1. High-power laser pulse propagation in silica
Extended Nonlinear Schrödinger Equation
in a reference frame moving at the group velocity
Group velocity dispersion
Higher-order dispersion
Diffraction
Multi-photon absorption
correction beyond the slowly varying envelope approximation (SVEA)
(1)
Kerr response
Plasma defocusing
Evolution of the electron density r in the conduction band
from the Keldysh theory
(2)

2. 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
Radius r (mm)
(a)
(b)
(c)
(d)
(e)
Self-focusing
Self-steepening
Plasma defocusing
3rd cone
1st cone
2nd cone 3 6 9 10 5
3700 mm
3800 mm
4000 mm
4500 mm
5000 mm
(f)
(g)
(h)
(i)
(j)
More than 10 cones
Intensity (1012 W/cm2) 5 10 15

3. Mechanism of the multiple-cone formation
1st cone
2nd cone
3300 mm
3400 mm
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 = 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.

4. 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)

5. Dependence on the 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) 5 10 15
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.

6. 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.