Simulating a complete experiment of pulse compression...

FilamentationSelf-compression of Femtosecond Pulses

Conclusion

Simulating a complete experiment of pulse

compression mediated by �lamentation

Luc BergéCEA-DIF - Bruyères-le-Châtel - 91297 Arpajon cedex - France

Stefan Skupin, Günter Steinmeyer

5th Conference on Solitons, Collapses and TurbulenceAugust 2-7, 2009

Luc Bergé Femtosecond Filaments


Conclusion

FilamentationMathematical modeling: Basic ingredientsDynamic spatial replenishmentModeling revisited: Towards quantitative comparison

Self-compression of Femtosecond PulsesSetup and compression mechanismSimulating the whole experimentTemporal self-healing mechanism

Conclusion



Conclusion

Mathematical modeling: Basic ingredientsDynamic spatial replenishmentModeling revisited: Towards quantitative comparison



Conclusion



Conclusion


Self-guided laser pulse propagating in air

S. Niedermeier, H. Wille, M. Rodriguez, J. Kasparian und R. SauerbreyFriedrich-Schiller-Universität Jena



Conclusion


Self-guided laser pulse propagating in air

peak power: 380 GW peak power: 2300 GW

G. Méjean, J. Yu, J. Kasparian, E. Salmon und J. P.WolfUniversité Cl. Bernard Lyon, LASIM



Conclusion


Simple propagation equations

∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I slowly-varying envelope of the optical �eld U in scalarapproximation (SVEA)

I propagation in z-direction



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I di�raction



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I optical Kerr-e�ect → self-focusing by nonlinear inducedrefractive index change

I two-dimensional Nonlinear Schrödinger Equation

I Pin > Pcr 'λ20

2πn0n2→ collapse for Gaussian input beam



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I group-velocity dispersion (GVD):k(ω0 + ω̄) = k0 + k

′ω̄ + k′′

2ω̄2 + . . .

I reference frame traveling with group-velocity

I sign of k ′′ in�uences collapse dynamics



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I increasing intensity leads to ionization of the medium→ free electron density ρ

I multi-photon ionization (MPI) involvingK = mod(Ui/~ω0) + 1 photons



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I interaction of the optical �eld with the generated plasma

I nonlinear induced refractive index ∆nnl = n2|U|2 − 12n0ρc ρ→ plasma balances Kerr self-focusing by defocusing

I critical plasma density ρc =ω20me�0

q2e



Conclusion



∂

∂zU =

i

2k0∇2⊥U − i

k ′′

2

∂2

∂t2U + i

ω0cn2|U|2U

− i k02n2

0ρcρU − β

(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

I nonlinear losses induced bymulti-photon-absorption (MPA)



Conclusion


Atmospheric propagation: single �lament 30 GW, 0.5 mm, 100 fs

I plasma-generation over ∼ 1 mI diameter of �lament ∼ 150 µmI robust against azimuthal perturbation (here 1% noise)



Conclusion


Atmospheric propagation: single �lament 30 GW, 0.5 mm, 100 fs

I focused time slices generate plasma channel

I �Dynamic spatial replenishment model�(Mlejnek, Wright, and Moloney - 1998)



Conclusion


Extended propagation equations

∂

∂zU =

i

2k0∇2⊥U

T−1

+ ik ′′

2

∂2

∂t2U

∑n≥2

k(n)

n!

(i∂

∂t

)n

+ iω0cn2 |U|2 U

∫

− i k02n2

0ρcρU

T−1

− β(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

+σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I the simple propagation model can be re�ned→ quantitative comparison with experiments



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2 |U|2 U

∫

− i k02n2

0ρcρU

T−1

− β(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

+σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I higher-order linear dispersion: Important in solids and liquids



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U

T − i ω0cn4|U|4UT

− i k02n2

0ρcρU

T−1

− β(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

+σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I Raman delayed part of the Kerr response:

h(t) = (1− xR)δ(t) + xRΘ(t)τ21+τ

22

τ1τ22e−t/τ2 sin(t/τ1)



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U

T

− i ω0cn4|U|4U

T

− i k02n2

0ρcρU

T−1

− β(K)

2|U|2(K−1)U

∂

∂tρ =

β(K)

K~ω0|U|2K

+σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I χ(5) nonlinearity



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U

T

− i ω0cn4|U|4U

T

− i k02n2

0ρcρU

T−1

− UiW (|U|2)(ρnt − ρ)

2|U|2U

− σ2ρU

β(K)

2

∂

∂tρ = W (|U|2)(ρnt − ρ)

+σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I improved modeling of ionization rate W : Keldysh (crystals);Perelomov, Popov and Terent'ev (atoms); Ammosov, Deloneand Krainov (tunnel regime)



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U

T

− i ω0cn4|U|4U

T

− i k02n2

0ρcρU

T−1

− UiW (|U|2)(ρnt − ρ)

2|U|2U − σ

2ρU

β(K)

2

∂

∂tρ = W (|U|2)(ρnt − ρ) +

σ

Uiρ|U|2

− f (ρ)

β(K)

K~ω0

I avalanche ionization and related nonlinear losses



Conclusion



∂

∂zU =

i

2k0∇2⊥U

T−1

+ i∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U

T

− i ω0cn4|U|4U

T

− i k02n2

0ρcρU

T−1

− UiW (|U|2)(ρnt − ρ)

2|U|2U − σ

2ρU

β(K)

2

∂

∂tρ = W (|U|2)(ρnt − ρ) +

σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I electron-ion recombination:f (ρ) ∼ ρ2 for gases; f (ρ) ∼ ρ for dielectrics



Conclusion



∂

∂zU =

i

2k0T−1∇2⊥U + i

∑n≥2

k(n)

n!

(i∂

∂t

)nU

+ iω0cn2T

∫h(t − t ′)

∣∣U(t ′)∣∣2 dt ′U − i ω0cn4T |U|4U

− i k02n2

0ρc

T−1ρU − UiW (|U|2)(ρnt − ρ)

2|U|2U − σ

2ρU

β(K)

2

∂

∂tρ = W (|U|2)(ρnt − ρ) +

σ

Uiρ|U|2 − f (ρ)

β(K)

K~ω0

I higher-order terms correction SVEA (Brabec and Krausz):Self-steepening and space-time focusing operatorT = (1 + iω0∂t)



Conclusion

Setup and compression mechanismSimulating the whole experimentTemporal self-healing mechanism



Conclusion



Conclusion


General considerations

I we want a single short pulse !

I no muliple �lamentation→ few critical powers only

I no post-compression

I no pulse trains→ only one peak should survive



Conclusion


Self-compression of femtosecond pulses inside laser �laments

G. Stibenz and G. SteinmeyerMax-Born-Institut Berlin



Conclusion



experiments

simulations




Conclusion



experiments simulations




Conclusion


Light bullet � simulation 800 nm, 1Pcr, 25 fs, 150 µm

I input intensity close to saturation→ slow di�raction

I temporal extent reduces to �stable�6-8 fs (λ0

c= 2.7 fs)

I characteristic spatio-temporalpro�le (time dependent radius)

I temporal asymmetry due toself-steepening



Conclusion


Light bullet � gas-glass-gas interface

experimental setup for pulse compression

I step I: Filamentation in argon

I step II: The pulse goes across a mm-thick silica window

I step III: Diagnostics are positioned at ∼ 1 m from the cell



Conclusion


Step I: Self-compression in Ar 800 nm, Pin = 1Pcr

I light bullet formation in Ar → 6-8 fs pulse duration



Conclusion


Step I: Self-compression in Ar 800 nm, Pin = 1Pcr

(loading movie)


selfcompression_fast_cinepak.aviMedia File (video/avi)


Conclusion


Step II: Pulse broadening in silica Pmax > 1000Pcr in glass

I GVD + Kerr → durations up to ≥ 30 fs in the window



Conclusion



I pulse duration exceeds

τlin(z) ' τ0√1 + 16(ln 2)2(k(2)∆z/τ2

0)2

I 8 fs Gaussian pulse broadens to ∼ 10 fs only over 0.5 mm

I two-scaled variational approach for beam radius R and pulseduration T :1

4R3Rzz = 1− p2T ;

1

4T 3Tzz = δ(δ +

Tp2R2

)

I p = Pin/Pcr and δ ≡ 2πn0R2ink(2)/λ0T 2in are largeI 8 fs Gaussian pulse (15 TW/cm2, 240 µm) broadens to∼ 25 fs over 0.5 mm



Conclusion



I At larger distances we do see collapse in the glass window



Conclusion



I What about spatial modulational instability (in 3D) ?

I MI is e�ciently suppressed by GVD!



Conclusion


Step III: Recompression in air Pmax ∼ 3.5Pcr in air

(II) (III)

I pulse compression is �self-restored� in air ∼ 10 fs durations.



Conclusion


Recompression works in vacuum as well

air (solid line, left �gure), vacuum (dashed line, right �gure)

I refocusing in second gas is not due to nonlinear self-focusing

I recompression di�erent from self-compression



Conclusion


Accumulated phase is important

phase after crossing the window (for 1 m argon)

I nonlinear phase curvature accumulated in glass acts like a lens

I each time slice has di�erent focal length→ new phase: eiRz r2/4R ∼ e−iω0r2w20 /2cf∗→ on-axis recompression



Conclusion


In�uence of glass window on far-�eld

after window, after window without phase, before window

I phase is crucial for recompression

I glass window has limited in�uence on far-�eld



Conclusion

Conclusion

I �Light bullets�: speci�c compression regime for reachingrobust, singly-peaked pulses with few-cycle durations

I Role of the exit glass window in a gas cell ? Important !

I New process: self-restoration of compressed light pulses

I S. Skupin et al., PRE 74, 056604 (2006)

LB et al., PRL 100, 113902 (2008); PRL 101, 213901 (2008); PRA 79, 033838 (2009)



Conclusion

NEW MEETING : W.L.M.I.

2nd International Workshop on Laser-Matter Interaction

I Porquerolles Island - France

I 13-17 September 2010 - Sponsors: CEA




Conclusion

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Simulating a complete experiment of pulse compression...

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