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)

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

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 = 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.

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)

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.

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.