142190, Moscow Region, Russia

R. A. Kraenkel and E. A. Makarova

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista–UNESP, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil 共Received 9 December 2002; published 25 April 2003兲

Using a variational approach we have studied the shape preserving coherent propagation of light pulses in a resonant dispersive medium in the presence of the Kerr nonlinearity. Within the framework of a combined nonintegrable system composed of one nonlinear Schro¨dinger and a pair of Bloch equations, we show the existence of a solitary wave. We have tested our analytical solution through numerical simulations confirming its solitary wave nature.

DOI: 10.1103/PhysRevE.67.046615 PACS number共s兲: 42.65.Tg, 42.50.⫺p, 42.81.Dp

I. INTRODUCTION

Investigation on coherent pulse propagation in a fiber waveguide with resonant impurities is a recurrent subject in the physical literature关1–3兴. It combines two quite different regimes of soliton propagation: resonant propagation related to the self-induced transparency共SIT兲 phenomenon and non-resonant propagation under the influence of weak dispersion and Kerr-like nonlinearity described by a mean field nonlin-ear Schro¨dinger equation 共NLS兲. Recently, 关4兴 periodic and soliton solutions describing the propagation of an optical field through a nonlinear dispersive medium embedded with two level atoms, were obtained within the inverse scattering transform共IST兲 method. However, the family of mixed soli-ton solutions presented in these studies is only possible for a particular choice of pulse power satisfying both the NLS condition for the fundamental soliton as well as the area theorem for the SIT soliton. More specifically, the power required to launch a fundamental fiber soliton should be equal to the power of a 2␲-SIT pulse, i.e., P_2␲⫽P_N⫽1. This condition on the interdependence of SIT and Kerr parameters is a physical consequence of the condition of complete inte-grability of the equations. Unfortunately this fact limits se-verely the usefulness of the SIT-Kerr soliton for practical applications 关5兴. Hence it is important to develop some ap-proximate alternative approach to the IST to deal with a non-integrable system so that this limitation is weakened or eliminated at all. This is a perfect scenario for a variational method used successfully in perturbation theory of solitons 关6兴, propagation in the presence of nonlinear dissipation 关7兴, nonintegrable systems such as cascaded ␹(2) nonlinearity 关8,9兴, etc. In these references, an approximate variational family of solitons was obtained, convergent to the known exact solutions for particular values of the physical param-eters. Usually, the variational result reproduces interpolation formulas for solitonlike pulses in a wide region of param-eters between limiting regions of values that admit exact solutions.

Based on the above discussion, we apply the variational method to investigate the existence of solitary wave solu-tions in media with combined Kerr nonlinearity and SIT resonance under conditions that do not necessarily satisfy the integrability requirement. Using an ansatz based on the exact result we have found a solitary wave solution whose new features are described by an extra term. This solution is shown to converge to the exact result obtained previously by the inverse scattering method. We show further that the ana-lytical result is confirmed by numerical simulations of the NLS-Bloch equations describing the system evolution.

II. RESONANT AND NONRESONANT LAGRANGIAN

We begin by writing the equations for a system with com-bined Kerr and SIT terms in a standard form

iEZ⫹ETT⫹2g兩E兩2E⫹d⫽0,

d_T⫹2i⌬d⫽⫺2i f En, 共1兲 nT⫽i f共d*E⫺dE*兲.

Here T⫽t⫺z. Z and t are the dimensionless spatial and time coordinates, respectively; E(Z,T) is the slowly varying elec-tromagnetic field envelope. d(Z,T) and n(Z,T) denote the normalized electric dipole moment of the transition and the population difference of the two-level atoms, respectively.⌬ is the detuning parameter between the atomic transition and the central frequency of the electromagnetic wave. The vari-ables d and n satisfy the normalization condition

兩d兩2_⫹n2_{⫽1, 共2兲} which reflects the conservation of probability in the sense that the total probability for an atom to be found either in the upper or lower levels is equal to unity.

Because of the condition set by Eq. 共2兲, the Lagrangian for the system cannot be written directly in terms of the

variables E, d, n. To circumvent this difficulty we introduce the real variables E1, E2, ␪, ␾ according to

E⫽E₁⫺iE₂, d⫽sin␪e⫺i␾, n⫽cos␪,

so that Eq. 共2兲 is satisfied automatically and the system of Eqs. 共1兲 takes the form

E1,Z⫺E2,TT⫺2g共E1 2_⫹E

2 2

兲E2⫺sin␪sin␾⫽0,

E2,Z⫹E1,TT⫹2g共E1 2_⫹E 2 2_兲E 1⫹sin␪cos␾⫽0, 共3兲 sin␪共␾T⫺2⌬兲⫺2 f cos␪共E1sin␾⫹E2sin␾兲⫽0,

␪T⫹2 f共⫺E1sin␾⫹E2cos␾兲⫽0.

The Lagrangian form of these last two equations was indi-cated in Ref. 关10兴. Using this result we obtain the full La-grangian of the system in the form

L⫽1₂共E1,ZE2⫺E1E2,Z兲⫹ 1 2共E1,T 2 _⫹E 2,T 2 _兲⫺g 2共E1 2_⫹E 2 2_兲 ⫺_{2 f}1 共␾T⫺2⌬兲cos␪⫺sin␪共E1cos␾⫹E2sin␾兲.

共4兲 It is easy to check that the Lagrange equations corresponding to the field Lagrangian Eq.共4兲 can be reduced to the system of equations共3兲. In original complex variables this Lagrang-ian becomes

L⫽⫺ i

2共EZE*⫺EEZ*兲⫹兩ET兩

2_⫺g兩E兩4_⫺Ed*⫺E*d

⫺1_f

冋

2兩d兩2 ⫺2⌬

册

It is known that the combined system composed of the NLS equation together with the SIT equations under the condition

g⫽ f2 共6兲

has an exact simultaneous soliton solution which can be writ-ten as关4兴

E共T,Z兲⫽2␥

f exp

冋

冉

2_⫺␥2_兲 ⫹ f共␣⫺⌬兲 共␣⫺⌬兲2_⫹␥2

冊

册

⫻exp

冋

冉

冊

册

1

V⫽4␣⫺

f

2关共␣⫺⌬兲2⫹␥2兴, 共8兲 with␣ and␥ being free parameters that determine the shape of the soliton and its velocity. Our aim is to generalize sys-tem 共7兲 for arbitrary values of g and f.

III. VARIATIONAL APPROACH TO SOLITON PROPAGATION

Following the variational approach, 共see Refs. 关6–9兴兲 we choose a realistic and yet simple set of trial functions of T and then calculate the ‘‘averaged’’ Lagrangian

L⫽

冕

⫺⬁ ⬁

LdT 共9兲

to determine the dependence of the parameters entering into the trial function on the evolution variable Z. It is well known that the success of this approach depends crucially on a proper choice of trial functions. For example, frequently used Gaussian pulses simplify the calculation of integrals in Eq. 共9兲, but they cannot reproduce exact solutions such as those in Eqs.共7兲. A natural choice here for the trial functions, is to use the exact solution with varying parameters such as the following: n共T,Z兲⫽ 2C 2 B2_⫹C2 1 cosh2_{关␬共T⫹␨兲兴}⫺1, d共T,Z兲⫽⫺ 2C B2_⫹C2兵B⫹iC tanh关␬共T⫹␨兲兴其 ⫻ exp共i␾兲 cosh关␬共T⫹␨兲兴, 共10兲 E共T,Z兲⫽ A exp共i␾兲 cosh关␬共T⫹␨兲兴, where ␾⫽␻共T⫹␨兲⫹␩,

and A(Z),B(Z),C(Z),␨(Z),␩(Z),␬(Z),␻(Z) are all consid-ered as functions of Z. The form of d and n is chosen so that Eq. 共2兲 is satisfied automatically.

Substituting these trial functions in the Lagrangian, one easily finds the exact soliton solution of the bare NLS equa-tion 共without SIT terms兲. However, a more extensive calcu-lation shows that the SIT soliton solution is not reproduced exactly by this choice of trial functions. Fortunately, the same calculation indicates that this deficiency may be fixed easily by a simple change of a constant factor in one of the terms in the averaged Lagrangian. As a result we obtain our Lagrangian final form

L⫽2A 2 ␬ 共␻␨Z⫹␩Z兲⫹ 2 3␬A 2_⫹2A 2_␻2 ␬ ⫺ 4 3 gA4 ␬ ⫹ 8ABC ␬共B2_⫹C2_兲⫹ 4 f ␻⫹2⌬ ␬ C2 B2⫹C2 ⫺4_f BC B2⫹C2⫹ 4 f arctan C B. 共11兲

To show that this Lagrangian is a correct starting point for the derivation of interpolation formulas for soliton solutions, we have to prove that it leads to exact soliton solutions in the special cases when such solutions exist. For the sake of clar-ity, let us consider them separately.

IV. NLS LIMIT

In this limit the Lagrangian described in Eq.共11兲 reduces to L⫽2A 2 ␬ 共␻␨Z⫹␩Z兲⫹ 2 3␬A 2_⫹2A 2_␻2 ␬ ⫺ 4 3 gA4 ␬ . 共12兲 Lagrange equations ⳵L ⳵A⫽ ⳵L ⳵␬⫽ ⳵L ⳵␻⫽0, ⳵ ⳵Z ⳵L ⳵␨Z ⫽ ⳵ ⳵Z ⳵L ⳵␩Z ⫽0 共13兲 yield to ␻␨Z⫹␩Z⫹␻2⫹ 1 3␬ 2_⫺4 3gA 2_{⫽0, 共14兲} ␻␨Z⫹␩Z⫹␻2⫺ 1 3␬ 2_⫺2 3gA 2_{⫽0, 共15兲} ␨Z⫽⫺2␻, 共16兲 ⳵ ⳵Z

冉

冊

冉

冊

The difference between Eqs.共14兲 and 共15兲 gives directly

A⫽ ␬

冑

and this relation together with Eq. 共17兲 show that A,␬, and ␻ are constants as expected. Then Eqs.共14兲 and 共16兲 give ␩Z⫽␬2⫹␻2, that is

␨⫽⫺2␻Z,␩⫽共␬2⫹␻2兲, ␻␨⫹␩⫽共␬2⫺␻2兲Z.

Thus, we can write the trial function Eq. 共10兲 as

E共T,Z兲⫽ ␬

冑

exp关i共␻T⫹共␬2⫺␻2兲Z兴

cosh关␬共T⫺2␻Z兲兴 , 共19兲

or after introducing new parameters

␬⫽2␥, ␻⫽⫺2␣, 共20兲

we arrive at the following expression:

E共T,Z兲⫽2␥

冑

exp关⫺2i␣T⫺4i共␣2⫺␥2兲Z兴

cosh关2␥共T⫹4␣Z兲兴 , 共21兲

which coincides exactly with the corresponding limit of Eq. 共7兲.

V. SIT LIMIT

In this limit Lagrangian 共11兲 takes the form

L⫽ 8ABC ␬共B2_⫹C2_兲⫹ 4 f ␻⫹2⌬ ␬ C2 B2⫹C2 ⫺4 f BC B2⫹C2⫹ 4 f arctan C B. 共22兲

Now, Lagrange equations ⳵L ⳵A⫽ ⳵L ⳵B⫽ ⳵L ⳵C⫽ ⳵L ⳵␬⫽ ⳵L ⳵␻⫽0, ⳵ ⳵Z ⳵L ⳵␨Z ⫽ ⳵ ⳵Z ⳵L ⳵␩Z ⫽0 共23兲 yield ␻␨Z⫹␩Z⫽⫺ 2BC A共B2_⫹C2_兲, 共24兲 f A共B2⫺C2兲⫹共␻⫹2⌬兲BC⫹␬C2⫽0, 共25兲 ␨Z⫽⫺ 2 f C2 A2_共B2_⫹C2_兲, 共26兲 f共␻␨Z⫹␩Z兲A2共B2⫹C2兲⫹4 f ABC⫹2共␻⫹2⌬兲C2⫽0, 共27兲 A2 ␬ ⫽const, ␻A2 ␬ ⫽const, 共28兲

where Eq. 共25兲 is obtained twice. Substitution of Eq. 共24兲 into Eq.共27兲 gives

AB

C ⫽⫺

␻⫹2⌬

f , 共29兲

and hence from Eq.共25兲 we get

A⫽␬

f . 共30兲

This equation together with Eq.共28兲 show again that A,␬,␻ are constant. Introducing␣ and␥ via Eq.共20兲, we find that the other equations are satisfied if

A⫽2␥

f , B⫽2共␣⫺⌬兲, C⫽2␥, 共31兲

and then the soliton velocity is equal to ␨Z⫽⫺

f

2关共␣⫺⌬兲2⫹␥2兴. 共32兲 Substitution of these parameters into the trial function 关Eq. 共10兲兴 produces the exact solution of the SIT equations which coincides with the corresponding limit of the combined so-lution关Eqs. 共7兲兴.

These calculations justify the above statement that the La-grangian together with the trial functions provide the basis to obtain approximate formulas for the soliton solution. After gaining this experience we can proceed to the general case with arbitrary values of g and f.

VI. COMBINED NLS AND SIT REGIME

In the general case of Lagrangian equation 共11兲, Hamil-ton’s principle provides the following set of Lagrange equa-tions: ␻␨Z⫹␩Z⫹␻2⫹ 1 3␬ 2_⫺4 3gA 2_⫹ 2BC A共B2⫹C2兲⫽0, 共33兲 f A共B2⫺C2兲⫹共␻⫹2⌬兲BC⫹␬C2⫽0, 共34兲 ␨Z⫽⫺2␻⫺ 2 f C2 A2共B2⫹C2兲, 共35兲 ␻␨Z⫹␩Z⫹␻2⫺ 1 3␬ 2_⫺2 3gA 2_⫹ 4BC A共B2⫹C2兲 ⫹2共␻⫹2⌬兲 f C2 A2共B2⫹C2兲⫽0, 共36兲

as well as Eq.共29兲. Again we introduce new constants␣and ␥, so that A,B, and C are determined by Eqs.共44兲 and

␻⫽⫺2␣⫹␥␮, 共37兲

where␮ is the parameter to be determined. To this end, we find on one hand that the difference between Eqs. 共33兲 and 共36兲 gives

␬2_⫽4g

f2 ␥

2_⫹ 3 f␥␮

4关共␣⫺⌬兲2⫹␥2兴, 共38兲 and on the other hand from Eq. 共34兲 we obtain

␬⫽2␥⫺共␣⫺⌬兲␮. 共39兲

Thus, we arrive at a quadratic equation for ␮. Note that␮ ⫽0 implies g⫽ f2 _{and this particular case reproduces the} exact solution. Thus we obtain

␮⫽ 1 2共␣⫺⌬兲2

再

冑

冉

冊

冉

冊

冎

At last, we find the expression for soliton’s velocity 1

V⫽␨Z⫽4␣⫺

f

2关共␣⫺⌬兲2⫹␥2兴⫺2␥␮, 共41兲 where Eqs.共37兲 and 共39兲 with␮defined by Eq.共40兲 give the width of the soliton and its velocity as functions of the ratio

g/ f2 provided the amplitude of the soliton is given by

A⫽2␥

f . 共42兲

VII. NUMERICAL RESULTS

To test our analytical solutions we have carried out nu-merical experiments. The basic idea is to show that using the

variational solution as an initial condition the resulting propagation exhibits a solitary wave behavior. To obtain the propagation behavior, we have numerically solved the set of Eqs. 共1兲, using a combination of the Runge-Kutta with the beam propagation method to determine both the dynamics of the atomic system and the optical field evolution, with the following set of parameters: ⌬⫽0.1, g⫽0.125. We choose the initial conditions for the atomic system so that the popu-lation is in the ground state to start with and we choose a sech temporal profile for the optical pulse with area 2␲, corresponding to a NLS soliton with amplitude A⫽0.5. In Fig. 1共a兲 we show the pulse propagation illustrating solitary wave behavior for the integrable case, i.e., for g⫽ f2_{. While} Fig. 2 illustrates the situation for the nonintegrable case, i.e.,

propagat-ing stably for several soliton periods.

VIII. CONCLUSIONS

Coherent soliton propagation in a resonant fiber wave-guide is demonstrated for a general system of equations that do not satisfy integrability conditions, opening the real pos-sibility of obtaining the SIT-Kerr soliton in practical condi-tions. Within a variational approach, we have obtained the full description of a family of approximate solutions that exhibits solitary wave properties beyond the mixed soliton state condition, that is, P2␲⫽PN⫽1. Based on the exact

re-sult for the integrable system, we were able to relax the dependence among the physical parameters allowing for re-alistic conditions to permit the experimental observation of a SIT-Kerr soliton. The variational solutions obtained depend on two free parameters and in the limiting cases are reduced

to the correspondent exact solution . Furthermore, we have confirmed the analytical results by numerical simulations, evidencing the solitary behavior of the light pulse. This SIT-Kerr solitonic behavior in doped fiber media may be proved extremely useful to the future of optical communications and photonic devices. It combines the convenient propagation properties of an optical fiber with the remarkable manipula-tion properties of a resonant active medium opening a new perspective for optical techniques.

ACKNOWLEDGMENTS

S.B.C. and J.M.H. thanks the Instituto do Mileˆnio de In-formac¸a˜o Quaˆntica, CAPES, CNPq, FAPEAL, PRONEX-NEON, ANP-CTPETRO for support. A.M.K. is grateful to FAPESP, FAPEAL, and RFBR 共Grant No. 01-01 00696兲. R.A.K. gratefully acknowledges the support of FAPESP.

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FIG. 1. Soliton propagation for the integrable case, g⫽ f2. _{FIG. 2. Soliton propagation for the nonintegrable case, g}

⫽2 f2