MultipleCone Formation during the FemtosecondLaser Pulse Propagation in Silica

1
Multiple-Cone Formation during the
Femtosecond-Laser Pulse Propagation in Silica

• Kenichi Ishikawa, Hiroshi Kumagai, and Katsumi
  Midorikawa
• Laser Technology Laboratory, RIKEN, Hirosawa 2-1,
  Wako-shi, Saitama 351-0198, Japan
• Present address Department of Quantum
  Engineering Systems Science, Graduate School of
  Engineering, University of Tokyo
• 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
• Email ishiken_at_q.t.u-tokyo.ac.jp

submitted to Phys. Rev. E
2
Abstract

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

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High-power regime
Intense laser pulse undergoes self- focusing due
to the refractive index distribution induced by
optical Kerr effect.
Threshold power for self-focusing Pcr 2.2 MW
for silica.
A few times Pcr has been used in existing studies
for gas and solid
Fig. 1 Focusing and propagation of high-power
femtosecond laser pulse in silica, considered in
the present study.
In the present study, we consider the input pulse
energy of 10 ? 150 mJ.
High-power regime
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Simulation model
Extended Nonlinear Schrödinger Equation
in a reference frame moving at the group velocity
(1)
(2)
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Numerical methods

• The couples equations (1) and (2) are solved with
  the following methods.
• Equation (1)
• Split-step Fourier method 1
• Diffraction term Peaceman-Rachford method 2
• Nonlinear terms (right-hand side) 4th-order
  Runge-Kutta method
• Equation (2)
• 4th-order Runge-Kutta method

1 G.P. Agrawal, Nonlinear Fiber Optics, 2nd ed.
(Academic, San Diego, 1995). 2 S.E. Koonin et
al., Phys. Rev. C15, 1359 (1977).
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Change of the spatio-temporal intensity profile
with propagation
input energy 135mJ
propagation distance
z 3200 mm
3300 mm
3400 mm
3500 mm
3600 mm
(d)
(e)
(a)
(c)
(b)
1st cone
2nd cone
3700 mm
3800 mm
4000 mm
4500 mm
5000 mm
(f)
(g)
(j)
(i)
(h)
More than 10 cones
Intensity (1012 W/cm2)
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Multiple cone-like structure formation
Self-focusing ? The pulse energy is concentrated
near the beam axis. Self-steepening ? The peak is
shifted toward the trailing edge.
(a)
As the self-focusing proceeds and the local
intensity increases, Multi-photon absorption ?
Conduction (plasma) electrons are produced.
Plasma formation has a negative contribution to
the refractive index ? Defocusing near the
trailing edge.
(b)
Dramatic new feature in the high-power regime !
(c)
Formation of a cone-like structure.
(d)?(j)
With pulse propagation, more and more cones are
formed. ? Formation of multiple-cone-like
structure.
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Mechanism of the multiple-cone formation
Fig. Radial distribution of intensity and
refractive index change Dn at t 44 fs.

• At z 3340 mm, the intensity decreases with
  increasing r in the range r 9 - 12 mm, while Dn
  is nearly flat there.
• Due to self-focusing, the first peak takes up
  much energy from its vicinity.
• At z 3360 mm, the second local maximum in Dn is
  formed around r 11.3 mm. ? The local
  self-focusing leads to the grow-up or the second
  cone.

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Fluence vs. Propagation distance
Self-focusing
Propagation
Propagation distance (micron)
Propagation distance (mm)
Radius (mm)
Plasma defocusing
Fluence (10-15 J/cm2)
Fluence (10-15 J/cm2)
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Multiple cone-like structure
Temporal profile integrated in r-direction.
FTOP signal
z 5000 mm from the silica surface Input energy
135 mJ
Integration in r
Propagation
Lateral profile
Propagation
Integration in time
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Dependence on the input energy
Input energy
135 mJ, z 4500 mm
45 mJ, z 5500 mm
15 mJ, z 7000 mm
Radius r (mm)
Radius r (mm)
Radius r (mm)
Intensity (1012 W/cm2)

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

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Uncertainty in plasma response and plasma
formation rate
Plasma response
Plasma formation rate
Intensity distribution at z 4000 mm obtained
with account of the saturation of conduction
electron drift velocity, by replacing r/rcr in
Eq. (1) by
Intensity distribution at z 3500 mm obtained
with a value of s6 which is 100 times smaller
than in Eq. (2).
where Ith 1012 W/cm2.

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

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Conclusion

• When the input power is several hundred times
  higher than Pcr, the pulse is split many times
  both temporally and spatially.
• As a result, the intensity distribution contains
  multiple cones. This is a new feature that
  emerges only in the high-power regime
• This structure is formed by the interplay of Kerr
  self-focusing and plasma defocusing.