Multiple-Cone Formation during the Multiple-Cone Formation during the Femtosecond-Laser Pulse Propagation Femtosecond-Laser Pulse Propagation in Silica in Silica Kenichi Ishikawa Kenichi Ishikawa * , Hiroshi Kumagai, and , Hiroshi Kumagai, and Katsumi Midorikawa Katsumi Midorikawa Laser Technology Laboratory, RIKEN, Hirosawa 2-1, Wako-shi, Saitama Laser Technology Laboratory, RIKEN, Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan 351-0198, Japan * Present address : Present address : Department of Quantum Engineering & Department of Quantum Engineering & Systems Science, Graduate School of Engineering, University of Tokyo Systems Science, Graduate School of Engineering, University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Email: Email: [email protected][email protected]submitted to Phys. Rev. E
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Multiple-Cone Formation during the Femtosecond-Laser Pulse Propagation in Silica
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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
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
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.
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.
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.
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
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.