On the role of conical waves in the self-focusing

and filamentation of ultrashort pulsesor

a reinterpretation of the spatiotemporal dynamics of ultrashort pulses in Kerr media

Miguel A. PorrasDepartamento de Física Aplicada. Universidad Politécnica de Madrid

Alberto Parola, Daniele Faccio, Paolo Di TrapaniUniversity of Insubria, Como, Italy

Arnaud CouaironCentre de Physique Théorique, CNRS, Palaiseau, France

1

1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media

ultrashort pulseTW/cm^2, 100 fs

self-focusing collapse filamentation

nonlinear Kerr sample

Self-focusing stage: fast nearly pure spatial dynamics self-similar compression towards a universal transversal profile: the Townes profile

Collapse region: enormous intensities (hundreds of TW/cm^2) onset of higher-order nonlinear phenomena birth of a full-spatiotemporal dynamics

2

temporal frequency

tran

sver

se f

requ

ency

Spatiotemporal spectrum

time

inte

nsity

AE

CE time splittingand narrowing

Filamentary regime:

recombination

ultrashort pulseTW/cm^2, 100 fs

self-focusing collapse filamentation

nonlinear Kerr sample

1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media

Pulse temporal splitting, narrowing and recombinationSeveral self-refocusing cycles, etc

Axial emission (spectral broadening, new temporal frequencies)Conical emission (new spatiotemporal frequencies)

3

k

ultrashort pulseTW/cm^2, 100 fs

self-focusing collapse filamentation

nonlinear Kerr sample

Spatial soliton ??? (balance Kerr focusing, plasma defocusing

and diffraction)

Axial emission(four-wave mixing amplification)

Conical emission (Cerenkov radiation ???)

Pulse splitting (arrest of collapse by GVD ???)

Alternate, unified interpretation of the basic features of the whole ST dynamicsfrom the self-focusing stage to the end of the filamentary regime

in terms of conical waves (Y-waves and wave-modes)

1. Spatiotemporal dynamics of femtosecond pulses in self-focusing Kerr media

4

2. From spatial self-focusing to the onset of spatiotemporal dynamics at collapse:Y-waves

spatialself-focusing

collapse

Blowing-upTransversal Townes profile

Incipient AE, CE and pulse splitting

Townes beammonochromatic light beama ()exp(i)ground state of the cubic NLSE0

5

spatiotemporal dynamics

• Spatiotemporal instability of the Townes profile

2. From spatial selfocusing to the onset of spatiotemporal dynamics at collapse:Y-waves

TP Spatiotemporal perturbation

• Two unstable modes: Y-wavesY-shaped spatiotemporal spectra of the 2 unstable modes

Townes beamfrequency

exp. growingST frequenciesexp. growing

ST frequencies

exp. growingST frequencies

6

• Unstable modes: Y-waves

up-shiftedaxial emission

2. From spatial self-focusing to the birth of spatiotemporal dynamics at collapse:Y-waves

up-shiftedconical emission

down-shiftedconical emission

down-shiftedaxial emission

firstY-wave

secondY-wave

• The ST instabilityof the TP can then explain the main ST phenomena (CE, AE and pulse splitting) observed immediately after collapse, from the features of the self-focusing beam (self-focusing into a TP).

superluminal subluminal

GVM =

• This establishes a casual connection between pre-collapse and post-collapse dynamics

7

2. From spatial self-focusing to the birth of spatiotemporal dynamics at collapse:Y-waves

self-focusing collapse

extremely localized and intense Townes profile

Growth of two Y-waves

Incipient AE, CE and pulse splitting

perturbations (higher-order, noise)seed ST instability

extremely localized and intense Townes profile

perturbations (higher-order, noise)seed ST instability

Growth of two Y-waves

Incipient AE, CE and pulse splitting

• Experimental observation of Y-waves upon collapse: 527 nm, 200 fs, fused silica

15 cm

E= 3 JE= 2 J

15 cm

8

3. Filament spatiotemporal dynamics: X-like wave-modes

filament

Water, 200 fs, 527 nm, 3 J

D. Faccio, M.A. Porras, A. Dubietis, F. Bragheri, A. Couairon, P. Di Trapani, Phys. Rev. Lett. 96 (2006) 193901

M. Kolesik, E.M.Wright, and J.V. Moloney, Phys. Rev. Lett. 94 (2004) 253901

We interpret this conical emission as the manifestation of the Kerr-driven formation of two X wave-modes (one for each split-off pulse)

9

3. Filament spatiotemporal dynamics: X wave-modes

• Wave-modes: spatiotemporal localized waves that can propagate in a linear dispersive medium without any temporal dispersion broadening

Bessel beamPulsed (polychromatic) Bessel beam

free parameters

10

3. Filament spatiotemporal dynamics: wave-modes

Transversal dispersion curve of WMs

X-wave:

Longitudinal wave number:

Transversal wave number:

water fs/mwater fs/m

11

0

K

pump

idlersignal

• intense, ST localized, pump wave (Y-wave)k +k (self-phase modulation), v =1/k´00 gNL 0

k + k0 NL

k + k0 NL

ki sk

i

• 2 weak, noncollinear, idler and signal waves at and –, propagation constants k = k( and k = k(-

s

k + k0 NL

k + k0 NL

ki sk

1) axial phase matching

k + k = 2k + 2kz,i z,s 0 NL

z

2) group matchingk - k

2 =1 v

z,i z,s

g0

WM results from the parametric amplification of new frequencies by a pump wave (a split-off pulse) via a Kerr-driven degenerate FWM interaction

3. Filament spatiotemporal dynamics: wave-modes

12

(two photons)

0

K

pump

idler

1) axial phase matching

2) group matching

k + k = 2k + 2kz,i z,s 0 NL

k = (k +k ) + / v z, i,s 0 NL 0 g linear with frequency

k - k

2 =1 v

z,i z,s

g0

Transversal dispersion curve

of a WM

K = k ([(k –) + (k’ –)()]i,s 0 0 0

2 2

signal

kNL

0k’ – 1/v0 g

• The ST frequencies preferentially amplified by a split-off pump pulse are just those forming wave-mode • The wave-mode travels at the same group velocity as the split-off pump pulse • For two split-off pump pulses, two WMs are amplified, each one accompanying each split-off pulse

133. Filament spatiotemporal dynamics: X-waves

Conclusory remarks

• We have introduced a interpretation of the spatiotemporal dynamics of ultrashort pulses in self-focusing Kerr media in terms of conical waves.

• In this interpretation the only essential nonlinearity is the Kerr nonlinearity. Higher-order effects play a secondary role.

• Conical waves (X-waves and Y-waves) are essentially linear waves, but are nonlinearly created (by Kerr-driven instability degenerate FWM).

• Y-waves can explain the excitation of the post-collapse filamentary regime from self-similar self-focusing prior to collapse.

• X-waves armonize the apparent stationarity of the filament with its actual complex spatiotemporal dynamics.

• Results are presented for media with normal dispersion but can be readily rewritten for anomalous dispersion.

14