Multiple-Cone Formation during the Multiple-Cone Formation during the Femtosecond-Laser Pulse Propagation in Femtosecond-Laser Pulse Propagation in

SilicaSilica

Kenichi IshikawaKenichi Ishikawa**, Hiroshi Kumagai, and Katsumi , Hiroshi Kumagai, and Katsumi MidorikawaMidorikawa

Laser Technology Laboratory, RIKEN, Hirosawa 2-1, Wako-shi, Saitama Laser Technology Laboratory, RIKEN, Hirosawa 2-1, Wako-shi, Saitama

351-0198, Japan351-0198, Japan**Present address : Present address : Department of Quantum Engineering & Systems Science, Department of Quantum Engineering & Systems Science,

Graduate School of Engineering, University of TokyoGraduate School of Engineering, University of Tokyo7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Email:Email: ishiken@q.t.u-tokyo.ac.jpishiken@q.t.u-tokyo.ac.jp

submitted to Phys. Rev. E

AbstractAbstract

We present a numerical study of the (2+1)-dimensional We present a numerical study of the (2+1)-dimensional propagation dynamics of femtosecond laser pulses in silica. Pulses propagation dynamics of femtosecond laser pulses in silica. Pulses whose power is tens to hundreds of times higher than the threshold whose power is tens to hundreds of times higher than the threshold for self-focusing is split into multiple cones during its propagation. for self-focusing is split into multiple cones during its propagation. This new structure is formed as a result of the interplay of strong This new structure is formed as a result of the interplay of strong Kerr self-focusing and plasma defocusing. The number of cones Kerr self-focusing and plasma defocusing. The number of cones increases with incident pulse energy. The uncertainty which may increases with incident pulse energy. The uncertainty which may be contained in the evaluation of plasma response and multi-phase be contained in the evaluation of plasma response and multi-phase band-to-band transition cross section does not affect our results band-to-band transition cross section does not affect our results much. much.

High-power regimeHigh-power regime

In the present study, we consider the input pulse In the present study, we consider the input pulse energy of 10 energy of 10 〜〜 150 150 J.J.

100 MW 100 MW 〜〜 1 GW1 GW

z

7.5mm

r

Silica glass

0

Hyperbolic-secant pulse (T0 = 130fs)Gaussian beam (r0 = 200m)

= 800 nm

Threshold power for self-focusing Threshold power for self-focusing PPcrcr = 2.2 MW for = 2.2 MW for silica.silica.

50 50 PPcrcr 〜〜 500 500 PPcrcr

A few times A few times PPcrcr has been used in existing studies has been used in existing studies for gas and solidfor gas and solid

High-power regimeHigh-power regime

Fig. 1 Focusing and propagation of high-power femtosecond laser pulse in silica, considered in the present study.

Intense laser pulse undergoes self- focusing due to Intense laser pulse undergoes self- focusing due to the refractive index distribution induced by optical the refractive index distribution induced by optical Kerr effect.Kerr effect.

€

E (z,r, t) = E(z,r, t)exp ik0z − iω0t( )

Simulation modelSimulation model

Extended Nonlinear Schrödinger EquationExtended Nonlinear Schrödinger Equation

Group velocity dispersionGroup velocity dispersion Higher-order dispersionHigher-order dispersion

Kerr responseKerr response

DiffractionDiffraction

Plasma defocusingPlasma defocusing

Multi-photon absorptionMulti-photon absorption

in a reference frame moving at the group velocityin a reference frame moving at the group velocity

correction beyond the slowly varying envelope approximation (SVEA)correction beyond the slowly varying envelope approximation (SVEA)

€

∂E∂z

+i2β2

∂2E∂t2 −i

6β3∂3E∂t3 − i

2n0k0

∂2

∂r2 +1r

∂∂r

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 1− i

ω0

∂∂t

⎛ ⎝ ⎜ ⎞

⎠ ⎟ E

=in2k0 1+ iω0

∂∂t

⎛ ⎝ ⎜ ⎞

⎠ ⎟ E 2E( )−ik0

2 1− iω0

∂∂t

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ρ

ρcrE

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −3σ 6 ρ0 −ρ( ) E 2

hω0

⎛ ⎝ ⎜ ⎞

⎠ ⎟

5

E

Evolution of the electron density Evolution of the electron density in the conduction band in the conduction band

€

∂ρ∂t

=σ 6E 2

hω0

⎛ ⎝ ⎜ ⎞

⎠ ⎟

6

ρ0 −ρ( )

€

ρ0 =2.23×1022 cm-3

€

σ6 =2.6×10−180 cm12s5 from the Keldysh theory

(1)

(2)

Numerical methodsNumerical methods

The couples equations (1) and (2) are solved with the following The couples equations (1) and (2) are solved with the following methods.methods.

Equation (1)Equation (1)– Split-step Fourier method [1]Split-step Fourier method [1]– Diffraction termDiffraction term : Peaceman-Rachford method [2] : Peaceman-Rachford method [2]– Nonlinear terms (right-hand side)Nonlinear terms (right-hand side) : 4th-order Runge-Kutta method : 4th-order Runge-Kutta method

Equation (2)Equation (2)– 4th-order Runge-Kutta method4th-order Runge-Kutta method

[1] G.P. Agrawal, [1] G.P. Agrawal, Nonlinear Fiber OpticsNonlinear Fiber Optics, 2nd ed. (Academic, San Diego, 1995)., 2nd ed. (Academic, San Diego, 1995).[2] S.E. Koonin [2] S.E. Koonin et al.et al., Phys. Rev. C, Phys. Rev. C1515, 1359 (1977)., 1359 (1977).

Change of the spatio-temporal intensity Change of the spatio-temporal intensity profile with propagationprofile with propagation

Intensity (1012 W/cm2)0 5 10 15

0 3 6 9 0 105

zz = 3200 = 3200 mm

200 100 0 -100 -200

0

25

50

75

100

Time (fs)

Rad

ius r

(m

)

3300 3300 mm 3400 3400 mm 3500 3500 mm 3600 3600 mm

3700 3700 mm 3800 3800 mm 4000 4000 mm 4500 4500 mm 5000 5000 mm

Self-focusingSelf-steepening Plasma defocusing

More than 10 cones

1st cone 2nd cone 3rd cone

input energy = 135mJinput energy = 135mJpropagation distance

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

Multiple cone-like structure formationMultiple cone-like structure formation

Self-focusing The pulse energy is concentrated near the beam axis.⇨Self-focusing The pulse energy is concentrated near the beam axis.⇨Self-steepening The peak is shifted toward the trailing edge.⇨Self-steepening The peak is shifted toward the trailing edge.⇨

(a)(a)

As the self-focusing proceeds and the local intensity increases, As the self-focusing proceeds and the local intensity increases, Multi-photon absorption Conduction (plasma) electrons are produced.⇨Multi-photon absorption Conduction (plasma) electrons are produced.⇨

Plasma formation has a negative contribution to the refractive indexPlasma formation has a negative contribution to the refractive index ⇨ ⇨ Defocusing near the trailing edge.Defocusing near the trailing edge.

(b)(b)

Formation of a cone-like structure.Formation of a cone-like structure.(c)(c)

With pulse propagation, more and more cones are formed.With pulse propagation, more and more cones are formed.⇨ ⇨ Formation of Formation of multiple-cone-like structure.multiple-cone-like structure.

(d) 〜 (j)

Dramatic new Dramatic new feature in the feature in the high-power high-power regime !regime !

Mechanism of the multiple-cone formation Mechanism of the multiple-cone formation

At At zz = 3340 = 3340 m, the intensity decreases with increasing m, the intensity decreases with increasing rr in the range in the range rr = 9 - 12 = 9 - 12 m, while m, while nn is nearly flat there. is nearly flat there.

Due to self-focusing, the first peak takes up much energy from its vicinity.Due to self-focusing, the first peak takes up much energy from its vicinity. At At zz = 3360 = 3360 m, the second local maximum in m, the second local maximum in nn is formed around is formed around rr = 11.3 = 11.3

m. → The local self-focusing leads to the grow-up or the second cone.m. → The local self-focusing leads to the grow-up or the second cone.

3300 3300 mm 3400 3400 mm

1st cone 2nd cone

Fig. Radial distribution of intensity and refractive index change Fig. Radial distribution of intensity and refractive index change nn at at tt = 44 fs. = 44 fs.

0 100 200 300 400

0

5000

Radius (micron)

Propagation distance (micron)

0.0e+00 2.0e+14 4.0e+14 6.0e+14Fluence (1e-15J/cm2)

Fluence vs. Propagation distanceFluence vs. Propagation distance

Self-focusing

Prop

agat

ion

Prop

agat

ion

Propagation

Propagation

distance (

distance (m)m)

Fluence (10Fluence (10-15-15 J/cm J/cm22))

Radius (m)

Fluence (10-15 J/cm2)

Plasma defocusingPlasma defocusing

Prop

agat

ion

dist

ance

(mi

cron

)

Multiple cone-like structureMultiple cone-like structure

zz = 5000 = 5000 m from the silica surfacem from the silica surfaceInput energy = 135 Input energy = 135 JJ

Propagation

Propagation

FTOP signalFTOP signal Temporal profile integrated Temporal profile integrated in in rr-direction.-direction.

Propagation

200 100 0 -100 -200

0

25

50

75

100

Time (fs)

Radius (micron)

0.0e+00 5.0e+12 1.0e+13 1.5e+13Intensity (W/cm2)

5000 microns

Integration in time

Integration in r

Lateral profileLateral profile

Dependence on the input energyDependence on the input energy

With decreasing input pulse energy,With decreasing input pulse energy, the number of cones decreases.the number of cones decreases. the cones are more parallel to the beam axis.the cones are more parallel to the beam axis.The multiple-cone formation ceases when we further decrease the input energy.The multiple-cone formation ceases when we further decrease the input energy.

Intensity (1012 W/cm2)0 5 10 15

135 135 JJ, z, z = 4500 = 4500 mm 45 45 JJ, z, z = 5500 = 5500 mm 15 15 JJ, z, z = 7000 = 7000 mm

200 100 0 -100 -200

0

25

50

75

100

Time (fs)200 100 0 -100 -200

0

25

50

75

100

Time (fs)200 100 0 -100 -200

0

25

50

75

100

Time (fs)

Input energyR

adiu

s r (m

)

Rad

ius

r (m

)

Rad

ius

r (m

)

Uncertainty in plasma response and plasma Uncertainty in plasma response and plasma formation rateformation rate

200 100 0 -100 -200

0

25

50

75

100

Time (fs)200 100 0 -100 -200

0

25

50

75

100

Time (fs)

Plasma responsePlasma response Plasma formation ratePlasma formation rate

Intensity distribution at Intensity distribution at zz = 4000 = 4000 m obtained m obtained with account of the saturation of conduction with account of the saturation of conduction electron drift velocity, by replacing electron drift velocity, by replacing //crcr in Eq. in Eq. (1) by (1) by

The uncertainty which may be contained in the evaluation of The uncertainty which may be contained in the evaluation of plasma response and multi-phase band-to-band transition cross plasma response and multi-phase band-to-band transition cross section does not affect the essential features of our results. section does not affect the essential features of our results.

where where IIthth = 10 = 101212 W/cm W/cm22..

Intensity distribution at Intensity distribution at zz = 3500 = 3500 m obtained m obtained with a value of with a value of 66 which is 100 times smaller which is 100 times smaller than in Eq. (2).than in Eq. (2).

ConclusionConclusion

When the input power is several hundred times When the input power is several hundred times higher than higher than PPcrcr, the pulse is split many times both , the pulse is split many times both temporally and spatially.temporally and spatially.

As a result, the intensity distribution contains As a result, the intensity distribution contains multiple conesmultiple cones. . This is a new feature that This is a new feature that emerges only in theemerges only in the high-power regimehigh-power regime

This structure is formed by the interplay of Kerr This structure is formed by the interplay of Kerr self-focusing and plasma defocusing.self-focusing and plasma defocusing.