Appl Phys B (2011) 103:563–570DOI 10.1007/s00340-011-4513-4

Role of the carrier-envelope phase in laser filamentation

L. Bergé · C.-L. Soulez · C. Köhler · S. Skupin

Received: 16 February 2011 / Revised version: 8 March 2011 / Published online: 10 May 2011© Springer-Verlag 2011

Abstract We numerically study the influence of the ini-tial carrier-envelope phase (CEP) on the filamentation ofultrashort laser pulses in noble gas. Emphasis is put onthe CEP-induced changes of pulses that reach their clamp-ing intensity during near-cycle self-compression. In otherpropagation regimes, the CEP does not significantly al-ter the pulse evolution. Our results indicate that third-harmonic generation, compared to plasma generation, isdominant in driving these changes. Finally, the stability ofthe filament CEP against shot-to-shot fluctuations is exam-ined.

1 Introduction

Modern laser sources can deliver intense, ultrashort pulsescontaining only a few optical cycles. This progress pavedthe way to the quickly-developing field of “extreme non-linear optics,” for which light–matter interactions stronglydepend on the laser carrier wave amplitude and classical en-velope models break down [1]. In those models, the com-plex envelope A(t, z) connects to the real laser electric fieldE(t, z) as E(t, z) = A(t, z)e−iω0t + c.c. (c.c. denotes com-plex conjugate), with carrier frequency ω0. The position

L. Bergé (�) · C.-L. SoulezCEA-DAM, DIF, 91297 Arpajon, Francee-mail: luc.berge@cea.fr

C. Köhler · S. SkupinMax Planck Institute for the Physics of Complex Systems,01187 Dresden, Germany

S. SkupinInstitute of Condensed Matter Theory and Optics,Friedrich Schiller University, 07743 Jena, Germany

of the carrier wave relative to the envelope, the so-calledcarrier-envelope phase (CEP), is encoded in the phase ofA(t, z). Slowly-varying envelope models are invariant un-der a change of CEP, i.e., the transformation A(t, z = 0) →A(t, z = 0)eiφ does not affect the propagation dynamics.This concept is expected to fail when the pulse durationbecomes comparable to the oscillation cycle of the carrierwave.

Xu et al. [2] first addressed the control of the absolutephase of few-cycle light pulses from mode-locked laser os-cillators, where the position of the carrier relative to theenvelope rapidly varies on each round trip in the laser cav-ity due to the difference of phase and group velocity. More-over, nonlinear (e.g., cubic) effects, including amplitude-to-phase conversion via self-steepening, are known to have anadditional impact on the CEP [3, 4]. From the early 2000s,phase-coherent locking techniques allowed direct control ofboth the repetition rate and the CEP of mode-locked laserpulses, resulting in reproducible optical waveform synthesisin time domain [5–7]. CEP-self-stabilized, few-cycle pulseswere furthermore produced using optical parametric am-plifiers [8]. Applications to laser-matter interaction experi-ments started almost simultaneously. Among those, we canrecall the light-phase-sensitive photo-emission from a metalsurface, varying with the timing of electric field oscillationswith respect to the pulse peak [9]. Unambiguous measure-ments of effects stemming from the absolute phase were ear-lier performed by exploiting the sensitivity of strong-fieldionization processes over subfemtosecond time scales [10].Another important topic is the higher-order harmonic gener-ation (HHG) [11] in the XUV and the production of isolatedattosecond pulses [12–14]. These works underline the highsensitivity of atomic currents and coherent X-ray emission

564 L. Bergé et al.

to the CEP of the driver pulse. Thereby, they emphasize theneed to stabilize the carrier absolute phase, from pulse-to-pulse and during propagation, in order to optimize phase-matched HHG.

On the other hand, experimental as well as numerical in-vestigations of intense femtosecond pulses reported the ca-pability of the latter to form narrow (micrometer-sized) fila-ments of light, when they nonlinearly propagate in a largevariety of transparent media [15]. Because the pulse du-ration can significantly shrink along the propagation axis,filaments are able to develop more than one-octave span-ning spectra and they nowadays serve as few-cycle opti-cal sources by themselves [16–22]. For instance, in kryp-ton, filaments at 800 nm center wavelength have been com-pressed down to 8 fs durations [16]. In helium having ahigher ionization potential, the possibility to create evensubcycle (<2 fs) infrared pulses has been suggested by nu-merical simulations [22]. Few-cycle pulses obtained by fil-amentary compression have interesting properties, e.g., theself-guiding process acts as a spatial filter producing excel-lent mode quality and a precise optical alignment is not re-quired. In [17], energy variations at the percent level wereobserved to keep the filament CEP rather stable, as mea-sured by f − 2f spectral interferometry. In [23], simulatedultrashort filaments were even shown to maintain a nearlyconstant CEP within variations ≤ π/8 over meter-range dis-tances, which was explained as a kinetic effect in terms ofdispersive X-waves.

Few investigations have been led so far on the impact ofthe CEP on the propagation dynamics when the pulse pro-file becomes distorted due to nonlinearities in filamentationregime. In this work, we examine the role of CEP in suchregimes by means of (3 + 1)-dimensional numerical simu-lations. Particular attention will be paid to infrared wave-lengths (λ0 = 0.7 − 2 µm) and pulses reaching one opticalcycle (τo.c. ≡ λ0/c). Emphasis is put on the nonlinear distor-tions of the rapidly-oscillating carrier wave and their varia-tions under different initial CEPs. It is shown that the inputphase offset does influence the filamentation, whenever thepulse duration reaches the single-cycle limit near intensityclamping. Among the key players supporting differences inthe pulse dynamics, we show that third-harmonic generation(THG) prevails over plasma generation. Outside the near-cycle regime, propagation does not significantly vary withthe input CEP. In agreement with [17], we find that fila-mentation mostly preserves the relative CEP, and this prop-erty holds for new frequencies appearing due to self-phase-modulation. However, shot-to-shot fluctuations, even lim-ited to the percent level, can affect the CEP stability of thesupercontinuum.

2 Model equation

Our reference model is the unidirectional pulse propagationequation (UPPE) [24, 25], that governs the forward compo-nent of linearly polarized pulses

∂zE = i

√k2(ω) − k2

x − k2yE + i

μ0ω2

2k(ω)FNL, (1)

where E(z, kx, ky,ω) is the Fourier transform of the laserelectric field with respect to x, y, and t . The first term onthe right-hand side of (1) describes linear dispersion anddiffraction of the pulse. The term FNL = P (3) + iJe/ω +iJloss/ω contains the third-order nonlinear polarizationP (3) = ε0χ

(3)E3, whose magnitude is characterized by theKerr index n2 = 3χ(3)/4n2

0cε0 [n0 = n(ω0)], the electroncurrent Je, and a loss term Jloss due to ionization (see, e.g.,[20, 21]). Compared with [25], the denominator of the non-linear term reduces to 2k(ω), as we consider optical wave-forms with transverse dimensions satisfying the paraxial

assumption k(ω) �√

k2x + k2

y . For practical convenience,

when going to time domain we transform to a referenceframe moving with the linear group velocity of the pulse,t → t + z/vg where v−1

g = ∂k/∂ω|ω=ω0 , keeping the pulsecenter at zero delay. Third-harmonic generation results fromthe χ(3) nonlinearity and higher-order odd harmonic cas-cading occurs from mixing pump and harmonic fields. Theplasma dynamics is described by the electron density

ρe(t) = ρnt[1 − e− ∫ t

−∞ W(t ′) dt ′] (2)

with neutral atom density ρnt. Because we are dealing withfew-cycle pulses, we use a field dependent quasistatic tun-neling ionization rate

W(E) = 4ωar2.5H

Ea

|E(t)|e−2r1.5H Ea/(3|E(t)|), (3)

where Ea = m2eq

5e /(4πε0)

3�

4 � 514 GV/m is the atomicelectric field strength, ωa = meq

4e /(4πε0)

2�

3 � 41.3 fs−1

denotes the atomic frequency unit and rH = Ei/Eh [26, 27].Here, Eh = 13.6 eV and Ei = 16 eV are the ionization po-tentials of hydrogen and the atom under consideration, i.e.,argon; me and qe are the electron mass and charge, respec-tively. Equation (3) applies to electric fields with maximumamplitude limited to 2r1.5

H Ea/3 ∼400 GV/m. It also discardsmultiphoton ionization, being dominant for field amplitudes<20 GV/m.

Using a purely field dependent rather than intensitydependent ionization rate leads to an additional couplingbetween envelope and carrier wave, apart from harmonicgeneration. We define the pulse intensity as I = 2n0

μ0c

| ∫ ∞0 E(ω)e−iωt dω|2, which gives for longer pulses the

usual time-averaged (over several optical cycles) absolute

Role of the carrier-envelope phase in laser filamentation 565

value of the Poynting vector. Note that, for few-cyclepulses, such time-averaged quantities make no sense. Withthe previous definition, we generalize the usual relationI ∼ |E |2, where E is the complex electric field. In thelong pulse limit, this model contains the classical enve-lope description, often known as the Nonlinear EnvelopeEquation (NEE) [1, 20]. The latter readily follows fromthe inverse Fourier transform of (1), after expanding k(ω)

around ω0 = 2πc/λ0, performing the envelope substitutionE(z, x, y, t) = U(z, x, y, t)e−iω0t + c.c. and discarding har-monic generation (P (3) ∼ |U |2U ). The ω-dependent con-tributions of the nonlinear terms in (1) consistently repro-duce the well-known phenomena of space-time focusing,i.e., (k2

x + k2y)E/k(ω) → −T −1∇2⊥U , and self-steepening,

i.e., ω2n2E/k(ω) → T n2|U |2U , where T ≡ 1 + (i/ω0)∂t .At fixed peak intensity, self-steepening nonlinearly lowersthe peak velocity compared with the (linear) pulse group ve-locity vg and is responsible for shock dynamics and carriersteepening [31]. NEE supplemented by third-harmonic gen-eration faithfully reproduces results obtained from direct in-tegration of Maxwell’s equations, as shown, e.g., by Gentyet al. for waveguides [32]. Equation (1) involves all thesephysical ingredients, but it discards dispersion of the non-linear refractive index n2, i.e., the frequency variations ofthe cubic susceptibility χ(3). As recently evaluated in [33],dispersion of the Kerr index contributes to less than 15%and 3% at the respective central wavelengths of 720 nm and2 µm, which are considered in the present work. Despitespectral broadening to high frequencies, UV componentsbeing more sensitive to nonlinear dispersion always remainof weak intensity (see Fig. 3 for instance) and they are ex-pected to undergo limited shifts in the pulse temporal profile[33, 34]. For these reasons, nonlinear dispersion is supposednegligible in the frequency ranges of interest.

Equations (1) and (2) are integrated numerically by usinginitially Gaussian pulses

E(z = 0, x, y, t)

=√

cμ0Pin

πn0w20

e−(x2+y2)/w20 e−(2 ln 2)t2/τ 2

p e−iω0t+iφ + c.c.,

(4)

with input power Pin, beam waist w0 and full width at half-maximum (FWHM) durations τp , in some cases focused bya converging lens of focal length f . To model the lens, weapply a frequency-dependent quadratic phase ensuring thatall frequency components focus at z = f in linear regimeand in the presence of space-time focusing [28]. Disper-sion curves for the refractive optical index n(ω) are takenfrom [29], while the Kerr index n2 is evaluated from [30]. Atatmospheric pressure, ρnt = 2.7 × 1019 cm−3 and the criti-cal power for self-focusing is defined by Pcr � λ2

0/2πn0n2.We carefully checked that in all simulations presented in this

paper no dc-fields (ω → 0) occur, as they are not describedby (1), and the pulse energy is preserved under any phaseshift φ.

3 Pulse dynamics versus initial CEP

Before proceeding with full 3D simulations, we wish to ar-gue on the CEP action by means of elementary field pro-files. Figures 1(a) and 1(b) show a couple of schematic

Gaussian electric fields E(t) = A cos (ω0t − φ)e−(2 ln 2)t2/τ 2p

operating at 720 nm in Ar at ambient pressure with inputCEP φ and amplitude A = 33 GV/m, a typical value ob-tained from numerical simulations in clamping regime. By“clamping regime,” we mean the propagation range alongwhich Kerr self-focusing is saturated and both pulse inten-sity and plasma density attain their maximum values. Wewant to stress that such field configurations are just extremeexamples to illustrate the CEP sensitive quantities ρe andP (3) and are, of course, not snapshots from our numeri-cal simulations. We verified that the dc-component of ourcosine ansatz is almost zero and that introducing a cor-rection factor assuring

∫E(t)dt = 0 [35] keeps Fig. 1 un-

changed. From this figure, it is seen right away that thelonger the pulse duration, the larger the number of ionizingsteps and the higher the ionization front calculated from (2).The phase offset φ locally alters the maxima of the fieldamplitude, so that the peak electron density can undergonoticeable modifications. The influence of the initial phasein the plasma response then disappears for pulse durationsτp ≥ 3τo.c.. With φ = 0, the field extrema are shifted totmax = (φ ± kπ)/ω0 with k integer, which causes a conse-quent temporal shift of the peak density ρmax. Figures 1(c)and 1(d) depict the behavior of the nonlinear polarizationP (3)(t), which mixes fundamental and third-harmonic con-tributions. Driven by THG, variations introduced by differ-ent phase offsets again become significant for pulse dura-tions less than 3τo.c.. Besides these nonlinear effects, wealso recall that the CEP evolves linearly during propagation,given by the difference between the pulse group and phasevelocity. Thus, at least for the less intense field oscillations,the phase offset is expected to shift by π/2 over the distance�zdrift � π/(2k0 − 2ω0∂k/∂ω|ω=ω0).

The examples shown in Fig. 1 do not, however, allow usto identify the relative impact of the CEP sensitive terms,THG and plasma generation, on the pulse propagation dy-namics. Therefore, we test the previous properties by di-rect numerical integrations of (1) for near-infrared pulses(λ0 = 2 µm). Figures 2(a) and 2(b) show the peak inten-sity and electron density for an initially Gaussian pulse withτp = 30 fs, w0 = 500 µm and Pin = 1.2 Pcr in an Ar gascell at 0.5 bar pressure. In Fig. 2(b), the decrease of theFWHM duration is indicated by cross symbols. Clearly,

566 L. Bergé et al.

Fig. 1 Schematic electric fieldsfor (a) single- and(b) triple-cycle 720-nmGaussian pulses with CEP φ = 0and φ = π/2 (blue and reddashed curves, respectively).Plasma responses obtained in Arare plotted as solid curves whichmust be read on the right-handside axis. (c, d) Correspondingcubic polarization P (3)

CEP-induced differences begin to occur as the pulse dura-tion becomes close to two optical cycles. The intensity I

then attains its clamping value at z � 0.7–0.8 m where max-imum plasma density is produced. Starting from ∼4.5 opti-cal cycles (τo.c. = 6.7 fs), the FWHM pulse duration short-ens down to 6 fs at z = 0.73 m with initial CEP φ = 0 andto 5 fs at z = 0.77 m with φ = π/2. In other words, thepulses become sub-cycled in terms of the initial carrier fre-quency. Figures 2(c) and (2d) show the peak intensity pro-files near maximum compression in the y = 0 plane whenφ = 0 (Fig. 2(c)) and φ = π/2 (Fig. 2(d)). The overall in-tensity distributions are comparable and mainly driven byself-steepening responsible for shock formation [36]. Inter-ferences between the pump pulse and (mostly) the third har-monic characterize solutions of the UPPE model. We recallin this regard that TH conversion efficiency may reach theorder of 10 % at near-infrared laser wavelengths, as numer-ically estimated in [37].

To visualize the differences induced by the CEP,Figs. 2(e) and 2(f) zoom the high-frequency structure ofthe on-axis pulse electric field (x = y = 0). Blue-shiftedhigh frequencies clearly develop in the trailing part of thepulse with oscillation periods down to ∼1 fs. These fieldcomponents support the maximum intensity. Even with theshortened FWHM duration, the pulse is not “subcycle” interms of the local carrier wave. In fact, the effective num-

ber of optical cycles does not change much during the pulseshortening process, except in the shock zone (t > 25 fs).Actually, the local cycle duration is modified. Hence, thepulse duration yielded by the FWHM intensity still coversa few oscillations with amplitudes exceeding ∼|Emax|/2.Here, a nonzero initial CEP retards the occurrence of plasmageneration. The local balance between all nonlinearitieschanges, which in turn affects the self-compression effi-ciency (5 vs. 6 fs). In Fig. 2(e), we can see that for lessintense oscillations the initial phase shift of π/2 is halvedat z ≈ �zdrift/2 � 0.7 m, as expected. Figures 2(e) and 2(f)suggest that as long as the filament path remains smaller thanthe linear phase drift �zdrift, the field oscillations mostlypreserve the phase of the input pulse [see solid and dashedcurves].

Corresponding on-axis spectra reveal that the trailingedges of the pulses contain a broad supercontinuum [seeFig. 3(a)]. These spectra do not noticeably vary in theirfirst decade at intensity clamping. Minor fluctuations onlyoccur at rather low spectral intensities beyond the secondharmonic region. A similar observation was earlier reportedin [32]. For comparison, Figs. 3(b)–3(d) illustrate alterna-tive filamentation scenarios. In Fig. 3(b), an almost single-cycle pulse (τp = 3.4 fs, w0 = 2.2 mm, Pin = 15Pcr) prop-agates in focused geometry (f = 1.5 m) in an argon cell at0.8 bar pressure, starting with φ = 0 or π/2 (�zdrift � 4 cm).

Role of the carrier-envelope phase in laser filamentation 567

Fig. 2 (a) Peak intensity and(b) electron density of a 30 fs,2 µm Gaussian pulse in Ar at0.5 bar (collimated geometry,Pin = 1.2Pcr) with initial CEPφ = 0 (solid blue curves) andφ = π/2 (solid red curves)computed from (1). In (b), blueand red cross symbols markevaluations of the FWHM pulsedurations along propagation,respectively. (c, d) Maximumintensity profiles in the y = 0plane at (c) z = 0.73 m withφ = 0 and (d) z = 0.77 m withφ = π/2. (e, f) On-axis electricfields (solid curves) and plasmadensity (dash-dotted curves) vs.time at the two distances shownin (c) and (d). Black dashedcurves recall the input field

At maximum intensity near the nonlinear focus zc � 1.3 m,the pulse dynamics appear identical whatever the input CEPmay be. The reason is that, at this distance, the pulse hasalready undergone strong dispersion and plasma-inducedpulse breakup, resulting in an effective FWHM durationof about ∼18 fs. Therefore, different input phase offsetshave no impact on the pulse dynamics. A second example isshown in Fig. 3(c), where a 30 fs, 720 nm “light bullet” [21]with 1.2 critical power and 150-µm waist self-channels in0.5 bar Ar without converging lens (�zdrift = 6.3 cm). Max-imum compression in time is reached at z � zc = 25 cmwhere the pulse duration attains 7 fs in terms of FWHM in-tensity, i.e., 2.9 initial optical cycles. The phase offset shouldhave no incidence in this case, which is again confirmed.Related on-axis spectra are shown in Fig. 3(d), displayingalmost no differences.

To identify the key player responsible for CEP-inducedvariations of the propagation dynamics, we modify the

nonlinear response FNL. Inspired by Loriot et al. mea-surements of nonlinear refractive indices [38], we exam-ine how differences in the initial phase can modify thepulse dynamics in the presence of a higher-order Kerr termweakening both the clamping intensity and production ofplasma. We introduce a quintic nonlinearity into (1), sim-ulating a saturating Kerr response, n2I → n2I − n4I

2, withn4 = −0.36 × 10−33 cm4/W2 in agreement with [39]. Notethat we do not address the role of higher order Kerr indices,since their relevance in reproducing quantitatively experi-mental measurements still remains questionable [40–43]. In-stead, we take advantage of reducing the plasma responsethrough the quintic saturation in order to let the overallKerr response prevail and highlight the phase-induced vari-ations driven by THG. In Fig. 4(a), we present results re-lated to the 2 µm case shown in Fig. 2, for which sig-nificant self-compression causes CEP dependence. As ex-pected, both the peak intensity and plasma density are di-

568 L. Bergé et al.

Fig. 3 Normalized on-axisspectra (a) at z = 0.2 m (dashedcurves; red and blue ones aresuperimposed) and z = 0.73 m(solid curves) for the pulsesshown in Fig. 2. (b) Electricfields (solid curves) and plasmaresponses (dash-dotted curves)at z = 1.3 m for a 720 nm,3.4-fs Gaussian pulse withw0 = 2.2 mm and Pin = 15Pcrfocused in Ar at 0.8 bar(f = 150 cm). (c) Samequantities at z = 0.25 m for a720 nm, 30-fs collimated pulsewith w0 = 150 µm andPin = 1.2Pcr in Ar at 0.5 barand (d) corresponding spectra.The black dotted curves recallthe input spectrum. Blue curves:φ = 0, red curves: φ = π/2

Fig. 4 Peak intensity (solidcurves, left-hand side axis) andelectron density (dashed curves,right-hand side axis) of the2 µm Gaussian pulse of Fig. 2with φ = 0 (blue curves) andφ = π/2 (red curves), computedfrom (1) (a) including quinticsaturation with coefficientn4 = −0.36 × 10−33 cm4/W2,and (b) with n4 = 0 and THGartificially removed

minished by the quintic saturation that promotes a longer fil-amentation range, as also reported in [42, 44]. Single-cyclepulses (∼6 fs) are achieved from z = 0.9 m, and strongCEP-induced changes occur in the nonlinear propagationdynamics. With electron densities decreased by one decade,plasma defocusing becomes irrelevant, which we verifiedby repeating this numerical run without plasma. Reversely,Fig. 4(b) exhibits the same quantities computed when n4 = 0and THG is artificially suppressed in our numerical code.The pulse attains subcycled FWHM durations (∼4 fs) fromz = 63 cm, but no significant modification of the peak in-tensity and plasma density occurs, despite a higher plasmayield. Thus, we can conclude that in filamentation regimeTHG, and more generally uneven harmonic generation pro-moted by the Kerr response, is the key mechanism responsi-ble for modifications in the pulse dynamics when consider-ing carrier waves with different initial CEPs.

4 CEP stability

An important issue is the CEP stability in filaments. Fig-ures 5(a) and 5(b) show a zoom into the trailing part of thepulses displayed in Figs. 3(c), 2(e) and 2(f). We clearly seethat the initial CEP difference of π/2 is mostly preserved,even for the newly generated high frequencies. By “mostlypreserved,” we mean that the CEP difference is maintainedalong each oscillation when the FWHM pulse duration re-mains above two laser cycles (Fig. 5(a)). In the oppositecase (Fig. 5(b)), due to a different balance between the non-linearities, high-frequency components with different phaseshifts may appear. This finding, which we checked for dif-ferent initial CEPs, is compatible with [17], where stabil-ity of CEP with small rms spread of 1.07 rad was experi-mentally reported from f − 2f measurements of filamentphases. Concerning the occurrence of a constant absolute

Role of the carrier-envelope phase in laser filamentation 569

Fig. 5 Zooms of the on-axiselectric fields at maximumcompression for the pulsesillustrated in (a) Fig. 3(c) and(b) Figs. 2(e), 2(f), showing therelative CEP preservation formost of all frequencies. The720 nm pulse with φ = 0 (bluecurve in (a)) is then compared tothe same configuration with (c)1% and (d) 5% random initialamplitude noise (green solidcurves). The CEPs of the newlygenerated high frequencies areclearly affected by the noise,whereas the pump CEP remainsalmost unchanged.(e) and (f) show a zoom of theon-axis electric field atz = 0.5 m and z = 0.73 m forthe 2 µm pulse illustrated in (b)(φ = 0) with 1% random initialamplitude noise. CEP of thenewly generated frequenciesgets more and more affectedduring nonlinear propagation

CEP over meter-range distances reported in [23], we couldnot find evidence of such a behavior, which would requirethat the nonlinearly induced change of the group velocityleads to a continuous matching with the phase velocity.

Another concern is the CEP stability against shot-to-shotvariations in filamentation regime. To test this property, weintroduced a random noise into the input field amplitude,from 0.1% up to 5% levels. Figure 5(c) presents results forthe 720 nm pulse of Fig. 3(c) with φ = 0. Near the pointof maximum compression, we can observe that the CEP ofthe original carrier wave is preserved, but the high frequencycomponents in the trailing edge, amplified by THG and self-steepening, feature additional phase-shifts. With 1% ampli-tude noise, the CEP shifts stay below 1 rad. Discrepanciesoccur with 2% random noise (not shown), while for 5% am-plitude noise CEP becomes “random” and locally shifts upto π (Fig. 5(d)). Hence, we can conclude that shot-to-shotfluctuations noticeably affect the CEP of the supercontin-uum frequencies, whereas the pump CEP remains rather sta-ble. These behaviors are again compatible with the filamentphase measurements performed in [17] at infrared wave-lengths. For comparison, Figs. 5(e) and 5(f) zoom the mostintense oscillations of the 2 µm pulse initially examined inFig. 2 with φ = 0 and now perturbed by a 1% amplituderandom noise. Already at this weak perturbation level, fluc-tuations impact the filament CEP and induce comparable π

phase-shifts over long distances. One reason for this behav-ior is the longer propagation range. Throughout our simu-

lations, we indeed recurrently observed that phase shifts in-duced by random noise perturbations become all the moreenhanced in the rear pulse as the self-channeling range islong.

5 Conclusion

In summary, changing the input phase offset alters the fila-mentation dynamics only when the pulse duration becomesnear-cycle during propagation. The FWHM intensity profilethen becomes close to one optical cycle of the input carrierwave, but can contain several intense, shorter oscillationsof the electric field, steepened by the nonlinearities. In thatcase, the CEP modifies the nonlinear polarization throughTHG and the plasma response. Among both these poten-tial players, third-harmonic generation was found to supportCEP-induced pulse variations which in turn can affect theself-compression process. Our numerical simulations indi-cate that it is possible to control the CEP in filamentationregime, provided that shot-to-shot power fluctuations remainat the percent level, and that FWHM durations cover a fewoptical cycles of the carrier wave. In the opposite case, lowerlevels of shot-to-shot noise is sufficient to jeopardize CEPconservation.

Acknowledgements This work was performed using HPC resourcesfrom GENCI-CCRT (Grant 20XX-x2010106003). The authors thankJ. Kasparian, O. Kosareva, and G. Steinmeyer for fruitful discussions.

570 L. Bergé et al.

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