Attosecond pulse generation from aligned molecules---dynamics and propagation in H2+

T h e o p e n – a c c e s s j o u r n a l f o r p h y s i c s

New Journal of Physics

Attosecond pulse generation from alignedmolecules—dynamics and propagation in H +

2

E Lorin 1,3, S Chelkowski 2 and A D Bandrauk 1,2

1 Centre de Recherches Mathématiques, Montréal, QC,H3T 1J4, Canada2 Laboratoire de chimie théorique Faculté des Sciences,Université de Sherbrooke, QC, J1K 2R1, Canada.E-mail:[email protected]

New Journal of Physics 10 (2008) 025033 (21pp)Received 17 September 2007Published 29 February 2008Online athttp://www.njp.org/doi:10.1088/1367-2630/10/2/025033

Abstract. The dynamics and propagation effects in attosecond (asec) pulsegeneration from high-order harmonic generation (HHG) of aligned one-dimensional (1D) H+2 molecules are investigated from numerical solutions offully coupled Maxwell and time-dependent Schrödinger equations (Maxwell-TDSEs), in the highly nonlinear nonperturbative regime of laser–moleculeinteraction. Density, laser-phase and propagation length effects are studied onthe total electric field and nonlinear polarization from the Maxwell-TDSE forintense few cycle (800 nm) laser pulses interacting with a 1D H+

2 gas. We showhow single and double asec pulses can be generated and propagated as a functionof the phase of individual harmonics created by ultrashort intense laser pulsesin aligned H+

2 molecules. We find furthermore extension of maximum HHGplateaux with increasing gas pressure.

3 Present address: Faculty of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North,Oshawa, ON, L1H 7Y4, Canada.

New Journal of Physics 10 (2008) 0250331367-2630/08/025033+21$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

mailto:[email protected]

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Contents

1. Introduction 22. Maxwell-TDSE model 33. Theoretical study of harmonic phase effects on asec pulse generation 44. HHG and asec pulse propagation 9

4.1. Harmonic dynamics and asec pulse propagation. . . . . . . . . . . . . . . . . 94.2. CEP dependence of HHG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5. Conclusion 17Acknowledgments 18Appendix 18References 20

1. Introduction

High-order harmonic generation (HHG) in atomic gases by high intensity ultrashort laser pulsesis the main method for producing coherent extreme ultraviolet and attosecond (asec) pulses [1].This is based on a universal model of electron-recollision with a maximum harmonic energy,

Nhω = Ip + 3.17Up, (1)

whereIp is the ionization potential of the atom,Up= e2E2/4mω2 is the ponderomotive energyof the electron in an oscillatory fieldE(t) of maximum intensityI = eE2/8π and frequencyω [2]–[7]. Molecules offer an interesting new medium as both ionization and recombinationsteps are dependent on the particular symmetry of the highest occupied molecular orbital(HOMO) and orientation [5]–[8]. Furthermore, at large distances, stretched or dissociatedmolecules offer the possibility of obtaining harmonics well beyond the 3Up cut-off law (1)[5, 9, 10]. The recombination model allows a full tomographic reconstruction of the HOMOto be performed (but a high degree of spatial alignment of the molecules is required [11]–[13]). The mathematical steps in structural retrieval from HHG are based upon the strong fieldapproximation (SFA), a single active electron model and a three-step process [2, 3]: (i) tunnelingionization with zero initial electron velocity; (ii) acceleration in the laser fieldE(t); and (iii)recombination back into the bound electronic state. This simple three-step model can be shownto always produce a maximum return energy given by (1) even with nonzero initial velocityupon ionization [4]. Nevertheless certain important issues remain, such as the influence of theintense laser field upon the bound electronic states upon recombination [14], depletion of theinitial ground state [15], the influence of the Coulomb potential upon the continuum electronstates [16], all effects neglected in SFA. Finally, one needs to consider macroscopic propagationeffects as these lead to interesting new phenomena such as filamentation [17, 18] with theconclusion that ionization dynamics can strongly influence the synthesis of isolated attosecondpulses [19]. In this paper, we address the problem of asec pulse generation and propagationby HHG in an aligned molecular medium. We focus our attention on the single electronH+

2 system which nevertheless involves coupled electron–nuclear motion beyond the Born–Oppenheimer approximation [20]. For this H+

2 system a previous time-dependent Schrödingerequation (TDSE) simulation with exact non-Born–Oppenheimer solutions leads to enhancedionization and HHG in the presence of an asec XUV and IR fs pulse with the resulting

New Journal of Physics 10 (2008) 025033 (http://www.njp.org/)

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efficient generation of new asec pulses [21]. An appropriate Maxwell-TDSE equation for thissystem was used based on a slowly varying envelope approximation (SVEA) leading to afirst-order partial differential equation for the coupled Maxwell-TDSE system [22]. Such anapproach neglects ground state depletion due to ionization, neglects backward propagation andis therefore appropriate for low field strengths. It was nevertheless found that initial asec pulsescould be shortened further in time through the resultant HHG asec pulses produced nonlinearlyin the presence of an intense IR fs pulse [23]. In the present paper, we extend the Maxwell-TDSEapproach beyond SVEA in order to include the above neglected effects. Particular attention willbe given to the harmonic phases to understand more precisely the asec pulse generation.

Section 2 is devoted to the presentation of the Maxwell–Schrödinger model initiallyintroduced in [24, 25]. In section3, we study the asec pulse generation mechanism usingin particular a model describing the transmitted electric field phase. Section4 is devoted tonumerical simulations for the propagation of electric fields and asec pulses inside an aligned H+

2gaseous medium. We also present a new phenomenon, a study of the driver phase effect on theHHG and laser pulse propagation.

2. Maxwell-TDSE model

We study the process of asec pulse generation by the analysis of harmonic phases, and theirbehavior depending on the driver phase, using a model that is a micro–macro Maxwell-TDSEapproach [24, 26]. It consists of the coupling of the Maxwell equations and TDSEs within orbeyond the Born–Oppenheimer approximation. The model is totally non-perturbative, vectorialand multidimensional, taking into account ionization, and high-order nonlinearities going farbeyond classical and semi-classical nonlinear Maxwell and Schrödinger models [25], [27]–[29].In the present work, we restrict ourselves to 1D TDSEs and 1D propagation in the fixed nucleiapproximation (Born–Oppenheimer), for which the equations are:∂t Bx(z′, t)= c∂z′Ey(z′, t),

∂t Ey(z′, t)= c∂z′Bx(z′, t)−4π∂t Py(z′, t),(2)

where

Py(z′, t)=−n(z′)

∫ψz′(y, t)yψ

∗

z′(y, t)dy=−n(z′)dz′(t). (3)

n(z′) is the molecular density taken equal to a constantn0 in the following,ψz′ the wavefunctionof a molecule located atz′ (the dipole approximation is used here). In the following, wewill denote by E the electric field componentEy, and by P the componentPy of the field-induced polarization. In the TDSEs, we suppose that the electric fieldE is constant in spaceat the molecular scale1y/λ� 1, so that the electric field inz′, E(z′, t), is rewrittenEz′(t).Alternatively one can use the acceleration of the electron as given in [14] (Fourier transforma(ω)= ω2d(ω)).

a(z′, t)=∫ψz′(y, t)

(−∂V(y, R0)

∂y+ E(t)

)ψ∗z′(y, t)dy. (4)

In the 1D framework, we can then rewrite equations (3) and (4) as the second-order waveequation

∂2t t E(z

′, t)− c2∂2z′z′E(z

′, t)=−4π∂2t t P(t). (5)


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The 1D electron wavefunction is solved by the following TDSE:

i∂tψz′(y, t)=

(−4

2+ Vc(y, R0)+ yEz′(t)

)ψz′(y, t). (6)

In the previous equation,R0 denotes the molecular internuclear distance,z′ the field propagationcoordinate andy the electric field polarization corresponding to the electron coordinate. Thefield model allows us to consider high-order nonlinearities as well as ionization. The modelis solved using a finite difference method described in detail in [24], where a full non-Born–Oppenheimer formulation is also presented. In the model, we solve one TDSE per Maxwell cell,using a classical Crank–Nicolson scheme, corresponding ton01z′ molecules. Each Maxwellspatial step1z′ is chosen such that1z′ < λmax/5, whereλmax is the largest internal wavelengthconsidered in the computation (allowing then to consider high-order harmonics). For details werefer to [24]. This model is then used to study asec pulse generation and propagation in thepresence of HHG.

In conclusion, as we solve one single TDSE per Maxwell cell1z′, a computation ofNTDSEs corresponds to a sample of gas of lengthl = N1z′ and containingN1z′n0 molecules,where the molecular densityn0 is supposed constant.

3. Theoretical study of harmonic phase effects on asec pulse generation

In order to understand the generation and dynamics of asec pulses, we propose a model toapproximate harmonic spectrum extrema. More precisely, we propose to approximate by acontinuous function the maxima and minima of the harmonic spectrum and to isolate one or twoasec pulses by Fourier transform filtering of these extrema. We then plan to model the frequency-dependent harmonic phase functionφ(ω)= arctan(Im(a(ω))/Re(a(ω)) of the accelerationa(ω)= |a(ω)|exp(iφ(ω)), wherea(ω) is the Fourier transform ofa(t) defined in (4). This willbe done near the cut-off frequency (a few-cycle laser pulse withI ∼ 1015 W cm−2).

The simplest model to approximate the dipole acceleration phase ofa(t) (4) consists ofintroducing the following functionfM, modeling a spectral maximum aroundω = 0 with apiecewise constant phase:

fM(ω)=− exp(−α1ω)exp(iφ1)1R−(ω)−exp(α2ω)exp(iφ2)1R+(ω). (7)

The constant coefficientφ1 corresponds to the left-phase (phase on the left of a maximum) andφ2 corresponds to the right-phase (phase on the right of a maximum). Coefficientsα1 andα2,assumed strictly positive, allow in particular to control the harmonic width. Finally, the function1R− is defined as 1 for negativeωs and 0 for positive ones, and1R+(ω) denotes the functionequal to 1 for positiveωs and 0 for negative ones (Heaviside functions).

To model a spectral minimum, we considerfm with a piecewise constant phase:

fm(ω)=− exp(α1ω)exp(iφ1)1R−(ω)−exp(−α2ω)exp(iφ2)1R+(ω). (8)

We next compute the Fourier transform offM and fm in order to determine if one or two asecpulses are created. Our goal is therefore to determine for which condition(φ1, φ2, α1, α2) onesingle asec pulse can be isolated.


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Intensity

ta−ta

ta−ta Time

Time

Intensity

φ − π < φ < φ 2

φ < φ < φ + π 2

1

21

2

Figure 1. Harmonic phasesφ1 and φ2 emitted at times−ta, ta producingnonsymmetric asec pulses (equation (13)).

From (7) and (8), we next prove the following:

1. If φ1 6= φ2 + kπ (k ∈ Z, whereZ is the set of negative and positive integers) two non-symmetric asec pulses are created. Ifφ1= φ2, there exists one single asec pulse andtwo symmetric asec pulses ifφ1= φ2 +π . Moreover, the asec pulse time durations areproportional to 1/α1 and 1/α2.

2. For two HHG emitted at timesta and−ta, then two situations occur depending on thephasesφ1 andφ2.

When two non-symmetric asec pulses are created we can identify if the more intense asec pulseis created before or after the less intense one according to figure1.First, we examine the Fourier transform offM which givesaM:

aM(t)=(α2− it)exp(iφ1)+ (α1 + it)exp(iφ2)

(α1 + it)(α2− it), (9)


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and

|aM(t)|2=at2 + bt + c

(α21 + t2)(α2

2 + t2),

a= 2(1− cos(φ1−φ2)),

b= 2(α1 +α2) sin(φ1−φ2),

c= 2α1α2 cos(φ1−φ2)+α21 +α2

2.

(10)

The function|aM(t)|2 is naturally symmetric if and only ifb= 0, i.e.φ1= φ2 + kπ, k ∈ Z. In theopposite case, we can directly conclude that this function is not symmetric, so that two pulsesare created. Now, ifφ1= φ2, we observe that|a(t)|2 degenerates into:

|aM(t)|2=

(α1 +α2)2

(α21 + t2)(α2

2 + t2). (11)

This allows us to conclude that onesingleasec pulse is created. Ifφ1= φ2 +π , then

|aM(t)|2=

4t2 + (α1 +α2)2

(α21 + t2)(α2

2 + t2). (12)

which gives two symmetric asec pulses.Classical Fourier analysis allows the conclusion that for smallα1 andα2 the two asec pulses

birth times are very close and naturally far for largeα1 andα2. Assuming thatα1= α2= α > 0,we sets= t/α andφ = φ1−φ2, then

|a(s)|2=(1− cos(φ))s2 + sin(φ)s+ 1 + cos(φ)

1 +s2. (13)

This then corresponds to the caseα = 1. The distance between the two asec pulse maxima isthen proportional to 1/α in accordance with statement 1 above.

For the second point, it is sufficient to study the ratioc := |a(ta)|/|a(−ta)|. Wesimply observe thatc> 1 if and only if b< 0 and only if sin(φ1−φ2) < 0 and only ifφ2−π < φ1 < φ2.

This simple analysis allows the transfer of the asec pulse generation conditions on thebehavior of the phase of the dipole acceleration spectruma(ω)= |a(ω)|exp(iφ(ω)). However,it is not sufficiently precise as the behavior of the acceleration harmonic phase is in fact morecomplex. We then consider a more general maximum defined by:

gM(ω)=−(1R−(ω)χ(−α1, ω, A)+ 1R+(ω)χ(α2, ω, A)

)exp(iφ(ω)). (14)

χ is a smooth function defined by

χ(α, ω, A)=

exp(αω)− A, if exp(αω)6 A,

0, otherwise.(15)

We denote by� the support ofχ (i.e.χ(x)= 0 for all x ∈�c) andA is a positive real constantrelated to the depth of the minima. In order to study the asec pulse generation around a minimumwe introduce similarly the more general amplitude,

gm(ω)=−(1R−(ω)χ(α1, ω, A)+ 1R+(ω)χ(−α2, ω, A)

)exp(iφ(ω)). (16)


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ω ω

π/2

−π/2

0

Phase

−π/2

π/2Phase

0

Time Time

|a( )|ω 2 |a( )|ω 2

Pulse Pulses

Figure 2. One or two asec pulse generation from maxima and minima in theHHG spectrum depending on the harmonic phaseφ (equation (12)).

For simplicity, we suppose that the phase is not piecewise constant but rather piecewise linear.This allows us to conclude that:

1. For gM defined in (14) with φ(ω)=−πω/2, one single asec pulse is generated.

2. Whereas forgm defined in (16) with φ(ω)=−(π/2)(ω + 1)1R−(ω)− (π/2)(ω + 1)1R−(ω),then two symmetric asec pulses are created.

To prove this, it is sufficient to observe that settingt ′ = t −π/2, we recover the caseφ(ω)= 0 studied above leading to one single attosecond pulse generation. The second case canbe proven in the same manner. It is sufficient to observe that settingt ′ = t +π/2, we recover thecaseφ(ω)=−π/2 for negativeω andφ(ω)= π/2 for positive ones, leading to two attosecondpulses being generated. This is illustrated and summarized in figure2.

In general, we can observe periodic behavior of the acceleration harmonic phaseφ(ω)=

arctan(Im(a(ω))/Re(a(ω))). The period is often observed to be numerically equal to 2ωl, withωl the frequency of the incident laser pulse. We then consider the following phase (whereZ is


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Phase

ω

−π/2

−π/2

π/2

π/2

Even harmonic

Phase

ωOdd harmonic

|a( )|ω 2

|a( )|ω 2

Figure 3. Mathematical modeling of a minimum and a maximum in continuumHHG spectrum|a(ω)|2 with phases of each harmonic.

the set of negative and positive integers):

φ(ω)=∑k∈2Z

π

2(k−ω)1[k−1,k+1[(ω). (17)

Note thatφ(ω) jumps from−π/2 to π/2 are due to Re(a(ω)) cancellations (from positive tozero, to negative values). We then consider the following phase function (see figure3). WeconsidergM at maxima andgm at minima defined in (14) and (16) with a phase given by (17).We suppose for the sake of simplicity thatα1= α2= α > 0. Then one obtains from (14) at amaximum harmonic and for a large enough or small enoughα, thatonly one single asec pulseis created. From (16) at a minimum and for large enough or small enoughα, two asec pulses


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0 5 10 15 20 25 30 35 40 4510

−22

10−20

10−18

10−16

10−14

10−12

10−10

Harmonic order

Tra

nsm

itted

ele

ctric

fiel

d (a

.u.)

Transmitted electric field harmonic — Short propagation

Filtering around a maximum

Filtering around a minimum

Figure 4. Harmonic spectrum filtering around a maximum and a minimum for2-cycle 800 nm,I = 1014 W cm−2 pulse with 5 nm propagation lengthl anddensityn= 1020 mol cm−3.

are generated. For both minima and maxima the above conclusions follow from application ofFourier transforms as proved further in the appendix.

These results allow us to deduce when one or two asec pulses will be created by analysis ofHHG spectra and their phases. We now give an illustration of these analytic results in figures4and 5. The example we propose corresponds to the propagation over a very short distance(∼100 au∼ 5 nm) of a 2-cycle laser pulse at 800 nm in a dense medium (∼1020 mol cm−3) atintensity 1014 W cm−2. We then filter around a minimum (32th harmonic), and a maximum (27thharmonic), figure4, the transmitted electric field HHG spectrum. As expected, respectively oneand two asec pulses (figure5) are created. The pulse duration is obviously proportional to thefiltering range (∼400 as in the example). This study allows us to develop precise conditions togenerate one or two asec pulses.

4. HHG and asec pulse propagation

In this section, we examine propagation effects on asec pulse generation studied in section3and the harmonic intensities. We then consider the interaction of an aligned 1D H+

2 moleculegas with intense ultrashort laser pulses. We will also present a new propagation effect at highintensity and density including discussion of driver phase effects.

4.1. Harmonic dynamics and asec pulse propagation

We study here the dynamics of 5-cycle Ti:S laser pulse at 800 nm inside a 1D aligned H+2-gas

of molecular density 3.5×1018 mol cm−3. The intensity of the laser is 1014 W cm−2 and the


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6.4 6.6 6.8 7 7.2 7.4 7.6

0.5

1

1.5

2

2.5

x 10−21

Time (femtosecond)

Inte

nsity

1 asec pulse — filtering around maximum at N~27

6.2 6.4 6.6 6.8 7 7.2 7.4

1

2

3

4

5

6

7

8

x 10−23

Time (femtosecond)

Inte

nsity

2 asec pulses — filtering around minimum at N~32

Figure 5. One and two asec pulse generation, withI = 1014 W cm−2, n0=

1020 mol cm−3 and propagation lengthl = 5 nm (2-cycle pulse 800 nm).


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internuclear distance is fixed (Born–Oppenheimer) atR0= 3.2 au. Simulations are here 1D andthe size of a Maxwell cell is equal to1z′ = 140 au∼ 7.4 nm.

Of particular interest here is the intensity of the first low harmonics and in the cut-offregime. From the Fourier transform of the transmitted electric fieldEz′(t) (see (3), (5) and (6))propagating in the gas at different lengths, we study the evolution of the first (up to 11)harmonic intensities. The harmonic spectrum of the transmitted field, denoted byET, possesses,in theory [18], a cut-off frequency given in (1) for recollision of the electron with the parention. Collisions with neighboring ions such as in molecules produce larger ordersN, due to largecollision energies exceeding 3.17Up [4]. We then investigate the harmonic intensities of thetransmitted electric field (ω, |ET(ω)|

2) for N = 3,7 and 11 at different distances. We observethat as physically expected, the quadratic scaling (see figure6) function of the propagationdistance (or molecular density) is well respected for small propagation distances. However, aftera sufficiently large propagation, nonlinear propagation effects appear (see in particular the 7thand 11th harmonics). In particular, we can observe in figure7(a) that the 3rd harmonic increases,then is absorbed after a sufficiently large propagation due to the fact that the internucleardistance has been fixed to 3.2 au leading to a 3-photon resonance. These results are comparablewith previous atomic calculations [20, 30] (note that a 7-photon resonance also occurs whenR0= 2 au).

We then define the total amount of energy of a selected asec pulse by:

Ec=

∫|Ec(t)|

2 dt, with Ec(t)=1

2π

∫ ωc+1ω

ωc−1ω

e−iωt ET(ω)dω, (18)

whereωc is the cut-off frequency and [ωc−4ω,ωc +4ω] the selected frequency range, toisolate one asec pulse as described in section3. As an example, we analyze a filtered (with1ω = 2,ωc= 33) asec pulse (time duration∼280 as) propagating over a distance ofl ∼ 40µm.We compute in figure7(b) its total amount of energyEc (18) (function of the propagationdistance) for which oscillating behavior appears. This result is to be compared to results of [22]where a much larger propagation distance is considered but however with a much simpler model,and where the polarization was deduced from one single atom response.

4.2. CEP dependence of HHG

We focus in this section on the effect of the incident laser pulse CEP, i.e. the carrier envelopephase, or laser driver phaseφ on the harmonic dynamics and consequently on the asecgeneration. We are especially interested in the dipole acceleration (4) and transmitted electricfield harmonic spectra but also in the classical electronic motion. As is well known, a simpleclassical model allows some very precise information on the electron classical behavior to beobtained, especially in the cut-off frequency of the harmonic spectra (1) [2]–[4]. First, we willintroduce a classical model to describe the electron motion. Using this model, we will firstexplain a phenomenon usually observed on harmonic spectra and asec pulse creation. Then,we will study the harmonic spectra depending on the driver phaseφ illustrating a new physicalphenomenon.

4.2.1. Classical recollision. Let us introduce the following Gaussian electric pulse:

E(ω, t)= E0 cos(ωt +φ)exp(−(t − t0)2). (19)


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0 1 2 3 4 5 6 7 8−22

−20

−18

−16

−14

−12

−10

−8

−6

Propagation distance (logscale)

)elacsgol( ytisnetni cinomrah drih

T

0 1 2 3 4 5 6 7 8−32

−30

−28

−26

−24

−22

−20

−18

−16


)elacsgol( ytisnetni cinomrah neve

S

Seventh harmonicy=2x

0 1 2 3 4 5 6 7 8−36

−34

−32

−30

−28

−26

−24

−22

−20


)elacsgol ( ytis netn i ci nomra h h tnev el

E

Eleventh harmonicy=2x

Figure 6. 3rd, 7th and 11th harmonic intensities (logscale), as function ofpropagation lengthl , up to 11µm, for R0= 3.2 au for 5-cycle 800 nm pulse(I = 1014 W cm−2, n= 3.5×1018 mol cm−3).


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0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−4

Propagation distance (µm)

ytisnetni cinomrah drih

T

0 10 20 30 40 50 60 70 80 900

2

4x 10

−15

Propagation distance (µm)

)–2

mcJ( ygrene lato

T

12800 TDSEs

6400 TDSEs

1 TDSE

Figure 7. (a) 3rd harmonic absorption (logscale) as function of propagationlength l , for R0= 3.2 au. (b) total energy in the filtered asec pulse dependingon the propagation length (up to 90µm) (harmonic range [25, 35]).

We present different return energy graphs, maxima of energy depending on the driverphaseφ. The analysis consists of solving Newton’s equations for one electron (in au):z(t)=−E(ω, t), z(t0)= z0, z(t0)= v0 from which are deduced the electronic trajectories and theelectronic return energy at timetf depending onφ [4]. We propose this simple analysis to explainwhy near the cut-off frequency, two trajectories separated by∼0.2 cycle are often observednumerically in quantum calculations [14, 31]. To our knowledge, no rigorous explanation has


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0.6 0.625 0.65 0.675 0.7 0.725 0.75 0.775 0.8 0.825

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3.1

Return phase (φ/2π)

Up

/E

r ygrene nrute

R

−0.5 0 0.5 1.0 1.50

0.5

1.0

1.5

2.0

2.5

3.0

t (cycle)

Up

/E

r nruter fo ygre n

E

Birth time t1Birth time t2Birth time t3

Figure 8. (a) Return energyEr/Up (Up ponderomotive energy) dependenceon the phaseφf of return (φf/2π ) for a monochromatic incident pulseE(t).(b) Return energyE(t), equation (19) for φ=0 and three different birth timesti .

been provided for this specific phenomenon. For the special case of monochromatic electricfields the energy of return (function of time) is represented in figure8(a), as obtained from theexact solution of Newton’s equations for initial velocityv0, maximum amplitudeE0, initial fieldphaseφ0, return phaseφf and initial electron positionz0 [4]:

z(tf)=(v0 +

E0

ωsin(φ0)

)φf −φ0

ω+ (cos(φ0)− cos(φf))

E0

ω2+ z0. (20)

We can then deduce the time delay between two trajectories; for instance this delay is0.1 cycle if the return energyEr is close to 2.95Up (Up is the ponderomotive energy (1)), see


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−2 −1.5 −1.0 −0.5t (cycle)

0.5 1.0 1.5 2.0−0.25

−0.2

−0.15

−0.1

−0.05

E(t)

0.05

0.1

0.15

0.2

0.25φ=0φ=π/4φ=π/3φ=π/2φ=π

Birth time t1

Birth time t2

Birth time t3

Figure 9. Electric fieldsE(t) equation (19) with driver phaseφ and birth timesti for a four cycle pulse (intensity (in au)/time (in cycles)).

figure 8(a). For shorter pulses, this exact solution is no longer valid as the envelope of thepulse has to be taken into account. We therefore solve numerically Newton’s equations for theenvelope given by exp(−α(t − tf/2)2) whereα > 0.

In figures8(b), we represent the maximum return energy dependence on time (hereφ isequal to 0, cos pulse) in equation (19), for birth times ti plotted in figure9. We recall thatφ = π/2 (sin pulse) corresponds to the smallest energy, whereasφ = 0 (andφ = π ) correspondsto the largest one, as is currently observed numerically and experimentally in harmonic spectra.In figure 10, we represent the harmonic spectrum dependence on the phaseφ [5]. We remarkthat, as expected, the driver phaseφ = π/2 gives the shortest cut-off frequency andφ = 0 thelongest one in accordance with the classical study presented above.

We finally present results with full propagation of a two- or four-cycle laser pulse within agas (again represented by many aligned TDSEs via (3) and (6)) and solved by the macroscopicMaxwell’s equations (2) with intensities 1015 W cm−2. We compute the transmitted electric fieldharmonics for 1 up to 12 800 TDSEs. These harmonics are obtained from the Fourier transformof the total transmitted field through the gas. In this part, we are especially interested in theharmonic spectra depending on the driver phase.

In figure 11, an interesting new phenomenon has to be noticed when comparing thetransmitted electric fieldEz′(t) (related to many molecules) spectrum with the single moleculedipole harmonics (3). The single molecule dipole spectrum shows deep minima near the cut-offfrequency. As we have seen earlier, around these minima two trajectories can often be isolated.By contrast around large continuous maxima (several harmonic orders) one single trajectoryreturns. We numerically observe that near the cut-off frequency, the deep minima observed in the


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1e-16

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

0.01

1.0

100 120 Harmonic order N 160 180 200

φ=0φ=π/2φ=π/3φ=π/4

Figure 10. HHG minima dependence on the laser driver phaseφ.

1e-20

1e-15

1e-10

1e-05

1

100 000

0 50 Harmonic order N 150 200

Total electric field |E(w)|2 (a.u.)Single molecule |a(w)|2 (a.u.)

Figure 11. Single molecule dipole acceleration harmonics (|a(ω)|2) andtransmitted electric field intensity (|E(ω)|2) (propagation length equal to 10 nm)harmonics. (Deep minima near the cut-off have disappeared after propagation,I = 1015 W cm−2 andn0= 1020 mol cm−3, φ = 0.)

dipole acceleration harmonics disappear in the transmitted electric field harmonics (propagationin the gas, less than 1µm). As a consequence, only one single trajectory seems to ‘survive’ (seesections3 and4.2).

For larger propagation distances and a driver phase equal to 0, figure12 shows surprisingbehavior for the 2-cycle pulse. Near the cut-off frequency, the HHG energy is larger than


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17

50 100 150 200 250 300 350 400 450

105

100

10–5

10–10

10–15

10–20

10–25

10–30

10–35

Harmonic order

|E(w

)|2 (

a.u.

)

Propagation 1 up to 12800 TDSE’s1 SE1280 SE12800 SE

Figure 12. 2-cycle—total electric field harmonic order with propagation for1,1280,12 800 TDSEs corresponding respectively to 7.4 nm, 7.4×1280 nm,7.4×12 800 nm (I = 1015 W cm−2, constant molecular density equal ton0=

1020 mol cm−3,800 nm, φ = 0).

the classical return energy 3.17Up(ωl)+ Ip after propagation. Furthermore we note then20

dependence of the efficiency of HHG with propagation, thus confirming it is a coherentcollective phenomenon depending on pairs of molecules.

We propose the following explanation for this phenomenon. For large densityn0 and short(2-cycles) and intense (I = 1015 W cm−2) laser pulses a spectral broadening due to Kerr-effectis usually observed [32]. Pulse compression is also operative at high densities (∼1 atm) [17].We note, however, that our result is opposite to the prediction of cut-off shortening due to non-adiabatic phase matching effects [28]. Therefore, during the propagation spreading occurs tofrequencies less than the incident laser frequencyωl (first harmonic). This produces an increaseof the ponderomotive energyUp (E0/4ω2

l ) whereωl < ωl. This results in the cut-off increasingwith propagation of the laser field in the gas. This phenomenon could be used, we suggest, toproduce shorter asec pulses.

5. Conclusion

Ultrashort intense pulse propagation effects on asec pulse generation have been investigated in a1D Maxwell-TDSE approach for aligned 1D H+

2 molecules as a function of various parameters:driver pulse phaseφ, propagation lengthl and molecule densityn0. The simulations, althoughlimited to the 1D model, take into account field quantum effects, such as ionization, HHG,interference, multiphoton resonance absorption and use a full second-order equation for thepropagated field (5).

The first new result as reported in figure12 is the increase of the HHG cut-off length withincreasing molecular density. We propose that this is due to pulse compression effects [17], thus


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18

broadening the spectrum of the driver (incident) pulse with the lower frequencies generatedincreasing the ponderomotive energyUp and the maximum cut-off law (1). The second result isthe possibility of creating double asec pulses (figure5) by filtering and recombining harmonicsaround sharp minima in the HHG spectrum. A detailed theory for the generation of such doubleasec pulses, their width and temporal separation is presented in section3 and the appendix.Finally, multiphoton resonance effects are shown to have a limiting effect on total asec pulseenergy as a function of propagation lengths (figures6, 7, etc). Further refinement of these effectsare being pursued in a full 3D Maxwell-TDSE treatment using a complete 3D numerical modelpresented in [24] in order to verify nonlinear absorption effects in propagation of intense pulsesin condensed media [33].

Acknowledgments

Funding of this research has been provided by the Centre de Recherches Mathematiques (CRM)and Canada Research Chair (CRC) program. Supercomputer facilities (more than 500 CPUs)were made available by RQCHP. Finally, we acknowledge ‘illuminating’ discussions with P BCorkum (NRC).

Appendix

To prove the existence of one or two asec pulse generation from HHG, we first begin by applyingthe Fourier transform to our model equation (14),

F(gM(t))=∫R

exp(iωt)gM(ω)dω =∑`∈2Z

∫ `+1

`−1exp(iωt)gM(ω)dω. (A.1)

Now (denoting byZ the set of negative and positive integers and� the support ofχ (seesection3))

F(gM(t))= −∑

`∈2Z∗−∩�

∫ `+1

`−1exp(iω(t −π/2))χ(−α, ω, A)exp(iπ`/2)dω

−

∑`∈2Z∗+∩�

∫ `+1

`−1exp(iω(t −π/2))χ(α, ω, A)exp(iπ`/2)dω

−

∫ 0

−1exp(iω(t −π/2))χ(−α, ω, A)exp(−iωπ/2)dω

−

∫ 1

0exp(iω(t −π/2))χ(α, ω, A)exp(−iωπ/2)dω. (A.2)

If we suppose thatα is large enough, the first two terms are negligible compared to the two lastasχ(α, ω, A) decrease quickly to zero. This then corresponds to the case of a constant phaseφ1= φ2= 0 studied above and leading to one single asec pulse.

Now as we consider thatχ has a compact support (χ is then zero outside a boundeddomain) a simple computation leads to (whereZ∗

−are the strictly negative integers andZ∗+


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19

the strictly positive ones):

F(gM(t))= −∑

`∈2Z∗−∩�

exp(i`π/2)exp((`+ 1)(it − iπ/2−α))−exp((`−1)(it − iπ/2−α))

it − iπ/2−α

−

∑`∈2Z∗+∩�

exp(i`π/2)exp((`+ 1)(it − iπ/2 +α))−exp((`−1)(it − iπ/2 +α))

it − iπ/2 +α

−

∫ 0

−1exp(iω(t −π/2))χ(−α, ω, A)exp(−iπ/2)dω

−

∫ 1

0exp(iω(t −π/2))χ(α, ω, A)exp(−iπ/2)dω + C(α, A).

(A.3)

In the previous formula,C(α, A) (denoted byC in the following) is a real constant that dependsonα andA sinceχ is defined as the sum of an exponential function and a real constantA. If weset`←−` in the first sum on the r.h.s., we obtain:

F(gM(t))= −∑

`∈2Z∗+∩�

exp(−i`π/2)

×exp((−`+ 1)(it − iπ/2−α))−exp((−`−1)(it − iπ/2−α))

it − iπ/2−α

−

∑`∈2Z∗+∩�

exp(i`π/2)exp((`+ 1)(it − iπ/2 +α))−exp((`−1)(it − iπ/2 +α))

it − iπ/2 +α

−

(1−exp(α + iπ/2− it)

it −α− iπ/2+

exp(α + it − iπ/2)−1

it +α− iπ/2

)+ C. (A.4)

Note also that ∈ 2Z∩� so that exp(i`π/2)= exp(−i`π/2). In the following, we willdenote�2+= 2Z∗+∩�

F(gM(t))= −∑`∈�2+

exp(i`π/2)

×

(exp((`−1)α)exp((`−1)(iπ/2− it))−exp((`+ 1)α)exp((`+ 1)(iπ/2− it))

it − iπ/2−α

+exp((`+ 1)α)exp((`+ 1)(it − iπ/2))−exp((`−1)α)exp((`−1)(it − iπ/2))

it − iπ/2 +α

)−

(1−exp(α + iπ/2− it)

it −α− iπ/2+

exp(α + it − iπ/2)−1

it +α− iπ/2

)+ C. (A.5)

Thus, if we setB= t −π/2

F(gM(t))= −∑`∈�2+

exp(`(iπ/2 +α))

(exp(−i`B)

(exp(−α)exp(iB)−exp(α)exp(−iB)

iB−α

)

+ exp(i`B)

(exp(α)exp(iB)−exp(−α)exp(−iB)

iB +α

))−

(1−exp(α− iB)

iB−α+

exp(α + iB)−1

iB +α

)+ C. (A.6)


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20

If we set now

N`(α)= exp(i`B)

(exp(α)exp(iB)−exp(−α)exp(−iB)

iB +α

),

(A.7)and M(α)=

exp(α + iB)−1

iB +α.

Then:

F(gM(t))=−∑`∈�2+

exp(`(iπ/2 +α))(N`(α)+N ∗` (α))− (M(α)+M(α)∗)+ C. (A.8)

It is easy to deduce for very smallα that

M(α)+M∗(α)= 2sin(t −π/2)

t −π/2+O(α) (A.9)

and that

N`(α)+N ∗` (α)= 4 cos(`(t −π/2))sin(t −π/2)

t −π/2+O(α) (A.10)

and then after simplification of the summation by Taylor expansion, we have

F(gM(t))=−2sin(t −π/2)

t −π/2

(1 + 2

∑`∈�2+

exp(`(iπ/2 +α)) cos(t −π/2) +O(α))

+ C. (A.11)

We then deduce that this model produces one main centered pulse and residual oscillation takingthe modulus ofF(gM(t)).

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Attosecond pulse generation from aligned molecules---dynamics and propagation in H2+

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