self-guiding of femtosecond light pulses in condensed

Physica D 220 (2006) 14–30www.elsevier.com/locate/physd

Self-guiding of femtosecond light pulses in condensed media: Plasmageneration versus chromatic dispersion

S. Skupina,b,∗, L. Bergea

a Departement de Physique Theorique et Appliquee, CEA/DAM Ile de France, B.P. 12, 91680 Bruyeres-le-Chatel, Franceb Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller-Universitat Jena, Max-Wien-Platz 1, 07743 Jena, Germany

Received 30 September 2005; received in revised form 20 April 2006; accepted 19 June 2006Available online 27 July 2006

Communicated by J. Lega

Abstract

We investigate the nonlinear self-guiding of ultrashort laser pulses in dielectric solids, such as fused silica. Emphasis is given to the role ofchromatic dispersion compared with plasma generation. A basic set of propagation equations is derived analytically and provides a nonlinearSchrodinger model accounting for high-order dispersion, space–time focusing, self-steepening and plasma generation. Three typical propagationregimes at the laser wavelengths of 790 nm, 1550 nm and 1275 nm, respectively promoting normal, anomalous and near-zero group velocitydispersions (GVD), are examined by means of theoretical arguments and numerical simulations. It is shown that normal GVD may favor asignificant self-guiding with inessential plasma generation, which does not occur in anomalous GVD regimes. Spectral broadening at these threelaser wavelengths is commented on, in the light of the temporal distortions undergone by the pulses.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Nonlinear Schrodinger equations; Optical self-focusing; Ultrashort pulse propagation; Nonlinear dispersive waves

1. Introduction

During the past decade, the improvement of ultrashort lasersources has made it possible to access optical powers farabove the self-focusing threshold in various media, such asgases, liquids and solids. This technical breakthrough allowedus to observe light pulses subject to Kerr-induced focusingincrease their intensity and keep it clamped at high levelsover distances far exceeding their classical diffraction length.This intriguing phenomenon was first reported from infraredfemtosecond pulses launched into the atmosphere [1] andwas later confirmed by several teams [2–4]. Although theseobservations suggested a self-guided mechanism maintainingan almost constant fluence, numerical simulations revealed thatthe atmospheric propagation is highly dynamic and causessevere distortions in the pulse temporal profile through the

∗ Corresponding author at: Departement de Physique Theorique etAppliquee, CEA/DAM Ile de France, B.P. 12, 91680 Bruyeres-le-Chatel,France.

E-mail address: [email protected] (S. Skupin).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.06.006

electron plasma created in the wake of the laser field [5].Despite this complex spatio-temporal dynamics, the beamdevelops a self-guided (self-confined) state, as far as thefluence profile is concerned. It is even possible to average themodel equations over time to describe the fluence evolutionseparately [6].

The same physics was also recovered when using cells filledwith noble gases like argon [7]. Recently, the possibility oftuning the plasma response by means of an accurate controlof the pressure in argon cells was proposed as a powerfultool for producing pulses compressed to a few femtosecondsonly, without external dispersion compensation [8]. Thisproperty was experimentally evidenced in [9,10]. Besides novelperspectives in pulse shortening techniques, the long-rangepropagation of femtosecond pulses in air opens promisingtrends in Lidar remote sensing and artificial triggeringof lightning [11]. In particular, remote elemental analysisnowadays privileges “femtolidar” set-ups for multipollutantdetection at high altitude, because ultrashort pulses producerobust filaments that survive inside aerosols [12] and theyundergo a huge spectral broadening spanning from 230 nm

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S. Skupin, L. Berge / Physica D 220 (2006) 14–30 15

(through third-harmonic generation [13]) to 4.5 µm [14], whichcovers a wide range of absorption rays.

Supercontinuum generation and white light emissionoccurring from femtosecond filaments are not limitedto gaseous media. From the numerical point of view,investigations on the optical breakdown threshold in wateralready emphasized the same physical ingredients as thosegoverning the propagation of ultrashort pulses through theatmosphere [15,16]. Laser-induced breakdown studies revealedthe possibility of self-channeling ultrafast light in condensedmedia with a strong involvement of multiphoton and avalancheionizations capable of delivering free electron densitiesexceeding ∼1020 cm−3. Below such high plasma densities, aself-guided propagation of infrared pulses in fused silica wasexperimentally reported a few years ago [17]. The self-guidingmechanism was claimed to be supported by multiphotonionization to the detriment of temporal dispersion. Morerecently, long-distance propagation has been explored in liquids[18,19]. Surprisingly, in [19] the authors claimed that lightfilaments were not sustained by a balance between Kerr-induced self-focusing and plasma defocusing, but instead bya spontaneous reshaping of the Gaussian input beam into aconical wave insuring minimal nonlinear losses of multiphotontype in the presence of normal GVD. This scenario was againdefended in Ref. [20]. In none of the previous experiments,however, was specific diagnostics for plasma measurementsemployed.

To clear up this phenomenon, numerical simulations basedon an unidirectional pulse propagation equation (UPPE) [21]were performed and confirmed the long-distance propagationof ultrashort pulses in water with a minor contribution fromplasma defocusing [22]. Here, the basic structure supportingthe propagation was proposed to be a nonlinear X wave, arisingfrom the combination of diffraction, normal group velocitydispersion (GVD) and self-focusing. Similar structures werenumerically evidenced a few years ago in Refs. [23,24]. Xwaves, reported experimentally and analyzed theoretically inRefs. [25,26], result from the nonlinear space–time localizationof conical Bessel-like optical beams in normally dispersivemedia. In Fourier space, the wave self-organizes around thehyperbola defined by the diffraction/dispersion operator (k2

⊥−

ω2). X-shaped waves were analyzed in terms of steady-state“soliton-like” structures with infinite energy (or power) [26].Because there is no hope of creating exact solitons withfinite power in this case (see, e.g., [27]), X waves maynevertheless attain quasi-stationary states slowly evolving alongthe propagation axis at specific power levels [28]. Althoughnormal GVD can be a key player in arresting the wave collapsein condensed media, it is, however, not guaranteed that thisscenario systematically holds whatever the input peak powermay be. Indeed, the pioneering works [29,30] proved that GVD-induced pulse splitting is intimately linked to the weight of thedispersion length compared with the ratio of input power overcritical. The smaller the dispersion length, the more powerfulthe beam that can be split by GVD. This relationship provides acurve plotting the dispersion coefficient versus the beam power,which was identified in [30] on the basis of a Fourier analysis

[31] and divides this plane into self-focusing (collapse) anddispersion/splitting regimes. Even if current models nowadaysinclude higher-order dispersion, space–time focusing and self-steepening, we can expect that, when chromatic dispersionarrests the collapse, the input beam parameters should satisfythe previous curve, up to limited deviations. This was the case inpropagation studies applied to sapphire samples [32] and fusedsilica [33], for which the input power, although close to critical,was high enough to trigger plasma generation with a weak roleof GVD, in spite of important pulse steepening effects. Withthis purpose, it is worth exploring how the frontier betweenGVD-dominated and plasma-dominated regimes evolves in thepresence of higher-order dispersion and pulse steepening. Thisissue will be addressed in the coming analysis.

Besides normal GVD, ultrashort pulses can propagatethrough a focusing Kerr medium, for which dispersion isanomalous with coefficient k′′

≡∂2k∂ω2 |

ω=ω0< 0. With

anomalous GVD, optical waves basically obey a nonlinearSchrodinger equation, which, in (3+1)-dimensional geometry,admits an elliptic dispersion relation in the form ∼(k2

⊥+ ω2),

that allows the concentration of the beam energy inside narrowregions of space and time. Although 3D stationary ground stateswith finite energy do exist in this case, they become highlyunstable to the wave collapse [34,35]. As a result, ultrashortpulses are subject to a 3D collapsing dynamics, continuouslytransferring the energy into the collapse region, which helpsthe pulse in covering several Rayleigh lengths throughmultiple collapse events associated with plasma emission. Thisdynamics was experimentally discovered in [36] and partlyreproduced numerically. Recent theoretical investigations [37]showed that the multiple collapse events are maintained by acontinuous motion of the wave-packet to the back of the pulse,that is simultaneously compressed both in space and time.This numerical study employed a nonlinear Schrodinger modelaccounting for chromatic dispersion, space–time focusing andself-steepening, following an earlier derivation by Brabecand Krausz [38]. It also described plasma generation, thatcontributes to further shortening of the pulse duration alongthe collapse cycles. The presence of fourth-order dispersion,expected to stabilize the wave blow-up [39] in (2 + 1)-dimensional media (as, e.g., in nonlinear waveguide arrays[40]), was found to play no significant role in (3 + 1) geometry.In spite of this, we will examine below the possibility ofmaking anomalous GVD and higher-order dispersion becomekey players in self-guiding regimes, according to the input beampower involved.

In this paper, we develop a comprehensive analysis ofpulses propagating in silica samples being normally dispersive,anomalously dispersive or close to the zero-dispersion pointat the laser wavelengths λ0 = 790 nm, 1550 nm and1275 nm, respectively. In Section 2, we perform a formalderivation of our propagation equations in the limits wherevectorial effects [41] as well as nonparaxial effects [42] areignored. Free electron generation is described by multiphoton[43] and avalanche ionizing sources in solids, together with anexponential decay due to electron recombination [44]. Mainlytriggered by multiphoton transitions, the threshold intensity

16 S. Skupin, L. Berge / Physica D 220 (2006) 14–30

for nonlinear ionization in dielectrics lies around 10 TW/cm2,in agreement with the most recent measurements in this field[45]. Section 3 specifies the collapse/no-collapse regimes forboth normal and anomalous GVD. Here, “collapse” impliesan extensive generation of plasma. By means of analyticalestimates [29,30,46], it is possible to map the zones of collapse(self-focusing) as a function of the dispersive coefficientsnormalized to the Rayleigh length versus the ratio of peakinput power over critical. Special attention is paid to transitionalzones, for which higher-order dispersion may help in stabilizingthe beam at large powers. Generic features in the self-guidingat the above three selected wavelengths are discussed inSections 4, 5 and 6, respectively. Supercontinuum generation(spectral broadening) for these three dispersion regimes isanalyzed in the same sections and compared with formeranalyses [47]. Section 7 discusses the influence of time-delayedRaman nonlinearities [48,49] for the previous wavelengths.

2. Theory

In this section we derive the model equations describingthe propagation of ultrashort optical pulses in transparent,condensed media. Maxwell’s equations

∇ · EE(Er , t) =ρ(Er , t)− ∇ · EP(Er , t)

ε0(1a)

∇ · EB(Er , t) = 0 (1b)

∇ × EE(Er , t) = −∂

∂tEB(Er , t) (1c)

∇ × EH(Er , t) = EJ (Er , t)+ ε0∂

∂tEE(Er , t)+

∂

∂tEP(Er , t) (1d)

are our starting point, where the electric field EE , the polarizationvector EP , the magnetic field EH , the magnetic induction vectorEB ' µ0 EH , the carrier density ρ and the current density EJ arereal valued.

2.1. Bound electron response

We consider isotropic, nonmagnetizable media with anonlinear polarization vector. The spectral range of any field isfar from any material resonance. Then, using the conventionaldescription of nonlinear optics, EP can be expressed as a powerseries in EE :

EP(Er , ω) = EP(1)(Er , ω)+ EP

(3)(Er , ω)+ EP

(5)(Er , ω)

+ EP(7)(Er , ω)+ · · · (2a)

P( j)µ (Er , ω) = ε0

∑α1...α j

∫· · ·

∫χ ( j)µα1...α j

(Er ,−ωσ ;ω1, . . . , ω j )

× Eα1(Er , ω1) . . . Eα j (Er , ω j )δ(ω − ωσ )dω1 . . . dω j (2b)

ωσ = ω1 + · · · + ω j , (2c)

with the Fourier transform defined as F(Er , ω) ≡1

2π

∫F(Er , t)

eiωt dt . In Eq. (2a), all susceptibility tensors↔χ( j)

with evenindex j vanish due to inversion symmetry. The subscripts

µ, α1, . . . , α j in Eq. (2b) indicate the field vector componentsin Cartesian coordinates, in such a way that α1, . . . , α j have tobe summed over x , y, and z. Because | EP(3)| | EP(1)|, we shallneglect here terms of order higher than three in the expansion(2a). Note in this respect that recent studies, however, pointedout to their possible influence in attenuating the saturationintensity of self-guided femtosecond filaments [13,50–52].

The linear polarization EP(1)

can be further simplified. In

homogeneous, isotropic media the tensor↔χ(1)

is diagonal anda single element remains: χ (1)µα = χ

(1)xx δµα . Hence, with the

convention χ (1)(ω) = χ(1)xx (−ω;ω), we have

EP(1)(Er , ω) = ε0χ

(1)(ω) EE(ω), (3)

and the scalar dielectric function is defined as

ε(ω) = 1 + χ (1)(ω). (4)

The third-order nonlinear polarization EP(3)

is determined by the

81 components of the general fourth-rank tensor↔χ(3)

. Again forisotropic, centro-symmetric media the well-known relation

χ (3)µα1α2α3= χ (3)xxyyδµα1δα2α3 + χ (3)xyyxδµα3δα1α2

+χ (3)xyxyδµα2δα1α3 (5)

holds. Assuming moreover media with homogeneous nonlin-earity and quasi-linearly polarized electric fields EE = Ex Eex ,only one relevant component of the tensor remains and settingχ (3) = χ

(3)xxxx = χ

(3)xxyy + χ

(3)xyyx + χ

(3)xyxy we have

EP(3)(Er , ω) = Eexε0

∫ ∫χ (3)(−ω;ω1, ω2, ω − ω1 − ω2)

× Ex (Er , ω1)Ex (Er , ω2)Ex (Er , ω − ω1 − ω2)dω1dω2. (6)

The linear refractive index is defined by n(ω) =√ε(ω)

and the wavenumber by k(ω) = ωn(ω)/c. For a pulsed beam,whose spectrum is centered around the operating frequencyω0 = 2π/λ0 with a sufficiently small spectral bandwidth, wecan expand k(ω) in a Taylor series around ω0:

k(ω) = k0 + k′ω +

∞∑n=2

1n!

k(n)ωn, (7a)

k2(ω) = k20 + 2k0k′ω + k′2ω2

+ 2(k0 + k′ω)

∞∑n=2

1n!

k(n)ωn

+

(∞∑

n=2

1n!

k(n)ωn

)2

, (7b)

where ω = ω − ω0 and k(n) ≡ ∂nk/∂ωn|ω=ω0has complex

values in general (imaginary parts account for linear losses).The complex slowly varying envelope function EE of the electricfield EE is next defined by

EE(Er , t) =

√ω0µ0

2k0

EE(x, y, z, t)ei(k0z−ω0t)+ c.c., (8)


where E is assumed to satisfy∣∣∣∣ ∂∂zE∣∣∣∣ k0 |E | ;

∣∣∣∣ ∂∂tE∣∣∣∣ ω0 |E | . (9)

If, in a first attempt, we neglect the frequency dependency of thenonlinear susceptibility, the nonlinear polarization (6) reducesto

EP(3)(Er , t)

= ε0

(ω0µ0

2k0

) 32 [

3χ (3)(−ω0;ω0,−ω0, ω0)|E |2 EEei(k0z−ω0t)

+ χ (3)(−3ω0;ω0, ω0, ω0)( EE · EE) EEei(3k0z−3ω0t)+ c.c.

],

(10)

which formally accounts for third-harmonic (3ω0) generation(THG). Because third-harmonic fields are usually weak witha large phase mismatch 1k = 3k0 − k(3ω0) leading todestructive interference after a propagation length of 1z ∼

π/1k, THG will here be discarded. Hence, with the definitionof the nonlinear refractive index

n2(ω0) =34χ (3)(−ω0;ω0,−ω0, ω0)

ε0cn20(ω0)

, (11)

the nonlinear polarization vector EP(3) can be simply expressedas

EP(3)(Er , t) = ε0

√ω0µ0

2k0

[2n0n2|E |

2 EEei(k0z−ω0t)+ c.c.

], (12)

whenever we suppose an instantaneous response of themedium, which leads us to neglect the contribution of molecularvibrations and rotations to χ (3) (Raman effect).

Strictly speaking, however, the computation of the cubicvector EP(3) for pulses requires the knowledge of the frequencydependencies of the nonlinear susceptibility over the relevantpart of the spectrum [see Eq. (6)]. For ultrashort pulsesdeveloping a wide spectrum, the Raman gain can amplifycertain frequencies to the detriment of the others. Thisphenomenon, intimately linked to the electronic and vibrationalresponse of the atoms, implies a delocalized Raman-delayedcontribution in the Kerr response of the medium. FollowingRef. [49], a full expression for the cubic nonlinearities mustfollow from

EP(3)(Er , t) = ε0

√ω0µ0

2k0

×

[2n0n2

∫R(t − t ′)|E(t ′)|2dt ′ EEei(k0z−ω0t)

+ c.c.]

(13a)

R(t) = (1 − xd K )δ(t)

+ xd K1 + Ω2τ 2

K

Ωτ 2K

Θ(t)e−tτK sin(Ω t), (13b)

where δ is the Dirac distribution in units of s−1 and Θ isthe Heaviside function. Expression (13) takes nonresonant andincoherent nonlinear effects into account. It possesses bothretarded and instantaneous components at the ratio xd K . Theinstantaneous part ∼δ(t) describes the response from the bound

electrons: Electronic response times are a few femtoseconds orless, and therefore considerably smaller than the duration ofthe initial pulse envelopes under consideration (≥ 50 fs). Theretarded part ∼ exp(−t/τK ) accounts for nuclear responses,namely, the rotational Raman contribution, in which the delaytime τK and the inverse resonance frequency 1/Ω lie in thesame range as the pulse durations selected below.

2.2. Free electron sources

High intensities require an accurate modeling of bothionization of atoms and feedback of the generated chargecarriers to the laser field. For this purpose, Keldysh theory[43] delivers a photo-ionization rate W(|E |) for condensedmedia (crystals). This rate encompasses ionization throughmultiphoton transitions as well as tunnel ionization. Forlaser intensities ≤1013 W/cm2, multiphoton ionization (MPI)dominates and its rate is expressed as

WMPI(|E |) = σK |E |2K

= σK I K (14)

σK '2ω0

9π

(m∗ω0

h

) 32

Φ[√

2 (K − Ui/hω0)]

e2K

×

(q2

e

8m∗ω20Uiε0n0

)K

,

where Φ[x] = e−x2 ∫ x0 ey2

dy, Ui is the energy gap separatingthe valence band from the conduction band, m∗ is the reducedmass of the electron of the electron/hole pair and qe is theelectron charge. Here, K = mod(Ui/hω0) + 1 is the numberof photons necessary for liberating one electron. For higherintensities, tunnel ionization starts to contribute. In the presentscope, however, considerable ionization preceding the damagethresholds takes place around 1013 W/cm2 only. In this rangeit is thus possible to employ the MPI limit, while keeping theinteraction physics still valid.

In addition, a second ionizing mechanism occurs byavalanche (cascade) ionization of free electrons accelerated bythe laser field. This mechanism is generally weaker than MPIat intensities <1014 W/cm2. The evolution equation for theelectron density ρe can finally be completed by inclusion of theelectronic recombination. For solids, we select an exponentialfall of the electron density over times τr ∼ 150 fs (see Refs.[44,17]). The equation governing free electron generation thentakes the form

∂

∂tρe(Er , t) = (WMPI +WAI) ρnt −

ρe(Er , t)

τr, (15)

where WAI is the avalanche ionization rate and ρnt the neutralmolecular density of the medium.

Neglecting the motions of the surrounding ion background,the free electron current density EJe(Er , t) is then governed by

∂

∂tEJe +

1τ0

EJe =q2

e

meρe EE, (16)

in which diffusion is also ignored because of the short pulse du-ration. Here me is the electron mass and τ0 ∼ 20 fs denotes theelectron collision time [53]. Free electron motion is dominated


by the fast oscillation of the optical field beating at ω0, and wethus introduce a complex slowly varying envelope EJe,

EJe(Er , t) =

√ω0µ0

2k0

EJe(x, y, z, t)ei(k0z−ω0t)+ c.c., (17)

analogous to Eq. (8). We can solve Eq. (16) formally as

EJe =q2

e

meω0

(−iT +

1τ0ω0

)−1

ρe EE, (18)

where T = 1 + i∂t/ω0. Since |∂tρe EE |/ω0 |ρe EE | while1/τ0ω0 1, Eq. (18) is simplified to

EJe =q2

e

meω0

(1

τ0ω0+ iT −1

)ρe EE . (19)

Furthermore, we have to consider a second contribution to thecurrent density: The generation of free carriers by MPI. Be-cause this term will lead to the expression for the multiphotonabsorption (MPA) in our final equations, we call it EJMPA, so that

EJ (Er , t) = EJe(Er , t)+ EJMPA(Er , t), (20)

with corresponding complex slowly varying envelopes EJ , EJeand EJMPA.

Via energy conservation law, self-consistent expressions forWAI and EJMPA can be derived. The temporal evolution of thelocal energy density w is determined by

ddtw(Er , t) = EJ (Er , t) · EE(Er , t), (21)

from which we can compute the energy transferred to themedium by the pulse at the position Er in a small volume 1V =

λ30, namely,

W1V (Er)

=

∫ λ0

0

∫ λ0

0

∫ λ0

0

∫∞

−∞

EJ (Er + Er ′, t) · EE(Er + Er ′, t)dtd3Er ′

= λ30

∫ µ0q2

e

meτ0ω0k0ρe(Er , t)|E(Er , t)|2

+ω0µ0

2k0

[EJMPA(Er , t) · EE∗(Er , t)+ c.c.

]dt. (22)

Note that terms containing iT −1 cancel via integration by parts,and fast oscillations proportional to exp[±i2(k0z −ω0t)] do notcontribute to the integral. Moreover, the choice of the volume1V is arbitrary, as its dimensions have just to be large againstthe scales of the electron movement and small compared tolength scales of the slowly varying envelopes.

Assuming that all energy losses due to electron–ioncollisions (∼1/τ0) contribute to the avalanche ionization,W1V (Er) is found to contain two components:

W1V (Er) = λ30

∫ [σK |E(Er , t)|2Kρnt K hω0

+ WAI(Er , t)ρntUi

]dt, (23)

which describe the energy consumption of MPI (one transitionneeds K hω0 in energy) and the energy consumption through

avalanche ionization (resp. Ui ). Equating the expressions (22)and (23) directly yields

EJMPA =k0β

(K )

ω0µ0|E |

2(K−1) EE, (24)

where β(K ) = σK K hω0ρnt, and

WAI =σ

Uiρntρe|E |

2, (25)

where σ = µ0q2e /meτ0ω0k0.

2.3. Scalar wave equation

In the following, we adopt a scalar modeling for thepropagation of ultrashort waves, under the current assumptionE∇ · EE = 0. This assumption discards the vectorial natureof light, which manifests itself in the form of a polarizationscrambling term ∼(EkEk/k2) · EPN L in the propagation equations[39,21]. This term induces nonlinear response gradients, whichbecome effective in arresting the collapse for wave-packetsbeing extremely shrunk in space with transverse wavenumbers

k⊥ ≡

√k2

x + k2y → k0. This sharp configuration applies to

tightly collimated pulses undergoing self-focusing to utmostcompression regimes. We omit such dynamics, since plasmageneration and/or temporal dispersion should stop self-focusingon larger spatial scales. Let us briefly justify that E∇ · EE can beneglected in the derivation of the wave equation: Eq. (1a) reads

∇ · EE =ρ

ε0ε−

∇ · EP(3)

ε0ε, (26)

and we want to show that |∇ · EE | |∂x Ex |. As |n2 I | n0 =

n(ω0); it is evident that |∇ · EP(3)|/ε0ε |∂x Ex |. On the otherhand, the carrier density ρ = qe(ρion − ρe) contains both ionand electron contributions. Discarding ion contributions to thecurrent density EJ because of their large mass, the continuityequation

∂

∂tρ + ∇ · EJ = 0 (27)

can then be used to substitute ρ in Eq. (26) and obtain(∂

∂t− iω0

)∇ · [ EEeik0z

] = −∇ · [ EJ eik0z

]

ε0ε(28)

for the complex envelopes. Employing the expressions for EJeand EJM P A, the relation

∇ · [ EEeik0z] = T −1

∇ ·

(T −1 ρe

n20ρc

EE − iσρe

k0

EE

− iβ(K )|E |

2(K−1)

k0

EE)

eik0z (29)

follows, where ρc = meε0ω20/q

2e is the critical plasma

density. Here, we can drop the operator T , since condition(9) guarantees T = 1 + i∂t/ω0 ∼ I. It turns out thatρe/ρc n2

0, σρe k0 and β(K )|E |2(K−1)

k0, when wetake into account peak values for intensity and electron density


remaining below their breakdown limits. Moreover, for smallavalanche ionization one readily finds∣∣∣∣E ∂∂x

ρe

∣∣∣∣ ∼ 2K

∣∣∣∣ρe∂

∂xE∣∣∣∣ ;∣∣∣∣E ∂∂x

|E |2(K−1)

∣∣∣∣ . 2(K − 1)|E |2(K−1)

∣∣∣∣ ∂∂xE∣∣∣∣ . (30)

Because the estimation ρ/ε0ε |∂x Ex | also holds, both termson the right-hand side of Eq. (26) are small compared to |∂x Ex |.Therefore, ∇ · EE = 0 is justified and we are able to derive thewave equation

1 EE −1

c2

∂2

∂t2EE = µ0

(∂2

∂t2EP +

∂

∂tEJ

). (31)

If we make use of Eqs. (2a), (3) and (13) together with Eqs.(19), (20) and (24), we immediately get(∂

∂z+ ik0

)2

E +1⊥ E +

∫k2(ω0 + ω)E(Er , ω)e−iωt dω

= −2k2

0n2

n0T 2∫R(t − t ′)|E(t ′)|2dt ′E +

k20

n20ρc

ρeE

− ik0T(σρeE + β(K )|E |

2(K−1)E)

(32)

where E denotes the Fourier transform of E . Reframing theprevious equation into coordinates moving with the pulse, t →

t − k′z and z → z, it can be expanded as

∂

∂z

(1 + i

k′

k0

∂

∂t−

i2k0

∂

∂z

)E

−i1⊥E2k0

− iD(

1 + ik′

k0

∂

∂t+

12k0

D

)E

=ik0n2

n0T 2∫R(t − t ′)|E(t ′)|2dt ′E −

ik0

2n20ρc

ρeE

−12

T(σρeE + β(K )|E |

2(K−1)E), (33)

where D =∑

∞

n=2(k(n)/n!)(i∂t )

n is the dispersion operator.We assume 1+ik′∂t/k0 ' 1+(i/ω0)∂t = T , which amounts

to considering |(k0 − ω0k′)/k0| 1, as in Ref. [38]. Using|∂zE | k0|E | we neglect the second derivative with respect toz. Knowing that the dispersion operator D is of second orderwith respect to derivatives in time, one discards the term ∼D2

[38]. After multiplying Eq. (34) with the inverse operator T −1,the final version of the nonlinear envelope equation (NEE) reads

∂

∂zE =

i2k0

T −1∇

2⊥E + iDE − i

k0

2n20ρc

T −1ρE −σ

2ρE

−β(K )

2|E |

2K−2E + iω0

cn2T

(1 − xd K ) |E |

2

+ xd K1 + Ω2τ 2

K

Ωτ 2K

∫ t

−∞

e−t−t ′τK sin[Ω(t − t ′)]|E(t ′)|2dt ′

E,

(34a)

∂

∂tρ = σKρnt|E |

2K+σ

Uiρ|E |

2−

1τrρ, (34b)

where we introduced the change ρe → ρ for notationalconvenience. With the appropriate material parameters, thismodel is generally applicable to femtosecond pulse propagationin any transparent media. With some small modifications,e.g., changing the delayed Kerr response and/or neglecting theoperators T , T −1, it is currently used to describe the pulseevolution in air [5,6], argon [7,8], water [15,22] and dielectricsolids [17,37].

Eqs. (34a) and (34b) describe wave focusing, plasma gen-eration, chromatic dispersion with a self-consistent action ofdeviations from the classical slowly varying envelope approx-imation through space–time focusing and self-steepening op-erators [i.e., (T −1

∇2⊥E) and (T |E |

2E), respectively]. They ig-nore vectorial effects ( E∇ · EE 6= 0) and nonparaxial devia-tions (∂2

z E 6= 0), which arrest the wave collapse only whenthe beam waist becomes shrunk to about the central laserwavelength [42]. Because more important key players (chro-matic dispersion, plasma generation) are expected to halt theself-focusing mechanism before such an extreme compressionregime, we legitimately neglect these higher-order effects. Asshown in [21], Eq. (34) including dispersive contributions k(n)

with n > 3 produces similar results to the UPPE model thatavoids any Taylor expansion of the frequency-dependent quan-tities.

In fact, both UPPE and NEE formulations turn aroundconsistent treatments of the linear wave equation [∂z +

iκ(kx , ky, ω)][∂z − iκ(kx , ky, ω)]E(kx , ky, ω) = 0, in which

the dispersion relation κ(kx , ky, ω) =

√k2(ω)− k2

x − k2y is

applied to forward running pulses, for which E is nonzero onlyfor k2

x + k2y R[k2(ω)]. Basically, the scalar version of the

UPPE prescription results in a wave equation of the form [21,22]

∂z E(kx , ky, ω) = iκ E(kx , ky, ω)+ iω

2cn0PNL(kx , ky, ω). (35)

It consists of the exact linear equation for the forwardwave component, supplemented by the nonlinearities (PNL)

projected along the same propagation direction. By making useof the previous envelope substitutions and of the expansion√

1 − x = −1

2√π

∑∞

n=0 xnΓ (n − 1/2)/Γ (n + 1), where |x | =

|(k2x + k2

y)/k2(ω)| < 1, the resulting equation restores theabove-derived NEE model. Discrepancies in solving the twomodels occur in the high-frequency range, in which the spectralintensity falls more rapidly with the UPPE [21]. Let us notice,however, that this high-frequency range superimposes with thatof THG, which may change the spectra in a more importantmanner than the small differences occurring between the UPPEand NEE wave models (see, e.g., Ref. [13] for the example ofatmospheric propagation).

2.4. Physical parameters

As a result of the previous analyses, the complex electricfield envelope E(r, t, z) is governed by a nonlinear Schrodinger


Table 1Parameter values for the three wavelengths λ0 = 790, 1275 and 1550 nm

λ0 (nm) k′′ (fs2/cm) k′′′ (fs3/cm) k(4) (fs4/cm) k(5) (fs5/cm)

790 370 270 −110 3101550 −279 1510 −4930 23 2451275 −1.93 742 −1640 6166

λ0 (nm) n2 (cm2/W) ρc (cm−3) σ (cm2) σK (s−1cm2K/WK) K

790 3.2 × 10−16 1.8 × 1021 5.5 × 10−19 1.3 × 10−55 51550 2.2 × 10−16 4.6 × 1020 2.1 × 10−18 1.9 × 10−120 101275 2.5 × 10−16 6.8 × 1020 1.4 × 10−18 1.7 × 10−108 9

(NLS) equation (34a) expressed in the frame moving withthe group velocity vg = 1/k′ (t → t − k′z) andcoupled with an equation that describes the growth of thefree electron density ρ(r, t, z) [Eq. (34b)]. In Eq. (34a),k0 = n0ω0/c is the central wavenumber in silica with n0 =

1.45. D =∑5

n=2(k(n)/n!)(i∂t )

n is the dispersion operator,whose coefficients k(n) ≡ ∂nk/∂ωn

∣∣ω=ω0

are computed froma Sellmeier formula for bulk fused silica given in Ref. [54].Tailoring of the dispersion relation is limited to the fifth order;the next orders were numerically checked to have no influenceon the pulse dynamics. For current applications, we consider aninitial neutral density ρnt = 2.1 × 1022 cm−3, a gap potentialof Ui = 7.8 eV and electron recombination with characteristictime τr = 150 fs [17]. The critical power for self-focusing isgiven by the formula Pcr = λ2

0/2πn0n2.For the wavelengths investigated here, namely, λ0 = 790,

1550, and 1275 nm, the values of the dispersion coefficients,together with the MPI rate σK , photon number K , Kerrnonlinear index n2, inverse bremsstrahlung cross-sections σ ,and the plasma critical density ρc employed in Eqs. (34a) and(34b) have been specified in Table 1. Note that the values ofσK for the highest numbers K may differ by only one decade,compared with other values proposed in the literature, whichdoes not change the self-guiding dynamics. They guaranteesaturation of the optical intensity near the ionization thresholdIth ∼ 10–20 TW/cm2, which was accurately measured in [45].

Except for the simulations shown in Section 7, we consideran instantaneous Kerr response only (xd K = 0). In theforthcoming numerical integration of Eqs. (34a) and (34b), weshall employ unchirped Gaussian pulses launched in parallel

geometry, E0 =

√2Pin/πw

20 e−r2/w2

0−t2/t2p , with input waist

w0, power Pin and half-width duration tp (=FWHM/√

2 ln 2)as initial conditions.

3. Collapse versus dispersion

Here, we investigate the conditions under which collapsemay be triggered until plasma generation occurs, as a functionof the power ratio p ≡ Pin/Pcr. The principal outlookof this section is on determining analytically the zones ofthe plane (2z0k′′/t2

p, p), for which a salient Kerr focusing(collapse) takes place, which will further be arrested by plasmageneration. For technical convenience, we limit this analysisto second-order dispersion. Ignoring plasma nonlinearities

together with k(n) where n ≥ 3, our theoretical study appliesalso to the limit case T → 1. We now introduce the rescaledvariables and field z → 4z0z, r → rw0, t → t tp, E →√

Pcr/4πw20ψ , from which Eq. (34a) simply reduces to the

nonlinear Schrodinger (NLS) equation

i∂zψ + r−1∂rr∂rψ − δ(2)∂2t ψ + |ψ |

2ψ = 0, (36)

where δ(2) ≡ 2z0k′′/t2p depends on both the dispersion length

t2p/|k

′′| and the Rayleigh distance z0 = πn0w

20/λ0. The mean

square extents in space and time of ψ are governed by thefunctional relations [27]

d2z

∫r2

|ψ |2dErdt = 8

∫| E∇⊥ψ |

2dErdt − 4∫

|ψ |4dErdt, (37)

d2z

∫t2

|ψ |2dErdt = 4δ(2)[2δ(2)

∫|∂tψ |

2dErdt

+12

∫|ψ |

4dErdt]. (38)

In the case of normal GVD [δ(2) > 0], we approach Eqs. (37)and (38) by means of the two-scaled self-similar substitution

ψ =

√I (z)

R(z)√

T (z)

× e−r2/2R2(z)−t2/2T 2(z)+iRz(z)r2/4R(z)−iTz(z)t2/4δ(2)T (z); (39)

ψ0 =√

8pe−r2−t2,

yielding the dynamical system for the transverse and temporalscales R(z) and T (z):

14

R3 Rzz = 1 −p

2T;

14

T 3Tzz = δ(2)[δ(2) +

T p

2R2

]. (40)

Temporal broadening induced by GVD is described by theequation 1

4 Tzz = [δ(2)]2(1 + B)/T 3. Assuming roughlyB ≡ pT/2R2δ(2) ' const. near the focus suppliesthe temporal length T (z) = T (0)

√1 + 16[δ(2)]2(1 + B)z2.

This relation yields a typical propagation distance, z∗,along which normal GVD is efficient enough to decreasethe pulse power below critical [T (0)p(0) = T (z∗)p(z∗)

with p(0) = p and p(z∗) = 1]. With the nonlinear

focus zc = 0.367/[4√(√

p − 0.852)2 − 0.0219] proposed by

Marburger [55], setting z∗ < zc leads to the critical value


Fig. 1. Collapse regions in the plane (±δ(2), p) for (a) normal GVD (λ0 = 790 nm) and (b) anomalous GVD (λ0 = 1550 nm). The meaning of the curves andsymbols is specified in the text.

of δ(2):

δ(2)crit =

√p2 − 1

√(√p − 0.852

)2− 0.0219

0.367√

1 + B, (41)

above which dispersion dominates self-focusing (and thusplasma generation). The inequality δ(2) > δ

(2)crit restores the

pioneering result of Chernev and Petrov [29]. In this reference,a “good fit” for the parameter B is given as

√1 + B ' 5.

Fig. 1(a) shows the corresponding curve (41) as a dashed line.A similar result was established later by Luther et al. [30] onthe basis of a Fourier analysis of Eq. (36) [49]. Expressed inour notation they found

δ(2)crit =

[√3.38 + 5.2(p2 − 1)− 1.84

]×

[(√p − 0.852

)2− 0.0219

]/p, (42)

plotted as the solid line in Fig. 1(a). The two curves lie closelyto each other and divide the plane (δ(2), p) into dispersion-dominated and self-focusing-dominated regimes. To verifythese boundaries, we performed numerical integration of thecomplete system (34) with different values of p and δ(2).Results are presented in Fig. 1(a). Open circles represent theinitial conditions that do not collapse, i.e., they spread outafter a possible stage of smooth intensity growth and pulsesplitting. In contrast, closed circles mark the input conditionsthat do collapse, i.e., the wave intensity increases by morethan one decade before producing a peak electron densityρmax > 1018 cm−3. As can be seen from Fig. 1(a), theinitial conditions decaying into a collapsing state are clearlyseparated from the spreading ones in the vicinity of thetheoretical curves (41) and (42). However, by comparison withLuther et al.’s simulations (see Ref. [30]), the influence ofhigher-order temporal derivatives, together with space–timefocusing and self-steepening, makes the transition betweendispersion/self-focusing regimes less sharp than in the absenceof these operators. We can thus target an intermediate regionof relatively high powers at large normalized GVD coefficients(δ(2) ≥ 3, p ≥ 5), for which plasma may not reach a maximal

value, while a “quasi-linear” self-guiding is possible. Suchconfigurations are represented by double-circle symbols.

We now investigate under which condition collapse withanomalous GVD [δ(2) < 0] may occur, or not. Again,“collapse” means extensive plasma generation triggered byself-focusing. Since higher-order dispersion cannot arrest thewave blow-up in (3 + 1) dimensions [39], we can considerthe academic 3D NLS equation to be a good approximationof the full system (34) in the self-focusing regime. Performingthe additional substitution t →

√−δ(2)t , Eq. (36) restores this

well-known equation to

i∂zψ + ∇2⊥ψ + ∂2

t ψ + |ψ |2ψ = 0, (43)

and admits the rescaled input ψ0 =√

8p exp(−r2+ δ(2)t2).

Let us recall that by combination of Eqs. (37) and (38), oneobtains d2

z

∫(r2

+ t2)|ψ |2dErdt = 8H − 2

∫|ψ |

4dErdt , whereH =

∫(|∇⊥ψ |

2+ |∂tψ |

2− |ψ |

4/2)dErdt is the Hamiltonianbounded from below by the function F , such that

H ≥ F ≡ X − 2√

N X3/2/(

3√

3Ns

), (44)

where X =∫(|∇⊥ψ |

2+ |∂tψ |

2)dErdt is the gradient norm,N =

∫|ψ0|

2dErdt the initial mass, while Ns = 18.94denotes the mass of the ground-state soliton for Eq. (43) [46].Following this latter reference, for given initial mass N =

(2π)3/2 p/√

−δ(2), gradient norm X0 =∫(|∇⊥ψ0|

2+

|∂tψ0|2)dErdt = N (2 − δ(2)) and Hamiltonian H = X0 −

12

∫|ψ0|

4dErdt = X0 − N√

2p, three characteristic regionscan be inferred by combining sufficient conditions for collapsed2

z

∫(r2

+ t2)|ψ |2dErdt < 0 and the exact bound H ≥ F : For

X0 > X ≡ 3N 2s /N , H < H ≡ N 2

s /N [46], collapse alwaysoccurs; for X0 < X , H < H , collapse never takes place; forthe complementary intervals collapse is optional, which simplymeans that the previous mathematical arguments cannot predictthe fate of the pulse. Fig. 1(b) shows these three regions in theplane (−δ(2), p), delimited by solid lines. Again we performeddirect numerical integrations of the full system (34a) and (34b)to check the validity of these boundaries and plotted the resultsin Fig. 1(b), using the same convention (closed/open circles) as


in Fig. 1(a). We observe that the blow-up regions inferred fromthe 3D NLS equation are still valid for the full model equations.Moreover, the dashed line (X = X0) approximately divides theplane into collapsing and noncollapsing regimes, and may beused as a first indication to evaluate their boundary. Several datawere thoroughly tested by discarding either third- and/or fourth-order dispersion, or by setting T, T −1 equal to unity. None ofthese modifications, however, altered the collapse points shownin Fig. 1(b), within a power increment of 1p = 0.5. Forexample, full/open squares locate initial conditions used in thepropagation model with no fourth-order dispersion. Full/openstars convey the same information when T = T −1

= 1, buthigh-order dispersion is retained. The diamonds mark initialconditions with T, T −1 reduced to unity, while dispersiveorders beyond k′′ have been set equal to zero. Because three-dimensional collapses are easier to trigger than critical ones[this follows from the contribution of one additional dimensionand from the virial identity yielding more stringent sufficientconditions for blow-up: d2

z

∫(r2

+ t2)|ψ |2dErdt ≤ 8H , where

the equal sign only applies to a critical (2D) collapse], theseparatrix between collapsing and noncollapsing sectors seemssharper with anomalous dispersion. Note, however, that thepower threshold for collapse still increases with −δ(2). Thisinteresting feature can be understood as follows. For δ(2) → 0,the diffraction length 2z0 is much shorter than the dispersionlength L D = t2

p/|k′′| and a 2D collapse dominates for p & 1

in the (x, y) plane. Conversely, for −δ(2) → +∞, diffractionbecomes negligible and the pulse tends to evolve with a 1Dnoncollapsing dynamics.

4. Self-guiding with normal GVD

4.1. Space–time dynamics

From now on, we analyze the self-guiding properties ofultrashort pulses at different laser wavelengths. We start withλ0 = 790 nm, which corresponds to a positive k′′ in silicaglasses, as given in Table 1. A beam waist of 71 µm andpulse duration of tp = 42.5 fs are selected, so that theGVD parameter takes the value δ(2) = 1.2 for a Rayleighdistance of z0 ' 3 cm. Fig. 2 depicts the peak intensity(left column) and related electron density (right column) ofthe self-guided beam, whose dynamics varies with the inputbeam power. With p = 3, the maximum intensity smoothlyincreases, but does not develop a collapse, as the maximalelectron density, ρmax ≡ maxt ρ, remains below 1015 cm−3

[Fig. 2(a)]. In contrast, a genuine collapse event, yielding asharp focusing/defocusing cycle, is observed from p = 4,for which ρmax exceeds 1018 cm−3 [Fig. 2(b)]. This propertyagrees with the results plotted in the map Fig. 1(a), followingwhich plasma generation is expected for power levels p ≥

3.5 when δ(2) = 1.2. By augmenting the power ratio, thenumber of collapse cycles regularly increases in proportionand still gives rise to free electron emission [Fig. 2(c), (d)].Focusing/defocusing events arise in the form of very localizedspikes staying below the Rayleigh distance, which fully agreeswith the experimental observations of Ref. [36]. The self-guiding distances with [Fig. 2(d)] or without [Fig. 2(a)] plasma

Fig. 2. Normal GVD: Maximum intensity maxt I (0, t, z) (left-hand sidecolumn) and peak electron density maxt ρ(0, t, z) (right-hand side column) of71 µm, 42.5 fs Gaussian pulses with power ratios (a) p = 3, (b) p = 4, (c)p = 6, and (d) p = 8 at the wavelength of 790 nm.

Fig. 3. Normal GVD: Spatial distribution of the pulse fluence (left-hand sidecolumn) and related temporal distributions (right-hand side column) versus zfor the power ratios specified in Fig. 2. The bottom inset shows the space–timedynamics of one focusing/defocusing event for the power p = 4.

generation remain comparable and cover a propagation rangeof about 2 cm.

For the above configurations, Fig. 3 details the fluencepatterns (F ≡

∫|E |

2dt) in the radial plane versus thepropagation direction (left-hand side column) and the temporal


Fig. 4. Normal GVD: (a) Averaged spectra and (b) on-axis spectral intensity for 71 µm, 42.5 fs pulses at 790 nm. Values of the corresponding powers and propagationdistances are indicated next to the curves.

intensity distributions of the same pulses along z (right-handside column). While the power ratio p = 3 leads to asmooth self-guiding limited in intensity, “collapsing” pulsesattained from p ≥ 4 develop much smaller spots. Thesespots follow a generic dynamics, according to which the beamself-compresses in space and becomes subject to a temporalreshaping dictated mainly by the interplay between chromaticdispersion and the plasma response: While it self-focuses, thefront pulse is directed to the back part, where it develops ashock profile. By “shock”, we mean a steep gradient in time,near which low-amplitude ripples may occur. Further, the pulseundergoes plasma defocusing that rejects optical defocusedcomponents to the front zone. This behavior yields a kindof “bow”-shaped motion in the (t, z) plane, which repeatedlyoccurs when increasing the beam power. Note that more thantwo spikes may form in the pulse distribution in time, if partsof two “bows” coincide at some distance z. The bottom inset ofFig. 3 illustrates this dynamics in the (r, t) plane for p = 4. Atz = 1.2 cm, the pulse strongly shrinks in space and lets a steeptrailing edge emerge as it reaches its maximum intensity, whichis caused by space–time focusing, self-steepening and third-order dispersion [32,33,37]. Later (z = 1.4 cm), the pulse onlykeeps a leading contribution before final spreading. Note thehyperbola around which the pulse self-organizes into a conical-like wave-packet for various propagation distances. This specialgeometry can be seen as the signature of a nonlinear X wavesupporting plasma generation.

4.2. Supercontinuum generation

In connection with the temporal distortions shown in Fig. 3,we make a few comments on the spectral evolution of pulseswith normal GVD. Fig. 4 displays the pulse spectrum forp = 4 at different z distances. For comparison, two versionsof these spectra are exhibited, because experiments do notcurrently access single-shot spectra measured at the center offemtosecond filaments. Fig. 4(a) shows the averaged spectralintensity, i.e., the pulse intensity I (r, λ) expressed in Fourierspace and integrated over the diffraction plane, denoted asdE/dλ = 2π

∫+∞

0 I (r, λ)rdr ; Fig. 4(b) depicts the spectralintensity I (r, λ) computed at the center r = 0. Apart fromdifferences appearing in the intensity levels of the satellitecontributions (analogous discrepancies have been reportedin [51]), both plotted quantities restore the same spectral

broadening, i.e., the pulse first develops a strong blueshift dueto the formation of a steep trailing edge. Next, owing to plasmageneration, a leading edge remains, which makes the pulseexperiences a final spectral redshift. Note that the blueshiftseems unstable along the pulse evolution, while the redshiftbecomes more amplified as the pulse spreads out. The seconddotted curve of Fig. 4(a) confirms that the spectrum exitingfrom the sample remains unchanged at higher powers (p = 8),even after developing several focusing/defocusing sequences.

4.3. Transient regimes

To end this section, we find it instructive to exploreintermediate regions of the plane (δ(2), p), where the pulseprogressively enters collapsing zones by increasing step bystep its power. The plasma response then grows up froma negligible ionization state (ρmax < 1015 cm−3) to asaturated ionization state (ρmax > 1018 cm−3). These piecesof information are provided in Fig. 5, which applies to a42.5 fs pulse with spatial waist of 130 µm, yielding δ(2) '

4. Fig. 5(a), (b) show maximal intensities and peak electrondensities reached at power levels equal to 5.5, 6 and 7 timesthe critical value Pcr. One can see that the collapse settlesdown little by little while raising the beam power, in agreementwith Fig. 1(a). Accordingly, the associated spectra plotted inFig. 5(c) become more and more redshifted at low spectralintensities. The spectrum remains almost symmetric when p =

5.5, for which self-phase modulation is the principal factorresponsible for spectral broadening, with a weak influence fromthe asymmetric operators (T, T −1, ∂3

t , . . ., plasma response).At higher powers, in spite of a blueshift emerging when thepulse starts to form a steep trail, the resulting spectrum ismarked by a redshift occurring from the ultimate steep frontformed in plasma-induced defocusing regimes. The temporaldistortions of the pulse are specified in the bottom inset. Theyemphasize the conical shape of the pulse in the (r, t) plane,passing from a regular, symmetric distribution at low powers(p = 5.5) to an asymmetric one in higher-power regimes.

5. Self-guiding with anomalous GVD

5.1. Temporal compression and dragging

Now, we analyze the self-guiding properties of “collapsing”pulses at 1550 nm. Fig. 6 shows the maximum optical intensity


Fig. 5. Normal GVD: (a) Maximal intensity and (b) peak electron density produced by 42.5 fs, 130 µm pulses in the power interval 5.5 ≤ p ≤ 7. (c) Correspondingaveraged spectra. The bottom inset illustrates characteristic spatio-temporal distributions. The middle subplot (p = 6) only shows the amplification of the backpulse, which precedes the plasma defocusing stage responsible for the occurrence of a dispersing leading edge, comparable with the one displayed for p = 7.

Fig. 6. Anomalous GVD: Maximum intensity maxt I (0, t, z) (left-hand sidecolumn) and peak electron density maxt ρ(0, t, z) (right-hand side column) of71 µm, 42.5 fs Gaussian pulses with power ratios (a) p = 1, (b) p = 3, (c)p = 4, and (d) p = 10 at the wavelength of 1550 nm.

(left-hand side column) and peak electron density (right-handside column) for different pulse powers. The input beam waist

is again w0 = 71 µm and tp = 42.5 fs, which leads to theRayleigh length z0 = 1.5 cm and −δ(2) = 0.46. Whereaspulses with p ≤ 0.7 were seen to spread out, starting fromp = 1 yields a single collapse sequence [Fig. 6(a)], which isagain compatible with Fig. 1(b). At increasing powers, the pulseundergoes more and more focusing/defocusing events while thefirst collapse point becomes closer to z = 0. Collapse cycles arerepeated [Fig. 6(b), (c)], until the pulse develops a large “block”or “segment” of high-intensity light, through which an extendedself-guiding occurs, characterized by a quasi-constant plateauof free electrons [Fig. 6(d)].

Fig. 7 specifies the fluence distributions (left-hand sidecolumn) and temporal evolutions (right-hand side column) forthe same pulses. At p = 1 [Fig. 7(a)], the pulse focuses inits back part and plasma defocusing prevents the occurrenceof further collapse events. At higher powers [Fig. 7(b), (c)],this dynamics is amplified and promotes a long “segment” oflight. The first plasma stage defocuses the back of the pulse,so that earlier time slices can refocus at larger z distances.As these slices increase, they still compress temporally andundergo both third-order dispersion and self-steepening, whichdrag them away to more positive instants. As a result, thepulse can propagate over long distances by emitting very shorttemporal structures at the end of the propagation range.


Fig. 7. Anomalous GVD: Spatial distribution of the pulse fluence (left-handside column) and related temporal distributions (right-hand side column) versusz for the power ratios specified in Fig. 6. The bottom inset shows the space–timedynamics of one focusing/defocusing event for the power p = 4.

This phenomenon is generic and can even be amplified witha larger waist (see Ref. [37]). Here, the first collapse regionwas found to be followed by a quasi-periodic emission of lightbursts shortened in time and pushed to the very back of thepulse. The bottom inset of Fig. 7 details temporal distortionswith p = 4. At z = 4 mm, the pulse self-focuses towardspositive times and develops a shock dynamics due to self-steepening. On its trailing edge, plasma defocusing comes intoplay and permits evacuation of the energy to earlier times[z = 6 mm]. The pulse is then pushed again to the extremerear region [z = 8 mm], where it confines into a highlyfocused state. Note in this regard that, unlike normal dispersionforming X-waves, anomalous dispersion forces the wave-packet to remain compressed within an O-shaped distribution.The same dynamics supports the primary long segment of light[Fig. 7(d)], maintained by powerful time slices located near t =

0. These central slices keep their intensity close to the ionizationthreshold over ∼1 cm, while later components (t > 0) sharplymove to the back of the pulse. For powers p = 10 the backcomponents of the pulse are rapidly dragged apart to positivetimes, while central time slices are still going on actively self-focusing, which results in a rather chaotic pattern in the (t, z)plane.

5.2. Long-range propagation versus laser wavelength

By augmenting the beam power, anomalous GVD makesit possible to keep the pulse localized in bounded regions ofspace and time and to repeat the dynamics shown in Fig. 6

to later distances. This feature, already notified in [37], isillustrated in Fig. 8, where the 71 µm pulse reaches the powerlevel p = 15. For this value, the primary block decays intobursts of collapsing structures that prolong the self-channelingover more than three Rayleigh lengths (solid curves in Fig. 8).The primary extended zone of collapse occupies the firstcm range along the optical path; the secondary bursts takeplace over ∼0.5 cm each. This dynamics is again in excellentagreement with the experimental data reported in Ref. [36].The same temporal evolution as that shown in the inset ofFig. 7 drives the emergence of the small cells emitted atlonger distances. The typical waist of the filament is ∼10 µm[Fig. 8(b)]. For comparison we computed the propagation rangefor the same beam operating at the wavelength of 790 nm andhaving a power increased to 20 times critical (dashed linesin Fig. 8). We can observe that even on increasing noticeablythe beam power, self-guiding with normal GVD cannot exceedthe Rayleigh distance (z0 ' 3 cm), whereas anomalousGVD, by keeping the pulse confined in time, may repeatseveral focusing/defocusing sequences at large z far beyond theRayleigh length (z0 ' 1.5 cm).

Because anomalous GVD favors time compression, it maybe used to produce light pulses with very short durations. Thisproperty was discussed in [37], where self-focusing/defocusingcycles provided optical wave-packets with a FWHM durationshrunk from 50 fs to the limit of two optical cycles (τo.c. =

λ0/c = 5 fs). The temporal centroid of these ultraconfinedpulses experienced strong backward motions, mainly driven bythird-order dispersion.


Fig. 9(a), (b) depict the spectral broadening undergoneby the previous pulses in the presence of anomalous GVD.Because of the continuous dragging motions of the most intensetime slices to the extreme back zones (see the bottom inset ofFig. 7), the pulse develops a prominent trailing peak with steepedge, near which a shock dynamics takes place. As a result, thespectra in wavelength should be shifted to the blue side [49].This expectation is verified in the couple of spectra, numericallycomputed in their averaged [Fig. 9(a)] and on-axis [Fig. 9(b)]versions. When the pulse sharply amplifies its trailing edge,this blueshift almost gives rise to a novel wavelength at around800 nm, whereas a dip occurs at intermediate wavelengths (thesame phenomenon happened in Ref. [47], where spectra ofpulses propagating in water at λ0 = 527 nm exhibited a similardip. Note, however, that the configuration of [47] concernedself-guiding regimes with normal GVD). More precisely, aplateau in wavelengths forms early at z = 4 mm between800 and 1550 nm, and falls down in this intermediate rangeof wavelengths at z = 8 mm. We attribute this spectralevolution to pulse steepening contributions and third-orderdispersion, responsible for confining the pulse to a steep trailand for sharply reinforcing the blue side of the spectrum.Although the spectra with anomalous GVD go on beingstrongly blueshifted at higher powers, this dip may not surviveover long propagation distances [see the second dotted curve inFig. 9(a)].


Fig. 8. (a) Peak intensity of high-power pulses with 71 µm waist and 42.5 fs duration for λ0 = 790 nm (dashed curve) and λ0 = 1550 nm (solid curve). Powerlevels are indicated next to the curves. (b) Corresponding spatial extents (FWHM of the fluence distribution).

Fig. 9. Anomalous GVD: (a) Averaged spectra and (b) on-axis spectral intensity for 71 µm, 42.5 fs pulses at 1550 nm. Values of the powers and propagationdistances are indicated next to the curves.

5.4. Transient regimes

Here, we examine some transitory zones in the plane(−δ(2), p), through which new propagation regimes involvingminor participation from multiphoton ionization could arise.We investigated two configurations, namely, pulses with theinput waists w0 = 199 µm yielding δ(2) = −3.6 andw0 = 210 µm yielding δ(2) = −4, for the same initialduration as before, i.e., tp = 42.5 fs. Results reportingoptical/electron peak evolutions, spectra and spatio-temporaldynamics are summarized in Fig. 10. Fig. 10(a), (b) show thepeak intensity and electron density of pulses propagating insidethe silica sample with the input powers p = 1.7 (dashedcurves), p = 2 (dash–dotted curves) and p = 2.5 (solidcurves), while δ(2) = −3.6. These quantities immediatelyincrease until their saturation levels, induced by full ionization,as soon as p = 2. The curve corresponding to the lower powerp = 1.7 is almost invisible in Fig. 1(a), as the maximal laserintensity never exceeds 0.05 TW/cm2 in this case, i.e., the wavepropagates linearly, and no free electrons are produced. Thesebehaviors are compatible with Fig. 1(b). However, within apower increment of 1p = 0.3, no transitional regime betweena slow linear guiding and a nonlinear self-channeling occurs,unlike in normally dispersive media where the raising of ρ cangradually develop inside power intervals1p ≥ 1.5 (see Fig. 5).This corroborates the theoretical expectations of Section 3.The same conclusion holds for other values of δ(2), such asδ(2) = −4, for which plasma is suddenly generated whenincrementing the power ratio from p = 2 to p = 2.2 only(not shown here). In close connection, the spectra evolve with

a weak self-phase modulation at low powers and with a strongblueshift amplifying a spike at around 800 nm for high powers,similarly to the spectral profiles shown in Section 5.3. Thebottom inset of Fig. 10 details some characteristic stages of thepulse dynamics in the (r, t) plane. Whereas low-power pulses(p = 1.7) do not significantly change, higher-power ones aresubject to a strong self-compression both in space and time andreach their maxima on their trailing part, where they developsharp temporal gradients. This dynamics is responsible for theblueshifts shown in Fig. 10(c).

6. Self-guiding with near-zero dispersion

6.1. Nonlinear dynamics

We finally investigate propagation regimes for which k′′

becomes zero, or at least two decades below the formerdispersive coefficients for normal and anomalous GVD [56].Tuning the wavelength as closely as possible to the zero-dispersion point for silica [49], we select λ0 = 1275 nm,providing k′′

= −1.93 fs2/cm. Numerical simulations wereperformed in this configuration, using the same Gaussian pulsesas in the former sections with the beam waist w0 = 71 µmand tp = 42.5 fs. In that case, the normalized dispersionparameter takes the value δ(2) = −3.85 × 10−3, i.e., closeto zero. Computation results plotted in Fig. 11 first emphasizea slight intensity growth with no plasma emission and finalbeam spreading at the power level p = 1 [Fig. 11(a)].Genuine collapse events happen from p ≥ 1.5 [Fig. 11(b),(c), (d)]. This behavior can be interpreted as a mixture of


Fig. 10. Anomalous GVD: (a) Maximal intensity and (b) peak electron density produced by 42.5 fs, 199 µm pulses within the power interval 1.7 ≤ p ≤ 2.5 [δ(2) =

−3.6]. (c) Corresponding averaged spectra. The bottom inset illustrates some characteristic spatio-temporal distributions.

Fig. 11. Near-zero GVD: Maximum intensity maxt I (0, t, z) (left-hand sidecolumn) and peak electron density maxt ρ(0, t, z) (right-hand side column) of71 µm, 42.5 fs Gaussian pulses with power ratios (a) p = 1, (b) p = 2, (c)p = 4, and (d) p = 10 at the wavelength of 1275 nm. Note the change of scalesbetween (a) and (b), (c), (d).

Fig. 1(a) and Fig. 1(b) in the limit of no second-order dispersion[δ(2) → 0]. The most suitable mapping for determining thecollapse/no-collapse regions in the plane (δ(2), p) seems tobe Fig. 1(a). However, the propagation dynamics suggestedby the peak intensities and electron densities resembles morethat characterizing anomalous GVD: As the pulse power isaugmented, focusing/defocusing cycles develop [Fig. 11(c)],until a quasi-constant plateau in maxt I and maxt ρ emerges,creating a light “block” sustaining a long waveguide over about1 cm.

In spite of these analogies, a detailed look at the fluenceand temporal distributions presented in Fig. 12 allows us tounderstand that, in fact, both normal and anomalous GVDdynamics are somewhat mixed in this peculiar regime. Indeed,the spikes attached to the collapse events belonging to thefirst “collapse block” are still discrete and very localized asin Fig. 3, while the temporal patterns cannot avoid the “bow-shaped” motions owing to the normal GVD propagation. Notethe asymmetry of each bow, along which anomalous dispersion,even small, pushes the pulses to the extreme back parts. Here,self-induced ionization generically defocuses the trailing edgeand pushes the spreading components of the pulse to its frontzone. This point features an important difference comparedwith anomalously GVD regimes examined in Section 5:Because the time dimension cannot fully contribute to anefficient 3D compression, the front pulse is able to survive


Fig. 12. Near-zero GVD: Spatial distribution of the pulse fluence (left-handside column) and related temporal distributions (right-hand side column) versusz for the power ratios specified in Fig. 11. Note the change of scales between(a) and (b), (c), (d). The bottom inset shows the space–time dynamics of onefocusing/defocusing event for the power p = 4.

after the plasma defocusing stage. This does not preventthe optical space–time distribution from developing a markedshock dynamics on the trail edge (see the bottom inset at z =

6 mm).


The spectra, shown in their averaged version for p = 4 inFig. 13, are broadened accordingly. A sharp blueshift occurswith the amplification of the trailing edge [z = 6 mm]. Atfurther distances [z ≥ 1 cm], this blueshift relaxes in widthto some extent, but not in intensity, while the red side of thespectrum becomes more prominent. At large distances [z =

2.8 cm], the spectrum has been enlarged in a way comparablewith the usual anomalous GVD spectra [see Fig. 10(a)],although it exhibits a more symmetric shape.

In summary, near-zero dispersion propagation regimesappear characterized by a “weaker” collapsing dynamics thanones operating at larger wavelengths with stronger anomalouscoefficient. Weakness in the “anomalous dispersion” makesthe pulse go back to its front zone after undergoing plasmadefocusing, similarly to normal GVD regimes. Also, Fig. 11(a)supplies evidence of the possibility for self-guiding such pulsesover long distances at power levels greater than or equal tocritical, with no plasma generation at all. In spite of thesefeatures resembling properties met in normally dispersive

Fig. 13. Near-zero GVD: Averaged spectra of 71 µm, 42.5 fs pulses at 1275 nm.Values of the powers and propagation distances are indicated next to the curves.

Fig. 14. Simulations including Raman scattering: (a) Peak intensities, (b) peakelectron densities, and temporal distributions for the (a) 790 nm pulse, (b)1550 nm pulse and (e) related averaged spectra. Solid, dotted and dash–dottedcurves refer to the pulse central wavelengths of 790 nm, 1275 nm, and 1550 nm,respectively. In (c), (d), horizontal axes refer to the propagation direction withsame labels and numbers as in (a), (b). In (e), computation distances areindicated next to the curves.

media, we can mention that there is still a strong tendencyof amplifying the trailing edge of the pulse, which makessupercontinuum generation marked by a persistent blueshift.

7. Influence of Raman scattering

Because Raman scattering may substantially alter theprevious spectra, we here briefly examine the consequences ofa delayed Kerr response on the propagation dynamics of theprevious pulses for the power level p = 4 at the differentwavelengths λ0 = 790, 1550 and 1275 nm. Following [48], wesimulate the Kerr function (13) employing the values xd K =

0.15, ΩτK = 4.2 with τK = 50 fs for the wavelength λ0 =

800 nm. For the other two wavelengths, we choose xd K = 0.18,ΩτK = 2.6 with τK = 32 fs [49].

Results are summarized in Fig. 14. Fig. 14(a), (b) showthe peak intensity and electron density reached along z for


these three configurations. Fig. 14(c), (d) detail the temporaldistributions in the (t, z) plane for the 790 nm and 1550 nmpulses, respectively. Fig. 14(e) illustrates the spectra for thethree previous wavelengths, computed at the same maximaldistances as those for which the influence of the Ramanresponse was formerly omitted. In the normal GVD regime(solid curves), the only relevant difference from computationsignoring the Raman response [Fig. 2(b)] lies in the peakintensity that decreases by the factor ∼(1 − xd K )

1/(K−1) withK = 5 for similar values of ρmax. Because the temporaldynamics looks unchanged [see Fig. 14(c) versus Fig. 3(b)], nonoticeable modification arises in the spectra [Fig. 14(e)]. Nearthe zero-dispersion point (dotted curves), similar conclusionshold. Discrepancies in the maximum intensity become moreattenuated with K = 9, but the effective weight of Ramanscattering [xd K (1 + Ω2τ 2

K )tp/Ωτ 2K ∼ 0.71] increases its value

compared with that for the wavelength of 790 nm [resp. ∼0.57].In the anomalous GVD regime, the peak intensity remains ofcomparable height (K = 10). However, the second focusingcycle at z ∼ 1.7 cm appears more prominent [Fig. 14(d) versusFig. 7(c)], as the instantaneous Kerr contribution is diminishedto some extent and since the Raman term shifts by ∼τK thetotal Kerr response to positive times (see, e.g., Ref. [57]).This causes a dip in the peak intensity [Fig. 14(a)], whereasthe same intensity is maintained to a nearly constant level∼5 TW/cm2 at similar distances if xd K = 0. Apart from theseminor differences (some of them were re-checked at higherpower levels), we do not observe any salient modification inthe nonlinear evolution of the femtosecond filaments whenintroducing Raman scattering for the above parameters.

8. Conclusion

In summary, we investigated the nonlinear self-channelingregimes of ultrashort laser pulses propagating in silica samplesat different central wavelengths making the group velocitydispersion normal, anomalous or close to zero. We analyticallydetermined the zones in the plane (2z0/L D, Pin/Pcr), wherethe collapse is strictly forbidden and where it triggers plasmageneration. Special care was taken in exploring transitionregions, for which the collapse can progressively take place byincreasing the beam power and monitoring the plasma response.

In normally dispersive regimes, high-power pulses capableof triggering ionization exhibit the same temporal dynamics,characterized by an amplified trailing edge caused by pulsesteepening and third-order dispersion. This stage is followedby plasma defocusing, that rejects the defocused opticalcomponent towards the front zone. The spectral dynamics isthus marked first by a blueshift, that further relaxes to thebenefit of a redshift. Importantly, normal GVD allows fortransitional regimes in which chromatic dispersion dominatesplasma generation in halting the collapse at powers far abovecritical. This property may explain similar conclusions drawnabout pulse self-guiding in condensed media, e.g., water [18,20,22].

In anomalously dispersive regimes, the beam is clampedover longer distances at its peak saturation intensity, because the

pulse temporal components are always compressed and shiftedto the back of the pulse through self-steepening and third-orderdispersion. This evolution favors the formation of narrow, self-compressed cells of light, whose typical duration can reach theoptical cycle limit. From the spectral point of view, the temporaldragging to the most extreme back regions and the resultingshock dynamics promote a prominent blueshift in the spectra.This blueshift is stable along the propagation axis, apart fromthe dip dug between a novel apparent wavelength emerging at∼800 nm and the fundamental at 1550 nm. The origin of thissecond spectral peak still remains unexplained. The transitionbetween spreading and collapsing regimes appears to be muchsharper than in normally dispersive media, which may be lessconvenient for monitoring the plasma response by varying thebeam power.

Near-zero dispersion regimes combine properties of bothkinds of dispersions. Temporally, they promote a “bow-shaped”dynamics in the (z, t) plane, which produces a shock dynamicson the trailing edge of the pulse, but keeps a significant leadingedge when the plasma is turned off. As a result, supercontinuumgeneration appears more symmetric than in the two previouspropagation regimes.

Accounting for a Raman-delayed Kerr response modifies theabove features by decreasing to a small extent the effectivenumber of critical powers contained in the earliest time slicesof the pulse. Apart from tiny variations in the temporal profiles,Raman scattering, at least for the cases investigated above, doesnot drastically modify the pulse dynamics and related spectra.

Acknowledgements

The authors thank Dr. Rachel Nuter for illuminatingdiscussions on ionization properties of silica. Numericalsimulations were performed on the COMPAQ alpha cluster ofthe CCRT at CEA-France and on the IBM p690 cluster JUMPof the Forschungs-Zentrum in Julich-Germany.

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self-guiding of femtosecond light pulses in condensed

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