PORRAS MAPAROLA, ALBERTOFACCIO, DANIELE FRANCO ANGELCOUAIRON ADI TRAPANI, PAOLOmostra contributor esterni nascondi contributor esterni 

Light-filament dynamics and the spatiotemporal instability of the Townes profile

Miguel A. Porras,

¹

Alberto Parola,

²

Daniele Faccio,

²

Arnaud Couairon,

³

and Paolo Di Trapani

^2,4

1

Departamento de Física Aplicada, Universidad Politécnica de Madrid, Rios Rosas 21, ES-28003, Spain

2

INFM and Department of Physics, University of Insubria, Via Valleggio 11, IT-22100 Como, Italy

3

Centre de Physique Théorique, CNRS, École Polytechnique, F-91128, Palaiseau, France

4

Department of Quantum Electronics, Vilnius University, Sauletekio 9, LT 01222, Vilnius, Lithuania

共Received 30 October 2006; published 12 July 2007

兲

The origin of the spatiotemporal filament dynamics of ultrashort pulses in nonlinear media, including axial-conical emission coupling, temporal splitting, and X waves, is explained by the spatiotemporal instability of spatially localized nonlinear modes. Our experiments support this interpretation.

DOI:

10.1103/PhysRevA.76.011803

PACS number共s兲: 42.65.Sf, 42.65.Jx

From optical pulses to Bose-Einstein condensates, from plasma instabilities to hydrodynamical or optical shocks, many nonlinear wave processes involve a dramatic increase of intensity along with compression into a strongly localized state, followed by relaxation into linear waves. In the case of ultrashort optical pulses in nonlinear self-focusing media, the dynamics generally develops in three dimensions 共time and two spatial dimensions 兲, but due to standard experimental conditions, the compression stage is dominated by a spatial self-focusing toward a strongly localized wave in space only.

In the subsequent filamentary and eventual relaxation re- gimes an apparent stationarity hides a rich, fully spatiotem- poral dynamics 关 1 兴. Temporal pulse splitting is accompanied by the emission of new temporal frequencies 共axial spectral broadening兲, and spatial frequencies 共off-axis, or conical emission兲, organized into X-shaped patterns in Fourier space 关2,3兴. A recent interpretation in terms of relaxation into mul- tiple spatiotemporal X waves harmonizes the ultimate sta- tionarity with some of these highly dynamic phenomena 关4兴.

In this Rapid Communication, we aim at clarifying the mechanisms responsible for the onset of the full spatiotem- poral dynamics in the filamentary regime. We build a unified interpretation based on the spatiotemporal instability of strongly localized waves in space, such as those formed upon Kerr self-focusing 关5兴, and that points out a common origin of temporal splitting, axial and conical emissions, and X waves. As a minimal model, we study the temporal instabil- ity with normal dispersion of the ground state of the cubic nonlinear Schrödinger equation 共NLSE兲 with two spatial di- mensions, or Townes profile 共TP兲 关6,7兴, which mimics the localized spatial profile generated at each refocusing event within an ultrashort filament 关 8 兴. Our numerical calculation of the unstable modes for all perturbation frequencies en- ables us to show for the first time that the spatiotemporal instability of a localized beam as the TP is featured by a couple of Y-shaped unstable modes that split each other in time, link axial and conical emission, and act as precursors of X waves. A four-wave mixing-based reinterpretation of the Y-shaped instability supports its generality for spatially localized waves, and its application to interpret filament dy- namics. Y-shaped unstable modes are observed at the initial stage of filamentation in our simulations and experiments.

We consider the simplest, cubic NLSE with normal time dispersion 关9,10兴:

⳵

z

A = i 2k

₀

⳵

r

共r ⳵

r

A兲 r − ik

₀

⬙

2 ⳵

t²⬘

A + i ␻

0

n

₂

c 兩A兩

²

A, 共1兲 共k

₀

⬙ ⬎0兲 describing the dynamics of a cylindrically symmet- ric wave-packet E = A共r,t,z兲exp共−i ␻

0

t + ik

₀

z兲 of carrier fre- quency ␻

0

. In 共1兲, r is the radial coordinate in a plane trans- verse to z, t ⬘ ^{= t − k}

0

⬘ ^{z, k}

₀^共n兲

⁼

兩dⁿ

k共 ␻ 兲/d ␻

ⁿ

兩

_␻0

, with k共 ␻ 兲 the propagation constant in the medium, n

₂

⬎0 is the nonlinear refractive index, and c is the speed of light in vacuum. We also consider the monochromatic, stationary, and exponen- tially localized solution A = I

^1/2

a

₀

共 ␳ 兲exp共i ␣␨ 兲 to 共1兲, where I is the peak intensity, ␣ ⯝0.2055, and a

0

共 ␳ 兲 is the TP 关see Fig.

1共a兲兴 关7兴. The scaled axial and radial coordinates are ␨

= k

_NL

z and ␳ ⁼ 共k

NL

k

₀

兲

^1/2

r, where k

_NL

= ␻

0

n

₂

I / c.

To perform the spatiotemporal instability analysis, we also introduce the dimensionless time ␶ ⁼ 共k

NL

/ k

₀

⬙ 兲

^1/2

t ⬘ ^and envelope a = A / I

^1/2

to rewrite 共1兲 in the form

(a)

(b)

(c)

(d)

FIG. 1. 共Color online兲 共a兲 Normalized TP. 共b兲 Real and imagi- nary parts of the relevant complex eigenvalue. Dashed curve:

⌰²/ 2 +␣. 共c兲 Modulus of the spatiotemporal spectrum pˆ_␬共Q,⌰兲 of

p

_␬. The amplitudes for different⌰ are arbitrarily chosen so that the energy兰0⬁兩pˆ␬兩²

Q dQ is independent of

⌰. 共d兲 Thick solid and dashed curves: Characterization of the spatiotemporal spectrum of the un- stable perturbations p_␬and p_−␬쐓by the dominant radial frequencies

Q

_u,v. Dotted curve: maximum gain curve of the plane wave. Thin solid and dashed curves: double-X spectrum.

PHYSICAL REVIEW A 76, 011803共R兲 共2007兲

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⳵

_␨

^{a = i} 2

⳵

_␳

共 ␳⳵

_␳

a兲

␳ ⁻

i

2 ⳵

_␶²

a + i兩a兩

²

a. 共2兲 Following a standard procedure 关11兴, we consider the spa- tially and temporally perturbed Townes beam

a = 兵a

0

共 ␳ 兲 + ⑀ 关u共 ␳ 兲e

^{−i⌰␶+i␬␨}

+ v

^쐓

共 ␳ 兲e

^{i⌰␶−i␬쐓␨}

兴其e

^i␣␨

, 共3兲 where ⑀ Ⰶ1, and where ⌰艌0 is the absolute temporal fre- quency shift 关in units of 共k

NL

/ k

₀

⬙ 兲

^1/2

兴 of the perturbation with respect to the Townes frequency. Note that u and v

^쐓

in 共3兲 are oppositely shifted by +⌰ and −⌰. The perturbation 共3兲 in the NLSE 共2兲 leads 共up to the first order in ⑀ 兲 to

冉 ^{− a H}

02

− H ^a

⁰²

冊 冉 ^{u v} 冊 ^{= ␬} 冉 ^{u v} 冊 ^{, 共4兲}

where H =

¹₂

关d

²

/ d ␳

²

⁺ 共1/ ␳ 兲d/d ␳ ⁺ ⌰

²

兴− ␣ + 2a

₀²

共 ␳ 兲. If this ei- genvalue problem for given ⌰ admits a solution 共u,v兲⫽0 with eigenvalue ␬ ^with ␬

I

⬅Im ␬ ⬍0, then the associated perturbation 共3兲 will grow exponentially with gain − ␬

I

, and the TP will be unstable under perturbations at frequency ⌰.

Note also that, if 共 ␬ , u , v 兲 is a solution of 共4兲 with gain − ␬

I

, then 共− ␬

^쐓

,v

^쐓

, u

^쐓

兲 is another solution with the same gain.

These solutions represent the two growing, physically dis- cernible perturbations

p

_␬

共 ␳ ^, ⌰兲 = 关u共 ␳ 兲e

^{−i⌰␶+i␬R␨}

+ v

^쐓

共 ␳ 兲e

^{i⌰␶−i␬R␨}

兴e

^{−␬I␨+i␣␨}

, p

_−␬쐓

共 ␳ ^, ⌰兲 = 关v

^쐓

共 ␳ 兲e

^{−i⌰␶−i␬R␨}

+ u 共 ␳ 兲e

^{i⌰␶+i␬R␨}

兴e

^{−␬I␨+i␣␨}

,

共5兲 since u and v

^쐓

are generally different and oppositely fre- quency shifted in p

_␬

and p

_−␬쐓

. Consequently, p

_␬

and p

_−␬쐓

may grow independently, depending on how the instability is seeded.

We have solved the problem 共4兲 numerically for each ⌰ 艌0 by discretization of the differential operators on a ␳ ^grid of finite size much larger than the Townes range d⬃1.5.

Increasing grid points and grid size allowed us to control the accuracy of the results. For ⌰=0, no complex eigenvalue is found. This is in agreement with the fact that the TP does not present exponential gain under spatial perturbations, but only algebraic instability. For each ⌰⬎0, only one pair of eigen- values ␬ ^{and −} ␬

^쐓

with negative imaginary part appear 关see Fig. 1共b兲 for ␬ 兴. The TP is then retrieved as modulationally unstable under spatiotemporal perturbations 关12兴, with gain

− ␬

I

limited to ⌰ⱗ1.5, but without an abrupt cutoff. Unlike previous generalizations of Zakharov’s results on the insta- bility of nonlinear waveguides 关 11 兴, our results are not lim- ited to long wavelengths. The study of the unstable perturba- tions presented below leads, moreover, to a unified interpretation of the filament dynamics.

Figure 1 共c兲 shows the results of the numerical calculation of the Hänkel transform p

_␬

共Q,⌰兲 of the unstable perturba- tion p

_␬

共 ␳ ^, ⌰兲 at each frequency shift ⌰, where Q is the radial frequency 关in units of 共k

NL

k

₀

兲

^1/2

兴. The reflection of Fig. 1共c兲 about ⌰=0 yields the Hänkel transform p

_−␬쐓

共Q,⌰兲 of p

_−␬쐓

共 ␳ ^, ⌰兲. These Q-⌰ spectra, or spatiotemporal spectra, are particularly useful in experiments with ultrashort pulses.

Their measurement 共in practice, measurement of the angu- larly resolved spectrum, displaying angles and wavelengths 兲 is a powerful diagnostic from which many aspects of the spatiotemporal dynamics can be inferred 关2兴. For the un- stable perturbation p

_␬

共Q,⌰兲 of the TP, the branch u at +⌰ is peaked about Q

u

⬃0 for all +⌰, and hence can be identified with an axial emission at upshifted frequencies. The branch v at − ⌰ presents instead a sharp maximum at increasing radial frequency with detuning, whose location fits well to Q

_v

⯝ 冑 2⌰, and can be identified with a conical emission at downshifted frequencies. For p

_−␬쐓

共Q,⌰兲, axial emission is instead associated with downshifted frequencies and conical emission with upshifted frequencies.

These features also follow from an asymptotic analysis.

Neglecting terms with a

₀²

in 共 4 兲 and H for ␳ →⬁, the problem 共4兲 becomes uncoupled for u and v, and admits an analytical solution. The bounded solutions for ␳ →⬁ are u ⬀H

₀^共1兲

关共Q

u

+ i ⌫

u

兲 ␳ 兴 and v⬀H

₀^共1兲

关共Q

v

+ i ⌫

v

兲 ␳ 兴, where H

₀^共1兲

is the Hänkel function of first class and zero order. The real quantities Q

_u,v

and ⌫

u,v

are defined by

Q

u

+ i⌫

u

= 冑 ⌰

²

− 2 ␣ ^{− 2} ␬ ^{, Q}

v

+ i⌫

v

= 冑 ⌰

²

− 2 ␣ ^{+ 2} ␬ ^, 共6兲 with the convention of taking square roots such that

⌫

u,v

艌0. Using H

₀^共1兲

共s兲⬃ 冑 ^共2/ ␲ s兲exp关i共s− ␲ / 4兲兴 for large 兩s兩, and ignoring constant factors and algebraical decay, u and v are found to be dominated at large ␳ by the damped oscillat- ing behavior exp 共−⌫

u,v

␳ ^{+ iQ}

u,v

␳ 兲, with radial frequencies Q

_u,v

. The thick curves in Fig. 1 共d兲 represents dominant radial frequencies Q

u

and Q

_v

, obtained from 共6兲, versus their re- spective frequency shifts, +⌰ and −⌰ for the perturbation p

_␬

共solid curve兲 and −⌰ and +⌰ for the perturbation p

_−␬쐓

共dashed curve兲. The dominant radial frequency Q

u

is close to zero, while Q

_v

is well described by Q

_v

⯝ 冑 2⌰, as obtained above from the numerical evaluation.

Let us compare the spatiotemporal instabilities of the TP and of the plane wave solution a = exp共i ␨ 兲 to the NLSE 关9兴.

The problem 共4兲 holds in the latter case if a

0

and ␣ ^are replaced by unity. The functions u and v associated with the most unstable perturbations p

_␬

and p

_−␬쐓

at each frequency ⌰ are two plane waves with the identical transverse wave num- bers Q

u

= Q

_v

= 冑 _⌰

²

+ 2, with ⌰-independent gain − ␬

I

= 1. The spectra p

_␬

共Q,⌰兲 and p

_−␬쐓

共Q,⌰兲 are then characterized by two identical hyperbolas, depicted in Fig. 1共d兲 as a dotted curve 共see Ref. 关9兴 for more details兲.

To summarize, if ⍀ denotes the physical 共positive or negative兲 temporal frequency shift, and K

_⬜

the physical ra- dial frequency, the instability spectrum of the plane wave is characterized by the hyperbola K

_⬜

= 冑 ^2k

0

共k

NL

+ k

₀

⬙ ⍀

²

/ 2兲 for arbitrary ⍀, and the instability spectrum of the TP by the two Y-shaped curves K

_⬜

⯝共− 冑 2k

₀

k

₀

⬙ ⍀ if ⍀⬍0,0 if ⍀⬎0兲 共and its reflection about ⍀=0兲, limited to 兩⍀兩ⱗ1.5 冑 k

_NL

/ k

₀

⬙ ^{. Note} the 冑 2-fold slope of the Y arm in comparison with those of the hyperbola. These Y-shaped spectra point out a common origin for axial and conical emission. Typical filament spec- tra, displaying on-axis and X-shaped off-axis radiation, are reinterpreted here as originating from two superimposed

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011803-2

Y-shaped spectra, each one linking axial to conical emission at opposite frequency bands.

The Y-shaped instability admits a simple interpretation and generalization as a four-wave-mixing interaction driven by the Kerr nonlinearity. Consider two intense, identical pump waves of frequency ⍀=0 propagating along the z di- rection, and amplifying two weak, noncollinear plane waves, u and v, at frequencies ± ⍀. The axial projection of the wave vector of each wave is expressed by k

z

= k

₀

+ k

₀

⬘ ⍀+k

₀

⬙ ⍀

²

/ 2

− K

_⬜²

/ 2k

₀

+ ⌬k

NL

up to second order in dispersion 关as in the NLSE 共 1 兲兴. The last two terms account for axial wave vector shortening due to noncollinearity 共for u and v兲 and lengthen- ing due to either self-phase 共for the pump p兲 or cross-phase modulation 共for u and v兲. If the pumps are plane waves, the conditions of axial and transverse phase matching 共k

z,u

+ k

_z,v

= 2k

_z,p

and K

_⬜,u²

= K

_⬜,v²

⬅K

_⬜²

兲, and that the cross- phase-modulation wave vector shift is twice the self-phase- modulation shift 关13兴 共⌬k

NL,u

= ⌬k

NL,v

= 2⌬k

NL,p

= 2k

_NL

兲, lead immediately to the hyperbolic curve K

_⬜

= 冑 ^2k

0

共k

NL

+ k

₀

⬙ ⍀

²

/ 2兲.

For transversally localized pump waves, however, effi- cient amplification is possible with transverse mismatch such that 兩K

_⬜,u

− K

_⬜,v

兩⬍ ␲ ^{/ d} 关14兴, where d⬃1.5/共k

NL

k

₀

兲

^1/2

is the width of the TP. We add here the hypothesis that, owing to pump localization, plane waves collinear to the pump are preferentially amplified. We then take u, e.g., as collinear 共K

_⬜,u

⯝0兲, with ⌬k

NL,u

= 2⌬k

NL,p

= 2 ␣ ^k

NL

. Axial phase matching is then seen to require v to be noncollinear 共K

_⬜,v

⫽0兲. It is then reasonable to take ⌬k

NL,v

⯝0 for its cross-phase modulation, 关4兴 since v does not remain in the localized interaction area of the TP. With these assumptions, the sole condition of axial phase matching 共k

z,u

+ k

_z,v

= 2k

_z,p

兲 leads to the Y arm K

_⬜,v

⯝− 冑 2k

₀

k

₀

⬙ ⍀. Axial phase matching is then responsible for axial-conical coupling, amplification in this configuration being possible due to the transverse lo- calization of the pump. Also, limitation of the gain band- width is a consequence of the maximum allowed transverse mismatch: if K

_⬜,u

⯝0, then 兩K

_⬜,v

兩ⱗ ␲ / d, which is satisfied for 兩⍀兩 ⱗ1.5 冑 k

_NL

/ k

₀

⬙ ^.

The independence of the four-wave-mixing analysis on the particular localized profile stresses the generality of the Y-based time-to-space coupling mechanism, which is ex- pected to be triggered when an intense wave remains tightly focused over a distance sufficient for a significant energy to be transferred to the phase-matched frequencies. Pulse tem- poral splitting is also accounted for by this mechanism. If the perturbation p

_␬

, e.g., is seeded coherently at different fre- quencies ⌰, its growth leads to the formation of a pulse.

Note that ␬

R

fits well to ⌰

²

/ 2 + ␣ around the values of ⌰ where the gain takes higher values 关dashed curve in Fig. 1共b兲兴. The axial wave vector shift ␬

R

+ ␣ of the branch u of the perturbation p

_␬

in 共5兲 is then approximated by

⌰

²

/ 2 + 2 ␣ , and the axial wave vector shift − ␬

R

+ ␣ ^{of the} branch v by −⌰

²

/ 2. The wave-mixing analysis yields the same values, which in physical units read k

_z,u

⯝k

0

+ k

₀

⬘ ⍀ + k

₀

⬙ ⍀

²

/ 2 + 2 ␣ ^k

NL

共for ⍀⬎0兲, and k

z,v

⯝k

0

+ k

₀

⬘ ⍀−k

₀

⬙ ⍀

²

/ 2 共for ⍀⬍0兲. This means that the axial part of the Y wave experiences the normal dispersion of the material, and the conical part the opposite anomalous dispersion as an effect

of its angular dispersion. Consequently, the 共inverse兲 group velocities of the axial part, v

_g⁻¹

= dk

_z,u

/ d⍀ evaluated at posi- tive ⍀, and of the conical part, v

g−1

= dk

_z,v

/ d⍀ evaluated at negative ⍀, are identical and equal to k

₀

⬘ ^{+ k}

0

⬙ 兩⍀兩. The Y wave then propagates as a whole at a well-defined group velocity.

For p

_−␬쐓

or a reflected Y wave, the group velocity is instead k

₀

⬘ ^{− k}

0

⬙ 兩⍀兩, and the group velocity mismatch between the two Y waves is 2k

₀

⬙ 兩⍀兩. At the stage of well-developed instability, the spectrum will exhibit sidebands at the maximum gain frequencies 兩⍀兩⯝0.4共k

NL

/ k

₀

⬙ 兲

^1/2

, and hence the group veloc- ity mismatch approaches 0.8共k

NL

k

₀

⬙ 兲

^1/2

= 0.8共 ␻

0

k

₀

⬙ ⁿ

2

I / c兲

^1/2

, in agreement with the dependence on pump intensity and ma- terial properties in filamentation experiments 关 2 兴.

It follows from our analysis that the onset of downshifted axial emission must be accompanied by upshifted conical emission, and vice versa, and that these two events may oc- cur independently. Our simulations and experiments support this interpretation. First, we perturbed asymmetrically a TP by making it pulsed 共i.e., nonmonochromatic兲 with unequal durations in its leading and trailing parts 共dimensionless du- rations ⌬ ␶ = 0.87 and 29兲. This perturbation, however, does not break the initial symmetry about ⌰=0 of the spectral intensity in the Q- ⌰ plane. Propagation under the NLSE 共 1 兲 accounts for dispersion and nonuniform spatial dynamics at different slices of the pulse, and results in an asymmetric spectrum, which, subtracted from the input one 共to visualize the newly generated frequencies 兲, yields the Y-shaped spec- trum of Fig. 2共a兲 共at ␨ = 7.6兲, with the off-axis tail fitting to the expected slope 冑 2 共thick line兲.

Y-shaped spectra are also formed at the earlier stage of filamentation. We simulated in real-world variables the fila- mentation in water of an input 共z=0 cm兲 Gaussian pulse of 200 fs duration 共full width at half maximum兲, 76 ␮ m Gauss- ian width, and energy 2 ␮ ^J 共peak intensity I

0

(a)

(b)

(c)

(d)

FIG. 2. 共Color online兲 Simulated spatiotemporal spectra 共in logarithmic scale, 10 decades plotted兲 of 共a兲 NLSE-propagated in- put pulsed TP共see text兲, with the input spectrum subtracted; 共b兲 filament in water at 1 mm beyond the collapse共see text兲. 共c兲, 共d兲 Measured angularly resolved spectra of filaments in fused silica at

E = 2

共c兲 and 共d兲 3␮J.

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⯝10

¹¹

W / cm

²

兲 at 527 nm carrier wave length. The material parameters and propagation equation, which includes Kerr nonlinearity, diffraction, dispersion at any order, nonlinear losses, and plasma defocusing, are thoroughly explained in Ref. 关 15 兴, where X-shaped spectra well beyond collapse were investigated. Collapse is regularized by non-Kerr effects from z = 1.1 to 1.2 mm, where strong localization along with nearly constant peak intensity I⯝7⫻10

¹²

W / cm

²

is ob- served. At 1.2 cm 关Fig. 2 共b兲兴 the spectrum displays half-axial and half-conical emission with the slope 冑 2k

₀

k

₀

⬙

⯝1.33 fs/ ␮ ^m 共thick line兲, while no particular spectral struc- ture was observed at 1.1 cm. The growth of the Y-shaped spectrum in about 1 mm is consistent with a characteristic gain length 共k

NL

兩 ␬

I

兩兲

⁻¹

⯝0.5 mm estimated from the peak in- tensity I in this region.

In the experiments, we used a 15-cm-long fused silica sample as Kerr medium. 200-fs-long pulses at 527 nm deliv- ered from a 10 Hz Nd:glass mode-locked and regeneratively amplified laser 共Twinkle, Light Conversion兲 were spatially filtered and focused with a lens of 50 cm focal length. The pulses then entered into the sample, whose input facet was placed at 52 cm from the lens, and formed a single filament for input energies E ⲏ2 ␮ J. Single-shot, angularly resolved spectra of the filament at the output facet were measured with an imaging spectrometer and a charge-coupled device camera, as described in detail in 关2兴. At 2 ␮ J in fused silica, the filament is formed just before the output facet of the sample, the Y-shaped 共blue axial, red conical兲 spectrum of Fig. 2共c兲 then being observed. At 3 ␮ J, the filament is formed closer to the input facet. The double-Y spectrum of Fig. 2共d兲 then corresponds to a longer filament path within the sample. The faster growth of one of the two unstable perturbations supposes some unbalancing in their seeds, aris- ing, as expected, from the higher-order effects that arrest collapse.

Extending our analysis, we may venture an explanation of the fact that two X waves are commonly observed in fila- ments at later stages of propagation 关3,4兴. Consider, within

the four-wave-mixing approach, the possible effects on the instability spectrum of the strong temporal localization of the pump, as may take place upon 共possibly multiple兲 splitting.

For a spatially and temporally localized pump of central fre- quency ⍀=0, axial wave vector k

z,p

= k

₀

+ ⌬k

NL

, and propa- gating at a group velocity v

g

different from that of a plane pulse at ⍀=0 共as for a Y wave兲, new plane waves u and v at opposite frequencies ±⍀ are expected to be preferentially amplified if in addition to axial phase matching 共k

z,u

+ k

_z,v

= 2k

p

兲, the velocity of the group formed by u and v matches the velocity of the pump 关i.e., the inverse beating group ve- locity verifies 共k

z,u

− k

_z,v

兲/2兩⍀兩=1/v

g

兴. These two conditions yield the linear relation k

_z

= k

₀

+ ⌬k

NL

+ ⍀/v

g

for u 共at ⍀⬎0兲 and for v 共at ⍀⬍0兲. Since the transversal and axial projec- tions of the wave vector are related by K

_⬜

= 冑 ^2k

0

关k共⍀兲−k

z

兴 关in the paraxial approximation involved in the NLSE 共 1 兲兴, where k共⍀兲=k

0

+ k

₀

⬘ ⍀+k

₀

⬙ ⍀

²

/ 2, we obtain K

_⬜

= 冑 ^2k

0

共 ␤ ⁺ ␥ ⍀+k

₀

⬙ ⍀

²

/ 2兲, which is the dispersion curve of a frequency-gap X-wave mode 关4,16兴 with ␤ = −⌬k

NL

and ␥

= k

₀

⬘ ^{− 1 / v}

g

. If this process takes place for the two split-off pulses with opposite group velocities, the two X waves of Fig. 1共d兲, the thin solid and dashed curves, are formed. On nonlinearity relaxation 共⌬k

NL

→0兲 at larger z, one branch of each X wave is seen to pass through 共K

_⬜

,⍀兲=0, as observed 关3,4兴.

In conclusion, Y waves, emerging from the spatiotempo- ral instability of spatially localized modes in Kerr dynamics, constitute the missing link between axial and conical emis- sion, and allow us to interpret temporal splitting and X-wave formation. These results provide a unified view of ultrashort pulse filamentation, and can find application to all nonlinear waves involving subdimensional localization, such as ␹

^共2兲

solitons, matter waves with anisotropic confinement, plasma waves 关17兴, etc.

M.A.P. acknowledges Project No. HI2004-078. P.D.T. ac- knowledges Marie Curie Chair project STELLA, Grant No.

MEXC-CT-2005-025710.

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