Derivation of a modified Korteweg–de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

Derivation of a modi fi ed Korteweg – de Vries model for few-optical-cycles soliton propagation from a general Hamiltonian

H. Triki

, H. Leblond

⁎ , D. Mihalache

aLUNAM Université, Université d'Angers, Laboratoire de Photonique d'Angers, EA 4464, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France

bRadiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria

cHoria Hulubei National Institute for Physics and Nuclear Engineering (IFIN-HH), 30 Reactorului, Magurele-Bucharest, 077125, Romania

dAcademy of Romanian Scientists, 54 Splaiul Independentei, Bucharest 050094, Romania

a b s t r a c t a r t i c l e i n f o

Article history:

Received 11 October 2011

Received in revised form 18 February 2012 Accepted 20 February 2012

Available online 3 March 2012 Keywords:

Few-cycle pulses Few-cycle solitons

Modified Korteweg–de Vries equation mKdV equation

Propagation of few-cycles optical pulses in a centrosymmetric nonlinear optical Kerr (cubic) type material described by a general Hamiltonian of multilevel atoms is considered. Assuming that all transition frequen- cies of the nonlinear medium are well above the typical wave frequency, we use a long-wave approximation to derive an approximate evolution model of modified Korteweg–de Vries type. The model derived by rigor- ous application of the reductive perturbation formalism allows one the adequate description of propagation of ultrashort (few-cycles long) solitons.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Since 1999, few-optical cycle pulses[1–4]with durations of only a few periods of the optical radiation have become the primary compo- nents in many important problems of dynamics of nonlinear optical waves. In particular, these ultrashort (femtosecond) pulses [5,6]

find applications in a wide variety of research areas, such as light– matter interactions at high field intensities, high-order harmonic generation, extreme nonlinear optics [7], and attosecond physics [8,9]. Notably, the theoretical modeling used to correctly describe the dynamics of such pulses in nonlinear optical media has also been developed in parallel to these incentive experimental studies.

It should be said that, the search for new ideas or even new mathe- matical concepts is of great interest as it is helpful to better under- stand the ultrashort pulse propagation in nonlinear optical media and the formation of robust few-optical-cycle solitons.

The continuing experimental progress in the study of wave dy- namics of few-cycle pulses (FCPs) in nonlinear optical media has paved the way for the development of new theoretical approaches to model their propagation in a lot of physical settings. Three classes of main dynamical models for FCPs have been put forward in the past years: (i) the quantum approach[10–14], (ii) the refinements within the framework of slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger-type envelope equations [15–24], and

(iii) the non-SVEA models[25–43]. The propagation of FCPs in Kerr media can be described beyond the SVEA by using the modified Korteweg–de Vries (mKdV)[31–33], sine-Gordon (sG)[34–36], or mKdV–sG equations [37–41]. The mKdV and sG equations are completely integrable by means of the inverse scattering transform method[44,45], whereas the mKdV–sG equation is completely inte- grable only if some condition between its coefficients is satisfied [46,47].

Other relevant works on few-cycle pulses deal with propagation and interaction of extremely short electromagnetic pulses in quadratic nonlinear media[48–51], the study of few-cycle light bullets created by femtosecondfilaments[52], the investigation of ultrashort spatiotem- poral optical solitons in quadratic nonlinear media[53], the ultrashort spatiotemporal optical pulse propagation in cubic (Kerr-like) media without the use of the slowly varying envelope approximation [54,55], the possibility of generating few-cycle dissipative optical solitons[56,57], generation of unipolar pulses from nonunipolar optical pulses in a quadratic nonlinear medium[58], and the existence of guided optical solitons of femtosecond duration and nanoscopic mode area, that is, femtosecond nanometer-sized optical solitons[59].

We also mention recent studies of ultrafast pulse propagation in mode-locked laser cavities in the few femtosecond pulse regime and the derivation of a master mode-locking equation for ultrashort pulses [60]. A relevant recent theoretical work presents a class of few-cycle elliptically polarized solitary waves in isotropic Kerr media, propose a method of producing multisolitons with different polarization states, and study their binary-collision dynamics[61].

Robust circularly polarized few-optical-cycle solitons in Kerr media

⁎Corresponding author. Tel.: + 33 2 41 73 54 31; fax: + 33 2 41 73 52 16.

E-mail address:[email protected](H. Leblond).

0030-4018/$–see front matter © 2012 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2012.02.045

Contents lists available atSciVerse ScienceDirect

Optics Communications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / o p t c o m

in both long-wave and short-wave approximation regimes were also studied in recent works[62–64].

So far, most of theoretical investigations concern only FCPs propagating in a nonlinear optical medium which is described by a Hamiltonian related to two-level atoms. However, to the best of our knowledge, studies based on very general Hamiltonians describing the wave dynamics of such ultrashort pulses have not yet been reported. It is, nevertheless, an interesting and potentially useful de- scription if FCPs can form and can be robust when we consider this more general physical setting. The aim of this paper is to investigate such a multilevel system in the framework of the reductive perturba- tion method (multiscale analysis). This fact allows one to extend the existing studies to a more general physical situation. Thus in this work we give a detailed mathematical derivation of the modified Korteweg–de Vries equation for a general Hamiltonian. We assume that the absorption spectrum of the nonlinear medium does not ex- tend below some cutoff frequency, and that the typical frequency of the FCP is much less than the latter. In other words, we assume that the transparency range of the medium is very large, and consider only the frequencies located in the ultraviolet spectral domain and further. The effect of the infrared transitions, which yield a sine- Gordon model in the case of two-level atoms, will be considered in a further study.

The present study can be considered from two different points of view: (i) a nonlinear cubic medium which has no transition in the infrared is actually described, and (ii) the most general medium con- taining two kinds of transitions, both in the infrared and in the ultra- violet domains, is expected to be described by a more general modified Korteweg–de Vries–sine-Gordon equation. However, we give here the detailed mathematical derivation for the modified Kor- teweg–de Vries part of the most general model.

This paper is organized as follows. In the next section we present in detail a derivation of the modified-Korteweg–de Vries equation as a rigorous formal asymptotics of the Maxwell–Bloch equations for the most general Hamiltonian for multilevel atoms. This evolution equation describes the propagation of ultrashort (a few femtosecond long) optical solitons in the so-called long-wave regime.

InSection 3we analyze both analytically and numerically the modi- fied Korteweg–de Vries breather, which is the prototype of few- optical-cycle solitons in such cubic nonlinear optical media. Finally, inSection 4we present our conclusions.

2. The derivation of a modified Korteweg–de Vries model from a general Hamiltonian

We consider a set of multilevel atoms described by a very general Hamiltonian:

H₀¼ℏ

ω1 0 ⋯ 0

0 ω2 ⋯ ⋯

⋯ ⋯ ⋯ ⋯

0 ⋯ ⋯ ωN

0 BB

@

1 CC

A: ð1Þ

The evolution of the density matrixρis governed by the Schrödinger– von Neumann equation

iℏ∂tρ¼½H;ρ; ð2Þ

in which the total Hamiltonian

H¼H₀−^→μ ⋅^→E ð3Þ

includes a term accounting for the coupling between the electricfield^→E and the atoms through a dipolar momentum operator^→μ.

The evolution of the electricfield is governed by Maxwell–Helmholtz wave equation

∂²z^→E¼ 1

c²∂²t ^→Eþ1 ε0

P→

; ð4Þ

in which the polarization densityP^→expresses as P→

¼NTrρ μ^→

: ð5Þ

We consider a cubic (Kerr) nonlinearity, i.e. we assume that the material is centrosymmetric, so that the second order susceptibility χ⁽²⁾vanishes. For the sake of simplicity, we will assume a linearly po- larized wave. Then, only the component of^→E along the direction of polarization, and the corresponding component ofμ^→will be involved.

Hence, we suppose that the operatorμ^→¼μ^→ex, where^→exis the unitary vector along thex-axis. Also, we have^→E¼E e^→x, and^→P¼P e^→x. Note that E→

andμ^→in Eq.(3)are replaced with scalar quantitiesEandμ, in which the matrix

μ¼ðμnmÞ_{ðn;mÞ∈½1;N}; ð6Þ is Hermitian, i.e.μmn=μnm∗ , where the star denotes the complex con- jugate. Due to centrosymmetry, sinceμmnare the matrix elements of an odd operator, the matrixμis off-diagonal.

We assume that the characteristic pulse frequencyωw has the same order of magnitude as the inverse of the pulse duration 1/t_w, and is very small with respect to any resonance frequency

Ωnm¼ωn−ωm ð7Þ

in the atomic spectrum, i.e.

1=t_w

˜

This motivates the use of the long-wave approximation[65]. We thus introduce the slow variables

τ¼ε t−z V

; ð9Þ

ζ¼ε³z; ð10Þ

so that we obtain

∂t¼ε∂τ; ð11Þ

∂z¼−ε

V∂_τþε³∂_ζ: ð12Þ

The scaling (Eqs.(9)–(10)) clearly assumes unidirectional propa- gation. Accurate studies [56,58]proved that this assumption may lead to erroneous results in non-homogeneous media, since some reflections which are negligible for long pulses are not for few-cycle pulses. This must be taken into account in the interpretation of our re- sults, the input wave should indeed be the pulse after it has entered the homogeneous nonlinear medium, which may appreciably differ from the incident pulse.

ThefieldE, the polarization densityPand the density matrixρare expanded in a power series of some small parameterεas

E¼∑

p≥1ε^pE_p; ð13Þ

P¼∑

p≥1ε^pP_p; ð14Þ

3180 H. Triki et al. / Optics Communications 285 (2012) 3179–3186

ρ¼∑

p≥0ε^pρ^{ð Þp}: ð15Þ

Ast→−∞,ρ⁽⁰⁾is assumed to be the density matrix at thermal equilibrium, whileEp,Pp, andρ^(p)vanish, for anyp≥1.

2.1. Order 0

The Schrödinger–von Neumann equation(2)at orderε⁰is 0¼hH₀;ρ^{ð Þ0}i

; ð16Þ

and H₀;ρ

½

ð Þ_nm¼ℏΩnmρnm: ð17Þ

We assume that each level is non-degenerated, and henceΩnm≠0 if n≠m. Then Eq.(16)yieldsρnm(0)= 0 forn≠m(i,j= 1, 2,..,N).

2.2. Order 1

The Schrödinger–von Neumann equation(2)at orderε¹is iℏ∂τρ^{ð Þ0} ¼½H₀;ρ1−E₁hμ;ρ^{ð Þ0}i

: ð18Þ

Since μ;ρ

½ ð Þ_nm¼X^N

ν¼1

μnνρνm−ρnνμ_νm

ð Þ; ð19Þ

and using Eq.(17), we obtain the equation

iℏ∂τρ^{ð Þ}nm⁰ ¼ℏΩnmρ^{ð Þ}nm¹−E₁X^N

ν¼1 μnνρ^{ð Þ}ν⁰m−ρ^{ð Þ}n⁰νμνm

; ð20Þ

(recall thatΩnm=ωn−ωm).

Sinceρ⁽⁰⁾is diagonal, Eq.(20)reduces to

ρ^{ð Þ}nm¹ ¼μnmE₁ ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩnm

ð21Þ forn≠m, while the diagonal componentsρnn0 are constant.

The polarization density is given by Eq.(5), i.e., at orderε¹, by P₁¼N∑

nmρ^{ð Þ}nm¹μmn: ð22Þ

Sinceμis off-diagonal, the sum in Eq.(22)extends overn≠monly.

Reporting Eq. (21) into Eq. (22), and using the fact that μ is Hermitian, we get

P₁¼NE₁∑

n≠m

μnm

2 ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩnm : ð23Þ

The Maxwell–Helmholtz wave equation(4)at leading orderε³is 1

V²∂²τE1¼ 1

c²∂²τ E1þP₁ ε0

: ð24Þ

Reporting Eq.(23)into Eq.(24)yields 1

V²∂²_τE₁¼ 1

c²∂²_τ E₁þNE₁ ε0ℏ ∑

n≠m

μnm

2 ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ωnm

2 4

3

5: ð25Þ

Eq.(25)admits a nonzero solution ifV=c/n0, in which the refractive indexn0is

n₀¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ N

ε0ℏ∑

n≠m

μnm

2 ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ωnm

vu ut

: ð26Þ

From Eq.(26), we can deduce the linear susceptibilityχ⁽¹⁾, which is related to the refractive index throughn0¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þχ^{ð Þ1} p , as

χ^{ð Þ1} ¼ N ε0ℏ∑

n≠m

μnm

2 ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ωnm : ð27Þ

Eq.(27)is worth being compared to the known expression of the lin- ear susceptibility, which is given in Ref.[66](Eq. 3.5.15, p. 163):

χ^{ð Þ}nm¹ ωp ¼ N ε0ℏ∑

nm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

μⁱmnμ^jnm

Ωnm−ωp−iγnm: ð28Þ Here we restrict to a linear polarization, hence χ⁽¹⁾=χxx(1) and μmn=μmnx . We neglect the damping, i.e. γnm= 0, and due to the long-wave approximation, the susceptibility must be evaluated as ωptends to zero. Then Eq.(27)coincides with Eq.(28).

2.3. Order 2

The Schrödinger–von Neumann equation(2)at orderε²is

iℏ∂_τρ^{ð Þ1} ¼hH₀;ρ^{ð Þ2}i

−E₂hμ;ρ^{ð Þ0}i

−E₁hμ;ρ^{ð Þ1}i

: ð29Þ

Inserting Eq.(17)into Eq.(29), we get the equation

iℏ∂τρ^{ð Þ}nm¹ ¼ℏΩnmρ^{ð Þ}nm²−E₂X^N

ν¼1

μnνρ^{ð Þ}νm⁰−ρ^{ð Þ}nν⁰μνm

−E₁X^N

ν¼1

μnνρ^{ð Þ}νm¹−ρ^{ð Þ}nν¹μνm

: ð30Þ

Sinceρ⁽⁰⁾is diagonal, Eq.(30)reduces to

iℏ∂τρ^{ð Þ}nm¹ ¼ℏΩnmρ^{ð Þ}nm²−E₂μnm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

−E₁X^N

ν¼1

μnνρ^{ð Þ}νm¹−ρ^{ð Þ}nν¹μ_νm

: ð31Þ Forn=m, by using Eq.(21), Eq.(31)becomes

iℏ∂_τρ^{ð Þ}nn¹ ¼−E²₁∑

ν≠n

μnνμ_νn ρ^{ð Þ}nn⁰−ρ^{ð Þ}νν⁰

ℏΩ_νn −μ_νnμnν ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

ℏΩnν

2 4

3 5: ð32Þ

The right-hand-side of Eq.(32)is zero, and henceρnn(1)= 0 for alln.

Consequently, taking into account the fact thatμis off-diagonal, we get from Eq.(31)

ρ^{ð Þ}nm² ¼ i

Ωnm∂τρ^{ð Þ}nm¹ þ E₂

ℏΩnmμnm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

þ E1

ℏΩnm ∑

ν≠n;m μnνρ^{ð Þ}νm¹−ρ^{ð Þ}nν¹μνm

;

ð33Þ

forn≠m, i.e., inserting Eq.(21)into Eq.(33)we get,

ρ^{ð Þ}nm² ¼iμnm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩ²nm

∂τE1þE₂μnm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩnm

− E²₁ ℏ²Ωnm

ν≠∑n;mμnνμ_νm

ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

Ωnν − ρ^{ð Þ}mm⁰ −ρ^{ð Þ}νν⁰

Ωνm

2 4

3 5:

ð34Þ

The polarization density at orderε²is thus

P2¼iN

ℏ∂τE1 ∑

n;m;n≠m

μnmμmn ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ω²nm

þNE₂

ℏ ∑

n;m;n≠m

μnmμmn ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ωnm

þNE²₁ ℏ² ∑

m;n;ν

m≠n≠ν

½

− ρ^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

μmnμ_νmμnν

ΩnmΩnν

ð35Þ

The second order nonlinear polarization is thus given by

P^NL₂ ¼ε0χ^{ð Þ2}ð ÞE₁ ²; ð36Þ

where the second order susceptibilityχ⁽²⁾is given by

χ^{ð Þ2} ¼ N ε0ℏ²

∑

m;n;ν m≠n≠ν

ρ^{ð Þ}mm⁰ −ρ^{ð Þ}νν⁰

μmnμ_nνμ_νm

ΩnmΩνm − ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

μmnμ_νmμ_nν ΩnmΩnν

:

ð37Þ The expression ofχ⁽²⁾can be found in Ref.[66](Eq. 3.6.14 p. 173), as

χ^{ð Þ}ijk² ωpþωq;ωq;ωp

¼ N 20ℏ²

∑

mnν

f

½

þ μⁱmnμ^knνμ^j_νm

Ωnm−ωp−ωq−iγnm

Ω_νm−ωq−iγ_νm

− ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

½

þ μⁱmnμ^k_νmμ^jnν

Ωnm−ωp−ωq−iγnm

Ωnν−ωq−iγnν

g

ð38Þ

According to the chosen polarization, we should have

χ^{ð Þ2} ¼χ^{ð Þ}xxx²ð2ω;ω;ωÞ; ð39Þ evaluated asωtends to zero, and neglecting the damping (γnm= 0). It is seen that Eqs.(36) and (38)coincide under these conditions, taking into account the fact thatμis off-diagonal. In the present study, we

assume a centrosymmetric medium, and henceχ⁽²⁾= 0. The coeffi- cient of∂τE1in Eq.(35)is

A¼iN∑

nm

μnmμmn ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

ℏΩ²nm

: ð40Þ

Permuting the dummy subscriptsmandnchanges the sign of the term in the latter sum, henceA= 0.

The Maxwell–Helmholtz wave equation(4)at orderε⁴is

1

V²∂²τE₂¼1

c²∂²τ E₂þP2

ε0

; ð41Þ

which is automatically satisfied if the expression (Eq.(26)) of the re- fractive index is taken into account.

2.4. Order 3

The Schrödinger–von Neumann equation(2)at orderε³is

iℏ∂_τρ^{ð Þ2} ¼hH₀;ρ^{ð Þ3}i

−E₃hμ;ρ^{ð Þ0}i

−E₂hμ;ρ^{ð Þ1}i

−E₁hμ;ρ^{ð Þ2}i

: ð42Þ

Using Eqs.(17) and (19)into Eq.(42), we obtain the equation

iℏ∂_τρ^{ð Þ}nm² ¼ℏΩnmρ^{ð Þ}nm³−E₃X^N

ν¼1

μnνρ^{ð Þ}_ν⁰m−ρ^{ð Þ}n⁰νμ_νm

−E₂X^N

ν¼1

μnνρ^{ð Þ}_ν¹m−ρ^{ð Þ}n¹νμ_νm

−E₁X^N

ν¼1

μnνρ^{ð Þ}_ν²m−ρ^{ð Þ}n²νμ_νm

: ð43Þ

Afirst step is to compute the diagonal terms ρnn(2). For m=n, Eq.(43)becomes

iℏ∂τρ^{ð Þ}nn² ¼−S_1nE₂−S_2nE₁; ð44Þ

where we have set S_jn¼∑

ν≠n μnνρ^{ð Þ}νn^j−ρ^{ð Þ}nν^jμ_νn

; ð45Þ

forj= 1, 2.S_1ncontains the same expression as the right-hand-side term of Eq.(32)above, and consequentlyS1n= 0. Using the expres- sion (34) ofρnν(2)forn≠ν,S2ncan be expanded as

S_2n¼F_1nE₂þF_2n∂τE₁þF_3nE²₁; ð46Þ where

F_1n¼μ_νnμnν

ℏ

ρ^{ð Þ}nn⁰−ρ^{ð Þ}_νν⁰

Ωνn

þ ρ^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

Ωνn

2 4

3

5¼0; ð47Þ

asS1n= 0 above, and

F2n¼∑

ν≠n

2iμnνμνn

ℏΩ²vn

ρ^{ð Þ}nn⁰−ρ^{ð Þ}vv⁰

; ð48Þ

3182 H. Triki et al. / Optics Communications 285 (2012) 3179–3186

F_3n¼∑^N

ν≠n

f

− μ_νn

ℏΩnν∑^N

l¼1μlνμnl

ρ^{ð Þ}νν⁰−ρ^{ð Þ}ll⁰

ℏΩlν − ρ^{ð Þ}ll⁰−ρ^{ð Þ}nn⁰

ℏΩnl

2 4

3

5

g

Then we get ρ^{ð Þ}nn² ¼iF_2n

2ℏ E²₁þiF_3n

ℏ ∫_−∞^τ E³₁dτ: ð50Þ

The off-diagonal terms ofρ⁽³⁾can also be computed. Sinceρ⁽⁰⁾is a diagonal matrix, andμand ρ⁽¹⁾are off-diagonal matrices, Eq.(43) yields, forn≠m,

ρ^{ð Þ}nm³ ¼ i

Ωnm∂_τρ^{ð Þ}nm² þE3μnm

ℏΩnm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

þ E2

ℏΩnm ∑

ν≠m;nμ_nνρ^{ð Þ}_ν¹m−ρ^{ð Þ}n¹νμ_νm þ E₁

ℏΩnm

ν≠∑m;n μnνρ^{ð Þ}_νm²−ρ^{ð Þ}_nν²μ_νm

þE₁μnm

ℏΩnm

ρ^{ð Þ}mm² −ρ^{ð Þ}nn²

:

ð51Þ Reporting Eqs.(21), (34) and (51)into Eq.(52), we compute the po- larization density as

P₃¼ε0χ^{ð Þ1}E₃þ2ε0χ^{ð Þ2}E₁E₂þε0χ^{ð Þ3}ð ÞE₁³

þA∂_τE₂þB∂²_τE₁þCE₁∂_τE₁þDE₁∫_−∞^τ E³₁dτ; ð52Þ in whichχ⁽¹⁾is given by Eq.(27), andχ⁽²⁾is given by Eq.(38). Here χ⁽³⁾=χR(3)+χS(3), with

χ^{ð Þ}R³ ¼ N ε0ℏ³∑

nmνl

½

−μmnμ_νmμ_lνμnlρ^{ð Þ}_νν⁰−ρ^{ð Þ}_ll⁰

ΩnmΩ_nνΩ_lν þμmnμ_νmμ_lνμnl ρ^{ð Þ}_ll⁰−ρ^{ð Þ}nn⁰

ΩnmΩ_nνΩnl

ð53Þ the sum being extended on all terms for which the denominators do not vanish (the‘regular’terms, motivating the subscript‘R’), and χ^{ð Þ}S³ ¼ N

ε0ℏ ∑

n;m;n≠m

μmnμnm

Ωnm

iF_2m 2ℏ −iF_2n

2ℏ

; ð54Þ

whereF2nis given by Eq.(48). The coefficientAis given by Eq.(40),

B¼−N ℏ ∑

nm

μnmμmn ρ^{ð Þ}mm⁰ −ρ^{ð Þ}nn⁰

Ω³nm

; ð55Þ

C¼−iN ℏ²∑

nmν

μmnμnνμ_νm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}νν⁰

ΩnmΩ²νm

−μmnμnνμ_νm ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

ΩnmΩ²nν

2 4

3 5 þ2iN

ℏ² ∑

nmν

μmnμ_νmμnν ρ^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

Ω²nmΩnν −μmnμ_νmμnνρ^{ð Þ}mm⁰ −ρ^{ð Þ}_νν⁰ Ω²nmΩνm

2 4

3 5;

ð56Þ

and D¼N

ℏ ∑

n;m;n≠m

μmnμnm

Ωnm

iF_3m 2ℏ −iF_3n

2ℏ

; ð57Þ

whereF3nis given by Eq.(49).

Due to the centrosymmetry, the coefficient Cis zero. Consider indeed the second order susceptibilityχxxx(2)(ωp+ωq;ωq,ωp) given by Eq.(38), forωp=ωq=ω, i.e.

χ^{ð Þ2}ð2ω;ω;ωÞ ¼ N

0ℏ²∑

mnν

½

− ρ^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

μmnμ_νmμnν

Ωnm−2ω

ð ÞðΩnν−ωÞ

ð58Þ

Taking the derivative with respect toω, and then the limit asωtends to zero yields

d

dωχ^{ð Þ2}ð2ω;ω;ωÞj

ω¼0¼ N 0ℏ²∑

mnν

½

− ρ^{ð Þ}_νν⁰−ρ^{ð Þ}nn⁰

μmnμ_νmμnν

ΩnmΩ²nν

þ 2N 0ℏ²∑

mnν

½

− ρ^{ð Þ}νν⁰−ρ^{ð Þ}nn⁰

μmnμνmμnν

Ω²nmΩnν

ð59Þ

It is seen that d

dωχ^{ð Þ2}ð2ω;ω;ωÞ

j

The medium being centrosymmetric,χ⁽²⁾(2ω;ω,ω)≡0, and hence C= 0. It has also been seen thatA= 0 andχ⁽²⁾= 0. The coefficientDis a nonlinear term of the 4th order. Since the order is even, it must be zero due to centrosymmetry. Notice that this feature cannot be justi- fied for the expression of the coefficient itself, as it was the case for theχ⁽²⁾coefficient. HenceD= 0 andfinally the polarization density reduces to

P₃¼ε0χ^{ð Þ1}E₃þε0χ^{ð Þ3}ð ÞE₁³þB∂²τE₁: ð61Þ Consider now the Maxwell–Helmholtz wave equation(4)at order ε⁵. It is

1

V²∂²τE3−2

V∂ζ∂τE1¼1

c²∂²τ E3þP₃ ε0

: ð62Þ

Reporting Eq.(51), and using the expression (26) of the velocityV, the terms involvingE₃vanish, and Eq.(62)reduces to

∂ζE1þγ∂τð ÞE1³þβ∂³τE1¼0; ð63Þ

which is exactly the mKdV equation.

The nonlinear coefficient is

γ¼ 1

2n₀cχ^{ð Þ3}; ð64Þ

withχ⁽³⁾given by Eqs.(53)–(54). An expression ofχ⁽³⁾can be found in Ref.[66](Eq. (3.7.10), p. 182), as

χ^{ð Þ}kjih³ ωpþωqþωr;ωr;ωq;ωp

¼ P_I χ~^{ð Þ}kjih³ ωpþωqþωr;ωr;ωq;ωp

; ð65Þ

in whichP_I represents an averaging over all permutations ofωr,ωq

andωp, and

χ~^{ð Þ}kjih³ ωpþωqþωr;ωr;ωq;ωp

¼ N ε_:ℏ³∑

nmνl

½

Ωνm−ωp−ωq

Ωlm−ωp

− ρ^{ð Þ}ll⁰−ρ^{ð Þ}_νν⁰

μ^kmnμ^jnνμⁱlmμ^h_νl

Ωnm−ωp−ωq−ωr

Ωνm−ωp−ωq

Ωνl−ωp

− ρ^{ð Þ}_νν⁰−ρ^{ð Þ}ll⁰

μ^kmnμ^j_νmμⁱnlμ^hlν

Ωnm−ωp−ωq−ωr

Ωnν−ωp−ωq

Ωlν−ωp

þ ρ^{ð Þ}ll⁰−ρ^{ð Þ}nn⁰

μ^kmnμ^j_νmμⁱlνμ^hnl

Ωnm−ωp−ωq−ωr

Ωnν−ωp−ωq

Ωnl−ωp

ð66Þ

in which we set the relaxation ratesγnmto zero.

The presentχ⁽³⁾should coincide withχxxxx(3)

(0; 0, 0, 0), however several terms in the sum(66)are singular as (ωr,ωq,ωp)→(0, 0, 0).

It is straightforwardly seen that the sum of the regular terms exactly coincides withχR(3)

as given by Eq.(53). Taking into account the fact thatμis off-diagonal, the singular terms in the sum(66)are:

• The 1st term forν=m:

Amnl¼ −K_mnl

Ωnm−ωp−ωq−ωr

ωpþωq

Ωlm−ωp

; ð67Þ

with

K_mnl¼ ρ^{ð Þ}mm⁰ −ρ^{ð Þ}ll⁰

μmnμnmμmlμlm: ð68Þ

• The 2nd term forν=m:

B_mnl¼ K_mnl

Ωnm−ωp−ωq−ωr

ωpþωq

Ωlmþωp

: ð69Þ

• The 3rd term forν=n:

C_mnl¼ Knml

Ωnm−ωp−ωq−ωr

ωpþωq

Ωln−ωp

: ð70Þ

• The 4th term forν=n:

D_mnl¼ −K_nml

Ωnm−ωp−ωq−ωr

ωpþωq

Ωlnþωp

: ð71Þ

Let us set (ωr,ωq,ωp) = (−ω+δωr,ω+δωq,ω+δωp), in whichδωr, δωqandδωpare infinitesimal quantities. After some calculations, the infinitesimal quantities simplify and we get

PIðAmnlþBmnlÞ ¼ −K_mnl 3ðΩnm−ωÞ

1 Ωlmþω

ð Þ²þ 1

Ωlm−ω

ð Þ²þ 1

Ω²lm−ω²

" #

: ð72Þ

It is comparable to the expression ofχ⁽³⁾found in Ref.[67], but with a slight sign change. Asω→0, it yields

P_IðA_mnlþB_mnlÞ ¼− ρ^{ð Þ}mm⁰ −ρ^{ð Þ}ll⁰

μmnμnmμmlμlm

ΩnmΩ²lm

: ð73Þ

P_IðC_mnlþD_mnlÞyields the same expression as Eq.(73)with permuted nandm. On the other hand, Eqs.(54) and (48)yield

χ^{ð Þ}S³ ¼−2N ε0ℏ³ ∑

n;m;ν;

n≠m;ν≠m

μmnμnmμ_νmμmν

ΩnmΩ²νm ρ^{ð Þ}mm⁰ −ρ^{ð Þ}_νν⁰

; ð74Þ

and hence χ^{ð Þ}S³ ¼ N

ε0ℏ³ ∑ n;m;ν;

n≠m;ν≠m

P_IðA_mnlþB_mnlþC_mnlþD_mnlÞ: ð75Þ

As a conclusion, taking into account both regular and singular terms, we see thatχ⁽³⁾exactly coincides withχxxxx(3)(0; 0, 0, 0).

The expression (Eq.(64)) of the nonlinear coefficient slightly dif- fers from the analogous expression found in Refs.[34,39]. A factor 4π is due to the system of units (i.e., CGS units versus SI units). In fact, the value ofχ⁽³⁾given by Eq.(78)in Ref.[67]is smaller by a factor of 1/3. The expression of the nonlinear coefficient in[34]was based on this value ofχ⁽³⁾, and this initial error resulted in the erroneous coefficient of 6 in Eq.(25)in[34]instead of 2 in Eq.(64)above.

The dispersion coefficient is β¼ 1

2ε0n₀cB; ð76Þ

whereBis given by Eq.(55). Taking twice the derivative of Eq.(28) with respect toωp, then setting ωp= 0, and comparing the result with Eq.(55)shows that

B¼−ε0

2

d²χ^{ð Þ}xx¹ð Þω

dω²

j

The wave vector isk(ω) =n0(ω)ω/c, and its third derivative for ω= 0 is

d³k

dω³

j

^′′

where we have set

n

′′

dω²

j

Taking twice the derivative ofn0¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þχ^{ð Þ1}

p with respect toωyields

d²n₀ dω²¼ 1

2n₀ d²χ^{ð Þ1}

dω² − 1 2n²₀

dχ^{ð Þ1} dω

!2

; ð80Þ

however,dχ⁽¹⁾/dωforω= 0 is proportional to the coefficientAgiven by Eq.(40), hence is zero. Finally, it is found that the dispersion coef- ficientβcan be written as

β¼−1 6

d³k

dω³j^ω¼0; ð81Þ

3184 H. Triki et al. / Optics Communications 285 (2012) 3179–3186

which exactly coincides with the expression found in Ref.[34]and generalizes the latter. The equivalent expression

β¼−n

′′

2c ; ð82Þ

evidences the fact thatβis not a third-order dispersion as it could been believed atfirst glance, but accounts in the present approxima- tion for the group velocity dispersion. It also may account for higher order dispersion terms, see Ref.[68].

3. The breather solution of the modified-Korteweg–de Vries equation

The mKdV equation(63)is completely integrable by means of the inverse scattering transform [69]. The N-soliton solution has been given by Hirota[70]. It is more convenient to write the mKdV equation (63)into the dimensionless form

∂Zuþ2∂Tu³þσ∂³Tu¼0; ð83Þ whereσ= ±1,uis a dimensionless electricfield, andZandTdimen- sionless space and time variables defined relative to the laboratory vari- ables as

u¼ E

E₀ ; Z¼z

L ; T¼t−z=V

t_w : ð84Þ

The reference time is thus chosen to be the pulse lengtht_w(in phys- ical units). Recall that the atomic resonance frequenciesΩnmhave been chosen above as zero order quantities in the perturbative scheme, whiletwis assumed to be formally large, of order 1/ε, with respect to the zero order times 1/Ωnm. The characteristic electric field and propagation distance are

E₀¼ 1 t_w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

−2σn₀n

′′

χ^{ð Þ3} vu

ut ; ð85Þ

L¼ 2ct³_w

−σn

′′

0

: ð86Þ

Ifχ⁽³⁾andn′′0have opposite sign, which is typically forχ⁽³⁾>0 and anomalous dispersion, thenσ=+1, and the mKdV equation(63)is a fo- cusing one. Else, typically forχ⁽³⁾>0 and normal dispersion,σ=−1,

Eq.(63)is a defocusing one and describes nonlinear dispersion[39]. In the focusing case, the mKdV equation admits real single-soliton solutions, andN-soliton and breather solutions. Integrating the mKdV equation(83)with respect toT, under the assumption thatu, i.e. the elec- tricfield, and its derivatives, vanishes at infinity, it is seen that the conser- vation law

∂Z∫_−∞^þ∞udT¼0 ð87Þ is satisfied. This is the expression in our situation of the general law of the conservation of the electric pulse area, as derived in[56,58]. Due to the Galilean transformation and the scaling (9-10), it is seen from Maxwell equations that the magnetic field is ^→B¼uE0=V e^→y; and that

∫^þ∞_−∞B_αdz∝∫^þ∞_−∞udT;hence the conservation law of the magnetic pulse area is also satisfied by the mKdV Eq.(63), since it does not differ from (87).

The two-soliton solution of the mKdV equation is[70]

u¼ e^η1þe^η2þ ^p_p^1−p2

1þp2

2 e^η1 4p²₁ þ_4p^eη22

2

e^η1þη2 1þ^e_4p^2η12

1

þ _p ²

1þp2

ð Þ² e^η1þη2þ^e_4p^2η22 2

þ ^p_p^1−p2

1þp2

4 e^2η1þ2η2

16p²₁p²₂

; ð88Þ

with

ηj¼pjT−p³jZ−γj; ð89Þ for j= 1 and 2. The parametersp1,p2,γ1, andγ2 are arbitrary. If p2=p1∗, where∗denotes the complex conjugate, andγ2=γ1∗, the ex- plicit solution (88) is an oscillating localized solution, calledbreather soliton, which actually adequately describes a FCP soliton.

An example of FCP soliton propagation is shown onFig. 1. The mKdV equation(63)is solved using the exponential time differencing 4th order Runge–Kutta scheme[71], for an input data (blue dotted line) of the form

u¼ Ae^−T2=w2sinðω0TþϕÞ; ð90Þ with the parametersw= 2.5,A ¼1:5,ω0= 0.6π, andϕ=π. The com- putation was run until Z≃80. The pulse evolves with very few changes in shape and width, apart from periodic oscillations. We chose a prop- agation distance (Z= 79.72) at which the carrier-envelope phase of thefinal FCP is the same as the initial one, moved to the initial posi- tion and plotted it inFig. 1(green thick solid line) for comparison.

Afit with the breather (88) is also shown (dashed red line): it is very close to the numerical result. The values of the parameters which yield the bestfit arep1= 0.875 + 1.7i,γ1= 0.24i(and a small shift in position).

Fig. 1.Propagation of a FCP according to the mKdV equation. Blue dotted line: initial input with Gaussian envelope. Green thick solid line: the FCP soliton observed after some propagation distance (Z= 79.72). Dashed red line:fit of the latter by the analytic breather.

T

Z -80

-60 -40 -20 0

0 2 4 6 8 10

Fig. 2.Propagation of a FCP according to the mKdV equation:uagainstTandZcomputed with the analytic formula (Eq.(88)); parameters are the same as thefit inFig. 1.