Half-cycle optical soliton in quadratic nonlinear media

Half-cycle optical soliton in quadratic nonlinear media

Hervé Leblond

Laboratoire POMA, FRE 2988, Université d’Angers, 2 Boulevard Lavoisier, 49000 Angers, France 共Received 9 November 2007; revised manuscript received 30 April 2008; published 8 July 2008兲 We show that a few-cycle pulse launched in a quadratic medium may result in a half-cycle soliton in the form of a single hump, with no oscillating tail. The analysis involves the derivation of a Korteweg-de Vries 共KdV兲equation from both a classical and a quantum mechanical simple model of matter-radiation interaction.

The sign of the electric field in the half-cycle KdV soliton is fully determined by the properties of the material, which definitely breaks the phase invariance.

DOI:10.1103/PhysRevA.78.013807 PACS number共s兲: 42.65.Tg, 42.65.Re

I. INTRODUCTION

Ultrafast optics has led recently to the production of pulses with durations down to a few optical cycles 共see the reviews 关1,2兴兲. The theoretical study of few-cycle pulses 共FCPs兲propagation is the matter of extensive research: cor- rections to the paraxial approximation 关1–3兴, the so-called short-pulse equation 共SPE兲 关4–6兴. Self-compression of the FCP and FCP soliton formation is a very important issue关7兴.

The propagation of FCP solitons in Kerr media can be de- scribed in a way completely free from the slowly varying envelope approximation, using completely integrable models as the modified Korteweg-de Vries 共mKdV兲, sine-Gordon 共sG兲or mKdV-sG equations关8–10兴.

However, a FCP still remains a wave packet, which pre- sents a large number of oscillations, even if only one or two of them have an important amplitude. A challenge is to pro- duce an optical pulse with a single oscillation, comparable in shape with the solitary waves which are well known in hy- drodynamics; in other words, to clean up the FCP by sup- pressing the oscillations around the main hump.

A second point is the phase invariance, which may appear as a fundamental symmetry of the optics. The FCP in some sense loses the phase invariance, and the importance of the carrier-envelope共CE兲phase has been emphasized关1兴. How- ever, although ways of stabilizing the CE phase have been proposed 关10,11兴, it remains a very difficult task. Further, even if a zero CE phase is realized, the existing setups cannot distinguish it from a phase␲: the polarity of the electric field remains random, and its mean value on a large number of FCPs remains zero.

Up to now, the study of FCP solitons is restricted to the case of a cubic 共Kerr兲 nonlinearity. For longer pulse dura- tions, when the slowly varying envelope approximation is valid, the study of optical solitons in quadratic nonlinear me- dia has shown many remarkable properties with respect to the cubic Kerr case 关12,13兴. First of all, the quadratic non- linearity, in principle, allows one to observe nonlinear effects with much lower input power than a cubic one. However, such an effect and the formation of envelope solitons in qua- dratic media, involve at least two components, a fundamental frequency and a second harmonic 关14兴. The interaction be- tween the two components is efficient if some phase- matching condition is satisfied; experimentally it can be achieved by different ways, e.g., by temperature adjustment

关15兴. It has been shown that a result of this interaction is the suppression of the collapse, which occurs in two-dimensions in a Kerr medium, leading to the formation of two- dimensional solitons 关16兴, and to the stabilization of many types of spatial and spatiotemporal solitons 关17兴. Far from the phase-matching, the so-called cascading effect leads to an effective cubic nonlinearity 关18兴. However, self- rectification and electro-optic effect remain, which are able to stop the collapse关19兴, and also can be used to control the pulse by means of adequately matched microwaves关20兴. The question, whether and how these properties might generalize to FCPs, naturally arises, and in a first stage, it is worth investigating the possibility to build a theory for FCP soliton propagation in quadratic media.

We show in the present paper that launching a FCP into a medium with an electronic quadratic nonlinearity allows one to produce half-cycle pulses, with no oscillating tail, a per- fectly zero CE phase, and a definite polarity of the field. The process breaks the phase invariance, even statistically, and yields an optical electric field with definite polarity, and non- zero mean value.

Our approach is based on the soliton theory. Indeed, soli- tons, i.e., fundamental solutions of integrable equations as Korteweg-de Vries 共KdV兲, mKdV, or sG involve solitary waves, i.e., single-oscillation, half-cycle pulses. Each pulse of this kind has a determined sign or polarity. Hence, due to phase invariance, it is believed that such pulses cannot be produced by an optical system, at least in the present state of knowledge. Breather solutions of mKdV, sG, and mKdV-sG have been given by the mathematical theory; they do not involve such a symmetry breaking. It has been shown that such objects were good candidates for the description of FCPs关9,10兴. We prove in the present paper that FCP soliton propagation in a quadratic medium can be described by the KdV equation itself, which is quadratic, and not by the cubic mKdV equation. Note that all solitons of KdV have the same sign, while mKdV admits both positive and negative soli- tons. Further KdV does not present breather solutions, and it will be shown that the KdV soliton itself, a half-cycle pulse, will arise from the propagation of an input FCP with mean value zero. The symmetry breaking in the wave, related to the existence or nonexistence of a breather, originates in the noncentrosymmetry of the quadratic medium.

II. KORTEWEG-DE VRIES MODEL FOR QUADRATIC FCP SOLITONS

A. Classical approach 1. Derivation of KdV

We consider first a classical model of an elastically bounded electron oscillating along the polarization direction xof the wave. The latter propagates in a directionzperpen- dicular tox, in such a way that the transverse variations can be neglected. The evolution of the position x of an atomic electron is described by an anharmonic oscillator 共damping is neglected兲

d²x

dt² +⍀²x+ax²= −e

m E, 共1兲

where ⍀ is the resonance frequency, −e is the electron charge, mis its mass, and E is the wave electric field com- ponent alongx. The evolution ofEis described by the Max- well equations, which reduce to

⳵_z²_E= ¹

c²⳵_t²_{共E+ 4}␲P兲, 共2兲 whereP= −Nexis the polarization density,Nbeing the den- sity of atoms.

Transparency implies that the characteristic frequency␻_w of the considered radiation 共in the optical range兲 strongly differs from the resonance frequency⍀of the atoms; hence it can be much higher or much lower. We consider here the latter case, i.e., we assume that␻_wis much smaller than ⍀.

This motivates the introduction of the slow variables

␶=␧

冉

冊

␧ being a small parameter. The delayed time ␶ involves propagation at some speedVto be determined. It is assumed to vary slowly in time, according to the assumption␻_wⰆ ⍀.

The pulse shape described by the variable ␶ is expected to evolve slowly in time, the corresponding scale being that of variable␨.

A weak amplitude assumption is needed in order that the nonlinear effects arise at the same propagation distance scale as the dispersion does. Hence we expand

E=␧²E₂+␧³E₃+␧⁴E₄+ ¯, 共4兲 as in the standard KdV-type expansions关21兴.Pis expanded in the same way. The leading order, i.e., Eq. 共1兲at order␧² and Eq.共2兲at order␧⁴, gives

P₂= Ne²

m⍀²E₂, 共5兲

and

V=c

冉

冊

Then it is seen thatE₃can be taken as zero. At next order关␧⁴ in Eq.共1兲兴, we get

P₄= Ne²

m⍀²E₄− Ne² m⍀⁴⳵

t

2E₂+ aNe³

m²⍀⁶共E₂兲². 共7兲 Reporting into Eq. 共2兲 at order ␧⁶, the terms in E₄ cancel using Eq.共6兲, and we obtain the KdV equation

⳵_␨E2=A⳵

␶

3E₂+B⳵_␶共E2兲²_{, 共8兲} with the coefficients

A=4␲VNe²

2mc²⍀⁴, B=− 4␲VaNe³

2m²c²⍀⁶ , 共9兲 for the dispersion and nonlinearity, respectively. We empha- size the fact that Eq.共8兲is satisfied by the electric field itself, and not by some envelope: there is no carrier. It is important to notice that, since Eq. 共8兲is not an envelope equation, the third order time derivative does not represent a third order dispersion. As shown in Ref. 关22兴, it includes at least the second and third order dispersions.

2. Coefficients of KdV

The dispersion relation of model共1-2兲is well known and easily computed. The refractive index is

n=

冉

⍀²−␻²

冊

We check that V=c/n. Defining the wave vector by k

=␻n/c, we check by straightforward computation that the dispersion coefficient is

A=1

6

冏

冏

冏

冏

Expression 共11兲 is exactly the same as in the case of the mKdV model accounting for FCP propagation in a Kerr me- dium 关9兴. Notice that, due to the long-wave approximation, the dispersion coefficientAonly involves the second deriva- tive of n, as the second order dispersiond²k/d␻²_共␻_兲 in the case of envelope solitons. The nonlinear susceptibility corre- sponding to model共1-2兲reads as关23兴

␹^共2兲_共2␻,␻,␻_兲= aNe³ m²

1

共⍀²−␻²兲²共⍀²− 4␻²兲 共12兲 共recall that the damping is neglected兲. Hence

B=− 2␲

nc 兩␹^共2兲_共2␻,␻,␻_兲兩␻=0. 共13兲 It is analogous to the expression obtained for the mKdV model关9兴, which involves␹^共3兲共␻;␻,␻, −␻兲, and is valid for a cubic Kerr nonlinearity.

B. Quantum mechanical approach 1. Second derivation of KdV

We consider a set of two-level atoms with the Hamil- tonian

H₀=ប

冉

冊

where⍀=␻_b−␻_a⬎0 is the frequency of the transition. The evolution of the electric fieldE共we restrict to one field com- ponent for the sake of simplicity兲 is described by the wave equation共2兲. The light propagation is coupled with the ma- terial by means of a dipolar electric momentum ␮ directed along the same direction xas the electric field, according to H=H₀−␮E, 共15兲 and the polarization density

P=NTr共␳␮_{兲 共16兲} along thexdirection,Nbeing the volume density of atoms, and␳ the matrix density. The latter obeys

iប⳵_t␳=关H,␳_兴+R, 共17兲 whereRis a phenomenological relaxation term. In the case of cubic nonlinearity, it has been shown that it was negligible 关9兴, and it will be neglected here. In the absence of perma- nent dipolar momentum,

␮=

冉

冊

is off-diagonal, and the quadratic nonlinearity is zero.

The quadratic nonlinearity can be phenomenologically ac- counted for as follows: it corresponds to a deformation of the electronic cloud induced by the field E, hence to a depen- dency of the energy of the excited levelbwith respect toE:

a Stark effect. The nonlinearity is quadratic if this depen- dency is linear, as

␻_b^→␻_b−␣E. 共19兲 This can be included phenomenologically in the Maxwell- Bloch equations by replacing the free HamiltonianH₀ with

H₀−␣E

冉

冊

The equations are exactly the same if the dipolar momentum

␮is replaced with

␮=

冉

冊

i.e., Eq.共20兲is equivalent to the assumption that the excited state bhas a nonzero permanent dipolar momentum ␣.

To simplify computations, we use the same rescaling as in Ref. 关9兴, so that the constantsc,N,ប, and 4␲ are replaced with 1. Then Tr共␳_兲 is not 1 anymore but ␯= 4␲Nបc. We denote the elements of a matrixM by indicesa,b,t,u accord- ing to

M=

冉

冊

We use the same slow variables and power expansion as for the classical model, with

␳₀=

冉

冊

Apart from this term the expansion begins at order␧². Equation共17兲at order ␧² gives

␳2,t=␯␮

⍀E₂, 共24兲

␳_2,a,b remaining free. We deduce P₂=2␯_兩␮_兩²

⍀ E₂. 共25兲 Reporting Eq.共25兲into Eq.共2兲at order␧⁴, we get the value of the velocity

V=

冉

冊

The expression ofVis exactly the same as in the mKdV case 关9兴, since the linear part of the model is exactly the same.

At order␧³, Eq.共17兲shows that␳_2,a=␳_2,b= 0 and

␳_3,t=␯␮

⍀E₃− i␯␮

⍀²⳵_{␶E2, 共27兲} from which we deduce

P₃=2␯_兩␮_兩²

⍀ E₃. 共28兲 Hence the wave equation 共2兲at order ␧⁵ does not give any further information.

Equation 共17兲 at order ␧³ gives useless expressions in- volving␳3,a,b and

␳_4,t=␯␮

⍀E₄−i␯␮

⍀²⳵_␶E3−␯␮

⍀³⳵_␶²_E2+␣␯␮

⍀² 共E₂兲², 共29兲 from which we deduce

P₄= 2␯兩␮_兩²

⍀ E₄−2␯兩␮_兩²

⍀³ ⳵

␶

2E₂+ 2␣␯_兩␮_兩²

⍀² 共E₂兲². 共30兲 Reporting Eq.共30兲into the wave equation共2兲at order␧⁶we get, after cancellation of the terms involving E₄, and one integration with respect to ␶ 共the field and derivatives are assumed to vanish at infinity兲,

⳵_␨E2=^V␯_兩␮_兩²

⍀³ ⳵_t³_E2−␣V␯_兩␮_兩²

⍀² ⳵_␶共E2兲²_{. 共31兲} Equation共31兲is exactly the KdV equation共8兲, with the dis- persion and nonlinear coefficients共in physical units兲

A= 4␲N兩␮_兩²

ncប⍀³ , B=− 4␲N␣_兩␮_兩²

ncប²⍀² , 共32兲 respectively, the refractive index being

n=

冑

The dispersion relation is the same as in the Kerr case 关9兴, and the same expression of A still holds; it coincides with Eq. 共11兲.

2. Coefficients of KdV in the quantum mechanical approach The dispersion relation is the same as in the Kerr case关9兴, and the same expression of A still holds; it coincides with Eq. 共11兲.

The second order susceptibility is computed as follows:

we report into Eqs.共15兲–共17兲the following harmonic electric field excitation,

E=␧E共e^i␻t+e^−i␻t兲, 共34兲 and expand the density matrix as

␳=␳₀+␧␳_1共t兲+␧²␳_2共t兲+ ¯, 共35兲 with the same␳₀as above. We obtain at first order␳_1,a,b= 0 and

␳_1,t=C₊e^i␻t+C₋e^−i␻t, 共36兲 with

C_⫾= ␮^E

ប共⍀ ⫿␻兲. 共37兲 At order␧², Eq.共17兲is easily soved to yield␳_2,a,b= 0 and

␳_2,t=K₊e^2i␻t+K₋e^−2i␻t+K₀, 共38兲 with

K_⫾= ␣␮^E2

ប²共⍀ ⫿2␻_兲共⍀ ⫿␻_兲^{. 共39兲} The second order susceptibility is related to the polarization, though

P=␧²␹^共2兲共2␻;␻,␻兲E²e^2i␻t+¯ 共40兲

=␧²NTr共K₊␮^ⴱ+K₋^ⴱ␮_兲e^2i␻t+ ¯, 共41兲 and hence

␹^共2兲_共2␻;␻,␻_兲=N␣_兩␮_兩^2E2

ប²

冉

+ 1

共⍀− 2␻_兲共⍀−␻_兲

冊

The nonlinear coefficientBof the KdV equation expresses as B=− 2␲

nc 兩␹^共2兲_共2␻;␻,␻_兲兩␻=0. 共43兲 We obtain exactly the same expression as for the classical model.

C. Generalization

It can be reasonably conjectured that the KdV model with the expression of the coefficients in terms of the linear re-

fractive index and second order susceptibilities is valid in more general and realistic situations than the simple models considered in this paper.

The latter assumes a single resonance line, at some fre- quency⍀. The long-wave approximation used to derive KdV assumes that the typical frequency␻, which is comparable to the inverse␻_⯝1/Tof the durationTof the solitary wave, is small with respect to ⍀. With ␻ belonging to the visible range, it means that the resonance line of the material be- longs to the uv.

Expression 共11兲 of the dispersion coefficient is expected to generalize as far as␻is far from any resonance line of the material. The expression共43兲 of the nonlinear coefficient is valid, in principle, if only one transition in the uv is involved by the quadratic nonlinearity. If several transitions are in- volved, all nonlinear terms will have the same form as in Eq.

共8兲, and the coefficients will combine together to yield Eq.

共43兲 again. Regarding the transition lines, which are far be- low ␻, it has been seen in the cubic case 关9兴 that they re- quired a much higher power input than the uv transitions to produce an observable nonlinear effect. Hence they are ex- pected to be negligible, at least in certain materials.

The values of the dispersion coefficient and␹^共2兲are taken at␻= 0. Due to the perturbative approach, this means in fact that they are taken at a value of␻well below the resonance line ⍀, as is assumed to be the inverse␻= 1/T of the pulse duration, typically in the visible range.

Notice that we have here a pure quadratic nonlinearity for a single wave, and that no effective third order nonlinearity due to cascaded second order ones is involved. This makes a sharp contrast with the nonlinear propagation of parametric solitons within the slowly varying envelope approximation.

In particular, no phase matching is required.

III. HALF-CYCLE QUADRATIC FCP SOLITON A. KdV soliton

The fundamental soliton of KdV is expressed in the present case as

E₂= k₃ncp²

− 2␲␹^共2兲sech

2

冉

冊

where we have set

k₃=

冏

冏

and

␹^共2兲=兩␹^共2兲_共2␻;␻,␻_兲兩␻=0. 共46兲 It is shown in Fig.1共c兲. It differs from the soliton of mKdV 关Fig. 1共a兲兴, which is a sech and not a sech². It can be ob- tained by direct integration关24兴, by the Hirota method关25兴, or by the inverse scattering transform共IST兲 关26,27兴. The sta- bility and robustness of the fundamental soliton 共44兲 is en- sured from the IST for any positive p. The corresponding solutions are proven to be the fundamental modes of the KdV equation, in the sense that any input decomposes into a combination of a finite number of solitons and Fourier-type

modes called “radiation,” which evolve separately, conserv- ing their characteristics all the propagation long.

For the sake of convenience, below we use the standard dimensionless form of the KdV equation共8兲,

u_Z+ 6uu_␶+u_␶␶␶= 0, 共47兲 which is obtained by means of he linear transform Z= −A␨, u=共B/6A兲E₂. Direct numerical simulation illustrates and confirms the mathematical result: for KdV, a FCP-type input decays into solitons, with a definite sign共Figs.2and3兲. For a short enough FCP, the number of solitons is only 1 or 2. In the latter case, the two solitons have different velocities, and are conserved when they interact 共Fig. 4兲. The number of emitted solitons depends on the CE phase of the initial FCP at the entrance of the quadratic medium共see Fig.5兲.

However, in every case 共i兲 single-oscillation 共half-cycle兲 pulses are produced and共ii兲the sign ofu, i.e., the sign of the electric field, is always the same. Indeed, the relative sign of u and E₂ is that of B/A and cannot be modified by a real linear transform. For the two-level model, we have B/A

= −␣⍀/ប, i.e., it is proportional to the dipolar momentum of the excited state. In the classical model, we can see that B/A= −2ae/共m⍀²兲, which has the same physical origin. In the general caseB/Ainvolves both the quadratic nonlinearity and the dispersion. Hence the polarity of the optical electric field is fully determined by the direction of the polarization of the atoms, and the dispersion.

B. mKdV equation possesses breathers while KdV does not We have shown that the fundamental FCP soliton in a quadratic medium is a sech² solitary wave, and not a wave packet. The latter are described in the frame of the mKdV, sG, and mKdV-sG models by breather solutions. Although the properties of breathers are well known, we believe that it is worth recalling them for the readers that are not specialists of soliton theory. Expression 共44兲, due to algebraic proper-

-5 -2.5 2.5 5 7.5 10

50

-50 -100

-150 -10

-5 5

10 0.5

0.4 0.3 0.2 0.1

-10 -5 5 10

-1.5 -1 -0.5 0.5 1 1.5

-4 -2 2 4

0.5 1 1.5

a) 2

b)

c)

d)

u

u u

τ

τ

τ τ

u

FIG. 1.共Color online兲 共a兲Soli- ton solution to mKdV. 共b兲 Breather solution to mKdV: the FCP soliton in Kerr media. 共c兲 Soliton solution to KdV: the FCP soliton in quadratic media.共d兲The breatherlike solution of KdV: can- not describe a FCP since it is singular.

0

0.5 1

−80

−60

−20 −40 20 0

−4

−2 0 2 4

u

τ

Z

FIG. 2. 共Color online兲 Evolution of an initial FCP into KdV solitons and dispersive waves.

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20

-4 -3 -2 -1 0 1 2 3 4

τ u

FIG. 3. 共Color online兲Evolution of an initial FCP into a KdV soliton and dispersive waves. The input共dashed blue line兲and out- put 共solid red line兲 wave profiles correspond to Fig.2. The lines represent the electric fieldE₂itself, and not an envelope.

ties, solves the KdV equation 共8兲for any complex value of the soliton parameterp. TheN-soliton solution has the same property, and an adequate choice of the parameters allows one to build a breather solution from the two-soliton solu- tion. However, neither realness, nor regularity, nor stability, nor robustness, is ensured this way. The breather solution to the mKdV equation关Fig.1共b兲兴is regular and describes FCP solitons 关8–10兴. KdV also possesses breatherlike solutions, constructed by the same procedure, but these solutions are singular关28兴 关Fig.1共d兲兴.

Recall that the equivalence between KdV and mKdV, due to the Miura transform 关29兴, holds if complex solutions are considered, but not if we consider real fields. The two mKdV equations

u_t+ 6␩u²u_x+u_xxx= 0, 共48兲 with␩equal to either +1 or −1, cannot be reduced one to the other if u is real, and the Miura transform is real for␩= −1 only. The mKdV equation valid for FCP soliton propagation in a two-level medium 关9兴, or more generally in a medium with focusing cubic Kerr nonlinearity, is the one with ␩

= + 1, which admits real regular breathers.

It has been proven 关30兴 that in the case of mKdV, a breather was preferentially formed from a “symmetric” ini- tial data, while a soliton arose if the input broke the symme- try. Hence FCP solitons 共of wave-packet type兲 form in a cubic focusing medium, and half-cycle single-oscillation solitons are very rare. Further, the mKdV soliton may have both signs with equal probabilities, since Eq.共48兲is invariant through u→−u. Hence a train of half-cycle solitons in a cubic medium would contain as much positive solitons as negative ones, and resemble an oscillating train.

As a consequence of the nonexistence of a regular breather to KdV, in a medium with quadratic optical nonlin-

earity, the fundamental soliton systematically arises, and al- ways has the same sign; that is, half-cycle single-oscillation solitons will be produced, and they always have the same polarity. This is closely related to the noncentrosymmetry required for quadratic nonlinearity.

Notice that the mKdV equation with ␩= −1 can describe FCP propagation in a medium with defocusing cubic Kerr nonlinearity. Since it does not admit any regular breather, the general input will decay into a train of solitons, i.e., half- cycle pulses. However, the solitons will have random sign, and the train remains close to an oscillating wave.

IV. CONCLUSION

Half-cycle optical pulses, which are KdV solitons, form when a FCP is launched into a quadratic nonlinear medium.

They consist of a single hump, without any satellite oscilla- tion. They always have an absolute phase zero, in the sense that the field polarity is completely determined by the prop- erties of the medium. As a consequence, the mean value of the optical electric field is not zero. This latter property might find applications in the frame of optical poling.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−10

−5 0 5 10 15

Z

τ

FIG. 4.共Color online兲An input FCP evolves into two half-cycle solitons and radiation. The two half-cycle solitons have different velocities, higher than that of the radiation. They interact keeping their characteristics. The figure is drawn for an input CE phase equal to 0.8␲.

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 -10

-8 -6 -4 -2 0 2 4 6 8 10

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 -10

-8 -6 -4 -2 0 2 4 6 8 10

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 -10

-8 -6 -4 -2 0 2 4 6 8 10

a)

b)

c)

τ u

u u

τ τ

FIG. 5. 共Color online兲The input 共dashed blue line兲and output 共solid red line兲 wave profiles for an input carrier-envelope phase equal to 0 共a兲, ␲/2 共b兲, and ␲ 共c兲. It is seen that the number of emitted solitons depends on the CE phase of the input FCP.

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